Book 4 of the Ray's Arithmetic shelf is now playable: Ray's New Higher Arithmetic (1880, Grades 7-8). It is the last and biggest of the four arithmetic books: about 400 pages, from numeration through fractions, circulating decimals, every table of measures, percentage and all its business uses, interest and annuities, partnership, alligation, roots, progressions and mensuration, ending with 106 miscellaneous problems. It runs in the same app as Books 1-3 at https://abcbooks.fyi/math/ , with the same "Reed" voice. This post covers what is new for Book 4 and how to rebuild it. The app, the server and the voice recorder are explained in the Book One post; the answer types for fractions are in the Book 2 post, and column sums and long numbers in the Book 3 post. Every block has a Copy button.

links redirect there). The files on CT 306 (abc-web) did not move; only the routes on CT 108 (Caddy reverse proxy) changed. Parents can now make a family account, so names and stars follow each learner between both apps and devices. The account container, the new routes and the changes to both apps are in the family accounts post. The files below are the current versions.

What is in Book 4

Each section of the book's table of contents became a lesson: 77 lessons, 1,932 questions and 2,871 answers.

Lessons 1-13: numeration (to decillions), the four rules with their shortcuts, the arithmetical signs, factoring, greatest common divisor, least common multiple, casting out nines, cancellation.
Lessons 14-26: common fractions, decimals (with "complex decimals" like .0154 1/4 and Oughtred's shortened multiplication and division), and circulating (repeating) decimals.
Lessons 27-36: money of four countries, every measure (including the metric system), reduction, the four rules with compound numbers, longitude and time, aliquot parts.
Lessons 37-39: ratio, simple and compound proportion.
Lessons 40-62: percentage and its uses: profit and loss, stocks and bonds, premium and discount, commission, stock investments, insurance, taxes and duties, simple interest, notes, partial payments (United States, Connecticut, Vermont and Mercantile rules), true and bank discount, exchange and arbitration, equation of payments, settlement of accounts, compound interest, annuities and life insurance.
Lessons 63-75: partnership and bankruptcy, alligation, involution and evolution (square, cube and higher roots, right triangles), arithmetical and geometrical progression, and mensuration of surfaces and solids, masonry, gauging, lumber, grain and hay.
Lessons 76-77: the book's 106 miscellaneous exercises (the first 50 to be done in the head).

New in the app for Book 4

Answers in several units. A question like "Add 2 rd. 9 ft. 7 1/4 in. ..." asks for the rods, then the feet, then the inches, one at a time. The build works each part out from the whole quantity, so the units always add up.
Complex decimals. Answers like .0154 1/4 are typed as a decimal, then and, then the fraction.
Repeating decimals. Circulates are written with the repeating figures in brackets: .45(3) means .4533... The Reed voice reads it "point 4 5, 3 repeating". Answers are given as fractions, and the circulate is shown after.
Very long numbers. The pad now takes up to 40 figures, for the numeration lessons (up to 36 figures).
Rounded answers. "To two places" accepts both the rounded and the cut-off figure, and money answers worked by the book's 6% method accept a cent either way, since the book's own answers differ from exact arithmetic by a cent here and there.
Fractions read in words. Any denominator is now read as a word: 14/23 is "14 twenty-thirds", not "23ths".

How the book was typed up

The free PDF (raysarithmetic.wordpress.com, "7-8 Ray's New Higher Arithmetic", 409 pages, and the separate answer key, 211 pages) is different from the earlier books: it has an old text layer, good enough for the words but not for figures, fractions or tables. So each page was rendered as a picture and read by eye, and the text layer used only to find things. The answer key has no text layer at all; it was read with Windows' text recognition (ocr-pdf.ps1 from the Book 2 post) and used to check the answers. PDF page = book page + 1.

Almost no answer was typed by hand. Each answer is written as the working that produces it, and the build works it out exactly:

q: Find the sum of the consecutive numbers from 119 to 131. {~(119 + 131) * 13 / 2}
q: Add .45(3), .06(8), .32(7), .94(6). {~q .45(3) + .06(8) + .32(7) + .94(6)}
q: Multiply 27.653 by 9.157, preserving three decimal places. {~r3 27.653 * 9.157}
q: Add 23.8 and 17 1/2 and .0256 and .41 2/3. {~c4 23.8 + 17_1/2 + .0256 + (41_2/3) / 100}
q: Find the simple interest of $354.80 for 3 yr. 7 mo. 19 da., at 6%. Dollars? {~m 354.80 * .06 * (3 + 7/12 + 19/360)}
q: Multiply 7 rd. 10 ft. 5 in. by 6. Rd.? {~p 16_1/2,12 # 0 # (7 * 198 + 125) * 6} Ft.? {~p # 1 # $} In.? {~p # 2 # $}
q: Nineteen quadrillion twenty trillion and five hundred billion. {=19020500000000000}

{~q ...} answer as a fraction; the circulate is added to the solution.
{~rN ...} rounded to N places (the cut-off value also counts).
{~cN ...} a complex decimal carried to N places.
{~m ...} money to the cent, a cent either way accepted.
{~p chain # i # expr} compound units: the chain says how many of each unit make the one above (rods to feet to inches: 16 1/2, 12), expr is the whole amount in the smallest unit, and i picks the unit asked. {~p # i # $} is the same amount, next unit.
{=digits} a whole number too long for the app to hold as a number; it is checked as typed.
{answer @ working} (from Book 3) was used wherever the key prints an answer, so a misread digit stops the build.

Where the book works a problem from its own tables (compound interest, annuities, life insurance) the key's printed answers are used. A few things found on the way:

Lesson 44: the canal company can declare 4 1/2%, not 4%, and still have $12,851 over (the key's answer).
Lesson 55: the New York note's proceeds are $1926.20 by the arithmetic; the key prints $1926.70.
Lesson 68: the key's 12th root problem matches 782,757,789,696 (96 to the 6th power), so that number is used.
Lesson 76: "how many days may 3 remain away" follows the key's reading (5 5/24 days).
Lesson 77: "how many inch balls in a box 10 in. square and 5 deep" is answered out loud: 500 laid square, more if each layer sits in the hollows of the one below.

The voice

The same recorder as Books 1-3: 4,930 lines, 4,238 of them new at each speed (lines the books share are recorded once).

Files

Set up the Book One post first. Book 4 adds or replaces the files below. Each block writes one file into /root/rays-rebuild/ when pasted into a Proxmox host shell (Proxmox web UI, pve, Shell). Then copy that folder to the PC and put:

higher.txt in C:\RaysArithmetic\source\
make-book.ps1, make-audio.ps1 in C:\RaysArithmetic\tools\
app.js, sw.js in C:\RaysArithmetic\public\ (they replace the earlier versions and run all four books)

Proxmox host shell:

mkdir -p /root/rays-rebuild

higher.txt: the whole of Book 4 with the answers.

cat > /root/rays-rebuild/higher.txt <<'EOF'
@book: higher | Ray's New Higher Arithmetic | 1880 - Book Four | #a0522d
# Ray's New Higher Arithmetic (1880), typed from source/pdf/rays-higher-arithmetic-7-8.pdf (book page N = PDF page N + 1;
# its text layer is old OCR, so numbers were read off the page images); answers checked against the Key
# (rays-higher-arithmetic-7-8-answer-key.pdf). Same format as practical.txt. {=digits} = a whole number too long for
# the app to hold as a number (more than 15 figures): it is checked as written.
# Each lesson here is one section of the book's Table of Contents.

== 1 | Numeration | Reading numbers
note: Begin at the right, and point the number into periods of three figures each. Commence at the left, and read in succession each period with its name.
q: Read out loud: 608921045. {open}
sol: The number, as divided into periods, is 608,921,045; and is read six hundred and eight million nine hundred and twenty-one thousand and forty-five.
q: Read out loud: 7; 40; 85; 278; 1345. {open}
sol: Seven; forty; eighty-five; two hundred and seventy-eight; one thousand three hundred and forty-five.
q: Read out loud: 4053; 7009; 12345; 70500; 165247. {open}
sol: Four thousand and fifty-three; seven thousand and nine; twelve thousand three hundred and forty-five; seventy thousand five hundred; one hundred and sixty-five thousand two hundred and forty-seven.
q: Read out loud: 204026; 500050; 730003. {open}
sol: Two hundred and four thousand and twenty-six; five hundred thousand and fifty; seven hundred and thirty thousand and three.
q: Read out loud: 1375482; 6030564. {open}
sol: One million three hundred and seventy-five thousand four hundred and eighty-two; six million thirty thousand five hundred and sixty-four.
q: Read out loud: 4300201; 29347283. {open}
sol: Four million three hundred thousand two hundred and one; twenty-nine million three hundred and forty-seven thousand two hundred and eighty-three.
q: Read out loud: 45004024. {open}
sol: Forty-five million four thousand and twenty-four.
q: Read out loud: 343827544. {open}
sol: Three hundred and forty-three million eight hundred and twenty-seven thousand five hundred and forty-four.
q: Read out loud: 830070320. {open}
sol: Eight hundred and thirty million seventy thousand three hundred and twenty.
q: Read out loud: 832,045,682,327,825,000,000,321. {open}
sol: Eight hundred and thirty-two sextillion forty-five quintillion six hundred and eighty-two quadrillion three hundred and twenty-seven trillion eight hundred and twenty-five billion three hundred and twenty-one.
q: Read out loud: 8,007,006,005,004,003,002,001,000,000. {open}
sol: Eight octillion seven septillion six sextillion five quintillion four quadrillion three trillion two billion one million.
q: Read out loud: 60,030,020,090,080,070,050,060,030,070. {open}
sol: Sixty octillion thirty septillion twenty sextillion ninety quintillion eighty quadrillion seventy trillion fifty billion sixty million thirty thousand and seventy.
q: Read out loud: 504,030,209,102,800,703,240,703,250,207. {open}
sol: Five hundred and four octillion thirty septillion two hundred and nine sextillion one hundred and two quintillion eight hundred quadrillion seven hundred and three trillion two hundred and forty billion seven hundred and three million two hundred and fifty thousand two hundred and seven.

== 2 | Numeration | Writing numbers
note: Begin at the left, and write each period in its proper place, filling the vacant orders with ciphers. Every period, except the highest, must have three figures.
q: Express in figures the number four million twenty thousand three hundred and seven. {4020307}
sol: Write 4 in millions period; then 20 in thousands period, with a cipher before it to make three figures; then 307 in units period. The number is 4020307.
q: Write in figures: Seventy-five. {75} One hundred and thirty-four. {134} Two hundred and four. {204} Three hundred and seventy. {370}
q: One thousand two hundred and thirty-four. {1234} Nine thousand and seven. {9007} Forty thousand five hundred and sixty-three. {40563}
q: Ninety thousand and nine. {90009} Two hundred and seven thousand four hundred and one. {207401}
q: Six hundred and forty thousand and forty. {640040} Seven hundred thousand and seven. {700007}
q: One million four hundred and twenty-one thousand six hundred and eighty-five. {1421685}
q: Seven million and seventy. {7000070} Ten million one hundred thousand and ten. {10100010}
q: Sixty million seven hundred and five thousand. {60705000}
q: Eight hundred and seven million forty thousand and thirty-one. {807040031}
q: Two billion and twenty million. {2020000000}
q: Nineteen quadrillion twenty trillion and five hundred billion. {=19020500000000000}
q: Ten quadrillion four hundred and three trillion ninety billion and six hundred million. {=10403090600000000}
q: Eighty octillion sixty sextillion three hundred and twenty-five quintillion and thirty-three billion. {=80000060325000000033000000000}
q: Nine hundred decillion seventy nonillion six octillion forty septillion fifty quadrillion two hundred and four trillion ten million forty thousand and sixty. {=900000070006040000050204000010040060}
note: In the English method six figures make a period: units, millions, billions, trillions. In the Roman notation I is one; V, five; X, ten; L, fifty; C, one hundred; D, five hundred; M, one thousand. A letter of less value before one of greater is taken from it; after it, added.
q: Read 210987654321 by the English method. {open}
sol: Two hundred and ten thousand nine hundred and eighty-seven million six hundred and fifty-four thousand three hundred and twenty-one. By the French method: two hundred and ten billion nine hundred and eighty-seven million six hundred and fifty-four thousand three hundred and twenty-one.
q: What number is XIV? {14} XIX? {19} XL? {40} XC? {90}
q: What number is CCCC? {400} DCCC? {800} MM? {2000}
q: What number is MDCCCLXXXI? {1881}

== 3 | Addition | Adding numbers
note: Write the numbers so that figures of the same order stand in the same column. Begin at the right and add each column, writing the units under it and carrying the tens to the next column.
q: What is the sum of 639, 82, and 543? [[+ 639 82 543]] {?}
sol: 3, 5, 14 units: write 4 and carry 1 ten. 1, 5, 13, 16 tens: write 6 and carry 1 hundred. 1, 6, 12 hundreds. The sum is 1264.
q: Find the sum of 76767; 7654; 50121; 775. [[+ 76767 7654 50121 775]] {?}
q: Find the sum of 97674; 686; 7676; 9017. [[+ 97674 686 7676 9017]] {?}
q: Find the sum of 971; 7430; 97476; 76734. [[+ 971 7430 97476 76734]] {?}
q: Find the sum of 999; 3400; 73; 47; 452; 11000; 193; 97; 9903; 42; and 5100. {~999 + 3400 + 73 + 47 + 452 + 11000 + 193 + 97 + 9903 + 42 + 5100}
q: Find the sum of four hundred and three; 5025; sixty thousand and seven; eighty-seven thousand; two thousand and ninety; and 100. {~403 + 5025 + 60007 + 87000 + 2090 + 100}
q: Find the sum of 20050; three hundred and seventy thousand two hundred; four million and five; two million ninety thousand seven hundred and eighty; one hundred thousand and seventy; 98002; seven million five thousand and one; and 70070. {~20050 + 370200 + 4000005 + 2090780 + 100070 + 98002 + 7005001 + 70070}
q: Find the sum of 609505; 90070; 90300420; 9890655; 789; 37599; 19962401; 5278; 2109350; 41236; 722; 8764; 29753; and 370247. {~609505 + 90070 + 90300420 + 9890655 + 789 + 37599 + 19962401 + 5278 + 2109350 + 41236 + 722 + 8764 + 29753 + 370247}
q: Find the sum of two hundred thousand two hundred; three hundred million six thousand and thirty; seventy million seventy thousand and seventy; nine hundred and four million nine thousand and forty; eighty thousand; ninety million nine thousand; six hundred thousand and sixty; five thousand seven hundred. {~200200 + 300006030 + 70070070 + 904009040 + 80000 + 90009000 + 600060 + 5700}
q: Find the sum of the consecutive numbers from 119 to 131, including these numbers. {~(119 + 131) * 13 / 2}
q: Find the sum of the consecutive numbers from 987 to 1001. {~(987 + 1001) * 15 / 2}
q: Find the sum of the consecutive numbers from 3267 to 3281. {~(3267 + 3281) * 15 / 2}
q: Find the sum of the consecutive numbers from 4197 to 4211. {~(4197 + 4211) * 15 / 2}
q: Find the sum of the consecutive numbers from 5397 to 5416. {~(5397 + 5416) * 20 / 2}
q: Find the sum of the consecutive numbers from 7815 to 7831. {~(7815 + 7831) * 17 / 2}
q: Find the sum of the consecutive numbers from 31989 to 32028. {~(31989 + 32028) * 40 / 2}
q: Paid for coffee, $375; for tea, $280; for sugar, $564; for molasses, $119; and for spices, $75: what did the whole amount to? Dollars? {1413 @ 375 + 280 + 564 + 119 + 75}
q: I bought three pieces of cloth: the first cost $87; the second, $25 more than the first; and the third, $47 more than the second: what did all cost? Dollars? {358 @ 87 + 112 + 159}
q: A man bought three bales of cotton. The first cost $325; the second cost $16 more than the first; and the third, as much as both the others: what sum was paid for the three bales? Dollars? {1332 @ 325 + 341 + 666}
q: A has $75; B has $19 more than A; C has as much as A and B, and $23 more; and D has as much as A, B, and C together: what sum do they all possess? Dollars? {722 @ 75 + 94 + 192 + 361}

== 4 | Subtraction | Subtracting and balancing accounts
note: Write the less number under the greater, units under units. Begin at the right; if any figure exceeds the one above it, add ten to the upper, and increase by 1 the next figure of the subtrahend. Proof: the remainder added to the less number gives the greater.
q: From 827 dollars take 534 dollars. [[- 827 534]] {?}
sol: 4 units from 7 units leave 3. 3 tens can not be taken from 2 tens: add 10 tens, and 3 from 12 leave 9 tens. Add 1 hundred to the 5 hundreds: 6 from 8 leave 2 hundreds. The remainder is 293.
q: From 30020037 take 50009. [[- 30020037 50009]] {?}
q: From 79685 take 30253. [[- 79685 30253]] {?}
q: From 1145906 take 39876. [[- 1145906 39876]] {?}
q: From 2900000 take 777888. [[- 2900000 777888]] {?}
q: From 71086540 take 64179730. [[- 71086540 64179730]] {?}
q: From 101067800 take 100259063. [[- 101067800 100259063]] {?}
q: How many years from the discovery of America in 1492, to the Declaration in 1776? {284 @ 1776 - 1492}
q: A farm that cost $7253, was sold at a loss of $395: for how much was it sold? Dollars? {6858 @ 7253 - 395}
q: The difference of two numbers is 19034, and the greater is 75421: what is the less? {56387 @ 75421 - 19034}
q: How many times can the number 285 be subtracted from 1425? {5 @ 1425 / 285}
q: Which is the nearer number to 920736; 1816045 or 25427? {open}
sol: 1816045 - 920736 = 895309, and 920736 - 25427 = 895309: they are equally near.
q: From a tract of land containing 10000 acres, the owner sold to A 4750 acres; and to B 875 acres less than A: how many acres had he left? {1375 @ 10000 - 4750 - 3875}
q: A, B, C, D, are 4 places in order in a straight line. From A to D is 1463 miles; from A to C, 728 miles; and from B to D, 1317 miles. How far is it from A to B? {146 @ 1463 - 1317} From B to C? {582 @ 728 - 146} From C to D? {735 @ 1463 - 728}
note: In an account, sums a person owes are his Debits (Dr.); sums owing to him are his Credits (Cr.). The difference between the sum of the Debits and the sum of the Credits is the Balance.
q: James Craig's account. Debits: $560.50, $24.90, $870.60, $320.10, $125.40. Credits: $550.50, $890.70, $310.80, $100.00. What is the sum of the debits? {1901.50|1901.5 @ 560.50 + 24.90 + 870.60 + 320.10 + 125.40} What is the balance? {49.50|49.5 @ 1901.50 - 550.50 - 890.70 - 310.80 - 100}
q: Balance Thomas Baldwin's account. Debits: Merchandise $810.30, Sundries $580.20, Cash $381.25, Merchandise $60.75. Credits: Sundries $1000.00, Cash $300.00, Merchandise $225.20. What is the balance? {307.30|307.3 @ 810.30 + 580.20 + 381.25 + 60.75 - 1000 - 300 - 225.20}

== 5 | Multiplication | Multiplying numbers
note: Begin with units; multiply each figure of the multiplicand by the multiplier, carrying as in Addition. When the multiplier exceeds 12, multiply by each figure of the multiplier in turn, placing the right-hand figure of each product under the figure that produced it, and add the partial products.
q: At the rate of 53 miles an hour, how far will a railroad car run in four hours? Miles? [[x 53 4]] {?}
sol: 4 times 3 are 12: write 2 and carry 1. 4 times 5 are 20, and 1 carried makes 21. The car runs 212 miles.
q: 195 x 3 = ? [[x 195 3]] {?}
q: 3823 x 4 = ? [[x 3823 4]] {?}
q: 8765 x 5 = ? [[x 8765 5]] {?}
q: 98374 x 6 = ? [[x 98374 6]] {?}
q: 64382 x 7 = ? [[x 64382 7]] {?}
q: 58765 x 8 = ? [[x 58765 8]] {?}
q: 837941 x 9 = ? [[x 837941 9]] {?}
q: 645703 x 10 = ? [[x 645703 10]] {?}
q: 407649 x 11 = ? [[x 407649 11]] {?}
q: If 4 men can perform a certain piece of work in 15 days, how long will it require 1 man? Days? {60 @ 4 * 15}
sol: One man must work four times as long as four men. 4 x 15 days = 60 days.
q: How many pages in a half-dozen books, each containing 336 pages? {2016 @ 6 * 336}
q: How far can an ocean steamer travel in a week, at the rate of 245 miles a day? Miles? {1715 @ 7 * 245}
q: What is the yearly expense of a cotton-mill, if $32053 are paid out every month? Dollars? {384636 @ 12 * 32053}
q: A receives from his business an average of $45 a day. He pays three clerks $3; three, $9; and three, $12 a week; other expenses amount to $4 a day: what are his profits for one week of 6 business days? Dollars? {174|215}
sol: 6 days at $45 = $270. The clerks get 3 x 3 + 3 x 9 + 3 x 12 = $72, and other expenses are 6 x $4 = $24. $270 - $72 - $24 = $174. (Counting 7 days: $315 - $72 - $28 = $215.)
q: Multiply 246 by 235. [[x 246 235]] {?}
sol: 246 x 5 = 1230; 246 x 3 tens = 738 tens; 246 x 2 hundreds = 492 hundreds. Adding the partial products, 57810.
q: 7198 x 216 = ? [[x 7198 216]] {?}
q: 8862 x 189 = ? [[x 8862 189]] {?}
q: 7575 x 7575 = ? [[x 7575 7575]] {?}
q: 15607 x 3094 = ? [[x 15607 3094]] {?}
q: 93186 x 4455 = ? [[x 93186 4455]] {?}
q: 135790 x 24680 = ? [[x 135790 24680]] {?}
q: 3523725 x 2583 = ? [[x 3523725 2583]] {?}
q: 4687319 x 1987 = ? [[x 4687319 1987]] {?}
q: 9264397 x 9584 = ? [[x 9264397 9584]] {?}
q: 9507340 x 7071 = ? [[x 9507340 7071]] {?}
q: 1644405 x 7749 = ? [[x 1644405 7749]] {?}
q: 1389294 x 8900 = ? [[x 1389294 8900]] {?}
q: 2778588 x 9867 = ? [[x 2778588 9867]] {?}
q: 204265 x 562402 = ? [[x 204265 562402]] {?}
q: In a mile are 63360 inches: how many inches are there in the circumference of the earth at the equator if the distance be 25000 miles? {~63360 * 25000}
q: The flow of the Mississippi at Memphis is about 434000 cubic feet a second: required the weight of water passing that point in one day of 86400 seconds, if a cubic foot of water weigh 62 pounds? Pounds? {~434000 * 86400 * 62}
q: John Sexton sold 25625 bushels of wheat, at $1.20 a bushel, and received in payment 320 acres of land, valued at $50 an acre; 60 head of horses, valued at $65 a head; 10 town lots, worth $150 each; and the remainder in money: how much money did he receive? Dollars? {~25625 * 1.20 - 320 * 50 - 60 * 65 - 10 * 150}
q: If light comes from the sun to the earth in 495 seconds, what is the distance from the earth to the sun, light moving 192500 miles a second? Miles? {~495 * 192500}
q: If 3702754400 cubic feet of solid matter is deposited in the Gulf of Mexico by the Mississippi every year, what is the deposit for 6000 years? Cubic feet? {~3702754400 * 6000}
q: The area of Missouri is 65350 square miles: how many acres are there in the State, allowing 640 acres to each square mile? {~65350 * 640}
q: In the United States, at the close of 1878, there were 81841 miles of railroad: if the average cost of building be $50000 a mile, what has been the total cost of building the railroads in this country? Dollars? {~81841 * 50000}
q: The number of pounds of tobacco produced in this country in 1870 was 260000000. If this were manufactured into plugs one inch wide and six inches long, and four plugs weigh a pound, what would be the length in inches of the entire crop? {~260000000 * 4 * 6}

== 6 | Multiplication | Bills and contractions
note: A Bill is an account of goods sold, with the price of each article; a Receipt is "Received Payment" signed at the foot of the bill. A Statement of Account shows a customer's debits and credits.
q: Joseph Allen bought 30 barrels of flour, at $3.60 a barrel; 48 barrels of mess pork, at $16.25 a barrel; 16 boxes of candles, at $3.50 a box; 23 barrels of molasses, at $28.75 a barrel; and 64 sacks of coffee, at $47.50 a sack. What does the flour come to? {108 @ 30 * 3.60} The pork? {780 @ 48 * 16.25} The candles? {56 @ 16 * 3.50} The molasses? {661.25 @ 23 * 28.75} The coffee? {3040 @ 64 * 47.50} The whole bill? {4645.25 @ 108 + 780 + 56 + 661.25 + 3040}
q: Chester Snyder bought 4 lb. of tea, at 40 ct.; 21 lb. of butter, at 21 ct.; 58 lb. of bacon, at 13 ct.; 16 lb. of lard, at 9 ct.; 30 lb. of cheese, at 12 ct.; 4 lb. of raisins, at 20 ct.; and 9 doz. of eggs, at 15 ct. What is the whole bill, in dollars? {20.74 @ (4 * 40 + 21 * 21 + 58 * 13 + 16 * 9 + 30 * 12 + 4 * 20 + 9 * 15) / 100}
sol: Tea $1.60, butter $4.41, bacon $7.54, lard $1.44, cheese $3.60, raisins $.80, eggs $1.35: in all $20.74. Received payment, Thomas Glenn.
q: John Smith's account: 525 McGuffey's Revised First Readers, @ 16c., and 50 Ray's New Higher Arithmetics, @ 75c.; he paid $20 cash and $12.75 in merchandise. What do the books come to? {121.50|121.5 @ 525 * .16 + 50 * .75} What does he still owe? {88.75 @ 121.50 - 20 - 12.75}
q: James Wilson & Co. bought of the Allegheny Coal Co. five hundred tons of coal, at $2.75 a ton, and sold the same Company 14 barrels of flour, at $6.55 a barrel; 6123 pounds of sugar, at 8c. a pound; they also paid them on account, cash, $687.50. What is the balance due the Allegheny Coal Co.? {105.96 @ 500 * 2.75 - 14 * 6.55 - 6123 * .08 - 687.50}
sol: To 500 tons coal, $1375.00. By flour $91.70, sugar $489.84, cash $687.50: $1269.04. Balance due, $105.96.
note: When the multiplier is a composite number, multiply by one factor, then this product by another. To multiply by 10, 100, 1000, annex as many ciphers. When ciphers are on the right of the factors, multiply without them and annex them to the product.
q: At 7 cents a piece, what will 6 melons cost? Cents? {42 @ 7 * 2 * 3}
sol: 3 times 2 times are 6 times: 7 cents x 2 = 14 cents, the cost of 2 melons; 14 x 3 = 42 cents, the cost of 6 melons.
q: At the rate of 37 miles a day, how far will a man walk in 28 days? Miles? {1036 @ 37 * 7 * 4}
q: Sound moves about 1130 feet per second: how far will it move in 54 seconds? Feet? {61020 @ 1130 * 6 * 9}
q: If an engine travel at an average speed of 25 miles an hour, how far can it travel in a week, or 168 hours? Miles? {4200 @ 25 * 168}
q: Multiply 743 by 10. {7430} Multiply 375 by 100. {37500} Multiply 207 by 1000. {207000}
q: Find the product of 5400 by 130. {702000 @ 5400 * 130}
sol: Find the product of 54 by 13, which is 702, and then annex three ciphers: 702000.
q: 15460 x 3200 = ? {~15460 * 3200}
q: 30700 x 5904000 = ? {~30700 * 5904000}
note: When the multiplier is a little less or greater than 10, 100, 1000, annex as many ciphers and subtract (or add) the multiplicand times the difference.
q: Multiply 3046 by 997. {3036862 @ 3046 * 997}
sol: 3046000 - 3 x 3046 = 3046000 - 9138 = 3036862.
q: 7023 x 99 = ? {695277 @ 7023 * 99}
q: 16642 x 996 = ? {16575432 @ 16642 * 996}
q: 372051 x 1002 = ? {372795102 @ 372051 * 1002}
q: Multiply 387295 by 216324. {83781203580 @ 387295 * 216324}
sol: 387295 x 3 = 1161885; that times 8 gives the product by 24, 9295080; that times 9 gives the product by 216, 83655720. Placed under the 3, the 4 and the 6, they add to 83781203580.
q: 38057 x 48618 = ? {1850255226 @ 38057 * 48618}
q: 267388 x 14982 = ? {4006007016 @ 267388 * 14982}
q: 481063 x 63721 = ? {30653815423 @ 481063 * 63721}
q: 66917 x 849612 = ? {56853486204 @ 66917 * 849612}
q: 102735 x 273162 = ? {28063298070 @ 102735 * 273162}
q: 536712 x 729981 = ? {391789562472 @ 536712 * 729981}

== 7 | Division | Long and short division
note: Find how often the divisor is contained in the left-hand figures of the dividend; multiply the divisor by this quotient figure, subtract, bring down the next figure, and divide as before. Proof: the divisor times the quotient, plus the remainder, is the dividend.
q: How many times is 24 cents contained in 73 cents? {3} What is the remainder, in cents? {1}
sol: 24 cents from 73 cents leaves 49; 24 from 49 leaves 25; 24 from 25 leaves 1. 24 cents is contained 3 times, with 1 cent remaining.
q: Divide $4225 equally among 13 men. Dollars each? {325 @ 4225 / 13}
sol: 13 in 42 hundreds, 3 hundreds times, 3 left; 13 in 32 tens, 2 tens times, 6 left; 13 in 65 units, 5 times. The quotient is 325.
q: 1004835 / 33. Quotient? {30449} Remainder? {18 @ 1004835 - 33 * 30449}
q: 5484888 / 67 = ? {81864 @ 5484888 / 67}
q: 4326422 / 961 = ? {4502 @ 4326422 / 961}
q: 1457924651 / 1204. Quotient? {1210900} Remainder? {1051 @ 1457924651 - 1204 * 1210900}
q: 65358547823 / 2789. Quotient? {23434402} Remainder? {645 @ 65358547823 - 2789 * 23434402}
q: 33333333333 / 5299. Quotient? {6290495} Remainder? {328 @ 33333333333 - 5299 * 6290495}
q: 245379633477 / 1263. Quotient? {194283161} Remainder? {1134 @ 245379633477 - 1263 * 194283161}
q: 555555555555 / 123456. Quotient? {4500028} Remainder? {98787 @ 555555555555 - 123456 * 4500028}
q: 555555555555 / 654321. Quotient? {849056} Remainder? {384579 @ 555555555555 - 654321 * 849056}
note: In the following, multiply A by itself, also B by itself, and divide the difference of the products as asked.
q: A is 2856 and B is 3765: divide the difference of their squares by the sum of A and B. {909 @ (3765 * 3765 - 2856 * 2856) / (2856 + 3765)}
q: A is 33698 and B is 42856: divide the difference of their squares by their sum. {9158 @ (42856 * 42856 - 33698 * 33698) / (33698 + 42856)}
q: A is 47932 and B is 152604: divide the difference of their squares by their sum. {104672 @ (152604 * 152604 - 47932 * 47932) / (47932 + 152604)}
q: A is 4986 and B is 5369: divide the difference of their squares by the difference of A and B. {10355 @ (5369 * 5369 - 4986 * 4986) / (5369 - 4986)}
q: A is 3973 and B is 4308: divide the difference of their squares by their difference. {8281 @ (4308 * 4308 - 3973 * 3973) / (4308 - 3973)}
q: A is 23798 and B is 59635: divide the difference of their squares by their difference. {83433 @ (59635 * 59635 - 23798 * 23798) / (59635 - 23798)}
q: A is 47329 and B is 65931: divide the difference of their squares by their difference. {113260 @ (65931 * 65931 - 47329 * 47329) / (65931 - 47329)}
q: If 25 acres produce 1825 bushels of wheat, how much is that per acre? Bushels? {73 @ 1825 / 25}
q: How many times 1024 in 1048576? {1024 @ 1048576 / 1024}
q: How many sacks, each containing 55 pounds, can be filled with 2035 pounds of flour? {37 @ 2035 / 55}
q: How many pages in a book of 7359 lines, each page containing 37 lines? Full pages? {198} Lines over? {33 @ 7359 - 37 * 198}
q: In what time will a vat of 10878 gallons be filled, at the rate of 37 gallons an hour? Hours? {294 @ 10878 / 37}
q: In what time will a vat of 3354 gallons be emptied, at the rate of 43 gallons an hour? Hours? {78 @ 3354 / 43}
q: The product of two numbers is 212492745; one is 1035; what is the other? {205307 @ 212492745 / 1035}
q: What number multiplied by 109, with 98 added to the product, will give 106700? {978 @ (106700 - 98) / 109}
note: In Short Division the work is done in the mind and only the quotient is written; a remainder is prefixed to the next figure.
q: How often is 2 cents contained in 652 cents? {326 @ 652 / 2}
sol: 2 in 6 hundreds, 3 hundreds times; 2 in 5 tens, 2 tens times, 1 ten left; 1 ten and 2 are 12, and 2 in 12, 6 times. The quotient is 326.
q: Divide 512653 by 5. Quotient? {102530} Remainder? {3}
q: Divide 534959 by 7. Quotient? {76422} Remainder? {5}
q: Divide 986028 by 8. Quotient? {123253} Remainder? {4}
q: Divide 986974 by 11. Quotient? {89724} Remainder? {10}
q: At $6 a head, how many sheep can be bought for $222? {37 @ 222 / 6}
q: At $5 a barrel, how many barrels of flour can be bought for $895? {179 @ 895 / 5}

== 8 | Division | Signs and contractions
note: The + or - before a number affects the whole result of the operations between it and the next + or -. When x and / come in succession, they take effect in the exact order in which they occur: 96 / 12 x 4 = 32.
q: Find the value of 4 x 3 + 7 x 2 - 9 x 3 + 6 x 4 - 3 x 3. {14 @ 4 * 3 + 7 * 2 - 9 * 3 + 6 * 4 - 3 * 3}
sol: 4 x 3 = 12, 7 x 2 = 14, 9 x 3 = 27 to be taken away, 6 x 4 = 24, 3 x 3 = 9 to be taken away. 12 + 14 + 24 = 50, and 27 + 9 = 36. 50 - 36 = 14.
q: 2 x 2 - 1 x 2 - 2 x 2 - 5 x 3 - 5 x 3 - 4 x 2 - 4 x 2 - 8 x 3 - 5 x 2 - 9 x 3 - 7 x 2 - 12 x 4 - 7 x 2 = ? Say what you get. {open}
sol: 4 - 2 - 4 - 15 - 15 - 8 - 8 - 24 - 10 - 27 - 14 - 48 - 14. Grouping, 4 - 189 = -185: there is 185 more to take away than to add.
q: 21 / 3 x 7 - 1 x 1 / 1 x 4 / 2 + 18 / 3 x 6 / (2 x 2) + (4 - 2 + 6 - 7) x 4 x 6 / 8 = ? {~21 / 3 * 7 - 1 * 1 / 1 * 4 / 2 + 18 / 3 * 6 / (2 * 2) + (4 - 2 + 6 - 7) * 4 * 6 / 8}
sol: 21 / 3 x 7 = 49; 1 x 1 / 1 x 4 / 2 = 2, taken away; 18 / 3 x 6 / 4 = 9; 1 x 4 x 6 / 8 = 3. 49 - 2 + 9 + 3 = 59.
q: 16 x 4 / 8 - 7 + 48 / 16 - 3 - 7 x 4 x 0 x 9 x 16 + 24 x 6 / 48 - 4 x 9 / 12 = ? {~16 * 4 / 8 - 7 + 48 / 16 - 3 - 7 * 4 * 0 * 9 * 16 + 24 * 6 / 48 - 4 * 9 / 12}
q: (16 / 16 x 96 / 8 - 7 - 5 + 3) x [(27 / 9) / 3 - 1] + (91 / 13 x 7 - 45 - 3) x 9 = ? {~(16 / 16 * 96 / 8 - 7 - 5 + 3) * ((27 / 9) / 3 - 1) + (91 / 13 * 7 - 45 - 3) * 9}
sol: The first parenthesis is 3, the bracket is 0, and the last parenthesis is 1. 3 x 0 + 1 x 9 = 9.
note: Multiplying or dividing both dividend and divisor by the same number does not change the quotient. To multiply by 33 1/3, 62 1/2, 16 2/3 and other simple parts of 100, multiply by 100 and take that part of the result.
q: Divide 217 by 15. Quotient? {14} True remainder? {7 @ 217 - 15 * 14}
sol: 15 = 3 x 5. 217 / 3 = 72 and 1 remainder; 72 / 5 = 14 and 2 remainder. The true remainder is 2 x 3 + 1 = 7.
q: Divide 23543 by 100. Quotient? {235} Remainder? {43}
q: Divide 3846 by 400. Quotient? {9} Remainder? {246 @ 3846 - 400 * 9}
q: Multiply 246 by 87 1/2. {21525 @ 246 * 87_1/2}
sol: 87 1/2 is 7/8 of 100: 24600 x 7 / 8 = 21525.
q: 122 x 33 1/3 = ? {~122 * 33_1/3}
q: 6564 x 62 1/2 = ? {~6564 * 62_1/2}
q: 10724 x 16 2/3 = ? {~10724 * 16_2/3}
q: Multiply 592643 by 66666. {39509138238 @ 592643 * 66666}
sol: 592643 x 99999 = 59263707357; 6 is 2/3 of 9, and 2/3 of this product is 39509138238.
q: 451402 x 3333 = ? {1504522866 @ 451402 * 3333}
q: 281257 x 555555 = ? {156253732635 @ 281257 * 555555}
q: 630224 x 4444000 = ? {2800715456000 @ 630224 * 4444000}
q: Divide 6903141128 by 21875. Quotient? {315572} True remainder? {3628 @ 6903141128 - 21875 * 315572}
sol: Multiply both by 8 and by 4: the divisor becomes 700000, the dividend 220900516096. The quotient is 315572, remainder 116096; divided by 8 and 4 this gives the true remainder, 3628.
q: 300521761 / 225. Quotient? {1335652} Remainder? {61 @ 300521761 - 225 * 1335652}
q: 1510337264 / 43750. Quotient? {34521} Remainder? {43514 @ 1510337264 - 43750 * 34521}
q: 22500712361 / 1406250. Quotient? {16000} Remainder? {712361 @ 22500712361 - 1406250 * 16000}
q: 620712480 / 20833 1/3. Quotient? {29794} Remainder? {~620712480 - 20833_1/3 * 29794}
q: 742851692 / 2916 2/3. Quotient? {254692} Remainder? {~742851692 - 2916_2/3 * 254692}

== 9 | Division | Miscellaneous exercises
q: A grocer gave 153 barrels of flour, worth $6 a barrel, for 54 barrels of sugar: what did the sugar cost per barrel? Dollars? {17 @ 153 * 6 / 54}
q: When the divisor is 35, quotient 217, and remainder 25, what is the dividend? {7620 @ 35 * 217 + 25}
q: What number besides 41 will divide 4879 without a remainder? {119 @ 4879 / 41}
q: Of what number is 103 both the divisor and the quotient? {10609 @ 103 * 103}
q: What is the nearest number to 53815, that can be divided by 375 without a remainder? {54000 @ 144 * 375}
sol: 53815 / 375 = 143 and 190 remainder. 144 times 375 = 54000, which is 185 more than 53815, while 143 times 375 is 190 less; so 54000 is the nearest.
q: A farmer bought 25 acres of land for $2675: what did 19 acres of it cost? Dollars? {2033 @ 2675 / 25 * 19}
q: I bought 15 horses, at $75 a head: at how much per head must I sell them to gain $210? Dollars? {89 @ 75 + 210 / 15}
q: A locomotive has 391 miles to run in 11 hours: after running 139 miles in 4 hours, at what rate per hour must the remaining distance be run? Miles an hour? {36 @ (391 - 139) / (11 - 4)}
q: A merchant bought 235 yards of cloth, at $5 per yard: after reserving 12 yards, what will he gain by selling the remainder at $7 per yard? Dollars? {386 @ 223 * 7 - 235 * 5}
q: A grocer bought 135 barrels of pork for $2295; he sold 83 barrels at the same rate at which he purchased, and the remainder at an advance of $2 per barrel: how much did he gain? Dollars? {104 @ (135 - 83) * 2}
q: A drover bought 5 horses, at $75 each, and 12 at $68 each; he sold them all at $73 each: what did he gain? Dollars? {50 @ 17 * 73 - 5 * 75 - 12 * 68} At what price per head must he have sold them to have gained $118? {77 @ (5 * 75 + 12 * 68 + 118) / 17}
q: A merchant bought 3 pieces of cloth of equal length, at $4 a yard; he gained $24 on the whole, by selling 2 pieces for $240: how many yards were there in each piece? {18 @ (240 - 24) / 3 / 4}
sol: $240 - $24 = $216, the whole cost; $216 / 3 = $72, the cost of 1 piece; $72 / $4 = 18 yards.
q: If 18 men can do a piece of work in 15 days, in how many days will one man do it? {270 @ 18 * 15}
q: If 13 men can build a wall in 15 days, in how many days can it be done if 8 men leave? {39 @ 13 * 15 / 5}
sol: 1 man can do it in 13 times 15 days, which is 195 days; and 5 men in 1/5 of 195 days, or 39 days.
q: If 14 men can perform a job of work in 24 days, in how many days can they perform it with the assistance of 7 more men? {16 @ 14 * 24 / 21}
q: A company of 45 men have provisions for 30 days: how many men must depart, that the provisions may last the remainder 50 days? {18 @ 45 - 45 * 30 / 50}
sol: For 1 day the provisions will support 30 times 45 men, or 1350 men; for 50 days, one fiftieth of 1350, or 27 men. 45 - 27 = 18 men.
q: A horse worth $85, and 3 cows at $18 each, were exchanged for 14 sheep and $41 in money: at how much each were the sheep valued? Dollars? {7 @ (85 + 3 * 18 - 41) / 14}
q: A drover bought an equal number of sheep and hogs for $1482: he gave $7 for a sheep, and $6 for a hog: what number of each did he buy? {114 @ 1482 / 13}
q: A trader bought a lot of horses and oxen for $1260; the horses cost $50, and the oxen $17, a head; there were twice as many oxen as horses: how many horses were there? {15 @ 1260 / 84} How many oxen? {30}
q: In a lot of silver change, worth 1050 cents, one seventh of the value is in 25-cent pieces; the rest is made up of 10-cent, 5-cent, and 3-cent pieces, of each an equal number: how many 25-cent pieces are there? {6 @ 1050 / 7 / 25} How many of each of the others? {50 @ (1050 - 150) / 18}
q: A speculator had 140 acres of land, which he might have sold at $210 an acre, and gained $6300; but after holding, he sold at a loss of $5600: how much an acre did the land cost him? Dollars? {165 @ 210 - 6300 / 140} How much an acre did he sell it for? {125 @ 165 - 5600 / 140}

== 10 | Properties of Numbers | Prime factors and divisors
note: A prime number can be exactly divided only by itself and 1. To find the prime factors, divide by the smallest prime number that will exactly divide the number, then the quotient in like manner, until the quotient is prime. Type the prime factors from the smallest.
q: Find the prime factors of 42. {2} times {3} times {7}
sol: 42 is divisible by 2, and 21 is divisible by 3 and 7; hence the prime factors of 42 are 2, 3, 7, and 2 x 3 x 7 = 42.
q: Find the prime factors of 45. {3} times {3} times {5}
q: Find the prime factors of 54. {2} times {3} times {3} times {3}
q: Find the prime factors of 72. {2} times {2} times {2} times {3} times {3}
q: Find the prime factors of 75. {3} times {5} times {5}
q: Find the prime factors of 96. {2} times {2} times {2} times {2} times {2} times {3}
q: Find the prime factors of 98. {2} times {7} times {7}
q: Factor 210. {2} times {3} times {5} times {7}
q: Factor 1155. {3} times {5} times {7} times {11}
q: Factor 10010. {2} times {5} times {7} times {11} times {13}
q: Factor 36414. {2} times {3} times {3} times {7} times {17} times {17}
q: Factor 58425. {3} times {5} times {5} times {19} times {41}
q: Find the prime factors common to 42 and 98. {2} times {7}
q: Find the prime factors common to 45 and 105. {3} times {5}
q: Find the prime factors common to 90 and 210. {2} times {3} times {5}
q: Find the prime factors common to 210 and 315. {3} times {5} times {7}
note: To find all the divisors of a number, resolve it into its prime factors and form all the different products of them: 42 = 2 x 3 x 7, and its divisors are 2, 3, 7, 6, 14, 21.
q: Find all the divisors of 70. {open}
sol: 70 = 2 x 5 x 7; its divisors are 2, 5, 7, 10, 14, 35 (and 1 and 70).
q: Find all the divisors of 196. {open}
sol: 196 = 2 x 2 x 7 x 7; its divisors are 2, 4, 7, 14, 28, 49, 98 (and 1 and 196).
q: Find all the divisors of 231. {open}
sol: 231 = 3 x 7 x 11; its divisors are 3, 7, 11, 21, 33, 77 (and 1 and 231).
q: Find all the divisors of 496, and name the properties of 496. {open}
sol: 496 = 2 x 2 x 2 x 2 x 31; its divisors are 1, 2, 4, 8, 16, 31, 62, 124, 248. Their sum is 496, so 496 is a perfect number; it is also even and composite.
q: What is the sum of all the divisors of 28, besides 28 itself? {28 @ 1 + 2 + 4 + 7 + 14} Is 28 perfect, abundant or defective? {open}
sol: 1 + 2 + 4 + 7 + 14 = 28: 28 is a perfect number.

== 11 | Properties of Numbers | Greatest common divisor
note: Resolve the numbers into their prime factors, and take the product of the factors common to all. Or: divide the greater number by the less, the divisor by the remainder, and so on, till nothing remains; the last divisor is the G. C. D.
q: Find the G. C. D. of 30 and 105. {15}
sol: 30 = 2 x 3 x 5; 105 = 3 x 5 x 7. The common factors are 3 and 5: 3 x 5 = 15.
q: Find the G. C. D. of 36, 63, 144, and 324. {9}
sol: Divide all by 3: 12, 21, 48, 108; again by 3: 4, 7, 16, 36, which have no common factor. 3 x 3 = 9.
q: Find the greatest common divisor of 30 and 42. {6}
q: Find the greatest common divisor of 42 and 70. {14}
q: Find the greatest common divisor of 63 and 105. {21}
q: Find the greatest common divisor of 66 and 165. {33}
q: Find the greatest common divisor of 90 and 150. {30}
q: Find the greatest common divisor of 60 and 84. {12}
q: Find the greatest common divisor of 90 and 225. {45}
q: Find the greatest common divisor of 112 and 140. {28}
q: Find the greatest common divisor of 30, 45, and 75. {15}
q: Find the greatest common divisor of 84, 126, and 210. {42}
q: Find the greatest common divisor of 16, 40, 88, and 96. {8}
q: Find the greatest common divisor of 21, 42, 63, and 126. {21}
q: Find the G. C. D. of 348 and 1024, by successive divisions. {4}
sol: 348 into 1024, 2 times, remainder 328; 328 into 348, once, remainder 20; 20 into 328, 16 times, remainder 8; 8 into 20, twice, remainder 4; 4 into 8, twice, nothing over. The G. C. D. is 4.
q: Find the greatest common divisor of 85 and 120. {5}
q: Find the greatest common divisor of 91 and 133. {7}
q: Find the greatest common divisor of 357 and 525. {21}
q: Find the greatest common divisor of 425 and 493. {17}
q: Find the greatest common divisor of 324 and 1161. {27}
q: Find the greatest common divisor of 589 and 899. {31}
q: Find the greatest common divisor of 597 and 897. {3}
q: Find the greatest common divisor of 825 and 1287. {33}
q: Find the greatest common divisor of 423 and 2313. {9}
q: Find the greatest common divisor of 18607 and 24587. {23}
q: Find the greatest common divisor of 105, 231, and 1001. {7}
q: Find the greatest common divisor of 165, 231, and 385. {11}
q: Find the greatest common divisor of 816, 1360, 2040, and 4080. {136}
q: Find the greatest common divisor of 1274, 2002, 2366, 7007, and 13013. {91}

== 12 | Properties of Numbers | Least common multiple
note: Resolve each number into its prime factors, and take the product of all the different prime factors, each the greatest number of times it occurs in any one number. Or: divide by any prime that will divide two or more of the numbers, and so on; the product of the divisors and the last line is the L. C. M.
q: Find the L. C. M. of 10, 12, and 15. {60}
sol: 10 = 2 x 5; 12 = 2 x 2 x 3; 15 = 3 x 5. The L. C. M. must contain 2 twice, 3 and 5: 2 x 2 x 3 x 5 = 60.
q: Find the L. C. M. of 8, 10, 15. {120}
q: Find the L. C. M. of 6, 9, 12. {36}
q: Find the L. C. M. of 12, 18, 24. {72}
q: Find the L. C. M. of 8, 14, 21, 28. {168}
q: Find the L. C. M. of 10, 15, 20, 30. {60}
q: Find the L. C. M. of 15, 30, 70, 105. {210}
q: Find the L. C. M. of 10, 20, 25, and 30. {300}
sol: Strike out 10, which is contained in 20. Divide by 2: 10, 25, 15; by 5: 2, 5, 3, prime to one another. 2 x 5 x 2 x 5 x 3 = 300.
q: Find the least common multiple of 6, 9, 20. {180}
q: Find the least common multiple of 15, 20, 30. {60}
q: Find the least common multiple of 7, 11, 13, 5. {5005}
q: Find the least common multiple of 35, 45, 63, 70. {630}
q: Find the least common multiple of 8, 15, 20, 25, 30. {600}
q: Find the least common multiple of 30, 45, 48, 80, 120, 135. {2160}
q: Find the least common multiple of 174, 485, 4611, 14065, 15423. {4472670}
q: Find the least common multiple of 498, 85988, 235803, 490546. {244291908}
q: Find the least common multiple of 2183, 2479, 3953. {146261}
sol: By successive divisions the common divisors are 37 and 67: 2183 = 37 x 59, 2479 = 37 x 67, 3953 = 59 x 67. The L. C. M. is 37 x 59 x 67 = 146261.
q: Find the least common multiple of 1271, 2573, 3403. {105493}

== 13 | Properties of Numbers | Casting out nines and cancellation
note: To cast the 9's out of a number, add its digits, dropping 9 whenever the sum reaches it; what is left is the excess. A number divided by 9 leaves the same remainder as the sum of its digits.
q: Find the excess of 9's in 768945. {3}
sol: 7 + 6 are 13; drop the 9: 4. 4 + 8 are 12; drop the 9: 3. 3 + 4 + 5 are 12; drop the 9: the excess is 3. The 9 in the number is not counted.
q: Find the excess of 9's in 835. {7} In 76. {4} In 63460. {1}
sol: 835 x 76 = 63460. The excesses of the factors, 7 and 4, multiplied give 28, whose excess is 1, the same as the excess of the product: the work is right.
q: Find the excess of 9's in 7352. {8} In 5834. {2} In 6241. {4} In 7302. {3} In their sum, 26729. {8}
sol: 7 + 3 + 5 + 2 = 17, excess 8; 5834, excess 2; 6241, excess 4; 7302, excess 3. 8 + 2 + 4 + 3 = 17, excess 8. The sum, 26729, also has excess 8 (2 + 6 + 7 + 2 + 9 = 26): the addition is right.
note: Cancel equal factors from dividend and divisor; multiply together the factors left in the dividend, and divide by the product of those left in the divisor.
q: Multiply 75, 153, and 28 together, and divide by the product of 63 and 36. {~75 * 153 * 28 / (63 * 36)}
sol: Cancel 4 out of 28 and 36, leaving 7 and 9; cancel 7 out of 7 and 63, leaving 9; cancel 9 out of 9 and 153, leaving 17; cancel 3 out of 9 and 75, leaving 25 and 3. 25 x 17 = 425, divided by 3 gives 141 2/3.
q: How many cows, worth $24 each, can I get for 9 horses, worth $80 each? {30 @ 9 * 80 / 24}
q: I exchanged 8 barrels of molasses, each containing 33 gallons, at 40 cents a gallon, for 10 chests of tea, each containing 24 pounds: how much a pound did the tea cost me? Cents? {44 @ 8 * 33 * 40 / (10 * 24)}
q: How many bales of cotton, of 400 pounds each, at 12 cents a pound, are equivalent to 6 hogsheads of sugar, 900 pounds each, at 8 cents a pound? {9 @ 6 * 900 * 8 / (400 * 12)}
q: Divide 15 x 24 x 112 x 40 x 10 by 25 x 36 x 56 x 90. {~15 * 24 * 112 * 40 * 10 / (25 * 36 * 56 * 90)}
== 14 | Fractions | Reducing fractions
note: A fraction is in its lowest terms when its terms are prime to each other: divide both terms by their greatest common divisor. Type a fraction with the / key (7/8); a mixed number with the "and" key (4 and 5/8).
q: Write in figures: Seven eighths. {=7/8} Four elevenths. {=4/11} Five thirteenths. {=5/13} One seventeenth. {=1/17} Three twenty-ninths. {=3/29}
q: Write in figures: Eight twenty-firsts. {=8/21} Nine forty-seconds. {=9/42} Nineteen ninety-thirds. {=19/93} Thirteen one-hundredths. {=13/100} Twenty-four one-hundred-and-fifteenths. {=24/115}
q: Reduce 20/30 to its lowest terms. {=2/3}
sol: Dividing both terms by 2, 10/15; dividing by 5, 2/3. Or dividing at once by 10, the greatest common divisor: 2/3.
q: Reduce to lowest terms: 30/45. {=2/3} 32/56. {=4/7} 42/189. {=2/9} 105/195. {=7/13} 154/210. {=11/15}
q: Reduce to lowest terms: 156/221. {=12/17} 253/414. {=11/18} 667/783. {=23/27} 1767/4557. {=19/49} 9702/18522. {=11/21}
q: Express in its simplest form: 923 / 1491. {=13/21} 890 / 1691. {=10/19} 2261 / 4123. {=17/31} 6160 / 40480. {=7/46}
q: Reduce 5/8 to fortieths. How many fortieths? {25 @ 5 * 40 / 8}
sol: 40 / 8 = 5; multiply both terms by 5: 5 x 5 over 8 x 5 = 25/40.
q: Reduce 3/11 and 4/33 to ninety-ninths. 3/11 = ? {=27/99} 4/33 = ? {=12/99}
q: Reduce 4/9, 3/7, and 3/21 to sixty-thirds. 4/9 = ? {=28/63} 3/7 = ? {=27/63} 3/21 = ? {=9/63}
q: Reduce 8/17, 9/19, and 11/21 to fractions having 6783 for a denominator. The numerators: {3192 @ 8 * 6783 / 17} {3213 @ 9 * 6783 / 19} {3553 @ 11 * 6783 / 21}
q: Reduce 3 3/4 to fourths. How many fourths? {15 @ 3 * 4 + 3}
sol: In 1 there are 4 fourths; in 3, 12 fourths; and 12 fourths + 3 fourths = 15 fourths.
q: In $7 3/8, how many eighths of a dollar? {59 @ 7 * 8 + 3}
q: In 19 3/4 gallons, how many fourths? {79 @ 19 * 4 + 3}
q: In 13 37/60 hours, how many sixtieths? {817 @ 13 * 60 + 37}
q: Reduce to improper fractions: 11 2/3. {=35/3} 15 8/11. {=173/11} 127 11/17. {=2170/17}
q: Reduce to improper fractions: 109 9/19. {=2080/19} 5 207/211. {=1262/211} 13 51/73. {=1000/73}
q: Reduce 8 to a fraction whose denominator is 7. {=56/7}
q: Reduce 13/5 of a dollar to dollars. {=2 3/5}
sol: 5 fifths make 1 dollar; 5 fifths are contained in 13 fifths 2 3/5 times: 2 3/5 dollars.
q: In 37/8 of a dollar, how many dollars? {=4 5/8}
q: In 137/4 of a bushel, how many bushels? {=34 1/4}
q: In 785/60 of an hour, how many hours? {=13 1/12}
q: Reduce to whole or mixed numbers: 23/23. {1} 1295/37. {35} 800/9. {=88 8/9}
q: Reduce to whole or mixed numbers: 1162/11. {=105 7/11} 4260/13. {=327 9/13} 15780/31. {=509 1/31}
note: To reduce a compound fraction (a fraction of a fraction), multiply the numerators together and the denominators together, canceling common factors. Mixed numbers are first made improper fractions.
q: Reduce 3/4 of 5/7 to a simple fraction. {~3/4 * 5/7}
sol: 1/4 of 1/7 = 1/28; 1/4 of 5/7 = 5/28; and 3/4 of 5/7 = 3 x 5 over 4 x 7 = 15/28.
q: Reduce 2/3 of 9/10 of 7/12 to a simple fraction. {~2/3 * 9/10 * 7/12}
sol: Indicate the work, 2 x 9 x 7 over 3 x 10 x 12, and cancel: 3 out of 9 and 3, leaving 3; 2 out of 2 and 10, leaving 5; 3 out of 3 and 12, leaving 4. 7 over 5 x 4 = 7/20.
q: Reduce to a simple fraction: 1/3 of 3/4 of 4/7. {~1/3 * 3/4 * 4/7}
q: Reduce to a simple fraction: 2/5 of 4/7 of 2 5/8. {~2/5 * 4/7 * 2_5/8}
q: Reduce to a simple fraction: 4/5 of 15/16 of 2 2/3. {~4/5 * 15/16 * 2_2/3}
q: Reduce to a simple fraction: 1/2 of 4/5 of 3 3/4. {~1/2 * 4/5 * 3_3/4}
q: Reduce to a simple fraction: 3/4 of 8/9 of 4/7 of 8 3/4. {~3/4 * 8/9 * 4/7 * 8_3/4}
q: Reduce to a simple fraction: 1/3 of 3/5 of 6/7 of 3/4 of 4 2/3. {~1/3 * 3/5 * 6/7 * 3/4 * 4_2/3}
q: Reduce to a simple fraction: 8/11 of 3/7 of 4/19 of 77/24 of 7 1/8. {~8/11 * 3/7 * 4/19 * 77/24 * 7_1/8}
q: Reduce to a simple fraction: 12/13 of 9/16 of 7/18 of 10/21 of 1 4/35. {~12/13 * 9/16 * 7/18 * 10/21 * 1_4/35}

== 15 | Fractions | Common denominators
note: To reduce fractions to a common denominator, multiply both terms of each fraction by the denominators of the other fractions. For the least common denominator, take the L. C. M. of the denominators, and multiply each numerator by the L. C. D. divided by its denominator.
q: Reduce 1/2, 2/3, and 3/4 to a common denominator, by multiplying by the other denominators. The denominator? {24} The numerators: {12} {16} {18}
sol: 2 x 3 x 4 = 24. The terms of 1/2 are multiplied by 3 x 4, of 2/3 by 2 x 4, of 3/4 by 2 x 3: 12/24, 16/24, 18/24.
q: Reduce 1/2, 3/5, and 6/7 to a common denominator. The denominator? {70} The numerators: {35} {42} {60}
q: Reduce 1/2, 2/3, 3/5 to a common denominator. The denominator? {30} The numerators: {15} {20} {18}
q: Reduce 1/4, 1/5, 1/6 to a common denominator, multiplying by the other denominators. The denominator? {120} The numerators: {30} {24} {20}
q: Reduce 2/3, 3/7, 5/8 to a common denominator. The denominator? {168} The numerators: {112} {72} {105}
q: Reduce 1/2, 3/5, 5/6, 7/8 to a common denominator, multiplying by the other denominators. The denominator? {480} The numerators: {240} {288} {400} {420}
q: Reduce 2/3, 1/2 of 3 1/2, and 2/3 of 3/5 to a common denominator. (First make them simple: 2/3, 7/4, 2/5.) The denominator? {60} The numerators: {40} {105} {24}
q: Reduce 2/3 of 6/7, 3/4 of 8/9, and 1/2 of 4/5 of 3/7 of 2 5/8 to a common denominator. (In lowest terms they are 4/7, 2/3, 9/20.) The denominator? {420} The numerators: {240} {280} {189}
q: Reduce 1/3 and 5/6 to a common denominator. 1/3 = how many sixths? {2}
sol: The largest denominator, 6, is a multiple of 3: multiply both terms of 1/3 by 6 / 3 = 2, and it is 2/6.
q: Reduce 1/2, 3/4, and 5/8 to a common denominator. The denominator? {8} The numerators: {4} {6} {5}
q: Reduce 2/3, 5/6 and 7/12 to a common denominator. The denominator? {12} The numerators: {8} {10} {7}
q: Reduce 3/4, 4/5, 9/10 and 11/20 to a common denominator. The denominator? {20} The numerators: {15} {16} {18} {11}
q: Reduce 5/8, 7/9, and 9/24 to the least common denominator. The L. C. D.? {72} The numerators: {45} {56} {27}
sol: The L. C. M. of 8, 9, and 24 is 72. 72 / 8 = 9, and 5 x 9 = 45; 72 / 9 = 8, and 7 x 8 = 56; 72 / 24 = 3, and 9 x 3 = 27.
q: Reduce 1/3, 3/4, 5/6 to the least common denominator. The L. C. D.? {12} The numerators: {4} {9} {10}
q: Reduce 1/2, 3/5, 9/10, 3/4 to the least common denominator. The L. C. D.? {20} The numerators: {10} {12} {18} {15}
q: Reduce 3/7, 5/8, 11/14 to the least common denominator. The L. C. D.? {56} The numerators: {24} {35} {44}
q: Reduce 3/4, 6/8, 9/12, 15/20 to the least common denominator, putting each in its simplest form first. What is each one in lowest terms? {=3/4}
sol: 6/8, 9/12 and 15/20 all reduce to 3/4: the four fractions are equal, and need no common denominator but 4.
q: Reduce 6/9, 9/12, 12/20, 7/10 to the least common denominator, in simplest form first. The L. C. D.? {60} The numerators: {40} {45} {36} {42}
q: Reduce 1 3/4, 3 2/3, and 3/10 of 3 4/7 to the least common denominator. The L. C. D.? {84} The numerators: {147} {308} {90}
sol: 1 3/4 = 7/4; 3 2/3 = 11/3; 3/10 of 25/7 = 15/14. The L. C. M. of 4, 3, 14 is 84: 147/84, 308/84, 90/84.

