Book 3 of the Ray's Arithmetic shelf is now playable: Ray's New Practical Arithmetic (Grades 5-6). This is the book of written arithmetic: long sums in columns, long division, fractions, decimals, weights and measures, percentage and interest, square and cube roots. It runs in the same app as Books 1 and 2 at https://abcbooks.fyi/math/ , with the same "Reed" voice. This post covers what is new for Book 3 and how to rebuild it. The app itself, the server and the voice recorder are explained in the Book One post; the answer types added for fractions are in the Book 2 post. Every block has a Copy button.
What is in Book 3
The book has no numbered lessons, only "articles", so each section of its table of contents became one lesson: 69 lessons, 2,430 questions and 3,273 answers.
- Lessons 1-14: notation and numeration (up to quadrillions), Roman numerals, and the four rules with long numbers: column addition, subtraction, multiplication and long division.
- Lessons 15-24: United States money and compound numbers: every table of measures (dry, liquid, weight, length, area, volume, time, Troy, apothecaries, angles), reduction, the four rules with them, the time between dates, and longitude and time.
- Lessons 25-37: factoring, greatest common divisor, least common multiple, cancellation, and all of fractions.
- Lessons 38-44: practice (aliquot parts of a dollar), decimals, and the metric system.
- Lessons 45-58: percentage and its uses: commission, discounts, profit and loss, stocks, interest (simple, compound, annual, partial payments), bank and true discount, exchange, insurance, taxes and duties.
- Lessons 59-69: ratio, proportion, partnership, averages, powers and roots, mensuration (areas and volumes), and progressions.
New in the app for Book 3
- Sums written in columns. Long additions, subtractions and multiplications are shown the book's way, the numbers lined up on the right with a rule under them. The column also checks the answer when the book is built.
- Long numbers. The number pad takes up to 16 digits and shows them with commas (654,321,987,654,321), in a smaller font so they fit on a phone.
- Answers given out loud, then shown. Questions like "Write 1492 in Roman letters" or "Read out loud: 41,582" are said to a grown-up. The first tap on Next ▶ shows and says the book's answer ("1492 is written MCCCCXCII"), so the grown-up can check; the second tap moves on.
- Money to the nearest cent. Interest and discount answers accept the nearest cent, and also the exact value where the book gives one ($255.58 or $255.5793).
- Any reader name. ABC Books now lets a grown-up add readers with any name, and Rays shows the same list (it reads ABC Books' saved settings on the same site and never changes them). On a new device with no readers yet, Rays shows a link to add names in ABC Books first. A name without a recorded greeting gets "Hi! Tap the book to start."
How the book was typed up
The free PDF (raysarithmetic.wordpress.com, with a separate answer key) is pictures of pages only. As with Book 2, each page was read with Windows' text recognition, then zoomed and read by eye wherever the scan was unclear (small fractions, column sums, tables), and typed into practical.txt. Two things worth knowing about the scan: PDF page 83 is a stray copy of book page 163, so from there on the PDF page is the book page plus one; and on page 45 the multipliers of six problems are cut off in the scan; they were recovered from the answer key.
Book 3 adds these to the source format:
q: Find the sum of 3745, 2831, 5983, and 7665. [[+ 3745 2831 5983 7665]] {20224}
q: Add: [[+ 184 216 135 320 413 101]] {?}
q: January has 31 days; ... how many days are there in these five months? {151 @ 31 + 28 + 31 + 30 + 31}
q: If 4 acres of land cost $92.25, how much is that an acre? {23.062|23.0625}[[+ ...]],[[- ...]],[[x ...]]is a sum in a column. With{?}the build works out the answer; with a typed answer it checks it.{151 @ 31 + 28 + ...}is the book's printed answer, with the working it came from. The build refuses to run if they differ, which catches a misread digit in either.{23.062|23.0625}accepts either form (here, mills or exact).{~d expr}works an answer out exactly and shows it as a decimal when it ends ($2.78, not 2 39/50).
Most of the long, regular sections (the measures tables, factoring, fractions, roots) were written by small helper scripts that work every answer out from the numbers in the question, so only the questions were typed, never the answers. Where the answer key prints an answer, it was compared.
A few answers worth knowing about:
- Lesson 51: "the interest of $1 for 3 yr. 7 mo. 20 da., at 8 per cent" is $0.291 1/9; .291, .29 and .2911 all count as right.
- Lesson 66: roots that do not come out even are asked to three places; both the truncated and the rounded figure count (the cube root of 2 is 1.259 or 1.26).
- Lesson 68: circle and sphere answers use 3.1416, as the book does.
The voice
The same recorder as Books 1 and 2: 4,531 new clips at each speed (lines the books share are recorded once), about 15 minutes on this PC.
Files
Set up the Book One post first. Book 3 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:
practical.txtinC:\RaysArithmetic\source\make-book.ps1,make-audio.ps1inC:\RaysArithmetic\tools\app.js,index.html,sw.js,prompts.jsoninC:\RaysArithmetic\public\(they replace the earlier versions and run all three books)
Proxmox host shell:
mkdir -p /root/rays-rebuildpractical.txt: the whole of Book 3 with the answers.
cat > /root/rays-rebuild/practical.txt <<'EOF'
@book: practical | Ray's New Practical Arithmetic | 1877 - Book Three | #2e7d4f
# Ray's New Practical Arithmetic (1877), typed from the clean scan source/pdf/rays-practical-arithmetic-5-6.pdf
# (book page N = PDF page N); answers checked against the Key (rays-practical-arithmetic-5-6-answer-key.pdf).
# Same format as intellectual.txt, plus: [[+ 184 216 135]] = a sum written in a column ([[- a b]], [[x a b]]);
# {?} after it is the column's result, a typed answer is checked against it. {1234 @ expr} = printed answer, checked.
# The book has no lessons, only articles; each lesson here is one section of its Table of Contents.
== 1 | Notation | Units, tens and hundreds
note: Write the number in figures. A unit is a single thing; ten units make a ten, ten tens make a hundred.
q: Write in figures: Twenty-three. {23} Twenty-four. {24} Twenty-five. {25} Twenty-six. {26} Twenty-seven. {27} Twenty-eight. {28} Twenty-nine. {29}
q: Thirty-seven. {37} Forty-two. {42} Fifty-six. {56} Sixty-nine. {69} Seventy-three. {73} Eighty-seven. {87} Ninety-four. {94}
q: Eighty-three. {83} Forty-five. {45} Ninety-nine. {99} Fifty-one. {51} Thirty-six. {36} Seventy-eight. {78} Sixty-two. {62}
q: Fifty-five. {55} Ninety-three. {93} Eighty-one. {81} Sixty-seven. {67} Forty-nine. {49} Seventy-four. {74} Thirty-eight. {38}
q: Seventy-six. {76} Forty-four. {44} Eighty-two. {82} Fifty-seven. {57} Thirty-five. {35} Ninety-one. {91} Sixty-three. {63}
q: Read these numbers out loud: 71; 58; 75; 39; 68; 32; 34; 43; 72; 77; 53; 79; 88; 54; 31; 84; 66; 61; 86; 89; 65; 41; 59; 47; 52; 46; 85; 33; 98; 96; 97; 92; 95; 64; 48. {open}
q: One hundred and thirty. {130} One hundred and forty. {140} One hundred and fifty. {150} One hundred and sixty. {160} One hundred and seventy. {170} One hundred and eighty. {180}
q: One hundred and twenty-three. {123} Four hundred and fifty-six. {456} Seven hundred and eighty-nine. {789} One hundred and forty-seven. {147} Two hundred and fifty-eight. {258} Three hundred and sixty-nine. {369}
q: One hundred and two. {102} Three hundred and forty-five. {345} Six hundred and seventy-eight. {678} Two hundred and thirty-four. {234} Five hundred and sixty-seven. {567} Eight hundred and ninety. {890}
q: Four hundred and fifty-three. {453} Seven hundred and eighty-six. {786} Nine hundred and twelve. {912} Two hundred and thirty. {230} Four hundred and fifty. {450} Six hundred and seventy. {670}
q: One hundred and fifty-three. {153} Four hundred and eighty-six. {486} Seven hundred and twenty-nine. {729} One hundred and three. {103} Four hundred and six. {406} Seven hundred and nine. {709}
q: Read these numbers out loud: 210; 213; 201; 543; 513; 320; 546; 435; 876; 846; 430; 879; 768; 192; 279; 540; 417; 324; 329; 301; 650; 528; 657; 548; 604; 760; 639; 980; 765; 907. {open}
== 2 | Notation | Thousands, millions and more
note: Begin at the left, and write each period as a number of hundreds, tens and units, filling the vacant orders with ciphers.
q: Write in figures: Six hundred and fifty-four trillion three hundred and twenty-one billion nine hundred and eighty-seven million six hundred and fifty-four thousand three hundred and twenty-one. {654321987654321}
q: Two thousand. {2000} Thirty thousand. {30000} Four hundred thousand. {400000}
q: Five million. {5000000} Sixty million. {60000000} Seven hundred million. {700000000}
q: Eight billion. {8000000000} Ninety billion. {90000000000} One hundred billion. {100000000000}
q: One thousand two hundred. {1200} Two thousand one hundred. {2100}
q: Three thousand four hundred and fifty. {3450} Six thousand seven hundred and eighty-nine. {6789}
q: Twelve thousand three hundred and forty-five. {12345}
q: Six hundred and seventy-eight thousand nine hundred and twelve. {678912}
q: One million three hundred and fifty-seven thousand nine hundred and twenty-four. {1357924}
q: Sixty-eight million one hundred and forty-three thousand seven hundred and ninety-two. {68143792}
q: One thousand and one. {1001} One thousand and ten. {1010} One thousand one hundred. {1100}
q: One thousand one hundred and one. {1101} One thousand one hundred and ten. {1110} One thousand one hundred and five. {1105}
q: Two thousand and three. {2003} Four thousand and fifty. {4050}
q: Forty-five thousand and twenty-six. {45026}
q: Eighty thousand two hundred and one. {80201}
q: Ninety thousand and one. {90001}
q: Four hundred and ten thousand two hundred and five. {410205}
q: One hundred thousand and ten. {100010}
q: Three million seventy thousand five hundred and nine. {3070509}
q: Forty-five million eighty-three thousand and twenty-six. {45083026}
q: Nine hundred and nine million ninety thousand. {909090000}
q: Seven hundred million ten thousand and two. {700010002}
q: Forty billion two hundred thousand and five. {40000200005}
q: Seven hundred and twenty-six billion fifty million one thousand two hundred and forty-three. {726050001243}
q: Eighty billion seven hundred and three million five hundred and four. {80703000504}
== 3 | Notation | Reading large numbers
note: Begin at the right, and point off the number into periods of three figures. Then begin at the left, and read each period, giving its name.
q: Read out loud: 654,321,987,654,321. {open}
sol: Six hundred and fifty-four trillion three hundred and twenty-one billion nine hundred and eighty-seven million six hundred and fifty-four thousand three hundred and twenty-one.
q: Read out loud: 41,582. {open}
sol: Forty-one thousand five hundred and eighty-two.
q: Read out loud: 763,491. {open}
sol: Seven hundred and sixty-three thousand four hundred and ninety-one.
q: Read out loud: 2,519,834. {open}
sol: Two million five hundred and nineteen thousand eight hundred and thirty-four.
q: Read out loud: 375,486,921. {open}
sol: Three hundred and seventy-five million four hundred and eighty-six thousand nine hundred and twenty-one.
q: Read out loud: 4,923,176,358. {open}
sol: Four billion nine hundred and twenty-three million one hundred and seventy-six thousand three hundred and fifty-eight.
q: Read out loud: 37,584,216,974. {open}
sol: Thirty-seven billion five hundred and eighty-four million two hundred and sixteen thousand nine hundred and seventy-four.
q: Read out loud: 432,685,729,145. {open}
sol: Four hundred and thirty-two billion six hundred and eighty-five million seven hundred and twenty-nine thousand one hundred and forty-five.
q: Read out loud: 6,253,971,438,267. {open}
sol: Six trillion two hundred and fifty-three billion nine hundred and seventy-one million four hundred and thirty-eight thousand two hundred and sixty-seven.
q: Read out loud: 1,300; 2,540; 6,070; 8,009; 13,200; 1,005. {open}
sol: One thousand three hundred; two thousand five hundred and forty; six thousand and seventy; eight thousand and nine; thirteen thousand two hundred; one thousand and five.
q: Read out loud: 682,300; 8,600,050; 3,040; 50,004; 704,208. {open}
sol: Six hundred and eighty-two thousand three hundred; eight million six hundred thousand and fifty; three thousand and forty; fifty thousand and four; seven hundred and four thousand two hundred and eight.
q: Read out loud: 7,085; 62,001; 400,009; 2,102,102; 9,001,003. {open}
sol: Seven thousand and eighty-five; sixty-two thousand and one; four hundred thousand and nine; two million one hundred and two thousand one hundred and two; nine million one thousand and three.
q: Read out loud: 130,670,921. {open}
sol: One hundred and thirty million six hundred and seventy thousand nine hundred and twenty-one.
q: Read out loud: 6,900,702,003. {open}
sol: Six billion nine hundred million seven hundred and two thousand and three.
q: Read out loud: 23,004,090,701. {open}
sol: Twenty-three billion four million ninety thousand seven hundred and one.
q: Read out loud: 9,420,163,070. {open}
sol: Nine billion four hundred and twenty million one hundred and sixty-three thousand and seventy.
q: Read out loud: 570,000,010,326,049. {open}
sol: Five hundred and seventy trillion ten million three hundred and twenty-six thousand and forty-nine.
q: Read out loud: 800,820,020,802,008. {open}
sol: Eight hundred trillion eight hundred and twenty billion twenty million eight hundred and two thousand and eight.
== 4 | Notation | The Roman system
note: I is one; V, five; X, ten; L, fifty; C, one hundred; D, five hundred; M, one thousand. IV is four, IX nine, XL forty, XC ninety; a smaller letter on the right is added.
q: Write the numbers from one to twenty in Roman letters. {open}
sol: I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII, XIII, XIV, XV, XVI, XVII, XVIII, XIX, XX.
q: Write the numbers from twenty to thirty in Roman letters. {open}
sol: XX, XXI, XXII, XXIII, XXIV, XXV, XXVI, XXVII, XXVIII, XXIX, XXX.
q: Write 30 in Roman letters. {open}
sol: 30 is written XXX.
q: Write 40 in Roman letters. {open}
sol: 40 is written XL.
q: Write 50 in Roman letters. {open}
sol: 50 is written L.
q: Write 60 in Roman letters. {open}
sol: 60 is written LX.
q: Write 70 in Roman letters. {open}
sol: 70 is written LXX.
q: Write 80 in Roman letters. {open}
sol: 80 is written LXXX.
q: Write 90 in Roman letters. {open}
sol: 90 is written XC.
q: Write 57 in Roman letters. {open}
sol: 57 is written LVII.
q: Write 29 in Roman letters. {open}
sol: 29 is written XXIX.
q: Write 61 in Roman letters. {open}
sol: 61 is written LXI.
q: Write 38 in Roman letters. {open}
sol: 38 is written XXXVIII.
q: Write 46 in Roman letters. {open}
sol: 46 is written XLVI.
q: Write 72 in Roman letters. {open}
sol: 72 is written LXXII.
q: Write 93 in Roman letters. {open}
sol: 93 is written XCIII.
q: Write 100 in Roman letters. {open}
sol: 100 is written C.
q: Write 101 in Roman letters. {open}
sol: 101 is written CI.
q: Write 106 in Roman letters. {open}
sol: 106 is written CVI.
q: Write 117 in Roman letters. {open}
sol: 117 is written CXVII.
q: Write 129 in Roman letters. {open}
sol: 129 is written CXXIX.
q: Write 168 in Roman letters. {open}
sol: 168 is written CLXVIII.
q: Write 199 in Roman letters. {open}
sol: 199 is written CXCIX.
q: Write 246 in Roman letters. {open}
sol: 246 is written CCXLVI.
q: Write 309 in Roman letters. {open}
sol: 309 is written CCCIX.
q: Write 482 in Roman letters. {open}
sol: 482 is written CCCCLXXXII.
q: Write 527 in Roman letters. {open}
sol: 527 is written DXXVII.
q: Write 693 in Roman letters. {open}
sol: 693 is written DCXCIII.
q: Write 734 in Roman letters. {open}
sol: 734 is written DCCXXXIV.
q: Write 859 in Roman letters. {open}
sol: 859 is written DCCCLIX.
q: Write 975 in Roman letters. {open}
sol: 975 is written DCCCCLXXV.
q: Write 1001 in Roman letters. {open}
sol: 1001 is written MI.
q: Write 1010 in Roman letters. {open}
sol: 1010 is written MX.
q: Write 1048 in Roman letters. {open}
sol: 1048 is written MXLVIII.
q: Write 1119 in Roman letters. {open}
sol: 1119 is written MCXIX.
q: Write 1285 in Roman letters. {open}
sol: 1285 is written MCCLXXXV.
q: Write 1326 in Roman letters. {open}
sol: 1326 is written MCCCXXVI.