== 16 | Fractions | Adding and subtracting fractions
note: Reduce the fractions to a common denominator, add (or subtract) their numerators, and write the result over the common denominator. Mixed numbers may be added separately and their sums united. Reduce the answer to its lowest terms.
q: What is the sum of 2/3, 5/8, and 7/12? {~2/3 + 5/8 + 7/12}
sol: Reduced to a common denominator, 16/24, 15/24, 14/24. Their sum is 45/24 = 1 7/8.
q: Add 1/6, 2/9, and 5/12. {~1/6 + 2/9 + 5/12}
q: Add 2/3, 3/5, 7/9, and 4/15. {~2/3 + 3/5 + 7/9 + 4/15}
q: Add 1 2/3 and 2 3/5. {~1_2/3 + 2_3/5}
q: Add 2 1/4, 3 2/7, and 4 5/6. {~2_1/4 + 3_2/7 + 4_5/6}
q: Add 6/10, 4/14, 8/12, and 2 1/3. {~6/10 + 4/14 + 8/12 + 2_1/3}
q: Add 1 1/2, 2 1/3, 3 1/4, and 4 1/5. {~1_1/2 + 2_1/3 + 3_1/4 + 4_1/5}
q: Add 2/3 of 4/5, and 3/7 of 5/8 of 2 1/3. {~2/3 * 4/5 + 3/7 * 5/8 * 2_1/3}
q: 3/8 + 1/22 + 7/24 + 29/88 = ? {~3/8 + 1/22 + 7/24 + 29/88}
q: 7/8 + 11/12 + 17/18 + 23/24 + 20/27 = ? {~7/8 + 11/12 + 17/18 + 23/24 + 20/27}
q: 4/7 of 96 1/4 + 8/9 of 11/12 of 5 1/6 = ? {~4/7 * 96_1/4 + 8/9 * 11/12 * 5_1/6}
q: 2/3 + 8/9 + 26/27 + 80/81 + 242/243 + 728/729 = ? {~2/3 + 8/9 + 26/27 + 80/81 + 242/243 + 728/729}
q: Find the difference between 5/6 and 8/15. {~5/6 - 8/15}
sol: 5/6 = 25/30 and 8/15 = 16/30; the difference is 9/30 = 3/10.
q: 4/5 - 5/12 = ? {~4/5 - 5/12}
q: 8/11 - 3/17 of 1/2 = ? {~8/11 - 3/17 * 1/2}
q: 11/54 - 3/28 of 5/6 = ? {~11/54 - 3/28 * 5/6}
q: 5/11 - 1/13 of 4 = ? {~5/11 - 1/13 * 4}
q: 11/14 - 4/63 = ? {~11/14 - 4/63}
q: 9/55 - 1/15 = ? {~9/55 - 1/15}
q: 7/45 - 11/75 = ? {~7/45 - 11/75}
q: 10/39 - 8/65 = ? {~10/39 - 8/65}
q: Subtract 23 15/16 from 31 3/4. {~31_3/4 - 23_15/16}
sol: 3/4 = 12/16. 15/16 can not be taken from 12/16: take a unit, 16/16, from 31 and add it, 28/16. 28/16 - 15/16 = 13/16; 30 - 23 = 7. The answer is 7 13/16.
q: 12 3/4 - 10 13/16 = ? {~12_3/4 - 10_13/16}
q: 12 23/28 - 9 27/35 = ? {~12_23/28 - 9_27/35}
q: 5 23/32 - 2 2/7 = ? {~5_23/32 - 2_2/7}
q: 7 5/12 - 3 1/2 = ? {~7_5/12 - 3_1/2}
q: 15 - 3/7 = ? {~15 - 3/7}
q: 18 - 5 3/8 = ? {~18 - 5_3/8}
q: 5/3 of 2 7/9 - 3 17/18 = ? {~5/3 * 2_7/9 - 3_17/18}
q: 3 1/3 - 4/5 of 1 7/8 = ? {~3_1/3 - 4/5 * 1_7/8}
q: 16/3 of 4 1/2 - 13/4 of 3 1/5 = what? {~16/3 * 4_1/2 - 13/4 * 3_1/5}
q: 11 2/3 + 8 7/9 - 9 19/22 = what? {~11_2/3 + 8_7/9 - 9_19/22}
q: A man owned 25/38 of a ship, and sold 3/5 of his share: how much had he left? {~25/38 - 3/5 * 25/38}
q: After selling 4/7 of 5/8 + 1/5 of 3/7 of a farm, what part of it remains? {~1 - (4/7 * 5/8 + 1/5 * 3/7)}
q: 3 1/4 + 4 2/5 - 5 1/2 + 16 5/8 - 7 11/24 + 10 - 14 5/6, is equal to what? {~3_1/4 + 4_2/5 - 5_1/2 + 16_5/8 - 7_11/24 + 10 - 14_5/6}
q: 5 1/5 - 2 5/6 + 13/2 - 3 3/10 + 3 1/12 + 8 1/9 - 16 1/4, is equal to what? {~5_1/5 - 2_5/6 + 13/2 - 3_3/10 + 3_1/12 + 8_1/9 - 16_1/4}
q: 1 - 3/8 of 5/6 - 2/3 of 3/7 = what? {~1 - 3/8 * 5/6 - 2/3 * 3/7}

== 17 | Fractions | Multiplying fractions
note: Multiply the numerators together for the numerator of the product, and the denominators for the denominator, canceling common factors. Mixed numbers are first made improper fractions.
q: Multiply 3/7 by 5/8. {~3/7 * 5/8}
sol: 1/8 times 3/7 is 3/56; 5/8 times 3/7 is 5 times 3/56 = 15/56.
q: Multiply 3/4 by 6. {~3/4 * 6}
q: Multiply 8 by 3/5. {~8 * 3/5}
q: 10/13 x 12 = ? {~10/13 * 12}
q: 11/24 x 18 = ? {~11/24 * 18}
q: 29/48 x 24 = ? {~29/48 * 24}
q: 9/16 x 28 = ? {~9/16 * 28}
q: 13/15 x 30 = ? {~13/15 * 30}
q: 3 2/3 x 5 = ? {~3_2/3 * 5}
sol: 5 x 3 = 15 and 5 x 2/3 = 3 1/3: in all 18 1/3. Or 3 2/3 = 11/3, and 11/3 x 5 = 55/3 = 18 1/3.
q: 45 x 7/9 = ? {~45 * 7/9}
q: 50 x 11/14 = ? {~50 * 11/14}
q: 25 x 3/4 = ? {~25 * 3/4}
q: 32 x 2 3/8 = ? {~32 * 2_3/8}
q: 28 x 3 2/3 = ? {~28 * 3_2/3}
q: 12/35 x 7/16 = ? {~12/35 * 7/16}
q: 15/16 x 24/25 = ? {~15/16 * 24/25}
q: 42/55 x 22/35 = ? {~42/55 * 22/35}
q: What will 3 1/3 yards of cloth cost, at $4 1/2 per yard? Dollars? {~3_1/3 * 4_1/2}
sol: 4 1/2 = 9/2; 3 1/3 = 10/3; 9/2 x 10/3 = 90/6 = $15. Or: 3 yards cost $13 1/2, and 1/3 of a yard costs $1 1/2: $15.
q: 6 2/3 x 4 1/2 = ? {~6_2/3 * 4_1/2}
q: 4 4/5 x 2 2/3 = ? {~4_4/5 * 2_2/3}
q: 12 3/8 x 3 3/11 = ? {~12_3/8 * 3_3/11}
q: 7 11/12 x 3 7/19 = ? {~7_11/12 * 3_7/19}
q: Multiply 1/5 of 8 by 1/4 of 10. {~1/5 * 8 * 1/4 * 10}
q: Multiply 2/3 of 5 2/5 by 3/5 of 3 1/3. {~2/3 * 5_2/5 * 3/5 * 3_1/3}
q: Multiply 3/4 of 2/3 of 5 3/5 by 3/7 of 3 3/8. {~3/4 * 2/3 * 5_3/5 * 3/7 * 3_3/8}
q: Multiply 5, 4 1/4, 2 1/3, and 3/7 of 4 4/9. {~5 * 4_1/4 * 2_1/3 * 3/7 * 4_4/9}
q: Multiply 4/5, 3/7, 5/11, 1/3 of 2 1/2, and 4/7 of 3 1/7. {~4/5 * 3/7 * 5/11 * 1/3 * 2_1/2 * 4/7 * 3_1/7}
q: Multiply 3/5, 7/9, 9/11, 3 1/3, and 3 1/7. {~3/5 * 7/9 * 9/11 * 3_1/3 * 3_1/7}
q: Multiply 3 1/2, 4 2/3, 5 3/5, 2/9 of 5/14, and 6 3/4. {~3_1/2 * 4_2/3 * 5_3/5 * 2/9 * 5/14 * 6_3/4}
q: At 7/8 of a dollar per yard, what will 25 yards of cloth cost? Dollars? {~7/8 * 25}
q: A quantity of provisions will last 25 men 12 3/4 days: how long will the same last one man? Days? {~25 * 12_3/4}
q: At 3 1/2 cents a yard, what will 2 3/4 yards of tape cost? Cents? {~3_1/2 * 2_3/4}
q: What must be paid for 3/5 of 2/3 of a lot of groceries that cost $18 3/4? Dollars? {~3/5 * 2/3 * 18_3/4}
q: K owns 5/8 of a ship, and sells 3/4 of his share to L: what part has he left? {~5/8 - 3/4 * 5/8}

== 18 | Fractions | Dividing fractions
note: Multiply the dividend by the divisor with its terms inverted. A complex fraction is reduced by dividing its numerator by its denominator.
q: Divide 5/6 by 4/7. {~5/6 / (4/7)}
sol: 1/7 is contained in 1 seven times; in 1/6, 7/6 times; in 5/6, 35/6 times; 4/7 is contained 1/4 as often, 35/24 = 1 11/24 times.
q: Divide 3 by 2/5. {~3 / (2/5)}
q: Divide 4/7 by 2. {~4/7 / 2}
q: Divide 2/9 by 3/5. {~2/9 / (3/5)}
sol: 2/9 = 10/45 and 3/5 = 27/45; 10/45 divided by 27/45 is 10 / 27 = 10/27. Or 2/9 x 5/3 = 10/27.
q: 9/16 / 3 = ? {~9/16 / 3}
q: 14/23 / 7 = ? {~14/23 / 7}
q: 3/5 / 8 = ? {~3/5 / 8}
q: 6 / 2/3 = ? {~6 / (2/3)}
q: 21 / 9/10 = ? {~21 / (9/10)}
q: 3/4 / 1/2 = ? {~3/4 / (1/2)}
q: 2/3 / 1/40 = ? {~2/3 / (1/40)}
q: 21/25 / 14/15 = ? {~21/25 / (14/15)}
q: 12/35 / 30/77 = ? {~12/35 / (30/77)}
q: 1 3/4 / 5 = ? {~1_3/4 / 5}
q: 8 1/6 / 2/3 = ? {~8_1/6 / (2/3)}
q: 19 1/2 / 1 7/8 = ? {~19_1/2 / 1_7/8}
q: 73 1/2 / 9 4/5 = ? {~73_1/2 / 9_4/5}
q: 54 43/48 / 25 5/6 = ? {~54_43/48 / 25_5/6}
q: Divide 1 1/2 by 1/2 of 3/5 of 7 1/2. {~1_1/2 / (1/2 * 3/5 * 7_1/2)}
q: Divide 3/10 of 3/16 of 5/12 by 7/24 of 12/35. {~3/10 * 3/16 * 5/12 / (7/24 * 12/35)}
q: Divide 3/4 of 1/2 by 1/4 of 2/3. {~3/4 * 1/2 / (1/4 * 2/3)}
q: Divide 7/9 of 3 3/5 by 13/14 of 7. {~7/9 * 3_3/5 / (13/14 * 7)}
q: Divide 1/3 of 4/5 x 39/320 by 2/75 of 3 1/4. {~1/3 * 4/5 * 39/320 / (2/75 * 3_1/4)}
q: Divide 2/7 of 5 1/2 by 2/5 of 9/14 of 3 1/3. {~2/7 * 5_1/2 / (2/5 * 9/14 * 3_1/3)}
q: Divide 1/3 of 2/7 of 4/11 by 2/5 of 1/3 of 4/7. {~1/3 * 2/7 * 4/11 / (2/5 * 1/3 * 4/7)}
q: Divide 1 7/8 times 4 2/3 by 1 7/11 times 3 8/9. {~1_7/8 * 4_2/3 / (1_7/11 * 3_8/9)}
q: Divide 3 1/7 by 4/9 of 8 1/4 times 5/11 of 3 3/10. {~3_1/7 / (4/9 * 8_1/4 * 5/11 * 3_3/10)}
q: Divide 9/11 of 2/3 of 27 1/2 by 4/9 of 3/17 of 5 1/2. {~9/11 * 2/3 * 27_1/2 / (4/9 * 3/17 * 5_1/2)}
q: What is 2 5/8 x 4/5 of 19 1/3 / (4 5/6 x 3/10 of 8)? {~2_5/8 * 4/5 * 19_1/3 / (4_5/6 * 3/10 * 8)}
q: Reduce the complex fraction 1 5/6 over 2 1/4 to a simple fraction. {~1_5/6 / 2_1/4}
sol: 1 5/6 = 11/6 and 2 1/4 = 9/4; 11/6 divided by 9/4 = 11/6 x 4/9 = 44/54 = 22/27.
q: Reduce to a simple fraction: 3/8 over 11/4. {~3/8 / (11/4)}
q: Reduce to a simple fraction: 1 3/4 over 2 4/5. {~1_3/4 / 2_4/5}
q: Reduce to a simple fraction: 4 2/3 over 2 11/12. {~4_2/3 / 2_11/12}
q: Reduce to a simple fraction: 2 5/14 over 2 2/21. {~2_5/14 / 2_2/21}
q: Reduce to a simple fraction: 12 3/8 over 18. {~12_3/8 / 18}
q: Reduce to a simple fraction: 62 over 16 10/11. {~62 / 16_10/11}
q: 7/10 x (4 1/4 over 8) = ? {~7/10 * 4_1/4 / 8}
q: 31/25 x (8 1/2 over 10 1/5) = ? {~31/25 * 8_1/2 / 10_1/5}
q: (5 1/3 over 9 1/5) x (11 11/12 over 24 1/5) = ? {~5_1/3 / 9_1/5 * 11_11/12 / 24_1/5}
q: (6 1/4 over 2 2/5) / (7 over 12 1/2) = ? {~6_1/4 / 2_2/5 / (7 / 12_1/2)}
q: (7 3/7 over 40 5/9) / (17 1/3 over 73) = ? {~7_3/7 / 40_5/9 / (17_1/3 / 73)}
q: (2 4/11 over 2 3/5) / (2 7/11 over 8 7/10) = ? {~2_4/11 / 2_3/5 / (2_7/11 / 8_7/10)}

== 19 | Fractions | Common divisors and multiples of fractions
note: The G. C. D. of fractions (in lowest terms) is the G. C. D. of the numerators divided by the L. C. M. of the denominators. The L. C. M. of fractions is the L. C. M. of the numerators divided by the G. C. D. of the denominators.
q: Find the G. C. D. of 5/8, 25/32, and 7/12. {=1/96}
sol: Since 5 and 7 are both prime, 1 is the G. C. D. of the numerators; 96 is the L. C. M. of 8, 32, and 12: the G. C. D. is 1/96.
q: Find the greatest common divisor of 83 1/3 and 268 3/4. {=25/12|2 1/12}
q: Find the greatest common divisor of 14 7/12 and 95 3/8. {=7/24}
q: Find the greatest common divisor of 59 1/9 and 735 14/15. {=133/45|2 43/45}
q: Find the greatest common divisor of 23 7/16 and 213 13/24. {=125/48|2 29/48}
q: Find the greatest common divisor of 418 3/5 and 1772 1/3. {=13/15}
q: Find the greatest common divisor of 261 13/14 and 652 11/21. {=193/42|4 25/42}
q: Find the greatest common divisor of 44 4/9, 546 2/3, and 3160. {=40/9|4 4/9}
q: A farmer sells 137 1/2 bushels of yellow corn, 478 1/8 bushels of white corn, and 2093 3/4 bushels of mixed corn: required the size of the largest sacks that can be used in shipping, so as to keep the corn from being mixed. Bushels? {=25/8|3 1/8} How many sacks of yellow corn? {44 @ 137_1/2 / (25/8)} Of white? {153 @ 478_1/8 / (25/8)} Of mixed? {670 @ 2093_3/4 / (25/8)}
q: A owns a tract of land, the sides of which are 134 3/4, 128 1/3, and 115 1/2 feet long: how many rails of the greatest length possible will be needed to fence it in straight lines, the fence to be 6 rails high, and the rails to lap 6 inches at each end? {354 @ (134_3/4 + 128_1/3 + 115_1/2) / (77/12) * 6}
sol: The G. C. D. of 539/4, 385/3, and 231/2 is 77/12 = 6 5/12 feet, the length of a panel; with the lap, each rail is 7 5/12 feet. The sides hold 21 + 20 + 18 = 59 panels, and 6 rails to each: 354 rails.
q: Find the L. C. M. of 1/3, 3/4, and 5/6. {15}
sol: The L. C. M. of the numerators 1, 3, 5 is 15; the G. C. D. of the denominators 3, 4, 6 is 1: 15/1, or 15.
q: Find the least common multiple of 2/3, 3/4, 4/5, 5/6, and 6/7. {60}
q: Find the least common multiple of 4 1/2, 6 3/4, 5 5/6, and 10 1/2. {~945 / 2}
q: Find the least common multiple of 3 1/3, 4 3/8, 5/12, 5 5/9, and 12 1/2. {350}
q: A can walk around an island in 14 2/7 hours; B, in 9 1/11 hours; C, in 16 2/3 hours; and D, in 25 hours. If they start from the same point, and at the same time, how many hours after starting till they are all together again? {100}
sol: 14 2/7 = 100/7, 9 1/11 = 100/11, 16 2/3 = 50/3, 25 = 25/1. The L. C. M. of 100, 100, 50, 25 is 100; the G. C. D. of 7, 11, 3, 1 is 1: 100 hours.
== 20 | Fractions | Promiscuous exercises
note: Problems marked mental in the book are to be solved in the head: hear the question once, answer, then tell how you got it.
q: What is the sum of 3 1/2, 4 1/3, 5 1/4, 3/4 of 7/8, and 1/2 of 1/3 of 5/8? {~3_1/2 + 4_1/3 + 5_1/4 + 3/4 * 7/8 + 1/2 * 1/3 * 5/8}
q: The sum of 1 1/26 and 1 over 1 4/9 is equal to how many times their difference? {5 @ (1_1/26 + 1 / 1_4/9) / (1_1/26 - 1 / 1_4/9)}
sol: 1 over 1 4/9 = 9/13 = 18/26, and 1 1/26 = 27/26. The sum is 45/26 and the difference 9/26; 45 contains 9 five times.
q: What is (2 3/4 + 5/2 of (7 over 3 4/5 - 1 2/3 over 2 1/2)) divided by 1 77/228? {~(2_3/4 + 5/2 * (7 / 3_4/5 - 1_2/3 / 2_1/2)) / 1_77/228}
q: Reduce to its simplest form: 4 4/15 of 2 5/8 over (5 1/5 - 4 1/2). {~4_4/15 * 2_5/8 / (5_1/5 - 4_1/2)} And 5/7 x (100 - 200/3 + 7 1/3 over 2 1/4). {~5/7 * (100 - 200/3 + 7_1/3 / 2_1/4)}
q: What is 1/4 of 5 1/4 - 1/3 of 3 7/8? {~1/4 * 5_1/4 - 1/3 * 3_7/8}
q: What is 32/51 x 85/112 x 189/207 x 23/36 equal to? {~32/51 * 85/112 * 189/207 * 23/36}
q: In your head: 1/2 x (1 - 1/2) over 2 x (2 - 1/2) over 3 = ? {~1/2 * (1 - 1/2) / 2 * (2 - 1/2) / 3}
q: 1/3 x (1 - 1/3) over 2 x (2 - 1/3) over 3 x (3 - 1/3) over 5 x (4 - 1/3) over 4 = ? {~1/3 * (1 - 1/3) / 2 * (2 - 1/3) / 3 * (3 - 1/3) / 5 * (4 - 1/3) / 4}
q: (2 + 1/5) divided by (3 + 1/7), all over (2 - 1/3) x (4 - 3 3/7) = what? {~(2 + 1/5) / (3 + 1/7) / ((2 - 1/3) * (4 - 3_3/7))}
q: (4 1/3 x 4 1/3 x 4 1/3 - 1) over (4 1/3 x 4 1/3 - 1) = what? {~(4_1/3 * 4_1/3 * 4_1/3 - 1) / (4_1/3 * 4_1/3 - 1)}
q: Add 2/3 of 3/4 of 7/8, 1/5 x 2/3 of 1 7/8, and 1/4. {~2/3 * 3/4 * 7/8 + 1/5 * 2/3 * 1_7/8 + 1/4}
q: 3/5 of 10/9 of what number, diminished by 3/10 over (2 1/6 + 7/30), leaves 25/64? {~(25/64 + 3/10 / (2_1/6 + 7/30)) / (3/5 * 10/9)}
sol: 3/10 over 2 1/6 + 7/30 = 3/10 over 2 2/5 = 1/8, the subtrahend; 25/64 + 1/8 = 33/64, the minuend. This is 3/5 of 10/9, or 2/3, of the number: the number is 3/2 of 33/64 = 99/128.
q: In your head: James's money equals 3/5 of Charles's money; and 3/4 of James's money + $33 equals Charles's money: how much has Charles? Dollars? {60} James? {36}
sol: 3/4 of James's money is 3/4 of 3/5, or 9/20, of Charles's; so 9/20 of Charles's + $33 is the whole of it, and 11/20 of it = $33. 1/20 is $3, the whole $60, and James has 3/5 of $60 = $36.
q: In your head: A leaves L for N at the same time that B leaves N for L. The two places are exactly 109 miles apart: A travels 7 1/2 miles per hour, and B, 8 1/4 miles per hour; in how many hours will they meet? {~109 / (7_1/2 + 8_1/4)} How far will A have traveled? Miles? {~109 / (7_1/2 + 8_1/4) * 7_1/2} And B? {~109 / (7_1/2 + 8_1/4) * 8_1/4}
q: What number multiplied by 5/9 of 3/7 of 3 11/15 will produce 2 1/2? {~2_1/2 / (5/9 * 3/7 * 3_11/15)}
q: What, divided by 1 3/5, gives 14 3/4? {~14_3/4 * 1_3/5}
q: What, added to 14 5/8, gives 29 23/35? {~29_23/35 - 14_5/8}
q: In your head: I spend 5/9 of my income in board, 1/6 of it in clothes, and save $60 a year: what is my income? Dollars? {216 @ 60 / (1 - 5/9 - 1/6)}
q: In your head: Divide 51 into two such parts that 2/3 of the first is equal to 3/4 of the second. The first? {27} The second? {24}
sol: 2/3 of the first = 3/4 of the second, so the whole first = 9/8 of the second, and 51 is 17/8 of the second. The second is 8/17 of 51 = 24; the first, 51 - 24 = 27.
q: In your head: 1/2 is what part of 2/3? {~1/2 / (2/3)}
q: Divide 7/9 of 3 3/5 by 13/14 of 7. {~7/9 * 3_3/5 / (13/14 * 7)} And 9/11 of 2/3 of 27 1/2 by 4/9 of 3/17 of 5 1/2. {~9/11 * 2/3 * 27_1/2 / (4/9 * 3/17 * 5_1/2)}
q: Multiply 7/11 of 2 1/2 by 3/13 of 19 1/2. {~7/11 * 2_1/2 * 3/13 * 19_1/2} And divide 7/8 of 5/9 of 14 1/7 by 3/11 of 2/7 of 13 4/9. {~7/8 * 5/9 * 14_1/7 / (3/11 * 2/7 * 13_4/9)}
q: (3131/4331 + 1470/2911 + 1931/2501) divided by 2 1/2 over 7 = what? {~(3131/4331 + 1470/2911 + 1931/2501) / (2_1/2 / 7)}
q: In your head: A bequeathed 11/20 of his estate to his elder son; the rest to his younger, who received $525 less than his brother. What was the estate? Dollars? {5250 @ 525 / (11/20 - 9/20)}
q: Find the sum of 3 7/8 and 2 1/9. {~3_7/8 + 2_1/9} The difference. {~3_7/8 - 2_1/9} The product. {~3_7/8 * 2_1/9} The quotient of their sum by the difference. {~(3_7/8 + 2_1/9) / (3_7/8 - 2_1/9)}
q: In your head: A cargo is worth 7 times the ship: what part of the cargo is 5/16 of the ship and cargo? {~5/16 * 8 / 7}
q: By what must the sum of 757/2747, 951/2479, and 517/1517 be multiplied to produce 1000? {~1000 / (757/2747 + 951/2479 + 517/1517)}
q: Multiply the sum of all the divisors of 8128, including 1 (but not 8128 itself), by the number of its prime factors excluding 1, and divide by 149 1/3. {381 @ 8128 * 7 / 149_1/3}
sol: 8128 = 2 x 2 x 2 x 2 x 2 x 2 x 127, seven prime factors. Its divisors 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064 add up to 8128, a perfect number. 8128 x 7 / 149 1/3 = 381.
q: 6 1/2 is what part of 10 7/11? {~6_1/2 / 10_7/11} Reduce to its simplest form 8/9 - 1/7 + 5/8 - 1 1/4. {~8/9 - 1/7 + 5/8 - 1_1/4}
q: Multiply 4 over 5 1/3, 14 1/7, 2 3/4 over 4, 5 over 7 1/3, 1 1/5 over 2 3/4, and 6. {~4 / 5_1/3 * 14_1/7 * 2_3/4 / 4 * 5 / 7_1/3 * 1_1/5 / 2_3/4 * 6}
q: 7/8 of 5/9 of what number equals 9 13/18? {~9_13/18 / (7/8 * 5/9)}
q: In your head: A 63-gallon cask is 5/8 full: 9 1/2 gallons being drawn off, how full will it be? {~(63 * 5/8 - 9_1/2) / 63}
q: If a person going 3 3/4 miles per hour, perform a journey in 14 3/4 hours, how long would he be, if he traveled 5 1/4 miles per hour? Hours? {~3_3/4 * 14_3/4 / 5_1/4}
q: A man buys 32 3/4 pounds of coffee, at 17 5/8 cents a pound: if he had got it 4 2/3 cents a pound cheaper, how many more pounds would he have received? {~32_3/4 * 17_5/8 / (17_5/8 - 4_2/3) - 32_3/4}
q: Henry spent 2/11 of his money and then received $65; he then lost 3/4 of all his money, and had in hand $10 less than at first. How much had he at first? Dollars? {33}
sol: After spending he had 9/11 of his money; with $65 more, then losing 3/4, he kept 1/4 of (9/11 of it + $65) = 9/44 of it + $16 1/4. This is $10 less than the whole, so 35/44 of it = $26 1/4, and the whole was $33.
== 21 | Decimals | Reading and writing decimals
note: A decimal is read as so many tenths, hundredths, thousandths, and so on, named by the place of its right-hand figure: .0325 is three hundred and twenty-five ten-thousandths. A complex decimal has a common fraction in its right-hand place: .03 1/3. On the pad, type a complex decimal as .03, and, 1/3.
q: Read the decimal .0325 out loud. {open}
sol: The numerator is 325; there are four decimal places, so the denomination is ten-thousandths: three hundred and twenty-five ten-thousandths.
q: Read out loud: .9; .6 2/3; .0 3/7; .035. {open}
sol: Nine tenths; six and two-thirds tenths; three-sevenths of a hundredth; thirty-five thousandths.
q: Read out loud: .7200; .5060; 1.008; 9.00 1/3. {open}
sol: Seven thousand two hundred ten-thousandths; five thousand and sixty ten-thousandths; one and eight thousandths; nine and one-third thousandths.
q: Read out loud: 105.0 7/8; .0003; 00.100; 180.010. {open}
sol: One hundred and five and seven-eighths hundredths; three ten-thousandths; one hundred thousandths; one hundred and eighty and ten thousandths.
q: Read out loud: 2030.0; 40.68031; 200.002; .0900001. {open}
sol: Two thousand and thirty; forty and sixty-eight thousand and thirty-one hundred-thousandths; two hundred and two thousandths; nine hundred thousand and one ten-millionths.
q: Read out loud: 61.010001; 31.0200703. {open}
sol: Sixty-one and ten thousand and one millionths; thirty-one and two hundred thousand seven hundred and three ten-millionths.
q: Read out loud: .000302501; .03672113. {open}
sol: Three hundred and two thousand five hundred and one billionths; three million six hundred and seventy-two thousand one hundred and thirteen hundred-millionths.
q: Write eighty-three thousand and one billionths. {.000083001}
sol: Write the numerator, 83001; fill four places with ciphers so that the final figure is in billionths' place: .000083001.
q: Write in figures: Five tenths. {.5} Twenty-two hundredths. {.22} One hundred and four thousandths. {.104} Two units and one hundredth. {2.01}
q: Write in figures: One thousand six hundred and five ten-thousandths. {.1605} Eighty-seven hundred-thousandths. {.00087}
q: Write in figures: Twenty-nine and one half ten-millionths. {.0000029 1/2}
q: Write in figures: Nineteen million and one billionths. {.019000001}
q: Write in figures: Seventy thousand and forty-two units and sixteen hundredths. {70042.16}
q: Write in figures: Two thousand units and fifty-six and one third millionths. {2000.000056 1/3}
q: Write in figures: Four hundred and twenty-one tenths. {42.1} Six thousand hundredths. {60} Forty-eight thousand three hundred and five thousandths. {48.305}
q: Write in figures: Eight units and one half a hundredth. {8.00 1/2} Thirty-three million ten-millionths. {3.3}
q: Write in figures: Four hundred thousandths. {.4} Four hundred-thousandths. {.00004}
q: Write in figures: One unit and one half a billionth. {1.000000000 1/2}
q: Write in figures: Sixty-six thousand and three millionths. {.066003} Sixty-six million and three thousandths. {66000.003}
q: Write in figures: Thirty-four and one third tenths. {3.4 1/3}

== 22 | Decimals | Reducing decimals
note: To reduce a decimal to a common fraction, write it as a common fraction and reduce it to its lowest terms. To reduce a common fraction to a decimal, annex ciphers to the numerator and divide by the denominator. A fraction whose denominator (in lowest terms) has any prime factor besides 2 and 5 can not be reduced exactly to a decimal: 1/12 = .08333+.
q: Reduce .24 to a common fraction. {=6/25}
sol: .24 = 24/100, which reduced is 6/25.
q: Reduce .12 1/2 to a common fraction. {=1/8}
sol: Write 12 1/2 over 100, and reduce the complex fraction: 25/2 over 100 = 25/200 = 1/8.
q: Reduce to common fractions: .25625 {=41/160} .15234375 {=39/256}
q: Reduce to common fractions: 2.125 {=2 1/8|17/8} 19.01750 {=19 7/400|7607/400}
q: Reduce to common fractions: 16.00 1/5 {=16 1/500|8001/500} 350.028 4/7 {=350 1/35|12251/35}
q: Reduce to common fractions: .666666 2/3 {=2/3} .003125 {=1/320} 11.0 5/9 {=11 1/18|199/18} .390625 {=25/64}
q: Reduce to common fractions: .1944 4/9 {=7/36} .24 4/9 {=11/45} .33 1/3 {=1/3} .66 2/3 {=2/3} .25 {=1/4} .75 {=3/4}
q: Reduce 7/8 to a decimal. {.875}
sol: 7 = 7.000; 1/8 of 7.0 = .8 with 6 tenths over; .60, 1/8 of it = .07 with 4 hundredths over; .040, 1/8 of it = .005. The answer is .875.
q: Reduce to decimals: 3/4 {~d 3/4} 1/8 {~d 1/8} 1/20 {~d 1/20} 15/32 {~d 15/32} 9/1600 {~d 9/1600}
q: Reduce to decimals: 4/5 {~d 4/5} 99/200 {~d 99/200} 5/64 {~d 5/64} 13/256 {~d 13/256} 1/1024 {~d 1/1024}
q: Reduce to decimals: 16 1/2 {~d 16_1/2} 42 3/16 {~d 42_3/16} .015 1/4 {~d (15_1/4) / 1000}
sol: The rule turns a mixed number into a mixed decimal and a complex decimal into a pure one: 16 1/2 = 16.5; 3/16 = .1875, so 42 3/16 = 42.1875; 1/4 = .25, annexed to .015 gives .01525.
q: Reduce to decimals: 101.01 3/4 {~d 101 + (1_3/4) / 100} 75119 3/80 {~d 75119_3/80} 2.00 1/320 {~d 2 + 1/320 / 100}

== 23 | Decimals | Adding and subtracting decimals
note: Write the numbers so that figures of the same order stand in the same column; add or subtract as in simple numbers, and put the point to the left of tenths. Complex decimals are carried out by division to the same place, and their fractions added or subtracted with the rest.
q: Add 23.8 and 17 1/2 and .0256 and .41 2/3. {~c4 23.8 + 17_1/2 + .0256 + (41_2/3) / 100}
sol: 17 1/2 = 17.5 and .41 2/3 = .4166 2/3. 23.8 + 17.5 + .0256 + .4166 2/3 = 41.7422 2/3.
q: Find the sum of 1 + .9475 {~d 1 + .9475}
q: Of 1.33 1/3 added to 2.66 2/3. {~1 + (33_1/3) / 100 + 2 + (66_2/3) / 100}
q: Of 14.034, 25, .000062 1/2, .0034. {~d 14.034 + 25 + (62_1/2) / 1000000 + .0034}
q: Of 83 thousandths, 2101 hundredths, 25 tenths, and 94 1/2 units. {~d .083 + 21.01 + 2.5 + 94.5}
q: Of .16 2/3, .37 1/2, 5, 3.4 3/8, .000 7/8. {~c3 (16_2/3) / 100 + (37_1/2) / 100 + 5 + 3 + (4_3/8) / 10 + 7/8 / 1000}
q: Of 4 units, 4 tenths, 4 hundredths. {~d 4.44}
q: Of .11 1/9 + .6666 2/3 + .222222 2/9. {~(11_1/9) / 100 + (6666_2/3) / 10000 + (222222_2/9) / 1000000}
q: Of .14 2/7, .018 3/5, 920, .0139 3/7. {~c4 (14_2/7) / 100 + (18_3/5) / 1000 + 920 + (139_3/7) / 10000}
q: Of 16.008 7/9, .0074 2/3, .2 5/6, .00019042 1/5. {16.299768199 7/9}
q: Of 675 thousandths, 2 millionths, 64 1/8, 3.489107, and .00089407. {~d .675 + .000002 + 64_1/8 + 3.489107 + .00089407}
q: Of four times 4.067 7/8 and .000 1/2. {~d 4 * (4 + (67_7/8) / 1000) + 1/2 / 1000}
q: Of 216.86301, 48.1057, .029, 1.3, 1000. {~d 216.86301 + 48.1057 + .029 + 1.3 + 1000}
q: Add 35 units, 35 tenths, 35 hundredths, 35 thousandths. {~d 35 + 3.5 + .35 + .035}
q: Add ten thousand and one millionths; four hundred-thousandths; 96 hundredths; forty-seven million sixty thousand and eight billionths. {~d .010001 + .00004 + .96 + .047060008}
q: From 6.8 subtract 2.057 {~d 6.8 - 2.057}
sol: Suppose ciphers annexed to the 8, and subtract as in whole numbers: 6.800 - 2.057 = 4.743.
q: From 13.256 5/9 subtract 6.77 1/3 {~c3 13 + (256_5/9) / 1000 - 6 - (77_1/3) / 100}
sol: 6.77 1/3 = 6.773 1/3; 13.256 5/9 - 6.773 1/3 = 6.483 2/9.
q: Subtract 8.00717 from 19.54 {~d 19.54 - 8.00717}
q: 3 thousandths from 3000. {~d 3000 - .003}
q: 72.0001 from 72.01 {~d 72.01 - 72.0001}
q: Subtract .93 2/35 from 1.169 3/7 {~c3 1 + (169_3/7) / 1000 - (93_2/35) / 100}
q: How much is 19 - 8.999 1/9? {~c3 19 - 8 - (999_1/9) / 1000}
q: How much less is .04 1/3 than .4? {~c2 .4 - (4_1/3) / 100}
q: How much is .65007 - 1/2? {~d .65007 - 1/2}
q: What is 2 3/4 - 1 4/5 in decimals? {~d 2_3/4 - 1_4/5}
q: Subtract 1 from 1.684 {~d 1.684 - 1}
q: 5/6 of a millionth from .000 4/9 {~c6 4/9 / 1000 - 5/6 / 1000000}
q: 1 1/2 hundredths from 49 3/8 tenths. {~d (49_3/8) / 10 - (1_1/2) / 100}
q: 10000 thousandths from 10 units. {0}
q: 24 1/2 tenths from 3701 thousandths. {~d 3.701 - (24_1/2) / 10}
q: 1 7/8 units from 1875 thousandths. {0}
q: 5/9 of a hundredth from 1/18 of a tenth. {0}
sol: 1/18 of a tenth = 1/180, and 5/9 of a hundredth = 5/900 = 1/180: the difference is nothing.
q: 64 1/6 hundredths from 100 units. {~c2 100 - (64_1/6) / 100}

== 24 | Decimals | Multiplying and dividing decimals
note: Multiply as in whole numbers, and point off as many decimal places in the product as there are in both factors. Divide as in whole numbers, and point off as many places in the quotient as those in the dividend exceed those in the divisor; when the division is not exact, annex ciphers to the dividend.
q: Multiply 2.56 by .184 {~d 2.56 * .184}
sol: 256/100 x 184/1000 = 47104/100000: hundredths by thousandths are hundred-thousandths, five places. The product is .47104.
q: 1 x .1 = ? {~d 1 * .1}
q: 16 x .03 1/3 = ? {~c2 16 * (3_1/3) / 100}
q: .01 x .1 1/2 = ? {~d .01 * (1_1/2) / 10}
q: .080 x 80 = ? {~d .080 * 80}
q: 37.5 x 82 1/2 = ? {~d 37.5 * 82_1/2}
q: 64.01 x .32 = ? {~d 64.01 * .32}
q: 48000 x 73 = ? {~48000 * 73}
q: 64.66 2/3 x 18 = ? {~(6466_2/3) / 100 * 18}
q: .56 1/4 x .03 1/16 = ? {~d (56_1/4) / 100 * (3_1/16) / 100}
q: 738 x 120.4 = ? {~d 738 * 120.4}
q: .0001 x 1.006 = ? {~d .0001 * 1.006}
q: 34 units x .193 = ? {~d 34 * .193}
q: 27 tenths x .4 1/5 = ? {~d 2.7 * (4_1/5) / 10}
q: 43.7004 x .008 = ? {~d 43.7004 * .008}
q: 21.0375 x 4.44 4/9 = ? {~d 21.0375 * (444_4/9) / 100}
q: 9300.701 x 251 = ? {~d 9300.701 * 251}
q: 430.0126 x 4000 = ? {~d 430.0126 * 4000}
q: .059 x .059 x .059 = ? {~d .059 * .059 * .059}
q: 42 units x 42 tenths = ? {~d 42 * 4.2}
q: 2 1/4 hundredths x 600 = ? {~d (2_1/4) / 100 * 600}
q: 7100 x 1/8 of a millionth = ? {~d 7100 * 1/8 / 1000000}
q: 26 millions x 26 millionths = ? {~26000000 * 26 / 1000000}
q: 2700 hundredths x 60 tenths = ? {~27 * 6}
q: 6.3029 x .03275 = ? {~d 6.3029 * .03275}
q: 135.027 x 1.00327 = ? {~d 135.027 * 1.00327}
note: Oughtred's method: when only a certain number of decimal places is wanted, multiply only such figures as produce one place more than required, beginning with the highest figure of the multiplier, and drop the last figure of the sum.
q: Multiply 3.8640372 by 1.2479603, retaining only seven decimal places. {~r7 3.8640372 * 1.2479603}
q: Multiply 27.653 by 9.157, preserving three decimal places. {~r3 27.653 * 9.157}
q: Multiply 43.2071 by 3.14159, preserving four decimal places. {~r4 43.2071 * 3.14159}
q: Multiply 3.62741 by 1.6432, preserving four decimal places. {~r4 3.62741 * 1.6432}
q: 9.012 x 48.75, preserving one place. {~r1 9.012 * 48.75}
q: 4.804136 x .010759, preserving six places. {~r6 4.804136 * .010759}
q: 814 53/719 x 26 35/44, preserving three places. {~r3 814_53/719 * 26_35/44}
q: 702.61 x 1.258 7/36, preserving three places. {~r3 702.61 * (1258_7/36) / 1000}
q: 849.93 3/4 x .0424444, preserving three places. {~r3 (84993_3/4) / 100 * .0424444}
q: 880.695 x 131.72 true to units. {~r0 880.695 * 131.72}
q: .025381 x .004907, preserving five places. {.00012|.000125|.000124}
q: 64.01082 x .03537, preserving six places. {~r6 64.01082 * .03537}
q: 1380.37 1/2 x .234 5/6, preserving two places. {~r2 (138037_1/2) / 100 * (234_5/6) / 1000}
q: Divide .50312 by .19 {~d .50312 / .19}
sol: Divide as in integers: 50312 / 19 = 2648. The quotient must have 5 - 2 = 3 places: 2.648.
q: Divide .36 by .008 {~.36 / .008}
sol: The dividend has fewer decimal places than the divisor: annex one cipher, .360; the places are equal and the quotient is an integer, 45.
q: Divide .002 19/40 by .06 3/5 {~d (2_19/40) / 1000 / ((6_3/5) / 100)}
sol: .002 19/40 = .002475 and .06 3/5 = .066. .002475 / .066 = .0375.
q: 63 / 4000 = ? {~d 63 / 4000}
q: 3.15 / 375 = ? {~d 3.15 / 375}
q: 1.008 / 18 = ? {~d 1.008 / 18}
q: 4096 / .032 = ? {~4096 / .032}
q: 9.7 / 97000 = ? {~d 9.7 / 97000}
q: .9 / .00075 = ? {~.9 / .00075}
q: 13 / 78.12 1/2 = ? {~d 13 / 78_1/8}
q: 12.9 / 8.256 = ? {~d 12.9 / 8.256}
q: 81.2096 / 1.28 = ? {~d 81.2096 / 1.28}
q: 1 / 100 = ? {~d 1 / 100}
q: 10.1 / 17 = ? To five places. {~r5 10.1 / 17}
q: .001 / 100 = ? {~d .001 / 100}
q: 12755 / 81632 = ? {~d 12755 / 81632}
q: 2401 / 21.4375 = ? {~2401 / 21.4375}
q: 21.13212 / .916 = ? {~d 21.13212 / .916}
q: 36.72672 / .5025 = ? {~d 36.72672 / .5025}
q: 2483.25 / 5.15625 = ? {~d 2483.25 / 5.15625}
q: 142.0281 / 9.2376 = ? {~d 142.0281 / 9.2376}
q: .08 1/3 / .12 1/2 = ? {~(8_1/3) / 100 / ((12_1/2) / 100)}
q: .0001 / .01 = ? {~d .0001 / .01}
q: 95.3 / .264 = ? To six places. {~r6 95.3 / .264}
q: 1000 / .001 = ? {~1000 / .001}
q: Ten / 1 tenth = ? {~10 / .1}
q: .000001 / .01 = ? {~d .000001 / .01}
q: .00001 / 1000 = ? {~d .00001 / 1000}
q: 16.275 / .41664 = ? {~d 16.275 / .41664}
q: 1 ten-millionth / 1 hundredth = ? {~d .0000001 / .01}
note: Oughtred's method in division: when the quotient need not contain figures below a certain place, take one figure more of the dividend than that place needs, and drop a figure of the divisor at each step.
q: Divide 84.27 by 1.27395607, securing a quotient true to four places of decimals. {~r4 84.27 / 1.27395607}
q: 1000 / .98, preserving two places. {~r2 1000 / .98}
q: 6215.75 / .99 1/2, preserving three places. {~r3 6215.75 / .995}
q: 28012 / .993, preserving two places. {~r2 28012 / .993}
q: 52546.35 / .99 3/4, preserving three places. {~r3 52546.35 / .9975}
q: 4840 / .9875, preserving two places. {~r2 4840 / .9875}
q: 2 / 1.4142136, preserving seven places. {~r7 2 / 1.4142136}
q: 9.869604401 / 3.14159265, preserving eight places. {~r8 9.869604401 / 3.14159265}
== 25 | Circulating Decimals | Repetends and common fractions
note: A circulate has figures repeated in the same order; here the repeating figures (the repetend) are written in brackets: .(3) = .333..., .2(083) = .2083083... The book marks them with a dot over the first and last figure. Every fraction whose denominator has a prime factor other than 2 and 5 gives a circulate.
q: Is 7/60 a terminate or a circulating decimal? Say why. {open}
sol: 60 = 2 x 2 x 3 x 5: the factor 3 is not 2 or 5, so the decimal will not end: 7/60 = .11(6).
q: Reduce 1/7 to a decimal. How many figures are in its repetend? {6}
sol: 1/7 = .(142857): after six figures the remainder is 1 again, and the figures repeat.
q: Change .(53) to a common fraction. {~.(53)}
sol: 100 times the repetend = 53.(53); once the repetend = .(53); so 99 times the repetend = 53, and once the repetend = 53/99.
q: Change .(456) to a common fraction. {~.(456)}
q: Change 25.(625) to a common fraction. {~25.(625)}
sol: Write the repetend for the numerator, and as many 9's as it has figures for the denominator: 625/999. The whole is 25 625/999.
q: Change .821(437) to a common fraction. {~.821(437)}
sol: Briefly: 821437 - 821 = 820616, over 999000; reduced, 102577/124875.
q: Change .04(8) to a common fraction. {~.04(8)}
sol: 48 - 4 = 44, over 900 (one 9 for the one repeating figure, two ciphers for the two figures before it): 44/900 = 11/225.
q: Reduce to common fractions: .(3) {~.(3)} .0(5) {~.0(5)} .(123) {~.(123)} 2.(63) {~2.(63)} .3(1) {~.3(1)}
q: Reduce to common fractions: .0(216) {~.0(216)} 48.(148) {~48.(148)} 1.(0011) {~1.(0011)}
sol: In the book these are written with dots over 4 and 1 of 48.1, and over the two 1's of 1.001: the repetends 481 and 1001, which are 48.148148... and 1.00110011...
q: Reduce to common fractions: .13(8) {~.13(8)} .208(3) {~.208(3)} 85.(714285) {~85.(714285)}
q: Reduce to common fractions: .(063492) {~.(063492)} .4(476190) {~.4(476190)} .0902(7) {~.0902(7)}

== 26 | Circulating Decimals | Working with circulates
note: Make the repetends similar (beginning and ending at the same places), then add, subtract or multiply as with other decimals, carrying what the continued circulates would carry. Or change the circulates to common fractions and work with those. Give your answers as common fractions or mixed numbers; the circulate is shown after.
q: Add .25(6), 5.34(72), 24.(815), and .9098. {~q .25(6) + 5.34(72) + 24.(815) + .9098}
sol: Made similar, the circulates are .2566(666666), 5.3472(727272), 24.8158(158158), .9098(000000). The first figures that would come next, 6, 7, 1, 0, make 14, carrying 1. The sum is 31.3295(552097).
q: Add .45(3), .06(8), .32(7), .94(6). {~q .45(3) + .06(8) + .32(7) + .94(6)}
q: Add 3.0(4), 6.45(6), 23.(38), .2(48). {~q 3.0(4) + 6.45(6) + 23.(38) + .2(48)}
q: Add .(25), .10(4), .(61), and .56(35). {~q .(25) + .10(4) + .(61) + .56(35)}
q: Add 1.(031), .25(7), 5.(04), 28.0445(245). {~q 1.(031) + .25(7) + 5.(04) + 28.0445(245)}
q: Add .(6), .13(8), .0(5), .09(72), .041(6). {~q .(6) + .13(8) + .0(5) + .09(72) + .041(6)}
q: Add 9.21(107), .(65), 5.00(4), 3.5(622). {~q 9.21(107) + .(65) + 5.00(4) + 3.5(622)}
q: Add .20(45), .(09), and .25 {~q .20(45) + .(09) + .25}
q: Add 5.0(770), .(24), and 7.(124943) {~q 5.0(770) + .(24) + 7.(124943)}
q: Add 3.(4884), 1.6(37), 130.8(1), .06(6). {~q 3.(4884) + 1.6(37) + 130.8(1) + .06(6)}
q: Subtract 9.3(156) from 12.90(21). {~q 12.90(21) - 9.3(156)}
sol: Made similar: 12.90(212121) and 9.31(561561). The next figure of the subtrahend, 5, is larger than the one above it, 2, so carry 1 to the right-hand figure of the subtrahend. The remainder is 3.58(650559).
q: Subtract .0(074) from .2(6) {~q .2(6) - .0(074)}
q: Subtract 9.(099) from 15.35(465) {~q 15.35(465) - 9.(099)}
q: Subtract 4.(514) from 18.23(673) {~q 18.23(673) - 4.(514)}
q: Subtract 37.01(28) from 100.73 {~q 100.73 - 37.01(28)}
q: Subtract 8.(27) from 10.0(563) {~q 10.0(563) - 8.(27)}
q: Subtract 190.(476190) from 199.6(428571) {~q 199.6(428571) - 190.(476190)}
q: Subtract 13.(637) from 104.(14) {~q 104.(14) - 13.(637)}
q: Multiply .375(4) by 17.4(3) {~q .375(4) * 17.4(3)}
sol: The repetend of the multiplier is 1/3, so 17.4(3) = 17.4 1/3. The partial products, carried forward until their repetends are similar and added, give 6.5452(481).
q: 4.7(35) x 7.349 = ? {~q 4.7(35) * 7.349}
q: .0(7067) x .9(432) = ? {~q .0(7067) * .9(432)}
q: 714.3(2) x 3.45(6) = ? {~q 714.3(2) * 3.45(6)}
q: 16.(204) x 32.(75) = ? {~q 16.(204) * 32.(75)}
q: 19.0(72) x .208(3) = ? {~q 19.0(72) * .208(3)}
q: 3.(7543) x 4.7157 = ? {~q 3.(7543) * 4.7157}
q: 1.(256784) x 6.420(81) = ? {~q 1.(256784) * 6.420(81)}
q: Divide .(154) by .(2) {~q .(154) / .(2)}
sol: .(154) = 154/999 and .(2) = 2/9. 154/999 divided by 2/9 = 154/999 x 9/2 = 77/111 = .(693).
q: .(75) / .(1) = ? {~q .(75) / .(1)}
q: 51.49(1) / 17 = ? {~q 51.49(1) / 17}
q: 681.55(98879) / 94 = ? {~q 681.55(98879) / 94}
q: 90.5(203749) / 6.7(54) = ? {~q 90.5(203749) / 6.7(54)}
q: 11.(068735402) / .(245) = ? {~q 11.(068735402) / .(245)}
q: 9.5(330663997) / 6.(217) = ? {~q 9.5(330663997) / 6.(217)}
q: 3.500(691358024) / 7.68(4) = ? {~q 3.500(691358024) / 7.68(4)}
== 27 | Compound Numbers | Measures of value
note: A compound denominate number expresses one kind of quantity in different denominations: 5 yards, 2 feet, and 8 inches. United States money is decimal: the dollar is the unit. In reading it, name the dollars, then the cents, then the mills: $9.124 is 9 dollars 12 cents 4 mills.
rule: 10 mills make 1 cent
rule: 10 cents make 1 dime
rule: 10 dimes make 1 dollar
rule: 10 dollars make 1 eagle
rule: 4 farthings make 1 penny
rule: 12 pence make 1 shilling
rule: 20 shillings make 1 pound sterling
rule: 10 centimes make 1 decime
rule: 10 decimes make 1 franc
rule: 100 pfennige make 1 mark
q: Read $9.124 out loud. {open}
sol: 9 dollars 12 cents 4 mills; or 9 dollars 124 thousandths of a dollar.
q: How many mills in a dollar? {1000} How many cents in an eagle? {1000}
q: How many farthings in a pound sterling? {960} How many pence in a pound? {240}
q: How many pence in a shilling? {12} How many farthings in a shilling? {48}
q: How many centimes in a franc? {100} How many millimes in a franc? {1000}
q: The pound sterling is worth $4.8665: what are 10 pounds worth, in dollars? {~d 10 * 4.8665}
q: The franc is worth 19.3 cents: what are 100 francs worth, in dollars? {~d 100 * .193}
q: The mark is worth 23.8 cents: what are 50 marks worth, in dollars? {~d 50 * .238}
q: A guinea is 21 shillings: how many pence is it? {252 @ 21 * 12}
q: A half-crown is 2s. 6d.: how many pence is it? {30 @ 2 * 12 + 6}

== 28 | Compound Numbers | Measures of weight
note: Troy weight is used for gold, silver, and jewels; Apothecaries' weight for mixing medicines; Avoirdupois weight for all ordinary articles. The Troy and Apothecaries' pound is 5760 grains; the Avoirdupois pound is 7000 grains.
rule: 24 grains make 1 pennyweight
rule: 20 pennyweights make 1 ounce Troy
rule: 12 ounces make 1 pound Troy
rule: 20 grains make 1 scruple
rule: 3 scruples make 1 dram
rule: 8 drams make 1 ounce
rule: 12 ounces make 1 pound Apothecaries'
rule: 16 ounces make 1 pound avoirdupois
rule: 25 pounds make 1 quarter
rule: 4 quarters make 1 hundred-weight
rule: 20 hundred-weight make 1 ton
q: How many grains in a pound Troy? {5760 @ 12 * 20 * 24} In an ounce Troy? {480 @ 20 * 24}
q: How many grains in a dram of Apothecaries' weight? {60 @ 3 * 20} In an ounce? {480 @ 8 * 60}
q: How many pounds in a ton? {2000 @ 20 * 4 * 25} How many ounces in a hundred-weight? {1600 @ 100 * 16}
q: How many more grains are there in the avoirdupois pound than in the Troy pound? {1240 @ 7000 - 5760}
q: In Great Britain the quarter is 28 lb.: how many pounds are in their hundred-weight of 4 quarters? {112} In their ton of 20 hundred-weight? {2240}
q: A stone, horseman's weight, is 14 lb.: how many stone in 112 lb.? {8 @ 112 / 14}
q: A carat grain is .792 Troy grains, and 4 carat grains make a carat: how many Troy grains in a carat? {~d 4 * .792}
q: Gold is said to be so many carats fine, out of 24. What part of 18-carat gold is pure gold? {~18 / 24}
q: An avoirdupois ounce is what part of a pound Troy? {~7000 / 16 / 5760}
sol: 1 oz. avoirdupois = 7000 / 16 = 437 1/2 grains; a Troy pound is 5760 grains: 437 1/2 / 5760 = 175/2304.

== 29 | Compound Numbers | Measures of extension
note: A line has length only; a surface, length and breadth; a solid, length, breadth and thickness. The area of a rectangle is its length multiplied by its breadth.
rule: 12 inches make 1 foot
rule: 3 feet make 1 yard
rule: 5 and 1 half yards make 1 rod
rule: 320 rods make 1 mile
rule: 100 links make 1 chain
rule: 80 chains make 1 mile
rule: 6 feet make 1 fathom
rule: 144 square inches make 1 square foot
rule: 9 square feet make 1 square yard
rule: 30 and 1 fourth square yards make 1 square rod
rule: 160 square rods make 1 acre
rule: 640 acres make 1 square mile
rule: 36 square miles make 1 township
rule: 1728 cubic inches make 1 cubic foot
rule: 27 cubic feet make 1 cubic yard
rule: 128 cubic feet make 1 cord
q: How many feet in a mile? {5280 @ 320 * 16_1/2} Inches in a mile? {63360 @ 5280 * 12}
q: How many feet in a rod? {~5_1/2 * 3} How many inches in a rod? {198}
q: How many inches long is a link of Gunter's chain? {~d 7.92} How many feet long is the chain? {66}
q: How many square feet in an acre? {43560 @ 160 * 30_1/4 * 9} Square yards in an acre? {4840 @ 160 * 30_1/4}
q: How many square feet in a square rod? {~30_1/4 * 9}
q: How many acres in a township? {23040 @ 36 * 640}
q: A rectangle is 4 inches long and 3 inches wide: how many square inches does it contain? {12 @ 4 * 3}
sol: On each inch of the length stands a square inch, making a row of 4; there are as many rows as inches in the width, 3. 4 x 3 = 12 square inches.
q: How many cubic inches in a cubic yard? {46656 @ 27 * 1728}
q: A perch of stone is 16 1/2 ft. long, 1 1/2 ft. wide, and 1 ft. high: how many cubic feet does it contain? {~16_1/2 * 1_1/2 * 1}
q: A pile of wood 4 ft. high, 4 ft. wide, and 8 ft. long is a cord: how many cubic feet in a cord foot, one foot of its length? {16 @ 4 * 4}
q: A board foot is 1 ft. long, 1 ft. wide, and 1 in. thick: how many cubic inches does it contain? {144 @ 12 * 12 * 1}
q: A nautical mile is about 1.15 statute miles, and 60 nautical miles make 1 degree on the equator, or 69.16 statute miles. A league is 3 nautical miles: how many statute miles is it? {3.458|3.45}
q: A cable-length is 720 feet: how many fathoms is it? {120 @ 720 / 6}

== 30 | Compound Numbers | Capacity, angles and time
note: The liquid gallon contains 231 cubic inches; the bushel, 2150.42 cubic inches. A circumference contains 360 degrees. Every year divisible by 4 is a leap year, unless it ends with two ciphers; then it must be divisible by 400.
rule: 4 gills make 1 pint
rule: 2 pints make 1 quart
rule: 4 quarts make 1 gallon
rule: 2 pints make 1 quart, dry
rule: 8 quarts make 1 peck
rule: 4 pecks make 1 bushel
rule: 60 minims make 1 fluid drachm
rule: 8 fluid drachms make 1 fluid ounce
rule: 16 fluid ounces make 1 pint
rule: 60 seconds make 1 minute
rule: 60 minutes make 1 degree
rule: 360 degrees make 1 circumference
rule: 60 seconds make 1 minute of time
rule: 60 minutes make 1 hour
rule: 24 hours make 1 day
rule: 7 days make 1 week
rule: 100 years make 1 century
q: How many gills in a gallon? {32 @ 4 * 2 * 4} How many pints in a bushel? {64 @ 4 * 8 * 2}
q: How many minims in a fluid ounce? {480 @ 8 * 60}
q: How many cubic inches in a liquid quart? {~231 / 4} In a dry quart? {~d 268.8 / 4}
q: A bushel of wheat legally weighs 60 lb.: what do 25 bushels weigh? Pounds? {1500 @ 25 * 60}
q: A bushel of oats legally weighs 32 lb., and of corn 56 lb.: how many bushels of oats weigh as much as 4 bushels of corn? {7 @ 4 * 56 / 32}
q: How many seconds in a circumference? {1296000 @ 360 * 60 * 60} How many minutes in a right angle of 90 degrees? {5400 @ 90 * 60}
q: The twelfth part of a circumference is a sign: how many degrees is it? {30}
q: How many seconds in a day? {86400 @ 24 * 60 * 60} In a week? {604800 @ 7 * 86400}
q: How many hours in a common year of 365 days? {8760 @ 365 * 24} In a leap year? {8784 @ 366 * 24}
q: Which of these are leap years: 1832, 1648, 1600, 2000, 1857, 1700, 1800, 1918? {open}
sol: 1832, 1648, 1600, and 2000 are leap years; 1857, 1700, 1800, and 1918 are not (1700 and 1800 end with two ciphers and are not divisible by 400).
q: The solar year is 365 da. 5 hr. 48 min. 46.05 sec.; the Julian year was 365 da. 6 hr. By how many seconds was the Julian year too long? {~d 6 * 3600 - (5 * 3600 + 48 * 60 + 46.05)}
sol: 6 hours = 21600 sec.; 5 hr. 48 min. 46.05 sec. = 20926.05 sec.; the difference is 673.95 sec., or 11 minutes 18.95 seconds.
q: The sun appears to pass round the earth, 360 degrees, in 24 hours: over how many degrees does it pass in 1 hour? {15 @ 360 / 24} Over how many minutes of a degree in 1 minute of time? {15}
q: A sheet folded into 8 leaves is an octavo: how many pages does it make? {16} A duodecimo has 12 leaves: how many pages? {24}
q: How many sheets of paper in a ream of 20 quires? {480 @ 20 * 24} How many things in a great gross? {1728 @ 12 * 144}
q: A barrel of flour is 196 lb., and a barrel of pork 200 lb.: what do 5 barrels of flour and 3 of pork weigh? Pounds? {1580 @ 5 * 196 + 3 * 200}

== 31 | Compound Numbers | The metric system
note: The meter (39.37 inches) is the unit of length; the ar (a square 10 meters on a side) of land; the liter (a cubic decimeter) of capacity; the gram (a cubic centimeter of water) of weight. Milli = .001, centi = .01, deci = .1, deka = 10, hekto = 100, kilo = 1000, myria = 10000. A number is changed to a lower denomination by moving the point to the right, to a higher by moving it to the left.
rule: 10 millimeters make 1 centimeter
rule: 10 centimeters make 1 decimeter
rule: 10 decimeters make 1 meter
rule: 10 meters make 1 dekameter
rule: 10 dekameters make 1 hektometer
rule: 10 hektometers make 1 kilometer
rule: 100 centars make 1 ar
rule: 100 ars make 1 hektar
rule: 10 liters make 1 dekaliter
rule: 10 dekaliters make 1 hektoliter
rule: 10 grams make 1 dekagram
rule: 1000 grams make 1 kilogram
rule: 1000 kilograms make 1 metric ton
q: 15.03 meters are how many dekameters? {~d 15.03 / 10} Hektometers? {~d 15.03 / 100} Decimeters? {~d 15.03 * 10} Centimeters? {~d 15.03 * 100}
sol: 15.03 m. is read 15 meters and 3 centimeters. Moving the point: 1.503 dekameters, .1503 hektometer, 150.3 decimeters, 1503 centimeters.
q: How many millimeters in a meter? {1000} How many meters in a kilometer? {1000}
q: How many square meters in an ar? {100 @ 10 * 10} In a hektar? {10000}
q: How many grams in a metric ton? {1000000}
q: A meter is 39.37 inches: how many inches in a kilometer? {~d 1000 * 39.37}
q: A kilometer is .62137 mile: how many miles are 10 kilometers? {~d 10 * .62137}
q: A liter is 1.0567 quarts: how many quarts are 20 liters? {~d 20 * 1.0567}
q: A kilogram is 2.2046 pounds: how many pounds are 50 kilograms? {~d 50 * 2.2046}
q: A hektar is 2.471 acres: how many acres are 12 hektars? {~d 12 * 2.471}
q: A gram is 15.432 grains Troy: how many grains in 5 grams? {~d 5 * 15.432}
== 32 | Compound Numbers | Reduction
note: From a higher denomination to a lower, multiply by the number of the lower that makes a unit of the higher, and add the lower denomination given. From a lower to a higher, divide by the number of its own denomination that makes a unit of the next higher. In the metric system, move the point.
q: Reduce 18 bushels to pints. {1152 @ 18 * 4 * 8 * 2}
sol: 18 bu. = 18 times 4 pk. = 72 pk.; 72 pk. = 72 times 8 qt. = 576 qt.; 576 qt. = 576 times 2 pt. = 1152 pt.
q: Reduce 236 inches to yards. {~236 / 36}
sol: 236 in. / 12 = 19 2/3 ft.; 19 2/3 ft. / 3 = 6 5/9 yd. Or, since 1 yd. = 36 in., 236 / 36 = 6 5/9 yd.
q: Reduce 3/8 gallons to pints. {~3/8 * 4 * 2}
q: Reduce 5 5/7 gr. to ounces, Apothecaries' weight. {~5_5/7 / 20 / 3 / 8}
sol: 5 5/7 gr. = 40/7 x 1/20 x 1/3 x 1/8 = 1/84 oz.
q: Reduce 9.375 acres to square rods. {~9.375 * 160}
q: Reduce 2000 seconds to hours. {~2000 / 3600}
q: Reduce 1238.73 hektograms to grams. {~1238.73 * 100}
q: How many yards in 880 meters? To three places. {~r3 39.37 * 880 / 36}
q: How many square rods in a rectangular field 18.22 chains long by 4.76 ch. wide? {~d 18.22 * 4.76 * 16}
q: Reduce 16.02 chains to miles. {~d 16.02 / 80}
q: How many bushels of wheat would it take to fill 750 hektoliters? {~d 750 * 2.8375}
q: Reduce 35.781 sq. yd. to sq. in. {~d 35.781 * 9 * 144}
q: Reduce 10240 sq. rd. to sq. ch. {640 @ 10240 / 16}
q: How many perches of masonry in a rectangular solid wall 40 ft. long by 7 1/2 ft. high, and 2 2/3 ft. average thickness? {~40 * 7_1/2 * 2_2/3 / 24.75}
q: How many ounces troy in the Brazilian Emperor's diamond, which weighs 1680 carats? {~d 1680 * 3.168 / 480}
q: Reduce 75 pwt. to drams. {30 @ 75 * 24 / 60}
q: Reduce 4/7 gr. to ounces, Apothecaries'. {~4/7 / 480}
q: Reduce 18 3/4 drams (Apothecaries') to oz. av. {~18_3/4 * 60 / 437.5}
q: Reduce 96 oz. av. to oz. troy. {~96 * 7000 / 16 / 480}
q: How many gal. in a tank 3 ft. long by 2 1/4 ft. wide and 1 1/2 ft. deep? {~3 * 2_1/4 * 1_1/2 * 1728 / 231}
q: How many bushels in a bin 9.3 ft. long by 3 5/8 ft. wide and 2 1/4 ft. deep? To one place. {~r1 9.3 * 3_5/8 * 2_1/4 * 1728 / 2150.4}
q: How many sters in 75 cords of wood? To two places. {~r2 75 / .2759}
q: Reduce 2 1/4 years (of 365 days) to seconds. {~2_1/4 * 31536000}
q: Forty-nine hours is what part of a week? {~49 / 168}
q: Reduce 90.12 kiloliters to liters. {~90.12 * 1000}
q: Reduce 25 seconds to the decimal of a degree. {~q 25 / 3600}
q: Reduce 192 sq. in. to sq. yd. {~192 / 1296}
q: Reduce 6 2/3 cu. yd. to cu. in. {~6_2/3 * 46656}
q: Reduce $117.14 to mills. {117140} Reduce 6.19 cents to dollars. {.0619} Reduce 1600 mills to dollars. {1.60|1.6}
q: Reduce $5 3/8 to mills. {5375 @ 5_3/8 * 1000}
q: Reduce 12 lb. av. to lb. troy. {~12 * 7000 / 5760}
q: How many grams in 6.45 quintals? {645000 @ 6.45 * 100 * 1000}
q: Reduce .216 gr. to oz. troy. {~d .216 / 480}
q: Reduce 47.3084 sq. mi. to sq. rd. {~d 47.3084 * 640 * 160}
q: Reduce 4 1/2 scruples to pounds, Apothecaries'. {~4_1/2 * 20 / 5760}
q: Reduce 7 1/9 oz. av. to cwt. {~7_1/9 / 1600}
q: Reduce 99 yd. to miles. {~99 / 1760}
q: How many acres in a rectangle 24.5 rd. long by 16.02 rd. wide? {~d 24.5 * 16.02 / 160}
q: How many cubic yards in a box 6 1/4 ft. long by 2 1/2 ft. wide and 3 ft. high? {~6_1/4 * 2_1/2 * 3 / 27}
q: Reduce 169 ars to square meters. {16900}
q: Reduce 2 1/2 fluid ounces to minims. {1200 @ 2_1/2 * 8 * 60}
q: If a piece of gold is 6/7 pure, how many carats fine is it? {~6/7 * 24}
q: In 18 3/4 carat gold, what part is pure? {~18_3/4 / 24} What part alloy? {~1 - 18_3/4 / 24}
q: How many square meters of matting are required to cover a floor, the dimensions of which are 6 m., 1 1/2 dm. by 5 m., 3 cm.? {~d 6.15 * 5.03}
q: How many cords of wood in a pile 120 ft. long, 6 1/2 ft. wide, and 8 3/4 ft. high? {~120 * 6_1/2 * 8_3/4 / 128}
q: How many sq. ft. in the four sides of a room 21 1/2 ft. long, 16 1/2 ft. wide, and 13 ft. high? {988 @ 2 * (21_1/2 + 16_1/2) * 13}
q: What will be the cost of 27 T. 18 cwt. 3 qr. 15 lb. 12 oz. of potash, at $48.20 a ton? Dollars, to cents. {~r2 (27 + (18 * 100 + 75 + 15_3/4) / 2000) * 48.20}
q: What is the value of a pile of wood 16 m., 1 dm., 5 cm. long, 1 m., 2 dm., 2 cm. wide, and 1 m., 6 dm., 8 cm. high, at $2.30 a ster? Dollars, to cents. {~r2 16.15 * 1.22 * 1.68 * 2.30}
q: What is the cost of a field 173 rods long and 84 rods wide, at $25.60 an acre? {~d 173 * 84 / 160 * 25.60}
q: If an open court contain 160 sq. rd. 85 sq. in., how many stones, each 5 inches square, will be required to pave it? {250909 @ (160 * 39204 + 85) / 25}
q: A lady had a grass-plot 20 meters long and 15 meters wide; after reserving two plots, one 2 meters square and the other 3 meters square, she paid 51 cents a square meter to have it paved with stones: what did the paving cost? Dollars? {~d (300 - 4 - 9) * .51}
q: A cubic yard of lead weighs 19,128 lb.: what is the weight of a block 5 ft. 3 1/4 in. long, 3 ft. 2 in. wide, and 1 ft. 8 in. thick? Pounds, to two places. {~r2 63_1/4 * 38 * 20 / 46656 * 19128}
sol: The block is 63 1/4 x 38 x 20 = 48070 cu. in.; a cubic yard is 46656 cu. in. 48070/46656 of 19128 lb. = 19707.71 lb., or 9 T. 17 cwt. 7 lb. 11.37 oz.
q: A lady bought a dozen silver spoons, weighing 3 oz. 4 pwt. 9 gr., at $2.20 an oz., and a gold chain weighing 13 pwt., at $1 1/4 a pwt.: required the total cost of the spoons and chain. Dollars, to cents. {~r2 (3 + 4/20 + 9/480) * 2.20 + 13 * 1_1/4}
q: A wagon-bed is 10 1/2 ft. long, 3 1/2 ft. wide, and 1 1/2 ft. deep, inside measure: how many bushels of corn will it hold, deducting one half for cobs? To three places. {~r3 10_1/2 * 3_1/2 * 1_1/2 * 1728 / 2150.4 / 2}
q: If a man weigh 160 lb. avoirdupois, what will he weigh by troy weight? Pounds? {~160 * 7000 / 5760}
sol: 160 x 7000 / 5760 = 194 4/9 lb. troy, or 194 lb. 5 oz. 6 pwt. 16 gr.
q: The fore-wheel of a wagon is 13 ft. 6 in. in circumference, and the hind wheel 18 ft. 4 in.: how many more revolutions will the fore-wheel make than the hind one in 50 miles? {5155|5156}
sol: 50 mi. = 264000 ft. 264000 / 13 1/2 = 19555 5/9; 264000 / 18 1/3 = 14400. The difference is 5155 whole revolutions.
q: An apothecary bought 5 lb. 10 oz. of quinine, at $2.20 an ounce, and sold it in doses of 9 gr., at 10 cents a dose: how much did he gain? Dollars, to cents. {~r2 70 * 480 / 9 * .10 - 70 * 2.20}
q: How many steps must a man take in walking from Kansas City to St. Louis, if the distance be 275 miles, and each step, 2 ft. 9 in.? {528000 @ 275 * 5280 / 2_3/4}
q: The area of Missouri is 65350 sq. mi.: how many hektars does it contain? To two places. {~r2 65350 * 640 / 2.471}
q: A school-room is 36 ft. long, 24 ft. wide, and 14 ft. high; required the number of gallons of air it will contain. To two places. {~r2 36 * 24 * 14 * 1728 / 231}
q: Allowing 8 shingles to the square foot, how many shingles will be required to cover the roof of a barn which is 60 feet long, and 15 feet from the comb to the eaves? {14400 @ 60 * 15 * 2 * 8}
q: A boy goes to bed 30 minutes later, and gets up 40 minutes earlier than his room-mate: how much time does he gain over his room-mate for work and study in the two years 1884 and 1885, deducting Sundays only? Hours? {~(731 - 104) * 70 / 60}
sol: The two years have 731 days. 1884 and 1885 have 52 Sundays each, so 627 days remain, each with a gain of 70 min.: 43890 min. = 731 hr. 30 min.
== 33 | Compound Numbers | Adding and subtracting compound numbers
note: Write units of the same denomination in the same column. Begin with the lowest; divide each sum by the number of its units that make one of the next higher, write the remainder and carry the quotient. In subtracting, when a lower number is larger, increase the upper by a unit of the next higher denomination and the next figure of the subtrahend by the same. Answer each denomination in turn.
q: Add 3 bu. 2 1/4 pk.; 1 pk. 1 1/2 pt.; 5 qt. 1 pt.; 2 bu. 1 1/3 qt.; and .125 pt. Bu.? {~p 4,8,2 # 0 # 3 * 64 + 2_1/4 * 16 + 16 + 1_1/2 + 5 * 2 + 1 + 2 * 64 + 1_1/3 * 2 + .125} Pk.? {~p # 1 # $} Qt.? {~p # 2 # $} Pt.? {~p # 3 # $}
sol: The right-hand column gives 3 7/24 pt. = 1 qt. 1 7/24 pt.; carrying 1 qt. makes 9 qt. = 1 pk. 1 qt.; carrying 1 pk. makes 4 pk. = 1 bu. The sum is 6 bu. 0 pk. 1 qt. 1 7/24 pt.
q: Add 2 rd. 9 ft. 7 1/4 in.; 13 ft. 5.78 in.; 4 rd. 11 ft. 6 in.; 1 rd. 10 2/3 ft.; 6 rd. 14 ft. 6 5/8 in. Rd.? {~p 16_1/2,12 # 0 # 2 * 198 + 9 * 12 + 7.25 + 13 * 12 + 5.78 + 4 * 198 + 11 * 12 + 6 + 198 + 10_2/3 * 12 + 6 * 198 + 14 * 12 + 6.625} Ft.? {~p # 1 # $} In.? {~p # 2 # $}
sol: The columns give 16 rd. 9 1/2 ft. 9.655 in. The 1/2 foot is 6 inches; 9.655 + 6 = 15.655 in. = 1 ft. 3.655 in. The answer is 16 rd. 10 ft. 3.655 in.
q: Add 2/7 mi.; 146 1/3 rd.; 10 mi. 14 rd. 7 ft. 6 in.; 209.6 rd.; 37 rd. 16 ft. 2 1/4 in.; 1 mi. 12 ft. 8.726 in. Mi.? {~p 320,16_1/2,12 # 0 # 2/7 * 63360 + 146_1/3 * 198 + 10 * 63360 + 14 * 198 + 7 * 12 + 6 + 209.6 * 198 + 37 * 198 + 16 * 12 + 2_1/4 + 63360 + 12 * 12 + 8.726} Rd.? {~p # 1 # $} Ft.? {~p # 2 # $} In.? {~p # 3 # $}
q: Add 6.19 yd.; 2 yd. 2 ft. 9 3/4 in.; 1 ft. 4.54 in.; 10 yd. 2.376 ft.; 3/4 yd.; 1 5/6 ft.; 7/8 in. Yd.? {~p 3,12 # 0 # 6.19 * 36 + 2 * 36 + 2 * 12 + 9_3/4 + 12 + 4.54 + 10 * 36 + 2.376 * 12 + 3/4 * 36 + 1_5/6 * 12 + 7/8} Ft.? {~p # 1 # $} In.? {~p # 2 # $}
q: Add 3 yd. 2 qr. 3 na. 1 1/2 in.; 1 qr. 2 2/5 na.; 6 yd. 1 na. 2.175 in.; 1.63 yd.; 2/3 qr.; 5/9 na. (cloth measure: 2 1/4 in. make 1 nail, 4 nails 1 quarter, 4 quarters 1 yard) Yd.? {~p 4,4,2_1/4 # 0 # 3 * 36 + 2 * 9 + 3 * 2_1/4 + 1_1/2 + 9 + 2_2/5 * 2_1/4 + 6 * 36 + 2_1/4 + 2.175 + 1.63 * 36 + 2/3 * 9 + 5/9 * 2_1/4} Qr.? {~p # 1 # $} Na.? {~p # 2 # $} In.? {~p # 3 # $}
q: If the volume of the earth is 1; Mercury, .06; Venus, .957; Mars, .14; Jupiter, 1414.2; Saturn, 734.8; Uranus, 82; Neptune, 110.6; the Sun, 1407124; and the Moon, .018, what is the volume of all? {~d 1 + .06 + .957 + .14 + 1414.2 + 734.8 + 82 + 110.6 + 1407124 + .018}
q: James bought a balloon for 9 francs and 76 centimes, a ball for 68 centimes, a hoop for one franc and 37 centimes, and gave to the poor 2 francs and 65 centimes, and had 3 francs and 4 centimes left. How much money did he have at first? Francs? {~p 100 # 0 # 976 + 68 + 137 + 265 + 304} Centimes? {~p # 1 # $}
q: Add 15 sq. yd. 5 sq. ft. 87 sq. in.; 16 1/2 sq. yd.; 10 sq. yd. 7.22 sq. ft.; 4 sq. ft. 121.6 sq. in.; 11/32 sq. yd. Sq. yd.? {~p 9,144 # 0 # 15 * 1296 + 5 * 144 + 87 + 16_1/2 * 1296 + 10 * 1296 + 7.22 * 144 + 4 * 144 + 121.6 + 11/32 * 1296} Sq. ft.? {~p # 1 # $} Sq. in.? {~p # 2 # $}
q: Add 101 A. 98.35 sq. rd.; 66 A. 74 1/2 sq. rd.; 20 A.; 12 A. 113 sq. rd.; 5 A. 13.33 1/3 sq. rd. A.? {~p 160 # 0 # 101 * 160 + 98.35 + 66 * 160 + 74_1/2 + 20 * 160 + 12 * 160 + 113 + 5 * 160 + 13_1/3} Sq. rd.? {~p # 1 # $}
q: Add 23 cu. yd. 14 cu. ft. 1216 cu. in.; 41 cu. yd. 6 cu. ft. 642.132 cu. in.; 9 cu. yd. 25.065 cu. ft.; 7/15 cu. yd. Cu. yd.? {~p 27,1728 # 0 # 23 * 46656 + 14 * 1728 + 1216 + 41 * 46656 + 6 * 1728 + 642.132 + 9 * 46656 + 25.065 * 1728 + 7/15 * 46656} Cu. ft.? {~p # 1 # $} Cu. in.? {~p # 2 # $}
q: Add 5/6 C.; 3/8 cu. ft.; 1000 cu. in. (a cord is 128 cu. ft.) C.? {~p 128,1728 # 0 # 5/6 * 128 * 1728 + 3/8 * 1728 + 1000} Cu. ft.? {~p # 1 # $} Cu. in.? {~p # 2 # $}
q: Add 2 lb. troy, 6 2/3 oz.; 1 3/4 lb.; 12.68 pwt.; 11 oz. 13 pwt. 19 1/5 gr.; 3/8 lb. 15/16 oz.; 5/9 pwt. Lb.? {~p 12,20,24 # 0 # 2 * 5760 + 6_2/3 * 480 + 1_3/4 * 5760 + 12.68 * 24 + 11 * 480 + 13 * 24 + 19_1/5 + 3/8 * 5760 + 15/16 * 480 + 5/9 * 24} Oz.? {~p # 1 # $} Pwt.? {~p # 2 # $} Gr.? {~p # 3 # $}
q: Add (Apothecaries' weight) 8 oz. 14.6 gr.; 4.18 oz.; 7 5/12 dr.; 2 dr. 2 sc. 18 gr.; 1 oz. 12 gr.; 1/5 sc. Oz.? {~p 8,3,20 # 0 # 8 * 480 + 14.6 + 4.18 * 480 + 7_5/12 * 60 + 2 * 60 + 2 * 20 + 18 + 480 + 12 + 1/5 * 20} Dr.? {~p # 1 # $} Sc.? {~p # 2 # $} Gr.? {~p # 3 # $}
q: Add 5/16 T.; 9 cwt. 1 qr. 22 lb.; 3.06 qr.; 4 T. 8.764 cwt.; 3 qr. 6 lb.; 7/12 cwt. T.? {~p 20,4,25,16 # 0 # 5/16 * 32000 + 9 * 1600 + 400 + 22 * 16 + 3.06 * 400 + 4 * 32000 + 8.764 * 1600 + 3 * 400 + 6 * 16 + 7/12 * 1600} Cwt.? {~p # 1 # $} Qr.? {~p # 2 # $} Lb.? {~p # 3 # $} Oz.? {~p # 4 # $}
q: Add .3 lb. av.; 5/9 oz. Ounces? {~.3 * 16 + 5/9}
q: Add 6 gal. 3 1/3 qt.; 2 gal. 1 qt. .83 pt.; 1 gal. 2 qt. 1/2 pt.; 3/5 gal.; 2/9 qt.; 7/8 pt. Gal.? {~p 4,2 # 0 # 6 * 8 + 3_1/3 * 2 + 2 * 8 + 2 + .83 + 8 + 4 + 1/2 + 3/5 * 8 + 2/9 * 2 + 7/8} Qt.? {~p # 1 # $} Pt.? {~p # 2 # $}
q: Add 4 gal. .75 pt.; 10 gal. 3 qt. 1 1/2 pt.; 8 gal. 2/3 pt.; 5.64 gal.; 2.3 qt.; 1.27 pt.; 3/20 pt. Gal.? {~p 4,2 # 0 # 4 * 8 + .75 + 10 * 8 + 6 + 1_1/2 + 8 * 8 + 2/3 + 5.64 * 8 + 2.3 * 2 + 1.27 + 3/20} Qt.? {~p # 1 # $} Pt.? {~p # 2 # $}
q: Add 1 bu. 1/2 pk.; 6/11 bu.; 3 pk. 5 qt. 1 1/4 pt.; 9 bu. 3.28 pk.; 7 qt. 1.16 pt.; 5/12 pk. Bu.? {~p 4,8,2 # 0 # 64 + 8 + 6/11 * 64 + 48 + 10 + 1_1/4 + 9 * 64 + 3.28 * 16 + 14 + 1.16 + 5/12 * 16} Pk.? {~p # 1 # $} Qt.? {~p # 2 # $} Pt.? {~p # 3 # $}
q: Add 3/8 bu.; 2/5 pk.; 7/9 qt.; 2/3 pt. Bu.? {~p 4,8,2 # 0 # 3/8 * 64 + 2/5 * 16 + 7/9 * 2 + 2/3} Pk.? {~p # 1 # $} Qt.? {~p # 2 # $} Pt.? {~p # 3 # $}
q: Add 6 fl. oz. 2 fl. dr. 25 minims; 2 1/2 fl. oz.; 7 fl. dr. 42 minims; 1 fl. oz. 2 2/3 fl. dr.; 3 fl. oz. 6 fl. dr. 51 minims. Fl. oz.? {~p 8,60 # 0 # 6 * 480 + 120 + 25 + 2_1/2 * 480 + 420 + 42 + 480 + 2_2/3 * 60 + 3 * 480 + 360 + 51} Fl. dr.? {~p # 1 # $} Minims? {~p # 2 # $}
q: Add 1/2 wk.; 1/2 da.; 1/2 hr.; 1/2 min.; 1/2 sec. Wk.? {~p 7,24,60,60 # 0 # 1/2 * 604800 + 1/2 * 86400 + 1/2 * 3600 + 1/2 * 60 + 1/2} Da.? {~p # 1 # $} Hr.? {~p # 2 # $} Min.? {~p # 3 # $} Sec.? {~p # 4 # $}
q: Add 3.26 yr. (365 da. each); 118 da. 5 hr. 42 min. 37 1/2 sec.; 63.4 da.; 7 4/5 hr.; 1 yr. 62 da. 19 hr. 24 4/5 min.; 73/192 da. Yr.? {~p 365,24,60,60 # 0 # 3.26 * 31536000 + 118 * 86400 + 5 * 3600 + 42 * 60 + 37_1/2 + 63.4 * 86400 + 7_4/5 * 3600 + 31536000 + 62 * 86400 + 19 * 3600 + 24_4/5 * 60 + 73/192 * 86400} Da.? {~p # 1 # $} Hr.? {~p # 2 # $} Min.? {~p # 3 # $} Sec.? {~p # 4 # $}
q: Add 27 degrees 14 minutes 55.24 seconds; 9 degrees 18 1/4 seconds; 1 degree 15 1/3 minutes; 116 degrees 44 minutes 23.8 seconds. Degrees? {~p 60,60 # 0 # 27 * 3600 + 14 * 60 + 55.24 + 9 * 3600 + 18_1/4 + 3600 + 15_1/3 * 60 + 116 * 3600 + 44 * 60 + 23.8} Minutes? {~p # 1 # $} Seconds? {~p # 2 # $}
q: Add $1/2; 1/4 ct.; 1/8 m. Dollars? {~p 100,10 # 0 # 1/2 * 1000 + 1/4 * 10 + 1/8} Cents? {~p # 1 # $} Mills? {~p # 2 # $}
q: Add 3 dollars 7 m.; 5 dollars 20 ct.; 100 dollars 2 ct. 6 m.; 19 dollars 1/8 ct. Dollars? {~p 100,10 # 0 # 3007 + 5200 + 100026 + 19000 + 1/8 * 10} Cents? {~p # 1 # $} Mills? {~p # 2 # $}
q: Add 21 pounds 6s. 3 1/2d.; 5 pounds 17 3/4s.; 9.085 pounds; 16s. 7 1/4d.; 5/12 of a pound. Pounds? {~p 20,12 # 0 # 21 * 240 + 6 * 12 + 3_1/2 + 5 * 240 + 17_3/4 * 12 + 9.085 * 240 + 16 * 12 + 7_1/4 + 5/12 * 240} Shillings? {~p # 1 # $} Pence? {~p # 2 # $}
q: Add 2/3 A.; 3/4 sq. rd.; 4/5 sq. ft. A.? {~p 160,30_1/4,9,144 # 0 # 2/3 * 6272640 + 3/4 * 39204 + 4/5 * 144} Sq. rd.? {~p # 1 # $} Sq. yd.? {~p # 2 # $} Sq. ft.? {~p # 3 # $} Sq. in.? {~p # 4 # $}
q: From 9 yd. 1 ft. 6 1/2 in. take 1 yd. 2.45 ft. Yd.? {~p 3,12 # 0 # 9 * 36 + 12 + 6_1/2 - 36 - 2.45 * 12} Ft.? {~p # 1 # $} In.? {~p # 2 # $}
sol: The minuend is 9 yd. 1 ft. 6.5 in.; the subtrahend, 1 yd. 2 ft. 5.4 in. 5.4 from 6.5 leaves 1.1 in. To take 2 ft. from 1 ft., add 3 ft. to the minuend and 1 yd. to the subtrahend: 2 ft. from 4 ft. leaves 2 ft., and 2 yd. from 9 yd. leaves 7 yd. The answer is 7 yd. 2 ft. 1.1 in.
q: From 2 sq. rd. 1 sq. ft. take 1 sq. rd. 30 sq. yd. 2 sq. ft. Sq. ft.? {~p 9,144 # 0 # 2 * 39204 + 144 - 39204 - 30 * 1296 - 2 * 144} Sq. in.? {~p # 1 # $}
sol: Add 1/4 sq. yd. (2 1/4 sq. ft.) to the 1 sq. ft. and take 2 sq. ft.: 1 1/4 sq. ft. Giving the same increase to the 30 sq. yd. makes it 30 1/4 sq. yd. = 1 sq. rd., which leaves nothing above feet. The answer is 1 1/4 sq. ft. = 1 sq. ft. 36 sq. in.
q: Subtract 2/5 mi. from 144.86 rd. Rd.? {~p 16_1/2,12 # 0 # (144.86 - 128) * 198} Ft.? {~p # 1 # $} In.? {~p # 2 # $}
q: Subtract 1.35 yd. from 4 yd. 2 qr. 1 na. 1 3/4 in. Yd.? {~p 4,4,2_1/4 # 0 # 4 * 36 + 18 + 2_1/4 + 1_3/4 - 1.35 * 36} Qr.? {~p # 1 # $} Na.? {~p # 2 # $} In.? {~p # 3 # $}
q: Subtract 2 sq. rd. 24 sq. yd. 91 sq. in. from 5 sq. rd. 16 sq. yd. 6 2/3 sq. ft. Sq. rd.? {~p 30_1/4,9,144 # 0 # 5 * 39204 + 16 * 1296 + 6_2/3 * 144 - 2 * 39204 - 24 * 1296 - 91} Sq. yd.? {~p # 1 # $} Sq. ft.? {~p # 2 # $} Sq. in.? {~p # 3 # $}
q: Subtract 384 A. 43.92 sq. rd. from 1.305 sq. mi. A.? {~p 160 # 0 # 1.305 * 102400 - 384 * 160 - 43.92} Sq. rd.? {~p # 1 # $}
q: Subtract 13 cu. yd. 25 cu. ft. 1204.9 cu. in. from 20 cu. yd. 4 cu. ft. 1000 cu. in. Cu. yd.? {~p 27,1728 # 0 # 20 * 46656 + 4 * 1728 + 1000 - 13 * 46656 - 25 * 1728 - 1204.9} Cu. ft.? {~p # 1 # $} Cu. in.? {~p # 2 # $}
q: Subtract 9.362 oz. troy from 1 lb. 15 pwt. 4 gr. Oz.? {~p 20,24 # 0 # 5760 + 15 * 24 + 4 - 9.362 * 480} Pwt.? {~p # 1 # $} Gr.? {~p # 2 # $}
q: Subtract 10/13 dram from 6/11 ounce (Apothecaries'). Dr.? {~p 3,20 # 0 # 6/11 * 480 - 10/13 * 60} Sc.? {~p # 1 # $} Gr.? {~p # 2 # $}
q: Subtract 56 T. 9 cwt. 1 qr. 23 lb. from 75.004 T. T.? {~p 20,4,25,16 # 0 # 75.004 * 32000 - 56 * 32000 - 9 * 1600 - 400 - 23 * 16} Cwt.? {~p # 1 # $} Qr.? {~p # 2 # $} Lb.? {~p # 3 # $} Oz.? {~p # 4 # $}
q: Subtract 5/24 lb. troy from 6/35 lb. avoirdupois. Grains? {~6/35 * 7000 - 5/24 * 5760}
sol: 6/35 lb. av. = 6/35 of 7000 = 1200 gr.; 5/24 lb. troy = 5/24 of 5760 = 1200 gr. The difference is nothing.
q: Subtract 12 gal. 1 qt. 3 gills from 31 gal. 1 1/2 pt. Gal.? {~p 4,2,4 # 0 # 31 * 32 + 6 - 12 * 32 - 8 - 3} Qt.? {~p # 1 # $} Pt.? {~p # 2 # $} Gi.? {~p # 3 # $}
q: Subtract .0625 bu. from 3 pk. 5 qt. 1 pt. Pk.? {~p 8,2 # 0 # 48 + 10 + 1 - .0625 * 64} Qt.? {~p # 1 # $} Pt.? {~p # 2 # $}
q: Subtract 1 fl. oz. 4 fl. dr. 38 minims from 4 fl. oz. 2 fl. dr. Fl. oz.? {~p 8,60 # 0 # 4 * 480 + 120 - 480 - 240 - 38} Fl. dr.? {~p # 1 # $} Minims? {~p # 2 # $}
q: Subtract 275 da. 9 hr. 12 min. 59 sec. from 2.4816 yr. (allowing 365 1/4 days to the year.) Yr.? {~p 365_1/4,24,60,60 # 0 # 2.4816 * 365_1/4 * 86400 - 275 * 86400 - 9 * 3600 - 12 * 60 - 59} Da.? {~p # 1 # $} Hr.? {~p # 2 # $} Min.? {~p # 3 # $} Sec.? {~p # 4 # $}
q: Subtract 43 degrees 18 minutes 57.18 seconds from a quadrant (90 degrees). Degrees? {~p 60,60 # 0 # 90 * 3600 - 43 * 3600 - 18 * 60 - 57.18} Minutes? {~p # 1 # $} Seconds? {~p # 2 # $}
q: Subtract 161 degrees 34 minutes 11.8 seconds from 180 degrees. Degrees? {~p 60,60 # 0 # 180 * 3600 - 161 * 3600 - 34 * 60 - 11.8} Minutes? {~p # 1 # $} Seconds? {~p # 2 # $}
q: Subtract 37/40 ct. from $3/32. Cents? {~p 10 # 0 # 3/32 * 1000 - 37/40 * 10} Mills? {~p # 1 # $}
q: Subtract 5 dollars 43 ct. 2 1/2 m. from 12 dollars 6 ct. 8 1/3 m. Dollars? {~p 100,10 # 0 # 12068_1/3 - 5432_1/2} Cents? {~p # 1 # $} Mills? {~p # 2 # $}
q: Subtract 9 pounds 18s. 6 1/2d. from 20 pounds. Pounds? {~p 20,12 # 0 # 4800 - 9 * 240 - 18 * 12 - 6_1/2} Shillings? {~p # 1 # $} Pence? {~p # 2 # $}
q: From 1/2 A. 10 sq. in. take 79 sq. rd. 30 sq. yd. 2 sq. ft. 30 sq. in. Sq. rd.? {~p 30_1/4,9,144 # 0 # 80 * 39204 + 10 - 79 * 39204 - 30 * 1296 - 2 * 144 - 30} Sq. yd.? {~p # 1 # $} Sq. ft.? {~p # 2 # $} Sq. in.? {~p # 3 # $}
q: From 3 sq. rd. 1 sq. ft. 1 sq. in. take 1 sq. rd. 30 sq. yd. 1 sq. ft. 140 sq. in. Sq. rd.? {~p 30_1/4,9,144 # 0 # 3 * 39204 + 145 - 39204 - 30 * 1296 - 144 - 140} Sq. yd.? {~p # 1 # $} Sq. ft.? {~p # 2 # $} Sq. in.? {~p # 3 # $}
q: From 3 rd. 2 in. take 2 rd. 5 yd. 1 ft. 4 in. Rd.? {~p 5_1/2,3,12 # 0 # 3 * 198 + 2 - 2 * 198 - 5 * 36 - 12 - 4} Yd.? {~p # 1 # $} Ft.? {~p # 2 # $} In.? {~p # 3 # $}
q: From 7 mi. 1 in. take 4 mi. 319 rd. 16 ft. 3 in. Mi.? {~p 320,16_1/2,12 # 0 # 7 * 63360 + 1 - 4 * 63360 - 319 * 198 - 16 * 12 - 3} Rd.? {~p # 1 # $} Ft.? {~p # 2 # $} In.? {~p # 3 # $}
q: From 13 A. 3 sq. rd. 5 sq. ft. take 11 A. 30 sq. yd. 8 sq. ft. 40 sq. in. A.? {~p 160,30_1/4,9,144 # 0 # 13 * 6272640 + 3 * 39204 + 5 * 144 - 11 * 6272640 - 30 * 1296 - 8 * 144 - 40} Sq. rd.? {~p # 1 # $} Sq. yd.? {~p # 2 # $} Sq. ft.? {~p # 3 # $} Sq. in.? {~p # 4 # $}
q: From 18 A. 3 sq. ft. 3 sq. in. take 15 A. 3 sq. rd. 30 sq. yd. 1 sq. ft. 142 sq. in. A.? {~p 160,30_1/4,9,144 # 0 # 18 * 6272640 + 3 * 144 + 3 - 15 * 6272640 - 3 * 39204 - 30 * 1296 - 144 - 142} Sq. rd.? {~p # 1 # $} Sq. yd.? {~p # 2 # $} Sq. ft.? {~p # 3 # $} Sq. in.? {~p # 4 # $}

q: Find the difference of time between Sept. 22d, 1855, and July 1st, 1856 (counting 30 days to the month). Months? {9} Days? {9}
sol: 1856 yr. 7 mo. 1 da. less 1855 yr. 9 mo. 22 da.: 22 da. from 31 da. leave 9 da.; 10 mo. from 18 mo. leave 8... with the carrying, 9 mo. 9 da.
q: Find the difference of time between December 31st, 1814, and April 1st, 1822. Years? {7} Months? {3} Days? {0|1}
sol: 1822 yr. 4 mo. 1 da. less 1814 yr. 12 mo. 31 da.: borrowing 30 days and 12 months, 7 yr. 3 mo. 0 da. (Counting December's 31 days, the last day is 1 day.)

== 34 | Compound Numbers | Multiplying and dividing compound numbers
note: Multiply the lowest denomination first; divide the product by the number of its units that make one of the next higher, write the remainder and carry the quotient. In dividing, begin with the highest denomination; reduce each remainder to the next lower denomination and add it in before dividing.
q: Multiply 9 hr. 14 min. 8.17 sec. by 10. Da.? {~p 24,60,60 # 0 # (9 * 3600 + 14 * 60 + 8.17) * 10} Hr.? {~p # 1 # $} Min.? {~p # 2 # $} Sec.? {~p # 3 # $}
sol: Ten times 8.17 sec. = 81.7 sec. = 1 min. 21.7 sec.; write 21.7 and carry 1 min. to the 140 min.: 141 min. = 2 hr. 21 min.; carrying 2 hr. gives 92 hr. = 3 da. 20 hr.
q: Multiply 12 A. 148 sq. rd. 28 3/8 sq. yd. by 84. A.? {~p 160,30_1/4 # 0 # (12 * 4840 + 148 * 30_1/4 + 28_3/8) * 84} Sq. rd.? {~p # 1 # $} Sq. yd.? {~p # 2 # $}
sol: 84 = 7 x 12. Multiplying by 7 gives 90 A. 82 sq. rd. 17 1/8 sq. yd.; that by 12 gives 1086 A. 30 sq. rd. 24 sq. yd.
q: Multiply 7 rd. 10 ft. 5 in. by 6. Rd.? {~p 16_1/2,12 # 0 # (7 * 198 + 125) * 6} Ft.? {~p # 1 # $} In.? {~p # 2 # $}
q: Multiply 1 mi. 14 rd. 8 1/4 ft. by 97. Mi.? {~p 320,16_1/2,12 # 0 # (63360 + 14 * 198 + 8_1/4 * 12) * 97} Rd.? {~p # 1 # $} Ft.? {~p # 2 # $} In.? {~p # 3 # $}
q: Multiply 5 sq. yd. 8 sq. ft. 106 sq. in. by 13. Sq. yd.? {~p 9,144 # 0 # (5 * 1296 + 8 * 144 + 106) * 13} Sq. ft.? {~p # 1 # $} Sq. in.? {~p # 2 # $}
q: Multiply 41 A. 146.1087 sq. rd. by 9.046 A.? {~p 160 # 0 # (41 * 160 + 146.1087) * 9.046} Sq. rd.? {~p # 1 # $}
q: Multiply 10 cu. yd. 3 cu. ft. 428.15 cu. in. by 67. Cu. yd.? {~p 27,1728 # 0 # (10 * 46656 + 3 * 1728 + 428.15) * 67} Cu. ft.? {~p # 1 # $} Cu. in.? {~p # 2 # $}
q: Multiply 7 oz. 16 pwt. 5 3/4 gr. (troy) by 174. Lb.? {~p 12,20,24 # 0 # (7 * 480 + 16 * 24 + 5_3/4) * 174} Oz.? {~p # 1 # $} Pwt.? {~p # 2 # $} Gr.? {~p # 3 # $}
q: Multiply 2 dr. 1 sc. 13 gr. (Apothecaries') by 20. Oz.? {~p 8,3,20 # 0 # (2 * 60 + 20 + 13) * 20} Dr.? {~p # 1 # $} Sc.? {~p # 2 # $} Gr.? {~p # 3 # $}
q: Multiply 16 cwt. 1 qr. 7.88 lb. by 11. T.? {~p 20,4,25 # 0 # (16 * 100 + 25 + 7.88) * 11} Cwt.? {~p # 1 # $} Qr.? {~p # 2 # $} Lb.? {~p # 3 # $}
q: Multiply 5 gal. 3 qt. 1 pt. 2 gills by 35.108 Gal.? {~p 4,2,4 # 0 # (5 * 32 + 3 * 8 + 4 + 2) * 35.108} Qt.? {~p # 1 # $} Pt.? {~p # 2 # $} Gi.? {~p # 3 # $}
q: Multiply 26 bu. 2 pk. 7 qt. .37 pt. by 10. Bu.? {~p 4,8,2 # 0 # (26 * 64 + 2 * 16 + 7 * 2 + .37) * 10} Pk.? {~p # 1 # $} Qt.? {~p # 2 # $} Pt.? {~p # 3 # $}
q: Multiply 3 fl. dr. 48 minims by 12. Fl. oz.? {~p 8,60 # 0 # (3 * 60 + 48) * 12} Fl. dr.? {~p # 1 # $} Minims? {~p # 2 # $}
q: Multiply 18 da. 9 hr. 42 min. 29.3 sec. by 16 7/11. Da.? {~p 24,60,60 # 0 # (18 * 86400 + 9 * 3600 + 42 * 60 + 29.3) * 16_7/11} Hr.? {~p # 1 # $} Min.? {~p # 2 # $} Sec.? {~p # 3 # $}
q: Multiply 215 pounds 16s. 2 1/4d. by 75. Pounds? {~p 20,12 # 0 # (215 * 240 + 16 * 12 + 2_1/4) * 75} Shillings? {~p # 1 # $} Pence? {~p # 2 # $}
q: Multiply 10 degrees 28 minutes 42 1/2 seconds by 2.754 Degrees? {~p 60,60 # 0 # (10 * 3600 + 28 * 60 + 42_1/2) * 2.754} Minutes? {~p # 1 # $} Seconds? {~p # 2 # $}
q: Divide 5 cwt. 3 qr. 24 lb. 14 3/4 oz. of sugar equally among 4 men. Cwt.? {~p 4,25,16 # 0 # (5 * 1600 + 3 * 400 + 24 * 16 + 14_3/4) / 4} Qr.? {~p # 1 # $} Lb.? {~p # 2 # $} Oz.? {~p # 3 # $}
sol: 4 into 5 cwt. gives 1 cwt., 1 cwt. over = 4 qr., making 7 qr.; 4 into 7 qr. gives 1 qr., 3 qr. over = 75 lb., making 99 lb.; 4 into 99 lb. gives 24 lb., 3 lb. over = 48 oz., making 62 3/4 oz.; 4 into 62 3/4 oz. gives 15 11/16 oz.
q: If $42 purchase 67 bu. 2 pk. 5 qt. 1 3/4 pt. of meal, how much will $1 purchase? Bu.? {~p 4,8,2 # 0 # (67 * 64 + 2 * 16 + 5 * 2 + 1_3/4) / 42} Pk.? {~p # 1 # $} Qt.? {~p # 2 # $} Pt.? {~p # 3 # $}
sol: 42 = 6 x 7. Dividing by 6 gives 11 bu. 1 pk. 0 qt. 1 23/24 pt.; that by 7 gives 1 bu. 2 pk. 3 qt. 1 23/168 pt.
q: Divide 16 mi. 109 rd. by 7. Mi.? {~p 320,16_1/2,12 # 0 # (16 * 63360 + 109 * 198) / 7} Rd.? {~p # 1 # $} Ft.? {~p # 2 # $} In.? {~p # 3 # $}
q: Divide 37 rd. 14 ft. 11.28 in. by 18. Rd.? {~p 16_1/2,12 # 0 # (37 * 198 + 14 * 12 + 11.28) / 18} Ft.? {~p # 1 # $} In.? {~p # 2 # $}
q: Divide 675 C. 114.66 cu. ft. by 83. C.? {~p 128 # 0 # (675 * 128 + 114.66) / 83} Cu. ft.? {~p # 1 # $}
q: Divide 10 sq. rd. 29 sq. yd. 5 sq. ft. 94 sq. in. by 17. Sq. rd.? {~p 30_1/4,9,144 # 0 # (10 * 39204 + 29 * 1296 + 5 * 144 + 94) / 17} Sq. yd.? {~p # 1 # $} Sq. ft.? {~p # 2 # $} Sq. in.? {~p # 3 # $}
q: Divide 6 sq. mi. 35 sq. rd. by 22 1/2. Sq. mi.? {~p 640,160 # 0 # (6 * 102400 + 35) / 22_1/2} A.? {~p # 1 # $} Sq. rd.? {~p # 2 # $}
q: Divide 1245 cu. yd. 24 cu. ft. 1627 cu. in. by 11.303 Cu. yd.? {~p 27,1728 # 0 # (1245 * 46656 + 24 * 1728 + 1627) / 11.303} Cu. ft.? {~p # 1 # $} Cu. in.? {~p # 2 # $}
q: Divide 3 oz. 7 dr. 18 gr. (Apothecaries') by 12. Oz.? {~p 8,3,20 # 0 # (3 * 480 + 7 * 60 + 18) / 12} Dr.? {~p # 1 # $} Sc.? {~p # 2 # $} Gr.? {~p # 3 # $}
q: Divide 600 T. 7 cwt. 86 lb. by 29.06 T.? {~p 20,4,25,16 # 0 # (600 * 2000 + 7 * 100 + 86) * 16 / 29.06} Cwt.? {~p # 1 # $} Qr.? {~p # 2 # $} Lb.? {~p # 3 # $} Oz.? {~p # 4 # $}
q: Divide 312 gal. 2 qt. 1 pt. 3.36 gills by 72 5/8. Gal.? {~p 4,2,4 # 0 # (312 * 32 + 2 * 8 + 4 + 3.36) / 72_5/8} Qt.? {~p # 1 # $} Pt.? {~p # 2 # $} Gi.? {~p # 3 # $}
q: Divide 19302 bu. by 6.215 Bu.? {~p 4,8,2 # 0 # 19302 * 64 / 6.215} Pk.? {~p # 1 # $} Qt.? {~p # 2 # $} Pt.? {~p # 3 # $}

== 35 | Compound Numbers | Longitude and time
note: The sun passes over 15 degrees of longitude in 1 hour, 15 minutes of longitude in 1 minute of time, and 15 seconds in 1 second. To find the difference of longitude, multiply the difference of time by 15 and mark it in degrees; to find the difference of time, divide the difference of longitude by 15 and mark it in hours. The place farther east has the later time. Longitudes used here: New York 74 0 24 W., Philadelphia 75 9 3 W., Detroit 83 3 0 W., Cincinnati 84 29 45 W., Chicago 87 35 0 W., Mobile 88 2 28 W., New Orleans 90 3 28 W., St. Louis 90 12 14 W., San Francisco 122 27 49 W., Honolulu 157 52 0 W., St. Helena 5 42 0 W., Rome 12 28 0 E., St. Petersburg 30 16 0 E., Sydney 151 11 0 E.
q: The difference of time between two places is 4 hr. 18 min. 26 sec.: what is their difference of longitude? Degrees? {~p 60,60 # 0 # (4 * 3600 + 18 * 60 + 26) * 15} Minutes? {~p # 1 # $} Seconds? {~p # 2 # $}
sol: Every hour of time is 15 degrees of longitude, every minute 15 minutes, every second 15 seconds: 4 hr. 18 min. 26 sec. x 15 = 64 degrees 36 minutes 30 seconds.
q: The difference of longitude between two places is 81 degrees 39 minutes 22 seconds: what is their difference of time? Hr.? {~p 60,60 # 0 # (81 * 3600 + 39 * 60 + 22) / 15} Min.? {~p # 1 # $} Sec.? {~p # 2 # $}
sol: 15 into 81 degrees gives 5 hr. and 6 degrees over; 6 degrees = 360 minutes, and with 39 makes 399; 15 into 399 gives 26 min. and 9 over; 9 minutes = 540 seconds, and with 22 makes 562; 15 into 562 gives 37 7/15 sec.
q: It is six o'clock A. M. at New York; what is the time at Cincinnati? Hr.? {~p 60,60 # 0 # 6 * 3600 - (10 * 3600 + 29 * 60 + 21) / 15} Min.? {~p # 1 # $} Sec.? {~p # 2 # $}
sol: The difference of longitude, 10 degrees 29 minutes 21 seconds, divided by 15 is 41 min. 57.4 sec. Cincinnati is west, so its time is earlier: 18 min. 2.6 sec. after 5 o'clock A. M.
q: The difference of time between Springfield, Ill., and Philadelphia being 58 min. 1 2/15 sec., what is the longitude of Springfield (west)? Degrees? {~p 60,60 # 0 # 75 * 3600 + 9 * 60 + 3 + (58 * 60 + 1_2/15) * 15} Minutes? {~p # 1 # $} Seconds? {~p # 2 # $}
q: At what hour must a man start, and how fast would he have to travel, at the equator, so that it would be noon for him for twenty-four hours? Say the hour. {open}
sol: He must start at noon, and travel west 15 degrees an hour; a degree on the equator is 69.16 statute miles, so 15 x 69.16 = 1037.4 miles an hour.
q: How many statute miles an hour is that, at 69.16 miles to the degree? {~d 15 * 69.16}
q: What is the relative time between Mobile and Chicago? How much is Chicago time ahead? Min.? {~p 60 # 0 # (27 * 60 + 28) / 15} Sec.? {~p # 1 # $}
q: A man travels from Halifax to St. Louis; on arriving, his watch shows 9 A. M. Halifax time. The time in St. Louis being 13 min. 32 4/15 sec. after 7 o'clock A. M., what is the longitude of Halifax (west)? Degrees? {~p 60,60 # 0 # 90 * 3600 + 12 * 60 + 14 - (9 * 3600 - 7 * 3600 - 13 * 60 - 32_4/15) * 15} Minutes? {~p # 1 # $} Seconds? {~p # 2 # $}
q: Noon occurs 46 min. 58 sec. sooner at Detroit than at Galveston, Texas: what is the longitude of the latter (west)? Degrees? {~p 60,60 # 0 # 83 * 3600 + 3 * 60 + (46 * 60 + 58) * 15} Minutes? {~p # 1 # $} Seconds? {~p # 2 # $}
q: When it is five minutes after four o'clock on Sunday morning at Honolulu, what is the hour and day of the week at Sydney, Australia? Minutes after midnight? {~p 60,60 # 1 # 4 * 3600 + 5 * 60 + (157 * 3600 + 52 * 60 + 151 * 3600 + 11 * 60) / 15 - 24 * 3600} Seconds? {~p # 2 # $}
sol: The difference of longitude is 157 degrees 52 minutes + 151 degrees 11 minutes = 309 degrees 3 minutes, or 20 hr. 36 min. 12 sec. of time; Sydney is that much later: 41 min. 12 sec. after 12 o'clock A. M., Monday.
q: What is the difference in time between St. Petersburg and New Orleans? Hr.? {~p 60,60 # 0 # (30 * 3600 + 16 * 60 + 90 * 3600 + 3 * 60 + 28) / 15} Min.? {~p # 1 # $} Sec.? {~p # 2 # $}
q: When it is one o'clock P. M. at Rome, it is 54 min. 34 sec. after 6 o'clock A. M. at Buffalo, N. Y.: what is the longitude of the latter (west)? Degrees? {~p 60,60 # 0 # (13 * 3600 - 6 * 3600 - 54 * 60 - 34) * 15 - 12 * 3600 - 28 * 60} Minutes? {~p # 1 # $} Seconds? {~p # 2 # $}
q: When it is six o'clock P. M. at St. Helena, what is the time (A. M.) at San Francisco? Hr.? {~p 60,60 # 0 # 18 * 3600 - (116 * 3600 + 45 * 60 + 49) / 15} Min.? {~p # 1 # $} Sec.? {~p # 2 # $}
q: A ship's chronometer, set at Greenwich, points to 4 hr. 43 min. 12 sec. P. M.: the sun being on the meridian, what is the ship's longitude (west)? Degrees? {~p 60,60 # 0 # (4 * 3600 + 43 * 60 + 12) * 15} Minutes? {~p # 1 # $} Seconds? {~p # 2 # $}

== 36 | Compound Numbers | Aliquot parts
note: An aliquot part is an exact divisor of a number. Multiply the rate by the number of the denomination it is given for, and find the value of the other parts as simple fractions of a value already found. 5 = 1/20 of 100; 6 1/4 = 1/16; 10 = 1/10; 12 1/2 = 1/8; 16 2/3 = 1/6; 20 = 1/5; 25 = 1/4; 33 1/3 = 1/3; 50 = 1/2. Also 37 1/2 = 3/8, 62 1/2 = 5/8, 75 = 3/4, 87 1/2 = 7/8.
q: Find the cost of 28 A. 145 sq. rd. 15 1/8 sq. yd. of land, at $16 per acre. Dollars? {462.55 @ (28 + 145/160 + 15_1/8 / 4840) * 16}
sol: 28 A. cost 28 x $16 = $448. 120 sq. rd. = 3/4 of an acre, $12; 20 sq. rd. = 1/6 of 120, $2; 5 sq. rd. = 1/4 of 20, $.50; 15 1/8 sq. yd. = 1/10 of 5 sq. rd., $.05. In all, $462.55.
q: A man travels 3 mi. 20 rd. 5 yd. in 1 hr.; how far will he go in 6 da. 9 hr. 18 min. 45 sec. (12 hr. to a day)? Mi.? {~p 320,5_1/2 # 0 # (3 * 1760 + 20 * 5_1/2 + 5) * (6 * 12 + 9 + 18/60 + 45/3600)} Rd.? {~p # 1 # $} Yd.? {~p # 2 # $}
q: If a man travel 2 mi. 105 rd. 6 1/6 ft. in 1 hour, how far can he travel in 30 hr. 29 min. 52 sec.? Mi.? {~p 320,16_1/2 # 0 # (2 * 5280 + 105 * 16_1/2 + 6_1/6) * (30 + 29/60 + 52/3600)} Rd.? {~p # 1 # $} Ft.? {~p # 2 # $}
q: An old record says that 694 A. 1 R. 22 P. of land, at $11.52 per acre, brought $8009.344; what is the true cost? {~d (694 + 1/4 + 22/160) * 11.52} What is the error in the calculation? Dollars too much? {10}
q: At $15.46 an oz., what will be the cost of 7 lb. 8 oz. 16 pwt. 11 gr. of gold? Dollars, to cents. {~r2 15.46 * (92 + 16/20 + 11/480)}
q: What is the cost of 88 gal. 3 qt. 1 pt. of vinegar, at 37 1/2 ct. a gallon? Dollars, to cents. {~r2 (88 + 3/4 + 1/8) * .375}
q: If the heart should beat 97920 times in each day, how many times would it beat in 8 da. 5 hr. 25 min. 30 sec.? {805494 @ 97920 * (8 + 5/24 + 25/1440 + 30/86400)}
q: At a cost of $8190.50 per mile over the plain, and at a rate of $84480 per mile of tunnel, what is the cost of a railway 17 mi. 150 rd. plain, and 70 rd. tunnel? Dollars, to cents. {~r2 8190.50 * (17 + 150/320) + 84480 * 70/320}
q: What is the value of 20 T. 1 cwt. 13 lb. of sugar, at $3 3/4 per cwt.? Dollars, to cents. {~r2 (401 + 13/100) * 3.75}
q: If 3 pounds 6s. silver weigh 1 lb. troy, how much will 17 lb. 11 oz. 16 pwt. 9 gr. be worth? Pounds? {~p 20,12 # 0 # 792 * (17 + 11/12 + 16/240 + 9/5760)} Shillings? {~p # 1 # $} Pence? {~p # 2 # $}
q: If a steam-ship could make 16 mi. 67 3/8 rd. in 1 hr., how far could it go in 24 da. 22 hr. 56 min. 12 sec.? Mi.? {~p 320,16_1/2 # 0 # (16 * 320 + 67_3/8) * 16_1/2 * (24 * 24 + 22 + 56/60 + 12/3600)} Rd.? {~p # 1 # $} Ft.? {~p # 2 # $}

== 37 | Ratio | Ratio
note: Ratio is the relation of one number to another of the same kind, found by dividing the first (the antecedent) by the second (the consequent): 6 : 4 = 6/4. Antecedent = consequent x ratio; consequent = antecedent / ratio. A compound ratio is the product of simple ratios; the inverse ratio is the consequent divided by the antecedent. Compound numbers are first reduced to the same denomination.
q: What is the ratio of 15 to 36? {~15 / 36}
sol: 15 : 36 = 15/36 = 5/12.
q: What is the ratio of 2 ft. 6 in. to 3 yd. 1 ft. 10 in.? {~30 / 130}
q: What is the ratio of 4 mi. 260 rd. to 1 mi. 96 rd.? {~1540 / 416}
q: What is the ratio of 13 A. 145 sq. rd. : 6 A. 90 sq. rd.? {~2225 / 1050}
q: What is the ratio of 3 lb. 10 oz. 6 pwt. 10 1/2 gr. : 2 lb. 14 3/4 pwt.? {~(3 * 5760 + 10 * 480 + 6 * 24 + 10_1/2) / (2 * 5760 + 14_3/4 * 24)}
q: What is the ratio of 10 gal. 1.54 pt. : 7 gal. 2 qt. .98 pt.? {~81.54 / 60.98}
q: What is the ratio of 56 bu. 2 pk. 1 qt. : 35 bu. 3 pk. 6.055 qt.? {~1809 / 1150.055}
q: If the antecedent is 7 and the ratio 1 1/2, what is the consequent? {~7 / 1_1/2}
q: If the consequent is 3/7 and the ratio 2/9, what is the antecedent? {~3/7 * 2/9}
q: What is the ratio of a yard to a meter (39.37 inches)? To four places. {~r4 36 / 39.37} And of a meter to a yard? {~r4 39.37 / 36}
q: What is the ratio of a pound avoirdupois to a pound troy? {~7000 / 5760}
q: Find the difference between the compound ratios (3 : 4 and 5 : 9) and (1 : 6 and 2 : 7). {~3 * 5 / (4 * 9) - 1 * 2 / (6 * 7)}
sol: 3 : 4 compounded with 5 : 9 is 15/36 = 5/12; 1 : 6 with 2 : 7 is 2/42 = 1/21. 5/12 - 1/21 = 31/84.
q: Find the difference between the ratio 4 2/3 : 7 1/2 and the inverse ratio. {~7_1/2 / 4_2/3 - 4_2/3 / 7_1/2}
q: If the consequent is 6 1/2, and the ratio is 2 1/4, what is the antecedent? {~6_1/2 * 2_1/4} What is the inverse ratio of the two numbers? {~6_1/2 / (6_1/2 * 2_1/4)}

== 38 | Proportion | Simple proportion
note: A proportion is an equality of ratios: 4 : 6 :: 8 : 12. The product of the means equals the product of the extremes. For the third term write the number of the same kind as the answer; for the second, the greater of the other two if the answer is to be greater, the less if less. Multiply the second and third terms and divide by the first.
q: If 6 horses cost $300, what will 15 horses cost? Dollars? {750 @ 300 * 15 / 6}
sol: 6 horses : 15 horses :: $300 : the answer. $300 x 15 / 6 = $750.
q: If 15 men do a piece of work in 9 3/5 da., how long will 36 men be in doing the same? Days? {~9_3/5 * 15 / 36}
sol: 36 men need less time than 15: 36 : 15 :: 9 3/5 : the answer. 48/5 x 15/36 = 4 days.
q: In your head: If I walk 10 1/2 mi. in 3 hr., how far will I go in 10 hr., at the same rate? Miles? {~10_1/2 * 10 / 3}
q: If the fore-wheel of a carriage is 8 ft. 2 in. in circumference, and turns round 670 times, how often will the hind-wheel, which is 11 ft. 8 in. in circumference, turn round in going the same distance? {469 @ 670 * 98 / 140}
q: If a horse trot 3 mi. in 8 min. 15 sec., how far can he trot in an hour, at the same rate? Miles? {~3 * 3600 / 495}
q: What is a servant's wages for 3 wk. 5 da., at $1.75 per week? Dollars? {~d 1.75 * 26 / 7}
q: What should be paid for a barrel of powder, containing 132 lb., if 15 lb. are sold for $5.43 3/4? Dollars? {~d 5.4375 * 132 / 15}
q: A body of soldiers are 42 in rank when they are 24 in file: if they were 36 in rank, how many in file would there be? {28 @ 24 * 42 / 36}
q: If a pulse beats 28 times in 16 sec., how many times does it beat in a minute? {105 @ 28 * 60 / 16}
q: If a cane 3 ft. 4 in. long, held upright, casts a shadow 2 ft. 1 in. long, how high is a tree whose shadow at the same time is 25 ft. 9 in.? Ft.? {~p 12 # 0 # 40 * 309 / 25} In.? {~p # 1 # $}
q: If a farm of 160 A. rents for $450, how much should be charged for one of 840 A.? Dollars? {~d 450 * 840 / 160}
q: A grocer has a false gallon, containing 3 qt. 1 1/2 pt.: what is the worth of the liquor that he sells for $240? Dollars? {225 @ 240 * 7_1/2 / 8} What is his gain by the cheat? {15}
q: If he uses 14 3/4 oz. for a pound, how much does he cheat by selling sugar for $27.52? Dollars? {~d 27.52 * 1_1/4 / 16}
q: An equatorial degree is 365000 ft.: how many ft. in 80 degrees 24 minutes 37 seconds of the same? {~365000 * 289477 / 3600}
q: If a pendulum beats 5000 times a day, how often does it beat in 2 hr. 20 min. 5 sec.? {~5000 * 8405 / 86400}
q: In your head: If it takes 108 days, of 8 1/2 hr., to do a piece of work, how many days of 6 3/4 hr. would it take? {136 @ 108 * 8_1/2 / 6_3/4}
q: A man borrows $1750, and keeps it 1 yr. 8 mon.: how long should he lend $1200 to compensate for the favor? (30 days to the month.) Yr.? {~p 12,30 # 0 # 1750 * 600 / 1200} Mon.? {~p # 1 # $} Da.? {~p # 2 # $}
q: A garrison has food to last 9 mon., giving each man 1 lb. 2 oz. a day: what should be a man's daily allowance, to make the same food last 1 yr. 8 mon.? Ounces? {~18 * 9 / 20}
q: A garrison of 560 men have provisions to last during a siege, at the rate of 1 lb. 4 oz. a day per man; if the daily allowance is reduced to 14 oz. per man, how large a reinforcement could be received? {240 @ 560 * 6 / 14}
q: A shadow of a cloud moves 400 ft. in 18 3/4 sec.: what was the wind's velocity per hour? Miles? {~400 * 3600 / 18_3/4 / 5280}
q: If 1 lb. troy of English standard silver is worth 3 pounds 6s., what is 1 lb. av. worth? Pounds? {~p 20,12 # 0 # 792 * 7000 / 5760} Shillings? {~p # 1 # $} Pence? {~p # 2 # $}
q: If I go a journey in 12 3/4 days, at 40 mi. a day, how long would it take me at 29 3/4 mi. a day? Days? {~12_3/4 * 40 / 29_3/4}
q: In your head: If 5/9 of a ship is worth $6000, what is the whole of it worth? Dollars? {10800 @ 6000 * 9 / 5}
q: If A, worth $5840, is taxed $78.14, what is B worth, who is taxed $256.01? Dollars, to cents. {~r2 5840 * 256.01 / 78.14}
q: In your head: What are 4 lb. 6 oz. of butter worth, at 28 ct. a lb.? Dollars? {~d 28 * 70 / 16 / 100}
q: If I gain $160.29 in 2 yr. 3 mon., what would I gain in 5 yr. 6 mon., at that rate? Dollars, to cents. {~r2 160.29 * 66 / 27}
q: If I gain $92.54 on $1156.75 worth of sugar, how much must I sell to gain $67.32? Dollars? {~d 1156.75 * 67.32 / 92.54}
q: If coffee costing $255 is now worth $318.75, what did $1285.20 worth cost? Dollars? {~d 1285.20 * 255 / 318.75}
q: A has cloth at $3.25 a yd., and B has flour at $5.50 a barrel. If, in trading, A puts his cloth at $3.62 1/2, what should B charge for his flour? Dollars, to cents. {~r2 5.50 * 3.625 / 3.25}
q: In your head: If a boat is rowed at the rate of 6 miles an hour, and is driven 44 feet in 9 strokes of the oar, how many strokes are made in a minute? {108 @ 9 * 528 / 44}
q: If I gain $7.75 by trading with $100, how much ought I to gain on $847.56? Dollars? {~d 7.75 * 847.56 / 100}
q: What is a pile of wood, 15 ft. long, 10 1/2 ft. high, and 12 ft. wide, worth, at $4.25 a cord? Dollars, to cents. {~r2 15 * 10_1/2 * 12 / 128 * 4.25}
note: Fahrenheit's thermometer marks freezing 32 degrees and boiling 212; the Centigrade, 0 and 100; Reaumur's, 0 and 80. So 180 degrees F. = 100 degrees C. = 80 degrees R.
q: Find the value of a degree of each thermometer in the degrees of the other two. 1 degree F. is how many degrees C.? {~100 / 180} R.? {~80 / 180} 1 degree C. is how many F.? {~180 / 100} R.? {~80 / 100} 1 degree R. is how many F.? {~180 / 80} C.? {~100 / 80}
q: Convert 108 degrees F. to degrees of the other two thermometers. C.? {~(108 - 32) * 100 / 180} R.? {~(108 - 32) * 80 / 180}
q: Convert 25 degrees R. to degrees of the other two thermometers. F.? {~25 * 180 / 80 + 32} C.? {~25 * 100 / 80}
q: Convert 46 degrees C. to degrees of the other two thermometers. F.? {~46 * 180 / 100 + 32} R.? {~46 * 80 / 100}
note: In machinery the available power is to the weight overcome, inversely as the distances they pass over in the same time. The available power is taken 2/3 of the whole power, 1/3 being allowed for friction.
q: If the whole power applied is 180 lb. and moves 4 ft., how far will it lift a weight of 960 lb.? Inches? {6 @ 2/3 * 180 * 48 / 960}
q: If 512 lb. be lifted 1 ft. 3 in. by a power moving 6 ft. 8 in., what is the power? Pounds? {144 @ 512 * 15 / 80 * 3/2}
q: A lifts a weight of 1440 lb. by a wheel and axle; for every 3 ft. of rope that passes through his hands the weight rises 4 1/2 in.: what power does he exert? Pounds? {270 @ 1440 * 4_1/2 / 36 * 3/2}
q: A man weighing 198 lb. lets himself down 54 ft. with a uniform motion, by a wheel and axle: if the weight at the hook rises 12 ft., how much is it? Pounds? {594 @ 2/3 * 198 * 54 / 12}
q: Two bodies free to move, attract each other with forces that vary inversely as their weights. If the weights are 9 lb. and 4 lb., and the smaller is attracted 10 ft., how far will the larger be attracted? Feet? {~10 * 4 / 9}
q: Suppose the earth and moon to approach each other in obedience to this law, their weights being 49147 and 123 respectively, how many miles would the moon move while the earth moved 250 miles? To the nearest mile. {~r0 250 * 49147 / 123}
q: Can this be solved by proportion: If 3 men mow 5 A. of grass in a day, how many men will mow 13 1/3 A. in a day? Men? {8 @ 3 * 13_1/3 / 5}
sol: Yes: the acres and the men are in direct ratio, the time being the same. 5 A. : 13 1/3 A. :: 3 men : 8 men.
q: In your head: Can this be solved by proportion: If 6 men build a wall in 7 da., how long would 10 men be in doing the same? Days? {~6 * 7 / 10}
sol: Yes, by an inverse ratio: the more men, the shorter the time. One man needs 42 days; 10 men, 4 1/5 days.
q: In your head: If I gain 15 cents each, by selling books at $4.80 a doz., what is my gain on each at $5.40 a doz.? Cents? {20 @ 15 + (540 - 480) / 12}
sol: This can not be solved by proportion, for the gains and prices are not in proportion, the cost remaining the same. The cost of each is 40 - 15 = 25 ct.; at 45 ct. the gain is 20 ct.
q: A clock which loses 5 minutes a day, was set right at 6 in the morning of January 1st: what will be the right time when that clock points to 11 on the 15th? Minutes after noon? {~p 60 # 0 # (14 * 86400 + 5 * 3600) * 1440 / 1435 - 14 * 86400 - 6 * 3600} Seconds? {~p # 1 # $}
sol: The clock shows 1435 minutes for every 1440 that pass. It shows 14 da. 5 hr. gone; the true time gone is 1440/1435 of that, 14 da. 6 hr. 11 min. 17.35 sec. So the right time is 11 min. 17.35 sec. past noon.
q: If water begin and continue running at the rate of 80 gal. an hour, into a cellar 12 ft. long, 8 ft. wide, and 6 ft. deep, while it soaks away at the rate of 35 gal. an hour, in what time will the cellar be full? Hours, to two places. {~r2 12 * 8 * 6 * 1728 / (231 * 45)}
q: Take the proportion of 4 : 9 :: 252 : a fourth term. If the third and fourth terms each be increased by 7, while the first remains unchanged, what multiplier is needed by the second to make a proportion? {~4 * (567 + 7) / (252 + 7) / 9}
sol: The fourth term is 567. With 7 added: 4 : ( ) :: 259 : 574; the missing term is 4 x 574 / 259 = 2296/259, and divided by 9 the multiplier is 328/333.
q: Prove that there is no number which can be added to each term of 6 : 3 :: 18 : 9 so that the resulting numbers shall stand in proportion. {open}
sol: Each ratio is 2. Adding the same number to both terms keeps the difference: the left ratio becomes 1 and 3 over (3 + the number), the right 1 and 9 over (9 + the number). For these to be equal, 9 + the number would have to be 3 times (3 + the number), that is 9 + the number = 9 + 3 times the number, which is impossible unless the number is nothing.
q: A certain number has been divided by one more than itself, giving a quotient 1/5: what is the number? {~1/4}
q: If 48 lb. of sea-water contain 1 1/2 lb. of salt, how much fresh water must be added to these 48 lb. so that 40 lb. of the mixture shall contain 1/2 lb. of salt? Pounds? {72 @ 1_1/2 * 40 / (1/2) - 48}
== 39 | Proportion | Compound proportion
note: Separate the quantities into two causes and their effects. The first cause is to the second cause as the first effect is to the second effect; the missing term is found by the principles of proportion, canceling where possible.
q: If 3 men mow 8 A. of grass in 4 da., how long would 10 men be in mowing 36 A.? Days? {~3 * 36 * 4 / (10 * 8)}
sol: 10 men : 3 men and 8 A. : 36 A. :: 4 da. : the answer. 3 x 36 x 4 over 10 x 8 = 27/5 = 5 2/5 days.
q: If 6 men, in 10 days of 9 hr. each, build 25 rd. of fence, how many hours a day must 8 men work to build 48 rd. in 12 days? {~6 * 10 * 9 * 48 / (8 * 12 * 25)}
sol: 6 men, 10 da., 9 hr. are the first cause, whose effect is 25 rd.; 8 men, 12 da., and the hours wanted are the second cause, whose effect is 48 rd. 6 x 10 x 9 x 48 over 8 x 12 x 25 = 10 4/5 hr.
q: If 18 pipes, each delivering 6 gal. per minute, fill a cistern in 2 hr. 16 min., how many pipes, each delivering 20 gal. per minute, will fill a cistern 7 1/2 times as large as the first, in 3 hr. 24 min.? {27 @ 18 * 6 * 136 * 7_1/2 / (20 * 204)}
q: The use of $100 for 1 year is worth $8: what is the use of $4500 for 2 yr. 8 mon. worth? Dollars? {960 @ 8 * 4500 * 32 / (100 * 12)}
q: If 12 men mow 25 A. of grass in 2 da. of 10 1/2 hr., how many hours a day must 14 men work to mow an 80 A. field in 6 days? {~10_1/2 * 12 * 2 * 80 / (14 * 6 * 25)}
q: If 4 horses draw a railroad car 9 miles an hour, how many miles an hour can a steam engine of 150 horse-power drive a train of 12 such cars, the locomotive and tender being counted 3 cars? {~9 * 150 / (4 * 15)}
q: If 12 men, working 20 days 10 hours a day, mow 247.114 hektars of timothy, how many men in 30 days, working 8 hours a day, will mow 1976912 centars of timothy of the same quality? {8 @ 12 * 20 * 10 * 197.6912 / (30 * 8 * 247.114)}
q: If the use of $3750 for 8 mon. is worth $68.75, what sum is that whose use for 2 yr. 4 mon. is worth $250? Dollars, to cents. {~r2 3750 * 250 * 8 / (68.75 * 28)}
q: If the use of $1500 for 3 yr. 8 mon. 25 da. is worth $336.25, what is the use of $100 for 1 yr. worth? (30 days to the month.) Dollars? {6 @ 336.25 * 360 * 100 / (1345 * 1500)}
q: A garrison of 1800 men has provisions to last 4 1/2 months, at the rate of 1 lb. 4 oz. a day to each: how long will 5 times as much last 3500 men, at the rate of 12 oz. per day to each man? Months? {~4_1/2 * 1800 * 5 * 20 / (3500 * 12)}
q: What sum of money is that whose use for 3 yr., at the rate of $4 1/2 for every hundred, is worth as much as the use of $540 for 1 yr. 8 mon., at the rate of $7 for every hundred? Dollars, to cents. {~r2 540 * 7 * 1_2/3 / (4_1/2 * 3)}
q: A man has a bin 7 ft. long by 2 1/2 ft. wide, and 2 ft. deep, which contains 28 bu. of corn: how deep must he make another, which is to be 18 ft. long by 1 7/8 ft. wide, in order to contain 120 bu.? Feet? {~2 * 7 * 2_1/2 * 120 / (18 * 1_7/8 * 28)}
q: If it require 4500 bricks, 8 in. long by 4 in. wide, to pave a court-yard 40 ft. long by 25 ft. wide, how many tiles, 10 in. square, will be needed to pave a hall 75 ft. long by 16 ft. wide? {1728 @ 4500 * 32 * 75 * 16 / (100 * 40 * 25)}
q: If 150000 bricks are used for a house whose walls average 1 1/2 ft. thick, 30 ft. high, and 216 ft. long, how many will build one with walls 2 ft. thick, 24 ft. high, and 324 ft. long? {240000 @ 150000 * 2 * 24 * 324 / (1_1/2 * 30 * 216)}
q: If 240 panes of glass 18 in. long, 10 in. wide, glaze a house, how many panes 16 in. long by 12 in. wide will glaze a row of 6 such houses? {1350 @ 240 * 6 * 18 * 10 / (16 * 12)}
q: If it require 800 reams of paper to publish 5000 volumes of a duodecimo book containing 320 pages, how many reams will be needed to publish 24000 copies of a book, octavo size, of 550 pages? {9900 @ 800 * 24000 * 12 * 550 / (5000 * 8 * 320)}
q: If 15 men cut 480 sters of wood in 10 days, of 8 hours each: how many boys will it take to cut 1152 sters of wood, only 2/5 as hard, in 16 days, of 6 hours each, provided that while working a boy can do only 3/4 as much as a man, and that 1/3 of the boys are idle at a time throughout the work? {24 @ 15 * 1152 * 2/5 * 10 * 8 / (480 * 16 * 6 * 3/4 * 2/3)}
== 40 | Percentage | Finding the percentage
note: Per cent means by the hundred: 4% = 4/100 = .04. The Base is the number on which the percentage is reckoned; the Rate, the number of hundredths taken; the Percentage, the part found; the Amount, base plus percentage; the Difference, base minus percentage. Base x Rate = Percentage.
q: If I have 160 sheep, and sell 35% of them: how many do I sell? {56 @ 160 * .35}
sol: 160 x .35 = 56 sheep. Or: 35% = 35/100 = 7/20, and 7/20 of 160 = 56.
q: Find 62 1/2% of 1664 men. {1040 @ 1664 * 5/8}
q: Find 35% of 2/7. {~35/100 * 2/7}
q: Find 9 3/8% of 48 mi. 256 rd. Mi.? {~p 320 # 0 # (48 * 320 + 256) * 9_3/8 / 100} Rd.? {~p # 1 # $}
q: Find 11 1/9% of $3283.47 {364.83 @ 3283.47 / 9}
q: Find 33 1/3% of 127 gal. 3 qt. 1 pt. Gal.? {~p 4,2 # 0 # (127 * 8 + 7) / 3} Qt.? {~p # 1 # $} Pt.? {~p # 2 # $}
q: Find 98% of 14 cwt. 2 qr. 20 lb. Cwt.? {~p 4,25 # 0 # 1470 * .98} Qr.? {~p # 1 # $} Lb.? {~p # 2 # $}
q: Find 40% of 6 hr. 28 min. 15 sec. Hr.? {~p 60,60 # 0 # 23295 * .4} Min.? {~p # 1 # $} Sec.? {~p # 2 # $}
q: Find 104% of 75 A. 75 sq. rd. A.? {~p 160 # 0 # 12075 * 1.04} Sq. rd.? {~p # 1 # $}
q: Find 15 5/8% of a book of 576 pages. {90 @ 576 * 15_5/8 / 100}
q: Find 56 1/4% of 144 cattle. {81 @ 144 * 9/16}
q: Find 16 2/3% of 1932 hogs. {322 @ 1932 / 6}
q: Find 1000% of $5.43 3/4 {54.375 @ 10 * 5.4375}
q: Find 2 1/2% of 4/5. {~2_1/2 / 100 * 4/5}
q: What part is 25% of a farm? {=1/4}
q: What part of a quantity is 18 3/4% of it? {=3/16} 31 1/4%? {=5/16} 37 1/2%? {=3/8} 43 3/4%? {=7/16} 56 1/4%? {=9/16} 62 1/2%? {=5/8}
q: What part of a quantity is 68 3/4% of it? {=11/16} 81 1/4%? {=13/16} 83 1/3%? {=5/6} 87 1/2%? {=7/8} 93 3/4%? {=15/16}
q: How much is 100% of a quantity? Times the quantity? {1} 125%? {~125 / 100} 250%? {~250 / 100} 675%? {~675 / 100} 1000%? {10} 9437 1/2%? {~9437_1/2 / 100}
q: In your head: A man owning 3/8 of a ship, sold 40% of his share: what part of the ship did he sell? {~3/8 * 2/5} What part did he still own? {~3/8 * 3/5}
q: In your head: A owed B a sum of money; at one time he paid him 40% of it; afterward he paid him 25% of what he owed; and finally he paid him 20% of what he then owed: what part of the debt does he still owe? {~3/5 * 3/4 * 4/5}
q: Out of a cask containing 47 gal. 2 qt. 1 pt., leaked 6 2/3%: how much was that? Gal.? {~p 8 # 0 # 381 / 15} Pt.? {~p # 1 # $}
q: A has an income of $1200 a year; he pays 23% of it for board; 10 3/5% for clothing; 6 3/4% for books; 7/12% for newspapers; 12 7/8% for other expenses: how much does he pay for board? {276 @ 1200 * .23} Clothing? {~d 1200 * 10_3/5 / 100} Books? {81 @ 1200 * 6_3/4 / 100} Newspapers? {7 @ 1200 * 7/12 / 100} Other expenses? {~d 1200 * 12_7/8 / 100} How much does he save at the end of the year? {~d 1200 - 1200 * (23 + 10_3/5 + 6_3/4 + 7/12 + 12_7/8) / 100}
q: Find 10% of 20% of $13.50 {~d .1 * .2 * 13.50}
q: Find 40% of 15% of 75% of $133.33 1/3 {6 @ .4 * .15 * .75 * 133_1/3}
q: A man contracts to supply dressed stone for a court-house for $119449, if the rough stone costs him 16 ct. a cu. ft.; but if he can get it for 15 ct. a cu. ft., he will deduct 3% from his bill: how many cu. ft. would be needed? {358347 @ 119449 * .03 / .01} What does he charge for dressing a cu. ft., in cents? {~(119449 / 358347) * 100 - 16}
sol: The deduction, 3% of $119449 = $3583.47, comes from saving 1 ct. on each cubic foot, so there are 358347 cu. ft. The whole cost of a foot is $119449 / 358347 = 33 1/3 ct.; less 16 ct. for the rough stone, the dressing is 17 1/3 ct.
q: 48% of brandy is alcohol; how much alcohol does a man swallow in 40 years (of 365 1/4 days), if he drinks a gill of brandy 3 times a day? Gal.? {~p 4,2,4 # 0 # 40 * 365_1/4 * 3 * .48} Qt.? {~p # 1 # $} Pt.? {~p # 2 # $} Gi.? {~p # 3 # $}
q: A had $1200; he gave 30% to a son, 20% of the remainder to his daughter, and so divided the rest among four brothers that each after the first had $12 less than the preceding: how much did the last receive? Dollars? {150 @ (1200 * .7 * .8 - 72) / 4}
q: What number increased by 20% of 3.5, diminished by 12 1/2% of 9.6, gives 3 1/2? {4 @ 3_1/2 + 1.2 - .7}