q: Write 1492 in Roman letters. {open}
sol: 1492 is written MCCCCXCII.
q: Write 1776 in Roman letters. {open}
sol: 1776 is written MDCCLXXVI.
q: Write 1861 in Roman letters. {open}
sol: 1861 is written MDCCCLXI.
q: Write 1900 in Roman letters. {open}
sol: 1900 is written MDCCCC.
q: What number is VI? {6} XVII? {17} LXXVIII? {78} CLXXXIX? {189} MDCCCLXXVII? {1877}
== 5 | Addition | Adding simple numbers
note: Write units of the same order in the same column. Begin at the right, add each column, set down the units and carry the tens.
table: +add 2 9 from 0
table: +add 3 9 from 0
table: +add 4 9 from 0
table: +add 5 9 from 0
table: +add 6 9 from 0
table: +add 7 9 from 0
table: +add 8 9 from 0
table: +add 9 9 from 0
q: If you have 2 cents and find 3 cents, how many will you then have? {5}
sol: Because 2 cents and 3 cents are 5 cents.
q: I spent 12 cents for a slate, and 5 cents for a copy-book: how many cents did I spend? {17}
q: John gave 6 cents for an orange, 7 cents for pencils, and 9 cents for a ball: how many cents did all cost? {22}
q: Joseph gave 5 cents for a daily paper, 10 cents for a weekly paper, 25 cents for a monthly magazine, 30 cents for a book of poems, and 40 cents for a novel: how much did he spend? Cents? {110}
q: I own 3 tracts of land: the first contains 240 acres; the second, 132 acres; the third, 25 acres: how many acres in all? [[+ 240 132 25]] {397}
sol: Begin at the right, and say 5 and 2 are 7 units; 2 and 3 are 5, and 4 are 9 tens; 1 and 2 are 3 hundreds: 397 acres.
q: I owe one man $210, another $142, and another $35: what is the sum of my debts? [[+ 210 142 35]] {387}
q: Find the sum of 4321, 1254, 3120. [[+ 4321 1254 3120]] {?}
q: Find the sum of 50230, 3105, 423. [[+ 50230 3105 423]] {53758}
q: Add the numbers 3415, 503, 1870, and 922. [[+ 3415 503 1870 922]] {6710}
sol: 2 and 3 are 5, and 5 are 10 units: write 0 and carry 1 ten. 1 and 2 are 3, and 7 are 10, and 1 are 11 tens: write 1, carry 1. 1 and 9 are 10, and 8 are 18, and 5 are 23, and 4 are 27 hundreds: write 7, carry 2. 2 and 1 are 3, and 3 are 6 thousands.
q: Find the sum of 3745, 2831, 5983, and 7665. [[+ 3745 2831 5983 7665]] {20224}
q: Add: [[+ 184 216 135 320 413 101]] {?}
q: Add: [[+ 204 302 401 311 109 43]] {?}
q: Add: [[+ 103 405 764 573 127 205]] {?}
q: Add: [[+ 495 207 185 825 403 325]] {?}
q: Add: [[+ 384 438 348 843 483 834]] {?}
q: Add: [[+ 1065 6317 5183 7102 3251 6044]] {?}
q: Add: [[+ 3725 5834 4261 7203]] {21023}
q: Add: [[+ 5943 6427 8204 7336]] {?}
q: Add: [[+ 82703 102 6005 759]] {89569}
q: Add: [[+ 987462 478345 610628 423158]] {2499593}
q: Add: [[+ 6840325 7314268 3751954 6287539]] {24194086}
q: 11 + 22 + 33 + 44 + 55 = how many? {?}
q: 23 + 41 + 74 + 83 + 16 = how many? {?}
q: 45 + 19 + 32 + 74 + 55 = how many? {?}
q: 51 + 48 + 76 + 85 + 4 = how many? {?}
q: 263 + 104 + 321 + 155 = how many? {?}
q: 94753 + 2847 + 93688 + 9386 + 258 + 3456 are how many? {?}
q: January has 31 days; February, 28; March, 31; April, 30; and May, 31: how many days are there in these five months? {151 @ 31 + 28 + 31 + 30 + 31}
q: June has 30 days; July, 31; August, 31; September, 30; October, 31: how many days in all? {153 @ 30 + 31 + 31 + 30 + 31}
q: The first 5 months have 151 days, the next 5 have 153 days, November has 30, and December, 31: how many days in the whole year? {365 @ 151 + 153 + 30 + 31}
q: I bought 4 pieces of muslin: the first contained 50 yards, the second, 65, the third, 42, the fourth, 89: how many yards in all? {246 @ 50 + 65 + 42 + 89}
q: I owe one man $245, another $325, a third $187, a fourth $96: how much do I owe? {853 @ 245 + 325 + 187 + 96}
q: General Washington was born A. D. 1732, and lived 67 years: in what year did he die? {1799}
q: Alfred the Great died A. D. 901; thence to the signing of Magna Charta was 314 years; thence to the American Revolution, 560 years: in what year did the American Revolution begin? {1775 @ 901 + 314 + 560}
q: A has 4 flocks of sheep; in the first are 65 sheep and 43 lambs; in the second, 187 sheep and 105 lambs; in the third, 370 sheep and 243 lambs; in the fourth, 416 sheep and 95 lambs: how many sheep and lambs has he? Sheep? {1038 @ 65 + 187 + 370 + 416} Lambs? {486 @ 43 + 105 + 243 + 95}
q: A man bought 30 barrels of pork for $285, 18 barrels for $144, 23 barrels for $235 and 34 barrels for $408: how many barrels did he buy, and how many dollars did he pay? Barrels? {105 @ 30 + 18 + 23 + 34} Dollars? {1072 @ 285 + 144 + 235 + 408}
q: The first of four numbers is 287; the second, 596; the third, 841; and the fourth, as much as the first three: what is their sum? {3448 @ (287 + 596 + 841) * 2}
q: The Pyramids of Egypt were built 1700 years before the founding of Carthage; Carthage was founded 47 years before and was destroyed 607 years after the founding of Rome, or 146 years before the Christian era. How many years before Christ were the Pyramids built? {2500 @ 1700 + 47 + 607 + 146}
q: Add three thousand and five; forty-two thousand six hundred and twenty-seven; 105; three hundred and seven thousand and four; 80079; three hundred and twenty thousand six hundred. [[+ 3005 42627 105 307004 80079 320600]] {753420}
q: Add 275432; four hundred and two thousand and thirty; three hundred thousand and five; 872026; four million two thousand three hundred and forty-seven. [[+ 275432 402030 300005 872026 4002347]] {5851840}
q: Add eight hundred and eighty million eight hundred and eighty-nine; 2002002; seventy-seven million four hundred and thirty-six thousand; two hundred and six million five thousand two hundred and seven; 49003; nine hundred and ninety million nineteen thousand nine hundred and nineteen. [[+ 880000889 2002002 77436000 206005207 49003 990019919]] {2155513020}
q: North America has an area of 8955752 square miles; South America, 6917246 square miles; and the West Indies, 94523 square miles: what is the area of the entire continent? {15967521 @ 8955752 + 6917246 + 94523}
q: A man pays $600 for a lot, $1325 for building materials, $30 for digging the cellar, $120 for stone-work, $250 for brick-work, $140 for carpenter-work, $120 for plastering, and $115 for painting: how much did his house and lot cost him? {2700 @ 600 + 1325 + 30 + 120 + 250 + 140 + 120 + 115}
q: A man bequeaths $7850 to his wife, $3275 to each of his two sons, and $2650 to each of his three daughters: what is the amount of his bequest? {22350 @ 7850 + 3275 + 3275 + 2650 + 2650 + 2650}
q: A merchant spent $8785 for dress goods, and $12789 for sheetings. He sold the dress goods at a profit of $878, and the sheetings at a profit of $1250: for how much did he sell the whole? {23702 @ 8785 + 12789 + 878 + 1250}
q: A merchant began business with $7000 cash, goods worth $12875, bank stock worth $5600, and other stocks worth $4785. In one year he gained $3500: what was he worth at its close? {33760 @ 7000 + 12875 + 5600 + 4785 + 3500}
q: A house has two parlors, each requiring 30 yards of carpet; four bed-rooms, each requiring 25 yards; a dining-room and sitting-room, each requiring 20 yards: how many yards are required to carpet the entire house? {200 @ 30 * 2 + 25 * 4 + 20 * 2}
== 6 | Addition | Adding two columns at once
note: Beginning with the units, add the tens of each number, then its units: 47, and 30 are 77, and 4 are 81...
q: Add: [[+ 7892 6779 4865 6234 9347]] {35117}
sol: Beginning with 47, add the 3 tens above, 77; then the 4 units, 81; then 6 tens, 141; 5 units, 146; 7 tens, 216; 9 units, 225; 9 tens, 315; 2 units, 317. Put down 17, carry 3 hundreds: 93 and 3 are 96, and 60 are 156, and 2 are 158, and 40 are 198, and 8 are 206, and 60 are 266, and 7 are 273, and 70 are 343, and 8 are 351.
q: Add: [[+ 3686 4724 6583 5798 6953]] {?}
q: Add: [[+ 9898 8989 4545 5454 6363]] {?}
q: Add: [[+ 4356 6342 7989 4878 6749]] {?}
q: Add: [[+ 893742 743698 437821 643567 892742]] {?}
q: Add: [[+ 234567 765432 987654 456789 778899]] {?}
q: Add: [[+ 5493275 6182463 9538719 2645834 8256386]] {?}
q: Add: [[+ 4819 9263 2752 8375 6498]] {?}
q: Add: [[+ 18356 49276 94678 36525 42983]] {?}
q: Add: [[+ 849627 532472 293784 468135 926547]] {?}
q: Add: [[+ 7421 6322 798 4352 547 674 2315 7218 1847 5721 6843 4722 5976 6843 1234]] {?}
q: Add: [[+ 6873 2196 583 79 684 4348 7896 233 594 6483 7542 3967 29 478 1717]] {?}
q: Add: [[+ 4729 6234 5781 3143 7182 6989 7222 6643 7859 6742 8982 3451 8692 7341 6822]] {?}
q: Add: [[+ 237285 64371 2143 842 55 789 4621 15115 647890 77442 84931 894623 446217 134162 192317]] {?}
q: Add: [[+ 884261 724353 416213 598624 784344 627517 843641 47821 52348 2932 4751 896 722 823344 874132]] {?}
== 7 | Subtraction | Subtracting simple numbers
note: Write the less number under the greater. Begin at the right; if the lower figure is greater, add ten to the upper figure, and carry one to the next lower figure.
table: +sub 2 10
table: +sub 3 10
table: +sub 4 10
table: +sub 5 10
table: +sub 6 10
table: +sub 7 10
table: +sub 8 10
table: +sub 9 10
q: If you have 9 apples, and give 4 away, how many will you have left? {5}
sol: Because 4 apples from 9 apples are 5 apples.
q: Frank had 15 cents; after spending 7, how many were left? {8}
q: If you take 8 from 13, how many are left? {5}
q: If I have 25 cents, and spend 10 of them for a lead-pencil, how much will I have left? Cents? {15}
q: Twelve from twenty leaves how many? {8}
q: A man having $135, spent $112: how much had he left? [[- 135 112]] {23}
sol: Begin at the right, and say 2 from 5 leaves 3; 1 from 3 leaves 2; 1 from 1 leaves 0, and the place is vacant.
q: A farmer having 245 sheep, sold 123: how many sheep had he left? [[- 245 123]] {?}
q: A man bought a farm for $751, and sold it for $875: how much did he gain? [[- 875 751]] {?}
q: What is the difference between 734 and 531? [[- 734 531]] {?}
q: Between 8752 and 3421? [[- 8752 3421]] {?}
q: Between 79484 and 25163? [[- 79484 25163]] {?}
q: Between 49528 and 16415? [[- 49528 16415]] {?}
q: James had 13 cents; after spending 5, how many cents had he left? [[- 13 5]] {8}
sol: 5 can not be subtracted from 3, but it can be from 13; 5 from 13 leaves 8.
q: From 73 subtract 45. [[- 73 45]] {28}
sol: 5 units can not be taken from 3 units. Take 1 ten from the 7 tens and add it to the 3 units, making 13 units; 5 from 13 leaves 8. 4 tens from 6 tens leave 2 tens. The difference is 28.
q: Find the difference between 805 and 637. [[- 805 637]] {168}
sol: 7 from 15 leaves 8; then, increasing the 3 by 1, 4 from 10 leaves 6; then, increasing 6 by 1, 7 from 8 leaves 1.
q: Subtract: [[- 7640 1234]] {6406}
q: Subtract: [[- 860012 430021]] {429991}
q: Subtract: [[- 4500120 2910221]] {1589899}
q: Subtract: [[- 3860000 120901]] {3739099}
q: Take 1234567 from 4444444. [[- 4444444 1234567]] {3209877}
q: Take 15161718 from 91516171. [[- 91516171 15161718]] {76354453}
q: Take 34992884 from 63046571. [[- 63046571 34992884]] {?}
q: 153425178 - 53845248 = ? [[- 153425178 53845248]] {99579930}
q: 100000000 - 10001001 = ? [[- 100000000 10001001]] {?}
q: Take 17 cents from 63 cents. {46}
q: A carriage cost $137, and a horse $65: how much more than the horse did the carriage cost? {72 @ 137 - 65}
q: A tree 75 feet high was broken; the part that fell was 37 feet long: how high was the stump? Feet? {38 @ 75 - 37}
q: America was discovered by Columbus in 1492: how many years had elapsed in 1837? {345 @ 1837 - 1492}
q: I deposited in the bank $1840, and drew out $475: how many dollars had I left? [[- 1840 475]] {1365}
q: A man has property worth $10104, and owes debts to the amount of $7426: when his debts are paid, how much will be left? [[- 10104 7426]] {2678}
q: A man having $100000, gave away $11: how many had he left? [[- 100000 11]] {99989}
q: Subtract 19019 from 20010. [[- 20010 19019]] {?}
q: Required the excess of nine hundred and twelve thousand and ten, above 50082. [[- 912010 50082]] {?}
q: Take 4004 from four million. [[- 4000000 4004]] {?}
q: Subtract 1009006 from two million twenty thousand nine hundred and thirty. [[- 2020930 1009006]] {1011924}
q: Subtract four hundred and five thousand and twenty-two from 2000687. [[- 2000687 405022]] {1595665}
q: What is the difference between thirteen million two hundred and one and 17102102? [[- 17102102 13000201]] {?}
q: A man invested in business $30,000; at the end of the first year he found that all his assets amounted to only $26,967; how much had he lost? [[- 30000 26967]] {?}
q: Take 9238715 from 18126402. [[- 18126402 9238715]] {?}
q: Take 9909090009 from 19900900900. [[- 19900900900 9909090009]] {?}
== 8 | Subtraction | Examples in addition and subtraction
q: 275 + 381 + 625 - 1098 = ? {183}
q: 6723 - 479 - 347 - 228 = ? {?}
q: In January, 1876, a merchant bought goods to the amount of $2675; in February, $4375; and in March, $1897; after making one payment of $3000, and another of $4947, how much did he still owe? {1000 @ 2675 + 4375 + 1897 - 3000 - 4947}
q: I owe three notes, whose sum is $1300; one note being for $250, and another for $650: what is the amount of the third note? {400 @ 1300 - 250 - 650}
q: Mr. Jones deposited $450 in bank on Monday; on Tuesday, $725; on Wednesday, $1235; on Thursday, $4675; and on Friday, $1727. On Saturday morning he drew out $5935, and Saturday afternoon, $877: how much money had he left in bank? {2000 @ 450 + 725 + 1235 + 4675 + 1727 - 5935 - 877}
q: At the end of one year I found I had spent $2300. Of this amount, $350 were paid for board, $125 for clothing, $375 for books, $150 for incidentals, and the remainder for two acres of ground: how much did the two acres cost? {1300 @ 2300 - 350 - 125 - 375 - 150}
q: A speculator bought three houses. For the first he gave $4875; for the second, $2250 more than for the first; and for the third he gave $3725. He afterward sold them all for $20838: how much did he gain? {5113 @ 20838 - (4875 + 4875 + 2250 + 3725)}
q: A man owns property valued at $49570, of which $16785 are in personal property, and $24937 in real estate; the remainder was deposited in bank: how much has he in bank? {7848 @ 49570 - 16785 - 24937}
q: A merchant bought a bill of goods for $7895, and paid $175 for freight, and $3 for drayage. He sold the goods for $10093: how much did he gain? {2020 @ 10093 - 7895 - 175 - 3}
q: A farmer invested $10000, as follows: in land, $5750; in horses, $925; in cattle, $1575; in hogs, $675; and the remainder in implements and tools: how much did he invest in implements and tools? {1075 @ 10000 - 5750 - 925 - 1575 - 675}
q: A speculator on Monday gained $4625; on Tuesday, $3785; on Wednesday he lost $6955; on Thursday he lost $895; on Friday he gained $985, and on Saturday he lost $1375: how much did he gain during the entire week? {170 @ 4625 + 3785 - 6955 - 895 + 985 - 1375}
q: The following is Mr. Brown's private account for two weeks: First week, received $50 for salary, and spent $25 for clothing, $7 for board, $2 for washing, and $5 for sundries. Second week, received $50 for salary, loaned $35 to Tom Jones, paid $7 for board, $2 for washing, and $8 for sundries. How much did Mr. Brown have at the end of the two weeks? {9 @ 50 - 25 - 7 - 2 - 5 + 50 - 35 - 7 - 2 - 8}
== 9 | Multiplication | Multiplying simple numbers
note: Write the multiplier under the multiplicand. Multiply by each figure of the multiplier, setting the right-hand figure of each product under that figure; then add the partial products.