== 41 | Percentage | Finding the rate and the base
note: The rate is the percentage divided by the base. The base is the percentage divided by the rate (as a decimal). Answer a rate as the number of per cent: 20 for 20%.
q: What per cent of 45 is 9? {20 @ 9 / 45 * 100}
sol: 9 is 1/5 of 45, and 1/5 of any number is 20% of it.
q: 15 ct. is how many % of $2? {~15 / 200 * 100}
q: 2 yd. 2 ft. 3 in. is how many % of 4 rd.? {~99 / 792 * 100}
q: 3 gal. 3 qt. is what % of 31 1/2 gal.? {~3.75 / 31_1/2 * 100}
q: 2/3 is how many % of 4/5? {~2/3 / (4/5) * 100}
q: 1/2 of 3/5 of 4/7 is what % of 1 7/10? {~1/2 * 3/5 * 4/7 / 1_7/10 * 100}
q: 2 1/2 over 3 is how many % of 3 3/4 over 10? {~2_1/2 / 3 / (3_3/4 / 10) * 100}
q: $5.12 is what % of $640? {~d 5.12 / 640 * 100}
q: $3.20 is what % of $2000? {~d 3.20 / 2000 * 100}
q: 750 men is what % of 12000 men? {~750 / 12000 * 100}
q: 3 qt. 1 1/2 pt. is what % of 5 gal. 2 1/2 qt.? {~7_1/2 / 45 * 100}
q: A's money is 50% more than B's; then B's money is how many % less than A's? {~50 / 150 * 100}
q: What % of a number is 8% of 35% of it? {~d 8 * 35 / 100}
q: What % of a number is 2 1/2% of 2 1/2% of it? {~d 2_1/2 * 2_1/2 / 100}
q: What % of a number is 40% of 62 1/2% of it? {25 @ 40 * 62_1/2 / 100}
q: 12% of $75 is what % of $108? {~12 * 75 / 108}
q: U. S. standard gold and silver are 9 parts pure to 1 part alloy: what % of alloy is that? {10}
q: What % of a meter (39.37 in.) is a yard? To two places. {~r2 36 / 39.37 * 100}
q: How many % of a township 6 miles square, does a man own who has 9000 acres? {~9000 / 23040 * 100}
q: How many % of a quantity is 40% of 25% of it? {10} 16% of 37 1/2% of it? {6} 4 1/6% of 120% of it? {5} 2% of 80% of 66 2/3% of it? {~2 * 80 / 100 * 66_2/3 / 100} 3/5% of 36% of 75% of it? {~3/5 * 36 / 100 * 75 / 100} 6 7/8% of 22 1/2% of 96% of it? {~6_7/8 * 22_1/2 / 100 * 96 / 100}
q: 30% of the whole of an article is how many % of 2/3 of it? {45 @ 30 / (2/3)}
q: 25% of 2/5 of an article is how many % of 3/4 of it? {~25 * 2/5 / (3/4)}
q: How many % of his time does a man rest, who sleeps 7 hr. out of every 24? {~7 / 24 * 100}
q: 95 is 5% of what number? {1900 @ 95 / .05}
sol: 5% = 95; 1% = 19; 100% = 1900. Or, 95 / .05 = 1900.
q: $3.80 is 5% of what sum? Dollars? {76 @ 3.80 / .05}
q: 2/11 is 80% of what number? {~2/11 / (4/5)}
q: 16 is 1 1/2% of what number? {~16 / 1_1/2 * 100}
q: 31 1/4 ct. is 15 5/8% of what? Dollars? {2 @ .3125 / .15625}
q: $10.75 is 3 1/3% of what? Dollars? {~d 10.75 / 3_1/3 * 100}
q: 162 men is 4 4/5% of how many men? {3375 @ 162 / .048}
q: $19.20 is 6/10% of what? Dollars? {3200 @ 19.20 / .006}
q: $189.80 is 104% of what? Dollars? {~d 189.80 / 1.04}
q: 16 gal. 1 pt. is 6 1/7% of what? Gal.? {~p 4 # 0 # 129 / 2 / (43/700)} Qt.? {~p # 1 # $}
q: 10 mi. 316 rd. is 75% of what? Mi.? {~p 320 # 0 # 3516 / .75} Rd.? {~p # 1 # $}
q: Thirty-six men of a ship's crew die, which is 42 6/7% of the whole: what was her crew? {84 @ 36 / (3/7)}
q: A stock-farmer sells 144 sheep, which is 12 4/5% of his flock: how many sheep had he? {1125 @ 144 / .128}
q: A merchant sells 35% of his stock for $6000: what is it all worth at that rate? Dollars, to cents. {~r2 6000 / .35}
q: I shot 12 pigeons, which was 2 2/3% of the flock: how many pigeons escaped? {438 @ 12 / (2_2/3 / 100) - 12}
q: A, owing B, hands him a $10 bill, and says, there is 6 1/4% of your money: what was the debt? Dollars? {160 @ 10 * 16}
q: $25 is 62 1/2% of A's money, and 41 2/3% of B's: how much has A? Dollars? {40 @ 25 / (5/8)} B? {60 @ 25 / (5/12)}
q: A found $5, which was 13 1/3% of what he had before: how much had he then? Dollars? {~d 5 / (2/15) + 5}
q: I drew 48% of my funds in bank, to pay a note of $150: how much had I left? Dollars? {~d 150 / .48 - 150}
q: A farmer gave his daughter at her marriage 65 A. 106 sq. rd. of land, which was 3% of his farm: how much land did he own? A.? {~p 160 # 0 # 10506 / .03} Sq. rd.? {~p # 1 # $}
q: A pays $13 a month for board, which is 20% of his salary: what is his yearly salary? Dollars? {780 @ 13 * 12 / .2}
q: Paid 40 ct. for putting in 25 bu. of coal, which was 11 3/7% of its cost: what did it cost a bu., in cents? {14 @ 40 / 25 / (4/35)}
q: 81 men is 5% of 60% of what? {2700 @ 81 / (.05 * .6)}
q: A, owning 60% of a ship, sells 7 1/2% of his share for $2500: what is the ship worth? Dollars, to cents. {~r2 2500 / (.6 * .075)}
q: A father, having a basket of apples, took out 33 1/3% of them; of these, he gave 37 1/2% to his son, who gave 20% of his share to his sister, who thus got 2 apples: how many apples were in the basket at first? {80 @ 2 / (1/3 * 3/8 * 1/5)}
q: B lost three dollars, which was 31 1/4% of what he had left: how much had he at first? Dollars? {~d 3 / .3125 + 3}
q: Bought 8000 bu. of wheat, which was 57 1/7% of my whole stock: how much had I before? Bushels? {6000 @ 8000 / (4/7) - 8000}
q: If 32% of 75% of 800% of a number is 1539, what is that number? {~1539 / (.32 * .75 * 8)}

== 42 | Percentage | Finding the base from the amount or difference
note: The base is the amount divided by 1 plus the rate, or the difference divided by 1 minus the rate. A = B + P; D = B - P.
q: A rents a house for $377, which is an advance of 16% on the rent of last year: what amount did he pay last year? Dollars? {325 @ 377 / 1.16}
sol: 100% = last year's rent; 116% = $377; 1% = $3.25; 100% = $325.
q: John has $136, which is 20% less than Joseph's money: how many dollars has Joseph? {170 @ 136 / .8}
q: $4.80 is 33 1/3% more than what? Dollars? {3.60|3.6 @ 4.80 / (4/3)}
q: 5/6 is 50% more than what? {~5/6 / 1_1/2}
q: 96 da. is 100% more than what? Days? {48}
q: 2576 bu. is 60% less than what? Bushels? {6440 @ 2576 / .4}
q: 87 1/2 ct. is 87 1/2% less than what? Dollars? {7 @ .875 / .125}
q: 42 mi. 60 rd. is 55% less than what? Mi.? {~p 320 # 0 # 13500 / .45} Rd.? {~p # 1 # $}
q: 2 lb. 9 29/96 oz. is 50% less than what number of pounds? {~(41_29/96) / 16 * 2}
q: 7/12 is 99 5/8% less than what? {~7/12 / (3/8 / 100)}
q: $920.93 3/4 is 337 1/2% more than what? Dollars? {210.50|210.5 @ 920.9375 / 4.375}
q: $4358.06 1/4 is 233 1/3% more than what? Dollars? {~4358.0625 / (10/3)}
q: In 64 1/2 gal. of alcohol, the water is 7 1/2% of the spirit: how many gal. of spirit? {60 @ 64_1/2 / 1.075} Of water? {~4_1/2}
q: In your head: A coat cost $32; the trimmings cost 70% less, and the making 50% less, than the cloth: what did the cloth cost? Dollars? {~32 / 1.8} The trimmings? {~32 / 1.8 * .3} The making? {~32 / 1.8 * .5}
q: In your head: If a bushel of wheat make 39 1/5 lb. of flour, and the cost of grinding be 4%, how many barrels (196 lb.) of flour can a farmer get for 80 bu. of wheat? {~80 * 39_1/5 / 196 / 1.04}
q: In your head: How many eagles, each containing 9 pwt. 16.2 gr. of pure gold, can I get for 455.6538 oz. pure gold at the mint, allowing 1 1/2% for expense of coinage? {928 @ 455.6538 * 480 / 232.2 / 1.015}
q: 2047 is 10% of 110% less than what number? {2300 @ 2047 / (1 - .11)}
q: 4246 1/2 is 6% of 50% of 466 2/3% more than what number? {3725 @ 4246_1/2 / 1.14}
q: A drew out of bank 40% of 50% of 60% of 70% of his money, and had left $1557.20: how much had he at first? Dollars? {1700 @ 1557.20 / .916}
q: I gave away 42 6/7% of my money, and had left $2: what had I at first? Dollars? {3.50|3.5 @ 2 / (4/7)}
q: In a school, 5% of the pupils are always absent, and the attendance is 570: how many on the roll? {600 @ 570 / .95} How many absent? {30}
q: A man dying, left 33 1/3% of his property to his wife, 60% of the remainder to his son, 75% of the remainder to his daughter, and the balance, $500, to a servant: what was the whole property? Dollars? {7500 @ 500 / (2/3 * 2/5 * 1/4)} The wife's share? {2500} The son's? {3000} The daughter's? {1500}
q: In a company of 87, the children are 37 1/2% of the women, who are 44 4/9% of the men: how many men? {54 @ 87 / (1 + 4/9 + 4/9 * 3/8)} Women? {24} Children? {9}
q: In your head: Our stock decreased 33 1/3%, and again 20%; then it rose 20%, and again 33 1/3%; we have thus lost $66: what was the stock worth at first? Dollars? {~66 / (1 - 2/3 * 4/5 * 6/5 * 4/3)}
q: In your head: A brewery is worth 4% less than a tannery, and the tannery 16% more than a boat; the owner of the boat has traded it for 75% of the brewery, losing thus $103: what is the tannery worth? Dollars? {~103 / (1 - .75 * .96 * 1.16) * 1.16}

== 43 | Profit and Loss | Profit and loss
note: The cost is the base; the per cent of profit or loss, the rate; the profit or loss, the percentage; the selling price, the amount (cost plus profit) or difference (cost minus loss). Answer a rate as the number of per cent.
q: Having invested $4800, my rate of profit is 13%: what is my profit? Dollars? {624 @ 4800 * .13}
q: If a man invests $1450 so as to gain 14 1/2%, what is his profit? Dollars? {210.25 @ 1450 * .145}
q: I bought $1760 worth of grain, and sold it so as to make 26 1/4% profit: what did I receive for it? Dollars? {2222 @ 1760 * 1.2625}
q: If a man invests $42540, and loses 11 2/3% of his capital: to what does his loss amount? Dollars? {4963 @ 42540 * 7/60} How much money has he left? {37577 @ 42540 - 4963}
q: A man buys 576 sheep, at $10 a head. If his flock increases 21 19/36 per cent, and he sells it at the same rate per head, how much money does he receive? Dollars? {7000 @ 5760 * (1 + 775/3600)}
q: The cost of publishing a book is 50 ct. a copy; if the expense of sale be 10% of this, and the profit 25%: what does it sell for by the copy? Cents? {~55 * 1.25}
q: A began business with $5000: the 1st year he gained 14 3/4%, which he added to his capital; the 2d year he gained 8%, which he added to his capital; the 3d year he lost 12%, and quit: how much better off was he than when he started? Dollars, to cents. {~r2 5000 * 1.1475 * 1.08 * .88 - 5000}
q: A bought a farm of government land, at $1.25 an acre; it cost him 160% to fence it, 160% to break it up, 80% for seed, 100% to plant it, 100% to harvest it, 112% for threshing, 100% for transportation; each acre produced 35 bu. of wheat, which he sold at 70 ct. a bushel: how much did he gain on every acre above all expenses the first year? Dollars? {~d 35 * .70 - 1.25 * 9.12}
q: For what must I sell a horse, that cost me $150, to gain 35%? Dollars? {202.50|202.5 @ 150 * 1.35}
q: Bought hams at 8 ct. a lb.; the wastage is 10%: how must I sell them, in cents a pound, to gain 30%? {~8 * 1.3 / .9}
q: I started in business with $10000, and gained 20% the first year, and added it to what I had; the 2d year I gained 20%, and added it to my capital; the 3d year I gained 20%: what had I then? Dollars? {17280 @ 10000 * 1.2 * 1.2 * 1.2}
q: I bought a cask of brandy, containing 46 gal., at $2.50 per gal.; if 6 gal. leak out, how must I sell the rest, a gallon, so as to gain 25%? Dollars? {~d 46 * 2.50 * 1.25 / 40}
q: A man bought part of a mine for $45000, and sold it for $165000: how many per cent profit did he make? {~120000 / 45000 * 100}
sol: The profit is $120000, which compared with the cost, $45000, is 8/3; and 8/3 of 100% = 266 2/3%.
q: If I buy at $1 and sell at $4, how many per cent do I gain? {300}
q: If I buy at $4 and sell at $1, how many per cent do I lose? {75}
q: If I sell 5/9 of an article for what the whole cost me, how many per cent do I gain? {80}
q: In your head: Paid $125 for a horse, and traded him for another, giving 60% additional money. For the second horse I received a third and $25; I then sold the third horse for $150: what was my per cent of loss? {~25 / 200 * 100}
q: In your head: A man bought a farm for $1635, which depreciated in value 25%. Selling out, he invested the proceeds so as to make 33 1/3% profit: what was his per cent of profit or loss on the entire transaction? {0}
sol: The value fell to 3/4 of itself; increased by 1/3 of itself it became 4/3 of 3/4, the whole. There was no gain, no loss.
q: It cost me $1536 to raise my wheat crop: if I sell it for $1728, what per cent profit is that per bushel? {~192 / 1536 * 100}
q: If I pay for a lb. of sugar (7000 gr.), and get a lb. troy (5760 gr.), what % do I lose? {~(175 - 144) / 175 * 100} What % does the grocer gain by the cheat? {~(175 - 144) / 144 * 100}
q: A, having failed, pays B $1750 instead of $2500, which he owed him: what % does B lose? {30}
q: An article has lost 20% by wastage, and is sold for 40% above cost: what is the gain per cent? {12}
q: If my retail profit is 33 1/3%, and I sell at wholesale for 10% less than at retail, what is my wholesale profit per cent? {20}
q: Bought a lot of glass; lost 15% by breakage: at what % above cost must I sell the remainder, to clear 20% on the whole? {~35 / 85 * 100}
q: If a bushel of corn is worth 35 ct. and makes 2 1/2 gal. of whisky, which sells at $1.14 a gal., what is the profit per cent of the distiller who pays a tax of 90 ct. a gallon? {~(2_1/2 * 1.14 - .35 - 2_1/2 * .9) / (.35 + 2_1/2 * .9) * 100}
q: I had a horse worth $80; sold him for $90; bought him back for $100: what % loss? {~10 / 80 * 100}
q: By selling a lot for 34 3/8% more than I gave, my profit is $423.50: what did it cost me? Dollars? {1232 @ 423.50 / .34375}
q: How large sales must I make in a year, at a profit of 8%, to clear $2000? Dollars? {25000 @ 2000 / .08}
q: I lost $50 by selling sugar at 22 1/2% below cost: what was the cost? Dollars, to cents. {~r2 50 / .225}
q: If I sell tea at 13 1/3% profit, I make 10 ct. a lb.: how much a pound did I give, in cents? {75 @ 10 / (2/15)}
q: I lost a 2 1/2 dollar gold coin, which was 7 1/7% of all I had: how much had I? Dollars? {35 @ 2_1/2 / (1/14)}
q: A and B each lost $5, which was 2 7/9% of A's and 3 1/3% of B's money: how much had A? Dollars? {180 @ 5 / (25/900)} B? {150 @ 5 / (1/30)} Which had the most, and by how much? {30}
q: I gained this year $2400, which is 120% of my gain last year, and that is 44 4/9% of my gain the year before: what were my profits last year? Dollars? {2000 @ 2400 / 1.2} The year before? {4500 @ 2000 / (4/9)}
q: The dogs killed 40 of my sheep, which was 4 1/6% of my flock: how many had I left? {920 @ 40 / (1/24) - 40}
q: Sold goods for $25.80, by which I gained 7 1/2%: what was the cost? Dollars? {24 @ 25.80 / 1.075}
q: Sold cloth at $3.85 a yard; my profit was 10%: how much a yard did I pay? Dollars? {3.50|3.5 @ 3.85 / 1.1}
q: Gold pens, sold at $5 apiece, yield a profit of 33 1/3%: what did they cost apiece? Dollars? {3.75 @ 5 / (4/3)}
q: Sold out for $952.82 and lost 12%: what was the cost? Dollars, to cents. {~r2 952.82 / .88} What would I have received if I had sold out at a profit of 12%? {~r2 952.82 / .88 * 1.12}
q: Sold my horse at 40% profit; with the proceeds I bought another, and sold him for $238, losing 20%: what did the second horse cost? Dollars? {297.50|297.5 @ 238 / .8} The first? {212.50|212.5 @ 297.50 / 1.4}
q: Sold flour at an advance of 13 1/3%; invested the proceeds in flour again, and sold this lot at a profit of 24%, realizing $3952.50: how much did the first lot cost me? Dollars? {2812.50|2812.5 @ 3952.50 / (17/15 * 1.24)} The second? {3187.50|3187.5 @ 3952.50 / 1.24}
q: An invoice of goods purchased in New York, cost me 8% for transportation, and I sold them at a gain of 16 2/3% on their total cost on delivery, realizing $1260: at what were they invoiced? Dollars? {1000 @ 1260 / (1.08 * 7/6)}
q: For 6 years my property increased each year, on the previous, 100%, and became worth $100000: what was it worth at first? Dollars? {1562.50|1562.5 @ 100000 / 64}

== 44 | Stocks and Premium | Stocks and bonds
note: Stock is the capital of a corporation, divided into shares, usually of $50 or $100. A Dividend is money paid to the stockholders, an Assessment money required of them, both in proportion to their stock. The stock is the base: Stock x Rate = Dividend or Assessment.
q: I own 18 shares, of $50 each, in the City Insurance Co., which has declared a dividend of 7 1/2%: what do I receive? Dollars? {67.50|67.5 @ 18 * 50 * .075}
q: I own 147 shares of railroad stock ($50 each), on which I am entitled to a dividend of 5%, payable in stock: how many additional shares do I receive? {7} And how many dollars of another share? {17.50|17.5 @ 147 * 50 * .05 - 7 * 50}
q: The Western Stage Co. declares a dividend of 4 1/2 per cent: if their whole stock is $150000, how much is distributed to the stockholders? Dollars? {6750 @ 150000 * .045}
q: The Cincinnati Gas Co. declares a dividend of 18%: what do I get on 50 shares ($100 each)? Dollars? {900 @ 5000 * .18}
q: A railroad company, whose stock account is $4256000, declared a dividend of 3 1/2%: what sum was distributed among the stockholders? Dollars? {148960 @ 4256000 * .035}
q: A telegraph company, with a capital of $75000, declares a dividend of 7%, and has $6500 surplus: what has it earned? Dollars? {11750 @ 75000 * .07 + 6500}
q: I own 24 shares of stock ($25 each) in a fuel company, which declares a dividend of 6%; I take my dividend in coal, at 8 ct. a bu.: how much do I get? Bushels? {450 @ 24 * 25 * .06 / .08}
q: My dividend on 72 shares of bank stock ($50 each) is $324: what was the rate of dividend? Per cent? {9 @ 324 / 3600 * 100}
q: A turnpike company, whose stock is $225000, earns during the year $16384.50: what rate of dividend can it declare, in whole per cent? {7} With what surplus? Dollars? {634.50|634.5 @ 16384.50 - 225000 * .07}
q: The receipts of a certain canal company, whose stock is $3650000, amount in one year to $256484; the outlay is $79383: what rate of dividend can the company declare, to the nearest half per cent below? {~4_1/2} Leaving how much over? Dollars? {12851 @ 256484 - 79383 - 3650000 * .045}
sol: The net earnings are $256484 - $79383 = $177101, a little more than 4.85% of the stock. A dividend of 4 1/2% takes $164250, leaving $12851.
q: I own 500 shares ($100) in a stock company. If I have to pay $250 on an assessment, what is the rate, in per cent? {~250 / 50000 * 100}
q: An insurance company earns $18000, and declares a 15% dividend: what is its stock account? Dollars? {120000 @ 18000 / .15}
q: A man gets $94.50 as a 7% dividend: how many shares of stock ($50 each) has he? {27 @ 94.50 / .07 / 50}
q: Received 5 shares ($50 each), and $26 of another share, as an 8% dividend on stock: how many shares had I? {69 @ (5 * 50 + 26) / .08 / 50}
q: Received 10% stock dividend, and then had 102 shares ($50 each), and $15 of another share: how many shares had I before the dividend? {93 @ (102 * 50 + 15) / 1.1 / 50}
q: Having received two dividends in stock, one of 5%, another of 8%, my stock has increased to 567 shares: how many had I at first? {500 @ 567 / (1.05 * 1.08)}

== 45 | Stocks and Premium | Premium and discount
note: The Par Value is the face; the Market Value the sum for which it sells. Premium is the excess of the market value over par; Discount, the excess of par over market value. The par value is the base. Stock "at 106%" sells for 106% of its face.
q: Bought 54 shares of railroad stock ($100 each) at 4% discount: find the discount. Dollars? {216 @ 5400 * .04} The cost? {5184 @ 5400 - 216}
q: Buy 18 shares of stock ($100 each) at 8% discount: find the discount. Dollars? {144 @ 1800 * .08} The cost? {1656}
q: Sell the same at 4 1/2% premium: find the premium. Dollars? {81 @ 1800 * .045} The price? {1881} The gain? {225 @ 1881 - 1656}
q: Bought 62 shares of railroad stock ($50 each) at 28% premium: what did they cost? Dollars? {3968 @ 62 * 50 * 1.28}
q: What is the cost of 47 shares of railroad stock ($50 each) at 30% discount? Dollars? {1645 @ 47 * 50 * .7}
q: Bought $150 in gold, at 3/4% premium: what is the premium? Dollars? {1.125|1.12|1.13} The cost? {151.125|151.12|151.13}
q: Sold a draft on New York of $2568.45, at 1/2% premium: what do I get for it? Dollars, to cents. {~r2 2568.45 * 1.005}
q: Sold $425 uncurrent money, at 3% discount: what did I get? Dollars? {412.25 @ 425 * .97} What did I lose? {12.75}
q: What is a $5 note worth, at 6% discount? Dollars? {4.70|4.7 @ 5 * .94}
q: Exchanged 32 shares of bank stock ($50 each), 5% premium, for 40 shares of railroad stock ($50 each), 10% discount, and paid the difference in cash: what was it? Dollars? {120 @ 40 * 50 * .9 - 32 * 50 * 1.05}
q: Bought 98 shares of stock ($50 each), at 15% discount; gave in payment a bill of exchange on New Orleans for $4000, at 5/8% premium, and the balance in cash: how much cash did I pay? Dollars? {140 @ 98 * 50 * .85 - 4000 * 1.00625}
q: Bought 56 shares of turnpike stock ($50 each), at 69%; sold them at 76 1/2%: what did I gain? Dollars? {210 @ 56 * 50 * .075}
q: Bought telegraph stock at 106%; sold it at 91%: what was my loss on 84 shares ($50 each)? Dollars? {630 @ 84 * 50 * .15}
q: What is the difference between a draft on Philadelphia of $8651.40, at 1 1/4% premium, and one on New Orleans for the same amount, at 1/2% discount? Dollars? {~r2 8651.40 * .0175}
q: Paid $2401.30 for a draft of $2360 on New York: what was the rate of premium? Per cent? {~d 41.30 / 2360 * 100}
q: Bought 112 shares of railroad stock ($50 each) for $3640: what was the rate of discount? Per cent? {35 @ 1960 / 5600 * 100}
q: If the stock in the last example yields 8% dividend, what is my rate of profit? Per cent? {~448 / 3640 * 100}
q: I sell the same stock for $5936: what rate of premium is that? Per cent? {6 @ 336 / 5600 * 100} What rate of profit? {~2296 / 3640 * 100}
q: If I count my dividend as part of the profit, what is my rate of profit? Per cent? {~(2296 + 448) / 3640 * 100}
q: Exchanged 12 Ohio bonds ($1000 each), 7% premium, for 280 shares of railroad stock ($50 each): what rate of discount were the latter? Per cent? {~(14000 - 12840) / 14000 * 100}
q: Gave $266.66 2/3 of notes, 4% discount, for $250 of gold: what rate of premium was the gold? Per cent? {~(266_2/3 * .96 - 250) / 250 * 100}
q: Bought 58 shares of mining stock ($50 each), at 40% premium, and gave in payment a draft on Boston for $4000: what rate of premium was the draft? Per cent? {~(58 * 50 * 1.4 - 4000) / 4000 * 100}
q: Received $4.60 for an uncurrent $5 note: what was the rate of discount? Per cent? {8}
q: Paid $2508.03 for 26 shares of stock ($100 each), and brokerage, $25.03: what is the rate of discount? Per cent? {~(2600 - 2483) / 2600 * 100}
q: Paid 36 ct. premium for gold 3/4% above par: how much gold was there? Dollars? {48 @ .36 / .0075}
q: Took stock at par; sold it for 2 1/4% discount, and lost $117: how many shares ($50 each) had I? {104 @ 117 / .0225 / 50}
q: The discount, at 7 1/2%, on stocks, was $93.75: how many shares ($50 each) were sold? {25 @ 93.75 / .075 / 50}
q: Buy stock at 4 1/2% premium; sell at 8 1/4% premium; profit, $345: how many shares ($100 each)? {92 @ 345 / .0375 / 100}
q: Buy stocks at 14% discount; sell at 3 1/2% premium; profit, $192.50: how many shares ($50 each)? {22 @ 192.50 / .175 / 50}
q: The premium on a draft, at 7/8%, was $10.36: what was the face? Dollars? {1184 @ 10.36 / .00875}
q: Buy stocks at 6% discount; sell at 42% discount; loss, $666: how many shares ($50 each)? {37 @ 666 / .36 / 50}
q: Bought stock at 10% discount, which rose to 5% premium, and sold for cash; paying a debt of $33, I invested the balance in stock at 2% premium, which, at par, left me $11 less than at first: how much money had I at first? Dollars? {148.50|148.5}
sol: Each 90 ct. bought $1 of stock, which sold for $1.05; and $1.02 buys $1 of the second stock. So without the debt my money would buy 105/(90 x 1.02) of itself in stock at par, or 175/153. The $33 paid away would have bought $32 6/17 of it, and the result is $11 less than my money: 22/153 of my money = $21 6/17, so the money was $148.50.
q: What is the face of a draft on Baltimore costing $2861.45, at 1 1/2% premium? Dollars, to cents. {~r2 2861.45 / 1.015}
q: Invested $1591 in stocks, at 26% discount: how many shares ($50 each) did I buy? {43 @ 1591 / .74 / 50}
q: Bought a draft on New Orleans, at 1/2% discount, for $6398.30: what was its face? Dollars, to cents. {~r2 6398.30 / .995}
q: Notes at 65% discount, 2% brokerage, cost $881.79: what is their face? Dollars, to cents. {~r2 881.79 / .37}
q: Exchanged 17 railroad bonds ($500 each) 25% below par, for bank stock at 6 1/4% premium: how many shares ($100 each) did I get? {60 @ 8500 * .75 / 1.0625 / 100}
q: How much gold, at 5/8% premium, will pay a check for $7567? Dollars? {7520 @ 7567 / 1.00625}
q: How much silver, at 1 1/4% discount, can be bought for $3172.64 of currency? Dollars? {3212.80|3212.8 @ 3172.64 / .9875}
q: How large a draft, at 1/4% premium, is worth 54 city bonds ($100 each), at 12% discount? Dollars, to cents. {~r2 5400 * .88 / 1.0025}
q: Exchanged 72 Ohio State bonds ($1000 each), at 6 1/4% premium, for Indiana bonds ($500 each), at 2% premium: how many of the latter did I get? {150 @ 72000 * 1.0625 / 1.02 / 500}

== 46 | Commission | Commission and brokerage
note: A commission-merchant sells or buys for another (the principal) and is paid a commission, a per cent of the sum handled; a broker's commission is brokerage. The sum bought or sold is the base. The net proceeds are what is left after all charges; in buying, the whole cost is the purchase plus commission and charges.
q: I collect for A $268.40, and have 5% commission: what does A get? Dollars? {254.98 @ 268.40 * .95}
q: I sell for B 650 barrels of flour, at $7.50 a barrel, 28 barrels of whisky, 35 gal. each, at $1.25 a gal.: what is my commission, at 2 1/4%? Dollars, to cents. {~r2 (650 * 7.50 + 28 * 35 * 1.25) * .0225}
q: Received on commission 25 hhd. sugar (36547 lb.), of which I sold 10 hhd. (16875 lb.), at 6 ct. a lb., and 6 hhd. (8246 lb.) at 5 ct. a lb., and the rest at 5 1/2 ct. a lb.: what is my commission, at 3%? Dollars, to cents. {~r2 (16875 * .06 + 8246 * .05 + 11426 * .055) * .03}
q: A lawyer charged 8% for collecting a note of $648.75: what is his fee? Dollars? {51.90|51.9 @ 648.75 * .08} The net proceeds? {596.85 @ 648.75 - 51.90}
q: A lawyer, having a debt of $1346.50 to collect, compromises by taking 80%, and charges 5% for his fee: what is his fee? Dollars, to cents. {~r2 1346.50 * .04} The net proceeds? {~r2 1346.50 * .76}
q: Bought for C, a carriage for $950, a pair of horses for $575, and harness for $120; paid charges for keeping, packing, shipping, etc., $18.25; freight, $36.50: what was my commission, at 3 1/3%? Dollars, to cents. {~r2 1645 / 30} What was the whole amount of my bill? {~r2 1645 * 31/30 + 18.25 + 36.50}
q: An architect charges 3 1/2% for designing and superintending a building, which cost $27814.60: to what does his fee amount? Dollars, to cents. {~r2 27814.60 * .035}
q: A factor has 2 3/4% commission, and 3 1/2% for guarantying payment: if the sales are $6231.25, what does he get? Dollars, to cents. {~r2 6231.25 * .0625}
q: Sold 500000 lb. of pork at 5 1/2 ct. a lb.: what is my commission at 1 1/4%? Dollars? {343.75 @ 500000 * .055 * .0125}
q: An architect charges 1 1/4% for plans and specifications, and 2 7/8% for superintending: what does he make, if the building costs $14902.50? Dollars, to cents. {~r2 14902.50 * .04125}
q: A sells a house and lot for me at $3850, and charges 5/8% brokerage: what is his fee? Dollars, to cents. {~r2 3850 * .00625}
q: I have a lot of tobacco on commission, and sell it through a broker for $4642.85: my commission is 2 1/2%, the brokerage 1 1/8%: what do I pay the broker? Dollars, to cents. {~r2 4642.85 * .01125} What do I keep? {~r2 4642.85 * .01375}
q: An auctioneer's commission for selling a lot was $50, and the sum paid the owner was $1200: what was the rate of commission, in per cent? {4 @ 50 / 1250 * 100}
q: A commission-merchant sells 800 barrels of flour, at $6.43 3/4 a barrel, and remits the net proceeds, $5021.25: what is his rate of commission, in per cent? {~(5150 - 5021.25) / 5150 * 100}
q: The cost of a building was $19017.92, including the architect's commission, which was $553.92: what rate did the architect charge, in per cent? {3 @ 553.92 / 18464 * 100}
q: Bought flour for A; my whole bill was $5802.57, including charges, $76.85, and commission, $148.72: find the rate of commission, in per cent. {~148.72 / (5802.57 - 76.85 - 148.72) * 100}
q: Charged $52.50 for collecting a debt of $1050: what was my rate of commission, in per cent? {5}
q: An agent gets $169.20 for selling property for $8460: what was his rate of brokerage, in per cent? {2}
q: My commission for selling books was $6.92, and the net proceeds, $62.28: what rate did I charge, in per cent? {10}
q: Paid $38.40 for selling goods worth $6400: what was the rate of brokerage, in per cent? {~38.40 / 6400 * 100}
q: Paid a broker $24.16, and retained as my part of the commission $42.28, for selling a consignment at $2416: what was the rate of brokerage, in per cent? {1} My rate of commission? {~(24.16 + 42.28) / 2416 * 100}
q: My commissions in 1 year, at 2 1/2%, are $3500: what were the sales? Dollars? {140000 @ 3500 / .025} The whole net proceeds? {136500}
q: An insurance agent's income is $1733.45, being 10% on the sums received for the company: what were the company's net receipts? Dollars? {15601.05 @ 1733.45 * 9}
q: A packing-house charged 1 1/2% commission, and cleared $2376.15, after paying out $1206.75 for all expenses of packing: how many pounds of pork were packed, if it cost 4 1/2 ct. a pound? {5308000 @ (2376.15 + 1206.75) / (.045 * .015)}
q: Paid $64.05 for selling coffee, which was 7/8% brokerage: what are the net proceeds? Dollars? {7255.95 @ 64.05 / .00875 - 64.05}
q: An agent purchased, according to order, 10400 bushels of wheat; his commission, at 1 1/4%, was $156, and charges for storage, shipping, and freight, $527.10: what did he pay a bushel? Dollars? {1.20|1.2 @ 156 / .0125 / 10400} What was the whole cost? {13163.10|13163.1 @ 12480 + 156 + 527.10}
q: Received produce on commission, at 2 1/4%; my surplus commission, after paying 1/2% brokerage, is $107.03: what was the amount of the sale? Dollars, to the nearest dollar. {6116 @ 107.03 / .0175} The brokerage? Dollars, to cents. {30.58} The net proceeds? {5978.39}
q: A lawyer collects a debt for a client, takes 4% for his fee, and remits the balance, $207.60: what was the debt? Dollars? {216.25 @ 207.60 / .96} The fee? {8.65}
q: Sent $1000 to buy a carriage, commission 2 1/2%: what must the carriage cost? Dollars, to cents. {~r2 1000 / 1.025}
sol: 100% + 2 1/2% = 102 1/2% = $1000; 1% = $9.756; 100% = $975.61.
q: A buys per order a lot of coffee; charges, $56.85; commission, 1 1/4%; the whole cost is $539.61: what did the coffee cost? Dollars? {476.80|476.8 @ (539.61 - 56.85) / 1.0125}
q: Buy sugar at 2 1/4% commission, and 2 1/2% for guaranteeing payment: if the whole cost is $1500, what was the cost of the sugar? Dollars, to cents. {~r2 1500 / 1.0475}
q: Sold 2000 hams (20672 lb.); commission, 2 1/2%, guaranty, 2 3/4%, net proceeds due consignor, $2448.34: what did the hams sell for a lb.? Cents? {~2584 / 20672 * 100}
q: Sold cotton on commission, at 5%; invested the net proceeds in sugar; commission, 2%; my whole commission was $210: what was the value of the cotton? Dollars? {3060 @ 210 / (.05 + .95 * 2/102)} The sugar? {2850 @ 3060 * .95 / 1.02}
q: Sold flour at 3 1/2% commission; invested 2/3 of its value in coffee, at 1 1/2% commission; remitted the balance, $432.50: what was the value of the flour? Dollars? {1500 @ 432.50 / (1 - .035 - 2/3 - 2/3 * .015)} The coffee? {1000} My commissions? Flour: {52.50|52.5} Coffee: {15}
q: Sold a consignment of pork, and invested the proceeds in brandy, after deducting my commissions, 4% for selling, and 1 1/4% for buying. The brandy cost $2304.00: what did the pork sell for? Dollars? {2430 @ 2304 * 1.0125 / .96} My commission for selling? {97.20|97.2} For buying? {28.80|28.8}
q: Sold 1400 barrels of flour, at $6.20 a barrel; invested the proceeds in sugar, as per order, reserving my commissions, 4% for selling and 1 1/2% for buying, and the expense of shipping, $34.16: how much did I invest in sugar? Dollars? {8176 @ (8680 * .96 - 34.16) / 1.015}
q: An agent sold my corn; and, after reserving his commission, invested all the proceeds in corn at the same price; his commission, buying and selling, was 3%, and his whole charge $12: for what was the corn first sold? Dollars? {206 @ 12 / (.03 + .97 * 3/103)}
q: My agent sold my flour at 4% commission; increasing the proceeds by $4.20, I ordered the purchase of wheat at 2% commission; after which, wheat declining 3 1/3%, my whole loss was $5: what was the flour worth? Dollars? {53 @ (5 - 4.2 * (1 - 290 / 306)) / (1 - .96 * 290 / 306)}

== 47 | Stocks and Premium | Stock investments
note: Stock "at 95" costs 95 cents for each dollar of its face. The par value bought = amount invested / market value; the income = par value x rate of dividend. The rate of income on the money = rate of dividend / market value. Brokerage is usually 1/8% on the par value.
q: A person invests $5652.50 in Mutual Insurance Company stock at 95 cents: what will be his income if the stock pay 10% dividend annually? Dollars? {595 @ 5652.50 / .95 * .1}
sol: As many dollars' worth of stock can be bought as $.95 is contained in $5652.50, which is 5950; 10% of $5950 is $595.
q: A invests $28000 in Lake Shore Railroad stock, at 70%. If the stock yields 8% annually, what is the amount of his income? Dollars? {3200 @ 28000 / .7 * .08}
q: B invests $100962 in U. S. cur. 6's, 1899, at 106 1/2%. If gold is at 1/8 premium, what does the government save by paying him his interest in greenbacks? Dollars, to cents. {~r2 100962 / 1.065 * .06 * .00125}
q: If I invest $10200 in Tennessee 6's, new, at 30%, what is my annual income? Dollars? {2040 @ 10200 / .3 * .06}
q: A broker invested $36000 in quicksilver preferred stock, at 40%: if the stock pays 4%, what is the income derived? Dollars? {3600 @ 36000 / .4 * .04}
q: Which is the better investment, stock paying 6 1/2% dividend, at a market value of 106 1/2%, or stock paying 4 1/2% dividend, at 104 1/8%? {open}
sol: A dollar in the first buys 1/1.065 of stock and yields 6 1/2 / 106 1/2 = 6.10 ct.; a dollar in the second yields 4 1/2 / 104 1/8 = 4.32 ct. The first is better.
q: Which is the more profitable, to invest $10000 in 6% stock purchased at 75%, or in 5% stock purchased at 60%, allowing brokerage 1/2%? Income from the 5% stock? Dollars, to cents. {~r2 10000 / .605 * .05} From the 6% stock? {~r2 10000 / .755 * .06} The 5% stock is better by how much? {~r2 10000 / .605 * .05 - 10000 / .755 * .06}
q: Invested $10132.50 in railroad stock at 105%, which pays me annually $965: what is the rate of dividend on the stock, in per cent? {10 @ 965 / (10132.50 / 1.05) * 100}
q: A has a farm, valued at $46000, which pays him 5% on the investment. Through a broker, who charges $56.50 for his services, he exchanges it for insurance stock at 9% premium, and this increases his annual income by $1072: what dividend does the stock pay, in per cent? {8 @ (46000 * .05 + 1072) / ((46000 - 56.50) / 1.09) * 100}
q: What dividend must stock pay, in per cent, in order that my rate of income on an investment of $64968.75 shall be 4 4/33%, provided the stock can be bought at 103 1/8%? {~64968.75 * 136/3300 / (64968.75 / 1.03125) * 100}
q: My investment was $9850, my income is $500, and the market value of the stock 108 1/10%, brokerage 1/4%: what is the rate of dividend, in per cent? {~500 / (9850 / 1.0835) * 100}
q: The sale of my farm cost me $500, but I gave the proceeds to a broker, allowing him 1/2%, to purchase railroad stock then in market at 102%; the farm paid a 5% income, equal to $2075, but the stock will pay $2025 more: what is the rate of dividend, in per cent? {~4100 / ((2075 / .05 - 500) / 1.025) * 100}
q: Howard has at order $122400, and can allow brokerage 1/2%, and buy insurance stock at 101 1/2%, yielding 4 1/6%; but if he send to the broker $100 more for investment, and buy rolling-mill stock at 103 1/2%, the income will only be half so large: what rate does the higher stock pay, in per cent? {~2500 / (122500 / 1.04) * 100}
q: If U. S. bonds, paying 5% interest, are selling at 108 1/2%, how much must be invested to secure an annual income of $2000? Dollars? {43400 @ 2000 / .05 * 1.085}
q: What amount is invested by A, whose canal stock, yielding 4%, brings an income of $300, but sells in market for 92%? Dollars? {6900 @ 300 / .04 * .92}
q: If I invest all my money in 5% furnace stock, salable at 75%, my income will be $180: how much must I borrow to make an investment in 6% state stock, selling at 102%, to have that income? Dollars? {360 @ 180 / .06 * 1.02 - 180 / .05 * .75}
q: If railroad stock be yielding 6%, and is 20% below par, how much would have to be invested to bring an income of $390? Dollars? {5200 @ 390 / .06 * .8}
q: A banker owns 2 1/2% stocks, at 10% below par, and 3% stocks, at 15% below par. The income from the former is 66 2/3% more than from the latter, and the investment in the latter is $11400 less than in the former: required the whole investment. Dollars? {31800} The whole income? {960}
sol: The first income is 2 1/2/90 of its investment, the second 3/85 of its. Incomes as 5 to 3 make the investments as 180 to 85; the difference, 95 parts, is $11400, so the investments are $21600 and $10200, in all $31800. Incomes: $600 and $360, in all $960.
q: Howard M. Holden sold $21600 U. S. 4's, 1907, registered, at 99 3/8%, and immediately invested a sufficient amount of the proceeds in Illinois Central Railroad stock, at 80%, which pays an annual dividend of 6%; he receives $840 from the railroad investment; with the remainder of his money he bought a farm at $30 an acre: required the amount invested in railroad stock. Dollars? {11200 @ 840 / .06 * .8} The number of acres in the farm? {~(21600 * .99375 - 11200) / 30}
q: W. T. Baird, through his broker, invested a certain sum of money in Philadelphia 6's, at 115 1/2%, and three times as much in Union Pacific 7's, at 89 1/2%, brokerage 1/2% in both cases: how much was invested in Philadelphia 6's if his annual income is $9920? Dollars? {34800 @ 9920 / (.06 / 1.16 + 3 * .07 / .9)} In Union Pacific 7's? {104400}
q: Thomas Reed bought 6% mining stock at 114 1/2%, and 4% furnace stock at 112%, brokerage 1/2%; the latter cost him $430 more than the former, but yielded the same income: what did the mining stock cost him? Dollars? {920 @ 430 / (1.125 / .04 / (1.15 / .06) - 1)} The furnace stock? {1350}
q: What per cent of his money will a man realize in buying 6% stock at 80%? {~6 / 80 * 100}
q: What is the rate of income on Pacific Mail 6's, bought at 30%? {20}
q: What is the rate of income on Union Pacific 6's, bought at 110%? {~6 / 110 * 100}
q: Which is the better investment: U. S. new 4's, registered, at 99 3/8%; or U. S. new 4 1/2's, coupons, at 106%? By how many per cent of income? {~4_1/2 / 106 * 100 - 4 / 99_3/8 * 100}
sol: The first yields 4 / 99 3/8 = 4.025%; the second 4 1/2 / 106 = 4.245%. The 4 1/2's are better.
q: Thomas Sparkler has an opportunity of investing $30000 in North-western preferred stock, at 76%, which pays an annual premium of 5%; in Panama stock, at 125%, which pays a premium annually of 8 1/2%; or he can lend his money, on safe security, at 6 1/2% per annum. What income would the loan bring? Dollars? {1950} The North-western stock? To cents. {~r2 30000 / .76 * .05} The Panama stock? {2040 @ 30000 / 1.25 * .085}
sol: The Panama stock, at $2040 a year, is the best investment.
q: Thomas Jackson bought 500 shares of Adams Express stock, at 105 1/2%, and paid 1 1/2% brokerage: what is the rate of income on his investment per annum if the annual dividend is 8%? {~8 / 107 * 100}
q: What must I pay for Lake Shore 6's ($100 a share), that the investment may yield 10%? Dollars a share? {60 @ 6 / .1}
q: What must I pay, in per cent, for Chicago, Burlington & Quincy Railroad stock that bears 6%, that my annual income on the investment may yield 5%? {120 @ 6 / 5 * 100}
q: Which is the best permanent investment: 4's at 70%, 5's at 80%, 6's at 90%, or 10's at 120%? What rate of income do the 10's give, in per cent? {~10 / 120 * 100}
sol: A dollar buys 4/70 = 5 5/7 ct. of income in the first, 6 1/4 ct. in the second, 6 2/3 ct. in the third, and 8 1/3 ct. in the 10's at 120%, the best.
q: The rate of income being 7% on the investment, and the dividend rate 4%, what is the market value of $3430 of the stock? Dollars? {1960 @ 3430 * 4 / 7}
q: In a mutual insurance company one capitalist has an investment paying 8%: what is the premium on the stock, in per cent, the dividend being 9%? {~(9 / 8 - 1) * 100}
q: Suppose 10% state stock 20% better in market than 4% railroad stock; if A's income be $500 from each, how much money has he paid for the state stock, the whole investment bringing 6 2/333%? Dollars? {5400} For the railroad stock? {11250}
sol: The whole income, $1000, is 6 2/333% of the whole cost, which is therefore $16650. The face of the state stock is $5000, of the railroad stock $12500; a dollar of state stock costs 6/5 of a dollar of railroad stock, so the costs are as 12 to 25: $5400 and $11250.
q: At what figure, in per cent, must be government 5 per cent's to make my purchase pay 9%? {~5 / 9 * 100}

== 48 | Insurance and Taxes | Insurance
note: The premium is the sum paid for insurance, a per cent of the amount insured: Amount insured x Rate = Premium. The amount insured is the base. Answer rates as the number of per cent.
q: Insured 5/8 of a vessel worth $24000, and 2/3 of its cargo worth $36000, the former at 2 1/4%, the latter at 1 1/8%: what is the premium? Dollars? {607.50|607.5 @ 15000 * .0225 + 24000 * .01125}
q: Insured a house for $2500, and furniture for $600, at 6/10%: what is the premium? Dollars? {18.60|18.6 @ 3100 * .006}
q: What is the premium on a cargo of railroad iron worth $28000, at 1 3/4%? Dollars? {490 @ 28000 * .0175}
q: Insured goods invoiced at $32760, for three months, at 8/10%: what is the premium? Dollars? {262.08 @ 32760 * .008}
q: My house is permanently insured for $1800, by a deposit of 10 annual premiums, the rate per year being 3/4%: how much did I deposit? Dollars? {135 @ 1800 * .0075 * 10} If, on terminating the insurance, I receive my deposit less 5%, how much do I get? {128.25 @ 135 * .95}
q: A shipment of pork, costing $1275, is insured at 5/9%, the policy costing 75 cents: what does the insurance cost? Dollars, to cents. {~r2 1275 * 5/900 + .75}
q: An insurance company having a risk of $25000, at 9/10%, re-insured $10000, at 4/5%, with another office, and $5000, at 1%, with another: how much premium did it clear above what it paid? Dollars? {95 @ 225 - 80 - 50}
q: Paid $19.20 for insuring 2/3 of a house, worth $4800: what was the rate? {~19.20 / 3200 * 100}
q: Paid $234, including cost of policy, $1.50, for insuring a cargo worth $18600: what was the rate? {~232.50 / 18600 * 100}
q: Bought books in England for $2468; insured them for the voyage for $46.92, including the cost of the policy, $2.50: what was the rate? To two places. {~r2 44.42 / 2468 * 100}
q: A vessel is insured for $42000; $18000 at 2 1/2%, $15000 at 3 4/5%, and the rest at 4 2/3%: what is the rate on the whole $42000? {~(450 + 570 + 420) / 42000 * 100}
q: I took a risk of $45000; re-insured at the same rate, $10000 each, in three offices, and $5000 in another; my share of the premium was $262.50: what was the rate? {~262.50 / 10000 * 100}
q: I took a risk at 1 1/2%; re-insured 2/5 of it at 2%, and 1/4 of it at 2 1/2%: what rate of insurance do I get on what is left? {~(1.5 - 2/5 * 2 - 1/4 * 2.5) / (1 - 2/5 - 1/4)}
q: Paid $118 for insuring, at 4/5%: what was the amount insured? Dollars? {14750 @ 118 / .008}
q: Paid $411.37 1/2 for insuring goods, at 1 1/2%: what was their value? Dollars? {27425 @ 411.375 / .015}
q: Paid $42.30 for insuring 5/8 of my house, at 9/10%: what is the house worth? Dollars? {7520 @ 42.30 / .009 / (5/8)}
q: Took a risk at 2 1/4%; re-insured 3/5 of it at 2 1/2%; my share of the premium was $197.13: how large was the risk? Dollars? {26284 @ 197.13 / (.0225 - .6 * .025)}
q: Took a risk at 1 3/5%; re-insured half of it at the same rate, and 1/3 of it at 1 1/2%; my share of the premium was $58.11: how large was the risk? Dollars? {19370 @ 58.11 / (.016 - .008 - .005)}
q: Took a risk at 2%; re-insured $10000 of it at 2 1/8%, and $8000 at 1 3/4%; my share of the premium was $207.50: what sum was insured? Dollars? {28000 @ (207.50 + 212.50 + 140) / .02}
q: The Mutual Fire Insurance Company insured a building and its stock for 2/3 of its value, charging 1 3/4%. The Union Insurance Company relieved them of 1/4 of the risk, at 1 1/2%. The building and stock being destroyed by fire, the Union lost forty-nine thousand dollars less than the Mutual: what amount of money did the owners of the building and stock lose? Dollars? {51750}
sol: The Mutual paid 2/3 of the value less the Union's 1/6, after receiving 1 3/4% of 2/3 and paying 1 1/2% of 1/6: a loss of .490833 of the value. The Union paid 1/6 and received 1 1/2% of it: .164167 of the value. The difference, .326667 of the value, is $49000, so the value was $150000. The owners lost 1/3 of it, $50000, and the premium, $1750: $51750.

== 49 | Insurance and Taxes | Taxes and duties
note: The rate of taxation is usually given in cents on $100, or mills on $1: 78 cents on $100 is 78/100 of 1%. A poll-tax is added to the property tax. Duties on imports are ad valorem (a per cent of the invoice price) or specific (so much a pound, gallon, yard).
q: The taxable property of a county is $486250, and the rate of taxation is 78 ct. on $100: what is the tax to be raised? Dollars? {3792.75 @ 486250 * .0078}
q: A's property is assessed at $3800; the rate of taxation is 96 ct. on $100: what is his whole tax, if he pays a poll-tax of $1? Dollars? {37.48 @ 3800 * .0096 + 1}
q: At a rate of 1 1/4% (125 ct. on $100), what will be the tax on property assessed at $25349? Dollars, to cents. {~r2 25349 * .0125}
sol: By the tax table: for $20000, $250; for $5000, $62.50; for $300, $3.75; for $40, $.50; for $9, $.1125. Added, $316.86.
q: Find the tax, at 1 1/4%, for $6815.30. Dollars, to cents. {~r2 6815.30 * .0125}
q: What is the tax, at 1 1/4%, on property assessed at $10424.50, and two polls, at $1.50 each? Dollars, to cents. {~r2 10424.50 * .0125 + 3}
q: A's property is assessed at $251350, and B's at $25135. At 1 1/4%, what is the difference in their taxes? Dollars, to cents. {~r2 (251350 - 25135) * .0125}
q: Property assessed at $2604, pays $19.53 tax: what is the rate of taxation, in cents on $100? {75 @ 19.53 / 2604 * 10000}
q: The taxable property in a town of 1742 polls, is $6814320; a tax of $66913 is proposed: if a poll-tax of $1.25 is levied, what should be the rate of taxation, in cents on $100 to the nearest cent? {~r0 (66913 - 1742 * 1.25) / 6814320 * 10000}
q: An estate of $350000 pays a tax of $5670: what is the rate of taxation, in cents on $100? {162 @ 5670 / 350000 * 10000}
q: A's tax is $50.46; he pays a poll-tax of $1.50, and owns $8704 taxable property: what is the rate of taxation, in cents on $100? {~(50.46 - 1.50) / 8704 * 10000}
q: What is the assessed value of property taxed $66.96, at 1 4/5%? Dollars? {3720 @ 66.96 / .018}
q: A corporation pays $564.42 tax, at the rate of 46 ct. on $100: find its capital. Dollars? {122700 @ 564.42 / .0046}
q: A is taxed $71.61 more than B; the rate is $1.32 on $100: how much is A assessed more than B? Dollars? {5425 @ 71.61 / .0132}
q: A tax of $4000 is raised in a town containing 1024 polls, by a poll-tax of $1, and a property-tax of 24 ct. on $100: what is the value of the taxable property in it? Dollars? {1240000 @ 2976 / .0024}
q: A's income is 16% of his capital; he is taxed 2 1/2% of his income, and pays $26.04: what is his capital? Dollars? {6510 @ 26.04 / (.16 * .025)}
q: A pays a tax of $1.35 on $100 on his capital, and has left $125127.66: what was his capital? Dollars? {126840 @ 125127.66 / .9865} His tax? {1712.34}
q: Sold a lot for $7599, which covered its cost and 2% beside, paid for tax: what was the cost? Dollars? {7450 @ 7599 / 1.02}
q: Import 24 trunks, at $5.65 each, and 3 doz. leather satchels, at $2.25 each; the rate is 35% ad valorem: what is the duty? Dollars, to cents. {~r2 (24 * 5.65 + 36 * 2.25) * .35}
q: What is the duty on 45 casks of wine, of 36 gal. each, invoiced at $1.25 a gal., at 40 ct. a gal. specific duty? Dollars? {648 @ 45 * 36 * .4}
q: There is a specific duty of $3 per gallon, and an ad valorem duty of 50% on cologne-water: what is the total amount of duty paid on 25 gallons, invoiced at $16.50 per gallon? Dollars? {281.25 @ 25 * 3 + 25 * 16.50 * .5}
q: What is the total duty on 36 boxes of sugar, each weighing 6 cwt. 2 qr. 18 lb., invoiced at 2 1/2 ct. per lb., the specific duty being 2 ct. per lb., and the ad valorem duty 25%? Dollars? {704.97 @ 26856 * (.02 + .025 * .25)}
q: What is the duty on 575 yards of broadcloth, weighing 1154 lb., invoiced at $2.56 per yd., the specific duty being 50 ct. per lb., and the ad valorem duty 35%? Dollars? {1092.20|1092.2 @ 575 * 2.56 * .35 + 1154 * .5} If the freight, charges, and losses amount to $160.80, how much a yard must I charge to gain 15%? {5.45 @ (1472 + 1092.20 + 160.80) * 1.15 / 575}
q: A merchant imported a ton (2240 lb.) of manilla, invoiced at 5 ct. per lb.; he paid a specific duty per ton, which, on this shipment, was equivalent to an ad valorem duty of 22 9/28%: what is the specific duty? Dollars? {25 @ 2240 * .05 * (22_9/28) / 100}
q: Received a shipment of 3724 lb. of wool, invoiced at 23 ct. per lb.; the duty is 10 ct. per lb., and 11% ad valorem, less 10%: what is the total amount of duty? Dollars, to cents. {~r2 (372.40 + 3724 * .23 * .11) * .9}
q: A dry-goods merchant imports 1120 yards of dress goods, 1 1/4 yd. wide, invoiced at 23 ct. a sq. yd.; there is a specific duty of 8 ct. per sq. yd., and an ad valorem duty of 40%: what must he charge per yard, cloth measure, to clear 25% on the whole? Cents? {~(1400 * .23 * 1.4 + 1400 * .08) * 1.25 / 1120 * 100}
q: If goods invoiced at $3684.50 pay a duty of $1473.80, what is the rate of duty? {40}
q: If laces invoiced at $7618.75, cost, when landed, $10285.31 1/4, what is the rate of duty? {~(10285.3125 - 7618.75) / 7618.75 * 100}
q: Forty hhd. (63 gal. each) of molasses, invoiced at 52 ct. a gallon, pay $453.60 duty. The specific duty is 5 ct. a gallon: what is the additional ad valorem duty? {25 @ (453.60 - 126) / 1310.40 * 100}
q: Paid $575.80 duty on watches, at 25%: at what were they invoiced? Dollars? {2303.20|2303.2 @ 575.80 / .25} What did they cost me in store? {2879}
q: The duty on 1800 yards of silk was $2970, at 60% ad valorem: what was the invoice price per yard? Dollars? {2.75 @ 2970 / .6 / 1800} What must I charge per yard to clear 20%? {5.28 @ (2970 + 4950) * 1.2 / 1800}
q: The duty on 15 gross, qt. bottles of porter, at a tax of 35 ct. a gallon, was $151.20: if this were equivalent to an ad valorem duty of 20 5/26% on the entire purchase, how many bottles were allowed for a gallon? {5 @ 2160 / (151.20 / .35)} At how much per bottle must the whole be sold to clear 20%? Cents? {50 @ (151.20 / (525/2600) + 151.20) * 1.2 / 2160 * 100}
q: 1000 boxes (100 each) of cigars, weighing 1200 lb., net, cost in store $13675. There is a specific duty of $2.50 per lb., an ad valorem duty of 25%, and an internal revenue tax of 60 ct. a box: freight and charges amount to $75; find the invoice price per thousand cigars. Dollars? {80 @ (13675 - 3000 - 600 - 75) / 1.25 / 100}
q: Supposing No. 1 pig-iron, American manufacture, to be of equal quality with Scotch pig-iron: at what price must the latter be invoiced, to compete in our markets, if American iron sells for $45 a ton; freight and charges amounting to $10 a ton, and the specific duty being equivalent, in this instance, to an ad valorem duty of 25%? Dollars a ton? {28 @ 35 / 1.25}
q: A marble-cutter imports a block of marble 6 ft. 6 in. long, 3 ft. wide, 2 ft. 9 3/5 in. thick; the whole cost to him being $130; he pays a specific duty of 50 ct. per cu. ft. and an ad valorem duty of 20%: freight and charges being $20.80, what was the invoice price per cu. ft.? Dollars? {1.25 @ (130 - 20.80 - 54.6 * .5) / 1.2 / 54.6}

== 50 | Interest | Simple interest
note: Interest = Principal x Rate x Time (in years). Unless otherwise said, 30 days make a month and 360 days a year. Six per cent method: take 6 cents for every year, 1/2 cent for every month, and 1/6 of a mill for every day; that is the interest on $1, which times the principal is the interest. Answer to the cent.
q: What is the interest of $320 for 3 yr. 5 mo. 18 da., at 4%? Dollars? {~m 320 * .04 * (3 + 5/12 + 18/360)}
sol: The interest of $320 for 1 year, at 4%, is $12.80; 5 mo. 18 da. = 7/15 of a year; for 3 7/15 years it is $12.80 x 3 7/15 = $44.37 1/3.
q: What is the simple interest of $354.80 for 3 yr. 7 mo. 19 da., at 6%? Dollars? {~m 354.80 * .06 * (3 + 7/12 + 19/360)}
sol: By aliquot parts: 1 yr., $21.288; 3 yr., $63.864; 6 mo. = 1/2, $10.644; 1 mo., $1.774; 18 da., $1.064; 1 da., $.059. The sum is $77.41.
q: What is the interest of $560 for 3 yr. 8 mo. 12 da., at 6%? Dollars? {124.32 @ 560 * .222}
sol: On $1: 18 ct. for 3 yr., 4 ct. for 8 mo., 2 mills for 12 da.: $.222. $560 x .222 = $124.32.
q: Find the simple interest of $178.63 for 2 yr. 5 mo. 26 da., at 7%. Dollars? {~m 178.63 * 7 / 100 * (2 + 5/12 + 26/360)}
q: Find the simple interest of $6084.25 for 1 yr. 3 mo., at 4 1/2%. Dollars? {~m 6084.25 * 4_1/2 / 100 * (1 + 3/12)}
q: Find the simple interest of $64.30 for 1 yr. 10 mo. 14 da., at 9%. Dollars? {~m 64.30 * 9 / 100 * (1 + 10/12 + 14/360)}
q: Find the simple interest of $1052.80 for 28 da., at 10%. Dollars? {~m 1052.80 * 10 / 100 * (28/360)}
q: Find the simple interest of $419.10 for 8 mo. 16 da., at 6%. Dollars? {~m 419.10 * 6 / 100 * (8/12 + 16/360)}
q: Find the simple interest of $1461.85 for 6 yr. 7 mo. 4 da., at 10%. Dollars? {~m 1461.85 * 10 / 100 * (6 + 7/12 + 4/360)}
q: Find the simple interest of $2601.50 for 72 da., at 7 1/2%. Dollars? {~m 2601.50 * 7_1/2 / 100 * (72/360)}
q: Find the simple interest of $8722.43 for 5 1/2 yr., at 6%. Dollars? {~m 8722.43 * 6 / 100 * (5_1/2)}
q: Find the simple interest of $326.50 for 1 mo. 8 da., at 8%. Dollars? {~m 326.50 * 8 / 100 * (1/12 + 8/360)}
q: Find the simple interest of $1106.70 for 4 yr. 1 mo. 1 da., at 6%. Dollars? {~m 1106.70 * 6 / 100 * (4 + 1/12 + 1/360)}
q: Find the simple interest of $10000 for 1 da., at 6%. Dollars? {~m 10000 * 6 / 100 * (1/360)}
q: Find the simple interest of $1532.45 for 9 yr. 2 mo. 7 da., at 12%. Dollars? {~m 1532.45 * 12 / 100 * (9 + 2/12 + 7/360)}
q: Find the simple interest of $78084.50 for 2 yr. 4 mo. 29 da., at 18%. Dollars? {~m 78084.50 * 18 / 100 * (2 + 4/12 + 29/360)}
q: Find the simple interest of $512.60 for 8 mo. 18 da., at 7%. Dollars? {~m 512.60 * 7 / 100 * (8/12 + 18/360)}
q: Find the simple interest of $6919.32 for 7 yr. 6 mo., at 6%. Dollars? {~m 6919.32 * 6 / 100 * (7_1/2)}
q: Find the simple interest of $990.73 for 9 mo. 19 da., at 7%. Dollars? {~m 990.73 * 7 / 100 * (9/12 + 19/360)}
q: Find the simple interest of $4642.68 for 5 mo. 17 da., at 15%. Dollars? {~m 4642.68 * 15 / 100 * (5/12 + 17/360)}
q: Find the simple interest of $13024 for 9 mo. 13 da., at 10%. Dollars? {~m 13024 * 10 / 100 * (9/12 + 13/360)}
q: Find the simple interest of $615.38 for 4 yr. 11 mo. 6 da., at 20%. Dollars? {~m 615.38 * 20 / 100 * (4 + 11/12 + 6/360)}
q: Find the simple interest of $2066.19 for 3 yr. 6 mo. 2 da., at 30%. Dollars? {~m 2066.19 * 30 / 100 * (3 + 6/12 + 2/360)}
q: Find the simple interest of $92.55 for 3 mo. 22 da., at 5%. Dollars? {~m 92.55 * 5 / 100 * (3/12 + 22/360)}
q: Find the simple interest of $1363.20 for 39 da., at 1 1/4% a month. Dollars? {~m 1363.20 * .0125 * 39 / 30}
q: Find the simple interest of $402.50 for 100 da., at 2% a month. Dollars? {~m 402.50 * .02 * 100 / 30}
q: Find the amount of $757.35 for 117 da., at 1 1/2% a month. Dollars? {~m 757.35 * (1 + .015 * 117 / 30)}
q: Find the amount of $1883 for 1 yr. 4 mo. 21 da., at 6%. Dollars? {~m 1883 * (1 + .06 * (1 + 4/12 + 21/360))}
q: Find the amount of $262.70 for 53 da., at 1% a month. Dollars? {~m 262.70 * (1 + .01 * 53 / 30)}
q: Find the amount of $584.48 for 133 da., at 7 1/2%. Dollars? {~m 584.48 * (1 + .075 * 133 / 360)}
q: Find the amount of $392.28 for 71 da., at 2 1/2% a month. Dollars? {~m 392.28 * (1 + .025 * 71 / 30)}
q: Find the interest of $7302.85 for 365 da., at 6%, counting 360 da. to a year. Dollars? {~m 7302.85 * .06 * 365 / 360}
q: If I borrow $1000000 in New York, at 7%, and lend it at 7% in Ohio, what do I gain in 180 da.? (In New York 365 days are counted a year instead of 360.) Dollars? {~m 1000000 * .07 * 180 * (1 / 360 - 1 / 365)}
q: Find the interest of $5064.30 for 7 mo. 12 da., at 7%, in New York (222 days, 365 to the year). Dollars? {~m 5064.30 * .07 * 222 / 365}
q: If I borrow $12500 at 6%, and lend it at 10%, what do I gain in 3 yr. 4 mo. 4 da.? Dollars? {~m 12500 * .04 * (3 + 4/12 + 4/360)}
q: If $4603.15 is loaned July 17, 1881, at 7%, what is due March 8, 1883 (1 yr. 7 mo. 19 da.)? Dollars? {~m 4603.15 * (1 + .07 * (1 + 7/12 + 19/360))}
sol: When the days in the subtrahend exceed those in the minuend, take the months one less than the count, and count the days from the month before the later date: 1 yr. 7 mo., and from February 17th to March 8th, 19 da.
q: Find the interest, at 8%, of $13682.45, borrowed from a minor 13 yr. 2 mo. 10 da. old, and retained till he is of age (21 years). Dollars? {~m 13682.45 * .08 * (7 + 9/12 + 20/360)}
q: In one year a broker loans $876459.50 for 63 da., at 1 1/2% a mo., and pays 6% on $106525.20 deposits: what is his gain? Dollars? {~m 876459.50 * .015 * 63 / 30 - 106525.20 * .06}
q: What is a broker's gain in 1 yr., on $100, deposited at 6%, and loaned 11 times for 33 da., at the rate of 2% a month? Dollars? {18.20|18.2 @ 100 * .02 * 33 / 30 * 11 - 6}
q: Find the simple interest of 493 pounds 16s. 8d. for 1 yr. 8 mo., at 6%. Pounds? {~p 20,12 # 0 # 118520 * .06 * 5/3} Shillings? {~p # 1 # $} Pence? {~p # 2 # $}
q: Find the simple interest of 24 pounds 18s. 9d. for 10 mo., at 6%. Pounds? {~p 20,12 # 0 # 5985 * .06 * 10/12} Shillings? {~p # 1 # $} Pence? {~p # 2 # $}
q: Of 25 pounds for 1 yr. 9 mo., at 5%. Pounds? {~p 20,12 # 0 # 6000 * .05 * 1_3/4} Shillings? {~p # 1 # $} Pence? {~p # 2 # $}
q: Of 648 pounds 15s. 6d. from June 2 to November 25 (176 days; in England the actual number of days is counted), at 5%. Pounds? {~p 20,12 # 0 # 155706 * .05 * 176 / 365} Shillings? {~p # 1 # $} Pence? {~p # 2 # $}

== 51 | Interest | Finding the time, rate and principal
note: Time = Interest / (Principal x Rate). Rate = Interest / (interest of the principal for the time at 1%). Principal = Interest / (Rate x Time), or Amount / (amount of $1 for the time). A part of a day is left out of the answer.
q: John Thomas loaned $480, at 5%, till the interest was $150; required the time. Years? {~150 / (480 * .05)}
q: In what time will $1200 amount to $1800, at 10%? Years? {5}
q: In what time will $415.50 amount to $470.90, at 10%? Yr.? {1} Mo.? {4}
q: In what time will $3703.92 amount to $4122.15, at 8%? Yr.? {1} Mo.? {4} Da.? {28}
q: In what time will any sum double itself by simple interest, at 4 1/2%? Years? {~100 / 4_1/2} At 6? {~100 / 6} At 7 1/2? {~100 / 7_1/2} At 9? {~100 / 9} At 10? {10} At 12? {~100 / 12} At 20? {5} At 25? {4} At 30? {~100 / 30}
q: In what time will any sum treble itself by simple interest, at 4%? Years? {50} At 10%? {20} At 12%? {~200 / 12}
q: How long must I keep on deposit $1374.50, at 10%, to pay a debt of $1480.78? Mo.? {9} Da.? {8}
q: How long will it take $3642.08 to amount to $4007.54, at 12%? Mo.? {10} Da.? {1}
q: How long would it take $175.12 to produce $6.43 interest, at 6%? Mo.? {7} Da.? {10}
q: How long would it take $415.38 to produce $10.69 interest in New York, at 7%? Days? {134}
q: At what rate per cent will $4800 gain $840 interest in 2 1/2 years? {7 @ 840 / (4800 * .01 * 2_1/2)}
q: At what rate per annum will any sum treble itself, at simple interest, in 5 years? Per cent? {40} In 10? {20} In 15? {~200 / 15} In 20? {10} In 25? {8} In 30? {~200 / 30}
q: At what rate of interest per annum will any sum quadruple itself, at simple interest, in 6 years? Per cent? {50} In 12? {25} In 18? {~300 / 18} In 24? {~300 / 24} In 30? {10}
q: What is the rate of interest when $35000 yields an income of $175 a month? Per cent? {6 @ 175 * 12 / 35000 * 100}
q: What is the rate of interest when $29200 produces $6.40 a day? Per cent? {8 @ 6.40 * 365 / 29200 * 100}
q: What is the rate of interest when $12624.80 draws $315.62 interest quarterly? Per cent? {~r0 315.62 * 4 / 12624.80 * 100}
q: Find the rate when stock, bought at 40% discount, yields a semi-annual dividend of 5%. Per cent? {~10 / 60 * 100}
q: A house that cost $8250, rents for $750 a year; the insurance is 6/10%, and the repairs 1/2%, every year: what rate of interest does it pay, to the nearest whole per cent? {~r0 (750 - 8250 * .011) / 8250 * 100}
q: A man receives $490.84 interest annually on a mortgage, at 7%: what is the amount of the mortgage? Dollars? {7012 @ 490.84 / .07}
q: What principal will produce $1500 a year, at 6%? Dollars? {25000}
q: What principal will produce $1830 in 2 yr. 6 mo., at 5%? Dollars? {14640 @ 1830 / .125}
q: What principal will produce $45 a mo., at 9%? Dollars? {6000 @ 45 * 12 / .09}
q: What principal will produce $17 in 68 da., at 1% a month? Dollars? {750 @ 17 / (.01 * 68 / 30)}
q: What principal will produce $656.25 in 9 mo., at 3 1/2%? Dollars? {25000 @ 656.25 / (.035 * 9/12)}
q: What principal will produce $86.15 in 9 mo. 11 da., at 10%? Dollars? {~m 86.15 / (.1 * 281 / 360)}
q: What principal will produce $313.24 in 112 da., at 7%? Dollars? {~m 313.24 / (.07 * 112 / 360)}
q: What principal will produce $146.05 in 7 mo. 14 da., at 6%? Dollars? {~m 146.05 / (.06 * 224 / 360)}
q: What principal will produce $58.78 in 1 yr. 3 mo. 20 da., at 4%? Dollars? {~m 58.78 / (.04 * (1 + 3/12 + 20/360))}
q: What principal will produce $79.12 in 5 mo. 25 da., at 7% in N. Y.? Dollars? {2325.16|2357.46|2357.45|2325.15|2325.17}
q: What is the par value of a bond which, in 8 yr. 8 mo., at 6%, will amount to $15200? Dollars? {10000 @ 15200 / 1.52}
q: What principal in 2 yr. 3 mo. 12 da., at 6%, will amount to $1367.84? Dollars? {~m 1367.84 / 1.137}
q: What principal in 10 mo. 26 da., will amount to $2718.96, at 10% interest? Dollars? {~m 2718.96 / (1 + .1 * 326 / 360)}
q: What principal, at 4 1/2%, will amount to $4613.36 in 3 yr. 1 mo. 7 da.? Dollars? {~m 4613.36 / (1 + .045 * (3 + 1/12 + 7/360))}
q: What principal, at 7%, will amount to $562.07 in 79 da. (365 da. to a year)? Dollars? {~m 562.07 / (1 + .07 * 79 / 365)}

== 52 | Interest | Notes and annual interest
note: A note payable a certain time after date is legally due on the last of three days of grace after the time named. Annual interest is interest on the principal, and on each annual interest after it is due, at simple interest.
q: $560.60. Chattanooga, Tenn., June 3, 1872. Sixty days after date, I promise to pay Madison Wilcox five hundred and sixty 60/100 dollars, with interest at 7%. On what day of August is it legally due? {5} What is its amount? Dollars? {~m 560.60 * (1 + .07 * 63 / 360)}
sol: 60 days and 3 days of grace from June 3 is August 5; the time is 2 mo. 3 da.
q: $430.00. Topeka, Kan., Jan. 30, 1879. Six months after date, I promise to pay Christine Ladd four hundred and thirty dollars, for value received, with interest at 12% per annum. On what day of August is it legally due? {2} What is its amount? Dollars? {~m 430 * (1 + .12 * (6/12 + 3/360))}
sol: Six months from January 30 gives July 30; 3 days of grace bring it to August 2. The time is 6 mo. 3 da.
q: $4650.80. St. Louis, Mo., August 10, 1875. Three months after date, I promise to pay Oliver Davis four thousand six hundred and fifty 80/100 dollars, with interest at the rate of 10% per annum. On what day of November is it legally due? {13} What is its amount? Dollars? {~m 4650.80 * (1 + .1 * (3/12 + 3/360))}
q: Find the amount of $10500 for 4 yr. 6 mo., interest 6%, payable annually. Dollars? {13637.40|13637.4 @ 10500 + 2835 + 302.40}
sol: The interest of $10500 for 4 1/2 yr. is $2835. The annual interests, $630 each, draw interest 3 1/2, 2 1/2, 1 1/2 and 1/2 years, the same as one for 8 years: $302.40. The amount is $13637.40.
q: Find the amount of a note for $1500, interest 6%, payable annually, given Sept. 3, 1870, and not paid till March 1, 1874. Dollars? {1838.12 @ 1500 + 314 + 24.12}
sol: 3 yr. 5 mo. 26 da.: interest $314.00. The annual interests of $90 draw interest for 2 yr. 5 mo. 28 da., 1 yr. 5 mo. 28 da. and 5 mo. 28 da., in all 4 yr. 5 mo. 24 da.: $24.12.
q: A gentleman holds six $1000 railroad bonds, due in 3 years, interest 6%, payable semi-annually: no interest having been paid, what amount is owing him when the bonds mature? Dollars? {7161 @ 6000 + 1080 + 180 * .03 * 15}
q: $2500. St. Paul, Minn., Jan. 11, 1869. For value received, I promise to pay Morgan Stuart, or order, twenty-five hundred dollars, with interest at 7%, payable annually. What was due on this note March 17, 1873, if the interest was paid the first two years? Dollars? {~m 2500 + 2500 * .07 * (2 + 2/12 + 6/360) + 175 * .07 * (1 + 4/12 + 12/360)}

== 53 | Interest | Partial payments
note: United States Rule: apply each payment first to the interest due; the surplus goes to the principal, and interest is then reckoned on the balance. If a payment is less than the interest, interest continues on the former principal until the payments together exceed the interest due. Find the time by years and months, then count the days.
q: $850. Cincinnati, April 29, 1880. Ninety days after date, I promise to pay Stephen Ludlow, or order, eight hundred and fifty dollars, with interest; value received. Charles K. Taylor. Indorsements: Oct. 13, 1880, $40; June 9, 1881, $32; Aug. 21, 1881, $125; Dec. 1, 1881, $10; March 16, 1882, $80. What was due Nov. 11, 1882, at 6%? Dollars? {679.43|679.42|679.44}
sol: Interest to Oct. 13, 1880 (5 mo. 14 da.), $23.233; less $40 leaves $833.233. Interest to June 9, 1881, $32.913, is more than the $32 paid, so interest runs on to Aug. 21: in all $10.912 due; with $833.233, less $125, leaves $719.145. The $10 of Dec. 1 does not meet the interest; to March 16, 1882 the interest is $14.571; less $80 leaves $653.716. Interest to Nov. 11, 1882, $25.713: due $679.43.
q: $304.75. Chicago, March 10, 1882. For value received, six months after date, I promise to pay G. Riley, or order, three hundred and four 75/100 dollars. H. McMakin. No payments. What was due Nov. 3, 1883 (interest at 6% from the day of maturity, Sept. 13, 1882)? Dollars? {325.63|325.62|325.64}
q: $429.30. Indianapolis, April 13, 1873. On demand, I promise to pay W. Morgan, or order, four hundred and twenty-nine 30/100 dollars, value received, with interest (6%). R. Wilson. Indorsed: Oct. 2, 1873, $10; Dec. 8, 1873, $60; July 17, 1874, $200. What was due Jan. 1, 1875? Dollars? {195.06|195.05|195.07}
q: $1750. New York, Nov. 22, 1872. For value received, two years after date, I promise to pay to the order of Spencer & Ward, seventeen hundred and fifty dollars, with interest at 7 per cent. Jacob Winston. Indorsed: Nov. 25, 1874, $500; July 18, 1875, $50; Sept. 1, 1875, $600; Dec. 28, 1875, $75. What was due Feb. 10, 1876? Dollars? {879.98|879.97|879.99}
q: A note of $312 given April 1, 1872, 8% from date, was settled July 1, 1874, the exact sum due being $304.98. Indorsed: April 1, 1873, $30.96; Oct. 1, 1873, $--; April 1, 1874, $20.40. Restore the lost figures of the second payment. Dollars? {11.08}
sol: The first payment brought the principal to $306. A second payment that met the interest would have left too small a sum at the end, so it was less than the interest and was applied with the $20.40 on April 1, 1874. The final principal was $304.98 less 3 months' interest: $299. $306 amounted then to $330.48, so $31.48 was paid in all, and $31.48 - $20.40 = $11.08.
q: There have been two equal annual payments on a 6% note for $175, given two years ago this day. The balance is $154.40: what was each payment? Dollars? {20.50|20.5}
sol: Each dollar put on the principal the first time saves 6 ct. of interest the second time, so $2.06 goes on the principal in all for each $1 the first time. $175 - $154.40 = $20.60, so $10 went on the principal first; the first interest was $10.50, and each payment $20.50.
q: A merchant gives his note, 10% from date, for $2442.04: what sum paid annually will have discharged the whole at the end of 5 years? Dollars? {644.20|644.204|644.2}
sol: Each $1 paid on the principal the first year lets $1.10 go on it the next, then $1.21, $1.331, $1.4641: $6.1051 in all. $2442.04 / 6.1051 = $400, the first payment on the principal; with the first year's interest, $244.204, the payment is $644.204.
note: Connecticut Rule: if a year or more has passed, add a year's interest to the principal and deduct the payment; if a payment comes before a year's interest has accrued, add a year's interest to the principal, and the interest on the payment to the end of that year to the payment, and deduct. A payment less than the interest then due draws no interest. Vermont Rule: on notes with interest payable annually, unpaid annual interests draw simple interest, and payments draw interest to the end of the year.
q: What is due on the Indianapolis note ($429.30, April 13, 1873) by the Connecticut rule? Dollars? {194.88|194.87|194.89}
q: What is due on the New York note ($1750, Nov. 22, 1872) by the Connecticut rule? Dollars? {879.22|879.21|879.23}
q: $1480. Woodstock, Vt., April 12, 1879. For value received, I promise to pay John Jay, or order, fourteen hundred and eighty dollars, with interest annually (6%). James Brown. Indorsed: July 25, 1879, $40; May 20, 1880, $50; June 3, 1881, $350. What was due April 12, 1882, by the Vermont rule? Dollars? {1291.95|1291.94|1291.96}
note: Mercantile Rule, for notes and accounts within a year: find the amount of the principal to the day of settlement, and the amount of each payment from its date; subtract the sum of the payments' amounts. Count the exact number of days.
q: $950.00. New York, Jan. 25, 1876. For value received, nine months after date, I promise to pay Peter Finley nine hundred and fifty dollars, with 7% interest. Thomas Hunter. The following payments were made: March 2, 1876, $225; May 5, 1876, $174.19; June 29, 1876, $187.50; Aug. 1, 1876, $79.15. Required the balance at settlement (Oct. 28, 1876). Dollars? {312.63|312.62|312.64}
q: A note for $600 was given June 12, 1865, 6% interest, and the following indorsements were made: Aug. 12, 1865, $100; Nov. 12, 1865, $250; Jan. 12, 1866, $120: what was due Feb. 12, 1866, counting by months instead of days? Dollars? {146.65 @ 600 * 1.04 - 100 * 1.03 - 250 * 1.015 - 120 * 1.005}

== 54 | Interest | True discount
note: The present worth of a debt is the sum which, at the given rate, will amount to the debt at its maturity. The true discount is the debt less its present worth. Present worth = debt / amount of $1 for the time and rate.
q: Find the present worth of $5101.75, due in 1 yr. 9 mo. 19 da., at 6%. Dollars? {~m 5101.75 / (1 + .06 * (1 + 9/12 + 19/360))} The true discount? {~m 5101.75 - 5101.75 / (1 + .06 * (1 + 9/12 + 19/360))}
sol: The amount of $1 for the time is $1.108 1/6; $5101.75 / 1.108 1/6 = $4603.775, the present worth; $5101.75 - $4603.775 = $497.975, the true discount.
q: Find the true discount on a debt for $5034.15 due in 3 yr. 5 mo. 20 da., without grace, at 7%. Dollars? {~m 5034.15 - 5034.15 / (1 + .07 * (3 + 5/12 + 20/360))}
q: What is the present worth of a note for $2500, bearing 6% interest, and payable in 2 yr. 6 mo. 15 da. (and 3 days of grace), discounted at 8%? Dollars? {~m 2500 * (1 + .06 * (2 + 6/12 + 18/360)) / (1 + .08 * (2 + 6/12 + 18/360))}

== 55 | Interest | Bank discount
note: Bank discount is simple interest on the face of a note, for the time it has to run plus three days of grace, paid in advance; the proceeds are the face less the discount. Count the exact days from the day of discount to the day of maturity.
q: What is the bank discount of $770 for 90 days, at 6%? Dollars? {11.935|11.94|11.93 @ 770 * .06 * 93 / 360} The proceeds? {758.065|758.07|758.06}
q: $792.50. Quincy, Ill., Jan. 3, 1870. Six months after date, I promise to pay to the order of Jones Brothers seven hundred and ninety-two 50/100 dollars, at the First National Bank, value received. Discounted Feb. 18, at 6%. Due July 3/6: how many days to run? {138} The proceeds? Dollars? {~m 792.50 * (1 - .06 * 138 / 360)}
q: $1962.45. New York, July 26, 1879. Value received, four months after date, I promise to pay B. Thoms, or order, at the Chemical National Bank. Discounted Aug. 26, at 7%. Due Nov. 26/29: how many days to run? {95} The proceeds? Dollars? {1926.20|1926.70|1926.19|1926.21}
sol: The bank discount of $1 for 95 days at 7% is $.018472; on $1962.45 it is $36.25; the proceeds are $1926.20. (The book prints $1926.70.)
q: $2672.18. Philadelphia, March 10, 1872. Nine months after date, for value received, I promise to pay Edward H. King. Discounted July 19, at 6%. Due Dec. 10/13: how many days to run? {148} The proceeds? Dollars? {~m 2672.18 * (1 - .06 * 148 / 360)}
q: $3886. St. Louis, Jan. 31, 1879. One month after date, we jointly and severally promise to pay C. McKnight. Discounted Jan. 31, at 1 1/2% a month. Due Feb. 28/Mar. 3: how many days to run? {32} The proceeds? Dollars? {~m 3886 * (1 - .015 * 32 / 30)}
sol: In Missouri the day of discount and the day of maturity are both counted: 32 days. The discount of $1 at 1 1/2% a month is $.016; on $3886, $62.18; proceeds $3823.82.
q: $2850. Austin, Tex., April 11, 1879. For value received, eight months after date, we promise to pay Henry Hopper, or order, twenty-eight hundred and fifty dollars, with interest from date, at six per cent per annum. Discounted June 15, at 6%. How many days to run? {182} The proceeds? Dollars? {~m 2850 * (1 + .06 * (8/12 + 3/360)) * (1 - .06 * 182 / 360)}
sol: The face to be discounted is $2850 with interest for 8 mo. 3 da.: $2965.425. The discount for 182 days is $89.951; proceeds $2875.47.
q: $737.40. Boston, Feb. 14, 1880. Value received, two months after date, I promise to pay to J. K. Eaton, or order. Discounted Feb. 23, at 10%. How many days to run? {54} The proceeds? Dollars? {~m 737.40 * (1 - .1 * 54 / 360)}
q: $4085.20. New Orleans, Nov. 20, 1875. Value received, six months after date. Discounted Dec. 31, 1875, at 5%. How many days to run? {144} The proceeds? Dollars? {~m 4085.20 * (1 - .05 * 144 / 360)}
q: For what sum must a 60 da. note be drawn, to yield $1000, when discounted, at 6% per annum? Dollars? {~m 1000 / (1 - .06 * 63 / 360)}
sol: The bank discount of $1 for 63 da., at 6%, is $.0105; the proceeds of $1, $.9895; $1000 / .9895 = $1010.61, the face of the note.
q: Find the face of a 30 da. note, which yields $1650, when discounted at 1 1/2% a mo. Dollars? {~m 1650 / (1 - .015 * 33 / 30)}
q: The face of a 60 da. note, which, discounted at 6% per annum, will yield $800. Dollars? {~m 800 / .9895}
q: The face of a 90 da. note, bought for $22.75 less than its face, discounted at 7%. Dollars? {~m 22.75 / (.07 * 93 / 360)}
q: The face of a 4 mo. note, which, discounted at 1% a month, yields $3375. Dollars? {~m 3375 / (1 - .01 * (4 + 3/30))}
q: The face of a 6 mo. note, which, discounted at 10% per annum, yields $4850. Dollars? {~m 4850 / (1 - .1 * (6/12 + 3/360))}
q: The face of a 60 da. note, discounted at 2% a month, to pay $768.25. Dollars? {~m 768.25 / (1 - .02 * 63 / 30)}
q: The face of a 40 da. note, which, discounted at 8%, yields $2072.60. Dollars? {~m 2072.60 / (1 - .08 * 43 / 360)}
q: The face of a 30 da. note, to net $1000 when discounted at 6%. Dollars? {~m 1000 / (1 - .06 * 33 / 360)} Of a 90 da. note? {~m 1000 / (1 - .06 * 93 / 360)}
note: The rate of interest a discount brings is the discount divided by the interest of the proceeds at 1% for the time. The rate of discount that brings a given rate of interest is the interest divided by the interest of the amount at 1% for the time.
q: What is the rate of interest when a 60 da. note is discounted at 2% a month? Per cent? {~4.20 / (95.80 * .00175)}
sol: On $100 for 63 days the discount is $4.20, the proceeds $95.80. The interest of $1 at 1% for the time is $.00175, of $95.80 it is $.16765. $4.20 / $.16765 = 25 25/479.
q: What is the rate of interest when a 30 da. note is discounted at 1% a month? Per cent? {~12 / (1 - .12 * 33 / 360)} At 1 1/4%? {~15 / (1 - .15 * 33 / 360)} At 1 1/2%? {~18 / (1 - .18 * 33 / 360)} At 2%? {~24 / (1 - .24 * 33 / 360)}
q: What is the rate of interest when a 60 da. note is discounted at 6% per annum? Per cent? {~6 / (1 - .06 * 63 / 360)} At 8%? {~8 / (1 - .08 * 63 / 360)} At 10%? {~10 / (1 - .1 * 63 / 360)}
q: What is the rate of interest when a 90 da. note is discounted at 2% a month? Per cent? {~24 / (1 - .24 * 93 / 360)} At 2 1/2%? {~30 / (1 - .3 * 93 / 360)} At 3%? {~36 / (1 - .36 * 93 / 360)}
q: What is the rate of interest when a note running 1 yr. without grace, is discounted at 5%? Per cent? {~5 / .95} 6%? {~6 / .94} 7%? {~7 / .93} 8%? {~8 / .92} 9%? {~9 / .91} 10%? {~10 / .9} 12%? {~12 / .88}
q: My note, which will be legally due in 1 yr. 4 mo. 20 da., is discounted by a banker, at 8%: what rate of interest does he receive? Per cent? {9 @ 8 / (1 - .08 * (1 + 4/12 + 20/360))}
q: What is the rate of discount when a 60 day note yields 2% interest a month? Per cent? {~4.20 / (104.20 * .00175)}
sol: On $100 for 63 days the interest is $4.20, the amount $104.20; the interest of the amount at 1% for the time is $.18235. $4.20 / $.18235 = 23 17/521.
q: What rate of discount on 30 da. notes yields 10% interest? Per cent? {~10 / (1 + .1 * 33 / 360)} 15%? {~15 / (1 + .15 * 33 / 360)} 20%? {~20 / (1 + .2 * 33 / 360)}
q: What rate of discount on 60 da. notes yields 6% interest? Per cent? {~6 / (1 + .06 * 63 / 360)} 8%? {~8 / (1 + .08 * 63 / 360)} 10%? {~10 / (1 + .1 * 63 / 360)}
q: What rate of discount on 90 da. notes yields 1% a month interest? Per cent a year? {~12 / (1 + .12 * 93 / 360)} 2% a month? {~24 / (1 + .24 * 93 / 360)} 4% a month? {~48 / (1 + .48 * 93 / 360)}
q: What rate of discount on notes running 1 yr. without grace yields 5% interest? Per cent? {~5 / 1.05} 6%? {~6 / 1.06} 7%? {~7 / 1.07} 8%? {~8 / 1.08} 9%? {~9 / 1.09} 10%? {~10 / 1.1}

== 56 | Exchange | Domestic and foreign exchange
note: Exchange is paying a debt in a distant place by transferring a credit, with a bill of exchange (a draft or check). The rate of exchange is a per cent premium or discount on the face. A time draft is also subject to bank discount for its time and three days of grace. Answer to the cent.
q: What is paid for $3805.40 sight exchange on Boston, at 1/2% premium? Dollars? {~m 3805.40 * 1.005}
q: What is the cost of a check on St. Louis for $1505.40, at 1/4% discount? Dollars? {~m 1505.40 * .9975}
q: What must be the face of a sight draft to cost $2000, at 5/8% premium? Dollars? {~m 2000 / 1.00625}
q: What must be the face of a sight draft to cost $4681.25, at 1 1/4% discount? Dollars? {~m 4681.25 / .9875}
q: What will a 30 day draft on New Orleans for $7216.85 cost, at 3/8% discount, interest 6%? Dollars? {~m 7216.85 * .99075}
sol: From the rate of exchange, $.99625, take the bank discount of $1 for 33 days, $.0055: $.99075 is the cost of exchange for $1. $7216.85 x .99075 = $7150.09.
q: What will a 60 day draft on New York for $12692.50 cost, at 3/4% premium, interest 6%? Dollars? {12654.42|12654.41|12654.43}
q: What must be the face of an 18 day draft, costing $5264.15, at 1/2% premium, interest 6%? Dollars? {5256.27|5256.26|5256.28}
q: What must be the face of a 21 day draft, costing $6836.75, at 7/8% discount, interest 6%? Dollars? {6925.04|6925.03|6925.05}
q: A commission merchant in St. Louis sold 5560 lb. of bacon, at 11 1/2 ct. a lb.; his commission is 2 1/2%, and the course of exchange 98 1/2%: what is the amount of the draft that he remits to his consignor? Dollars? {~m 639.40 * .975 / .985}
q: A grain merchant in Toledo sold 11875 bu. of corn, at 40 cents a bushel; deducting 3% as commission, he purchased a 60 day draft with the proceeds, at 2% premium; required the face of the draft. Dollars? {~m 4607.50 / 1.0095}
q: Sold lumber to the amount of $20312.50, charging 1 1/2% commission, and remitted the proceeds to my consignor by draft; required the face of the draft, exchange 1/2% discount. Dollars? {~m 20312.50 * .985 / .995}
q: If a 45 day draft for $5500 costs $5538.50, find the rate of exchange, in per cent premium. {~(5538.50 + 44 - 5500) / 5500 * 100}
sol: The bank discount of $5500 for 48 days at 6% is $44. $5538.50 + $44 - $5500 = $82.50 premium; $82.50 / $5500 = 1 1/2%.
q: A father sent to his son, at school, a draft for $250, at 3 mo., interest 6%, paying $244.25 for it; find the rate of exchange, in per cent discount. {~(250 - 3.875 - 244.25) / 250 * 100}
q: An agent owing his principal $1011.84, bought a draft with this sum and remitted it; the principal received $992: find the rate of exchange, in per cent premium. {2 @ (1011.84 - 992) / 992 * 100}
q: Find the cost of a bill of exchange on London, at 3 days sight, for 530 pounds 12s., exchange being quoted at $4.88. Dollars? {~m 4.88 * 530.6}
q: Find the cost of a bill of exchange on London, at sight, for 625 pounds 10s. 10d., when exchange is $4.87 a pound sterling. Dollars? {~m 4.87 * (625 + 130/240)}
q: What is the cost of a bill on Paris for 1485 francs, $1 = 5.15 francs? Dollars? {~m 1485 / 5.15}
q: What is the cost of a bill on Amsterdam for 4800 guilders, quoted at 41 1/2 cents, brokerage 1/2%? Dollars? {2001.96 @ 4800 * .415 * 1.005}
q: What will a draft in St. Petersburg on New York for $5000 cost, if a rouble be worth $.74? Roubles? {~m 5000 / .74}
q: What will a draft on Charleston for $4500 cost at Rio Janeiro (milreis = $.54), discount 2%? Milreis? {~m 4500 / .54 * .98}
q: A gentleman sold a 60 day draft, which was drawn on Amsterdam for 1000 guilders; he discounted it at 6%, and brokerage was 1/8%: what did he get for it, a guilder being valued at 40 1/4 cents? Dollars? {397.77|397.76|397.78}

== 57 | Exchange | Arbitration of exchange
note: Circular exchange draws through one or more places instead of directly. By the chain rule, write equations so that the right-hand quantity of each and the left-hand quantity of the next are of the same kind; divide the product of the full column by the product of the other.
q: A merchant wishes to remit $2240 to Lisbon: buying a bill directly on Lisbon, at 1 milreis = $1.10, how many milreis does he get? To three places. {~r3 2240 / 1.1} Through London and Paris, at 1 pound = $4.88 = 25.20 francs, and 1 milreis = 5.5 francs? {~r3 2240 * 25.20 / (4.88 * 5.5)} What is the gain, in milreis? {66.765|66.76|66.77}
q: A merchant in Memphis wishes to remit $8400 to New York; exchange on Chicago is 1 1/2% premium, between Chicago and Detroit 1% discount, and between Detroit and New York 1/2% discount: what is the value of the draft in New York, if sent through Chicago and Detroit? Dollars? {~m 8400 / (1.015 * .99 * .995)}
q: A, of Galveston, has $6000 to pay in New York. The direct exchange is 1/2% premium; but exchange on New Orleans is 1/4% premium, and from New Orleans to New York is 1/4% discount. By circular exchange, how much will pay his debt? Dollars? {~m 6000 * 1.0025 * .9975} What is his gain? {~m 6030 - 6000 * 1.0025 * .9975}
q: A merchant of St. Louis wishes to remit $7165.80 to Baltimore. Exchange on Baltimore is 1/4% premium; but, on New Orleans it is 1/8% premium; from New Orleans to Havana, 1/8% discount; from Havana to Baltimore, 1/4% discount. What will be the value in Baltimore directly? Dollars? {~m 7165.80 / 1.0025} By the circular method? {~m 7165.80 / (1.00125 * .99875 * .9975)} How much better is the circular? {35.84|35.83|35.85}
q: A Louisville merchant has $10000 due him in Charleston. Exchange on Charleston is 1/8% premium. Instead of drawing directly, he advises his debtor to remit to his agent in New York at 1/5% premium, on whom he immediately draws at 12 da., and sells the bill at 3/4% premium, interest off at 6%. What does he realize by drawing directly? Dollars? {10012.50|10012.5} By the circular way? {10029.94|10029.93|10029.95} What gain over the direct exchange? {17.44|17.43|17.45}
q: A Cincinnati manufacturer receives, April 18th, an account of sales from New Orleans; net proceeds $5284.67, due June 4/7. He advises his agent to discount the debt at 6%, and invest the proceeds in a 7 day bill on New York, interest off at 6%, at 1/2% discount, and remit it to Cincinnati. The agent does this, April 26. The bill reaches Cincinnati May 3, and is sold at 1/4% premium. What is the proceeds? Dollars? {5296.10|5296.09|5296.11} How much greater than if a bill had been drawn May 3, on New Orleans, due June 7, sold at 1/8% premium, and interest off at 6%? {35.65|35.64|35.66}
q: A merchant in Louisville wishes to pay $10000, which he owes in Berlin. He can buy a bill of exchange in Louisville on Berlin at the rate of $.96 for 4 reichmarks; or he is offered a circular bill through London and Paris, brokerage at each place 1/8%, at the following rates: 1 pound = $4.90 = 25.38 francs, and 5 francs = 4 reichmarks. What is the difference in the cost? Dollars? {81.35|81.34|81.36}

== 58 | Exchange | Equation of payments
note: The equated time is when several debts, due at different times, may be paid at once without loss. Multiply each debt by its term of credit and divide the sum of the products by the sum of the debts. With debits and credits, take a focal date; divide the difference of the products by the difference of the items, and count that many days from the focal date. Answer a date by giving its day of the month.
q: A buys an invoice of $250 on 3 mo. credit; $800 on 2 mo.; and $1000 on 4 mo.; and gives his note for the whole amount: how long should the note run? Months? {3} Days? {3}
sol: 250 x 3 = 750; 800 x 2 = 1600; 1000 x 4 = 4000; in all 6350 for $2050. 6350 / 2050 = 3 2/41 months = 3 mo. 3 da.
q: I buy of a wholesale dealer, at 3 mo. credit: Jan. 7, an invoice of $600; Jan. 11, $240; Jan. 13, $400; Jan. 20, $320; Jan. 28, $1200; I give a note at 3 mo. for the whole amount: on what day of January is it dated? {19}
sol: Counting from Jan. 7: 240 x 4 + 400 x 6 + 320 x 13 + 1200 x 21 = 32720; 32720 / 2760 = 11.8 days, or 12 nearly: Jan. 19.
q: A in acc't with B. Debits: Jan. 14, invoice 2 mo., $400; Feb. 1, $200; Mar. 10, $500; Mar. 25, $800; May 16, $300 (all at 2 mo.). Credits: Mar. 1, cash, $500; Apr. 15, remittance May 1, $250; May 20, cash, $375; June 4, remittance June 25, $450. What is the balance? Dollars? {625} On what day of June is it due? {2}
sol: From March 1, the debit products are 155500, the credit products 97450. 58050 / 625 = 93 days after March 1 = June 2.
q: What is the mean time of these invoices at 4 months, 1877: May 15, $800; June 1, $700; June 10, $900; July 20, $600; Aug. 1, $500; Aug. 15, $1000? Day of October? {30}
q: E in acc't current with F. Debits: Feb. 4, Mar. 20, Apr. 1, Apr. 5, invoices at 3 mo., $550, $260, $150, $325. Credits: May 8, cash, $150; May 26, remittance June 5, $420; June 3, note 1 month, $340; July 1, note 2 months, $170. What does E owe F? Dollars? {205} Due on what day of February? {26}
q: H. Wright to Mason & Giles, 1876: Feb. 1, invoice 3 months, $900; Feb. 20, 3 months, $700; Mar. 10, 3 months, $600; Apr. 8, cash, $500; May 10, 3 months, $900; June 15, 3 months, $400. When due? Day of June? {13}
q: A in acc't current with B. Debits, invoices cash: Mar. 19, $900; Apr. 20, $800; June 15, $700; May 10, $600. Credits: Feb. 20, cash, $400; Mar. 5, remittance received Mar. 15, $300; June 20, cash, $200; July 10, cash, $500. What balance does A owe B? Dollars? {1600} Due on what day of April? {23}
q: C in acc't current with D. Debits: Jan. 4, invoice 2 mo., $250; Feb. 3, 1 mo., $140; Feb. 15, 2 mo., $450; Apr. 2, cash, $100. Credits: Mar. 10, cash, $350; Mar. 21, cash, $200; Apr. 4, note 2 mo., $240; May 20, remittance May 25, $120; June 16, acceptance 16 days sight, $500. What balance is C credited with? Dollars? {470} Due on what day of August? {12}
q: I owe $912, due Oct. 16, and $500, due Dec. 20. If I pay the first, Oct. 1, 15 days before due, when should I pay the last? Days after Dec. 20? {27} Day of January? {16}
q: Oct. 3, I had two accounts, amounting to $375, one due Dec. 6, and the other Nov. 6, but equated for Nov. 16: what was the one due Dec. 6? Dollars? {125} The other? {250}
sol: One was due in 64 days, the other in 34, and both together in 44. 64 times one and 34 times the other = 44 x 375 = 16500; 34 times both = 12750; so 30 times the first = 3750, and the first is $125.
q: A owes $840, due Oct. 3; he pays $400, July 1; $200, Aug. 1: when will the balance be due? Days after Oct. 3? {~(400 * 94 + 200 * 63) / 240} Day of April? {30}
q: I owe $3200, Oct. 25; I pay $400, Sept. 15; $800, Sept. 30: when should the balance be paid? Days after Oct. 25? {18 @ (400 * 40 + 800 * 25) / 2000} Day of November? {12}
q: An account of $2500 is due Sept. 16; $500 are paid Aug. 1; $500, Aug. 11; $500, Aug. 21: when will the balance be due? Days after Sept. 16? {54 @ (500 * 46 + 500 * 36 + 500 * 26) / 1000} Day of November? {9}
q: Exchange the five following notes for six others, each for the same amount, and payable at equal intervals: one of $1200, due in 41 days; one of $1500, due in 72 days; one of $2050, due in 80 days; one of $1320, due in 110 days; one of $1730, due in 125 days. What is each new note? Dollars? {1300} The interval, in days? {25}
sol: The equated products are 682650. Six notes of $1300 at 1, 2, ... 6 intervals give 27300 dollar-intervals; 682650 / 27300 = 25 days. The notes run 25, 50, 75, 100, 125 and 150 days.
q: Burt owed in two accounts $487; neither was to draw interest till after due, one standing a year, and the other two years. He paid both in 1 yr. 5 mo., finding the true discount of the second, at 6%, exactly equal to the interest of the first: what difference of time would the common rule have made? Days? {3}
sol: The first drew interest 5 months, 1/40 of itself; the second was discounted 7 months, 7/207 of itself. So the first is 280 and the second 207. By the common rule (280 x 1 + 207 x 2) / 487 = 1.42505 yr. = 1 yr. 5 mo. 3 da.: 3 days different.

== 59 | Exchange | Settlement of accounts
note: In an account current each item draws interest from its date to the day of settlement. The difference of the interests on the two sides is the interest due; with the difference of the items it is the cash balance. An account sales shows the goods sold (credits), the charges (debits), and the net proceeds. Storage is charged by the month of 30 days on each barrel: add the barrels on hand each day.
q: F. H. Willis in acc't with E. S. Kennedy. Debits, checks 1879: Jan. 6, $170; Jan. 13, $480; Jan. 16, $96; Jan. 25, $500; Jan. 28, $50. Credits: Dec. 31, 1878, balance forward, $325; Jan. 7, cash deposit, $800; Jan. 20, $175; Jan. 30, $240. Interest to Jan. 31, at 6 per cent. What interest is due? Dollars? {2.33|2.326|2.32|2.34} What is the cash balance due Willis? {246.33 @ 1540 - 1296 + 2.33}
sol: The interests on the debits come to $2.913, on the credits $5.239: $2.33 is due Willis. The items differ by $244; $244 + $2.33 = $246.33.
q: Henry Hammond, 1876. Debits: Jan. 4, Mdse. at 3 mo., $900.10; Feb. 1, at 4 mo., $400.00; Feb. 18, at 3 mo., $700.50; Mar. 7, at 4 mo., $600.40; April 8, cash, $500.20; May 10, Mdse. at 30 da., $400.00; June 1, at 1 mo., $100.60. Find the equated time: on what day of May? {15} The cash balance July 7, 1876, interest 6%? Dollars? {3633.62|3633.61|3633.63} The cash balance April 30, 1876? {3592.80|3592.79|3592.81}
sol: From April 4: the products come to 147732 for $3602; 41 days after April 4 is May 15. $3601.80 held 53 days more to July 7 amounts to $3633.62; paid 15 days early, April 30, it is worth $3592.80.
q: William Smith, 1876, money worth 10%. Debits: Jan. 1, Mdse. $800.00; Jan. 16, at 30 da., $180.30; Feb. 14, at 60 da., $400.60; Mar. 25, $500.00; April 1, cash, $800.00; May 7, $600.00; May 21, at 60 da., $700.00; June 10, $200.00; June 15, at 90 da., $2000.00; July 12, $500.00; Aug. 4, at 2 mo., $1000.10. Credits: Jan. 10, cash, $400; Jan. 28, $200; Feb. 15, bills rec. at 60 da., $180.30; Feb. 28, cash, $100; Mar. 30, bills rec. at 90 da., $450; April 14, cash, $400.60; May 1, $500; May 15, bills rec. at 1 mo., $680; June 16, cash, $300; July 19, $700; Aug. 10, bills rec. at 20 da., $200; Sept. 1, cash, $150; Oct. 3, $1100. What is the balance of the items? Dollars? {2320.10|2320.1} Equated time: on what day of June? {18} The cash balance Oct. 4, 1876? {2389.70|2389.69|2389.71}
q: George Cummings. Debits, 1876: Jan. 2, cash, $300; Jan. 21, $194; Mar. 4, $150. Credits: Dec. 1, 1875, Mdse. at 1 mo., $583; Feb. 3, 1876, at 1 mo., $40; Mar. 30, at 1 mo., $130. Find the equated time: on what day of February? {11} The cash balance March 31, 1876, interest 10% per annum? Dollars? {110.48|110.47|110.49}
q: A. L. Morris in acc't with T. J. Fisher & Co. Debits, checks 1871: Jan. 13, $350; Jan. 22, $275; Feb. 25, $100; May 1, $400; June 23, $108.25. Credits: Dec. 31, 1870, balance from old acc't, $813.64; Feb. 4, cash, $120; Mar. 17, $500; May 31, $84.50. Interest to June 30, at 6 per cent. What interest is due Morris? Dollars? {13.34|13.33|13.35} What is the balance due him? {298.23|298.22|298.24}
q: Wm. White in acc't with Beach & Berry. Debits, checks 1875: July 2, $212.50; July 20, $66; Aug. 7, $235; Aug. 25, $300; Sept. 5, $110; Sept. 11, $46.40; Sept. 27, $454.25. Credits: June 30, balance from old acc't, $1102.50; July 6, cash deposit, $50; July 15, $95; Aug. 9, $168.75; Sept. 18, $32; Oct. 3, $79.90. Interest to Oct. 12, at 10 per cent. What is the interest due? Dollars? {19.68|19.67|19.69} The balance due? {123.68|123.67|123.69}
q: Charles Maynard's Consignment, 1876. Sales: Aug. 12, J. Barnes, at 10 da., $50.60; Aug. 14, cash, $800; Aug. 24, bills rec. at 30 da., $850; Aug. 29, cash, $210; Aug. 30, George Hand, $4900; Aug. 31, bills rec. at 20 da., $1400; in all $8210.60. Charges: Aug. 10, freight, $75; Aug. 31, storage, $10; insurance, $10.26; commission 2 1/2%, $205.25; in all $300.51. Net proceeds due Maynard, $7910.09. On what day of September are the proceeds due? {4}
q: William Thomas sold on account of B. F. Jonas 2000 bu. wheat July 8, 1876, for $2112.50; July 11, 300 bu. wheat, and took a 20 day note for $362.50; paid freight July 6, $150.00; July 11, storage, $6; drayage, $5; insurance, $4; commission, at 2 1/2%, $61.87; loss and gain for his net gain, $11.57. What are the net proceeds? Dollars? {2236.56 @ 2112.50 + 362.50 - 150 - 6 - 5 - 4 - 61.87 - 11.57} On what day of July is the balance due? {12}
q: Mdse. Co. B, 1876. Sales: July 18, note at 20 da., $120; July 24, note at 25 da., $60; July 30, cash, $55. Charges: July 15, freight, $33; drayage, $4; July 30, insurance, $1.50; storage, $3; commission at 2 1/4%, $5.29; C. V. Carnes's net proceeds, $62.73; in all $109.52, less our 1/3 net loss, $3.93: $105.59. Find the equated time: on what day of August? {19} Find C. V. Carnes's net proceeds if paid Jan. 1, 1877, money being worth 10% per annum. Dollars? {65.05|65.04|65.06}
q: Storage to Jan. 31, at 5 ct. a bbl. per month. Received, 1876, January: 2, 200 bbl.; 5, 150; 7, 30; 10, 120; 14, 80; 17, 150; 20, 75; 24, 60; 28, 200. Delivered, January: 10, 110; 13, 90; 17, 20; 20, 115; 25, 140; 27, 72; 30, 100. What is the storage? Dollars? {19.25} How many barrels are on hand? {418}
q: Storage to Feb. 20, at 5 ct. a bbl. per month. Received: Jan. 31, 418 bbl.; Feb. 4, 250; Feb. 9, 120; Feb. 12, 100; Feb. 16, 30. Delivered, February: 5, 100; 10, 80; 12, 220; 14, 140; 18, 90; 20, 288. What is the storage? Dollars? {15.85} How many barrels are on hand? {0}

== 60 | Interest | Compound interest
note: Compound interest adds each interval's interest to the principal. The amount of $1 for one interval, taken as a factor once for each interval, times the principal, is the compound amount; less the principal, the compound interest. A part of an interval is reckoned at simple interest. The book's tables give the amount of $1 for up to 55 intervals; answers are to the cent.
q: Find the compound amount of $1000, in 4 years, at 2% per annum. Dollars? {~m 1000 * 1.02 * 1.02 * 1.02 * 1.02}
sol: $1000 x 1.02 = $1020, the first year's amount; using the factor 1.02 four times gives $1082.43216.
q: Find the compound amount of $3850, for 4 yr. 7 mo. 16 da., at 5%, payable annually. Dollars? {4826.59|4826.58|4826.60} The compound interest? {976.59|976.58|976.60}
q: The compound interest of $13062.50, for 1 yr. 10 mo. 12 da., at 8%, payable quarterly. Dollars? {2082.25|2082.24|2082.26}
q: The compound amount of $1000, for 3 yr., at 10%, payable semi-annually. Dollars? {~m 1000 * 1.05 * 1.05 * 1.05 * 1.05 * 1.05 * 1.05}
q: What sum, at simple interest, 6%, for 2 yr. 5 mo. 27 da., amounts to the same as $2000, at compound interest, for the same time and rate, payable semi-annually? Dollars? {2016.03|2016.02|2016.04}
q: What is the difference between the annual and the compound interest of $5000 in 6 years, 6% per annum? Dollars? {22.60|22.59|22.596|22.61}
q: Required the amount of $1000, at compound interest, 21%, for 2 yr. 6 mo. Dollars? {~m 1000 * 1.21 * 1.21 * 1.105}
q: Find the compound amount of $750 for 17 yr., at 6%, payable annually. Dollars? {2019.58|2019.57|2019.59}
q: Of $5428 for 33 yr., 5% annually. Dollars? {27157.31|27157.30|27157.32}
q: The compound interest of $1800 for 14 yr., at 8%, payable semi-annually. Dollars? {3597.67|3597.66|3597.68}
q: If $1000 is deposited for a child, at birth, and draws 7% compound interest, payable semi-annually, till it is of age (21 yr.), what will be the amount? Dollars? {4241.26|4241.25|4241.27}
q: Find the amount of $9401.50, at compound interest for 19 yr. 4 mo., 9%, payable semi-annually. Dollars? {51576.68|51576.67|51576.69}
q: Find the compound amount of $1000 for 100 yr., at 10%, payable annually. Dollars? {13780612.34|13780612.33|13780612.35}
q: The compound interest of $3600 for 15 yr., at 8%, payable quarterly. Dollars? {8211.71|8211.70|8211.72}
q: The compound interest of $4000 for 40 yr., at 5%, payable semi-annually. Dollars? {24838.27|24838.26|24838.28}
q: The compound interest of $1200 for 27 yr. 11 mo. 4 da., at 12%, payable quarterly. Dollars? {31404.74|31404.73|31404.75}
note: To find the time, divide the amount by the principal; find the number next less in the table under the rate; divide its deficiency by its gain for one interval, and annex the fraction to the number of intervals.
q: Find the time in which $750 will amount to $2000, the interest being 8%, payable semi-annually. Yr.? {12} Mo.? {6} Da.? {1}
sol: $2000 / 750 = 2.6666667, the amount of $1 at 4%. The table gives 2.6658363 for 25 intervals, short by .0008303; the gain in an interval is .1066335; .0008303 / .1066335 = 1/128 nearly. 25 1/128 intervals of 6 mo. = 12 yr. 6 mo. 1 da.
q: In what time, at compound interest, will $8000 amount to $12000, at 6%? Yr.? {6} Mo.? {11} Da.? {15}
q: $5200 amount to $6508, 6%, payable semi-annually? Yr.? {3} Mo.? {9} Da.? {16}
q: $12500 gain $5500, 10%, payable quarterly? Yr.? {3} Mo.? {8} Da.? {9}
q: $1 gain $1, at 6%? Yr.? {11} Mo.? {10} Da.? {21} At 8%? Yr.? {9} Mo.? {0} Da.? {2} At 10%? Yr.? {7} Mo.? {3} Da.? {5}
q: $9862.50 amount to $22576.15, 12%, payable semi-annually? Yr.? {7} Mo.? {1} Da.? {7}
note: To find the rate, divide the amount by the principal and look for the quotient in the table opposite the number of intervals.
q: At what rate will $1000 amount to $2411.714 in 20 years? Per cent? {~4_1/2}
q: At what rate, by compound interest, will $1000 amount to $1593.85 in 8 yr.? Per cent? {6}
q: $3600 amount to $9932.51 in 15 yr.? Per cent? {7}
q: $13200 amount to $48049.58, in 26 yr. 5 mo. 21 da.? Per cent? {5}
q: $2813.50 amount to $13276.03, in 17 yr. 7 mo. 14 da., interest payable semi-annually? Per cent a year? {9}
q: $7652.18 gain $17198.67, interest payable quarterly, in 11 yr. 11 mo. 3 da.? Per cent a year? {10}
q: At what rates will any sum double itself in 10, 15, 20 yr.? {open}
sol: Looking for 2 in the tables: in 10 years, between 7 and 8%; in 15 years, not quite 5%; in 20 years, a little over 3 1/2%.
note: To find the principal, divide the given interest or amount by the interest or amount of $1 for the time and rate. The principal for an amount due later is its present worth; the difference is the compound discount.
q: What principal will yield $31086.78 compound interest in 40 yr., at 8%? Dollars? {1500}
q: What principal, at compound interest, will yield $52669.93 in 25 yr., 6%? Dollars? {16000.00|15999.99|16000.01}
q: Will gain $1625.75 in 6 yr. 2 mo., 7%, payable semi-annually? Dollars? {3075.00|3074.99|3075.01}
q: Will yield $3598.61 in 3 yr. 6 mo. 9 da., 10%, payable quarterly? Dollars? {8640.00|8639.99|8640.01}
q: Will yield $31005.76 in 9 1/2 yr., at 8%, payable semi-annually? Dollars? {28012.63|28012.62|28012.64}
q: Will amount to $27062.85 in 7 yr., at 4%? Dollars? {20565.54|20565.53|20565.55}
q: What is the present worth of $14625.70, due in 5 yr. 9 mo., at 6% compound interest, payable semi-annually? Dollars? {10409.77|10409.76|10409.78}
q: What is the compound discount on $8767.78, due in 12 yr. 8 mo. 25 da., 5%? Dollars? {4058.87|4058.86|4058.88}

== 61 | Interest | Annuities
note: An annuity is a sum payable at regular intervals. The initial value of a perpetuity is the payment divided by the interest of $1 for one interval. A deferred perpetuity is worth its initial value discounted at compound interest for the time deferred. An annuity for a term is the difference of two perpetuities, one beginning when it begins and one when it ends. The final (forborne) value is the compound interest of the perpetuity's initial value for the time it runs.
q: What is the initial value of a perpetual lease of $250 a year, allowing 6% interest? Dollars? {~m 250 / .06}
q: What is the initial value of a perpetual leasehold of $300 a year, allowing 6% interest? Dollars? {5000 @ 300 / .06}
q: What must I pay for a perpetual lease of $756.40 a year, to secure 8% interest? Dollars? {9455 @ 756.40 / .08}
q: Ground rents on perpetual lease, yield an income of $15642.90 a year: what is the present value of the estate, allowing 7% interest? Dollars? {223470 @ 15642.90 / .07}
q: What is the initial value of a perpetual leasehold of $1600 a year, payable semi-annually, allowing 5% interest, payable annually? Dollars? {32400 @ 1620 / .05}
sol: The yearly payment is $1620, allowing 5% interest for half a year on the first half-yearly payment.
q: What is the initial value of a perpetual leasehold of $2500 a year, payable quarterly, interest 6% payable semi-annually? Dollars? {41979.17|41979.16|41979.18} 6% payable annually? {42604.17|42604.16|42604.18} 6% payable quarterly? {41666.67|41666.66|41666.68}
q: Find the present value of a perpetuity of $250 a year, deferred 8 yr., allowing 6% interest. Dollars? {2614.22|2614.21|2614.23}
q: Find the present value of a perpetuity of $780 a year, to commence in 12 yr., int. 5%. Dollars? {8686.66|8686.65|8686.67}
q: Of a perpetual lease of $160 a year, to commence in 3 yr. 4 mo., int. 7%. Dollars? {1823.28|1823.27|1823.29}
q: Of the reversion of a perpetuity of $540 a year, deferred 10 yr., int. 6%. Dollars? {5025.55|5025.54|5025.56}
q: Of an estate which, in 5 yr., is to pay $325 a year forever: int. 8%, payable semi-annually. Dollars? {2690.67|2690.66|2690.68}
q: Of a perpetuity of $1000 a year, payable quarterly, to commence in 9 yr. 10 mo. 18 da., int. 10%, payable semi-annually. Dollars? {3858.88|3858.87|3858.89}
q: Find the present value of an immediate annuity of $250 continuing 8 years, 6% interest. Dollars? {1552.45|1552.44|1552.46}
sol: Present value of an immediate perpetuity of $250, $4166.67; of one deferred 8 years, $2614.22; the difference, $1552.45.
q: The present value of an annuity of $680, to commence in 7 yr. and continue 10 yr., 5% int. Dollars? {3731.63|3731.62|3731.64}
q: Find the present value of an annuity of $125, to commence in 12 yr. and run 12 yr., int. 7%. Dollars? {440.83|440.82|440.84}
q: The present value of an immediate annuity of $400, running 15 yr. 6 mo., int. 8%. Dollars? {3484.41|3484.40|3484.42}
q: The present value of an annuity of $826.50, to commence in 3 yr. and run 13 yr. 9 mo., int. 6%, payable semi-annually. Dollars? {6324.69|6324.68|6324.70}
q: The present value of an annuity of $60, deferred 12 yr. and to run 9 yr., int. 4 1/2%. Dollars? {257.17|257.16|257.18}
q: Sold a lease of $480 a year, payable quarterly, having 8 yr. 9 mo. to run, for $2500: do I gain or lose, int. 8%, payable semi-annually? How much do I lose? Dollars? {509.96|509.95|509.97}
q: Find the final or forborne value of an annuity of $250, continuing 8 yr., int. 6%. Dollars? {2474.37|2474.36|2474.38}
q: Find the forborne value of an immediate annuity of $300, running 18 yr., int. 5%. Dollars? {8439.72|8439.71|8439.73}
q: A pays $25 a year for tobacco: how much better off would he have been in 40 yr. if he had invested it at 10% per annum? Dollars? {11064.81|11064.80|11064.82}
q: Find the forborne value of an annuity of $75, to commence in 14 yr., and run 9 yr., int. 6%. Dollars? {861.85|861.84|861.86}
q: A pays $5 a year for a newspaper: if invested at 9%, what will his subscription have produced in 50 yr.? Dollars? {4075.42|4075.41|4075.43}
q: An annuity, at simple interest 6%, in 14 yr., amounted to $116.76: what would have been the difference had it been at compound interest 6%? Dollars? {9.33|9.32|9.34}
q: A boy just 9 yr. old, deposits $35 in a bank: if he deposit the same each year hereafter, and receive 10%, compound interest, what will be the entire amount when he is of age? Dollars? {858.30|858.29|858.31}
q: An immediate annuity running 11 yr., can be purchased for $6000: what is the payment, int. 6%? Dollars? {760.76|760.75|760.77}
q: How much a year should I pay, to secure $15000 at the end of 17 yr., int. 7%? Dollars? {486.38|486.37|486.39}
q: What is the payment of an annuity, deferred 4 yr., running 16 yr., and worth $4800, int. 6%? Dollars? {599.64|599.63|599.65}
q: In what time will a debt of $10000, drawing interest at 6%, be paid by installments of $1000 a year? Full years? {15} The balance then due? Dollars? {689.61|689.60|689.62}
q: In how many years can a debt of $1000000, drawing interest at 6%, be discharged by a sinking fund of $80000 a year? Full years? {23} The balance due at the end? Dollars? {60083.43|60083.42|60083.44}
q: In how many years can a debt of $30000000, drawing interest at 5%, be paid by a sinking fund of $2000000? Full years? {28} The balance? Dollars? {798709|798708|798710}
q: In how many years can a debt of $22000, drawing 7% interest, be discharged by a sinking fund of $2500 a year? Full years? {14} The balance? Dollars? {351.53|351.52|351.54}
q: Let the conditions be the same as those of the $10000 example, and let each $1000 payment be itself a year's accumulation of simple interest: what would be the whole time required to discharge the debt? Yr.? {15} Mo.? {8} Da.? {19}
q: Suppose the national debt $2000000000, and funded at 4%: how many years would be required to pay it off, by a sinking fund of $100000000 a year? Full years? {41}
q: If $9000 is paid for an immediate annuity of $750, to run 20 yr., what is the rate, to the nearest per cent? {~5_1/2}
sol: $9000 / 750 = 12, the present value of $1 for 20 years: about half way between 12.46 at 5% and 11.47 at 6%, so about 5 1/2%.
q: If an immediate annuity of $80, running 14 yr., sells for $650, what is the rate, to the nearest per cent? {8}

== 62 | Interest | Life annuities and life insurance
note: The value of a life annuity of $1 for each age and rate comes from the Carlisle table: for example at age 72, 7%, 5.162; at 55, 6%, 9.524; at 21, 6%, 13.769; at 26, 6%, 13.368; at 43, 5%, 12.957; at 58, 7%, 8.153; at 47, 5%, 12.301; at 38, 6%, 12.239; at 62, 7%, 7.403. A widow's dower is a life estate in one third of the real estate. The reversion is the value of the property less the life estate. Life insurance premiums per $1000 for life policies at age 40, $31.30; at 50, $47.18; at 20 to 25, $19.80. Endowment premiums in 10 years: at 28, $104.29; at 50, $112.68.
q: What must be paid for a life-annuity of $650 a year, by a person aged 72 yr., int. 7%? Dollars? {3355.30|3355.3 @ 650 * 5.162}
q: What is the life-estate in $25000, age 55 yr., int. 6%? Dollars? {14286 @ 1500 * 9.524} The reversion? {10714}
q: The dower and reversion in $46250, age 21 yr., int. 6%? The dower, dollars? {12736.33|12736.32|12736.34} The reversion? {2680.34|2680.33|2680.35}
q: How large an annuity can be purchased for $500, age 26 yr., int. 6%? Dollars? {37.40|37.39|37.41}
q: For $1200, age 43, int. 5%? Dollars? {92.61|92.60|92.62}
q: For $840, age 58, int. 7%? Dollars? {103.03|103.02|103.04}
q: Find the present value of the reversion of a perpetuity of $500 a year, after the death of a person aged 47, int. 5%. Dollars? {3849.50|3849.5 @ 500 * (20 - 12.301)}
q: Of the reversion of an annuity of $165 a year for 30 yr., after the death of a person 38 yr. old, int. 6%. Dollars? {730.57|730.56|730.565}
q: Of the reversion of a lease of $1600 a year, for 40 yr., after the death of A, aged 62, int. 7%. Dollars? {11012.34|11012.33|11012.35}
q: W. R. Hamilton, aged 40 years, took a life policy for $5000. Required the annual premium. Dollars? {156.50|156.5 @ 31.30 * 5}
q: Conditions as above, how much would he have paid out in premiums, his death having occurred after he was 53? Dollars? {2191 @ 156.50 * 14}
q: Conditions the same, what did the premiums amount to, interest 6%? Dollars? {3045.49 @ 2191 + 156.50 * .06 * 91}
q: James Bragg, aged 50 years, took out an endowment policy for $20000, payable in 10 years, and died after making 6 payments: how much less would he have paid out by taking a life policy for the same amount, the premium payable annually? Dollars? {7860 @ (112.68 - 47.18) * 20 * 6}
q: Thomas Winn, 28 years of age, took out an endowment policy for $10000, payable in 10 years; he died in 18 months: what was the gain, interest on the premiums at 6%? Dollars? {7789.05|7789.052|7789.04} How much greater would the profit have been had he taken a life policy, premiums payable annually? {1755.57|1755.56|1755.58}
q: P. Darling took out a life policy at the age of 40 years, and died just after making the tenth payment; his premiums amounted to $3975.10, interest 6%; required the amount of his policy. Dollars? {10000 @ 3975.10 / (313 + 31.30 * .06 * 45) * 1000}
q: R. C. Storey took out an endowment policy for $10000 for 15 years; he lived to pay all of the premiums; but had he put them, instead, at 6% interest as they fell due, they would have amounted to $15426.78: what was his age? {40}
q: Allen Wentworth had his life insured at the age of twenty, on the life plan, for $8000, premium payable annually: how old must he be, that the sum of the premiums may exceed the policy? {70}
q: T. B. Bullene, aged 40 years, took out a life policy for $30000, payments to cease in 5 years, the rate being $9.919 on the $100; his death occurred two months after he had made the third payment: what was gained over and above the premiums, interest 6%? Dollars? {20448|20448.00|20448.003}
q: F. M. Harrington took out an endowment policy for $11000 when he was 42; at its maturity he had paid in premiums $635.80 more than the face of the policy: what was the period of the endowment, in years? {20 @ 11635.80 / 11 / 52.89}