table: +mul 1 12
table: +mul 2 12
table: +mul 3 12
table: +mul 4 12
table: +mul 5 12
table: +mul 6 12
table: +mul 7 12
table: +mul 8 12
table: +mul 9 12
table: +mul 10 12
table: +mul 11 12
table: +mul 12 12
q: If 1 orange cost 2 cents, what will 3 oranges cost? Cents? {6}
sol: Three oranges will cost 3 times as much as one orange; that is, 2 cents taken 3 times: 2 + 2 + 2 = 6.
q: If 1 lemon cost 3 cents, how many cents will 4 lemons cost? {12}
q: In an orchard there are 4 rows of trees, and in each row, 21 trees: how many trees in the orchard? [[x 21 4]] {84}
sol: Write 21 once, place 4 underneath, and say, 4 times 1 are 4, which put in units' place; then, 4 times 2 are 8, which put in tens' place.
q: How many yards of cloth are there in 3 pieces, each containing 123 yards? [[x 123 3]] {369}
q: What will 2 houses cost at $231 each? [[x 231 2]] {?}
q: What will 3 horses cost at $132 each? [[x 132 3]] {?}
q: What is the product of 201 x 4? [[x 201 4]] {?}
q: What is the product of 2301 x 3? [[x 2301 3]] {?}
q: At $43 an acre, what will 5 acres of land cost? [[x 43 5]] {215}
sol: Say, 5 times 3 are 15; write the 5, and carry the 1; then, 5 times 4 are 20, and 1 carried make 21.
q: Multiply: [[x 5142 5]] {?}
q: Multiply: [[x 4184 6]] {?}
q: Multiply: [[x 3172 5]] {?}
q: Multiply: [[x 41834 7]] {?}
q: Multiply 49 by 3. [[x 49 3]] {?}
q: Multiply 57 by 4. [[x 57 4]] {?}
q: Multiply 128 by 5. [[x 128 5]] {?}
q: Multiply 367 by 6. [[x 367 6]] {?}
q: Multiply 1427 by 7. [[x 1427 7]] {?}
q: Multiply 19645 by 8. [[x 19645 8]] {?}
q: Multiply 44386 by 9. [[x 44386 9]] {?}
q: Multiply 708324 by 7. [[x 708324 7]] {?}
q: Multiply 96432 by 10. [[x 96432 10]] {?}
q: Multiply 46782 by 11. [[x 46782 11]] {?}
q: Multiply 86458 by 12. [[x 86458 12]] {?}
q: What is the product of 43 x 25? [[x 43 25]] {1075}
sol: Multiply by the 5 units: 215. Then by the 2 tens: 86, set under the tens. Add: 1075, because 5 times 43 and 20 times 43 equal 25 times 43.
q: Multiply 2345 by 123. [[x 2345 123]] {288435}
q: Multiply 327 by 203. [[x 327 203]] {66381}
sol: When there is a cipher in the multiplier, leave it, and multiply by the other figures, placing the right-hand figure of each partial product under the multiplying figure.
q: [[x 235 13]] {3055}
q: [[x 346 19]] {6574}
q: [[x 425 29]] {12325}
q: [[x 518 34]] {17612}
q: [[x 279 37]] {10323}
q: [[x 869 49]] {42581}
q: [[x 294 57]] {16758}
q: [[x 429 62]] {26598}
q: [[x 485 76]] {36860}
q: [[x 624 85]] {53040}
q: [[x 976 97]] {94672}
q: [[x 342 364]] {124488}
q: [[x 376 526]] {197776}
q: [[x 476 536]] {255136}
q: [[x 2187 215]] {470205}
q: [[x 3489 276]] {962964}
q: [[x 1646 365]] {600790}
q: [[x 8432 635]] {5354320}
q: Multiply 6874 by 829. [[x 6874 829]] {5698546}
q: Multiply 2873 by 1823. [[x 2873 1823]] {5237479}
q: Multiply 4786 by 3497. [[x 4786 3497]] {16736642}
q: Multiply 87603 by 9865. [[x 87603 9865]] {864203595}
q: Multiply 83457 by 6835. [[x 83457 6835]] {570428595}
q: Multiply 31624 by 7138. [[x 31624 7138]] {225732112}
q: What will 126 barrels of flour cost, at $6 a barrel? {756 @ 126 * 6}
q: What will 823 barrels of pork cost, at $12 a barrel? {9876 @ 823 * 12}
q: What will 675 pounds of cheese cost, at 13 cents a pound? Cents? {8775 @ 675 * 13}
q: What will 496 bushels of oats cost, at 24 cents a bushel? Cents? {11904 @ 496 * 24}
q: If a man travel 28 miles a day, how many miles will he travel in 152 days? {4256 @ 152 * 28}
q: There are 1760 yards in one mile: how many yards are there in 209 miles? {367840 @ 1760 * 209}
q: There are 24 hours in a day, and 365 days in a year: if a ship sail 8 miles an hour, how far will she sail in a year? Miles? {70080 @ 24 * 365 * 8}
q: Multiply two thousand and twenty-nine by one thousand and seven. [[x 2029 1007]] {2043203}
q: Multiply eighty thousand four hundred and one by sixty thousand and seven. [[x 80401 60007]] {4824622807}
q: Multiply one hundred and one thousand and thirty-two by 20001. [[x 101032 20001]] {2020741032}
q: A grocer bought 2 barrels of sugar, each weighing 215 pounds, for 8 cents a pound: how much did he pay for the sugar? Cents? {3440 @ 2 * 215 * 8}
q: A grocer bought a barrel of molasses, containing 36 gallons, for 45 cents a gallon; and sold it for 55 cents a gallon: how much did he gain? Cents? {360 @ 36 * 55 - 36 * 45}
q: A commission merchant sold 2650 bushels of wheat for a farmer, at 95 cents a bushel, and charged him 2 cents a bushel for selling: how much money was due the farmer? Cents? {246450 @ 2650 * (95 - 2)}
q: A farmer bought 6 horses of one man for 75 dollars each, and 5 horses of another for 125 dollars each, and sold them all for 150 dollars each: how many dollars did he gain? {575 @ 11 * 150 - (6 * 75 + 5 * 125)}
q: A merchant bought one box of goods for 250 dollars, two more for 325 dollars each, and three more for 175 dollars each; he sold them all so as to gain 356 dollars: for how much did he sell them? {1781 @ 250 + 2 * 325 + 3 * 175 + 356}
q: A farmer bought 24 sheep, at 5 dollars a head; 36 hogs, at 14 dollars a head; and 9 cows, at 45 dollars a head: when he sold them all, he lost 275 dollars: for how much did he sell them? {754 @ 24 * 5 + 36 * 14 + 9 * 45 - 275}
q: To 75 x 37 add 85 x 54, and subtract 5284. {2081 @ 75 * 37 + 85 * 54 - 5284}
q: To 69 x 53 add 48 x 27, and subtract 4279. {674 @ 69 * 53 + 48 * 27 - 4279}
q: I bought 50 bags of coffee, averaging 63 pounds in a bag, paying 34 cents a pound: how much did it cost? Cents? {107100 @ 50 * 63 * 34}
== 10 | Multiplication | Contractions in multiplication
note: When the multiplier can be separated into factors, multiply by one factor, and that product by another. To multiply by 10, 100, 1000, annex as many ciphers as there are in the multiplier.
q: What will 15 oranges cost, at 8 cents each? Cents? {120}
sol: Since 15 is 3 times 5: the cost of 5 oranges is 8 cents x 5 = 40 cents; the cost of 15 oranges is 40 cents x 3 = 120 cents.
q: What will 24 acres of land cost, at $124 an acre? {2976 @ 124 * 24}
q: How far will a ship sail in 56 weeks, at the rate of 1512 miles per week? Miles? {84672 @ 1512 * 56}
q: How many pounds of iron are there in 54 loads, each weighing 2873 pounds? {155142 @ 2873 * 54}
q: Multiply 2874 by 72. {206928 @ 2874 * 72}
q: Multiply 8074 by 108. {871992 @ 8074 * 108}
q: Multiply 245 by 100. {24500 @ 245 * 100}
q: Multiply 138 by 1000. {138000 @ 138 * 1000}
q: Multiply 428 by 10000. {4280000 @ 428 * 10000}
q: Multiply 872 by 100000. {87200000 @ 872 * 100000}
q: Multiply 9642 by 1000000. {9642000000 @ 9642 * 1000000}
q: Multiply 10045 by 1000000. {10045000000 @ 10045 * 1000000}
q: Find the product of 625 by 500. {312500}
sol: Multiplying by 5, the product is 3125; and multiplying this by 100 is the same as annexing two ciphers: 312500.
q: Find the product of 2300 x 170. {391000}
sol: Multiply 23 by 17, which is 391, then annex 3 ciphers, as there are 3 ciphers at the right of both factors.
q: Multiply 2350 by 60. {141000 @ 2350 * 60}
q: Multiply 80300 by 450. {36135000 @ 80300 * 450}
q: Multiply 10240 by 3200. {32768000 @ 10240 * 3200}
q: Multiply 9600 by 2400. {23040000 @ 9600 * 2400}
q: Multiply 18001 by 26000. {468026000 @ 18001 * 26000}
q: Multiply 8602 by 1030. {8860060 @ 8602 * 1030}
q: Multiply 3007 by 9100. {27363700 @ 3007 * 9100}
q: Multiply 80600 by 7002. {564361200 @ 80600 * 7002}
q: Multiply 70302 by 80300. {5645250600 @ 70302 * 80300}
q: Multiply 904000 by 10200. {9220800000 @ 904000 * 10200}
== 11 | Division | Short division
note: When there is a remainder after dividing the last figure, place the divisor under it, and annex it to the quotient: 743 divided by 3 is 247 2/3 (type 247 and 2/3).
table: +div 1 12
table: +div 2 12
table: +div 3 12
table: +div 4 12
table: +div 5 12
table: +div 6 12
table: +div 7 12
table: +div 8 12
table: +div 9 12
table: +div 10 12
table: +div 11 12
table: +div 12 12
q: If you divide 6 apples equally between 2 boys, how many will each boy have? {3}
sol: It will require 2 apples to give each boy 1. Hence, each boy will have as many apples as 2 is contained times in 6, which are 3.
q: If you divide 8 peaches equally between 2 boys, how many will each have? {4}
q: How many times 2 in 10? {5}
q: A boy has 8 cents: how many lemons can he buy, at 2 cents each? {4}
sol: He can buy 4, because 4 lemons, at 2 cents each, will cost 8 cents.
q: How many times is 2 contained in 468? {234 @ 468 / 2}
sol: 2 in 4 (hundreds), 2 times; 2 in 6 (tens), 3 times; 2 in 8 (units), 4 times.
q: How many times 3 in 693? {~693 / 3}
q: How many times 4 in 848? {~848 / 4}
q: How many times 2 in 4682? {~4682 / 2}
q: How many times 4 in 8408? {~8408 / 4}
q: How many times 3 in 36936? {12312 @ 36936 / 3}
q: How many times 2 in 88468? {44234 @ 88468 / 2}
q: How many times is 3 contained in 129? {43 @ 129 / 3}
sol: 3 is not contained in 1; but 3 is contained in 12 (tens), 4 times; 3 in 9 (units), 3 times.
q: How many times is 3 contained in 735? {245 @ 735 / 3}
sol: 3 is contained in 7 hundreds, 2 times, and 1 hundred over; the 1 hundred with the 3 tens makes 13 tens, in which 3 is contained 4 times and 1 ten left; this 1 ten with the 5 units makes 15 units, in which 3 is contained 5 times.
q: How many times is 3 contained in 618? {206 @ 618 / 3}
sol: 3 is contained in 6 hundreds, 2 times; as the 1 in tens' place will not contain 3, a cipher is placed in tens' place; the 1 ten is added to the 8 units, making 18, and 6 is placed in units' place.
q: How many times is 3 contained in 609? {203 @ 609 / 3}
q: How many times is 3 contained in 743? {247 2/3 @ 743 / 3}
sol: After dividing, there is 2 left, which is written over the divisor: 247 2/3, read 247, and two divided by three.
q: How many times 3 in 462? {~462 / 3}
q: How many times 5 in 1170? {~1170 / 5}
q: How many times 4 in 948? {~948 / 4}
q: Divide 653 cents by 3. {217 2/3 @ 653 / 3}
sol: Proof: 217 x 3 = 651 cents divided; add the remainder 2; 653 = the dividend.
q: Divide 454212 by 6. {75702 @ 454212 / 6}
q: Divide 874293 by 7. {124899 @ 874293 / 7}
q: Divide 3756031 by 8. {469503 7/8 @ 3756031 / 8}
q: Divide 8652 by 2. {~8652 / 2}
q: Divide 406235 by 3. {~406235 / 3}
q: Divide 675043 by 4. {~675043 / 4}
q: Divide 984275 by 5. {~984275 / 5}
q: Divide 258703 by 6. {~258703 / 6}
q: Divide 8643275 by 7. {~8643275 / 7}
q: Divide 6032520 by 8. {~6032520 / 8}
q: Divide 9032706 by 9. {~9032706 / 9}
q: Divide 1830024 by 10. {~1830024 / 10}
q: Divide 603251 by 11. {~603251 / 11}
q: Divide 41674008 by 12. {~41674008 / 12}
q: If oranges cost 3 cents each, how many can be bought for 894 cents? {~894 / 3}
q: If 4 bushels of apples cost 140 cents, how much is that a bushel? Cents? {~140 / 4}
q: If flour cost $4 a barrel, how many barrels can be bought for $812? {~812 / 4}
q: A carpenter receives $423 for 9 months' work: how much is that a month? {~423 / 9}
q: There are 12 months in 1 year: how many years are there in 540 months? {~540 / 12}
q: There are 4 quarts in 1 gallon: how many gallons are there in 321276 quarts? {~321276 / 4}
q: At $8 a barrel, how many barrels of flour can be bought for $1736? {~1736 / 8}
q: There are 7 days in one week: how many weeks are there in 734566 days? {~734566 / 7}
q: A number has been multiplied by 11, and the product is 495: what is the number? {~495 / 11}
q: The product of two numbers is 3582: one of the numbers is 9: what is the other? {~3582 / 9}
q: Find one-half of 56. {~56 / 2} One-half of 3725. {~3725 / 2}
q: Find one-third of 147. {~147 / 3} One-fourth of 500. {~500 / 4} One-fifth of 1945. {~1945 / 5}
q: Find one-sixth of 4476. {~4476 / 6} One-seventh of 2513. {~2513 / 7} One-eighth of 5992. {~5992 / 8}
q: Find one-ninth of 8793. {~8793 / 9} One-tenth of 1090. {~1090 / 10} One-eleventh of 4125. {~4125 / 11} One-twelfth of 5556. {~5556 / 12}
q: I divided 144 apples equally among 4 boys; the eldest boy gave one-third of his share to his sister: what number did the sister receive? {12 @ 144 / 4 / 3}
q: James found 195 cents, and gave to Daniel one-fifth of them: Daniel gave one-third of his share to his sister: how many cents did she receive? {13 @ 195 / 5 / 3}
q: One-eleventh of 275 is how much greater than one-eighth of 192? {1 @ 275 / 11 - 192 / 8}
== 12 | Division | Long division
note: Find how many times the divisor is contained in the fewest left-hand figures that will contain it; multiply, subtract, bring down the next figure, and divide again.
q: Divide 3465 dollars equally among 15 men. Dollars each? {231 @ 3465 / 15}
sol: 15 in 34 hundreds, 2 times (30 hundreds); 4 hundreds left with the 6 tens make 46 tens; 15 in 46, 3 times (45 tens); 1 ten left with the 5 units make 15; 15 in 15, once. Each man receives 231 dollars.
q: In 147095 days, how many years, each of 365 days? {403 @ 147095 / 365}
sol: 1470 contains 365 four times (1460); bringing down the 9, 109 will not contain the divisor, so write a cipher in the quotient; the last partial dividend, 1095, contains the divisor three times.
q: Divide 4056 by 13. {312 @ 4056 / 13}
q: Divide 78994 by 319. {247 201/319 @ 78994 / 319}
sol: 319 in 789, 2 times: 638, leaves 151; bring down 9: 1519 contains 319 4 times: 1276, leaves 243; bring down 4: 2434 contains 319 7 times: 2233, leaves 201 remainder.
q: Divide 11577 by 14. {~11577 / 14}
q: Divide 48690 by 15. {~48690 / 15}
q: Divide 1110960 by 23. {~1110960 / 23}
q: Divide 122878 by 67. {~122878 / 67}
q: Divide 12412 by 53. {~12412 / 53}
q: Divide 146304 by 72. {~146304 / 72}
q: Divide 47100 by 54. {~47100 / 54}
q: Divide 71104 by 88. {~71104 / 88}
q: Divide 43956 by 66. {~43956 / 66}
q: Divide 121900 by 99. {~121900 / 99}
q: Divide 25312 by 112. {~25312 / 112}
q: Divide 381600 by 123. {~381600 / 123}
q: Divide 105672 by 204. {~105672 / 204}
q: Divide 600000 by 1234. {~600000 / 1234}
q: Divide 1234567 by 4321. {~1234567 / 4321}
q: Divide 50964242 by 7819. {~50964242 / 7819}
q: Divide 48905952 by 9876. {~48905952 / 9876}
q: Divide 4049160 by 12345. {~4049160 / 12345}
q: Divide 552160000 by 973. {~552160000 / 973}
q: At $15 an acre, how many acres of land can be bought for $3465? {~3465 / 15}
q: If a man travel 26 miles a day, in how many days will he travel 364 miles? {~364 / 26}
q: If $1083 be divided equally among 19 men, how many dollars will each have? {~1083 / 19}
q: A man raised 9523 bushels of corn on 107 acres: how much was that on one acre? Bushels? {~9523 / 107}
q: In 1 hogshead there are 63 gallons: how many hogsheads in 14868 gallons? {~14868 / 63}
q: The President receives $50000 a year (365 days): how much is that a day? {~50000 / 365}
q: The yearly income from a railroad is $379600: how much is that a day? (365 days = 1 year.) {~379600 / 365}
q: The product of two numbers is 6571435; one of the factors is 1235: what is the other? {~6571435 / 1235}
q: Divide one million two hundred and forty-seven thousand four hundred by 405. {~1247400 / 405}
q: Divide 10 million four hundred and one thousand by one thousand and six. {~10401000 / 1006}
q: A colony of 684 men bought a tract of land, containing 109440 acres: if equally divided, to how many acres was each man entitled? {~109440 / 684}
q: A farmer raised 8288 bushels of corn, averaging 56 bushels to the acre: how many acres did he plant? {~8288 / 56}
q: The capital of a joint-stock company is $262275, and is divided into 269 shares: what is the value of each share? {~262275 / 269}
q: The earth, at the equator, is about 24899 miles in circumference, and turns on its axis once in 24 hours: how many miles an hour does it turn? {~24899 / 24}
q: A railroad 238 miles long, cost $3731840: what was the cost per mile? {~3731840 / 238}
q: A fort is 27048 feet distant from a city; the flash of a cannon was seen 24 seconds before the sound was heard: how many feet a second did the sound travel? {~27048 / 24}
q: Light travels at the rate of 11520000 miles a minute: how many minutes does it require for the light of the sun to reach the earth, the sun being 92160000 miles distant? {~92160000 / 11520000}
q: Subtract 86247 from 94231 and divide the remainder by 16. {~(94231 - 86247) / 16}
q: Divide the sum of 46712 and 6848 by 104. {~(46712 + 6848) / 104}
q: Divide the product of 497 x 583 by 71. {~497 * 583 / 71}
q: To the difference between 2832 and 987 add 678, and divide the sum by 87. {~(2832 - 987 + 678) / 87}
q: Multiply the difference between 4896 and 2384 by 49, and divide the product by 112. {~(4896 - 2384) * 49 / 112}
q: Multiply the sum of 228 + 786 by 95, and divide the product by 114. {~(228 + 786) * 95 / 114}
q: Multiply the sum of 478 and 296 by their difference, and divide the product by 387. {~(478 + 296) * (478 - 296) / 387}
q: A horse-dealer received $7560 for horses; he sold a part of them for $3885; if he sold the rest for $175 apiece, how many horses did he sell the second time? {~(7560 - 3885) / 175}
q: A farmer expended at one time $7350 for land, and at another, $4655, paying $49 an acre each time: how many acres did he buy in both purchases? {~(7350 + 4655) / 49}
q: A refiner bought 58 hogsheads of sugar, at $77 a hogshead, and afterward sold them for $5742: how much did he gain on each hogshead? {~(5742 - 58 * 77) / 58}
q: A man bought 240 acres of land, at $26 an acre, giving in payment a house valued at $2820, and horses valued at $180 apiece: how many horses did he give? {~(240 * 26 - 2820) / 180}
q: A speculator bought 25 acres of land for $10625, and after dividing it into 125 village lots, sold each lot for $250: how much did he gain on the whole? {~125 * 250 - 10625} On each acre? {~(125 * 250 - 10625) / 25} On each lot? {~(125 * 250 - 10625) / 125}
== 13 | Division | Contractions in division
note: When the divisor can be separated into factors, divide by one factor, then the quotient by the other. To divide by 10, 100, 1000, cut off as many figures from the right of the dividend as there are ciphers in the divisor: they are the remainder.
q: A man paid $255 for 15 acres of land: how much was that per acre? {17}
sol: 15 acres are 3 times 5 acres; dividing $255 by 3 gives $85, the value of 5 acres; dividing $85 by 5 gives $17, the value of 1 acre.
q: Find the quotient of 37, divided by 14. {2 9/14 @ 37 / 14}
sol: Dividing by 2, the quotient is 18 twos and 1 unit remaining. Dividing by 7, the quotient is 2, with a remainder of 4 twos; the whole remainder then is 4 twos plus 1, or 9.
q: Divide 2583 by 63 (63 = 7 x 9). {~2583 / 63}
q: Divide 6976 by 32 (32 = 4 x 8). {~6976 / 32}
q: Divide 2744 by 28 (28 = 7 x 4). {~2744 / 28}
q: Divide 6145 by 42 (42 = 6 x 7). {~6145 / 42}
q: Divide 19008 by 132. {~19008 / 132}
q: Divide 7840 by 64. {~7840 / 64}
q: Divide 14771 by 72. {~14771 / 72}
q: Divide 10206 by 81. {~10206 / 81}
q: Divide 81344 by 121. {~81344 / 121}
q: Divide 98272 by 108. {~98272 / 108}
q: Divide 34872 by 100. Quotient? {348} Remainder? {72}
sol: Cut off the two right-hand figures: 348 is the quotient, 72 the remainder.
q: Divide 2682 by 10. Quotient? {268} Remainder? {2}
q: Divide 4700 by 100. Quotient? {47} Remainder? {0}
q: Divide 37201 by 100. Quotient? {372} Remainder? {1}
q: Divide 46250 by 100. Quotient? {462} Remainder? {50}
q: Divide 18003 by 1000. Quotient? {18} Remainder? {3}
q: Divide 4072 by 800. Quotient? {5} Remainder? {72}
sol: Regard 800 as 100 x 8: cut off the two right-hand figures for the remainder, 72; then 40 divided by 8 is 5.
q: Divide 77939 by 2400. {32 1139/2400 @ 77939 / 2400}
sol: Cut off the two right-hand figures, 39; divide 779 by 24: 32, remainder 11. Annex the figures cut off to the last remainder: the true remainder is 1139.
q: Divide 62700 by 2500. {~62700 / 2500}
q: Divide 73005 by 4000. {~73005 / 4000}
q: Divide 36001 by 9000. {~36001 / 9000}
q: Divide 1078000 by 11000. {~1078000 / 11000}
q: Divide 40167 by 180. {~40167 / 180}
q: Divide 907237 by 2100. {~907237 / 2100}
q: Divide 364006 by 6400. {~364006 / 6400}
q: Divide 76546037 by 250000. {~76546037 / 250000}
q: Divide 43563754 by 63400. {~43563754 / 63400}
q: Let 24 be a dividend, and 6 the divisor: the quotient is 4. If the dividend be multiplied by 2, what is the quotient? {8} If instead the divisor be multiplied by 2? {2} If both be multiplied by 2? {4}
sol: If the dividend be multiplied, or the divisor be divided, the quotient will be multiplied. If the dividend be divided, or the divisor be multiplied, the quotient will be divided. If both be multiplied or divided by the same number, the quotient will not be changed.
== 14 | Division | Promiscuous examples
q: In 4 bags are $500; in the first, $96; in the second, $120; in the third, $55: what sum in the 4th bag? {~500 - 96 - 120 - 55}
q: Four men paid $1265 for land: the first paid $243; the second $61 more than the first; the third $79 less than the second: how much did the fourth man pay? {~1265 - 243 - (243 + 61) - (243 + 61 - 79)}
q: I have five apple trees: the first bears 157 apples; the second, 264; the third, 305; the fourth, 97; the fifth, 123: I sell 428, and 186 are stolen: how many apples are left? {~157 + 264 + 305 + 97 + 123 - 428 - 186}
q: In an army of 57068 men, 9503 are killed; 586 join the enemy; 4794 are prisoners; 1234 die of wounds; 850 are drowned: how many return? {~57068 - 9503 - 586 - 4794 - 1234 - 850}
q: On the first of the year a speculator is worth $12307: during the year he gains $8706; in January he spends $237; in February, $301; in each of the remaining ten months he spends $538: how much had he at the end of the year? {~12307 + 8706 - 237 - 301 - 10 * 538}
q: The Bible has 31173 verses: in how many days can I read it, by reading 86 verses a day? {362 41/86 @ 31173 / 86}
q: I bought 28 horses for $1400: 3 died; for how much each must I sell the rest to incur no loss? {~1400 / 25}
q: How many times can I fill a 15 gallon cask, from 5 hogsheads of 63 gallons each? {~5 * 63 / 15}
q: A certain dividend is 73900; the quotient 214; the remainder, 70: what is the divisor? {~(73900 - 70) / 214}
q: Multiply the sum of 148 and 56 by their difference; divide the product by 23. {~(148 + 56) * (148 - 56) / 23}
q: How much woolen cloth, at $6 a yard, will it take to pay for 8 horses at $60 each, and 14 cows at $45 each? Yards? {~(8 * 60 + 14 * 45) / 6}
q: Two men paid $6000 for a farm: one man took 70 acres at $30 an acre, the other the remainder at $25 an acre: how many acres in all? {~70 + (6000 - 70 * 30) / 25}
q: My income is $1800 a year. If I spend $360 a year for provisions, $300 for rent, $150 for clothing, $100 for books, and $90 for incidentals, in how many years can I save $10400? {~10400 / (1800 - 360 - 300 - 150 - 100 - 90)}
q: A man bought 40 acres of ground at $15 an acre, and 80 acres at $25 an acre. He sold 90 acres for $4500, and the remainder at $60 an acre: for how much did he sell the whole land? {~4500 + 30 * 60} How much did he gain? {~4500 + 30 * 60 - (40 * 15 + 80 * 25)}
q: A merchant bought 275 yards of cloth at $4 a yard; he sold 250 yards at $5 a yard, and the remainder at $6 a yard: how much did he gain? {~250 * 5 + 25 * 6 - 275 * 4}
q: A broker buys 125 shares of stock for $85 a share, and 75 shares at $115 a share. He invests it all in other stock at $175 a share: how many shares does he get by the last purchase? {~(125 * 85 + 75 * 115) / 175}
q: A farmer sends to a dealer 20 horses and 15 mules to be sold. The dealer sells the horses for $125 each, and the mules for $150 each, charging $95 for selling. The farmer then buys 50 head of cattle at $45 each, with part of the money, and deposits the remainder in bank: how much does he deposit in bank? {~20 * 125 + 15 * 150 - 95 - 50 * 45}
== 15 | Compound Numbers | United States money
note: 10 mills make 1 cent; 10 cents, 1 dime; 10 dimes, 1 dollar; 10 dollars, 1 eagle. The figures left of the point are dollars; the next two, cents; the third, 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
q: Write in figures, in dollars: Twelve dollars seventeen cents eight mills. {12.178} Six dollars six cents six mills. {6.066} Seven dollars seven mills. {7.007}
q: Forty dollars fifty-three cents five mills. {40.535} Two dollars three cents. {2.03} Twenty dollars two cents two mills. {20.022}
q: One hundred dollars ten cents. {100.10} Two hundred dollars two cents. {200.02} Four hundred dollars one cent eight mills. {400.018}
q: Read out loud: $18.625; $20.324; $79.05; $46.00; $70.015; $100.28; $150.05; $100.00. {open}
sol: 18 dollars 62 cents 5 mills; 20 dollars 32 cents 4 mills; 79 dollars 5 cents; 46 dollars; 70 dollars 1 cent 5 mills; 100 dollars 28 cents; 150 dollars 5 cents; 100 dollars.
q: Read out loud: $6.12; $3.06; $4.31; $5.43; $29.00; $100.03; $20.05; $40.125. {open}
sol: 6 dollars 12 cents; 3 dollars 6 cents; 4 dollars 31 cents; 5 dollars 43 cents; 29 dollars; 100 dollars 3 cents; 20 dollars 5 cents; 40 dollars 12 cents 5 mills.
q: Reduce 17 ct. to mills. {170} 28 ct. to mills. {280}
q: Reduce 43 ct. and 6 m. to mills. {436} 70 ct. and 6 m. to mills. {706}
q: Reduce 106 m. to cents. Cents? {10} Mills? {6}
q: Reduce 490 mills to cents. {49}
q: Reduce 9 dollars to cents. {900} 14 dollars to cents. {1400} 104 dollars to cents. {10400}
q: Reduce $60 and 13 ct. to cents. {6013} $40 and 5 ct. to cents. {4005}
q: Reduce 375 ct. to dollars. {3.75} 9004 ct. to dollars. {90.04}
q: Reduce 4 dollars to mills. {4000} $14 and 2 ct. to mills. {14020}
q: Reduce 2465 mills to dollars. {2.465} 3007 mills to dollars. {3.007}
q: Reduce 3187 cents to dollars. {31.87} 10375 mills to dollars. {10.375}
q: Add together 4 dollars 12 cents 5 mills; 7 dollars 6 cents 2 mills; 20 dollars 43 cents; 10 dollars 5 mills; 16 dollars 87 cents 5 mills. [[+ 4.125 7.062 20.430 10.005 16.875]] {58.497}
sol: Write the numbers and add as in simple numbers; place the decimal point in the sum under the decimal points above.
q: What is the sum of 17 dollars 15 cents; 23 dollars 43 cents; 7 dollars 19 cents; 8 dollars 37 cents; and 12 dollars 31 cents? [[+ 17.15 23.43 7.19 8.37 12.31]] {68.45}
q: Add 18 dollars 4 cents 1 mill; 16 dollars 31 cents 7 mills; 100 dollars 50 cents 3 mills; and 87 dollars 33 cents 8 mills. [[+ 18.041 16.317 100.503 87.338]] {222.199}
q: William had the following bills for collection: $43.75; $29.18; $17.63; $268.95; and $718.07: how much was to be collected? [[+ 43.75 29.18 17.63 268.95 718.07]] {1077.58}
q: Bought a gig for $200; a watch for $43.87; a suit of clothes for $56.93; a hat for $8.50; and a whip for $2.31: what was the amount? {311.61 @ 200 + 43.87 + 56.93 + 8.50 + 2.31}
q: A person has due him, five hundred and four dollars six cents; $420.19; one hundred and five dollars fifty cents; $304; $888.47: what is the whole amount due him? [[+ 504.06 420.19 105.50 304.00 888.47]] {2222.22}
q: Add five dollars seven cents; thirty dollars twenty cents three mills; one hundred dollars five mills; sixty dollars two cents; seven hundred dollars one cent one mill; $1000.10; forty dollars four mills; and $64.587. [[+ 5.07 30.203 100.005 60.02 700.011 1000.10 40.004 64.587]] {2000}
q: From one hundred dollars five cents three mills, take eighty dollars twenty cents and seven mills. [[- 100.053 80.207]] {19.846}
q: From $29.342 take $17.265. [[- 29.342 17.265]] {12.077}
q: From $46.28 take $17.75. [[- 46.28 17.75]] {28.53}
q: From $20.05 take $5.50. [[- 20.05 5.50]] {14.55}
q: From $3, take 3 ct. {2.97}
q: From $10, take 1 mill. {9.999}
q: From $50, take 50 ct. 5 mills. {49.495}
q: From one thousand dollars, take one dollar one cent and one mill. {998.989}
q: B owes 1000 dollars 43 cents; if he pay nine hundred dollars sixty-eight cents, how much will he still owe? {99.75 @ 1000.43 - 900.68}
q: What will 13 cows cost, at 47 dollars 12 cents 5 mills each? [[x 47.125 13]] {612.625}
sol: Multiply as in simple numbers; put the decimal point in the same place in the product as it is in the multiplicand.