== 63 | Partnership | Partnership and bankruptcy
note: Gains and losses are shared in proportion to the sums invested and the times of investment. When the times differ, multiply each partner's stock by the time it is used, and divide the gain in proportion to the products. A bankrupt pays on each dollar such a part of a dollar as his property (less expenses) is of his debts.
q: A, B, and C are partners, with $3000, $4000, and $5000 stock, respectively; if they gain $5400, what is A's share? Dollars? {1350 @ 5400 * 3 / 12} B's? {1800} C's? {2250}
q: A and B gain in one year $3600; their store expenses are $1500. If A's stock is $2500 and B's $1875, how much does A gain? Dollars? {1200 @ 2100 * 2500 / 4375} B? {900}
q: A, B, and C are partners; A puts in $5000, B $6400, C $1600. C is allowed $1000 a year for personal attention to the business; their store expenses for one year are $800, and their gain, $7000. Find A's gain. Dollars? {2000 @ 5200 * 50 / 130} B's gain? {2560} C's income? {1640}
q: A, B, and C form a partnership; A puts in $24000, B $28000, C $32000; they lose 1/6 of their stock by a fire, but sell the remainder at 3/5 more than cost: if all expenses are $8000, what is the gain of A? Dollars? {~m 20000 * 24 / 84} Of B? {~m 20000 * 28 / 84} Of C? {~m 20000 * 32 / 84}
q: A, B, and C are partners; A's stock is $5760, B's, $7200; their gain is $3920, of which C has $1120: what is C's stock? Dollars? {5184} A's gain? {~m 3920 * 5760 / 18144} B's gain? {~m 3920 * 7200 / 18144}
q: A, B, and C are partners; A's stock, $8000; B's, $12800; C's, $15200; A and B together gain $1638 more than C: what is the gain of A? Dollars? {2340 @ 8000 * 1638 / 5600} Of B? {3744} Of C? {4446}
q: A, B, and C have a joint capital of $27000; none of them draw from the firm, and when they quit A has $20000; B, $16000; C, $12000: how much did A contribute? Dollars? {11250 @ 27000 * 20 / 48} B? {9000} C? {6750}
q: A, B, C, and D gain 30% on the stock; A, B, and C gain $1150; A, B, D, $1650; B, C, D, $1000; A, C, D, $1600: what was A's stock? Dollars? {~m 800 / .3} B's? {~m 200 / .3} C's? {~m 150 / .3} D's? {~m 650 / .3}
sol: The four sums add to $5400, three times the whole gain; so the whole gain is $1800. D's gain is $1800 - $1150 = $650, and his stock $650 / .30 = $2166.67; in the same way C gained $150, A $800, B $200.
q: A, B, and C are partners; A puts in $2500 for 8 mo.; B, $4000 for 6 mo.; C, $3200 for 10 mo.; their net gain is $4750: what is A's share? Dollars? {1250 @ 4750 * 20000 / 76000} B's? {1500} C's? {2000}
q: A begins business with $6000; at the end of 6 mo. he takes in B, with $10000; 6 mo. after, their gain is $3300: what is A's share? Dollars? {1800 @ 3300 * 72 / 132} B's? {1500}
q: A and B are partners; A's stock is to B's, as 4 to 5; after 3 mo., A withdraws 2/3 of his, and B 3/4 of his: divide their year's gain, $1675. A? Dollars? {800} B? {875}
q: A, B, and C join capitals, which are as 1/2, 1/3, 1/4; after 4 mo., A takes out 1/2 of his; after 9 mo. more, their gain is $1988: divide it. A? Dollars? {714} B? {728} C? {546}
q: A and B are partners; A puts in $2500; B, $1500; after 9 mo., they take in C with $5000; 9 mo. after, their gain is $3250: what is A's gain? Dollars? {1250} B's? {750} C's? {1250}
q: A and B are partners, each contributing $1000; after 3 months, A withdraws $400, which B advances; the same is done after 3 months more; their year's gain is $800: what should A get? Dollars? {200} B? {600}
q: A, B, and C are employed to empty a cistern by two pumps of the same bore; A and B go to work first, making 37 and 40 strokes respectively a minute; after 5 minutes, each makes 5 strokes less a minute; after 10 minutes, A gives way to C, who makes 30 strokes a minute until the cistern is emptied, which was in 22 minutes from the start: divide their pay, $2. A's, cents? {46} B's? {106} C's? {48}
q: A and B are partners; A's capital is $4200; B's, $5600: after 4 months, how much must A put in, to entitle him to 1/2 the year's gain? Dollars? {2100 @ (5600 - 4200) * 12 / 8}
q: A and B go into partnership, each with $4500. A draws out $1500, and B $500, at the end of 3 mo., and each the same sum at the end of 6 and 9 mo.; at the end of 1 yr. they quit with $2200: how must they settle? B takes the $2200; how much does A still owe B? Dollars? {300}
sol: They put in $9000 and drew out $4500 and $1500, leaving $3000 to show for it; quitting with $2200 they lost $800. A had $27000 for a month, B $45000, so A bears 3/8 of the loss, $300, and B 5/8, $500. A has nothing left, so B takes the $2200 and A owes him $300.
q: A, B, C, and D go in partnership; A owns 12 shares of the stock; B, 8 shares; C, 7 shares; D, 3 shares. After 3 mo., A sells 2 shares to B, 1 to C, and 4 to D; 2 mo. afterward, B sells 1 share to C, and 2 to D; 4 mo. afterward, A buys 2 from C and 2 from D. Divide the year's gain ($18000). A? Dollars? {4650} B? {4650} C? {4700} D? {4000}
q: A, with $400; B, with $500; and C, with $300, joined in business; at the end of 3 mo. A took out $200; at the end of 4 mo. B drew out $300, and after 4 mo. more, he drew out $150; at the end of 6 mo. C drew out $100; at the end of the year they close; A's gain was $225: what was the whole gain? Dollars? {675}
q: A has a lot worth $8000, good notes $2500, and cash $1500; his debts are $20000: what can he pay on $1, in cents? {60 @ 12000 / 20000 * 100} What will A receive, whose claim is $4500? Dollars? {2700}
q: My assets are $2520; I owe A $1200; B, $720; C, $600; D, $1080: what is paid on each dollar, in cents? {70} What does A get? Dollars? {840} B? {504} C? {420} D? {756}
q: A bankrupt's estate is worth $16000; his debts, $47500; the assignee charges 5%: what is paid on $1, in cents? {32 @ 15200 / 47500 * 100} What does A get, whose claim is $3650? Dollars? {1168}

== 64 | Alligation | Alligation medial
note: To find the average value of a mixture, find the values of the definite parts, and divide the sum of the values by the sum of the parts. The specific gravity of a body is its weight compared with the weight of an equal bulk of water.
q: If 3 lb. of sugar, at 5 ct. a lb., and 2 lb., at 4 1/2 ct. a lb., be mixed with 9 lb., at 6 ct. a lb., what per lb. is the mixture worth? Cents? {~78 / 14}
q: Find the average price of 6 lb. tea, at 80 ct.; 15 lb., at 50 ct.; 5 lb., at 60 ct.; 9 lb., at 40 ct. Cents? {54 @ (6 * 80 + 15 * 50 + 5 * 60 + 9 * 40) / 35}
q: The average price of 40 hogs, at $8 each; 30, at $10 each; 16, at $12.50 each; 54, at $11.75 each. Dollars? {~m (40 * 8 + 30 * 10 + 16 * 12.50 + 54 * 11.75) / 140}
q: How fine is a mixture of 5 pwt. of gold, 16 carats fine; 2 pwt., 18 carats fine; 6 pwt., 20 carats fine; and 1 pwt. pure gold? Carats? {~(5 * 16 + 2 * 18 + 6 * 20 + 24) / 14}
q: Find the specific gravity of a compound of 15 lb. of copper, specific gravity, 7 3/4; 8 lb. of zinc, specific gravity, 6 7/8; and 1/4 lb. of silver, specific gravity, 10 1/2. To three places. {~r3 23_1/4 / (15 / 7_3/4 + 8 / 6_7/8 + 1/4 / 10_1/2)}
sol: The copper has the bulk of 15 / 7.75 = 1.93548 lb. of water; the zinc of 1.16364 lb.; the silver of .02381 lb. In all 3.12293 lb. of water has the bulk of 23 1/4 lb. of the compound: 23.25 / 3.12293 = 7.445.
q: If a piece of metal weighs 1 3/4 oz. in air, but in water only 1 1/2 oz., what is its specific gravity? {7 @ 1_3/4 / (1_3/4 - 1_1/2)}
q: What per cent of alcohol in a mixture of 9 gal., 86% strong; 12 gal., 92% strong; 10 gal., 95% strong; and 11 gal., 98% strong? {93 @ (9 * 86 + 12 * 92 + 10 * 95 + 11 * 98) / 42}
q: At a teacher's examination, where the lowest passable average grade was 50, an applicant received the following grades: In Orthography, 50; Reading, 25; Writing, 50; Arithmetic, 60; Grammar, 55; Geography, 55. What was his average? {~295 / 6} Did he succeed, or did he fail? {open}
sol: 295 / 6 = 49 1/6, below 50: he failed.

== 65 | Alligation | Alligation alternate
note: To find the proportions for a mixture of a given average value, set each value below the average against one above it: each kind is taken in the amount of the other's difference from the average. When one quantity or the whole amount is fixed, multiply the balanced columns to suit. Many of these have more than one right answer; say one, then hear the book's.
q: What relative quantities of sugar, at 9 ct. a lb. and 5 ct. a lb., must be used for a compound, at 6 ct. a lb.? Pounds at 5 ct. for each pound at 9 ct.? {3}
sol: 1 lb. at 9 ct. sold for 6 ct. loses 3 ct.; 1 lb. at 5 ct. gains 1 ct.; 3 such lb. gain 3 ct. So 3 lb. at 5 ct. with 1 lb. at 9 ct.: 4 lb. worth 24 ct., 6 ct. a lb.
q: What relative numbers of hogs, at $3, $5, $10 per head, can be bought at an average value of $7 per head? {open}
sol: The $3 hogs gain 4, the $5 gain 2, the $10 lose 3. Balance 3 of the first with 4 of the third, and 3 of the second with 2 of the third: 3, 3, 6, or 1, 1, 2.
q: What relative quantities of tea, worth 25, 27, 30, 32, and 45 ct. per lb., must be taken for a mixture worth 28 ct. per lb.? {open}
sol: One answer: 6, 17, 3, 3, 1 lb.; another, 17, 6, 1, 1, 3 lb.
q: What of sugar, at 5, 5 1/2, 6, 7, and 8 ct. per lb., must be taken for a mixture worth 6 3/4 ct. per lb.? {open}
sol: One answer: 5 lb. at 5 ct., 1 at 5 1/2, 1 at 6, 8 at 7, and 7 at 8 ct.: 22 lb. worth 148 1/2 ct., or 6 3/4 ct. a lb.
q: What relative quantities of alcohol, 84, 86, 88, 94, and 96% strong, must be taken for a mixture 87% strong? {open}
sol: One answer: 1, 16, 3, 1, 1 gal.: 1914 per cent-gallons in 22 gal., 87%.
q: What of gold and silver, whose specific gravities are 19 1/4 and 10 1/2, will make a compound whose specific gravity shall be 16.84? How many pounds of gold to 723 lb. of silver? {3487}
sol: Compare bulks: a pound of silver has the bulk of 2/21 lb. of water, of gold 4/77, of the compound 1/16.84. Balancing the bulks gives 723 lb. of silver to 3487 lb. of gold.
q: What of silver 3/4 pure, and 9/10 pure, will make a mixture 7/8 pure? Parts of the 9/10 silver for each part of the 3/4? {5}
q: What of pure gold, and 18 carats, and 20 carats fine, must be taken to make 22 carat gold? {open}
sol: One answer: 1 part of 18 carats, 1 of 20 carats, and 3 of pure gold: (18 + 20 + 72) / 5 = 22.
q: How many whole bushels of each of two kinds of wheat, worth respectively $1.20 and $1.40, per bushel, will, with 14 bushels, at $1.90 per bushel, make a combination whose average value is $1.60 per bushel? {open}
sol: Ten answers: 1 and 19, 2 and 17, 3 and 15, ... 10 and 1 bushels, with the 14; for example 7, 7, 14.
q: How many railroad shares, at 50%, must A buy, who has 80 shares that cost him 72%, in order to reduce his average to 60%? {96 @ 80 * 12 / 10}
q: How many bushels of hops, worth respectively 50, 60, and 75 ct. per bu., with 100 bu., at 40 ct. per bu., will make a mixture worth 65 ct. a bu.? {open}
sol: One answer: 10 bu. at 50 ct., 10 at 60 ct., and 270 at 75 ct.: 390 bu. worth $253.50, 65 ct. a bushel.
q: How much water (0 per cent) will dilute 3 gal. 2 qt. 1 pt. of acid 91% strong, to 56%? Gal.? {~p 4,2 # 0 # 29 * 91 / 56 - 29} Qt.? {~p # 1 # $} Pt.? {~p # 2 # $}
q: A jeweller has 3 pwt. 9 gr. of old gold, 16 carats fine: how much U. S. gold, 21 3/5 carats fine, must he mix with it, to make it 18 carats fine? Pwt.? {1} Gr.? {21}
q: How much water, with 3 pt. of alcohol, 96% strong, and 8 pt., 78%, will make a mixture 60% strong? Pints? {~(3 * 96 + 8 * 78) / 60 - 11}
q: I mixed 1 gal. 2 qt. 1/2 pt. of water with 3 qt. 1 1/2 pt. of pure acid; the mixture has 15% more acid than desired: how much water will reduce it to the required strength? Gal.? {~p 4,2 # 0 # 7_1/2 / .225 - 20} Qt.? {~p # 1 # $} Pt.? {~p # 2 # $}
sol: 12 1/2 pt. of water and 7 1/2 pt. of acid make 20 pt., 37 1/2% acid. 22 1/2% is desired: 7 1/2 / .225 = 33 1/3 pt.; 13 1/3 pt. more water.
q: How much lead, specific gravity 11, with 1/2 oz. copper, sp. gr. 9, can be put on 12 oz. of cork, sp. gr. 1/4, so that the three will just float, that is, have a sp. gr. (1) the same as water? Ounces? {~(48 + 1/18 - 12_1/2) * 11 / 10}
q: How many shares of stock, at 40%, must A buy, who has bought 120 shares, at 74%, 150 shares, at 68%, and 130 shares, at 54%, so that he may sell the whole at 60%, and gain 20%? {610 @ (120 * 24 + 150 * 18 + 130 * 4) / 10}
q: A buys 400 bbl. of flour, at $7.50 each, 640 bbl., at $7.25, and 960 bbl., at $6.75: how many must he buy at $5.50, to reduce his average to $6.50 per bbl.? {1120 @ 400 * 1 + 640 * .75 + 960 * .25}
q: If a man pay $16 for each cow, $3 for each hog, and $2 for each sheep, how many of each kind may he purchase so as to have 100 animals for $600? {open}
sol: One answer: 25 sheep, 50 hogs, and 25 cows; others are 12, 64, 24; 38, 36, 26; 51, 22, 27; 64, 8, 28.
q: What quantities of sugar, at 3 ct. per lb. and 7 ct. per lb., with 2 lb. at 8 ct., and 5 lb. at 4 ct. per lb., will make 16 lb., worth 6 ct. per lb.? Lb. at 3 ct.? {~3/4} At 7 ct.? {~8_1/4}
q: How many bbl. flour, at $8 and $8.50, with 300 bbl. at $7.50, and 800 at $7.80, and 400 at $7.65, will make 2000 bbl. at $7.85 a bbl.? At $8? {200} At $8.50? {300}
q: What quantities of tea, at 25 ct. and 35 ct. a lb., with 14 lb. at 30 ct., and 20 lb. at 50 ct., and 6 lb. at 60 ct., will make 56 lb. at 40 ct. a lb.? At 25 ct.? {10} At 35 ct.? {6}
q: How much copper, specific gravity 7 3/4, with silver, specific gravity 10 1/2, will make 1 lb. troy, of specific gravity 8 3/5? Ounces of copper? {~(12 / 8_3/5 - 12 / 10_1/2) / (1 / 7_3/4 - 1 / 10_1/2)} Of silver? {~12 - (12 / 8_3/5 - 12 / 10_1/2) / (1 / 7_3/4 - 1 / 10_1/2)}
q: How much gold 15 carats fine, 20 carats fine, and pure, will make a ring 18 carats fine, weighing 4 pwt. 16 gr.? {open}
sol: One answer: 64 gr. of 15 carats, 24 gr. of 20 carats, and 24 gr. of pure gold: (960 + 480 + 576) / 112 = 18.
q: A dealer in stock can buy 100 animals for $400, at the following rates: calves, $9; hogs, $2; lambs, $1: how many may he take of each kind? (One of nine different answers.) {open}
sol: Taking one dollar for each of the 100 animals leaves $300, and each calf costs $8 over a dollar, each hog $1 over: 8 times the calves and the hogs make 300. One answer: 30 calves, 60 hogs, 10 lambs.
q: Hiero's crown, sp. gr. 14 5/8, was of gold, sp. gr. 19 1/4, and silver, sp. gr. 10 1/2; it weighed 17 1/2 lb.: how much gold was in it? Pounds? {~(17_1/2 / 10_1/2 - 17_1/2 / 14_5/8) / (1 / 10_1/2 - 1 / 19_1/4)}

== 66 | Powers and Roots | Involution
note: A power is the product of equal factors: 5 x 5 is the 2d power of 5, or its square; 5 x 5 x 5 the 3d power, or cube. Multiply the number by itself till it has been used as a factor as many times as the exponent shows. A power of a fraction is that power of the numerator over that power of the denominator. A power of a decimal has as many decimal places as the root, times the exponent.
q: Show by involution that (5) squared equals what? {25} 14 cubed? {2744} 6 to the 5th power? {7776} 192 squared? {36864} 1 to the 10th power? {1}
q: (3/5) to the 4th power equals? {~3/5 * 3/5 * 3/5 * 3/5} (2 1/4) cubed? {~9/4 * 9/4 * 9/4} (7/8) to the 5th power? {~7/8 * 7/8 * 7/8 * 7/8 * 7/8}
q: (.02) cubed equals? {.000008} (5 to the 4th) squared? {390625} (.046) cubed? {.000097336}
q: (1/9) to the 7th power equals? {~1 / 4782969} 2056 squared? {4227136} (7.62 1/2) squared? {58.140625}
q: To obtain the 9th power of 2, multiply its 6th power by its 3d power: what is it? {512 @ 64 * 8}
q: Find the square of 64 by the principle: the square of the tens, plus twice the tens times the units, plus the square of the units. {4096 @ 3600 + 480 + 16}
sol: 64 = 60 + 4. 60 squared = 3600; twice 60 x 4 = 480; 4 squared = 16. In all, 4096.
q: Square these numbers, each as the sum of two parts: 19. {361} 29. {841} 4. {16} 40. {1600} 125. {15625} 59. {3481}
q: Find the cube of 64: the cube of the tens, plus three times the square of the tens by the units, plus three times the tens by the square of the units, plus the cube of the units. {262144 @ 216000 + 43200 + 2880 + 64}
q: Cube these numbers, each as made of two parts: 19. {6859} 29. {24389} 4. {64} 40. {64000} 125. {1953125} 216. {10077696}

== 67 | Powers and Roots | Square root
note: Point off the number into periods of two figures, from the units. Find the greatest square in the left period, write its root, subtract, and bring down the next period. Double the root found for a trial divisor; see how often it goes in the dividend without its right-hand figure, annex that figure to the root and the divisor, multiply, subtract, and go on. A root that does not end is given to the places asked.
q: What is the length of the side of a square containing 576 sq. in.? Inches? {24}
sol: The root is more than 2 tens and less than 3. 400 taken from 576 leaves 176. Twice 20 is 40; 40 goes into 176 four times. 44 x 4 = 176: the side is 24 in.
q: Find the square root of 1764. {42}
q: Square root of 2809. {53} Of 1444. {38} Of 11881. {109}
q: Square root of 185640625. {13625} Of 6203794, to two places. {2490.74}
q: Square root of 3444736. {1856} Of 57600. {240} Of 16499844. {4062} Of 49098049. {7007}
q: Square root of 80012304, to two places. {8944.96|8944.95} Of 73005, to two places. {270.19|270.2} Of 386 cubed, to two places. {7583.70|7583.69|7583.7}
q: Find the square root of 3, true to the 7th decimal place. {1.7320508}
q: Find the square root of 9.869604401089358, true to the 7th decimal place. {3.1415926|3.1415927}
q: Square root of .030625 x square root of 40.96 x square root of .00000625 = what? {.0028 @ .175 * 6.4 * .0025}
q: Square root of (126 squared x 58 squared x 604 squared) = ? {4414032 @ 126 * 58 * 604}
q: Square root of 12.96 x square root of 75/108 = ? To four places. {3.2863|3.2864}
note: The square root of a fraction is the root of the numerator over the root of the denominator; it is best to multiply both terms first to make the denominator a square.
q: Square root of 6/7, to five places. {.92582} Of 34 5/8, to four places. {5.8843|5.8842} Of 4/7, to four places. {.7559|.7558}
q: Square root of 272 3/12. {~33 / 2} Of 6 2/5, to four places. {2.5298}
q: Square root of (28/57 x 392/2527 x 35/38) x square root of 3 = ? To five places. {.45886|.45885}
q: Square root of (123.454321 x .81) = ? {9.9999}
q: Square root of (1.728 x 4.8 x 3/7) = ? To four places. {1.8854}

== 68 | Powers and Roots | Cube root and higher roots
note: Point off the number into periods of three figures from the units. Find the greatest cube in the left period and subtract it. For a trial divisor, square the root found and multiply by 300; the quotient figure, times 30 times the root found, and its square, added to the trial divisor, give the complete divisor. Any root whose index is a composite number can be found by successive roots: the 6th root is the square root of the cube root. Horner's method gives any root.
q: What is the edge of a cube whose solid inches number 13824? Inches? {24}
sol: 8000 is the greatest cube of tens; 5824 remains. 2 squared x 300 = 1200 goes in 5824 four times. 1200 + 20 x 4 x 3 + 16 = 1456; 1456 x 4 = 5824: the edge is 24 in.
q: Find the cube root of 74088. {42}
q: Cube root of 512. {8} Of 19683. {27} Of 7301384. {194} Of 94818816. {456} Of 1067462648. {1022}
q: Cube root of 5.088448 {1.72} Of 22188.041 {28.1}
q: Cube root of 32.65, to six places. {3.196154} Of .0079, to seven places. {.1991632|.1991631} Of 3.0092, to six places. {1.443722|1.443724}
q: Cube root of 23/729, to five places. {.31599|.31598} Of 25, to six places. {2.924018|2.924017} Of 11, to six places. {2.223980}
q: Cube root of 2/3, to five places. {.87358} Of 4/15, to five places. {.64366|.64365} Of 171.416328875 {5.555}
q: Cube root of 7011, to seven places. {19.1393267} Of 48/4394, to seven places. {.2218846|.2218845} Of 2/3 of 4/11, to seven places. {.6235319}
q: Extract the fourth root of 68719476736. {512}
sol: By Horner's method in four columns; or, as the 4th root is the square root of the square root: the square root is 262144, and its square root 512.
q: Extract the cube root of 44.6 to six decimals. {3.546323|3.546322}
q: Extract the square root of 15625. {125} The cube root of 68719476736. {4096} The fifth root of 14348907. {27}
q: The cube root of 151, to six places. {5.325074} The 4th root of 97.41, to four places. {3.1416|3.1415}
q: The 4th root of 1.08, to five places. {1.01943|1.01942} The 5th root of 5/12, to five places. {.83938|.83937} The 5th root of 35.2, to five places. {2.03849|2.03848}
q: The 12th root of 782757789696, to eight places. {9.79795897|9.79795898} The 9th root of 1367631, to seven places. {4.8058955}

== 69 | Powers and Roots | Applications of roots
note: In a right-angled triangle the square on the hypotenuse equals the sum of the squares on the other two sides: hypotenuse = square root of (base squared + perpendicular squared). Areas of similar figures are as the squares of their like dimensions; solidities of similar solids as the cubes.
q: Find the length of a ladder reaching 12 ft. into the street, from a window 30 ft. high. Feet, to two places. {32.31}
q: What is the diagonal, or line joining the opposite corners, of a square whose side is 10 ft.? Feet, to three places. {14.142}
q: What is saved by following the diagonal instead of the sides, 69 rd. and 92 rd., of a rectangle? Rods? {46 @ 69 + 92 - 115}
q: A boat in crossing a river 500 yd. wide, drifted with the current 360 yd.; how far did it go? Yards, to the nearest yard. {616}
q: Integers for the sides of right-angled triangles: take any two unequal numbers; the sum of their squares is the hypotenuse, the difference of their squares one side, and double their product the other. From 4 and 1, what is the hypotenuse? {17} The sides? {15} {8}
q: One square is 12 1/4 times another: how many times does the side of the 1st contain the side of the 2d? {~7 / 2}
q: The diagonals of two similar rectangles are as 5 to 12: how many times does the larger contain the smaller? {~144 / 25}
q: The lengths of two similar solids are 4 in. and 50 in.; the 1st contains 16 cu. in.: what does the 2d contain? Cu. in.? {31250 @ 16 * 125000 / 64}
q: The solidities of two balls are 189 cu. in. and 875 cu. in.; the diameter of the 2d is 17 1/2 in.; find the diameter of the 1st. Inches? {~21 / 2}
sol: 875 : 189 :: 17.5 cubed : the cube of the diameter: 1157.625, whose cube root is 10 1/2.
q: In extracting the square root of a perfect power, the last complete dividend was found 4725: what was the power? {225625}
sol: The right-hand period ended in 25, so the last figure of the root was 5; the last divisor was 4725 / 5 = 945, so 94 was double the root found: 47. The root was 475, and the power 225625.
q: What number multiplied by 2/7 of itself makes 504? {42 @ 6 * 7}
q: Separate 91252 1/2 into three factors which are as the numbers 1, 2 1/2, and 3. The smallest? {23} The others? {~57_1/2} {69}
q: What integer multiplied by the next greater, makes 1332? {36}
q: The length and breadth of a ceiling are as 6 and 5; if each dimension were one foot longer, the area would be 304 sq. ft.: what is the length? Feet? {18} The breadth? {15}
q: In extracting the cube root of a perfect integral power, the operator found the last complete dividend 241984: what was the power? {2985984 @ 144 * 144 * 144}
q: If we cut from a cubical block enough to make each dimension one inch shorter, it will lose 1657 cubic inches: what is the solidity? Cu. in.? {13824 @ 24 * 24 * 24}
q: A hall standing east and west, is 46 ft. by 22 ft., and 12 1/2 ft. high: what is the length of the shortest path a fly can travel, by walls and floor, from a southeast lower corner to a northwest upper corner? Feet? {~57_1/2}
sol: Turn the end wall down level with the floor: the path is the diagonal of a rectangle 46 ft. by 34 1/2 ft., 57 1/2 ft. (Turning the side wall gives 58 1/2 by 22, a diagonal of 62+ ft.)
q: How many stakes can be driven down upon a space 15 ft. square, allowing no two to be nearer each other than 1 1/2 ft.? {128} How many allowing no two to be nearer than 1 1/4 ft.? {180}
q: What integer is that whose square root is 5 times its cube root? {15625}
q: If the true annual rate of interest be 10%, what would be the true rate for each 73 days, if the interest be compounded through the year? Per cent? {1.924|1.92}
q: If a field be in the form of an equilateral triangle whose altitude is 4 rods, what would be the cost of fencing it in, at 75 ct. a rod? Dollars? {10.39}

== 70 | Series | Arithmetical progression
note: In an arithmetical progression each term differs from the one before by the common difference. Last term = first term + (number of terms - 1) x common difference (less, if descending). Sum = (sum of the extremes) x (number of terms) / 2.
q: Find the 20th term of the arithmetical series 1, 4, 7, 10, etc. {58 @ 1 + 19 * 3}
sol: The common difference is taken once in forming the 2d term, twice in the 3d, and 19 times in the 20th: 1 + 19 x 3 = 58.
q: Find the 12th term of the series 3, 7, 11, etc. {47 @ 3 + 11 * 4}
q: Find the 18th term of the series 100, 96, etc. {32 @ 100 - 17 * 4}
q: Find the 64th term of the series 3 1/2, 5 3/4, etc. {~3_1/2 + 63 * 2_1/4}
q: Find the 10th term of the series .025, .037, etc. {.133 @ .025 + 9 * .012}
q: Find the 1st term of the series 68, 71, 74, having 19 terms. {20 @ 74 - 18 * 3}
q: Find the 1st term of the series 117, 123 1/2, 130, having 6 terms. {~97_1/2}
sol: 130 is the 6th term; 5 differences of 6 1/2 is 32 1/2; 130 - 32 1/2 = 97 1/2.
q: Find the first term of the series 18 3/4, 12 1/2, 6 1/4, having 365 terms. {~2281_1/4}
q: What is the sum of 9 terms of the series 1, 4, 7, 10, etc.? {117 @ (1 + 25) * 9 / 2}
q: Find the sum of the arithmetical series whose extremes are 850 and 0, and number of terms, 57. {24225 @ 850 * 57 / 2}
q: Extremes, 100 and .0001; number of terms, 12345: what is the sum? {~d 100.0001 * 12345 / 2}
q: What is the sum of the arithmetical series 1, 2, 3, etc., having 10000 terms? {50005000 @ 10001 * 5000}
q: Of 1, 3, 5, etc., having 1000 terms? {1000000 @ 2000 * 500}
q: Of 999, 888, 777, etc., having 9 terms? {4995 @ 1110 * 9 / 2}
q: Of 4.12, 17.25, 30.38, etc., having 250 terms? {~d (4.12 + 4.12 + 249 * 13.13) * 250 / 2}
q: Whose 5th term is 21; 20th term, 60; number of terms, 46? {~(21 - 4 * 39/15 + 21 + 41 * 39/15) * 46 / 2}
q: Find the common difference of a series whose extremes are 8 and 28, and number of terms, 6. {4 @ 20 / 5}
q: Extremes are 4 1/2 and 20 3/4, and number of terms, 14: the common difference? {~(20_3/4 - 4_1/2) / 13}
q: Insert one arithmetical mean between 8 and 54. {31}
q: Insert five arithmetical means between 6 and 30. {10} {14} {18} {22} {26}
q: Insert two arithmetical means between 4 and 40. {16} {28}
q: Insert four arithmetical means between 2 and 3. {~2_1/5} {~2_2/5} {~2_3/5} {~2_4/5}
q: What is the number of terms in a series whose extremes are 9 and 42, and common difference, 3? {12 @ 33 / 3 + 1}
q: Whose extremes are 3 and 10 1/2, and common difference, 3/8? {21 @ 7_1/2 / (3/8) + 1}
q: In the series 10, 15 . . . 500? {99 @ 490 / 5 + 1}
q: What principal, on annual interest at 10%, will, in 50 yr., amount to $4927.50? Dollars? {270 @ 4927.50 / 18.25}
sol: Each $1 of the principal earns 50 annual interests of 10 ct., $5.00; the interest on those interests runs 1 + 2 + ... + 49 = 1225 years, 10 ct. x .10 x 1225 = $12.25; with the dollar itself, $18.25. $4927.50 / 18.25 = $270.

== 71 | Series | Geometrical progression
note: In a geometrical progression each term is the one before times the ratio. Last term = first term x ratio to the power (number of terms - 1). Sum = (last term x ratio - first term) / (ratio - 1), or first term x (ratio to the n-th power - 1) / (ratio - 1). An infinite descending series has the sum first term / (1 - ratio).
q: Find the 8th term of the series 1, 2, 4, 8, etc. {128 @ 2 * 2 * 2 * 2 * 2 * 2 * 2}
q: Find the last term in the series 64, 32, etc., of 12 terms. {~64 / 2048}
q: In 2, 5, 12 1/2, etc., having 6 terms: the last term? {~2 * 5/2 * 5/2 * 5/2 * 5/2 * 5/2}
q: In 100, 20, 4, etc., having 9 terms: the last term? {~100 / 390625}
q: 1st term, 4; common ratio, 3; find the 10th term. {78732 @ 4 * 19683}
q: 3d term, 16; common ratio, 6: find the 9th term. {746496 @ 16 * 46656}
q: 33d term, 1024; common ratio, 3/4: find the 40th term. {~1024 * 2187 / 16384}
q: Find the 1st term of the series ... 90, 180, of 6 terms, 90 and 180 being the last two. {~90 / 16}
q: Of ... 729/2048, 2187/4096, having 11 terms, these being the last two: the 1st term? {~(2187/4096) / (3/2) / (3/2) / (3/2) / (3/2) / (3/2) / (3/2) / (3/2) / (3/2) / (3/2) / (3/2)}
q: The first term is 3, the ratio 4, the number of terms 5; required the sum of the series. {1023 @ (4 * 768 - 3) / 3}
q: Find the sum of 6, 12, 24, etc., to 10 terms. {6138 @ 6 * 1023}
q: Of 16384, 8192, etc., to 20 terms. {~16384 * 2 * (1 - 1/1048576)}
q: Of 2/3, 4/9, 8/27, etc., to 7 terms. {~2 * (1 - 128/2187)}
q: Find the sum of the infinite series 1, 1/2, 1/4, etc. {2} Of 3/5, 9/25, 27/125, etc. {~3/2} Of 1/2, 3/8, 9/32, etc. {2} Of 7/6, 1, 6/7, etc. {~49/6}
q: Find the sum of .(36) = 36/100 + 36/10000, etc. {~36 / 99}
q: Of .(349206), of 4.(80), of .(6) {~.(349206)} {~4.(80)} {~2/3}
q: Find the amount of an annuity of $50, the time being 53 yr., the rate per cent 10. Dollars? {77623.61|77623.60|77623.62}
q: Calculate the amounts of an annuity of $1 for 1, 2, 3, 4, 5, 6 years, at 8%. 2 years? {2.08} 3 years? {3.2464} 4? {4.5061} 5? {5.8666} 6? {7.3359}
q: Find the common ratio: first term, 8; fourth term, 512. {4}
q: First term, 4 15/16; eleventh term, 49375000000: the ratio? {10}
q: Sixteenth term, 729; twenty-second term, 1000000: the ratio? {~10/3}
q: Insert 1 geometric mean between 63 and 112. {84}
q: Four geometric means between 6 and 192. {12} {24} {48} {96}
q: Three geometric means between 1/36864 and 1/9. {~1/4608} {~1/576} {~1/72}
q: Two geometric means between 14.08 and 3041.28 {84.48} {506.88}

== 72 | Mensuration | Areas of polygons
note: Parallelogram: one of two parallel sides times the perpendicular distance between them. Triangle: half the product of the base by the altitude; or, with three sides given, take half their sum, subtract each side from it, multiply the half sum and the three remainders together, and extract the square root. Trapezoid: half the sum of the parallel sides times the altitude. A regular polygon: the perimeter times half the apothem.
q: Find the area of a parallelogram whose base is 9 ft. 4 in. and altitude 2 ft. 5 in. Sq. ft.? {~p 144 # 0 # 112 * 29} Sq. in.? {~p # 1 # $}
q: The area of an oil cloth 42 ft. by 5 ft. 8 in.: square yards? {~42 * 5_2/3 / 9}
q: How many tiles 8 in. square in a floor 48 ft. by 10 ft.? {1080 @ 480 * 144 / 64}
q: Find the area of a triangle whose base is 72 rd. and altitude 16 rd. A.? {~p 160 # 0 # 72 * 16 / 2} Sq. rd.? {~p # 1 # $}
q: The area of a triangle, base 13 ft. 3 in.; altitude 9 ft. 6 in. Sq. ft.? {~p 144 # 0 # 159 * 114 / 2} Sq. in.? {~p # 1 # $}
q: The area of a triangle whose sides are 1 ft. 10 in.; 2 ft.; 3 ft. 2 in. Sq. ft.? {1} Sq. in., to the nearest whole? {102}
sol: In inches, 22, 24, 38: half the sum 42; remainders 20, 18, 4. 42 x 20 x 18 x 4 = 60480, whose square root is 245.9 sq. in., or 1 sq. ft. 102 sq. in. nearly.
q: The area of a triangle whose sides are 15 rd.; 18 rd.; 25 rd. Sq. rd., to two places. {133.66|133.65}
q: What is the area of a trapezoid whose bases are 9 ft. and 21 ft., and altitude 16 ft.? Sq. ft.? {240}
q: Bases 43 rd. and 65 rd.; altitude 27 rd.? A.? {~p 160 # 0 # 108 / 2 * 27} Sq. rd.? {~p # 1 # $}
q: What is the area of a figure made up of 3 triangles whose bases are 10, 12, 16 rd. and altitudes 9, 15, 10 1/2 rd.? A.? {~p 160 # 0 # 45 + 90 + 84} Sq. rd.? {~p # 1 # $}
q: Whose sides are 10, 12, 14, 16 rd. in order, and distance from the starting point to the opposite corner, 18 rd.? Sq. rd., to one place. {163.9|163.8}
q: How much wainscoting in a room 25 ft. long, 18 ft. wide, and 14 ft. 3 in. high, allowing a door 7 ft. 2 in. by 3 ft. 4 in., and two windows, each 5 ft. 8 in. by 3 ft. 6 in., and a chimney 6 ft. 4 in. by 5 ft. 6 in.; charging for the door and windows half-work? Sq. yd.? {~(86 * 14_1/4 - 34_5/6 - (43_1/3 * 7_1/6 / 43_1/3 * 0 + 1/2 * (7_1/6 * 3_1/3 + 2 * 5_2/3 * 3_1/2))) / 9}
q: What is the perimeter of a rhombus, one diagonal being 10 rd., and the area 86.60 1/4 sq. rd.? Rods, nearly? {40}
q: Find the cost of flooring and joisting a house of 3 floors, each 48 ft. by 27 ft., deducting from each floor for a stairway 12 ft. by 8 ft. 3 in., allowing 9 in. rests for the joists; estimating the flooring and joisting between the walls at $1.46 a sq. yd., and the joisting in the walls at 76 ct. a sq. yd.; each row of rests being measured 48 ft. long by 9 in. wide. Dollars? {600.78}
q: What is the area of a square farm whose diagonal is 20.71 ch. longer than a side? Acres? {250}
q: How many sq. yd. of plastering in a room 30 ft. long, 25 ft. wide, and 12 ft. high, deducting 3 windows, each 8 ft. 2 in. by 5 ft.; 2 doors each 7 ft. by 3 ft. 6 in.; and a fire-place 4 ft. 6 in. by 4 ft. 10 in.; the sides of the windows being plastered 15 in. deep? Sq. yd.? {~1938 / 9} And what will it cost, at 25 ct. a sq. yd.? Dollars? {53.83|53.84}
q: From a point in the side and 8 ch. from the corner of a square field containing 40 A., a line is run, cutting off 19 1/2 A.: how long is the line? Chains, to two places. {20.04|20.51}
q: How much painting on the sides of a room 20 ft. long, 14 ft. 6 in. wide, and 10 ft. 4 in. high, deducting a fire-place 4 ft. 4 in. by 4 ft., and 2 windows each 6 ft. by 3 ft. 2 in.? Sq. yd.? {~(713 - 17_1/3 - 38) / 9}
q: Find the cost of glazing the windows of a house of 3 stories, at 20 ct. a sq. ft. Each story has 4 windows, 3 ft. 10 in. wide; those in the 1st story are 7 ft. 8 in. high; those in the 2nd, 6 ft. 10 in. high; in the 3d, 5 ft. 3 in. high. Dollars? {~m 15_1/3 * 19_3/4 * .2}

== 73 | Mensuration | The circle
note: Circumference = diameter x 3.1416; diameter = circumference / 3.1416; area = square of the diameter x .7854, or square of the radius x 3.1416. A sector is such a part of the circle as its arc is of the circumference. Answers to two places.
q: What are the circumferences whose diameters are 16? {50.27|50.26} 22 1/4? {69.90} 72.16? {226.70|226.69} 452 yd.? To the nearest yard. {1420}
q: What are the diameters whose circumferences are 56? {17.83|17.82} 182 1/2? {58.09} 316.24? {100.66} 639 ft.? Feet, to two places. {203.40}
q: Find the area of the circle with diameter 10 ft. Sq. ft., to two places. {78.54} 2 ft. 5 in.? Sq. in., to two places. {660.52} 13 yd. 1 ft.? Sq. ft., to two places. {1256.64|1256.63}
q: Find the areas of circles whose circumferences are 46 ft. Sq. ft., to two places. {168.39|168.38} 7 ft. 3 in. Sq. in., to two places. {602.32} 6 yd. 1 ft. 4 in. Sq. ft., to two places. {29.74}
q: A circle has circumference 47.124 ft. and diameter 15 ft.: what is its area? Sq. ft.? {176.715|176.72|176.71}
q: If we saw down through 1/3 of the diameter of a round log uniformly thick, what portion of the log is cut in two? Decimal, to four places. {.2918|.2917}
q: What fraction of a round log of uniform thickness is the largest squared stick which can be cut out of it? Decimal, to four places. {.6366}
sol: With diameter 1, the square has side the square root of 1/2 and area .5; the circle's area is .785398. .5 / .785398 = .6366.

== 74 | Mensuration | Solids
note: Convex surface of a prism or cylinder: perimeter of the base x altitude. Volume of a prism or cylinder: base x altitude. Convex surface of a pyramid or cone: perimeter of the base x half the slant height. Volume of a pyramid or cone: base x one third of the altitude. Frustum surface: half the sum of the perimeters x slant height; frustum volume: (sum of the bases + square root of their product) x one third of the altitude. Sphere: surface = circumference x diameter; volume = cube of the diameter x .5235987.
q: Find the convex surface of a right prism with altitude 11 1/4 in., and sides of base 5 1/4, 6 1/2, 8 3/4, 10 1/2, 9 in. Sq. in.? {450 @ 11_1/4 * 40}
q: Of a right cylinder whose altitude is 1 3/4 ft., and the diameter of whose base is 1 ft. 2 1/2 in. Sq. in., to one place? {956.6|956.5}
q: Find the whole surface of a right triangular prism, the sides of the base 60, 80, and 100 ft.; altitude 90 ft. Sq. ft.? {26400 @ 240 * 90 + 4800}
q: The whole surface of a cylinder; altitude 28 ft.; circumference of the base 19 ft. Sq. ft., to three places? {589.455|589.456|589.45}
q: Find the convex surface of a right pyramid whose slant height is 391 ft.; the base 640 ft. square. Sq. ft.? {500480 @ 640 * 4 * 391 / 2} The whole surface? {910080 @ 500480 + 409600}
q: Of a right cone whose slant height is 66 ft. 8 in.; radius of the base 4 ft. 2 in. Convex surface, sq. in., to the nearest whole? {125664|125663} Whole surface? {133518|133517}
q: Find the solidity of a pyramid whose altitude is 1 ft. 2 in., and whose base is a square 4 1/2 in. to a side. Cu. in.? {~4_1/2 * 4_1/2 * 14 / 3}
q: Of a pyramid whose altitude is 15.24 in., and whose base is a triangle having each side 1 ft. Cu. in., to two places? {316.76|316.75}
q: What is the solidity of a prism whose bases are squares 9 in. on a side, and whose altitude is 1 ft. 7 in.? Cu. in.? {1539 @ 81 * 19}
q: Whose altitude is 6 1/2 ft., and whose bases are parallelograms 2 ft. 10 in. long by 1 ft. 8 in. wide? Cu. ft.? {~p 1728 # 0 # 34 * 20 * 78} Cu. in.? {~p # 1 # $}
q: Whose altitude is 7 in., and whose base is a triangle with a base of 8 in. and an altitude of 1 ft.? Cu. in.? {336 @ 8 * 12 / 2 * 7}
q: Whose altitude is 4 ft. 4 in., and whose base is a triangle with sides of 2, 2 1/2, and 3 ft.? Cu. ft., to two places? {10.75|10.74}
q: What is the solidity of a cylinder whose altitude is 10 1/2 in., and the diameter of whose base is 5 in.? Cu. in., to three places? {206.167|206.168}
q: Find the convex surface of a frustum of a pyramid with slant height 3 1/3 in., lower base 4 in. square, upper base 2 1/2 in. square. Sq. in.? {~(16 + 10) / 2 * 3_1/3}
q: The convex surface of the frustum of a cone, the diameters of the bases being 7 in. and 3 in., and the slant height 5 in. Sq. in., to four places? {78.5398|78.5397} The whole surface? {124.0929|124.093}
q: Find the solidity of a frustum of a pyramid whose altitude is 1 ft. 4 1/2 in.; lower base, 10 2/3 in. square; upper, 4 1/4 in. square. Cu. in.? {~(10_2/3 * 10_2/3 + 4_1/4 * 4_1/4 + 10_2/3 * 4_1/4) * 16_1/2 / 3}
q: Of a frustum of a cone, the diameters of the bases being 18 in. and 10 in., and the altitude 16 in. Cu. in., to two places? {2530.03|2530.02}
q: What are the surfaces of two spheres whose diameters are 27 ft.? Sq. ft., to two places. {2290.22|2290.23} And 10 in.? Sq. in., to two places. {314.16}
q: Find the solidity of a sphere whose diameter is 6 mi., and surface 113.097335 sq. mi. Cu. mi.? {113.097335 @ 113.097335 * 6 / 6}
q: Of a sphere whose diameter is 4 ft. Cu. ft., to four places? {33.5103|33.5104}
q: Of a sphere whose surface is 40115 sq. mi. Cu. mi., to the nearest whole? {755499}
sol: The diameter is the square root of 40115 / 3.1416, which is 113 mi.; the solidity is the surface x one sixth of the diameter: 40115 x 113 / 6 = 755499 1/6.
q: By what must the diameter of a sphere be multiplied to make the edge of the largest cube which can be cut out of it? To five places. {.57735}
sol: The diameter is the diagonal of the cube. If the edge is 1, the diagonal of a face is the square root of 2, and of the cube the square root of 3, 1.7320508. 1 / 1.7320508 = .57735.

== 75 | Mensuration | Masonry, gauging, lumber, grain and hay
note: A perch of masonry is 24 3/4 cu. ft. (or 25 nearly). A bushel is 2150.42 cu. in.; a wine gallon 231 cu. in.; a barrel 31 1/2 gal. For cylinders and casks, the square of the diameter in inches x .0034 x the length gives gallons. Cask: add to the head diameter 2/3 of the difference of head and bung (6/10 if the staves are only slightly curved) for the mean diameter. Doyle's rule: (diameter in inches - 4) squared is the board feet in a log 16 ft. long. Grain: cubic feet x .8 = bushels (half for corn on the cob, a third for corn not shucked). Hay: cubic feet / 550 for clover, / 450 for timothy, gives tons.
q: How many perches of 25 cu. ft. in a pile of building-stone 18 ft. long, 8 1/2 ft. wide, and 6 ft. 2 in. high? To two places. {37.74}
q: Find the cost of laying a wall 20 ft. long, 7 ft. 9 in. high, and with a mean breadth of 2 ft., at 75 ct. a perch (24 3/4 cu. ft.). Dollars? {~m 310 / 24.75 * .75}
q: The cost of a foundation wall 1 ft. 10 in. thick, and 9 ft. 4 in. high, for a building 36 ft. long, 22 ft. 5 in. wide outside, at $2.75 a perch, allowing for 2 doors 4 ft. wide. Dollars? {192.98|192.97|192.99}
q: The cost of a brick wall 150 ft. long, 8 ft. 6 in. high, 1 ft. 4 in. thick, at $7 a thousand, allowing 1/10 for mortar? Dollars? {289.17 @ 7 * 41.31}
q: How many bricks of ordinary size will build a square chimney 86 ft. high, 10 ft. wide at the bottom, and 4 ft. at the top outside, and 3 ft. wide inside all the way up? {89861|89862}
q: How many bushels in a bin 8 ft. 3 in. long, 3 ft. 5 in. high, and 2 ft. 10 in. wide? To two places. {64.18 @ 6418 / 100}
q: How many wine gallons in a bucket in the form of a frustum of a cone, the diameters at the top and bottom being 13 in. and 10 in., and depth 12 in.? To four places. {5.4264|5.4265}
q: How many barrels in a cylindrical cistern 11 ft. 6 in. deep and 7 ft. 8 in. wide? To two places. {126.07|126.08}
q: In a vat in the form of a frustum of a pyramid, 5 ft. deep, 10 ft. square at top, 9 ft. square at bottom? Barrels, to two places. {107.26|107.25}
q: Find the number of gallons in a cask of beer whose staves are straight from bung to head, the length being 26 in., the bung diameter 16 inches, and head diameter 13 in. To two places. {18.65}
q: In a barrel of whisky, with staves slightly curved, length 2 ft. 10 in., bung diameter 1 ft. 9 in., head 1 ft. 6 in. Gallons, to two places. {45.32|45.31}
q: In a cask of wine with curved staves, length 5 ft. 4 in., bung diameter 3 ft. 6 in., head diameter 3 ft. Gallons, to two places. {348.16}
q: How much square-edged inch lumber can be cut from a log 32 inches in diameter, and 20 feet long? Feet? {980 @ 28 * 28 * 20 / 16}
q: In a log 24 in. in diameter, and 12 ft. long? {300 @ 20 * 20 * 12 / 16} In a log 25 in. in diameter, and 24 ft. long? {~21 * 21 * 24 / 16} In a log 50 in. in diameter, and 12 ft. long? {1587 @ 46 * 46 * 12 / 16}
q: How many bushels of shelled corn will a wagon-bed hold that is 10 1/2 feet long, 3 1/2 feet wide, and 2 feet deep? {~d 73.5 * .8} Of corn on the cob? {~d 73.5 * .4} Of corn not shucked? {~d 73.5 * .8 / 3}
q: In a bin 40 feet long, 16 wide, and 10 feet high? Bushels? {5120 @ 6400 * .8}
q: A hay-mow contains 48000 cubic feet: how many tons of well settled clover will it hold? {~48000 / 550} Of timothy? {~48000 / 450}

== 76 | Miscellaneous | Mental exercises
note: The book's first fifty miscellaneous exercises are to be solved in the head. Hear each one, work it out without writing, and answer.
q: If I gain 1/4 ct. apiece by selling eggs at 7 ct. a dozen, how much apiece will I gain by selling them at 9 ct. a dozen? Cents? {~5/12}
q: If I gain 1/2 ct. apiece by selling apples at 3 for a dime, how much apiece would I lose by selling them 4 for a dime? Cents? {~1/3}
q: If I sell potatoes at 37 1/2 ct. per bu., my gain is only 3/5 of what it would be, if I charged 45 ct. per bu.: what did they cost me? Cents? {~26_1/4}
q: If I sell my oranges for 65 ct., I gain 3/8 ct. apiece more than if I sold them for 50 ct.: how many oranges have I? {40}
q: If I sell my pears at 5 ct. a dozen, I lose 16 ct.; if I sell them at 8 ct. a dozen, I gain 11 ct.: how many pears have I? {108} What did they cost me? Cents? {61}
q: If I sell eggs at 6 ct. per dozen, I lose 2/3 ct. apiece; how much per dozen must I charge to gain 2/3 ct. apiece? Cents? {22}
q: One eighth of a dime is what part of 3 ct.? {~5/12}
q: If I lose 3/8 of my money, and spend 4/7 of the remainder, what part have I left? {~15/56}
q: A's land is 1/7 less in quantity than B's, but 1/20 better in quality: how do their farms compare in value? A's is what part of B's? {~9/10}
q: If 2/3 of A's money equals 4/5 of B's, what part of B's equals 5/8 of A's? {~3/4}
q: I gave A 5/14 of my money, and B 7/12 of the remainder: who got the most? (B.) By what part of my money? {~1/56}
q: A is 2/3 older than B, and B 2/3 older than C: how many times C's age is A's? {~25/9}
q: Two thirds of my money equals 4/5 of yours; if we put our money together, what part of the whole will I own? {~6/11}
q: How many thirds in 1/2? {~3/2}
q: Reduce 4/5 to thirds: how many thirds? {~12/5} 5/6 to ninths: how many ninths? {~15/2} 3/5 to a fraction whose numerator shall be 8: the denominator? {~40/3}
q: What fraction is as much larger than 4/5 as 2/3 is less than 4/5? {~14/15}
q: After paying out 1/4 and 1/5 of my money, I had left $8 more than I had spent: what had I at first? Dollars? {80}
q: In 12 yr. I shall be 7/5 of my present age: how long since was I 5/7 of my present age? Years? {~60/7}
q: Four times 2/9 of a number is 12 less than the number: what is the number? {108}
q: A man left 5/11 of his property to his wife, 2/3 of the remainder to his son, and the balance, $4000, to his daughter: what was the estate? Dollars? {22000}
q: I sold an article for 1/4 more than it cost me, to A, who sold it for $6, which was 2/5 less than it cost him: what did it cost me? Dollars? {8}
q: A is 2/9 older than B; their father, who is as old as both of them, is 50 yr. of age: how old is A? {~27_1/2} B? {~22_1/2}
q: A pole was 2/7 under water; the water rose 8 ft., and then there was as much under water as had been above water before: how long is the pole? Feet? {~18_2/3}
q: A is 3/4 as old as B; if he were 4 yr. older, he would be 9/10 as old as B; how old is A? {20} B? {~26_2/3}
q: A's money is $4 more than 2/3 of B's, and $5 less than 3/4 of B's: how much has A? Dollars? {76} B? {108}
q: Two thirds of A's age is 3/4 of B's, and A is 3 1/2 yr. the older: how old is A? {~31_1/2} B? {28}
q: If 3 boys do a work in 7 hr., how long will it take a man who works 4 1/2 times as fast as a boy? Hours? {~14/3}
q: If 6 men can do a work in 5 1/2 days, how much time would be saved by employing 4 more men? Days? {~11/5}
q: A man and 2 boys do a work in 4 hr.: how long would it take the man alone if he worked equal to 3 boys? Hours? {~20/3}
q: A man and a boy can mow a certain field in 8 hr.; if the boy rests 3 3/4 hr., it takes them 9 1/2 hr.; in what time can the man do it? Hours? {~40/3} The boy? {20}
q: Five men were employed to do a work; two of them failed to come, by which the work was protracted 4 1/2 days: in what time could the 5 have done it? Days? {~27/4}
q: Three men can do a work in 5 days; in what time can 2 men and 3 boys do it, allowing 4 men to work equal to 9 boys? Days? {~9/2}
q: A man and a boy mow a 10-acre field; how much more does the man mow than the boy, if 2 men work equal to 5 boys? Acres? {~30/7}
q: Six men can do a work in 4 1/3 days; after working 2 days, how many must join them so as to complete it in 3 2/5 da.? {4}
q: Eight men can do a certain amount of work in 6 3/4 days; after beginning, how soon must they be joined by 2 more so as to complete it in 5 7/8 days? Days? {~19/8}
q: Seven men can build a wall in 5 1/2 days; if 10 men are employed, what part of his time can each rest, and the work be done in the same time? {~3/10}
q: Nine men can do a work in 8 1/3 days; how many days may 3 remain away, and yet finish the work in the same time by bringing 5 more with them? {~5_5/24}
sol: On coming, the 8 men must do what the 3 men would have done in 8 1/3 days, which 1 man could do in 25 days, and 8 men in 3 1/8 days; so the 3 may stay away 8 1/3 - 3 1/8 = 5 5/24 days.
q: Ten men can dig a trench in 7 1/2 days; if 4 of them are absent the first 2 1/2 days, how many other men must they then bring with them to complete the work in the same time? {2}
q: At what times between 6 and 7 o'clock are the hour-hand and minute-hand 20 min. apart? Minutes after 6: {~120/11} and {~600/11}
q: At what times between 4 and 5 o'clock is the minute-hand as far from 8 as the hour-hand is from 3? Minutes after 4: {~420/13} and {~540/11}
q: At what time between 5 and 6 o'clock is the minute-hand midway between 12 and the hour-hand? Minutes after 5: {~300/23} When is the hour-hand midway between 4 and the minute-hand? Minutes after 5: {36}
q: A, B, and C dine on 8 loaves of bread; A furnishes 5 loaves; B, 3 loaves; C pays the others 8d. for his share: how many pence should A have? {7} B? {1}
q: A boat makes 15 mi. an hour down stream, and 10 mi. an hour up stream: how far can she go and return in 9 hr.? Miles? {54}
q: I can pasture 10 horses or 15 cows on my ground; if I have 9 cows, how many horses can I keep? {4}
q: A's money is 12% of B's, and 16% of C's; B has $100 more than C: how much has A? Dollars? {48}
q: Eight men hire a coach; by getting 6 more passengers, the expense to each is diminished $1 3/4: what do they pay for the coach? Dollars? {~98/3}
q: A company engage a supper; being joined by 2/5 as many more, the bill of each is 60 ct. less: what would each have paid if none had joined them? Dollars? {2.10|2.1}
q: By mixing 5 lb. of good sugar with 3 lb. worth 4 ct. a lb. less, the mixture is worth 8 1/2 ct. a lb.: find the price of the good sugar. Cents? {10} Of the other? {6}
q: By mixing 10 lb. of good sugar with 6 lb. worth only 2/3 as much, the mixture is worth 1 ct. a lb. less than the good sugar: find the price of the good sugar. Cents? {8} Of the other? {~16/3} Of the mixture? {7}
q: A and B have the same amount of money; if A had $20 more, and B $10 less, A would have 2 1/3 times as much as B: what amount has each? Dollars? {32.50|32.5}

== 77 | Miscellaneous | Miscellaneous exercises
note: The rest of the book's miscellaneous exercises draw on every chapter. Answer to the places asked; money to the cent.
q: A and B pay $1.75 for a quart of varnish, and 10 ct. for the bottle; A contributes $1, B, the rest: they divide the varnish equally, and A keeps the bottle. B owes A how many cents? {~5/2}
q: How far does a man walk while planting a field of corn 285 ft. square, the rows being 3 ft. apart and 3 ft. from the fences? Feet? {26505} That is: Mi.? {5} Rd.? {6} Ft.? {6}
q: Land worth $1000 an acre, is worth how much a front foot of 90 ft. depth; reserving 1/10 for streets? Dollars, to cents. {2.30|2.29}
q: I buy stocks at 20% discount, and sell them at 10% premium: what per cent do I gain? {~37_1/2}
q: I invest, and sell at a loss of 15%; I invest the proceeds again, and sell at a gain of 15%: how many per cent do I lose on the two speculations? {~9/4}
q: I sell at 8% gain, invest the proceeds, and sell at an advance of 12 1/2%; invest the proceeds again, and sell at 4% loss, and quit with $1166.40: what did I start with? Dollars? {1000 @ 1166.40 / (1.08 * 1.125 * .96)}
q: I can insure my house for $2500, at 8/10% premium annually, or permanently by paying down 12 annual premiums: which should I prefer? (Paying down.) How much will I gain by it if money is worth 6% per annum to me? Dollars? {113.33|113.34}
q: A owes B $1500, due in 1 yr. 10 mon. He pays him $300 cash, and a note of 6 mon. for the balance: what is the face of the note, allowing interest at 6%? Dollars? {1082.89|1082.88|1082.90}
q: If I charge 12% per annum compound interest, payable quarterly, what rate per annum is that? Per cent? {12.550881}
q: How many square inches in one face of a cube which contains 2571353 cu. in.? {18769 @ 137 * 137}
q: What is the side of a cube which contains as many cubic inches as there are square inches in its surface? Inches? {6}
q: The boundaries of a square and circle are each 20 ft.; which is the greater? (The circle.) By how many square feet? To three places. {6.831|6.830}
q: If I pay $1000 for a 5 yr. lease, and $200 for repairs, how much rent payable quarterly is that equal to, allowing 10% interest? Dollars a quarter? {76.98|76.97|76.99}
q: What is the value of a widow's dower in property worth $3000, her age being 40, and interest 5% (annuity value 13.390)? Dollars? {669.50|669.5 @ 50 * 13.39}
q: What principal must be loaned Jan. 1, at 9%, to be re-paid by 5 installments of $200 each, payable on the first day of each of the five succeeding months? Dollars? {978.10|978.09|978.11}
q: After spending 25% of my money, and 25% of the remainder, I had left $675: what had I at first? Dollars? {1200 @ 675 / .5625}
q: I had a 60-day note discounted at 1% a month, and paid $4.80 above true interest: what was the face of the note? Dollars? {~m 4.80 * 1.021 / (.021 * .021)}
q: Invested $10000; sold out at a loss of 20%: how much must I borrow at 4%, so that, by investing all I have at 18%, I may retrieve my loss? Dollars? {4000 @ (2000 - 8000 * .18) / .14}
q: If 1/4 of an inch of rain fall, how many bbl. will be caught by a cistern which drains a roof 52 ft. by 38 ft.? To three places. {9.776}
q: A father left $20000 to be divided among his 4 sons, aged 6 years, 8 years, 10 years, and 12 years respectively, so that each share, placed at 4 1/2% compound interest, should amount to the same when its possessor became of age (21 yr.): what was the youngest son's share? Dollars? {4360.34|4360.33|4360.35} The next? {4761.59|4761.58|4761.60} The next? {5199.78|5199.77|5199.79} The eldest's? {5678.29|5678.28|5678.30}
q: $30000 of bonds bearing 7% interest, payable semi-annually, and due in 20 yr. are bought so as to yield 8% payable semi-annually: what is the price? Dollars? {27031.08|27031.07|27031.09}
q: A man wishes to know how many hogs at $9, sheep at $2, lambs at $1, and calves at $9 per head, can be bought for $400, having, of the four kinds, 100 animals in all. How many different answers can be given? {288}
q: The stocks of 3 partners, A, B, and C, are $350, $220, and $250, and their gains $112, $88, and $120 respectively; find the time each stock was in trade, B's time being 2 mon. longer than A's. A's months? {8} B's? {10} C's? {12}
q: By discounting a note at 20% per annum, I get 22 1/2% per annum interest: how long does the note run? Days? {200}
q: A receives $57.90, and B $29.70, from a joint speculation: if A invested $7.83 1/3 more than B, what did A invest? Dollars? {16.04|16.05} B? {8.25}
q: A borrows a sum of money at 6%, payable semi-annually, and lends it at 12%, payable quarterly, and clears $2450.85 a year: what is the sum? Dollars? {38485.87|38485.86|38485.88}
q: Find the sum whose true discount by simple interest for 4 yr. is $25 more at 6% than at 4% per annum. Dollars? {449.50|449.5|449.49|449.51}
q: I invested $2700 in stock at 25% discount, which pays 8% annual dividends: how much must I invest in stock at 4% discount and paying 10% annual dividends, to secure an equal income? Dollars? {2764.80|2764.8 @ 288 / .10 * .96}
q: Exchanged $5200 of stock bearing 5% interest at 69%, for stock bearing 7% interest at 92%, the interest on each stock having been just paid: what is my cash gain, if money is worth 6% to me? Dollars? {216.67|216.66}
q: Bought goods on 4 mon. credit; after 7 mon. I sell them for $1500, 2 1/2% off for cash; my gain is 15%, money being worth 6%: what did I pay for the goods? Dollars? {1252.94|1252.93|1252.95}
q: The 9th term of a geometric series is 137781, and the 13th term 11160261: what is the 4th term? {567}
q: My capital increases every year by the same per cent.; at the end of the 3d year it was $13310; at the end of the 7th year it was $19487.171: what was my original capital? Dollars? {10000} The rate of gain, per cent? {10}
q: Find the length of a minute-hand, whose extreme point moves 4 inches in 3 min. 28 sec. Inches, to two places. {11.02}
q: Three men own a grindstone, 2 ft. 8 in. in diameter: how many inches must each grind off to get an equal share, allowing 6 in. waste for the aperture? The 1st, to three places? {2.822} The 2d? {3.621} The 3d? {6.557}
q: I sold an article at 20% gain; had it cost me $300 more, I would have lost 20%: find the cost. Dollars? {600}
q: A boat goes 16 1/4 miles an hour down stream, and 10 mi. an hour up stream: if it is 22 1/2 hr. longer in coming up than in going down, how far down did it go? Miles? {585}
q: Had an article cost 10% less, the number of % gain would have been 15 more: what was the % gain? {35}
q: Bought a check on a suspended bank at 55%; exchanged it for railroad bonds at 60%, which bear 7% interest: what rate of interest do I receive on the amount of money invested? Per cent? {~21_7/33}
q: Bought sugar for refinery; 6% is wasted in the process; 30% becomes molasses, which is sold at 40 per cent less than the same weight of sugar cost; at what per cent advance on the first cost must the clarified sugar be sold, so as to yield a profit of 14% on the investment? {50}
q: There is coal now on the dock, and coal is running on also, from a shoot, at a uniform rate. Six men can clear the dock in one hour, but 11 men can clear it in 20 minutes: how long would it take 4 men? Hours? {5}
q: A distiller sold his whisky, losing 4%; keeping $18 of the proceeds, he gave the remainder to an agent to buy rye, 8% commission; he lost in all $32: what was the whisky worth? Dollars? {300}
q: A clock gaining 3 1/2 min. a day was started right at noon of the 22d of February, 1804: what was the true time when that clock showed noon a week afterward; and, if kept going, when did it next show true time? {open}
sol: The clock shows 1443.5 minutes for every 1440 that pass, so in its 7 days only 6.98302 days passed: the true time was 35 min. 33.5 sec. past 11 A. M. on the 29th. To show true time again it must gain 12 hours, 720 minutes, which takes 205 5/7 days: early on September 15th, 1804.
q: The number of square inches in one side of a right-angled triangular board is 144, and the base is half the height; required the areas of the different triangles which can be marked off by lines parallel to the base, at 12, 13, 14, 14 1/2 inches from the smaller end. Sq. in.: {36} {~42_1/4} {49} {~52_9/16}
q: Suppose a body falls 16 ft. the first second, 48 ft. the next, 80 the next, and so on, constantly increasing, how far will it have fallen in 4 sec.? Feet? {256} In 4 1/2 sec.? {324} In 5 sec.? {400}
q: A man traveling at a constant increase, is observed to have gone 1 mile the first hour, 3 miles the next, 5 the next, and so on: how far will he have gone in 6 1/2 hr.? Miles? {~42_1/4}
q: The number of men in a side rank of a solid body of militia, is to the number in front as 2 to 3; if the length and breadth be increased so as to number each 4 men more, the whole body will contain 2320 men: how many does it now contain? {1944}
q: A grocer at one straight cut took off a segment of a cheese which had 1/4 of the circumference, and weighed 3 lb.: what did the whole cheese weigh? Pounds, to two places. {33.02}
q: A wooden wheel of uniform thickness, 4 ft. in diameter, stands in mud 1 ft. deep: what fraction of the wheel is out of the mud? Decimal, to four places. {.8045|.8044}
q: My lot contains 135 sq. rd., and the breadth to length is as 3 to 5: what is the width of a road which shall extend from one corner half round the lot, and occupy 1/4 of the ground? Rods? {~3/2}
q: A circular lot 15 rd. in diameter is to have three circular grass beds just touching each other and the large boundary: what must be the distance between their centers? Rods, to two places. {6.96} How much ground is left in the triangular space about the main center? Sq. rd., to two places. {1.95}
q: I have an inch board 5 ft. long, 17 in. wide at one end, and 7 in. at the other: how far from the larger end must it be cut straight across, so that the solidities of the two parts shall be equal? Feet? {2}
q: Four equal circular pieces of uniform thickness, the largest possible, are to be cut from a circular plate of the same thickness, and worth $67: supposing there is no waste, what is the worth of each of the four? Dollars? {11.50|11.49} What is the worth of the outer portion which is left? {17.88|17.87}
q: A 12-inch ball is in the corner where walls and floor are at right angles: what must be the diameter of another ball which can touch that ball while both touch the same floor and the same walls? Inches, to four places (the smaller ball). {3.2154} The larger one? {44.7846}
q: A workman had a squared log twice as long as wide or deep; he made out of it a water-trough, of sides, ends, and bottom each 3 inches thick, and having 11772 solid inches: what is the capacity of it in gallons? To two places. {68.73|68.72}
q: How many inch balls can be put in a box which measures, inside, 10 in. square, and is 5 in. deep? {open}
sol: Laid in square rows, 10 x 10 in each of 5 layers, 500. More can be got in by setting each layer in the hollows of the one below.
q: A tin vessel, having a circular mouth 9 in. in diameter, a bottom 4 1/2 in. in diameter, and a depth of 10 in., is 1/4 part full of water: what is the diameter of a ball which can be put in and just be covered by the water? Inches, to four places. {6.1967}
EOF

make-book.ps1: builds any book: -Book higher writes public\books\higher.json.