q: Multiply $7.835 by 8. {62.68 @ 7.835 * 8}
q: Multiply $12, 9 ct. 3 m. by 9. {108.837 @ 12.093 * 9}
q: Multiply $23, 1 ct. 8 m. by 16. {368.288 @ 23.018 * 16}
q: Multiply $35, 14 ct. by 53. {1862.42 @ 35.14 * 53}
q: Multiply $125, 2 ct. by 62. {7751.24 @ 125.02 * 62}
q: Multiply $40, 4 ct. by 102. {4084.08 @ 40.04 * 102}
q: Multiply 12 ct. 5 m. by 17. In dollars? {2.125 @ .125 * 17}
q: Multiply $3.28 by 38. {124.64 @ 3.28 * 38}
q: What cost 338 barrels of cider, at 1 dollar 6 cents a barrel? {358.28 @ 338 * 1.06}
q: Sold 38 cords of wood, at 5 dollars 75 cents a cord: to what did it amount? {218.50 @ 38 * 5.75}
q: At 7 ct. a pound, what cost 465 pounds of sugar? {32.55 @ 465 * .07}
sol: Instead of multiplying 7 cents by 465, multiply 465 by 7, which gives the same product; but remember that 7 cents is the true multiplicand: $32.55.
q: What cost 89 yards of sheeting, at 34 ct. a yard? {30.26 @ 89 * .34}
q: What will 24 yards of cloth cost, at $5.67 a yard? {136.08 @ 24 * 5.67}
q: I have 169 sheep, valued at $2.69 each: what is the value of the whole? {454.61 @ 169 * 2.69}
q: If I sell 691 bushels of wheat, at $1.25 a bushel, what will it amount to? {863.75 @ 691 * 1.25}
q: I sold 73 hogsheads of molasses, of 63 gallons each, at 55 ct. a gallon: what is the sum? {2529.45 @ 73 * 63 * .55}
q: What cost 4 barrels of sugar, of 281 pounds each, at 6 cents 5 mills a pound? {73.06 @ 4 * 281 * .065}
q: Bought 35 bolts of tape, of 10 yards each, at 1 cent a yard: what did it cost? {3.50 @ 35 * 10 * .01}
q: If I earn 13 ct. an hour, and work 11 hours a day, how much will I earn in 312 days? {446.16 @ .13 * 11 * 312}
q: I sold 18 bags of wheat, of 3 bushels each, at $1.25 a bushel: what is the amount? {67.50 @ 18 * 3 * 1.25}
q: What cost 150 acres of land, at 10 dollars 1 mill per acre? {1500.15 @ 150 * 10.001}
q: What cost 17 bags of coffee, of 51 pounds each, at 24 cents 7 mills per pound? {214.149 @ 17 * 51 * .247}
q: How much cloth, at 7 cents a yard, will $1.75 buy? Yards? {25}
sol: As many yards as 7 cents is contained times in 175 cents, which are 25.
q: How much rice, at 9 cents a pound, can be bought for 72 cents? Pounds? {8}
q: How many towels, at 37 cents and 5 mills apiece, can be bought for $6? {16 @ 6 / .375}
q: How many yards of calico, at 8 cents a yard, can be bought for $2.80? {35 @ 2.80 / .08}
q: How many yards of ribbon, at 25 cents a yard, can be purchased for $3? {12 @ 3 / .25}
q: At $8.05 a barrel, how many barrels of flour will $161 purchase? {20 @ 161 / 8.05}
q: At 7 cents 5 mills each, how many oranges can be bought for $1.20? {16 @ 1.20 / .075}
q: At $1.125 per bushel, how many bushels of wheat can be purchased for $234? {208 @ 234 / 1.125}
q: A man worked 3 days for $3.75, what were his daily wages? {1.25}
q: A farmer sold 6 bushels of wheat for $9: how much a bushel did he get? {1.50}
sol: $9 divided by 6 gives $1, with a remainder $3 = 300 cents; 300 cents divided by 6 gives 50 cents: $1.50.
q: Divide 65 dollars equally among 8 persons. {8.125 @ 65 / 8}
q: A farmer received $29.61 for 23 bushels of wheat: how much was that per bushel? {1.287|1.28739}
q: If 4 acres of land cost $92.25, how much is that an acre? {23.062|23.0625}
q: Make an equal division of $57.50 among 8 persons. {7.187|7.1875}
q: A man received $25.76 for 16 days' work: how much was that a day? {1.61 @ 25.76 / 16}
q: I bought 755 bushels of apples for $328.425: what did they cost a bushel? {.435 @ 328.425 / 755}
q: My salary is $800 a year: how much is that a day, there being 313 working days in the year? {2.555}
q: Divide ten thousand dollars equally among 133 men: what is each man's share? {75.187}
q: A man purchased a farm of 154 acres, for two thousand seven hundred and five dollars and 1 cent: what did it cost per acre? {17.565 @ 2705.01 / 154}
q: I sold 15 kegs of butter, of 25 pounds each, for $60: how much was that a pound? {.16 @ 60 / 375}
q: I bought 8 barrels of sugar, of 235 pounds each, for $122.20: what did 1 pound cost? {.065 @ 122.20 / 1880}
== 16 | Compound Numbers | Money examples and bills
q: I owe A $47.50; B, $38.45; C, $15.47; D, $19.43: what sum do I owe? {120.85 @ 47.50 + 38.45 + 15.47 + 19.43}
q: A owes $35.25; B, $23.75; C, as much as A and B, and $1 more: what is the amount? {119 @ 35.25 + 23.75 + 35.25 + 23.75 + 1}
q: A paid me $18.38; B, $81.62; C, twice as much as A and B: how much did I receive? {300 @ 3 * (18.38 + 81.62)}
q: I went to market with $5; I spent for butter 75 cents, for eggs 35 cents, for vegetables 50 cents, for flour $1.50: how much money was left? {1.90 @ 5 - .75 - .35 - .50 - 1.50}
q: A lady had $20; she bought a dress for $8.10, shoes for $5.65, eight yards of delaine at 25 cents a yard, and a shawl for $4: what sum was left? {.25 @ 20 - 8.10 - 5.65 - 8 * .25 - 4}
q: I get $50 a month, and spend $30.50 of it: how much will I have left in 6 months? {117 @ (50 - 30.50) * 6}
q: A farmer sold his marketing for $21.75: he paid for sugar $3.85, for tea $1.25, for coffee $2.50, for spices $1.50: how much had he left? {12.65 @ 21.75 - 3.85 - 1.25 - 2.50 - 1.50}
q: I owe A $37.06; B, $200.85; C, $400; D, $236.75, and E $124.34; my property is worth $889.25: how much do I owe more than I am worth? {109.75 @ 37.06 + 200.85 + 400 + 236.75 + 124.34 - 889.25}
q: Bought 143 pounds of coffee, at 23 cents a pound: after paying $12.60, what was due? {20.29 @ 143 * .23 - 12.60}
q: A owed me $400: he paid me 435 bushels of corn, at 45 cents a bushel: what sum is due? {204.25 @ 400 - 435 * .45}
q: If B spend 65 cents a day, how much will he save in 365 days, his income being $400? {162.75 @ 400 - 365 * .65}
q: Bought 21 barrels of apples, of 3 bushels each, at 35 cents a bushel: what did they cost? {22.05 @ 21 * 3 * .35}
q: What cost four pieces of cambric, each containing 19 yards, at 23 cents a yard? {17.48 @ 4 * 19 * .23}
q: If 25 men perform a piece of work for $2000, and spend, while doing it, $163.75, what will be each man's share of the profits? {73.45 @ (2000 - 163.75) / 25}
q: If 16 men receive $516 for 43 days' work, how much does each man earn a day? {.75 @ 516 / 43 / 16}
q: C earned $90 in 40 days, working 10 hours a day: how much did he earn an hour? {.225 @ 90 / 400}
q: A merchant failing, has goods worth $1000, and $500 in cash, to be equally divided among 22 creditors: how much will each receive? {68.181|68 2/11}
q: Bought 9 pounds Coffee, at $0.32 per lb.; 4 pounds Tea, at 1.25; 45 pounds Sugar, at .09; 17 pounds Cheese, at .20: what is the amount of my bill? {15.33 @ 9 * .32 + 4 * 1.25 + 45 * .09 + 17 * .20}
q: Bought 22 yards Silk, at $1.75 per yd.; 18 yards Muslin, at .15; 25 yards Linen, at .65; 6 yards Gingham, at .18: what is the whole amount? {58.53 @ 22 * 1.75 + 18 * .15 + 25 * .65 + 6 * .18}
q: Bought 4 pounds Prunes, at $0.18 per lb.; 8 pounds Peaches, at .23; 7 pounds Rice, at .11; 6 pounds Oat-meal, at .09; 13 pounds Java Coffee, at .35; 26 pounds Sugar, at .12: what is the whole amount? {11.54 @ 4 * .18 + 8 * .23 + 7 * .11 + 6 * .09 + 13 * .35 + 26 * .12}
q: Bought 43 yards Muslin, at $0.13 per yd.; 28 yards Calico, at .09; 23 yards Alpaca, at .23: what is the whole amount? {13.40 @ 43 * .13 + 28 * .09 + 23 * .23}
== 17 | Compound Numbers | Dry and liquid measure
note: To reduce from a higher to a lower denomination, multiply by the number of units of the lower that make one of the higher. To reduce to a higher, divide.
rule: 2 pints make 1 quart
rule: 8 quarts make 1 peck
rule: 4 pecks make 1 bushel
q: How many pints in 2 quarts? {4} In 4? {8} In 6? {12} In 8? {16} In 10? {20}
q: How many quarts in 3 pk.? {24} In 5? {40} In 7? {56} In 9? {72}
q: How many pecks in 9 bu.? {36} In 11? {44} In 13? {52} In 15? {60} In 17? {68} In 19? {76}
q: How many quarts in 10 bu.? {320} In 12? {384} In 14? {448} In 18? {576} In 25? {800} In 56? {1792}
q: How many pecks in 16 qt.? {2} In 24? {3} In 32? {4} In 40? {5} In 48? {6} In 64? {8}
q: How many bushels in 32 qt.? {1} In 64? {2} In 96? {3}
q: How many pints in 1 bu.? {64} In 2? {128} In 5? {320}
q: Reduce 3 bushels to pints. {192}
sol: To reduce bu. to pk. multiply by 4; pk. to qt., by 8; qt. to pt., by 2: 3 bu. = 12 pk. = 96 qt. = 192 pt.
q: Reduce 192 pints to bushels. {3}
sol: To reduce pt. to qt. divide by 2; qt. to pk., by 8; pk. to bu., by 4: 192 pt. = 96 qt. = 12 pk. = 3 bu.
q: Reduce 7 bu. 3 pk. 6 qt. 1 pt. to pints. {509}
sol: Multiply the bu. by 4, making 28 pk., and add the 3 pk. Then multiply the 31 pk. by 8 and add the 6 qt.; multiply the 254 qt. by 2 and add the 1 pt.: 509 pt.
q: Reduce 509 pt. to bushels. Bu.? {7} Pk.? {3} Qt.? {6} Pt.? {1}
q: Reduce 4 bu. 2 pk. 1 qt. to pints. {290}
q: Reduce 7 bu. 3 pk. 7 qt. 1 pint to pints. {511}
q: Reduce 3 bu. 1 pt. to pints. {193}
q: Reduce 384 pt. to bushels. Bu.? {6}
q: Reduce 47 pt. to pecks. Pk.? {2} Qt.? {7} Pt.? {1}
q: Reduce 95 pt. to bushels. Bu.? {1} Pk.? {1} Qt.? {7} Pt.? {1}
q: Reduce 508 pt. to bushels. Bu.? {7} Pk.? {3} Qt.? {6}
rule: 4 gills make 1 pint
rule: 2 pints make 1 quart
rule: 4 quarts make 1 gallon
q: Reduce 17 gal. to pints. {136}
q: Reduce 13 gal. to gills. {416}
q: Reduce 126 gal. to pints. {1008}
q: Reduce 1260 gal. to gills. {40320}
q: Reduce 1120 gi. to gallons. Gal.? {35}
q: How many gallons in 1848 cubic inches? (A gallon contains 231 cubic inches.) {8 @ 1848 / 231}
q: How many gallons in a vessel containing 138138 cubic inches? {598 @ 138138 / 231}
== 18 | Compound Numbers | Weight and long measure
rule: 16 ounces make 1 pound
rule: 100 pounds make 1 hundred-weight
rule: 20 cwt. make 1 ton
q: Reduce 2 cwt. to pounds. {200}
q: Reduce 3 cwt. 75 lb. to pounds. {375}
q: Reduce 1 T. 2 cwt. to pounds. {2200}
q: Reduce 3 T. 75 lb. to pounds. {6075}
q: Reduce 4 cwt. 44 lb. to pounds. {444}
q: Reduce 5 T. 90 lb. to pounds. {10090}
q: Reduce 2 cwt. 77 lb. 12 oz. to ounces. {4444}
q: Reduce 2 cwt. 17 lb. 3 oz. to ounces. {3475}
q: Reduce 1 T. 6 cwt. 4 lb. 2 oz. to ounces. {41666}
q: Reduce 4803 lb. to cwt. Cwt.? {48} Lb.? {3}
q: Reduce 22400 lb. to tons. T.? {11} Cwt.? {4}
q: Reduce 2048000 oz. to tons. T.? {64}
q: Reduce 64546 oz. to cwt. Cwt.? {40} Lb.? {34} Oz.? {2}
q: Reduce 97203 oz. to tons. T.? {3} Cwt.? {0} Lb.? {75} Oz.? {3}
q: Reduce 544272 oz. to tons. T.? {17} Cwt.? {0} Lb.? {17}
q: What is the total weight of 52 parcels, each containing 18 lb.? Cwt.? {9} Lb.? {36}
q: What is the weight of 180 iron castings, each weighing 75 lb.? T.? {6} Cwt.? {15}
rule: 12 inches make 1 foot
rule: 3 feet make 1 yard
rule: 5 1/2 yards make 1 rod
rule: 320 rods make 1 mile
q: Reduce 2 yd. 2 ft. 7 in. to inches. {103}
q: Reduce 7 yd. 11 in. to inches. {263}
q: Reduce 12 mi. to rods. {3840}
q: Reduce 7 mi. 240 rd. to rods. {2480}
q: Reduce 9 mi. 31 rd. to rods. {2911}
q: Reduce 133 in. to yards. Yd.? {3} Ft.? {2} In.? {1}
q: Reduce 181 in. to yards. Yd.? {5} Ft.? {0} In.? {1}
q: Reduce 2240 rd. to miles. {7}
q: Reduce 2200 rd. to miles. Mi.? {6} Rd.? {280}
q: Reduce 1 mi. to yards. {1760}
q: Reduce 1 mi. to feet. {5280}
== 19 | Compound Numbers | Square and cubic measure
note: The area of a rectangle is the length multiplied by the breadth; the solid contents of a box are the length, breadth and thickness multiplied together.
rule: 144 square inches make 1 square foot
rule: 9 square feet make 1 square yard
rule: 30 1/4 square yards make 1 square rod
rule: 160 square rods make 1 acre
rule: 640 acres make 1 square mile
q: Reduce 8 sq. yd. to square inches. {10368}
q: Reduce 4 A. to square rods. {640}
q: Reduce 1 sq. mi. to square rods. {102400}
q: Reduce 2 sq. yd. 3 sq. ft. to sq. in. {3024}
q: Reduce 5 A. 100 sq. rd. to sq. rd. {900}
q: Reduce 960 sq. rd. to acres. {6}
q: Reduce 3888 sq. in. to square yards. {3}
q: Reduce 20000 sq. rd. to acres. {125}
q: Reduce 515280 sq. rd. to square miles. Sq. mi.? {5} A.? {20} Sq. rd.? {80}
q: Reduce 4176 sq. in. to sq. yd. Sq. yd.? {3} Sq. ft.? {2}
q: How many square inches in a board 4 inches long and 3 inches wide? {12}
sol: In each of the longer rows there are 4 squares, and there are 3 such rows: 4 x 3 = 12.
q: In a floor 16 feet long and 12 feet wide, how many square feet? {192}
q: How many square yards of carpeting will cover a room 5 yards long and 4 yards wide? {20}
q: How many square yards of carpeting will cover two rooms, one 18 feet long and 12 feet wide, the other 21 feet long and 15 feet wide? {59 @ (18 * 12 + 21 * 15) / 9}
q: How many square yards in a ceiling 18 feet long and 14 feet wide? {28 @ 18 * 14 / 9}
q: In a field 35 rods long and 32 rods wide, how many acres? {7 @ 35 * 32 / 160}
q: How much will it cost to carpet two rooms, each 18 feet long and 15 feet wide, if the carpet costs $1.25 per square yard? {75 @ 2 * 18 * 15 / 9 * 1.25}
q: What will it cost to plaster a ceiling 21 feet long and 18 feet wide, at 17 cents per square yard? {7.14 @ 21 * 18 / 9 * .17}
q: A floor containing 132 square feet, is 11 feet wide: what is its length? Feet? {12}
sol: If the area of a rectangle be divided by either side, the quotient will be the other side: 132 divided by 11 is 12.
q: A floor is 18 feet long, and contains 30 square yards: what is its width? Feet? {15 @ 30 * 9 / 18}
q: A field containing 9 acres, is 45 rods in length: what is its width? Rods? {32 @ 9 * 160 / 45}
q: A field 35 rods wide, contains 21 acres: what is its length? Rods? {96 @ 21 * 160 / 35}
rule: 1728 cubic inches make 1 cubic foot
rule: 27 cubic feet make 1 cubic yard
rule: 128 cubic feet make 1 cord
q: Reduce 2 cu. yd. to cubic inches. {93312}
q: Reduce 28 cords of wood to cu. ft. {3584}
q: Reduce 34 cords of wood to cu. in. {7520256}
q: Reduce 1 cord of wood to cu. in. {221184}
q: Reduce 63936 cu. in. to cu. yd. Cu. yd.? {1} Cu. ft.? {10}
q: How many cubic feet in a rectangular solid, 8 ft. long, 5 ft wide, 4 ft. thick? {160}
q: How many cubic yards of excavation in a cellar 8 yd. long, 5 yd. wide, 2 yd. deep? {80}
q: How many cubic yards in a cellar, 18 feet long, 15 feet wide, 7 feet deep? {70 @ 18 * 15 * 7 / 27}
q: In a pile of wood 40 feet long, 12 feet wide, and 8 feet high, how many cords? {30 @ 40 * 12 * 8 / 128}
q: What will be the cost of a pile of wood 80 feet long, 8 feet high, and 4 feet thick, at $5.50 per cord? {110 @ 80 * 8 * 4 / 128 * 5.5}
q: What will be the cost of excavating a cellar 24 ft. long, 15 ft. wide, and 6 ft. deep, at $1.25 per cubic yard or load? {100 @ 24 * 15 * 6 / 27 * 1.25}
== 20 | Compound Numbers | Time and other tables
note: In finding a time in months here, 4 weeks make 1 month.
rule: 60 seconds make 1 minute
rule: 60 minutes make 1 hour
rule: 24 hours make 1 day
rule: 7 days make 1 week
rule: 365 days and 6 hours make 1 year
rule: 100 years make 1 century
q: Reduce 2 hr. to seconds. {7200}
q: Reduce 7 da. to minutes. {10080}
q: Reduce 1 da. 3 hr. 44 min. 3 sec. to seconds. {99843}
q: Reduce 9 wk. 6 da. 10 hr. 40 min. to minutes. {100000}
q: Reduce 4 wk. 3 da. 4 min. to minutes. {44644}
q: Reduce 10800 seconds to hours. {3}
q: Reduce 432000 seconds to days. {5}
q: Reduce 7322 seconds to hours. Hr.? {2} Min.? {2} Sec.? {2}
q: Reduce 4323 minutes to days. Da.? {3} Hr.? {0} Min.? {3}
q: Reduce 20280 minutes to weeks. Wk.? {2} Da.? {0} Hr.? {2}
q: Reduce 41761 min. to months. Mo.? {1} Wk.? {0} Da.? {1} Hr.? {0} Min.? {1}
rule: 24 grains make 1 pennyweight
rule: 20 pennyweights make 1 ounce
rule: 12 ounces make 1 pound
q: Reduce 5 lb. 4 oz. Troy to ounces. {64}
q: Reduce 9 lb. 3 oz. 5 pwt. to pwt. {2225}
q: Reduce 8 lb. 9 oz. 13 pwt. 17 gr. to grains. {50729}
q: Reduce 805 pwt. to pounds. Lb.? {3} Oz.? {4} Pwt.? {5}
q: Reduce 12530 gr. to pounds. Lb.? {2} Oz.? {2} Pwt.? {2} Gr.? {2}
note: Apothecaries weight: 20 grains make 1 scruple; 3 scruples, 1 dram; 8 drams, 1 ounce; 12 ounces, 1 pound.
q: Reduce 4 lb. 5 ounces 2 grains (apothecaries) to grains. {25442}
q: Reduce 7 lb. 2 ounces 1 scruple to grains. {41300}
q: Reduce 431 drams to pounds. Lb.? {4} Oz.? {5} Dr.? {7}
q: Reduce 975 scruples to pounds. Lb.? {3} Oz.? {4} Dr.? {5}
q: Reduce 6321 gr. to pounds (apothecaries). Lb.? {1} Oz.? {1} Dr.? {1} Sc.? {1} Gr.? {1}
q: Reduce 4 congii 7 fluid ounces to fluid drams. {4152}
q: Reduce 5 pints (O.) 6 fluid ounces 3 fluid drams to minims. {41460}
q: Reduce 2469 fluid drams to gallons (congii). Cong.? {2} O.? {3} F. oz.? {4} F. dr.? {5}
q: Reduce 3 yd. to barleycorns. (3 barleycorns make 1 inch.) {324}
q: How many lines in 1 foot 6 inches? (12 lines make 1 inch.) {216}
q: What is the height of a horse of 16 1/2 hands? (4 inches make 1 hand.) Feet? {5} Inches? {6}
q: A field measures 24 chains in length and 15 chains in breadth: how many acres in it? (A chain is 4 rods.) {36 @ 24 * 4 * 15 * 4 / 160}
q: A cistern contains 267 cubic feet 624 cubic inches: how many gallons does it hold? (231 cubic inches make 1 gallon.) {2000 @ (267 * 1728 + 624) / 231}
q: Reduce 8 degrees 41 minutes 45 seconds to seconds. {31305}
q: Reduce 61 degrees 59 minutes 28 seconds to seconds. {223168}
q: Reduce 915 minutes (of a degree) to degrees. Deg.? {15} Min.? {15}
q: Reduce 3661 seconds (of a degree) to degrees. Deg.? {1} Min.? {1} Sec.? {1}
q: What cost 6 gross of screws at 5 cents a dozen? {3.60 @ 6 * 12 * .05}
q: A man is 4 score and 10: how old is he? {90}
q: At 18 cents a quire, what will 3 bundles of paper cost? (24 sheets make 1 quire; 20 quires, 1 ream; 2 reams, 1 bundle.) {21.60 @ 3 * 2 * 20 * .18}
q: How many sheets of paper will be required for a 12mo. book of 336 pages? (A 12mo. sheet is folded in 12 leaves.) {14 @ 336 / 2 / 12}
q: An octavo work in 5 volumes has 512 pages in Vol. 1, 528 in Vol. 2, 528 in Vol. 3, 512 in Vol. 4, and 496 in Vol. 5: how much paper was used for one copy of the whole work? Quires? {6} Sheets? {17}
sol: 2576 pages are 1288 leaves; an octavo sheet makes 8 leaves: 161 sheets, which are 6 quires 17 sheets.
== 21 | Compound Numbers | Examples in reduction
q: What cost 2 bu. of plums, at 5 ct. a pint? {6.40 @ 2 * 64 * .05}
q: What cost 3 bu. 2 pk. of peaches, at 50 ct. a peck? {7 @ 14 * .5}
q: What cost 3 pk. 3 qt. of barley, at 3 ct. a pint? {1.62 @ 54 * .03}
q: At 15 ct. a peck, how many bushels of apples can be bought for $3? {5 @ 3 / .15 / 4}
q: If salt cost 2 ct. a pint, how much can be bought with $1.66? Bu.? {1} Pk.? {1} Qt.? {1} Pt.? {1}
q: I put 91 bu. of wheat into bags containing 3 bu. 2 pk. each: how many bags were required? {26 @ 91 * 4 / 14}
sol: Reduce both quantities to pecks, and then divide: 364 pk. divided by 14 pk. is 26.
q: How many spikes, weighing 4 oz. each, are in a parcel weighing 15 lb. 12 oz.? {63 @ (15 * 16 + 12) / 4}
q: I bought 44 cwt. 52 lb. of cheese; each cheese weighed 9 lb. 15 oz.: how many cheeses did I buy? {448 @ 4452 * 16 / 159}
q: How many kegs, of 84 lb. each, can be filled from a hogshead of sugar weighing 14 cwt. 28 lb.? {17 @ 1428 / 84}
q: How many boxes, containing 12 lb. each, can be filled from 7 cwt. 56 lb. of tobacco? {63 @ 756 / 12}
q: If a family use 3 lb. 13 oz. of sugar a week, how long will 6 cwt. 10 lb. last them? Weeks? {160 @ 610 * 16 / 61}
q: What will 2 acres 125 square rods of land cost, at 20 cents a square rod? {89 @ 445 * .2}
q: A farmer has a field of 16 A. 53 sq. rd. to divide into lots of 1 A. 41 sq. rd. each: how many lots will it make? {13 @ 2613 / 201}
q: How many cu. in. in a block of marble 2 ft. long, 2 ft. high, 2 ft. wide? {13824}
q: One cu. ft. of water weighs 1000 oz. avoirdupois: what do 5 cu. ft. weigh? Lb.? {312} Oz.? {8}
q: What is the weight of a quantity of water occupying the space of 1 cord of wood, each cubic foot of water weighing 1000 ounces avoirdupois? T.? {4}
q: A cubic foot of oak weighs 950 oz. avoirdupois: what do 2 cords of oak weigh? T.? {7} Cwt.? {12}
q: Find the cost of 63 gallons of wine, at 20 cents a pint. {100.80 @ 63 * 8 * .2}
q: Find the cost of 5 barrels of molasses, each containing 31 gal. 2 qt., at 10 cents a quart. {63 @ 5 * 126 * .1}
q: At 5 cents a pint, what quantity of molasses can be bought for $2? Gallons? {5 @ 2 / .05 / 8}
q: How many dozen bottles, each bottle holding 3 qt. 1 pt., can be filled from 63 gal. of cider? {6 @ 504 / 84}
q: How many kegs, of 4 gal. 3 qt. 1 pt. each, can be filled from 58 gal. 2 qt.? {12 @ 468 / 39}
q: If a human heart beat 70 times a minute, how many times will it beat in a day? {100800 @ 70 * 1440}
q: How many seconds in the month of February, 1876? {2505600 @ 29 * 86400}
sol: 1876 is a leap year, because it is exactly divisible by 4; hence, February has 29 days.
q: If a ship sail 8 miles an hour, how many miles will it sail in 3 wk. 2 da. 3 hr.? {4440 @ 8 * (23 * 24 + 3)}
q: A horse is fed 1 peck of oats daily. If oats cost 44 cents a bushel, how much will it cost to feed him a year of 365 days? {40.15 @ 365 * .11}
q: A flour dealer bought 40 barrels of flour for 3 ct. a pound, and sold it for 5 ct. a pound: how much did he gain? (196 pounds make a barrel of flour.) {156.80 @ 40 * 196 * .02}
== 22 | Compound Numbers | Adding and subtracting compound numbers
note: Write units of the same denomination in the same column. Begin with the lowest; divide its sum by the number that makes one of the next higher, write the remainder, and carry the quotient.
q: A farmer sold three lots of wheat: the first lot contained 25 bu. 3 pk.; the second, 14 bu. 2 pk.; the third, 32 bu. 1 pk.: how much did he sell? Bu.? {72} Pk.? {2}
sol: Beginning with pecks, the sum is 6, which divided by 4 gives 1 bushel and 2 pecks; write the 2 pk., and carry the 1 bu. to the bushels: 72 bu. 2 pk.
q: Add: 3 bu. 2 pk. 0 qt. 1 pt.; 4 bu. 0 pk. 6 qt. 1 pt.; 1 bu. 3 pk. 7 qt. 1 pt. Bu.? {9} Pk.? {2} Qt.? {6} Pt.? {1}
q: Add: 7 bu. 3 pk. 7 qt. 1 pt.; 6 bu. 2 pk. 0 qt. 0 pt.; 9 bu. 2 pk. 4 qt. 1 pt. Bu.? {24} Pk.? {0} Qt.? {4}
q: Add: 4 bu. 3 pk. 7 qt.; 5 bu. 2 pk. 2 qt.; 7 bu. 1 pk. 6 qt. Bu.? {17} Pk.? {3} Qt.? {7}
q: Add: 8 bu. 1 pk. 7 qt. 1 pt.; 7 bu. 3 pk. 2 qt. 1 pt.; 9 bu. 2 pk. 7 qt. 1 pt. Bu.? {26} Pk.? {0} Qt.? {1} Pt.? {1}
q: Add: 7 qt. 1 pt. 3 gi.; 6 qt. 0 pt. 2 gi.; 9 qt. 1 pt. 3 gi. Qt.? {24}
q: Add: 40 gal. 3 qt. 1 pt. 3 gi.; 16 gal. 1 qt. 0 pt. 2 gi.; 71 gal. 2 qt. 1 pt. 2 gi. Gal.? {128} Qt.? {3} Pt.? {1} Gi.? {3}
q: Add: 45 T. 3 cwt. 53 lb. 10 oz.; 14 T. 14 cwt. 75 lb. 15 oz.; 19 T. 17 cwt. 18 lb. 13 oz. T.? {79} Cwt.? {15} Lb.? {48} Oz.? {6}
q: Add: 16 cwt. 85 lb. 14 oz.; 15 cwt. 90 lb. 13 oz.; 18 cwt. 74 lb. 12 oz. Cwt.? {51} Lb.? {51} Oz.? {7}
q: Add: 28 mi. 129 rd.; 64 mi. 280 rd.; 17 mi. 275 rd. Mi.? {111} Rd.? {44}
q: Add: 4 yd. 2 ft. 11 in.; 3 yd. 1 ft. 9 in.; 5 yd. 1 ft. 8 in. Yd.? {14} Ft.? {0} In.? {4}
q: Add: 41 A. 51 sq. rd.; 64 A. 104 sq. rd.; 193 A. 155 sq. rd. A.? {299} Sq. rd.? {150}
q: Add: 15 sq. yd. 8 sq. ft. 115 sq. in.; 20 sq. yd. 7 sq. ft. 109 sq. in.; 14 sq. yd. 5 sq. ft. 137 sq. in. Sq. yd.? {51} Sq. ft.? {4} Sq. in.? {73}
q: Add: 13 C. 28 cu. ft. 390 cu. in.; 15 C. 90 cu. ft. 874 cu. in.; 20 C. 67 cu. ft. 983 cu. in. C.? {49} Cu. ft.? {58} Cu. in.? {519}
q: Add: 50 cu. yd. 18 cu. ft. 900 cu. in.; 45 cu. yd. 17 cu. ft. 828 cu. in.; 46 cu. yd. 20 cu. ft. 990 cu. in. Cu. yd.? {143} Cu. ft.? {2} Cu. in.? {990}
q: Add: 16 da. 18 hr. 28 min. 47 sec.; 13 da. 15 hr. 49 min. 59 sec.; 19 da. 16 hr. 53 min. 42 sec. Da.? {50} Hr.? {3} Min.? {12} Sec.? {28}
q: Add: 3 mo. 0 wk. 0 da. 23 hr. 51 min. 40 sec.; 1 mo. 2 wk. 4 da. 19 hr. 30 min. 37 sec.; 3 mo. 1 wk. 5 da. 13 hr. 27 min. 18 sec. Mo.? {8} Wk.? {0} Da.? {4} Hr.? {8} Min.? {49} Sec.? {35}
q: Five loads of wheat measured thus: 21 bu. 3 pk.; 14 bu. 1 pk.; 23 bu. 2 pk.; 18 bu. 1 pk.; 22 bu. 1 pk.: how many bushels in all? Bu.? {100}
q: A farmer raised of oats 200 bu. 3 pk.; barley, 143 bu. 1 pk.; corn, 400 bu. 3 pk.; wheat, 255 bu. 1 pk.: how much in all? Bu.? {1000}
q: A grocer sold 5 hogsheads of sugar: the first weighed 8 cwt. 36 lb.; the second, 4 cwt. 64 lb.; the third, 5 cwt. 19 lb.; the fourth, 7 cwt. 75 lb.; the fifth, 7 cwt. 84 lb.: what did all weigh? Cwt.? {33} Lb.? {78}
q: Add 13 lb. 11 oz.; 17 lb. 13 oz.; 14 lb. 14 oz.; 16 lb.; 19 lb. 7 oz.; and 17 lb. 9 oz. Lb.? {99} Oz.? {6}
q: Two men depart from the same place: one travels 104 mi. 50 rd. due east; the other, 95 mi. 270 rd. due west: how far are they apart? Mi.? {200}
q: A man has 3 farms: in the first are 186 A. 134 sq. rd.; in the second, 286 A. 17 sq. rd.; in the third, 113 A. 89 sq. rd.: how much in all? A.? {586} Sq. rd.? {80}
q: Add 17 sq. yd. 3 sq. ft. 119 sq. in.; 18 sq. yd. 141 sq. in.; 23 sq. yd. 7 sq. ft.; 29 sq. yd. 5 sq. ft. 116 sq. in. Sq. yd.? {88} Sq. ft.? {8} Sq. in.? {88}
q: A has 4 piles of wood: in the first, 7 C. 78 cu. ft.; the second, 16 C. 24 cu. ft.; the third, 35 C. 127 cu. ft.; the fourth, 29 C. 10 cu. ft.: how much in all? C.? {88} Cu. ft.? {111}
q: I sold 4642 gal. 3 qt. 1 pt. of wine to A; 945 gal. to B; 1707 gal. 1 pt. to C; 10206 gal. 1 qt. to D: how many hogsheads of 63 gal. each did I sell? Hhd.? {277} Gal.? {50} Qt.? {1}
q: I have 67 bu. 2 pk. of wheat: how much will remain after selling 34 bu. 3 pk.? Bu.? {32} Pk.? {3}
q: From 12 bu. 0 pk. 1 qt. 0 pt. take 8 bu. 2 pk. 1 qt. 1 pt. Bu.? {3} Pk.? {1} Qt.? {7} Pt.? {1}
q: From 5 bu. 0 pk. 0 qt. 0 pt. take 1 bu. 0 pk. 0 qt. 1 pt. Bu.? {3} Pk.? {3} Qt.? {7} Pt.? {1}
q: From 17 gal. 2 qt. 1 pt. take 13 gal. 3 qt. 0 pt. Gal.? {3} Qt.? {3} Pt.? {1}
q: From 43 gal. 1 qt. 1 pt. 2 gi. take 23 gal. 3 qt. 1 pt. 3 gi. Gal.? {19} Qt.? {1} Pt.? {1} Gi.? {3}
q: From 14 T. 12 cwt. 50 lb. take 10 T. 13 cwt. 75 lb. T.? {3} Cwt.? {18} Lb.? {75}
q: From 16 T. 7 cwt. 18 lb. 14 oz. take 5 T. 6 cwt. 75 lb. 15 oz. T.? {11} Cwt.? {0} Lb.? {42} Oz.? {15}
q: From 18 mi. 198 rd. take 11 mi. 236 rd. Mi.? {6} Rd.? {282}
q: From 4 yd. 1 ft. 10 in. take 2 yd. 1 ft. 11 in. Yd.? {1} Ft.? {2} In.? {11}
q: From 327 A. 148 sq. rd. take 77 A. 155 sq. rd. A.? {249} Sq. rd.? {153}
q: From 19 sq. yd. 6 sq. ft. 72 sq. in. take 16 sq. yd. 6 sq. ft. 112 sq. in. Sq. yd.? {2} Sq. ft.? {8} Sq. in.? {104}
q: From 28 C. 116 cu. ft. take 19 C. 119 cu. ft. C.? {8} Cu. ft.? {125}
q: From 18 cu. yd. 7 cu. ft. 927 cu. in. take 9 cu. yd. 15 cu. ft. 928 cu. in. Cu. yd.? {8} Cu. ft.? {18} Cu. in.? {1727}
q: From 18 hr. 43 min. 27 sec. take 17 hr. 51 min. 45 sec. Min.? {51} Sec.? {42}
q: From 245 da. 17 hr. 40 min. 37 sec. take 190 da. 11 hr. 44 min. 42 sec. Da.? {55} Hr.? {5} Min.? {55} Sec.? {55}
q: If 2 bu. 1 pk. 1 qt. be taken from a bag containing 4 bushels of hickory nuts, what quantity will remain? Bu.? {1} Pk.? {2} Qt.? {7}
q: From 100 bu. take 24 bu. 1 pt. Bu.? {75} Pk.? {3} Qt.? {7} Pt.? {1}
q: I bought 46 lb. 4 oz. of rice: after selling 19 lb. 8 oz., how much remained? Lb.? {26} Oz.? {12}
q: A wagon loaded with hay weighs 32 cwt. 66 lb.; the wagon alone weighs 8 cwt. 67 lb.: what is the weight of the hay? Cwt.? {23} Lb.? {99}
q: It is 24899 miles round the earth: after a man has traveled 100 mi. 41 rd. what distance will remain? Mi.? {24798} Rd.? {279}
q: I had a farm containing 146 A. 80 sq. rd. of land. I gave my son 86 A. 94 sq. rd.: how much was left? A.? {59} Sq. rd.? {146}
q: From 8 C. 50 cu. ft. of wood, 3 C. 75 cu. ft. are taken: how much is left? C.? {4} Cu. ft.? {103}
q: A cask of wine containing 63 gal. leaked; only 51 gal. 1 qt. 2 gi. remained: how much was lost? Gal.? {11} Qt.? {2} Pt.? {1} Gi.? {2}
q: From 5 da. 10 hr. 27 min. 15 sec. take 2 da. 4 hr. 13 min. 29 sec. Da.? {3} Hr.? {6} Min.? {13} Sec.? {46}
note: In finding the time between two dates, consider 30 days 1 month, and 12 months 1 year.