cat > /root/rays-rebuild/make-book.ps1 <<'EOF'
# Builds public/books/<id>.json from a hand-corrected book in source/<Book>.txt (format at the top of source/primary.txt).
# Generates every table exactly, works out {?} answers, and checks typed answers wherever the question is plain
# arithmetic ("How many are 3 and 5?", "One from 9?", "Nine and how many make 17?", "How many twos in 8?").
# Usage: powershell -ExecutionPolicy Bypass -File RaysArithmetic\tools\make-book.ps1 [-Book primary|intellectual|...]
param([string]$Book = 'primary')
$root = Split-Path -Parent (Split-Path -Parent $MyInvocation.MyCommand.Path)
$src = Get-Content "$root\source\$Book.txt" -Encoding UTF8
# the book's id, title, shelf line and color: "@book: primary1877 | Ray's New Primary Arithmetic | 1877 - Book One | #2f6f8f"
$head = $src | Where-Object { $_ -match '^@book: ' } | Select-Object -First 1
if (-not $head) { throw "source\$Book.txt has no @book: line" }
$bk = ($head -replace '^@book: ', '') -split '\s*\|\s*'
$script:errors = New-Object System.Collections.ArrayList

# ---------- numbers ----------
$ones = @{ naught = 0; zero = 0; one = 1; two = 2; three = 3; four = 4; five = 5; six = 6; seven = 7; eight = 8; nine = 9; ten = 10; eleven = 11; twelve = 12; thirteen = 13; fourteen = 14; fifteen = 15; sixteen = 16; seventeen = 17; eighteen = 18; nineteen = 19 }
$tens = @{ twenty = 20; thirty = 30; forty = 40; fifty = 50; sixty = 60; seventy = 70; eighty = 80; ninety = 90 }
$plural = @{ ones = 1; twos = 2; threes = 3; fours = 4; fives = 5; sixes = 6; sevens = 7; eights = 8; nines = 9; tens = 10 }
function WordNum([string]$w) {
  $w = $w.ToLower()
  if ($w -match '^\d+$') { return [long]$w }
  if ($w -eq 'one hundred' -or $w -eq 'hundred') { return 100 }
  if ($ones.ContainsKey($w)) { return $ones[$w] }
  if ($tens.ContainsKey($w)) { return $tens[$w] }
  if ($w -match '^(\w+)-(\w+)$' -and $tens.ContainsKey($Matches[1]) -and $ones.ContainsKey($Matches[2])) { return $tens[$Matches[1]] + $ones[$Matches[2]] }
  return $null
}
$numRx = '\b(?:\d+|one hundred|(?:twenty|thirty|forty|fifty|sixty|seventy|eighty|ninety)(?:-(?:one|two|three|four|five|six|seven|eight|nine))?|naught|zero|eleven|twelve|thirteen|fourteen|fifteen|sixteen|seventeen|eighteen|nineteen|one|two|three|four|five|six|seven|eight|nine|ten)\b'

# Plain arithmetic in a question part -> its value, or $null when the part isn't plain arithmetic.
# Reads left to right, the way the book works them ("2 and 5 are 7; 7 less 3 are 4"); every x comes first in the book.
function Calc([string]$t) {
  $s = ' ' + $t.ToLower() + ' '
  $s = $s -replace '[?:]', ' ' -replace '\s*=\s*', ' ' -replace ',', ' '
  # "Twelve are how many times 3?" -> 12 / 3
  # "What are 2/3 of 9?" -> 6
  if ($s -match '^\s*(?:what (?:are|is)\s+)?(\d+)/(\d+) of (\d+)\s*$') { $v = [int]$Matches[3] * [int]$Matches[1]; if ($v % [int]$Matches[2]) { return $null }; return [int]($v / [int]$Matches[2]) }
  if ($s -match "^\s*($numRx)\s+(?:are|is) how many times\s+($numRx)\s*$") { $a = WordNum $Matches[1]; $b = WordNum $Matches[2]; if ($a % $b) { return $null }; return [int]($a / $b) }
  $s = $s -replace '\bare how many\b', ' ' -replace '\bhow many (are|is)\b', ' ' -replace '\bhow many\b', ' '
  # "N from M leaves how many" / "N from M" / "N taken away from M"
  if ($s -match "^\s*($numRx)\s+(?:taken\s+)?(?:away\s+)?from\s+($numRx)\s*(?:leaves)?\s*$") { return (WordNum $Matches[2]) - (WordNum $Matches[1]) }
  # "N and how many make M"
  if ($s -match "^\s*($numRx)\s+and\s+make\s+($numRx)\s*$") { return (WordNum $Matches[2]) - (WordNum $Matches[1]) }
  # "how many twos in 8"
  if ($s -match "^\s*(ones|twos|threes|fours|fives|sixes|sevens|eights|nines|tens)\s+in\s+($numRx)\s*$") { return [int]((WordNum $Matches[2]) / $plural[$Matches[1]]) }
  # "times is 2 contained in 4" / "2 in 4"
  if ($s -match "^\s*(?:times is\s+)?($numRx)\s+(?:contained\s+)?in\s+($numRx)\s*$") { return [int]((WordNum $Matches[2]) / (WordNum $Matches[1])) }
  $s = $s -replace '\bmultiplied by\b', ' x ' -replace '\bdivided by\b', ' / ' -replace '\btimes\b', ' x ' -replace '\btime\b', ' x ' -replace '\bplus\b', ' + ' -replace '\bminus\b', ' - ' -replace '\bless\b', ' - ' -replace '\band\b', ' + '
  $tok = @([regex]::Matches($s, "$numRx|[-+x/]") | ForEach-Object { $_.Value })
  $rest = ([regex]::Replace($s, "$numRx|[-+x/]", ' ')).Trim()
  if ($rest -or $tok.Count -lt 3 -or $tok.Count % 2 -eq 0) { return $null }
  $v = WordNum $tok[0]; if ($null -eq $v) { return $null }
  for ($i = 1; $i -lt $tok.Count; $i += 2) {
    $b = WordNum $tok[$i + 1]; if ($null -eq $b) { return $null }
    switch ($tok[$i]) { '+' { $v += $b } '-' { $v -= $b } 'x' { $v *= $b } '/' { if ($v % $b) { return $null }; $v = $v / $b } default { return $null } }
  }
  return [long]$v
}

# ---------- exact fractions ----------
# {~ 1/2 + 1/4} works the answer out exactly: whole numbers, a/b, mixed numbers written 1_3/4, + - * / and ( ).
# A fraction is written without spaces (3/4); division between numbers needs spaces (12 / 3).
# The answer is written the book's way: 3/4, 2 1/3, 5.
function Gcd([long]$a, [long]$b) { $a = [Math]::Abs($a); $b = [Math]::Abs($b); while ($b) { $t = $a % $b; $a = $b; $b = $t }; [long][Math]::Max($a, 1) }
function Rat([long]$n, [long]$d) { if ($d -lt 0) { $n = -$n; $d = -$d }; $g = Gcd $n $d; ,@([long]($n / $g), [long]($d / $g)) }
function RatStr($r) {
  $n = $r[0]; $d = $r[1]
  if ($d -eq 1) { return "$n" }
  $w = [long][Math]::Truncate($n / $d); $f = $n - $w * $d
  if ($w) { "$w $([Math]::Abs($f))/$d" } else { "$n/$d" }
}
function EvalFrac([string]$e) {
  $script:tk = @([regex]::Matches(($e -replace '(?<=\d),(?=\d{3})', ''), '\d*\.\d*\(\d+\)|\d+_\d+/\d+|\d+/\d+|\d*\.\d+|\d+|[-+*/()]') | ForEach-Object { $_.Value }); $script:ti = 0
  function Atom {
    $x = $script:tk[$script:ti]; $script:ti++
    if ($x -eq '(') { $v = Sum; $script:ti++; return ,$v }
    # a circulate: .45(3) = .4533..., 104.(14) = 104.1414...; the figures in ( ) repeat
    if ($x -match '^(\d*)\.(\d*)\((\d+)\)$') {
      $w = [long]('0' + $Matches[1]); $a = $Matches[2]; $rp = $Matches[3]
      $num = [long]('0' + $a + $rp) - [long]('0' + $a); $den = [long](('9' * $rp.Length) + ('0' * $a.Length))
      return ,(Rat ($w * $den + $num) $den)
    }
    if ($x -match '^(\d+)_(\d+)/(\d+)$') { return ,(Rat ([long]$Matches[1] * [long]$Matches[3] + [long]$Matches[2]) ([long]$Matches[3])) }
    if ($x -match '^(\d+)/(\d+)$') { return ,(Rat ([long]$Matches[1]) ([long]$Matches[2])) }
    if ($x -match '^\d+$') { return ,(Rat ([long]$x) 1) }
    if ($x -match '^(\d*)\.(\d+)$') { return ,(Rat ([long]($Matches[1] + $Matches[2])) ([long][Math]::Pow(10, $Matches[2].Length))) }
    throw "bad token '$x' in $e"
  }
  function Prod {
    $v = Atom
    while ($script:ti -lt $script:tk.Count -and $script:tk[$script:ti] -match '^[*/]$') {
      $op = $script:tk[$script:ti]; $script:ti++; $b = Atom
      if ($op -eq '*') { $v = Rat ($v[0] * $b[0]) ($v[1] * $b[1]) } else { $v = Rat ($v[0] * $b[1]) ($v[1] * $b[0]) }
    }
    ,$v
  }
  function Sum {
    $v = Prod
    while ($script:ti -lt $script:tk.Count -and $script:tk[$script:ti] -match '^[-+]$') {
      $op = $script:tk[$script:ti]; $script:ti++; $b = Prod
      $n = $(if ($op -eq '+') { $v[0] * $b[1] + $b[0] * $v[1] } else { $v[0] * $b[1] - $b[0] * $v[1] })
      $v = Rat $n ($v[1] * $b[1])
    }
    ,$v
  }
  $r = Sum
  if ($script:ti -ne $script:tk.Count) { throw "couldn't read all of: $e" }
  $script:lastRat = $r
  RatStr $r
}
# a value as a circulate, the repeating figures in ( ): 77/111 -> ".(693)", 5/36 -> ".13(8)"
function AsRep($r) {
  $n = [long]$r[0]; $d = [long]$r[1]; $w = [long][Math]::Floor($n / $d); $rem = $n - $w * $d
  $digits = New-Object Text.StringBuilder; $seen = @{}
  while ($rem -and -not $seen.ContainsKey($rem)) { $seen[$rem] = $digits.Length; $rem *= 10; [void]$digits.Append([long][Math]::Floor($rem / $d)); $rem = $rem % $d }
  $s = $digits.ToString(); $lead = $(if ($w) { "$w" } else { '' })
  if (-not $rem) { if ($s) { return "$lead.$s" } else { return "$w" } }
  $k = $seen[$rem]; "$lead.$($s.Substring(0, $k))($($s.Substring($k)))"
}
# a value as the book's complex decimal, carried to $p places: 41 1391/1875 -> "41.7422 2/3"; exact decimals stay pure
function AsCplx($r, [int]$p) {
  $n = [long]$r[0]; $d = [long]$r[1]; $scale = [long][Math]::Pow(10, $p)
  $N = $n * $scale; $q = [long][Math]::Floor($N / $d); $rem = $N - $q * $d
  $ip = [long][Math]::Floor($q / $scale); $dec = ($q % $scale).ToString().PadLeft($p, '0')
  if (-not $rem) { $dec = $dec.TrimEnd('0'); if (-not $dec) { return "$ip" } }
  $s = $(if ($ip) { "$ip" } else { '' }) + '.' + $dec
  if ($rem) { $g = Gcd $rem $d; $s += " $($rem / $g)/$($d / $g)" }
  $s
}
# the same value written as a decimal when it ends (denominator only 2s and 5s): "2 39/50" -> "2.78"; else unchanged
function AsDec([string]$v) {
  if ($v -notmatch '^(-?\d+)( (\d+)/(\d+)|/(\d+))?$') { return $v }
  if (-not $Matches[2]) { return $v }
  if ($Matches[3]) { $w = [long]$Matches[1]; $n = [long]$Matches[3]; $d = [long]$Matches[4] } else { $w = 0; $n = [long]$Matches[1]; $d = [long]$Matches[5] }
  $t = $d; while ($t % 2 -eq 0) { $t /= 2 }; while ($t % 5 -eq 0) { $t /= 5 }; if ($t -ne 1) { return $v }
  $s = ([decimal]$w + [decimal]$n / [decimal]$d).ToString('0.##########################')
  if ($s.StartsWith('0.')) { $s.Substring(1) } else { $s }
}

# ---------- the book's tables ----------
function Table([string]$op, [int]$n) {
  $f = New-Object System.Collections.ArrayList
  # t = the book's line, q = the same fact asked as a question (the app's drill and the voice use both)
  function F($t, $a, $b, $r) {
    $q = switch ($op) { 'add' { "How many are $a and $b?" } 'sub' { "$a from $b leaves how many?" } 'mul' { if ($a -eq 1) { "How many is 1 time $b?" } else { "How many are $a times $b?" } } 'div' { "How many times is $a contained in $b?" } }
    [void]$f.Add([ordered]@{ t = $t; q = $q; a = $a; b = $b; r = $r })
  }
  switch ($op) {
    'add' { foreach ($k in 1..10) { F "$n and $k are $($n + $k)" $n $k ($n + $k); if ($k -ne $n) { F "$k and $n are $($n + $k)" $k $n ($n + $k) } } }
    'sub' { F "$n from $n leaves 0" $n $n 0
            foreach ($k in 1..9) { $m = $n + $k; F "$n from $m leaves $k" $n $m $k; if ($k -ne $n) { F "$k from $m leaves $n" $k $m $n } } }
    'mul' { if ($n -eq 1) { foreach ($k in 2..10) { F "1 time $k is $k" 1 $k $k; F "$k times 1 are $k" $k 1 $k } }
            else { F "$n times 1 are $n" $n 1 $n; foreach ($k in 2..10) { F "$k times $n are $($k * $n)" $k $n ($k * $n); if ($k -ne $n) { F "$n times $k are $($k * $n)" $n $k ($k * $n) } } } }
    'div' { F "$n in $n, 1 time" $n $n 1
            foreach ($k in 2..10) { $m = $n * $k; F "$n in $m, $k times" $n $m $k; if ($k -ne $n) { F "$k in $m, $n times" $k $m $n } } }
  }
  return ,$f.ToArray()
}

# ---------- algebra (Book 6) ----------
# Math in a text is written between backticks in plain ASCII: `3a^2b - 4(x+y)/2`. It is shown the book's way
# (3a²b − 4(x+y)/2) and said for the voice ("3 A squared B minus 4 times the quantity X plus Y, over 2").
# An answer {$ expr} is typed on the algebra pad; {$ expr @ same} checks it has the value of `same`, {$~ (a+b)(c+d)} multiplies it out itself,
# {$ 12 @x: 6x+2x = 3x+60} checks it solves the equation for x, {15 @= 5^2-2*5} checks a plain number.
# chk: x=5, y=4 | x+y=9 | x-y=1   checks a pair of answers against the equations.
Add-Type -Path "$root\tools\alg.cs"
function MathSplit([string]$t, [scriptblock]$plain, [scriptblock]$math) {
  $bits = $t -split '`'; $o = ''
  for ($i = 0; $i -lt $bits.Count; $i++) { $o += $(if ($i % 2) { & $math $bits[$i] } elseif ($bits[$i]) { & $plain $bits[$i] } else { '' }) }
  $o
}
$SUPS = @{ '0' = [char]0x2070; '1' = [char]0x00B9; '2' = [char]0x00B2; '3' = [char]0x00B3; '4' = [char]0x2074; '5' = [char]0x2075; '6' = [char]0x2076; '7' = [char]0x2077; '8' = [char]0x2078; '9' = [char]0x2079; '-' = [char]0x207B }
function AlgPretty([string]$m) {
  $m = [regex]::Replace($m, '\^(-?\d+)', { param($x) -join ($x.Groups[1].Value.ToCharArray() | ForEach-Object { $SUPS[[string]$_] }) })
  $m.Replace('-', [string][char]0x2212).Replace('*', [string][char]0x00D7)
}
function AlgSay([string]$m) {
  $tk = @([regex]::Matches(($m -replace [char]0x2212, '-'), '\d+\.?\d*|\.\d+|[A-Za-z]|\^-?\d+|[-+*/()=<>,;:.]|\S') | ForEach-Object { $_.Value })
  $w = New-Object System.Collections.ArrayList; $prev = ''
  for ($i = 0; $i -lt $tk.Count; $i++) {
    $t = $tk[$i]; $nx = $(if ($i + 1 -lt $tk.Count) { $tk[$i + 1] } else { '' })
    if ($t -match '^\^(-?\d+)$') { $n = [int]$Matches[1]; [void]$w.Add($(switch ($n) { 0 { 'to the zero power' } 2 { 'squared' } 3 { 'cubed' } default { "to the $(if ($n -lt 0) { 'minus ' })$(OrdWord ([Math]::Abs($n))) power" } })) }
    elseif ($t -match '^[a-z]$') { [void]$w.Add($t.ToUpper()) }
    elseif ($t -match '^[\d.]+$') {
      # a plain number fraction 3/4 (not 3/4x) is said "3 fourths"
      if ($nx -eq '/' -and $i + 2 -lt $tk.Count -and $tk[$i + 2] -match '^\d+$' -and [int]$tk[$i + 2] -gt 1 -and ($i + 3 -ge $tk.Count -or $tk[$i + 3] -notmatch '^([a-z(]|\^)') -and $prev -notmatch '^[a-z)]$') { [void]$w.Add((FracWords $t $tk[$i + 2])); $i += 2; $t = 'frac' }
      else { if ($prev -match '^[a-z)]$') { [void]$w.Add('times') }; [void]$w.Add($t) }
    }
    elseif ($t -eq '(') { [void]$w.Add($(if ($prev -match '^([\d.]+|[a-z)]|\^-?\d+)$') { 'times the quantity' } else { 'the quantity' })) }
    elseif ($t -eq ')') { [void]$w.Add(',') }
    else { [void]$w.Add($(switch ($t) { '+' { 'plus' } '-' { 'minus' } '*' { 'times' } '/' { 'over' } '=' { 'equals' } '^' { 'to the power' } '<' { 'is less than' } '>' { 'is greater than' } ([string][char]0x221A) { $(if ($prev -match '^([\d.]+|[a-z)]|\^-?\d+)$') { 'times the square root of' } else { 'the square root of' }) } ([string][char]0x00F7) { 'divided by' } default { $t } })) }
    $prev = $t
  }
  (($w -join ' ') -replace ' ,', ',' -replace ',(\s*,)+', ',' -replace ',\s*$', '').Trim()
}
# the pad's extra keys: the letters used, and a root sign when there is one
function AlgLetters([string]$s) { (-join (@([regex]::Matches(($s -replace '\^-?\d+', ''), '[a-z]') | ForEach-Object { $_.Value }) | Sort-Object -Unique)) + $(if ($s.Contains([string][char]0x221A)) { [string][char]0x221A } else { '' }) }

# ---------- questions ----------
# display: the book's x and / become real signs; spoken: signs become words for the voice
$TIMES = " $([char]0x00D7) "; $DIV = " $([char]0x00F7) "; $MINUS = " $([char]0x2212) "   # plain-ASCII script: no BOM needed
function Display([string]$t) { if ($t.Contains('`')) { return MathSplit $t { param($x) Display $x } { param($x) AlgPretty $x } }; ($t -replace '(?<=[\d)]) x (?=[\d.(])', $TIMES -replace '(?<=[\d)]) / (?=[\d.(])', $DIV -replace '(?<=[\d)]) - (?=[\d.(])', $MINUS) }
# fractions said the book's way: "3/4" -> "3 fourths", "2 1/2" -> "2 and 1 half"
$FRN = @{ 2 = 'half'; 3 = 'third'; 4 = 'fourth'; 5 = 'fifth'; 6 = 'sixth'; 7 = 'seventh'; 8 = 'eighth'; 9 = 'ninth'; 10 = 'tenth'; 11 = 'eleventh'; 12 = 'twelfth'; 13 = 'thirteenth'; 14 = 'fourteenth'; 15 = 'fifteenth'; 16 = 'sixteenth'; 17 = 'seventeenth'; 18 = 'eighteenth'; 19 = 'nineteenth'; 20 = 'twentieth'; 100 = 'hundredth' }
# any other denominator as an ordinal word: 23 -> twenty-third, 115 -> one hundred and fifteenth
function OrdWord([int]$d) {
  $o = 'zero one two three four five six seven eight nine ten eleven twelve thirteen fourteen fifteen sixteen seventeen eighteen nineteen' -split ' '
  $t = ',,twenty,thirty,forty,fifty,sixty,seventy,eighty,ninety' -split ','
  $oo = @{ one = 'first'; two = 'second'; three = 'third'; five = 'fifth'; eight = 'eighth'; nine = 'ninth'; twelve = 'twelfth' }
  function Card([int]$n) { if ($n -lt 20) { return $o[$n] }; $w = $t[[Math]::Floor($n / 10)]; if ($n % 10) { $w += '-' + $o[$n % 10] }; $w }
  $words = @()
  if ($d -ge 1000) { $words += (Card ([Math]::Floor($d / 1000))) + ' thousand'; $d = $d % 1000 }
  if ($d -ge 100) { $words += (Card ([Math]::Floor($d / 100))) + ' hundred'; $d = $d % 100 }
  if ($d) { if ($words.Count) { $words += 'and' }; $words += (Card $d) }
  $s = $words -join ' '
  if ($s -match '^(.*?)(\w+)$') { $head = $Matches[1]; $last = $Matches[2] }
  $last = $(if ($oo[$last]) { $oo[$last] } elseif ($last -match 'y$') { $last -replace 'y$', 'ieth' } else { $last + 'th' })
  $head + $last
}
function FracWords([int]$n, [int]$d) {
  $name = $FRN[$d]; if (-not $name) { $name = OrdWord $d }
  if ($n -gt 1) { $name = $(if ($name -eq 'half') { 'halves' } else { $name + 's' }) }
  "$n $name"
}
function SayFractions([string]$t) {
  $t = [regex]::Replace($t, '(?<![\d/])(\d+) (\d+)/(\d+)(?![\d/])', { param($m) "$($m.Groups[1].Value) and $(FracWords $m.Groups[2].Value $m.Groups[3].Value)" })
  [regex]::Replace($t, '(?<![\d/])(\d+)/(\d+)(?![\d/])', { param($m) FracWords $m.Groups[1].Value $m.Groups[2].Value })
}
function SayRep([string]$t) { [regex]::Replace($t, '(\d*)\.(\d*)\((\d+)\)', { param($m) $w = $m.Groups[1].Value; $a = ($m.Groups[2].Value.ToCharArray() -join ' '); $r = ($m.Groups[3].Value.ToCharArray() -join ' '); (($(if ($w) { "$w " } else { '' }) + 'point ' + $(if ($a) { "$a, " } else { '' }) + "$r repeating")) }) }
function Spoken([string]$t) { if ($t.Contains('`')) { return MathSplit $t { param($x) Spoken $x } { param($x) AlgSay $x } }; $t =($t -replace '(?<=[\d)]) x (?=[\d.])', ' times ' -replace '(?<=[\d)]) / (?=[\d.])', ' divided by ' -replace '(?<=[\d)]) - (?=[\d.])', ' minus ' -replace '(?<=[\d)]) \+ (?=[\d.])', ' plus ' -replace '= \?', 'equals what?'); SayFractions (SayRep $t) }
function Question([string]$q, [string]$where) {
  $parts = New-Object System.Collections.ArrayList; $script:reps = New-Object System.Collections.ArrayList
  $pos = 0
  foreach ($m in [regex]::Matches($q, '\{([^}]*)\}')) {
    $seg = $q.Substring($pos, $m.Index - $pos).Trim(); $pos = $m.Index + $m.Length
    $ans = $m.Groups[1].Value.Trim(); if ($ans -notmatch '^~p\s') { $ans = $ans -replace '(?<=\d),(?=\d{3})', '' }   # 1,234 -> 1234 (not in a compound chain: 9,144)
    $algE = $null; if ($ans -match '^\$\s*(.+)$') { $algE = $Matches[1].Trim(); $ans = '$' }   # algebra answer, below
    # {15 @= 5^2-2*5}: a plain number checked with the algebra reader (powers, letters' values written in)
    # {12 @x: 6x+2x = 3x+60}: a plain number that must solve the equation(s) for x (several split by ;)
    if ($ans -match '^([^@]+?)\s*@([a-z]):\s*(.+)$') { $ans = $Matches[1].Trim(); $r = [RaysAlgCheck]::Holds("$($Matches[2])=" + ($ans -replace '^(\d+) (\d+)/(\d+)$', '($1+$2/$3)'), @($Matches[3] -split '\s*;\s*')); if ($r) { [void]$script:errors.Add("$where : $r") } }
    if ($ans -match '^([^@]+?)\s*@=\s*(.+)$') { $ans = $Matches[1].Trim(); $r = [RaysAlgCheck]::Same(($ans -replace '^(\d+) (\d+)/(\d+)$', '($1+$2/$3)'), $Matches[2]); if ($r) { [void]$script:errors.Add("$where : $r") } }
    # {1234 @ 567 + 667}: the book's printed answer, checked against the sum worked out from the question (catches misread digits)
    if ($ans -match '^([^@]+?)\s*@\s*(.+)$') {
      $ans = $Matches[1].Trim(); $chk = $Matches[2]
      try { $want = EvalFrac $chk; $got = EvalFrac (($ans -replace '^(\d+) (\d+/\d+)$', '$1_$2') -replace '\|.*$', '')
            if ($want -ne $got) { [void]$script:errors.Add("$where : printed answer $ans, but $chk = $want") } } catch { [void]$script:errors.Add("$where : $_") }
    }
    # [[+ 184 216 135]] / [[- 5000 1234]] / [[x 345 27]]: a sum written in a column, the book's way; its result checks the answer
    $cv = $null; $colSay = $null
    if ($seg -match '\[\[([-+x])\s+([\d,. ]+)\]\]') {
      $op = $Matches[1]; $ns = @($Matches[2].Trim() -split '\s+'); $seg = $seg.Replace($Matches[0], '').Trim()
      $expr = ($ns | ForEach-Object { $_ -replace ',', '' }) -join $(switch ($op) { '+' { ' + ' } '-' { ' - ' } 'x' { ' * ' } })
      try { $cv = EvalFrac $expr } catch { [void]$script:errors.Add("$where : $_") }
      $colSay = switch ($op) { '+' { 'Add ' + (($ns | Select-Object -SkipLast 1) -join ', ') + ', and ' + $ns[-1] + '.' } '-' { "From $($ns[0]) take $($ns[1])." } 'x' { "Multiply $($ns[0]) by $($ns[1])." } }
      if ($ans -eq '?') { $ans = "$cv" } elseif ($ans -match '^[\d.]+( \d+/\d+)?$' -and $ans -ne $cv -and (EvalFrac ($ans -replace ' ', '_')) -ne $cv) { [void]$script:errors.Add("$where : the column comes to $cv, not $ans") }
    }
    $p = [ordered]@{ t = (Display $seg) }
    if ($null -ne $cv) { $p.col = [ordered]@{ op = $op; n = $ns } }
    if ($ans -match '^(-?\d+)\.\.(-?\d+) by (-?\d+)$') {
      $a = [int]$Matches[1]; $b = [int]$Matches[2]; $st = [int]$Matches[3]
      $seq = @(); for ($v = $a; ($st -gt 0 -and $v -le $b) -or ($st -lt 0 -and $v -ge $b); $v += $st) { $seq += $v }
      if ($seq[-1] -ne $b) { [void]$script:errors.Add("$where : count by $st from $a never lands on $b") }
      $p.seq = $seq
    } elseif ($ans -match '^\d+(\|\d+)+$') {   # more than one right answer, e.g. {100|1} = 100 cents or 1 dollar
      $all = @($ans -split '\|' | ForEach-Object { [int]$_ }); $p.a = $all[0]; $p.alt = @($all | Select-Object -Skip 1)
    } elseif ($ans -match '^[\d./ ]+(\|[\d./ ]+)+$') {   # either form, e.g. money to mills or exact: {23.062|23.0625}
      $all = @($ans -split '\|' | ForEach-Object { $_.Trim() }); $p.f = $all[0]; $p.falt = @($all | Select-Object -Skip 1)
    } elseif ($ans -match '^~\s*(.+)$') {   # worked out exactly: {~ 1/2 + 1/4} -> 3/4
      # {~d expr} or an expression with a decimal in it: show a decimal answer when it ends (2.78, not 2 39/50)
      # {~c4 expr}: the answer as a complex decimal carried to 4 places (.0154 1/4), the book's way with complex decimals
      # {~r3 expr}: rounded to 3 decimal places ("preserving three places"); the cut-off value is taken too
      $ex = $Matches[1]
      # {~m expr}: money to the cent; the book works interest by aliquot parts carried to mills, so a cent either way is right too
      if ($ex -match '^m\s+(.+)$') {
        try { [void](EvalFrac $Matches[1]); $r = $script:lastRat; $x = [decimal]$r[0] / [decimal]$r[1]
              $c = [Math]::Round($x * 100, [MidpointRounding]::AwayFromZero)
              $p.f = ($c / 100).ToString('0.00'); $p.falt = @((($c - 1) / 100).ToString('0.00'), (($c + 1) / 100).ToString('0.00')) } catch { [void]$script:errors.Add("$where : $_") }
        $ex = $null
      }
      elseif ($ex -match '^r(\d+)\s+(.+)$') {
        $rp = [int]$Matches[1]; $rex = $Matches[2]
        try { [void](EvalFrac $rex); $r = $script:lastRat; $sc = [decimal][Math]::Pow(10, $rp); $x = [decimal]$r[0] / [decimal]$r[1]
              $rnd = [Math]::Round($x * $sc, [MidpointRounding]::AwayFromZero) / $sc; $cut = [Math]::Truncate($x * $sc) / $sc
              $fmt = { param($v) $s = $v.ToString('0.' + ('0' * $rp)); if ($rp -eq 0) { $s = $v.ToString('0') }; if ($s.StartsWith('0.')) { $s.Substring(1) } else { $s } }
              $p.f = & $fmt $rnd; if ($cut -ne $rnd) { $p.falt = @(& $fmt $cut) } } catch { [void]$script:errors.Add("$where : $_") }
        $ex = $null
      }
      # {~p 320,16_1/2,12 # 2 # expr}: a compound number. expr is the whole quantity in the lowest unit (inches);
      # the chain gives how many of each unit make one of the unit above (mi.>rd.>ft.>in.); # 2 asks for the 3rd unit (ft.).
      # Every unit but the last is a whole number; the last takes the remainder (decimal when it ends, else a fraction).
      if ($null -ne $ex -and $ex -match '^p\s+([^#]*)#\s*(\d+)\s*#\s*(.+)$') {
        $chain = @($Matches[1].Trim() -split '\s*,\s*'); $idx = [int]$Matches[2]; $pex = $Matches[3].Trim()
        if ($pex -eq '$') { $pex = $script:lastPex; $chain = $script:lastChain } else { $script:lastPex = $pex; $script:lastChain = $chain }   # {~p # 1 # $}: same quantity, next unit
        try {
          [void](EvalFrac $pex); $rem = $script:lastRat
          for ($k = 0; $k -le $chain.Count; $k++) {
            $sz = $(if ($k -lt $chain.Count) { [void](EvalFrac (($chain[$k..($chain.Count - 1)]) -join ' * ')); $script:lastRat } else { ,@([long]1, [long]1) })
            if ($k -eq $chain.Count) { $part = $rem } else {
              $qq = [long][Math]::Floor(([decimal]$rem[0] * $sz[1]) / ([decimal]$rem[1] * $sz[0]))
              $part = Rat $qq 1; $rem = Rat ($rem[0] * $sz[1] - $qq * $sz[0] * $rem[1]) ($rem[1] * $sz[1]) }
            if ($k -eq $idx) { $ps = RatStr $part; if ($ps -match '^\d+$') { $p.a = [long]$ps } else { $p.f = $(if ($pex -match '\d\.\d') { AsDec $ps } else { $ps })
                # the book often gives the last unit "nearly": accept it rounded or cut to 0-4 places too
                $x = ([decimal]($part[0])) / ([decimal]($part[1])); $alts = @()
                foreach ($dp in 0..4) { $sc = [decimal][Math]::Pow(10, $dp); foreach ($v in @(([Math]::Round($x * $sc, [MidpointRounding]::AwayFromZero) / $sc), ([Math]::Truncate($x * $sc) / $sc))) { $vs = $v.ToString('0.####'); if ($vs.StartsWith('0.')) { $vs = $vs.Substring(1) }; if ($alts -notcontains $vs) { $alts += $vs } } }
                if ($pex -match '\d\.\d' -or $part[1] -gt 100) { $p.falt = $alts } }; break }
          }
        } catch { [void]$script:errors.Add("$where : $_") }
        $ex = $null
      }
      if ($null -ne $ex) {
      # {~q expr}: as a common fraction, with the circulate shown after the question (.45(3))
      $wantRep = $ex -match '^q\s'; $ex = $ex -replace '^q\s+', ''
      $cp = $(if ($ex -match '^c(\d+)\s') { [int]$Matches[1] } else { -1 }); $ex = $ex -replace '^c\d+\s+', ''
      $wantDec = $ex -match '^d\s' -or $ex -match '\d*\.\d'; $ex = $ex -replace '^d\s+', ''
      try { $v = EvalFrac $ex; if ($wantRep) { [void]$script:reps.Add((AsRep $script:lastRat)) }
            if ($cp -ge 0) { $c = AsCplx $script:lastRat $cp; if ($c -match '^\d+$') { $p.a = [long]$c } else { $p.f = $c } }
            elseif ($v -match '^\d+$') { $p.a = [long]$v } else { $p.f = $(if ($wantDec) { AsDec $v } else { $v }) } } catch { [void]$script:errors.Add("$where : $_") }
      }
    } elseif ($ans -match '^=(.+)$') {   # exact written form, e.g. {=1/2} for "reduce to lowest terms"; {=3/6|1/2} more than one
      $ex = @($Matches[1] -split '\|' | ForEach-Object { $_.Trim() }); $p.f = $ex[0]; $p.ex = $ex
    } elseif ($ans -eq '$') {         # algebra: typed on the algebra pad, checked by alg.js (same value, written as simply)
      $e = $algE; $chk = $null
      # {$~ (a+b)(c+d)}: multiplied out exactly, written the book's way
      $expand = { param($x) try { [RaysAlgCheck]::Expand($x) } catch { [void]$script:errors.Add("$where : $($_.Exception.InnerException.Message) in $x"); '0' } }
      if ($e -match '^~\s*(.+)$') { $e = & $expand $Matches[1] }
      else {
        if ($e -match '^(.+?)\s+@\s*(.+)$') { $e = $Matches[1].Trim(); $chk = $Matches[2].Trim() }
        # {$ x^2(x+a) | ~x^2(x+a)}: other ways that count too (~ = multiplied out)
        $alts = @($e -split '\s*\|\s*'); $e = $alts[0]
        if ($alts.Count -gt 1) { $p.ealt = @($alts[1..($alts.Count - 1)] | ForEach-Object { if ($_ -match '^~\s*(.+)$') { & $expand $Matches[1] } else { $_ } }) }
      }
      # @gcd: A; B  = the answer divides each evenly;  @lcm: A; B  = each divides the answer evenly
      if ($chk -match '^(gcd|lcm):\s*(.+)$') {
        $mode = $Matches[1]
        foreach ($qq in @($Matches[2] -split '\s*;\s*')) {
          try { [void][RaysAlgCheck]::Expand($(if ($mode -eq 'gcd') { "($qq)/($e)" } else { "($e)/($qq)" })) } catch { [void]$script:errors.Add("$where : $e is not a $(if ($mode -eq 'gcd') { 'divisor' } else { 'multiple' }) of $qq") }
        }
        $chk = $null
      }
      foreach ($x in @($p.ealt)) { if ($x) { $r = [RaysAlgCheck]::Same($x, $e); if ($r) { [void]$script:errors.Add("$where : alternative $r") } } }
      $p.e = $e; $p.es = AlgSay $e
      # pad letters: the answer's and the question's (so they don't give the answer away); none for a plain number
      $p.v = $(if ($e -match "[a-z$([char]0x221A)]") { AlgLetters ($e + ' ' + (@([regex]::Matches($seg, '`([^`]*)`') | ForEach-Object { $_.Groups[1].Value }) -join ' ')) } else { '' })
      if ($chk) {
        # @sq: radicand = the answer squared gives it (a square root)
        $r = $(if ($chk -match '^sq:\s*(.+)$') { [RaysAlgCheck]::Same("($e)^2", $Matches[1]) } elseif ($chk -match '^([a-z]):\s*(.+)$') { [RaysAlgCheck]::Holds("$($Matches[1])=$e", @($Matches[2] -split '\s*;\s*')) } else { [RaysAlgCheck]::Same($e, $chk) })
        if ($r) { [void]$script:errors.Add("$where : $r") }
      } else { $r = [RaysAlgCheck]::Same($e, $e); if ($r) { [void]$script:errors.Add("$where : can't read answer: $r") } }
    } elseif ($ans -eq 'open') {      # answered out loud to a grown-up (e.g. "Give two numbers which make 10")
      $p.open = $true
    } elseif ($ans -match '^(\d+ )?\d+/\d+$' -or $ans -match '^\d*\.\d+$' -or $ans -match '^\d*\.\d+ \d+/\d+$') {   # fraction, mixed number, decimal, or complex decimal (.0154 1/4)
      $p.f = $ans
    } elseif ($ans -eq '?') {
      $v = Calc $seg
      if ($null -eq $v) { [void]$script:errors.Add("$where : can't work out {?} for '$seg'") } else { $p.a = $v }
    } elseif ($ans -match '^\d+$') {
      $p.a = [long]$ans
      $v = Calc $seg
      if ($null -ne $v -and $v -ne $p.a) { [void]$script:errors.Add("$where : '$seg' is $v, not $ans") }
    } else { [void]$script:errors.Add("$where : bad answer {$ans}") }
    if ($colSay) { $p.s = ((Spoken $seg) + ' ' + $colSay).Trim() } elseif ((Spoken $seg) -ne $seg) { $p.s = Spoken $seg }
    [void]$parts.Add($p)
  }
  # "x = {A} or {B}": either order (the two roots of a quadratic); app.js swaps them when B comes first
  for ($k = 1; $k -lt $parts.Count; $k++) { if ($parts[$k].t -match '^or\b') { $parts[$k - 1].pr = 1 } }
  $tail = $q.Substring($pos).Trim()
  if ($tail) { if ($parts.Count) { $parts[$parts.Count - 1].t += ' ' + $tail } else { [void]$script:errors.Add("$where : question has no {answer}") } }
  $res = [ordered]@{ parts = $parts.ToArray() }
  if ($script:reps.Count) { $res.sol = 'As a circulate: ' + ($script:reps -join '; ') + '.' }
  return $res
}

# ---------- lessons ----------
$lessons = New-Object System.Collections.ArrayList
$L = $null; $lastQ = $null; $qn = 0
foreach ($line in $src) {
  if ($line -match '^\s*(#|$)' -or $line -match '^@book: ') { continue }
  if ($line -match '^== (\d+) \| ([^|]+) \| (.+)$') {
    $L = [ordered]@{ n = [int]$Matches[1]; part = $Matches[2].Trim(); title = $Matches[3].Trim(); kind = 'problems' }
    $L.review = $L.part -eq 'Review'
    [void]$lessons.Add($L); $lastQ = $null; $lastArt = $null; $qn = 0; continue
  }
  if (-not $L) { continue }
  $where = "Lesson $($L.n)"
  if ($line -match '^kind: (count|numbers) (\d+) (\d+)$') { $L.kind = $Matches[1]; $L.from = [int]$Matches[2]; $L.to = [int]$Matches[3] }
  elseif ($line -match '^kind: (read|write)$') { $L.kind = $Matches[1] }
  elseif ($line -match '^list: (.+)$') { $L.list = @($Matches[1] -split '\s+' | ForEach-Object { [int]$_ }) }
  elseif ($line -match '^note: (.+)$') { $L.note = $Matches[1] }
  elseif ($line -match '^table: (add|sub|mul|div) (\d+)$') { $L.op = $Matches[1]; $L.table = Table $Matches[1] ([int]$Matches[2]) }
  elseif ($line -match '^table: \+(add|sub|mul|div) (\d+) (\d+)( from 0)?$') {   # Book Two style "2 + 1 = 3" ... "2 + 12 = 14"; several add up
    $L.op = $Matches[1]; $op = $Matches[1]; $n = [int]$Matches[2]; $max = [int]$Matches[3]; $k0 = $(if ($Matches[4]) { 0 } else { 1 })   # Book Three: "2 + 0 = 2"
    $rows = New-Object System.Collections.ArrayList; if ($L.table) { [void]$rows.AddRange(@($L.table)) }
    foreach ($k in $k0..$max) {
      switch ($op) {
        'add' { $r = $n + $k; [void]$rows.Add([ordered]@{ t = "$n + $k = $r"; q = "How many are $n and $k?"; a = $n; b = $k; r = $r }) }
        'sub' { $m = $n + $k - 1; $r = $k - 1; [void]$rows.Add([ordered]@{ t = "$m - $n = $r"; q = "$n from $m leaves how many?"; a = $n; b = $m; r = $r }) }
        'mul' { $r = $n * $k; [void]$rows.Add([ordered]@{ t = "$k x $n = $r"; q = "How many are $k times $n?"; a = $k; b = $n; r = $r }) }   # the book's columns: 1 x 4, 2 x 4 ... 12 x 4
        'div' { $m = $n * $k; [void]$rows.Add([ordered]@{ t = "$m / $n = $k"; q = "How many times is $n contained in $m?"; a = $n; b = $m; r = $k }) }
      }
    }
    foreach ($f in $rows) { $f.t = Display $f.t }
    $L.table = $rows.ToArray()
  }
  elseif ($line -match '^rule: (\d+) (.+) make 1 (.+)$') {
    if (-not $L.rules) { $L.rules = New-Object System.Collections.ArrayList }
    $r = [ordered]@{ t = "$($Matches[1]) $($Matches[2]) make 1 $($Matches[3])"; q = "How many $($Matches[2]) make 1 $($Matches[3])?"; n = [int]$Matches[1]; a = $Matches[2]; b = $Matches[3] }
    if ($r.a -match '\d|^and ') { $r.quiz = $false }   # "365 days and 6 hours": show it, don't quiz it
    [void]$L.rules.Add($r)
  }
  elseif ($line -match '^q: (.+)$') {
    $qn++; if (-not $L.qs) { $L.qs = New-Object System.Collections.ArrayList }
    $lastQ = Question $Matches[1] "$where q$qn"; [void]$L.qs.Add($lastQ)
  }
  elseif ($line -match '^sol: (.+)$') { if ($lastQ) { $lastQ.sol = $Matches[1] } else { [void]$script:errors.Add("$where : sol without a question") } }
  # Book Five (Geometry): the book's articles to study, each with its figure cut from the scan, then a quiz
  elseif ($line -match '^art: (\d+) \| ([^|]*) \| (.+)$') {
    if (-not $L.arts) { $L.arts = New-Object System.Collections.ArrayList }
    $lastArt = [ordered]@{ n = [int]$Matches[1]; h = $Matches[2].Trim(); t = $Matches[3].Trim() }; [void]$L.arts.Add($lastArt)
    if ("$($lastArt.h)$($lastArt.t)".Contains('`')) { $lastArt.s = "$($lastArt.n). $(if ($lastArt.h) { "$(Spoken $lastArt.h). " })$(Spoken $lastArt.t)"; $lastArt.h = Display $lastArt.h; $lastArt.t = Display $lastArt.t }
  }
  elseif ($line -match '^chk: ([^|]+)\|(.+)$') {   # chk: x=5, y=4 | x+y=9 | x-y=1
    $r = [RaysAlgCheck]::Holds($Matches[1].Trim(), @($Matches[2] -split '\s*\|\s*' | ForEach-Object { $_.Trim() }))
    if ($r) { [void]$script:errors.Add("$where : $r") }
  }
  elseif ($line -match '^fig: (\d+) ([\d.]+) ([\d.]+)( [\d.]+ [\d.]+)?$') {   # page top bottom [left right], fractions of the page
    if (-not $lastArt) { [void]$script:errors.Add("$where : fig without an art"); continue }
    $fk = "p$($Matches[1])-" + [int]([double]$Matches[2] * 1000)
    $lastArt.figs = @($lastArt.figs | Where-Object { $_ }) + "fig/$($bk[0])/$fk.png"   # an article may have several
  }
  elseif ($line -match '^mc: (.+)$') {
    $parts = @($Matches[1] -split '\s*\|\s*'); $ch = @($parts[1..($parts.Count - 1)])
    $right = @(for ($i = 0; $i -lt $ch.Count; $i++) { if ($ch[$i] -like '[*]*') { $i } })
    if ($ch.Count -lt 2 -or $right.Count -ne 1) { [void]$script:errors.Add("$where : mc needs 2+ choices and one *right one: '$line'"); continue }
    if (-not $L.quiz) { $L.quiz = New-Object System.Collections.ArrayList }
    $cs = @($ch | ForEach-Object { $_ -replace '^\*', '' }); $item = [ordered]@{ q = $parts[0]; c = $cs; a = $right[0] }
    if ($line.Contains('`')) { $item.q = Display $parts[0]; $item.c = @($cs | ForEach-Object { Display $_ }); $item.qs = Spoken $parts[0]; $item.cs = @($cs | ForEach-Object { Spoken $_ }) }   # algebra: shown the book's way, and what the voice says
    [void]$L.quiz.Add($item)
  }
  elseif ($line -match '^tf: (.+) \{([TF])\}$') {
    if (-not $L.quiz) { $L.quiz = New-Object System.Collections.ArrayList }
    $tq = $Matches[1]; $item = [ordered]@{ q = $tq; c = @('True', 'False'); a = $(if ($Matches[2] -eq 'T') { 0 } else { 1 }); tf = $true }
    if ($tq.Contains('`')) { $item.q = Display $tq; $item.qs = Spoken $tq }
    [void]$L.quiz.Add($item)
  }
  else { [void]$script:errors.Add("$where : can't read line '$line'") }
}
foreach ($L in $lessons) {
  if ($L.rules) { $L.rules = $L.rules.ToArray() }
  if ($L.qs) { $L.qs = $L.qs.ToArray() }
  if ($L.arts) { $L.arts = $L.arts.ToArray() }
  if ($L.quiz) { $L.quiz = $L.quiz.ToArray() }
  if ($L.kind -eq 'problems' -and -not $L.qs -and -not $L.rules -and -not $L.table -and -not $L.arts) { [void]$script:errors.Add("Lesson $($L.n) : no questions") }
}
$nums = @($lessons | ForEach-Object { $_.n })
for ($i = 0; $i -lt $nums.Count; $i++) { if ($nums[$i] -ne $i + 1) { [void]$script:errors.Add("lesson numbers out of order at $($nums[$i])"); break } }

if ($script:errors.Count) { "PROBLEMS:"; $script:errors; exit 1 }
$out = [ordered]@{ id = $bk[0]; title = $bk[1]; sub = $bk[2]; color = $bk[3]; lessons = $lessons.ToArray() }
New-Item -ItemType Directory -Force "$root\public\books" | Out-Null
[IO.File]::WriteAllText("$root\public\books\$($bk[0]).json", ($out | ConvertTo-Json -Depth 10 -Compress), (New-Object Text.UTF8Encoding $false))
$q = ($lessons | ForEach-Object { @($_.qs).Count } | Measure-Object -Sum).Sum
$a = ($lessons | ForEach-Object { foreach ($x in @($_.qs)) { if ($x) { $x.parts.Count } } } | Measure-Object -Sum).Sum
"$($lessons.Count) lessons, $q questions, $a answers -> public\books\$($bk[0]).json"
EOF

make-audio.ps1: records every line of a book with Eloquence: -Id higher.

cat > /root/rays-rebuild/make-audio.ps1 <<'EOF'
# Records everything Ray's Arithmetic says with the PC's ETI-Eloquence (Reed, the same voice as ABC Books) at each
# speed, then writes public/audio/<id>.json (what the app asks for -> clip id). WAVs land in tools/out/s<speed>/,
# shared by every book (a line two books both say is recorded once).
# Usage: powershell -ExecutionPolicy Bypass -File RaysArithmetic\tools\make-audio.ps1 [-Id primary1877|intellectual|...]
# The keys must match app.js: prompts, "lesson:N", "n:N", "find:N", "ans:N", and the book's own texts.
param([string]$Id = 'primary1877', [int[]]$Speeds = @(60, 75))
$root = Split-Path -Parent (Split-Path -Parent $MyInvocation.MyCommand.Path)
$book = Get-Content "$root\public\books\$Id.json" -Raw -Encoding UTF8 | ConvertFrom-Json
$P = Get-Content "$root\public\prompts.json" -Raw -Encoding UTF8 | ConvertFrom-Json
$kids = 'Layla', 'Lavon', 'Lilly'

$say = New-Object System.Collections.Specialized.OrderedDictionary ([StringComparer]::Ordinal)
function Add($key, $spoken) { if ($key -and -not $say.Contains([string]$key)) { $say[[string]$key] = $(if ($spoken) { $spoken } else { [string]$key }) } }
foreach ($pr in $P.PSObject.Properties) { if ($pr.Value -like '*{kid}*') { foreach ($k in $kids) { Add ($pr.Value -replace '\{kid\}', $k) } } else { Add $pr.Value } }
foreach ($k in $kids) { Add "Great job, $k!" }
foreach ($n in 0..100) { Add "n:$n" "$n."; Add "find:$n" "Find $n." }
# "2 1/2" -> "2 and 1 half", "3/4" -> "3 fourths": how the book says fractions
# any denominator as an ordinal word: 23 -> twenty-third, 115 -> one hundred and fifteenth
function OrdWord([int]$d) {
  $o = 'zero one two three four five six seven eight nine ten eleven twelve thirteen fourteen fifteen sixteen seventeen eighteen nineteen' -split ' '
  $t = ',,twenty,thirty,forty,fifty,sixty,seventy,eighty,ninety' -split ','
  $oo = @{ one = 'first'; two = 'second'; three = 'third'; five = 'fifth'; eight = 'eighth'; nine = 'ninth'; twelve = 'twelfth' }
  function Card([int]$n) { if ($n -lt 20) { return $o[$n] }; $w = $t[[Math]::Floor($n / 10)]; if ($n % 10) { $w += '-' + $o[$n % 10] }; $w }
  $words = @()
  if ($d -ge 1000) { $words += (Card ([Math]::Floor($d / 1000))) + ' thousand'; $d = $d % 1000 }
  if ($d -ge 100) { $words += (Card ([Math]::Floor($d / 100))) + ' hundred'; $d = $d % 100 }
  if ($d) { if ($words.Count) { $words += 'and' }; $words += (Card $d) }
  $s = $words -join ' '
  if ($s -match '^(.*?)(\w+)$') { $head = $Matches[1]; $last = $Matches[2] }
  $last = $(if ($oo[$last]) { $oo[$last] } elseif ($last -match 'y$') { $last -replace 'y$', 'ieth' } else { $last + 'th' })
  $head + $last
}
function Frac([string]$f) {
  $den = @{ 2 = 'half'; 3 = 'third'; 4 = 'fourth'; 5 = 'fifth'; 6 = 'sixth'; 7 = 'seventh'; 8 = 'eighth'; 9 = 'ninth'; 10 = 'tenth'; 11 = 'eleventh'; 12 = 'twelfth'; 20 = 'twentieth'; 100 = 'hundredth' }
  if ($f -match '^\d*\.\d+$' -or $f -match '^\d+$') { return $f }   # decimals, and whole numbers too long for a number ({=digits})
  if ($f -match '^(\d*\.\d+) (\d+/\d+)$') { return "$($Matches[1]) and $(Frac $Matches[2])" }   # complex decimal: .0154 1/4
  $w = ''; if ($f -match '^(\d+) (.+)$') { $w = "$($Matches[1]) and "; $f = $Matches[2] }
  $n, $d = $f -split '/'; $name = $den[[int]$d]; if (-not $name) { $name = OrdWord ([int]$d) }
  if ($name -eq 'half' -and [int]$n -gt 1) { $name = 'halve' }
  return "$w$n $name$(if ([int]$n -gt 1) { 's' })"
}
function GeoSay([string]$s) {
  $s = $s -replace '\s*\(\d+(, ?\d+)*\)', ''
  [regex]::Replace($s, '\b[A-Z]{2,6}\b', { param($m) ($m.Value.ToCharArray() -join ' ') })
}
# must match artKey() in app.js
function ArtKey($a) { "$($a.n). $(if ($a.h) { "$($a.h). " })$($a.t)" }
$answers = New-Object 'System.Collections.Generic.SortedSet[long]'
foreach ($L in $book.lessons) {
  Add "lesson:$($L.n)" "Lesson $($L.n). $($L.title)."
  foreach ($v in @($L.list)) { if ($null -ne $v) { [void]$answers.Add([long]$v); if ([long]$v -gt 100) { Add "n:$v" "$v." } } }
  foreach ($f in @($L.table)) { if ($f) { Add $f.t; Add $f.q; [void]$answers.Add([long]$f.r) } }
  foreach ($r in @($L.rules)) { if ($r) { Add $r.t; Add $r.q } }
  foreach ($q in @($L.qs)) {
    if (-not $q) { continue }
    foreach ($p in $q.parts) {
      $s = if ($p.s) { $p.s } else { $p.t }; Add $s
      if ($null -ne $p.a) { [void]$answers.Add([long]$p.a) }
      if ($p.f) { Add "ans:$($p.f)" "The answer is $(Frac $p.f)." }
      if ($null -ne $p.e) { Add "ans:$($p.e)" "The answer is $($p.es)." }   # algebra: es = the answer in words
      foreach ($v in @($p.seq)) { if ($null -ne $v) { [void]$answers.Add([long]$v); if ($v -gt 100) { Add "n:$v" "$v." } } }
    }
    if ($q.sol) { Add $q.sol }
  }
  # Geometry: figure letters are said one by one ("AOE" -> "A O E"), article references "(103)" are not read
  # Algebra: arts and quiz with math carry what to say (s, qs, cs) since "3a²" can't be read off the screen text
  foreach ($a in @($L.arts)) { if ($a) { $k = ArtKey $a; Add $k $(if ($a.s) { $a.s } else { GeoSay $k }) } }
  foreach ($x in @($L.quiz)) {
    if (-not $x) { continue }
    $qs = $(if ($x.qs) { $x.qs } else { GeoSay $x.q })
    if ($x.tf) { Add $x.q "True or false? $qs" } else { Add $x.q $qs }
    for ($i = 0; $i -lt @($x.c).Count; $i++) { $c = @($x.c)[$i]; Add "ansc:$c" $(if ($x.cs) { "The answer is: $(@($x.cs)[$i])." } else { GeoSay "The answer is: $c." }) }
  }
}
foreach ($a in $answers) { Add "ans:$a" "The answer is $a." }
Add "Answer out loud, then tap Next."
"$($say.Count) clips per speed"

$sha = [Security.Cryptography.SHA1]::Create()
function Id($s) { (($sha.ComputeHash([Text.Encoding]::UTF8.GetBytes($s)) | Select-Object -First 6 | ForEach-Object { $_.ToString('x2') }) -join '') }
$items = New-Object System.Collections.Specialized.OrderedDictionary ([StringComparer]::Ordinal)
foreach ($k in $say.Keys) { $items[$k] = Id $say[$k] }
$ps32 = 'C:\Windows\SysWOW64\WindowsPowerShell\v1.0\powershell.exe'
foreach ($sp in $Speeds) {
  $dir = "$root\tools\out\s$sp"; New-Item -ItemType Directory -Force $dir | Out-Null
  $jobs = @(); $seen = @{}
  foreach ($k in $say.Keys) { $cid = $items[$k]; if (-not $seen[$cid] -and -not (Test-Path "$dir\$cid.wav")) { $seen[$cid] = 1; $jobs += [ordered]@{ text = $say[$k]; out = "$dir\$cid.wav" } } }
  "speed $sp : $($jobs.Count) to record"
  for ($i = 0; $i -lt $jobs.Count; $i += 400) {   # batches, so a long run shows progress
    $jf = "$dir\jobs.json"; ConvertTo-Json @($jobs[$i..([Math]::Min($i + 399, $jobs.Count - 1))]) | Set-Content -Encoding utf8 $jf
    & $ps32 -NoProfile -ExecutionPolicy Bypass -File "$root\tools\eloq-render.ps1" -Jobs $jf -Speed $sp | Out-Null
    "  $([Math]::Min($i + 400, $jobs.Count)) / $($jobs.Count)"
  }
  Remove-Item "$dir\*.wav.json" -ErrorAction SilentlyContinue   # word times aren't used here
}
$out = [ordered]@{ speeds = $Speeds; items = $items }
New-Item -ItemType Directory -Force "$root\public\audio" | Out-Null
[IO.File]::WriteAllText("$root\public\audio\$Id.json", ($out | ConvertTo-Json -Depth 4 -Compress), (New-Object Text.UTF8Encoding $false))
"manifest: $root\public\audio\$Id.json"
EOF

app.js: the game, for the whole shelf.

cat > /root/rays-rebuild/app.js <<'EOF'
// Ray's Arithmetic: gamified Ray's New Primary Arithmetic (1877). Progress is saved per kid in localStorage ('rays').
// Same look and voice as ABC Books; the kid picked in ABC Books (same site) is picked here too.
const Rays = (() => {
  const $ = id => document.getElementById(id);
  // Readers come from ABC Books on this site ('abcbooks'.readers: names a grown-up added there). Same rule as ABC
  // Books: no 'abcbooks' (or no kid and no progress) = no readers yet; an old device without .readers = the three
  // original kids plus any names in its progress.
  const NAME_OK = n => typeof n === 'string' && /^\p{L}[\p{L}' .-]{0,15}$/u.test(n.trim());
  function readers() {
    let c = null; try { c = JSON.parse(localStorage.getItem('abcbooks')); } catch (e) {}
    let r = [];
    if (c && Array.isArray(c.readers)) r = c.readers.filter(NAME_OK);
    else if (c && (c.kid || (c.progress && Object.keys(c.progress).length))) r = ['Layla', 'Lavon', 'Lilly'].concat(Object.keys(c.progress || {}).filter(n => NAME_OK(n) && !['Layla', 'Lavon', 'Lilly'].includes(n)));
    if (S && S.kid && r.length && !r.includes(S.kid)) r = r.concat(S.kid);
    return r;
  }
  // Ray's series in order (raysarithmetic.wordpress.com/rays-free-arithmetic). A book is playable once books/<id>.json exists.
  const SHELF = [
    { id: 'primary1877', title: "Ray's New Primary Arithmetic", sub: 'Book 1 · Grades 1-2', color: '#2f6f8f' },
    { id: 'intellectual', title: "Ray's New Intellectual Arithmetic", sub: 'Book 2 · Grades 3-4', color: '#6a4c93' },
    { id: 'practical', title: "Ray's New Practical Arithmetic", sub: 'Book 3 · Grades 5-6', color: '#2e7d4f' },
    { id: 'higher', title: "Ray's New Higher Arithmetic", sub: 'Book 4 · Grades 7-8', color: '#a0522d' },
    { id: 'geometry', title: "Ray's Geometry and Trigonometry", sub: 'Book 5 · Grades 9-10', color: '#7a6a1f' },
    { id: 'algebra1', title: "Ray's New Elementary Algebra", sub: 'Book 6 · Grades 9-10', color: '#8f2f4f' },
    { id: 'algebra2', title: "Ray's New Higher Algebra", sub: 'Book 7 · Grade 11', color: '#3b4a8f', soon: true }
  ];
  const BOOKS = {};
  let P = {};
  const STICKERS = ['🦊', '🐢', '🦉', '🐝', '🦋', '🐞', '🐸', '🐧', '🦄', '🐬', '🦁', '🐼', '🐨', '🐙', '🦀', '🐳', '🦒', '🦓', '🐘', '🦔', '🐿️', '🦜', '🐠', '🐤', '🐰', '🐻', '🐯', '🐮', '🐷', '🐵', '🌻', '🌈', '⭐', '🚂', '🚀', '⛵', '🎈', '🍎', '🍓', '🍩'];
  const STEPS = {
    count: { name: 'Count', icon: '●' }, numbers: { name: 'Numbers', icon: '#' }, find: { name: 'Find the number', icon: '👂' },
    say: { name: 'Read the numbers', icon: '💬' }, write: { name: 'Write the numbers', icon: '✎' }, trace: { name: 'Trace the numbers', icon: '1' },
    learn: { name: 'Learn the table', icon: '📖' }, rules: { name: 'Learn the table', icon: '📖' }, drill: { name: 'Practice', icon: '⚡' },
    problems: { name: 'Story problems', icon: '?' }, study: { name: 'Study', icon: '📐' }, quiz: { name: 'Quiz', icon: '✔' }
  };

  // ---------- storage ----------
  let S = load();
  function load() {
    let s = null; try { s = JSON.parse(localStorage.getItem('rays')); } catch (e) {}
    s = s || {};
    s.progress = s.progress || {}; s.pos = s.pos || {}; s.best = s.best || {}; s.stickers = s.stickers || {};
    if (!s.speed) s.speed = 60;
    if (!s.kid) {   // the kid picked in ABC Books or BlockCraft on this site
      for (const k of ['abcbooks', 'blockcraft_cfg']) { try { const c = JSON.parse(localStorage.getItem(k)); const n = c && (k === 'abcbooks' ? c.kid : c.name); if (NAME_OK(n)) { s.kid = n.trim(); break; } } catch (e) {} }
    }
    return s;
  }
  function save() { try { localStorage.setItem('rays', JSON.stringify(S)); } catch (e) {} }
  const kidMap = obj => (obj[S.kid] = obj[S.kid] || {});
  const prog = b => (kidMap(S.progress)[b] = kidMap(S.progress)[b] || {});

  // ---------- sound + voice (recorded Eloquence clips, else the device voice) ----------
  let AC;
  function beep(good) {
    try {
      AC = AC || new (window.AudioContext || window.webkitAudioContext)();
      const o = AC.createOscillator(), g = AC.createGain(), t = AC.currentTime;
      o.type = 'triangle'; o.connect(g); g.connect(AC.destination);
      const notes = good ? [523, 659, 784] : [220, 180];
      notes.forEach((f, i) => o.frequency.setValueAtTime(f, t + i * 0.1));
      g.gain.setValueAtTime(0.25, t); g.gain.exponentialRampToValueAtTime(0.001, t + notes.length * 0.1 + 0.2);
      o.start(t); o.stop(t + notes.length * 0.1 + 0.25);
    } catch (e) {}
  }
  const hasTTS = 'speechSynthesis' in window;
  let voice = null;
  function pickVoice() { if (!hasTTS) return; const v = speechSynthesis.getVoices().filter(v => /^en/i.test(v.lang)); voice = v.find(v => /female|samantha|zira|aria|jenny|google us/i.test(v.name)) || v[0] || null; }
  if (hasTTS) { pickVoice(); speechSynthesis.onvoiceschanged = pickVoice; }
  let AUDIO = null;   // audio/primary1877.json: spoken text -> clip id (tools/make-audio.ps1)
  const clip = new Audio(); clip.preload = 'auto';
  let clipDone = null, tok = 0, unlocked = false;
  const blobs = new Map();
  function clipURL(url) {   // fetched whole and played from memory: works offline and on Safari
    if (!blobs.has(url)) blobs.set(url, fetch(url).then(r => { if (!r.ok) throw new Error(r.status); return r.blob(); }).then(b => URL.createObjectURL(b)));
    const p = blobs.get(url); p.catch(() => blobs.delete(url)); return p;
  }
  const speedOf = () => AUDIO.speeds.includes(S.speed) ? S.speed : AUDIO.speeds[0];
  const urlFor = key => { const c = AUDIO && AUDIO.items[key]; return c ? `audio/s${speedOf()}/${c}.mp3` : null; };
  function stopAll() { tok++; clip.pause(); if (clipDone) { const d = clipDone; clipDone = null; d(); } if (hasTTS) speechSynthesis.cancel(); }
  function playURL(u, my) {
    return new Promise(res => {
      clipURL(u).then(b => {
        if (my !== tok) return res();
        clipDone = res; clip.muted = false; clip.src = b;
        clip.onended = clip.onerror = () => { if (clipDone) { const d = clipDone; clipDone = null; d(); } };
        clip.play().catch(() => clip.onended());
      }).catch(() => res());
    });
  }
  // speak(...keys): one clip per key, played one after the other; a key with no clip uses the device voice
  async function speak(...keys) {
    stopAll(); const my = tok;
    for (const k of keys) {
      if (my !== tok || k == null || k === '') continue;
      const u = urlFor(String(k));
      if (u) await playURL(u, my); else await tts(String(k).replace(/^(n|find|ans|lesson):/, ''));
    }
  }
  function tts(text) {
    return new Promise(res => {
      if (!hasTTS) return setTimeout(res, 300);
      const u = new SpeechSynthesisUtterance(text); if (voice) u.voice = voice; u.rate = 0.85;
      u.onend = u.onerror = res; speechSynthesis.speak(u);
    });
  }
  const sayNum = n => speak('n:' + n);
  window.addEventListener('pointerdown', () => {   // iOS: unlock audio inside the first tap
    if (AC && AC.state === 'suspended') AC.resume();
    const u = urlFor(P.yes); if (unlocked || !u) return; unlocked = true;
    clipURL(u).then(b => { if (!clip.src) { clip.muted = true; clip.src = b; clip.play().then(() => { clip.pause(); clip.muted = false; }).catch(() => { clip.muted = false; }); } }).catch(() => {});
  }, { passive: true });

  // ---------- helpers ----------
  function toast(t, ms) { const e = $('toast'); e.textContent = t; e.style.display = 'block'; clearTimeout(toast.t); toast.t = setTimeout(() => e.style.display = 'none', ms || 1100); }
  function el(tag, cls, html) { const e = document.createElement(tag); if (cls) e.className = cls; if (html != null) e.innerHTML = html; return e; }
  function btn(label, fn, cls) { const b = el('button', cls, label); b.onclick = fn; return b; }
  const shuffle = a => { a = a.slice(); for (let i = a.length - 1; i > 0; i--) { const j = Math.floor(Math.random() * (i + 1)); [a[i], a[j]] = [a[j], a[i]]; } return a; };
  const range = (a, b) => Array.from({ length: b - a + 1 }, (_, i) => a + i);
  const starStr = n => '★'.repeat(n) + '☆'.repeat(3 - n);
  const starsFor = (miss, n) => miss === 0 ? 3 : miss <= Math.max(2, Math.round(n * 0.15)) ? 2 : 1;
  const wait = ms => new Promise(r => setTimeout(r, ms));
  function go(id) { document.querySelectorAll('.screen').forEach(s => s.classList.toggle('on', s.id === id)); if (id === 'home') home(); if (id === 'stickers') stickers(); }
  const ONES = ['zero', 'one', 'two', 'three', 'four', 'five', 'six', 'seven', 'eight', 'nine', 'ten', 'eleven', 'twelve', 'thirteen', 'fourteen', 'fifteen', 'sixteen', 'seventeen', 'eighteen', 'nineteen'];
  const TENS = ['', '', 'twenty', 'thirty', 'forty', 'fifty', 'sixty', 'seventy', 'eighty', 'ninety'];
  const words = n => n < 20 ? ONES[n] : n === 100 ? 'one hundred' : TENS[Math.floor(n / 10)] + (n % 10 ? '-' + ONES[n % 10] : '');

  // pictures: dots for a fact (a and b in two colors, crossed out when taken away, rows for times, groups for division)
  function dotsFor(op, a, b) {
    const box = el('div', 'dots');
    const grp = (n, cls, boxed) => { const g = el('div', 'grp' + (boxed ? ' box' : '')); for (let i = 0; i < n; i++) g.appendChild(el('i', 'dot ' + (cls || ''))); return g; };
    if (op === 'add') { box.append(grp(a), el('span', 'op', '+'), grp(b, 'b')); }
    else if (op === 'sub') { const g = el('div', 'grp'); for (let i = 0; i < b; i++) g.appendChild(el('i', 'dot ' + (i >= b - a ? 'x' : ''))); box.appendChild(g); }
    else if (op === 'mul') { if (a * b > 100) return box; for (let i = 0; i < a; i++) box.appendChild(grp(b, i % 2 ? 'b' : '', true)); }
    else if (op === 'div') { if (b > 100) return box; for (let i = 0; i < b / a; i++) box.appendChild(grp(a, i % 2 ? 'b' : '', true)); }
    return box;
  }
  const SIGN = { add: '+', sub: '−', mul: '×', div: '÷' };
  const expr = (op, f) => op === 'sub' ? `${f.b} − ${f.a}` : op === 'div' ? `${f.b} ÷ ${f.a}` : `${f.a} ${SIGN[op]} ${f.b}`;
  // a picture for a story problem, from the first thing it counts
  const PICS = [['cent|dime|money', '💰'], ['dollar', '💵'], ['apple', '🍎'], ['pear', '🍐'], ['peach', '🍑'], ['orange', '🍊'], ['lemon', '🍋'], ['cherr', '🍒'], ['plum|raisin|fig', '🍇'], ['melon', '🍈'], ['pine-apple', '🍍'], ['strawberr|berr', '🍓'], ['nut|chestnut|pecan|walnut', '🌰'],
    ['marble', '🔵'], ['top', '🎯'], ['kite', '🎏'], ['ball', '⚽'], ['doll', '🧸'], ['bird|swallow|crow|pigeon', '🐦'], ['owl', '🦉'], ['bat', '🦇'], ['duck', '🦆'], ['geese|goose', '🦢'], ['chicken|hen', '🐔'], ['fish', '🐟'], ['horse', '🐴'], ['cow|calf|calves', '🐄'], ['sheep', '🐑'], ['pig|hog', '🐖'], ['rabbit', '🐇'], ['cat|kitten', '🐈'], ['mice|mouse', '🐭'],
    ['egg', '🥚'], ['cake|candy', '🍬'], ['milk', '🥛'], ['book|primer|reader|geograph', '📚'], ['pen|pencil|slate', '✏️'], ['ribbon|tape|cloth|calico|muslin|silk', '🎀'], ['tree|orchard', '🌳'], ['flower|rose', '🌹'], ['leaves|leaf', '🍃'], ['ship|boat', '⛵'], ['car', '🚃'], ['mile|travel', '🛤️'], ['hour|minute', '⏰'], ['day|week|month|year', '📅'], ['wood|cord', '🌲'], ['flour|wheat|corn|oats|seed', '🌾'],
    ['boy|girl|pupil|scholar|men|man|persons', '🧒'], ['window|pane', '🏠'], ['chair|bench', '💺'], ['foot|feet|inch|yard|rod', '📏'], ['pound|ounce|ton', '⚖️'], ['quart|pint|gallon|gill|peck|bushel', '🍶']];
  function picFor(text) { const t = text.toLowerCase(); let best = null, at = 1e9; for (const [rx, e] of PICS) { const m = t.match(new RegExp('\\b(' + rx + ')')); if (m && m.index < at) { at = m.index; best = e; } } return best || '🧮'; }

  // ---------- spoken answers ----------
  // The kid can say the answer. First choice: our own Vosk server (CT 307, /api/recognize, shared with ABC Books),
  // told to listen only for number words. Fallback: the browser's own recognizer. No mic at all: the button hides.
  const NUM_WORDS = 'zero oh one two three four five six seven eight nine ten eleven twelve thirteen fourteen fifteen sixteen seventeen eighteen nineteen twenty thirty forty fifty sixty seventy eighty ninety hundred thousand and a';
  const SR = window.SpeechRecognition || window.webkitSpeechRecognition;
  const GUM = !!(window.isSecureContext && navigator.mediaDevices && navigator.mediaDevices.getUserMedia);
  let micOK = GUM || !!SR, useServer = GUM;
  const W1 = { zero: 0, oh: 0, one: 1, two: 2, three: 3, four: 4, for: 4, five: 5, six: 6, seven: 7, eight: 8, ate: 8, nine: 9, ten: 10, eleven: 11, twelve: 12, thirteen: 13, fourteen: 14, fifteen: 15, sixteen: 16, seventeen: 17, eighteen: 18, nineteen: 19 };
  const W10 = { twenty: 20, thirty: 30, forty: 40, fifty: 50, sixty: 60, seventy: 70, eighty: 80, ninety: 90 };
  // "nine thousand six hundred" -> 9600; "four two" (digits one at a time) -> 42; "7" -> 7
  function wordsToNumber(words) {
    // "a hundred" = one hundred; a stray "a" (Vosk often hears it for "one"/"and" in fast speech) alone is one
    const w = words.join(' ').toLowerCase().replace(/-/g, ' ').split(/\s+/).filter(x => x && x !== 'and').map((x, i, all) => x === 'a' ? (all.length === 1 || /^(hundred|thousand)$/.test(all[i + 1] || '') ? 'one' : '') : x).filter(Boolean);
    if (!w.length) return null;
    if (w.every(x => /^\d+$/.test(x))) return parseInt(w.join(''), 10);
    if (w.length > 1 && w.every(x => x in W1 && W1[x] < 10)) return parseInt(w.map(x => W1[x]).join(''), 10);
    let total = 0, cur = 0, any = false;
    for (const x of w) {
      if (/^\d+$/.test(x)) { cur += parseInt(x, 10); any = true; }
      else if (x in W1 || x in W10) {   // "nineteen twenty" (said like a year) -> 1920; "twenty four" -> 24
        const v = x in W1 ? W1[x] : W10[x];
        cur = v >= 10 && cur % 100 ? cur * 100 + v : cur + v; any = true;
      }
      else if (x === 'hundred') { cur = (cur || 1) * 100; any = true; }
      else if (x === 'thousand') { total += (cur || 1) * 1000; cur = 0; any = true; }
      else return null;   // a word that isn't a number: didn't understand
    }
    return any ? total + cur : null;
  }
  let stopListening = () => {};
  async function hearNumber() {
    if (useServer) {
      try { const w = await listenServer(7000); return w ? wordsToNumber(w) : null; }
      catch (e) { useServer = false; if (!SR) { micOK = false; return null; } }
    }
    const alts = await listenBrowser(7000);
    for (const a of alts) { const n = wordsToNumber(a); if (n != null) return n; }
    return null;
  }
  // records until the kid stops talking (1.2 s quiet) or the time runs out (same as ABC Books)
  async function listenServer(timeoutMs) {
    let stream;
    try { stream = await navigator.mediaDevices.getUserMedia({ audio: { echoCancellation: true, noiseSuppression: true } }); }
    catch (e) { micOK = false; return null; }
    const ctx = new (window.AudioContext || window.webkitAudioContext)();
    try { await ctx.resume(); } catch (e) {}
    const src = ctx.createMediaStreamSource(stream), proc = ctx.createScriptProcessor(4096, 1, 1);
    const chunks = []; let spoke = false, quiet = 0; const t0 = performance.now();
    await new Promise(res => {
      let done = false; const fin = () => { if (!done) { done = true; res(); } };
      stopListening = fin;
      proc.onaudioprocess = e => {
        const d = e.inputBuffer.getChannelData(0); chunks.push(new Float32Array(d));
        let s = 0; for (let i = 0; i < d.length; i++) s += d[i] * d[i];
        if (Math.sqrt(s / d.length) > 0.015) { spoke = true; quiet = 0; } else if (spoke) quiet += d.length / ctx.sampleRate * 1000;
        if ((spoke && quiet > 1200) || performance.now() - t0 > timeoutMs) fin();
      };
      src.connect(proc); proc.connect(ctx.destination);
    });
    stopListening = () => {};
    proc.disconnect(); src.disconnect(); stream.getTracks().forEach(t => t.stop());
    const rate = ctx.sampleRate; ctx.close();
    if (!spoke) return null;
    const r = await fetch('/api/recognize?near=0&words=' + encodeURIComponent(NUM_WORDS), { method: 'POST', headers: { 'Content-Type': 'application/octet-stream' }, body: to16k(chunks, rate) });
    if (!r.ok) throw new Error('speech ' + r.status);
    const j = await r.json();
    return j.words || [];
  }
  function to16k(chunks, rate) {   // Float32 at the device rate -> 16 kHz 16-bit PCM (what Vosk wants)
    const n = chunks.reduce((a, c) => a + c.length, 0), all = new Float32Array(n); let o = 0;
    for (const c of chunks) { all.set(c, o); o += c.length; }
    const ratio = rate / 16000, m = Math.floor(n / ratio), out = new Int16Array(m);
    for (let i = 0; i < m; i++) {
      const a = Math.floor(i * ratio), b = Math.min(n, Math.max(a + 1, Math.floor((i + 1) * ratio))); let s = 0;
      for (let k = a; k < b; k++) s += all[k];
      out[i] = Math.max(-1, Math.min(1, s / (b - a))) * 32767;
    }
    return out.buffer;
  }
  function listenBrowser(timeoutMs) {
    return new Promise(res => {
      if (!SR) return res([]);
      const r = new SR(); r.lang = 'en-US'; r.interimResults = false; r.maxAlternatives = 5;
      let best = [], done = false;
      const fin = () => { if (done) return; done = true; try { r.stop(); } catch (e) {} res(best); };
      stopListening = fin;
      r.onresult = e => { best = [...e.results[0]].map(a => a.transcript.split(/\s+/)); fin(); };
      r.onerror = e => { if (/not-allowed|service-not-allowed/.test(e.error || '')) micOK = false; fin(); };
      r.onend = fin;
      try { r.start(); } catch (e) { micOK = false; return fin(); }
      setTimeout(fin, timeoutMs);
    });
  }

  // number pad (big buttons; a computer keyboard works too; 🎤 to say the answer)
  function numPad(onOK) {
    const wrap = el('div', 'col'); wrap.style.height = 'auto'; wrap.style.gap = '1vmin';
    const entry = el('div', 'entry', '&nbsp;'); let val = '';
    // long whole numbers are shown with commas, the book's way (4,576,791)
    const show = () => { entry.textContent = (algMode ? RaysAlg.pretty(val) : /^\d{5,}$/.test(val) ? val.replace(/\B(?=(\d{3})+$)/g, ',') : val) || ' '; entry.classList.toggle('long', val.length > 9); entry.classList.remove('no', 'ok'); };
    const pad = el('div', 'pad');
    let fracMode = false;   // answers like 3/4 or 2 1/2: extra keys, and OK hands back the text
    let algMode = false;    // Algebra: + − ( ) / power and the question's letters; OK hands back the text (alg.js checks it)
    const press = k => {
      if (k === '⌫') val = val.slice(0, -1);
      else if (k === 'OK') { if (val.trim()) onOK(fracMode || algMode ? val.trim() : parseInt(val, 10)); return; }
      else if (algMode) { if (val.length < 200 && !(k === '^' && !/[\da-z)]$/.test(val))) val += k === '−' ? '-' : k; }
      // "and" after a decimal makes a complex decimal (.0154 1/4); the fraction after it may then have a /
      else if (k === 'and') { if (fracMode && val && !/[ /]/.test(val) && !val.endsWith('.')) val += ' '; }
      else if (k === '/') { if (fracMode && val && !val.endsWith(' ') && !val.includes('/') && (!val.includes('.') || /\.\d+ \d+$/.test(val))) val += '/'; }
      else if (k === '.') { if (fracMode && !/[ /.]/.test(val)) val += '.'; }
      else if (val.length < 40) val += k;   // the book's long numbers (up to 36 figures) are checked as written
      show();
    };
    ['1', '2', '3', '4', '5', '6', '7', '8', '9', '⌫', '0', 'OK'].forEach(k => pad.appendChild(btn(k, () => press(k), k === 'OK' ? 'okb' : '')));
    const fracRow = el('div', 'pad'); fracRow.style.display = 'none';
    fracRow.append(btn('/', () => press('/')), btn('and', () => press('and')), btn('.', () => press('.')));
    const algRow = el('div', 'pad alg'), letRow = el('div', 'pad alg'); algRow.style.display = letRow.style.display = 'none';
    ['+', '−', '(', ')', '/', '^'].forEach(k => algRow.appendChild(btn(k === '^' ? 'xⁿ' : k, () => press(k))));
    const onKey = e => { if (algMode && /^[a-z+\-()/^*]$/.test(e.key)) press(e.key); else if (/^[0-9]$/.test(e.key)) press(e.key); else if (e.key === '/' || e.key === '.') press(e.key); else if (e.key === ' ' && fracMode) { e.preventDefault(); press('and'); } else if (e.key === 'Backspace') press('⌫'); else if (e.key === 'Enter') press('OK'); };
    window.addEventListener('keydown', onKey);
    const mic = btn('🎤 Say it', async () => {
      if (mic.classList.contains('live')) return stopListening();
      stopAll(); mic.classList.add('live'); mic.textContent = '🎤 Listening…';
      const n = await hearNumber();
      mic.classList.remove('live'); mic.textContent = '🎤 Say it';
      if (!micOK) { mic.style.display = 'none'; return toast('This device cannot listen. Type the answer.', 2000); }
      if (n == null) return toast('I did not catch that. Say it again!', 1500);
      val = String(n); show(); onOK(n);
    }, 'micb');
    if (!micOK) mic.style.display = 'none';
    wrap.append(entry, pad, fracRow, algRow, letRow, mic);
    return { el: wrap, entry, frac(on) { fracMode = !!on; fracRow.style.display = on ? '' : 'none'; mic.style.display = on || !micOK ? 'none' : ''; },
      alg(keys) {   // keys = the question's letters ("abx"); null turns the algebra keys off
        algMode = keys != null; wrap.classList.toggle('algmode', algMode); algRow.style.display = letRow.style.display = algMode ? '' : 'none'; letRow.innerHTML = '';
        [...(keys || '')].forEach(c => letRow.appendChild(btn(c, () => press(c), 'let')));
        letRow.style.display = algMode && keys ? '' : 'none';
        if (algMode) { fracMode = false; fracRow.style.display = 'none'; mic.style.display = 'none'; }
      }, show(on) { wrap.style.display = on ? '' : 'none'; }, clear() { val = ''; show(); }, set(v, cls) { val = String(v); show(); if (cls) entry.classList.add(cls); }, bad() { entry.classList.remove('no'); void entry.offsetWidth; entry.classList.add('no'); }, stop() { stopListening(); window.removeEventListener('keydown', onKey); } };
  }
  // fractions compare by value: 5/2 = 2 1/2 = 2.5
  function fracVal(s) {
    s = String(s).trim(); let m;
    if ((m = s.match(/^(\d+) (\d+)\/(\d+)$/))) return +m[1] + m[2] / m[3];
    if ((m = s.match(/^(\d+)\/(\d+)$/))) return m[1] / m[2];
    if ((m = s.match(/^(\d*)\.(\d+) (\d+)\/(\d+)$/))) return parseFloat((m[1] || '0') + '.' + m[2]) + m[3] / m[4] / Math.pow(10, m[2].length);   // complex decimal
    return /^\d*\.?\d+$/.test(s) ? parseFloat(s) : NaN;
  }
  // relative check, so tiny decimals (.00000002) and large ones (2900.000029) both compare exactly
  const sameVal = (a, b) => { const x = fracVal(a), y = fracVal(b); return Math.abs(x - y) <= 1e-13 + 1e-10 * Math.abs(y); };
  // counters tray: tap beads to count with
  function initTray() {
    const f = $('frame'); f.innerHTML = '';
    for (let i = 0; i < 30; i++) f.appendChild(btn('', e => { e.currentTarget.classList.toggle('on'); count(); }));
    const count = () => { $('trayN').textContent = f.querySelectorAll('.on').length; };
    $('trayClear').onclick = () => { f.querySelectorAll('.on').forEach(b => b.classList.remove('on')); count(); };
    $('trayClose').onclick = () => $('tray').classList.remove('on');
  }

  // "Show your work": a pencil button that opens scratch paper beside the question (scratch.js). The paper comes
  // back open on the next question and the next lesson if the kid left it open.
  function workPad() {
    const w = RaysWork.Pad($('stage'), S.kid, bookId, toast);
    const b = btn('', () => { label(w.toggle()); });
    const label = on => { b.textContent = on ? '✏️ Put paper away' : '✏️ Show work'; b.classList.toggle('on', on); };
    label(w.start());
    return { btn: b, page: w.page, result: w.result, stop: w.stop };
  }
  // grown-ups: every page of work one kid has in one book, newest first, grouped by lesson
  // With a family account, the pages saved there from other devices are added (fetched after the first draw).
  let workBack = 'map';
  const gone = new Set();   // deleted in this visit: the account may not have heard yet
  function showWork(from, fromAcct) {
    workBack = from; const id = bookId || 'primary1877', B = BOOKS[id], kid = S.kid;
    const pages = RaysWork.list(kid, id), box = $('workList'); box.innerHTML = '';
    if (fromAcct) {
      const at = new Map(pages.map((pg, i) => [pg.key, i]));
      fromAcct.filter(pg => !gone.has(id + '|' + pg.key) && !gone.has(id + '|*')).forEach(pg => {
        if (!at.has(pg.key)) pages.push(pg); else if ((pg.t || 0) > (pages[at.get(pg.key)].t || 0)) pages[at.get(pg.key)] = pg;
      });
    } else RaysWork.remote(kid, id).then(r => { if (r.length && $('work').classList.contains('on') && S.kid === kid && (bookId || 'primary1877') === id) showWork(from, r); });
    // lessons with the newest work first; inside a lesson, the pages in question order
    const newest = {}; pages.forEach(pg => { newest[pg.l] = Math.max(newest[pg.l] || 0, pg.t || 0); });
    pages.sort((a, b) => newest[b.l] - newest[a.l] || a.key.localeCompare(b.key, undefined, { numeric: true }));
    $('workTitle').textContent = `${S.kid}: work in ${B ? B.title : ''} (${pages.length} page${pages.length === 1 ? '' : 's'})`;
    $('workDel').style.display = pages.length ? '' : 'none';
    $('workDel').onclick = () => { if (confirm(`Delete all of ${S.kid}'s work in this book?`)) { RaysWork.remove(S.kid, id); gone.add(id + '|*'); showWork(from); } };
    if (!pages.length) box.appendChild(el('p', 'note', 'No work saved yet. In Story problems and Practice, the ✏️ Show work button opens scratch paper; each page is kept here with its question.'));
    const tw = Math.min(300, Math.floor((window.innerWidth - 60) / 2));
    let shown = 0, lastL = null;
    function more() {
      pages.slice(shown, shown + 40).forEach(pg => {
        if (pg.l !== lastL) { lastL = pg.l; const Ls = B && B.lessons.find(x => x.n === pg.l); box.appendChild(el('h3', 'wles', '')).textContent = `Lesson ${pg.l}${Ls ? ': ' + Ls.title : ''}`; }
        const card = el('button', 'wpage'); card.appendChild(RaysWork.picture(pg, tw));
        const cap = el('div', 'wcap'); cap.textContent = pg.q;
        const res = el('div', 'wres' + (pg.r === 'miss' ? ' miss' : pg.r === 'ok' ? ' ok' : ''));
        res.textContent = (pg.a != null ? `Answer: ${pg.a} ` + (pg.r === 'ok' ? '✓' : pg.r === 'miss' ? '(missed a try)' : '') : 'Not answered') + ' · ' + new Date(pg.t).toLocaleString([], { month: 'short', day: 'numeric', hour: 'numeric', minute: '2-digit' });
        card.append(cap, res); card.onclick = () => bigPage(pg, id, from); box.appendChild(card);
      });
      shown += 40; const m = box.querySelector('.wmore'); if (m) m.remove();
      if (shown < pages.length) box.appendChild(btn('Show more', more, 'wmore'));
    }
    more(); go('work');
  }
  function bigPage(pg, id, from) {
    const v = $('workBig'); v.innerHTML = ''; v.classList.add('on');
    const q = el('div', 'wcap'); q.textContent = `Lesson ${pg.l}: ${pg.q}`;
    const w = Math.min(window.innerWidth - 40, 1000), h = window.innerHeight - 190;
    v.append(q, RaysWork.picture(pg, Math.round(Math.min(w, h / (pg.h || 0.75)))),
      el('div', 'row'));
    v.lastChild.append(btn('Close', () => v.classList.remove('on'), 'big'),
      btn('Delete this page', () => { if (confirm('Delete this page?')) { RaysWork.remove(S.kid, id, pg.key); gone.add(id + '|' + pg.key); v.classList.remove('on'); showWork(from); } }, 'ghost parent'));
  }
  $('mapWork').onclick = () => showWork('map');
  $('stkWork').onclick = () => { if (S.kid) showWork('stickers'); };
  $('workBack').onclick =() => { $('workBig').classList.remove('on'); go(workBack); };

  // ---------- home, map, stickers ----------
  let addingReader = false;
  function home() {
    const k = $('kids'); k.innerHTML = '';
    // names go in as text (never innerHTML); a name with no recorded greeting gets the line without a name
    const rs = readers();
    const signed = window.ABCAccount && window.ABCAccount.signedIn;
    if (!rs.length && S.kid) { S.kid = null; save(); }
    $('whoHint').textContent = rs.length ? '' : 'Grown-ups: add each learner (a first name or nickname).';
    rs.forEach(n => { const b = btn('', () => { S.kid = n; save(); home(); const g = P.greet.replace('{kid}', n); speak(urlFor(g) ? g : P.hello); }, S.kid === n ? 'sel' : ''); b.textContent = n; k.appendChild(b); });
    if (signed) {   // a family account: names are added and removed on the account page
      if (!rs.length) $('whoHint').innerHTML = 'Grown-ups: add your learners on the <a href="/account/">account page</a>.';
      else { const a = el('a', 'parent'); a.href = '/account/'; a.textContent = 'Add or remove learners'; k.appendChild(a); }
    } else if (addingReader || !rs.length) {
      // the names live in 'abcbooks'.readers, shared with ABC Books, so a learner added here is in both
      const inp = el('input'); inp.maxLength = 16; inp.placeholder = 'Name'; inp.autocomplete = 'off';
      const add = () => {
        const n = inp.value.trim().replace(/\s+/g, ' ');
        if (!NAME_OK(n)) return toast('Letters only, please');
        if (rs.some(x => x.toLowerCase() === n.toLowerCase())) return toast(`${n} is already here`);
        let c = null; try { c = JSON.parse(localStorage.getItem('abcbooks')); } catch (e) {}
        c = c || { kid: null, progress: {}, boards: {} };
        c.readers = rs.concat(n); if (!c.kid) c.kid = n;
        try { localStorage.setItem('abcbooks', JSON.stringify(c)); } catch (e) {}
        S.kid = n; addingReader = false; save(); home();
      };
      inp.onkeydown = e => { if (e.key === 'Enter') add(); };
      k.append(inp, btn('Add', add, 'big'));
      if (rs.length) k.appendChild(btn('Cancel', () => { addingReader = false; home(); }, 'ghost parent'));
      setTimeout(() => inp.focus(), 50);
    } else if (rs.length < 8) k.appendChild(btn('＋', () => { addingReader = true; home(); }, 'add'));
    const sb = $('speedBtn'); sb.textContent = 'Voice: ' + (S.speed === 75 ? 'faster' : 'normal');
    sb.onclick = () => { S.speed = S.speed === 75 ? 60 : 75; save(); home(); speak(P.yes); };
    const sh = $('shelf'); sh.innerHTML = '';
    // the kid's current book glows: the first open book they haven't finished
    const doneIn = id => S.kid && BOOKS[id] ? Object.values(prog(id)).filter(x => x && x.stars).length : 0;
    const cur = S.kid && SHELF.find(b => BOOKS[b.id] && doneIn(b.id) < BOOKS[b.id].lessons.length);
    SHELF.forEach(b => {
      const B = BOOKS[b.id], n = B ? B.lessons.length : 0, d = doneIn(b.id);
      const pr = !B ? 'Coming soon' : S.kid ? (d >= n ? '★ Finished!' : d ? `Lesson ${d + 1} of ${n}` : `Lessons: ${n}`) : '';
      const e = btn(`<b>${b.title}</b><span class="bp">${pr}</span><small>${b.sub}</small>`, () => {
        if (!B) return toast('Coming soon!');
        if (!S.kid) return toast('Pick your name first!');
        openBook(b.id);
      }, 'book' + (!B ? ' soon' : '') + (cur && cur.id === b.id ? ' cur' : ''));
      e.style.background = b.color; sh.appendChild(e);
    });
  }
  let book, bookId;
  let skipArm = null;   // a locked tile tapped once: a 2nd tap within 4 s skips ahead to it
  function openBook(id) {
    bookId = id; book = BOOKS[id]; $('mapTitle').textContent = `${S.kid}: ${book.title}`;
    const t = $('tiles'); t.innerHTML = ''; const p = prog(id);
    // Lessons are open up to one past the furthest one finished, so after a skip the kid carries on from there.
    let far = -1; book.lessons.forEach((L, i) => { if ((p[L.n] || {}).stars) far = i; });
    const paidAt = i => window.ABCAccount && window.ABCAccount.locked('math', i);
    book.lessons.forEach((L, i) => {
      const st = (p[L.n] || {}).stars || 0, open = i <= far + 1, isNext = i === far + 1;
      // past the free lessons without a subscription (/account/sync.js; only when the paywall is switched on)
      const paid = paidAt(i);
      const tile = btn(`${L.n}<small>${L.part}</small><span class="st">${st ? starStr(st) : paid ? '🔒' : ''}</span>`, () => {
        if (paid) return window.ABCAccount.unlockScreen('math');
        if (!open && !st && !(skipArm && skipArm.i === i && Date.now() - skipArm.at < 4000)) {
          skipArm = { i, at: Date.now() };
          return toast(`Lesson ${L.n} is ahead. Tap it again to skip to it.`, 3500);
        }
        skipArm = null; startLesson(L);
      }, 'tile' + (isNext ? ' next' : '') + (!open && !st ? ' locked' : '') + (L.review ? ' rev' : ''));
      tile.title = `Lesson ${L.n}: ${L.title}` + (st ? ', done' : !open ? ', ahead' : '');
      tile.setAttribute('aria-label', tile.title); t.appendChild(tile);
    });
    // "Go to lesson": every lesson by title, grouped by part, starting on the next one; Go starts it.
    const sel = $('jumpSel'); sel.innerHTML = ''; let grp = null;
    book.lessons.forEach((L, i) => {
      if (!grp || grp.label !== L.part) { grp = document.createElement('optgroup'); grp.label = L.part || ''; sel.appendChild(grp); }
      const o = el('option'); o.textContent = `Lesson ${L.n}: ${L.title}` + ((p[L.n] || {}).stars ? ' ★' : '');
      o.value = i; if (i === Math.min(far + 1, book.lessons.length - 1)) o.selected = true; grp.appendChild(o);
    });
    $('jumpGo').onclick = () => { const i = +sel.value; paidAt(i) ? window.ABCAccount.unlockScreen('math') : startLesson(book.lessons[i]); };
    go('map'); speak(P.map);
    const nx = t.querySelector('.next'); if (nx) nx.scrollIntoView({ block: 'center' });
  }
  const stkKey = (b, n) => b === 'primary1877' ? n : `${b}:${n}`;   // Book One keeps its first keys
  function stickers() {   // the sticker book for the book the kid is in (Book One when none is open yet)
    const g = $('stkList'); g.innerHTML = ''; const got = kidMap(S.stickers), id = bookId || 'primary1877', B = BOOKS[id];
    const n = B ? B.lessons.filter(L => got[stkKey(id, L.n)]).length : 0;
    $('stkTitle').textContent = `${S.kid || ''} sticker book: ${B ? B.title : ''}, ${n} of ${B ? B.lessons.length : 0}`;
    (B ? B.lessons : []).forEach(L => { const v = got[stkKey(id, L.n)]; const s = el('span', v ? 'got' : '', v || ''); s.title = 'Lesson ' + L.n; g.appendChild(s); });
  }
  $('toStickers').onclick = () => { if (!S.kid) return toast('Pick your name first!'); go('stickers'); };

  // ---------- lesson flow ----------
  let L, steps, stepIdx, scores, cleanup = null, sayAgain = null, stepTok = 0;
  function stepsFor(L) {
    switch (L.kind) {
      case 'count': return ['count', 'find', 'trace'];
      case 'numbers': return ['numbers', 'find', 'trace'];
      case 'read': return ['say', 'find'];
      case 'write': return ['write', 'trace'];
    }
    const s = [];
    if (L.arts) s.push('study'); if (L.quiz) s.push('quiz');
    if (L.table) s.push('learn', 'drill');
    if (L.rules) s.push('rules'); if (L.rules && L.rules.some(r => r.quiz !== false) && !L.table) s.push('drill');
    if (L.qs) s.push('problems');
    return s;
  }
  function startLesson(lesson) {
    L = lesson; steps = stepsFor(L); scores = {}; stepIdx = 0;
    $('lesTitle').textContent = `Lesson ${L.n}: ${L.title}`;
    go('lesson'); runStep(true);
  }
  function leave() { stopStep(); openBook(bookId); }
  function stopStep() { stopAll(); $('tray').classList.remove('on'); if (cleanup) { cleanup(); cleanup = null; } }
  function runStep(first) {
    stopStep();
    const id = steps[stepIdx];
    const dots = $('steps'); dots.innerHTML = '';
    steps.forEach((x, i) => dots.appendChild(el('i', i < stepIdx ? 'done' : i === stepIdx ? 'cur' : '', STEPS[x].icon)));
    if (!id) return finish();
    $('stepTitle').textContent = STEPS[id].name; $('stage').innerHTML = ''; $('controls').innerHTML = ''; $('hint').textContent = P[id] || '';
    sayAgain = null; const my = ++stepTok;
    const intro = (first ? speak('lesson:' + L.n, P[id]) : speak(P[id]));
    $('sayBtn').onclick = () => sayAgain ? sayAgain() : speak(P[id]);
    ({ count: stepCount, numbers: stepNumbers, find: stepFind, say: stepSay, write: stepWrite, trace: stepTrace, learn: stepLearn, rules: stepRules, drill: stepDrill, problems: stepProblems, study: stepStudy, quiz: stepQuiz })[id](intro.then(() => my === stepTok));
  }
  function next(stars) { scores[steps[stepIdx]] = stars; stepIdx++; runStep(); }
  const afterIntro = (intro, fn) => intro.then(ok => { if (ok) fn(); });

  // Lesson 1: count the balls, one tap per ball
  function stepCount(intro) {
    let n = L.from;
    const st = $('stage'); const col = el('div', 'col'); st.appendChild(col);
    function show() {
      col.innerHTML = ''; let c = 0;
      const balls = el('div', 'balls'); const big = el('div', 'big-n', '&nbsp;'); const w = el('div', 'word', '&nbsp;');
      for (let i = 0; i < n; i++) balls.appendChild(btn('', e => {
        const b = e.currentTarget; if (b.classList.contains('on')) return;
        b.classList.add('on'); b.textContent = ++c; sayNum(c);
        if (c === n) { big.textContent = n; w.textContent = words(n); setTimeout(() => { beep(true); speak('n:' + n).then(() => wait(700)).then(() => { if (++n > L.to) next(3); else show(); }); }, 650); }
      }, 'ball'));
      col.append(balls, big, w);
      $('hint').textContent = `Tap each ball and count. (${n - L.from + 1} of ${L.to - L.from + 1})`;
    }
    sayAgain = () => speak(P.count);
    show();
  }
  // Lessons 2-4: each number with its tens and ones
  function stepNumbers(intro) {
    let n = L.from; const st = $('stage'); const col = el('div', 'col'); st.appendChild(col);
    let auto = false;
    function show() {
      col.innerHTML = '';
      col.append(el('div', 'big-n', n), el('div', 'word', words(n)));
      const rods = el('div', 'rods');
      for (let t = 0; t < Math.floor(n / 10); t++) { const r = el('div', 'rod'); for (let i = 0; i < 10; i++) r.appendChild(el('i')); rods.appendChild(r); }
      const ones = el('div', 'grp'); for (let i = 0; i < n % 10; i++) ones.appendChild(el('i', 'dot')); rods.appendChild(ones);
      col.appendChild(rods);
      col.appendChild(el('div', 'note', `${Math.floor(n / 10)} tens and ${n % 10} ones`));
      $('hint').textContent = `${n - L.from + 1} of ${L.to - L.from + 1}`;
      return sayNum(n);
    }
    async function playAll() { auto = true; playB.disabled = true; while (auto && n <= L.to) { await show(); await wait(700); if (!auto) break; if (n < L.to) n++; else break; } auto = false; playB.disabled = false; nextB.disabled = false; }
    const prevB = btn('◀', () => { auto = false; if (n > L.from) { n--; show(); } });
    const fwdB = btn('▶', () => { auto = false; if (n < L.to) { n++; show(); } else nextB.disabled = false; });
    const playB = btn('▶ Play all', playAll, 'big'), nextB = btn('Next step ▶', () => next(3));
    nextB.disabled = true; $('controls').append(prevB, playB, fwdB, nextB);
    sayAgain = show; cleanup = () => { auto = false; };
    col.append(el('div', 'big-n', n), el('div', 'word', words(n)));
    afterIntro(intro, playAll);
  }
  // hear a number, tap it
  function stepFind(intro) {
    const pool = L.list || range(L.from, L.to);
    const rounds = shuffle(pool).slice(0, 8); let r = 0, miss = 0, busy = false, target;
    const st = $('stage'); const col = el('div', 'col'); st.appendChild(col);
    const grid = el('div', 'choices'); col.appendChild(grid);
    function round() {
      if (r >= rounds.length) { toast('Great job! ⭐'); return setTimeout(() => next(starsFor(miss, rounds.length)), 900); }
      target = rounds[r++]; grid.innerHTML = '';
      const opts = shuffle([target, ...shuffle(pool.filter(x => x !== target)).slice(0, 3)]);
      opts.forEach(v => grid.appendChild(btn(v, e => {
        if (busy) return; const b = e.currentTarget;
        if (v === target) { busy = true; beep(true); b.classList.add('ok'); setTimeout(() => { busy = false; round(); }, 800); }
        else { miss++; beep(false); b.classList.remove('no'); void b.offsetWidth; b.classList.add('no'); }
      })));
      $('hint').textContent = `Find the number you hear (${r} of ${rounds.length})`;
      sayAgain = () => speak('find:' + target); sayAgain();
    }
    afterIntro(intro, round);
  }
  // Lesson 5: numbers to be read, out loud to a grown-up
  function stepSay(intro) {
    const g = el('div', 'grid'); $('stage').appendChild(g);
    L.list.forEach(v => g.appendChild(btn(v, e => { e.currentTarget.classList.add('seen'); sayNum(v); })));
    $('controls').appendChild(btn('Done ✓', () => next(3), 'big'));
  }
  // Lesson 6: numbers to be written: hear it, type it
  function stepWrite(intro) {
    const items = shuffle(L.list).slice(0, 12); let i = 0, miss = 0, tries = 0;
    const col = el('div', 'col'); $('stage').appendChild(col);
    const pad = numPad(v => {
      if (v === items[i]) { beep(true); pad.set(v, 'ok'); tries = 0; setTimeout(() => { i++; ask(); }, 700); }
      else if (++tries < 2) { miss++; beep(false); pad.bad(); speak(P.tryAgain, 'n:' + items[i]); }
      else { miss++; pad.set(items[i], 'ok'); speak('ans:' + items[i]); tries = 0; setTimeout(() => { i++; ask(); }, 1800); }
    });
    col.appendChild(pad.el);
    function ask() {
      if (i >= items.length) return next(starsFor(miss, items.length));
      pad.clear(); $('hint').textContent = `Write the number you hear (${i + 1} of ${items.length})`;
      sayAgain = () => sayNum(items[i]); sayAgain();
    }
    cleanup = () => pad.stop();
    afterIntro(intro, ask);
  }
  // trace the figures (same engine as ABC Books)
  function stepTrace(intro) {
    const nums = L.kind === 'count' ? ['1 2 3', '4 5 6', '7 8 9', '10'] : L.kind === 'write' ? ['0 1 2 3 4', '5 6 7 8 9', '10 20 30'] : (() => { const r = range(L.from, L.to), out = []; for (let i = 0; i < r.length && out.length < 4; i += 4) out.push(r.slice(i, i + 4).join(' ')); return out; })();
    let i = 0, misses = 0, tracer;
    const box = el('div', 'grow paper'); box.style.height = '100%'; $('stage').appendChild(box);
    const cv = el('canvas'); box.appendChild(cv);
    ABCTrace.Tracer.onMiss = why => { misses++; beep(false); $('hint').textContent = why === 'start' ? 'Start at the green dot!' : 'Follow the arrow all the way.'; };
    function page() {
      if (tracer) tracer.stop();
      $('hint').textContent = `Page ${i + 1} of ${nums.length}: start at the green dot and follow the arrow.`;
      tracer = ABCTrace.Tracer(cv, nums[i], 'print', () => { beep(true); toast('⭐', 700); setTimeout(() => { if (++i < nums.length) page(); else next(misses < 3 ? 3 : misses < 8 ? 2 : 1); }, 800); });
    }
    $('controls').appendChild(btn('Skip page', () => { if (++i < nums.length) page(); else next(1); }, 'ghost parent'));
    setTimeout(page, 50);
    const onRes = () => page(); window.addEventListener('resize', onRes);
    cleanup = () => { tracer && tracer.stop(); window.removeEventListener('resize', onRes); };
  }

  // the book's table, one line at a time with its picture
  function stepLearn(intro) { learnList(L.table.map(f => ({ t: f.t, pic: () => dotsFor(L.op, f.a, f.b), ex: `${expr(L.op, f)} = ${f.r}` })), intro); }
  function stepRules(intro) { learnList(L.rules.map(r => ({ t: r.t, pic: () => el('div', 'pic', picFor(r.t)) })), intro); }
  function learnList(items, intro) {
    const col = el('div', 'col'); $('stage').appendChild(col);
    const face = el('div', 'col'); face.style.height = 'auto'; const chips = el('div', 'chips'); col.append(face, chips);
    let i = 0, auto = false; const seen = new Set();
    const chipEls = items.map((it, k) => { const c = btn(it.t, () => { auto = false; i = k; show(); }, 'chip'); chips.appendChild(c); return c; });
    function show() {
      face.innerHTML = ''; const it = items[i];
      const f = el('div', 'fact', it.t); if (it.ex) f.appendChild(el('span', 'sm', it.ex));
      face.append(f, it.pic());
      chipEls.forEach((c, k) => c.classList.toggle('cur', k === i)); chipEls[i].classList.add('seen'); seen.add(i);
      chipEls[i].scrollIntoView({ block: 'nearest' });
      if (seen.size === items.length) nextB.disabled = false;
      return speak(it.t);
    }
    async function playAll() { auto = true; playB.disabled = true; for (i = 0; i < items.length && auto; i++) { await show(); await wait(600); } i = Math.min(i, items.length - 1); auto = false; playB.disabled = false; nextB.disabled = false; }
    const playB = btn('▶ Read to me', playAll, 'big'), nextB = btn('Next ▶', () => next(3));
    nextB.disabled = true; $('controls').append(playB, nextB);
    sayAgain = show; cleanup = () => { auto = false; };
    face.appendChild(el('div', 'fact', items[0].t));
    $('hint').textContent = 'Tap ▶ Read to me, or tap each line.';
    afterIntro(intro, playAll);
  }

  // practice: every fact in the table (or every rule), timed; a missed one comes back later
  function stepDrill(intro) {
    const items = L.table ? L.table.map(f => ({ q: f.q, ex: expr(L.op, f) + ' = ?', a: f.r, say: f.t, pic: () => dotsFor(L.op, f.a, f.b) }))
      : L.rules.filter(r => r.quiz !== false).map(r => ({ q: r.q, ex: `? ${r.a} = 1 ${r.b}`, a: r.n, say: r.t, pic: () => el('div', 'pic', picFor(r.t)) }));
    let queue = shuffle(items), done = 0, miss = 0, tries = 0, streak = 0, t0 = 0, cur;
    const key = `${bookId}:${L.n}`, best = kidMap(S.best);
    const col = el('div', 'qa'); $('stage').appendChild(col);
    const top = el('div', 'row'); const timer = el('div', 'timer'); const fire = el('div', 'timer'); top.append(timer, fire);
    const ex = el('div', 'expr'); const qd = el('div', 'note'); const picBox = el('div', 'picbox'); picBox.style.minHeight = '0';
    const pad = numPad(check);
    col.append(top, ex, qd, picBox, pad.el);
    const showB = btn('👀 Show me', () => { picBox.innerHTML = ''; picBox.appendChild(cur.pic()); }, 'ghost');
    const work = workPad();
    $('controls').append(showB, work.btn);
    const tick = setInterval(() => { if (t0) timer.textContent = `⏱ ${Math.round((Date.now() - t0) / 1000)}s` + (best[key] ? `  (best ${best[key]}s)` : ''); }, 500);
    function ask() {
      if (!queue.length) {
        const secs = Math.round((Date.now() - t0) / 1000);
        let msg = `${secs} seconds!`;
        if (!best[key] || secs < best[key]) { if (best[key]) msg += ' New best time! 🏆'; best[key] = secs; save(); }
        toast(msg, 1800); return setTimeout(() => next(starsFor(miss, items.length)), 1800);
      }
      cur = queue[0]; tries = 0; pad.clear(); picBox.innerHTML = '';
      ex.textContent = cur.ex; ex.className = 'expr' + (cur.ex.length > 12 ? ' long' : ''); qd.textContent = cur.q;
      work.page(`${L.n}:d:${cur.ex}`, { q: `${cur.ex}   ${cur.q}`, l: L.n });
      $('hint').textContent = `${done + 1} of ${items.length}`;
      sayAgain = () => speak(cur.q); sayAgain();
      if (!t0) t0 = Date.now();
    }
    function check(v) {
      work.result(v, v === cur.a);
      if (v === cur.a) {
        beep(true); pad.set(v, 'ok'); streak++; fire.textContent = streak >= 3 ? `🔥 ${streak}` : '';
        queue.shift(); done++; setTimeout(ask, 600);
      } else if (++tries < 2) { beep(false); pad.bad(); miss++; streak = 0; fire.textContent = ''; speak(P.tryAgain); }
      else {   // show it, say the book's line, and ask it again later
        pad.set(cur.a, 'ok'); picBox.innerHTML = ''; picBox.appendChild(cur.pic());
        const it = queue.shift(); queue.splice(Math.min(3, queue.length), 0, it);
        speak(cur.say).then(() => wait(600)).then(ask);
      }
    }
    cleanup = () => { clearInterval(tick); pad.stop(); work.stop(); };
    afterIntro(intro, ask);
  }

  // story problems: the book's questions, one answer at a time; two misses shows the answer and the book's solution
  function stepProblems(intro) {
    const qs = L.qs, posKey = `${bookId}:${L.n}`, pos = kidMap(S.pos);
    let qi = pos[posKey] || 0, pi = 0, tries = 0, miss = 0, si = 0, busy = false;
    if (qi >= qs.length) qi = 0;
    const col = el('div', 'qa'); $('stage').appendChild(col);
    const pic = el('div', 'pic'); const qd = el('div', 'q'); const seq = el('div', 'seq'); const sol = el('div', 'sol');
    const pad = numPad(check);
    col.append(pic, qd, seq, sol, pad.el);
    initTray();
    const work = workPad();
    if (!/^algebra/.test(bookId)) $('controls').append(btn('🧮 Counters', () => $('tray').classList.toggle('on')));   // no counting beads in Algebra
    $('controls').append(work.btn, btn('Finish later', () => { pos[posKey] = qi; save(); leave(); }, 'ghost parent'));
    if (L.note) $('hint').textContent = L.note;
    const Q = () => qs[qi], Pt = () => Q().parts[pi];
    function render() {
      const q = Q(); qd.innerHTML = ''; qd.classList.toggle('has-stack', q.parts.some(p => p.col));
      q.parts.forEach((p, k) => {
        qd.appendChild(el('span', 'pt' + (k === pi ? ' cur' : k < pi ? '' : ''), p.t + ' '));
        if (p.col) {   // a sum written in a column, the book's way: numbers right-aligned, the sign on the last line, a rule under
          const st = el('div', 'stack'), sign = { '+': '+', '-': '−', x: '×' }[p.col.op];
          p.col.n.forEach((v, j) => st.appendChild(el('div', '', (j === p.col.n.length - 1 ? `<b>${sign}</b>` : '') + v)));
          if (k < pi) st.appendChild(el('div', 'sum', String(p.f || p.a))); qd.appendChild(st);
        }
        if (k < pi) qd.appendChild(el('span', 'got' + (p.missed ? ' miss' : ''), p.seq || p.open ? '✓' : p.e != null ? RaysAlg.pretty(p.e) : (p.f || p.a)));
        if (k < q.parts.length - 1) qd.append(' ');
      });
      pic.textContent = picFor(q.parts[0].t);
      seq.innerHTML = '';
      const p = Pt(); if (p && p.seq) p.seq.forEach((v, k) => seq.appendChild(el('span', k < si ? 'got' : k === si ? 'cur' : '', k < si ? v : '?')));
      const cur = seq.querySelector('.cur'); if (cur) cur.scrollIntoView({ block: 'nearest' });
    }
    function ask(sayIt) {
      if (qi >= qs.length) { pos[posKey] = 0; save(); return next(starsFor(miss, qs.reduce((n, q) => n + q.parts.length, 0))); }
      tries = 0; pad.clear(); sol.textContent = ''; if (pi === 0) Q().shown = false;
      const p = Pt(); if (p.seq && si === 0) si = 1;   // the first number of a count is given
      pad.frac(p.f); pad.alg(p.e != null ? p.v || '' : null); pad.show(!p.open); openB.style.display = p.open ? '' : 'none';
      render();
      work.page(`${L.n}:${qi}`, { q: Q().parts.map(x => x.t).join(' '), l: L.n });   // one page for all the parts of a question
      $('hint').textContent = `Question ${qi + 1} of ${qs.length}`;
      sayAgain = () => speak(p.s || p.t, p.open ? 'Answer out loud, then tap Next.' : null);
      if (sayIt !== false) sayAgain();
    }
    function advance() {
      busy = false; const q = Q();
      if (++pi < q.parts.length) { si = 0; return ask(); }
      if (q.sol && !q.shown) { q.shown = true; sol.textContent = q.sol; }
      qi++; pi = 0; si = 0; pos[posKey] = qi; save();
      setTimeout(() => ask(), q.sol && sol.textContent ? 600 : 0);
    }
    function check(v) {
      if (busy) return; const p = Pt();
      const norm = s => String(s).trim().replace(/\s+/g, ' ');
      const isRight = (p, v) => { const want = p.seq ? p.seq[si] : p.e != null ? p.e : p.f || p.a;
        return p.e != null ? RaysAlg.ok(v, p.e) || (p.ealt || []).some(x => RaysAlg.ok(v, x)) : p.ex ? p.ex.includes(norm(v)) : p.f ? sameVal(v, p.f) || (p.falt || []).some(x => sameVal(v, x)) : v === want || (!p.seq && p.alt && p.alt.includes(v)); };
      let right = isRight(p, v);
      // "x = {A} or {B}" (the two roots of a quadratic): B typed first counts too; the next box then wants A
      const nx = Q().parts[pi + 1];
      if (!right && p.pr && nx && isRight(nx, v)) {
        for (const k of ['e', 'es', 'ealt', 'a', 'alt', 'f', 'falt', 'v']) { const t = p[k]; if (nx[k] === undefined) delete p[k]; else p[k] = nx[k]; if (t === undefined) delete nx[k]; else nx[k] = t; }
        right = true;
      }
      const want = p.seq ? p.seq[si] : p.e != null ? p.e : p.f || p.a;
      if (!p.seq || !right) work.result(v, right);
      if (right) {
        beep(true); pad.set(v, 'ok'); tries = 0;
        if (p.seq) { if (++si >= p.seq.length) { busy = true; render(); return setTimeout(() => { si = 0; advance(); }, 700); } setTimeout(() => { pad.clear(); render(); }, 350); return; }
        busy = true; speak(P.yes); setTimeout(advance, 900); return;
      }
      miss++;
      if (++tries < 2) { beep(false); pad.bad(); speak(P.tryAgain); return; }
      // second miss: show the answer, say it, and the book's solution when there is one
      busy = true; pad.set(want, 'ok'); tries = 0;
      const q = Q(), last = !p.seq && pi === q.parts.length - 1;
      if (last && q.sol) { sol.textContent = q.sol; q.shown = true; }
      if (p.seq) { si++; if (si >= p.seq.length) { si = 0; p.missed = true; return speak('ans:' + want).then(() => advance()); } return speak('n:' + want).then(() => { busy = false; pad.clear(); render(); }); }
      p.missed = true;
      speak('ans:' + want, last && q.sol ? q.sol : null).then(() => wait(800)).then(advance);
    }
    // a question answered out loud to a grown-up: no typing, just Next. When the book gives the answer (sol:), the first
    // Next shows and says it (e.g. "57 is written LVII."), the second moves on.
    const openB = btn('Next ▶', () => {
      if (busy) return; const q = Q();
      if (pi === q.parts.length - 1 && q.sol && !q.shown) { q.shown = true; sol.textContent = q.sol; speak(q.sol); return; }
      busy = true; beep(true); setTimeout(advance, 300);
    }, 'big'); openB.style.display = 'none';
    col.appendChild(openB);
    cleanup = () => { pad.stop(); work.stop(); };
    afterIntro(intro, () => ask());
  }

  // Book Five: the book's articles, each beside its figure cut from the scan (same key as make-audio.ps1 ArtKey)
  const artKey = a => `${a.n}. ${a.h ? a.h + '. ' : ''}${a.t}`;
  function stepStudy(intro) {
    const arts = L.arts; let i = 0, run = 0; const seen = new Set();
    const stop = () => { run++; playB.disabled = false; };   // a tap cancels reading-to-me
    const col = el('div', 'col study'); $('stage').appendChild(col);
    const face = el('div', 'art'); const chips = el('div', 'chips'); col.append(face, chips);
    const chipEls = arts.map((a, k) => { const c = btn('§' + a.n, () => { stop(); i = k; show(); }, 'chip'); chips.appendChild(c); return c; });
    function draw() {
      const a = arts[i]; face.innerHTML = '';
      if (a.figs) { const fb = el('div', 'figs'); a.figs.forEach(f => { const im = el('img', 'fig'); im.src = f; im.alt = 'Figure for article ' + a.n; fb.appendChild(im); }); face.appendChild(fb); }
      face.appendChild(el('p', '', `<b>${a.n}.</b> ${a.h ? `<b>${a.h}.</b> ` : ''}${a.t}`));
      chipEls.forEach((c, k) => c.classList.toggle('cur', k === i));
    }
    function show() {
      const a = arts[i]; draw(); chipEls[i].classList.add('seen'); seen.add(i);
      chipEls[i].scrollIntoView({ block: 'nearest' }); face.scrollTop = 0;
      $('hint').textContent = `Part ${i + 1} of ${arts.length}`;
      if (seen.size === arts.length) nextB.disabled = false;
      return speak(artKey(a));
    }
    async function playAll() {   // from the part on screen to the end
      const my = ++run; playB.disabled = true;
      for (; i < arts.length; i++) { await show(); if (my !== run) return; await wait(700); if (my !== run) return; }
      i = arts.length - 1; playB.disabled = false; nextB.disabled = false;
    }
    const prevB = btn('◀', () => { stop(); if (i > 0) { i--; show(); } });
    const fwdB = btn('▶', () => { stop(); if (i < arts.length - 1) { i++; show(); } else nextB.disabled = false; });
    const playB = btn('▶ Read to me', playAll, 'big'), nextB = btn('Next step ▶', () => next(3));
    nextB.disabled = true; $('controls').append(prevB, playB, fwdB, nextB);
    sayAgain = show; cleanup = () => { run++; };
    draw(); afterIntro(intro, () => { if (!seen.size) playAll(); });
  }
  // pick the answer (or True / False); a miss comes back at the end, two misses show the answer
  function stepQuiz(intro) {
    let queue = shuffle(L.quiz.map(x => ({ ...x }))), done = 0, miss = 0, tries = 0, busy = false, cur;
    const total = queue.length;
    const col = el('div', 'qa'); $('stage').appendChild(col);
    const qd = el('div', 'q'); const grid = el('div', 'mc'); col.append(qd, grid);
    function ask() {
      if (!queue.length) { toast('Great job! ⭐'); return setTimeout(() => next(starsFor(miss, total)), 900); }
      cur = queue[0]; tries = 0; busy = false; grid.innerHTML = '';
      qd.textContent = (cur.tf ? 'True or false? ' : '') + cur.q;
      const order = cur.tf ? [0, 1] : shuffle(cur.c.map((_, k) => k));
      order.forEach(k => grid.appendChild(btn(cur.c[k], e => pick(k, e.currentTarget))));
      $('hint').textContent = `${done + 1} of ${total}`;
      sayAgain = () => speak(cur.q); sayAgain();
    }
    function pick(k, b) {
      if (busy) return;
      if (k === cur.a) { busy = true; beep(true); b.classList.add('ok'); queue.shift(); done++; speak(P.yes); return setTimeout(ask, 900); }
      miss++; beep(false); b.classList.remove('no'); void b.offsetWidth; b.classList.add('no'); b.disabled = true;
      if (++tries < 2 && !cur.tf) return speak(P.tryAgain);
      busy = true; [...grid.children].forEach(x => { if (x.textContent === cur.c[cur.a]) x.classList.add('ok'); });
      const it = queue.shift(); it.again = true; queue.push(it);   // asked again at the end
      speak('ansc:' + cur.c[cur.a]).then(() => wait(700)).then(ask);
    }
    afterIntro(intro, ask);
  }

  function finish() {
    const vals = Object.values(scores), stars = Math.max(1, Math.round(vals.reduce((a, b) => a + b, 0) / Math.max(1, vals.length)));
    const p = prog(bookId), old = (p[L.n] || {}).stars || 0;
    p[L.n] = { stars: Math.max(old, stars), at: Date.now() };
    const got = kidMap(S.stickers), sk = stkKey(bookId, L.n); const sticker = got[sk] || STICKERS[(L.n * 7 + SHELF.findIndex(b => b.id === bookId) * 3) % STICKERS.length]; got[sk] = sticker;
    save();
    $('stepTitle').textContent = `${S.kid}, you finished Lesson ${L.n}!`;
    $('stage').innerHTML = `<div class="col"><div class="stars">${starStr(stars)}</div><div class="stk">${sticker}</div><div class="note">A new sticker for your sticker book!</div></div>`;
    $('hint').textContent = ''; $('controls').innerHTML = '';
    $('controls').append(btn('Lessons', () => openBook(bookId), 'big'), btn('Sticker book', () => go('stickers')));
    const gj = `Great job, ${S.kid}!`; beep(true); speak(urlFor(gj) ? gj : P.greatJob, P.done);
  }

  // ?preview opens the "Coming soon" books that already have a book file (testing a book before it goes live)
  const ready = b => !b.soon || /[?&]preview\b/.test(location.search);
  Promise.all([
    fetch('prompts.json').then(r => r.json()).then(j => { P = j; }),
    ...SHELF.filter(ready).map(b => fetch(`books/${b.id}.json`).then(r => r.ok ? r.json() : null).then(j => { if (j) BOOKS[b.id] = j; })),
    ...SHELF.filter(ready).map(b => fetch(`audio/${b.id}.json`).then(r => r.ok ? r.json() : null).then(j => {
      if (!j) return; if (!AUDIO) AUDIO = { speeds: j.speeds, items: {} }; Object.assign(AUDIO.items, j.items);
    }).catch(() => {}))
  ]).then(home, () => toast('Could not load the book. Check the internet.', 4000));
  // the account answer can arrive after the first draw: redraw the home screen
  if (window.ABCAccount) window.ABCAccount.ready.then(() => { if ($('home').classList.contains('on') && Object.keys(BOOKS).length) home(); });
  return { go, leave, debug: () => ({ S, L, steps, stepIdx, scores }), toStep: id => { const i = steps.indexOf(id); if (i >= 0) { stepIdx = i; runStep(); } }, wordsToNumber };
})();
EOF

sw.js: offline support (now keeps Book 4 too).

cat > /root/rays-rebuild/sw.js <<'EOF'
// Offline support. App files: network first (so updates show), cached copy when offline.
// Voice clips (audio/*.mp3) never change for a given name: cache first, saved the first time they play.
const SHELL = 'rays-shell-v19', CLIPS = 'rays-clips-v1';
const FILES = ['./', 'index.html', 'app.js', 'prompts.json', 'trace.js', 'scratch.js', 'alg.js', 'hershey.js', 'books/primary1877.json', 'audio/primary1877.json', 'books/intellectual.json', 'audio/intellectual.json', 'books/practical.json', 'audio/practical.json', 'books/higher.json', 'audio/higher.json', 'books/geometry.json', 'audio/geometry.json', 'books/algebra1.json', 'audio/algebra1.json', 'icon-192.png', 'manifest.webmanifest'];
self.addEventListener('install', e => { e.waitUntil(caches.open(SHELL).then(c => Promise.all(FILES.map(f => c.add(f).catch(() => {})))).then(() => self.skipWaiting())); });
self.addEventListener('activate', e => {
  e.waitUntil(caches.keys().then(ks => Promise.all(ks.filter(k => k.startsWith('rays-') && k !== SHELL && k !== CLIPS).map(k => caches.delete(k)))).then(() => self.clients.claim()));
});
self.addEventListener('fetch', e => {
  const r = e.request; if (r.method !== 'GET' || new URL(r.url).origin !== location.origin) return;
  if (r.url.endsWith('.mp3')) {
    e.respondWith(caches.open(CLIPS).then(c => c.match(r).then(hit => hit || fetch(r).then(res => { if (res.ok && res.status === 200) c.put(r, res.clone()); return res; }))));
    return;
  }
  e.respondWith(fetch(r).then(res => { if (res.ok) caches.open(SHELL).then(c => c.put(r, res.clone())); return res; }).catch(() => caches.match(r)));
});
EOF

Rebuild, step by step

1. Build the book data

PC (PowerShell). It prints "77 lessons, 1932 questions, 2871 answers", or a list of problems:

powershell -ExecutionPolicy Bypass -File C:\RaysArithmetic\tools\make-book.ps1 -Book higher

2. Record the voice

PC (PowerShell), about 15 minutes:

powershell -ExecutionPolicy Bypass -File C:\RaysArithmetic\tools\make-audio.ps1 -Id higher

3. Put it on the server

PC (PowerShell). -Audio sends only the recordings CT 306 (abc-web) doesn't have yet. It ends with "RAYS-DONE":

powershell -ExecutionPolicy Bypass -File C:\RaysArithmetic\deploy.ps1 -Audio

4. Check it

Open https://abcbooks.fyi/math/ , tap a name, and tap the brown Book 4. Lesson 1 asks you to read 608921045 out loud. Lessons open one after another.

Troubleshooting

Book 4 says "Coming soon": app.js still has soon: true on the higher line of the shelf, or books/higher.json is missing on the server.
Book 4 is silent: check that its voice list exists. Proxmox web UI, pve, Shell:

pct exec 306 -- ls -l /var/www/abcbooks/rays/audio/higher.json

and count the MP3s (all four books together are about 16,000 at each speed):

pct exec 306 -- ls /var/www/abcbooks/rays/audio/s60 | wc -l

make-book.ps1 says "printed answer X, but ... = Y": the printed answer and the working disagree; one of them has a misread digit. Look at the page again.
make-book.ps1 says "couldn't read all of": the working uses a form the calculator doesn't know, such as 29/30.6; write it with whole numbers (290 / 306).
An old version shows after an update: change rays-shell-v10 in sw.js to the next number, deploy, and reload the page twice.

Notes

Stickers are kept per book and per reader, so Book 4 has its own sticker page.
Next on the shelf: Ray's Geometry and Trigonometry, and the two algebras, which will need new answer types.

Ray's New Higher Arithmetic as a game (Ray's Arithmetic Book 4 rebuild guide) 12OCT2026

Book 4 of Ray's, Grades 7-8: the whole of the Higher Arithmetic, from numeration to annuities, roots, mensuration and the miscellaneous exercises, read aloud in the same Eloquence voice