q: A note, dated April 14, 1875, was paid February 12, 1877: find the time between these dates. Years? {1} Months? {9} Days? {28}
sol: February is the 2d month and April the 4th; from 1877 yr. 2 mo. 12 da. subtract 1875 yr. 4 mo. 14 da.: 1 yr. 9 mo. 28 da.
q: The Independence of the United States was declared July 4, 1776: what length of time had elapsed on the 1st of September, 1876? Years? {100} Months? {1} Days? {27}
q: The first crusade ended July 15, 1099; the third crusade, July 12, 1191: find the difference of time between these dates. Years? {91} Months? {11} Days? {27}
q: Magna Charta was signed June 15, 1215; Mary, Queen of Scots, was beheaded February 8, 1587: find the difference of time between these dates. Years? {371} Months? {7} Days? {23}
q: The battle of Hastings was fought Oct. 14, 1066; William, Prince of Orange, landed at Tor Bay Nov. 5, 1688: what was the difference of time between the two events? Years? {622} Months? {0} Days? {21}
q: The battle of Austerlitz was fought December 2, 1805; the battle of Waterloo, June 18, 1815: find the difference of time. Years? {9} Months? {6} Days? {16}
q: Find the number of days from May 10 to Oct. 21. {164}
sol: Of May, there remain 31 - 10 = 21 days; 30 in June, 31 in July, 31 in August, 30 in September, and 21 in October: 164.
q: Find the number of days from March 17 to September 12. {179}
q: A note dated April 18, 1877, is due June 20, 1877: how many days does it run? {63}
q: A note dated Sept. 5, 1877, is due Dec. 7, 1877: how many days does it run? {93}
q: Find the number of days from Oct. 12, 1877, to May 25, 1878. {225}
q: Find the number of days from Aug. 20, 1875, to March 8, 1876. {201}
== 23 | Compound Numbers | Multiplying and dividing compound numbers
note: Multiply the lowest denomination first, and carry to the next. To divide, begin with the highest denomination; reduce each remainder to the next lower, and add it before dividing.
q: A farmer takes to mill 5 bags of wheat, each containing 2 bu. 3 pk.: how much had he in all? Bu.? {13} Pk.? {3}
q: Multiply 2 bu. 1 pk. 1 qt. 1 pt. by 6. Bu.? {13} Pk.? {3} Qt.? {1}
q: Multiply 2 bu. 2 pk. 2 qt. by 9. Bu.? {23} Pk.? {0} Qt.? {2}
q: If 4 bu. 3 pk. 3 qt. 1 pt. of wheat make 1 bl. of flour, how much will make 12 bl.? Bu.? {58} Pk.? {1} Qt.? {2}
q: Find the weight of 9 hogsheads of sugar, each weighing 8 cwt. 62 lb. T.? {3} Cwt.? {17} Lb.? {58}
q: How much hay in 7 loads, each weighing 10 cwt. 89 lb.? T.? {3} Cwt.? {16} Lb.? {23}
q: If a ship sail 208 mi. 176 rd. a day, how far will it sail in 15 days? Mi.? {3128} Rd.? {80}
q: Multiply 23 cu. yd. 9 cu. ft. 228 cu. in. by 12. Cu. yd.? {280} Cu. ft.? {1} Cu. in.? {1008}
q: Multiply 16 cwt. 74 lb. by 119. T.? {99} Cwt.? {12} Lb.? {6}
q: Multiply 47 gal. 3 qt. 1 pt. by 59. Gal.? {2824} Qt.? {2} Pt.? {1}
q: A travels 27 mi. 155 rd. in 1 day: how far will he travel in one month of 31 days? Mi.? {852} Rd.? {5}
q: In 17 piles of wood, each pile containing 7 C. 98 cu. ft.: what is the quantity of wood? C.? {132} Cu. ft.? {2}
q: Multiply 2 wk. 4 da. 13 hr. 48 min. 39 sec. by 75. Wk.? {199} Da.? {0} Hr.? {3} Min.? {48} Sec.? {45}
q: A planter sold 75 hogsheads of sugar, each weighing 10 cwt. 84 lb., to a refiner, for 6 ct. a pound. The refiner sold the sugar for 8 ct. a pound; how much did he gain? {1626 @ 75 * 1084 * .02}
q: A cotton-factor sold 425 bales of cotton, each weighing 4 cwt. 85 lb., for 13 ct. a pound. He paid $24735 for the cotton: how much did he gain? {2061.25 @ 425 * 485 * .13 - 24735}
q: Divide 14 bu. 2 pk. 1 qt. by 3. Bu.? {4} Pk.? {3} Qt.? {3}
q: Divide 33 bu. 2 pk. 6 qt. by 7. Bu.? {4} Pk.? {3} Qt.? {2}
q: Divide 17 da. 12 hr. 56 min. 15 sec. by 5. Da.? {3} Hr.? {12} Min.? {11} Sec.? {15}
q: Divide 67 bu. 3 pk. 4 qt. 1 pt. by 5. Bu.? {13} Pk.? {2} Qt.? {2} Pt.? {1}
q: Eleven casks of sugar weigh 35 cwt. 44 lb. 12 oz.: what is the average weight of each? Cwt.? {3} Lb.? {22} Oz.? {4}
q: I traveled 39 mi. 288 rd. in 7 hr.: at what rate per hour did I travel? Mi.? {5} Rd.? {224}
q: Divide 69 A. 64 sq. rd. by 16. A.? {4} Sq. rd.? {54}
q: Divide 490 bu. 2 pk. 4 qt. by 100. Bu.? {4} Pk.? {3} Qt.? {5}
q: Divide 265 lb. 10 oz. by 50. Lb.? {5} Oz.? {5}
q: Divide 45 T. 18 cwt. by 17. T.? {2} Cwt.? {14}
q: Divide 114 da. 22 hr. 45 min. 18 sec. by 54. Da.? {2} Hr.? {3} Min.? {5} Sec.? {17}
q: Divide 10 cwt. 27 lb. 13 oz. by 23. Lb.? {44} Oz.? {11}
q: Divide 309 bu. 2 pk. 2 qt. by 78. Bu.? {3} Pk.? {3} Qt.? {7}
q: Divide 127 gal. 3 qt. 1 pt. 3 gi. by 63. Gal.? {2} Qt.? {0} Pt.? {0} Gi.? {1}
q: Divide 788 mi. 169 rd. by 319. Mi.? {2} Rd.? {151}
q: A farmer has two farms, one of 104 A. 117 sq. rd.; the other, 87 A. 78 sq. rd. He reserves 40 A. 40 sq. rd., and divides the remainder equally among his 3 sons: what is the share of each son? A.? {50} Sq. rd.? {105}
q: A farmer's crop consisted of 5000 bu. 3 pk. of corn one year, and 7245 bu. 2 pk. the year following. He sold 8022 bu. 1 pk. and placed the remainder in 8 cribs, each crib containing an equal amount: how many bushels in each crib? Bu.? {528}
q: A speculator bought 6 adjoining pieces of land, each containing 4 A. 80 sq. rd. He divided the whole into 54 lots, and sold them at $5 a sq. rd.: how much did he get for each lot? {400 @ 6 * 720 * 5 / 54}
q: Add 35 lb. 9 oz., 75 lb. 14 oz., 85 lb. 15 oz.; from the sum take 186 lb. 14 oz.; multiply the remainder by 8; divide the product by 64: what is the result? Lb.? {1} Oz.? {5}
== 24 | Compound Numbers | Longitude and time
note: 15 degrees of longitude make 1 hour of time; 15 minutes of longitude, 1 minute of time; 15 seconds of longitude, 1 second of time. A place east has later time; west, earlier.
q: How many hr. min. and sec. of time correspond to 18 degrees 25 minutes 30 seconds of longitude? Hr.? {1} Min.? {13} Sec.? {42}
sol: Degrees of longitude divided by 15 give hours of time; minutes give minutes; seconds give seconds.
q: The difference of longitude between two places is 30 degrees: what is their difference of time? Hr.? {2}
q: The difference of longitude between two places is 71 degrees 4 minutes: what is the difference of time? Hr.? {4} Min.? {44} Sec.? {16}
q: The difference of longitude between New York and Cincinnati is 10 degrees 35 minutes: what is the difference of time? Min.? {42} Sec.? {20}
q: The difference of time between Cincinnati and Philadelphia is 37 min. 20 sec.: what is the difference of longitude? Deg.? {9} Min.? {20}
q: The difference of time between New York and St. Louis is 1 hr. 4 min. 56 sec.: what is the difference of longitude? Deg.? {16} Min.? {14}
q: The difference of time between London and Washington is 5 hr. 8 min. 4 sec.: what is the difference of longitude? Deg.? {77} Min.? {1}
q: When it is noon at Cincinnati, what is the time at Philadelphia (37 min. 20 sec. later)? Hour? {12} Min.? {37} Sec.? {20}
q: When it is 11 o'clock A. M. at New York, what is the time in longitude 30 degrees east of New York? O'clock? {1}
sol: 30 degrees east is 2 hours later: 1 o'clock P. M.
q: When 12 o'clock (noon) at Philadelphia, what is the time at Cincinnati? Hour? {11} Min.? {22} Sec.? {40}
q: When it is 11 o'clock A. M. at New York, what is the time at St. Louis? Hour? {9} Min.? {55} Sec.? {4}
q: Wheeling, W. Va., is in longitude 80 degrees 42 minutes west: the mouth of the Columbia river, in longitude 124 degrees west: when it is 1 o'clock P. M. at Wheeling, what is the time at the mouth of Columbia river? Hour? {10} Min.? {6} Sec.? {48}
sol: The difference of longitude is 43 degrees 18 minutes, which is 2 hr. 53 min. 12 sec. of time; the river is west, so its time is earlier: 10 hr. 6 min. 48 sec. A. M.
== 25 | Factoring | Prime factors
note: A prime number has no factors. Two is a factor of every even number; three, when the sum of the digits is 3 or a multiple of 3; five, when the unit figure is 0 or 5. Type the prime factors from the smallest.
q: Find the prime numbers from 1 to 100, by writing the odd numbers and erasing every 3d after 3, every 5th after 5, every 7th after 7. {open}
sol: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
q: Resolve 30 into its prime factors. {2} times {3} times {5}
sol: 2 is a factor of 30; dividing 30 by 2, the quotient is 15. 3 is a factor of 15; dividing 15 by 3, the quotient is 5, a prime number. Then 2, 3 and 5 are the prime factors of 30.
q: Resolve 4 into its prime factors. {2} times {2}
q: Resolve 8 into its prime factors. {2} times {2} times {2}
q: Resolve 9 into its prime factors. {3} times {3}
q: Resolve 10 into its prime factors. {2} times {5}
q: Resolve 12 into its prime factors. {2} times {2} times {3}
q: Resolve 14 into its prime factors. {2} times {7}
q: Resolve 15 into its prime factors. {3} times {5}
q: Resolve 16 into its prime factors. {2} times {2} times {2} times {2}
q: Resolve 18 into its prime factors. {2} times {3} times {3}
q: Resolve 20 into its prime factors. {2} times {2} times {5}
q: Resolve 22 into its prime factors. {2} times {11}
q: Resolve 24 into its prime factors. {2} times {2} times {2} times {3}
q: Resolve 25 into its prime factors. {5} times {5}
q: Resolve 26 into its prime factors. {2} times {13}
q: Resolve 27 into its prime factors. {3} times {3} times {3}
q: Resolve 28 into its prime factors. {2} times {2} times {7}
q: Resolve 32 into its prime factors. {2} times {2} times {2} times {2} times {2}
q: Resolve 34 into its prime factors. {2} times {17}
q: Resolve 35 into its prime factors. {5} times {7}
q: Resolve 36 into its prime factors. {2} times {2} times {3} times {3}
q: Resolve 38 into its prime factors. {2} times {19}
q: Resolve 39 into its prime factors. {3} times {13}
q: Resolve 40 into its prime factors. {2} times {2} times {2} times {5}
q: Resolve 42 into its prime factors. {2} times {3} times {7}
q: Resolve 44 into its prime factors. {2} times {2} times {11}
q: Resolve 45 into its prime factors. {3} times {3} times {5}
q: Resolve 46 into its prime factors. {2} times {23}
q: Resolve 48 into its prime factors. {2} times {2} times {2} times {2} times {3}
q: Resolve 49 into its prime factors. {7} times {7}
q: Resolve 50 into its prime factors. {2} times {5} times {5}
q: Resolve 70 into its prime factors. {2} times {5} times {7}
q: Resolve 77 into its prime factors. {7} times {11}
q: Resolve 91 into its prime factors. {7} times {13}
q: Resolve 105 into its prime factors. {3} times {5} times {7}
q: Resolve 119 into its prime factors. {7} times {17}
q: Resolve 133 into its prime factors. {7} times {19}
q: Resolve 154 into its prime factors. {2} times {7} times {11}
q: Resolve 210 into its prime factors. {2} times {3} times {5} times {7}
q: Resolve 231 into its prime factors. {3} times {7} times {11}
q: Resolve 330 into its prime factors. {2} times {3} times {5} times {11}
q: Resolve 462 into its prime factors. {2} times {3} times {7} times {11}
q: Resolve 2310 into its prime factors. {2} times {3} times {5} times {7} times {11}
== 26 | Factoring | Common factors and the greatest common divisor
note: Write the numbers in a line; divide by any prime number that will exactly divide all of them, and so on. The divisors are the common factors; their product is the G. C. D.
q: What prime factors are common to 30 and 42? {2} times {3}
sol: 2 is a prime factor of both 30 and 42; dividing by 2, the quotients are 15 and 21. 3 is a factor of both; dividing by 3, the quotients 5 and 7 are prime to each other. Then 2 and 3 are the common factors.
q: What prime factors are common to 60 and 90? {2} times {3} times {5}
q: What prime factors are common to 56 and 88? {2} times {2} times {2}
q: What prime factors are common to 72 and 84? {2} times {2} times {3}
q: What prime factors are common to 54 and 90? {2} times {3} times {3}
q: What prime factors are common to 81 and 108? {3} times {3} times {3}
q: What prime factors are common to 80 and 100? {2} times {2} times {5}
q: What prime factors are common to 84 and 126? {2} times {3} times {7}
q: What prime factors are common to 52, 68 and 76? {2} times {2}
q: What prime factors are common to 66, 78 and 102? {2} times {3}
q: What prime factors are common to 63, 99 and 117? {3} times {3}
q: What prime factors are common to 50, 70 and 110? {2} times {5}
q: What prime factors are common to 45, 75 and 105? {3} times {5}
q: What prime factors are common to 75, 125 and 175? {5} times {5}
q: What prime factors are common to 42, 70 and 98? {2} times {7}
q: What prime factors are common to 33, 55, 77 and 121? {11}
q: What prime factors are common to 39, 65, 91 and 104? {13}
q: What prime factors are common to 34, 51, 85 and 102? {17}
q: What prime factors are common to 38, 57, 95 and 114? {19}
q: What prime factors are common to 46, 69, 92 and 115? {23}
q: Find the G. C. D. of 30 and 42. {6}
sol: First method: the prime factors common to 30 and 42 are 2 and 3; their product is 6. Second method: dividing 42 by 30, the remainder is 12; dividing 30 by 12, the remainder is 6; dividing 12 by 6, the remainder is 0. Then 6 is the G. C. D.
q: Find the greatest common divisor of 16, 24 and 40. {8}
q: Find the greatest common divisor of 24, 36 and 60. {12}
q: Find the greatest common divisor of 36, 54 and 90. {18}
q: Find the greatest common divisor of 40, 60 and 100. {20}
q: Find the greatest common divisor of 54, 81 and 108. {27}
q: Find the greatest common divisor of 60, 90 and 120. {30}
q: Find the greatest common divisor of 32, 48, 80 and 112. {16}
q: Find the greatest common divisor of 48, 72, 96 and 120. {24}
q: Find the greatest common divisor of 72, 108, 144 and 180. {36}
q: Find the greatest common divisor of 62 and 93. {31}
q: Find the greatest common divisor of 78 and 130. {26}
q: Find the greatest common divisor of 161 and 253. {23}
q: Find the greatest common divisor of 247 and 323. {19}
q: Find the greatest common divisor of 391 and 697. {17}
q: Find the greatest common divisor of 2145 and 3471. {39}
q: Find the greatest common divisor of 16571 and 38363. {227}
q: Find the greatest common divisor of 72, 120 and 132. {12}
q: Find the greatest common divisor of 75, 125 and 165. {5}
q: Find the greatest common divisor of 64, 96, 112 and 136. {8}
== 27 | Factoring | The least common multiple
note: Divide by any prime number that will exactly divide two or more of the numbers; bring down the quotients and undivided numbers, and continue till they are prime to each other. The product of the divisors and the last line is the L. C. M.
q: Find the L. C. M. of 4, 6, 9 and 12. {36}
sol: The prime factors of 4 are 2 and 2; of 6, 2 and 3; of 9, 3 and 3; of 12, 2, 2 and 3. The L. C. M. contains 2, 2, 3, 3, and no other factor: 2 x 2 x 3 x 3 = 36.
q: Find the least common multiple of 4, 6 and 8. {24}
q: Find the least common multiple of 6, 9 and 12. {36}
q: Find the least common multiple of 4, 8 and 10. {40}
q: Find the least common multiple of 6, 10 and 15. {30}
q: Find the least common multiple of 6, 8, 9 and 12. {72}
q: Find the least common multiple of 10, 12, 15 and 20. {60}
q: Find the least common multiple of 9, 15, 18 and 30. {90}
q: Find the least common multiple of 12, 18, 27 and 36. {108}
q: Find the least common multiple of 15, 25, 30 and 50. {150}
q: Find the least common multiple of 14, 21, 30 and 35. {210}
q: Find the least common multiple of 15, 20, 21 and 28. {420}
q: Find the least common multiple of 20, 24, 28 and 30. {840}
q: Find the least common multiple of 45, 30, 35 and 42. {630}
q: Find the least common multiple of 36, 40, 45 and 50. {1800}
q: Find the least common multiple of 42, 56 and 63. {504}
q: Find the least common multiple of 78, 104 and 117. {936}
q: Find the least common multiple of 125, 150 and 200. {3000}
q: Find the least common multiple of 10, 24, 25, 32 and 45. {7200}
q: Find the least common multiple of 2, 3, 4, 5, 6, 7, 8 and 9. {2520}
q: Find the least common multiple of 16, 27, 42 and 108. {3024}
q: Find the least common multiple of 13, 29, 52 and 87. {4524}
q: Find the least common multiple of 120, 360, 144, 720 and 72. {720}
== 28 | Factoring | Cancellation
note: Cancel the factors common to both the dividend and divisor; then divide the product of the factors left in the dividend by the product of those left in the divisor.
q: I bought 3 oranges at 5 cents each, and paid for them with pears at 3 cents each: how many pears did it take? {5}
sol: Indicate the multiplication and division, 5 x 3 divided by 3; cancel the multiplier 3 and the divisor 3: the result is 5.
q: If I buy 10 pears at 3 cents each, and pay for them with oranges at 5 cents each: how many oranges will it take? {6}
sol: 5 is a factor of 10; cancel the divisor 5 and the factor 5 in 10, leaving 2. The product of the remaining factors, 2 x 3, is 6.
q: Divide 15 x 21 by 14 x 10. {~15 * 21 / (14 * 10)}
sol: Cancel 5 from 15 and 10, leaving 3 and 2; cancel 7 from 21 and 14, leaving 3 and 2. 9 divided by 4 is 2 1/4.
q: How many barrels of molasses, at $13 a barrel, will pay for 13 barrels of flour, at $4 a barrel? {~13 * 4 / 13}
q: Multiply 17 by 18, and divide by 6. {~17 * 18 / 6}
q: In 15 times 8, how many times 4? {~15 * 8 / 4}
q: In 24 times 4, how many times 8? {~24 * 4 / 8}
q: In 37 times 15, how many times 5? {~37 * 15 / 5}
q: Multiply 36 by 40, and divide the product by 30 multiplied by 8. {~36 * 40 / (30 * 8)}
q: In 36 times 5, how many times 15? {~36 * 5 / 15}
q: Multiply 42, 25, and 18 together, and divide the product by 21 x 15. {~42 * 25 * 18 / (21 * 15)}
q: I sold 23 sheep, at $10 each, and was paid in hogs, at $5 each: how many did I receive? {~23 * 10 / 5}
q: How many yards of flannel, at 35 cents a yard, will pay for 15 yards of calico, at 14 cents? {~15 * 14 / 35}
q: What is the quotient of 21 x 11 x 6 x 26, divided by 13 x 3 x 14 x 2? {~21 * 11 * 6 * 26 / (13 * 3 * 14 * 2)}
q: The factors of a dividend are 21, 15, 33, 8, 14, and 17; the divisors, 20, 34, 22 and 27: required the quotient. {~21 * 15 * 33 * 8 * 14 * 17 / (20 * 34 * 22 * 27)}
q: I bought 21 kegs of nails of 95 pounds each, at 6 cents a pound; paid for them with pieces of muslin of 35 yards each, at 9 cents a yard: how many pieces of muslin did I give? {~21 * 95 * 6 / (35 * 9)}
q: What is the quotient of 35 x 39 x 40 divided by 26 x 30 x 42? {~35 * 39 * 40 / (26 * 30 * 42)}
q: What is the quotient of 26 x 33 x 35 divided by 4 x 9 x 25? {~26 * 33 * 35 / (4 * 9 * 25)}
q: What is the quotient of 6 x 9 x 15 x 21 divided by 4 x 6 x 10 x 14? {~6 * 9 * 15 * 21 / (4 * 6 * 10 * 14)}
q: What is the quotient of 21 x 24 x 28 x 35 divided by 14 x 18 x 20 x 25? {~21 * 24 * 28 * 35 / (14 * 18 * 20 * 25)}
== 29 | Fractions | Fractions and improper fractions
note: The denominator shows into how many equal parts the unit is divided; the numerator, how many of the parts are taken. Type a fraction with the / key: 7/2. A mixed number: 3, and, 1/2.
q: When a unit is divided into four equal parts, what are three of the parts called? {=3/4}
q: Is 5/4 a proper or an improper fraction? Say it out loud. {open}
sol: Improper: its value is greater than 1, because the numerator is greater than the denominator.
q: In 3 apples, how many halves? {6}
sol: In 1 apple there are 2 halves; then, in 3 apples there are 3 x 2 halves = 6 halves.
q: In 4 apples, how many halves? {8}
q: In 2 apples, how many thirds? {6}
q: In 3 apples, how many fourths? {12}
q: In 4 apples, how many fifths? {20}
q: In 6 inches, how many tenths? {60}
q: In 8 feet, how many twelfths? {96}
q: Reduce 4 to sevenths. {=28/7}
q: Reduce 8 to ninths. {=72/9}
q: Reduce 19 to thirteenths. {=247/13}
q: Reduce 25 to twentieths. {=500/20}
q: Reduce 37 to twenty-thirds. {=851/23}
q: In 3 1/2 apples, how many halves? {7}
sol: In 3 apples there are 3 x 2 halves = 6 halves. 6 halves and 1 half are 7 halves.
q: In 4 1/2 apples, how many halves? {9}
q: In 2 1/3 apples, how many thirds? {7}
q: In 2 2/3 apples, how many thirds? {8}
q: In 5 3/4 dollars, how many fourths? {23}
q: Reduce 8 3/4 to an improper fraction. {=35/4}
q: Reduce 12 3/5 to an improper fraction. {=63/5}
q: Reduce 15 5/6 to an improper fraction. {=95/6}
q: Reduce 26 13/24 to an improper fraction. {=637/24}
q: Reduce 3 17/55 to an improper fraction. {=182/55}
q: Reduce 46 5/8 to an improper fraction. {=373/8}
q: Reduce 21 117/583 to an improper fraction. {=12360/583}
q: Reduce 1 999/1000 to an improper fraction. {=1999/1000}
q: Reduce 14 6/71 to an improper fraction. {=1000/71}
q: Reduce 10 1/111 to an improper fraction. {=1111/111}
q: In 6/2 of an apple, how many apples? {3}
sol: There are 2 halves in 1 apple; then, in 6 halves, there are 6 divided by 2 = 3 apples.
q: In 9/4 of a dollar, how many dollars? {~9/4}
sol: There are 4 fourths in 1 dollar; then, in 9 fourths, there are 9 divided by 4 = 2 1/4 dollars.
q: In 6/3 of an apple, how many apples? {~6/3}
q: In 12/4 of an apple, how many apples? {~12/4}
q: In 15/4 of a dollar, how many dollars? {~15/4}
q: In 17/5 of a dollar, how many dollars? {~17/5}
q: In 19/7 of a bushel, how many bushels? {~19/7}
q: In 23/10 of a dollar, how many dollars? {~23/10}
q: In 25/3 of an ounce, how many ounces? {~25/3}
q: In 53/4 of a dollar, how many dollars? {~53/4}
q: Reduce 75/4 to a mixed number. {~75/4}
q: Reduce 125/8 to a mixed number. {~125/8}
q: Reduce 611/24 to a mixed number. {~611/24}
q: Reduce 3000/75 to an integer. {~3000/75}
q: Reduce 775/25 to an integer. {~775/25}
q: Reduce 171/12 to a mixed number. {~171/12}
q: Reduce 509/11 to a mixed number. {~509/11}
q: Reduce 6437/298 to a mixed number. {~6437/298}
q: Reduce 7536/125 to a mixed number. {~7536/125}
q: Reduce 3781/19 to an integer. {~3781/19}
q: Reduce 1325/101 to a mixed number. {~1325/101}
== 30 | Fractions | Higher and lowest terms
note: Multiplying or dividing both terms of a fraction by the same number does not change its value. A fraction is in its lowest terms when the terms are prime to each other.
q: Reduce 4/5 to thirtieths. {=24/30}
sol: 30 divided by 5 is 6. Multiplying both terms of 4/5 by 6, the result is 24/30.
q: Reduce 1/2 to fourths. {=2/4}
q: Reduce 2/3 to sixths. {=4/6}
q: Reduce 3/4 to twelfths. {=9/12}
q: Reduce 5/6 to twenty-fourths. {=20/24}
q: Reduce 5/7 to twenty-eighths. {=20/28}
q: Reduce 4/21 to eighty-fourths. {=16/84}
q: Reduce 7/8 to seventy-seconds. {=63/72}
q: Reduce 3/5 to sixtieths. {=36/60}
q: Reduce 9/10 to hundredths. {=90/100}
q: Reduce 9/20 to a fraction whose denominator is 720. {=324/720}
q: Reduce 13/14 to a fraction whose denominator is 2016. {=1872/2016}
q: Reduce 22/43 to a fraction whose denominator is 1935. {=990/1935}
q: Reduce 35/41 to a fraction whose denominator is 8118. {=6930/8118}
q: Reduce 16/17 to a fraction whose denominator is 5134. {=4832/5134}
q: Reduce 77/81 to a fraction whose denominator is 23328. {=22176/23328}
q: Reduce 13/21 to a fraction whose denominator is 2541. {=1573/2541}
q: Reduce 24/30 to its lowest terms. {=4/5}
sol: 2 is a common factor of 24 and 30: dividing both terms by 2, the result is 12/15. 3 is a common factor of 12 and 15: dividing by 3, the result is 4/5. Or divide both terms by their greatest common divisor, 6.
q: Reduce 18/30 to its lowest terms. {=3/5}
q: Reduce 60/90 to its lowest terms. {=2/3}
q: Reduce 12/18 to its lowest terms. {=2/3}
q: Reduce 30/45 to its lowest terms. {=2/3}
q: Reduce 60/150 to its lowest terms. {=2/5}
q: Reduce 42/70 to its lowest terms. {=3/5}
q: Reduce 96/112 to its lowest terms. {=6/7}
q: Reduce 60/125 to its lowest terms. {=12/25}
q: Reduce 126/198 to its lowest terms. {=7/11}
q: Reduce 182/196 to its lowest terms. {=13/14}
q: Reduce 615/915 to its lowest terms. {=41/61}
q: Reduce 873/1067 to its lowest terms. {=9/11}
q: Reduce 777/1998 to its lowest terms. {=7/18}
q: Reduce 909/2323 to its lowest terms. {=9/23}
q: Reduce 391/667 to its lowest terms. {=17/29}
q: Reduce 585/1287 to its lowest terms. {=5/11}
q: Reduce 796/14129 to its lowest terms. {=4/71}
q: Reduce 1457/5921 to its lowest terms. {=47/191}
q: Reduce 6465/7335 to its lowest terms. {=431/489}
== 31 | Fractions | The least common denominator
note: Find the L. C. M. of the denominators for the least common denominator; then reduce each fraction to another having this denominator. Mixed numbers must first be made improper, and each fraction put in its lowest terms.
q: Reduce 3/4, 5/6, 8/9, and 11/12 to their least common denominator. {=27/36} {=30/36} {=32/36} {=33/36}
sol: The least common multiple of the denominators 4, 6, 9 and 12 is 36. Each fraction must be reduced to thirty-sixths: 3/4 = 27/36, 5/6 = 30/36, 8/9 = 32/36, and 11/12 = 33/36.
q: Reduce 1/2, 2/3, 3/4 to their least common denominator. {=6/12} {=8/12} {=9/12}
q: Reduce 2/3, 5/6, 7/9 to their least common denominator. {=12/18} {=15/18} {=14/18}
q: Reduce 1/2, 3/4, 4/5 to their least common denominator. {=10/20} {=15/20} {=16/20}
q: Reduce 3/8, 4/5, 9/10 to their least common denominator. {=15/40} {=32/40} {=36/40}
q: Reduce 2/3, 3/4, 7/8 to their least common denominator. {=16/24} {=18/24} {=21/24}
q: Reduce 3/4, 5/8, 5/9 to their least common denominator. {=54/72} {=45/72} {=40/72}
q: Reduce 1/2, 3/4, 7/8 to their least common denominator. {=4/8} {=6/8} {=7/8}
q: Reduce 2/3, 5/6, 7/12 to their least common denominator. {=8/12} {=10/12} {=7/12}
q: Reduce 3/4, 5/8, 11/16 to their least common denominator. {=12/16} {=10/16} {=11/16}
q: Reduce 1/2, 2/3, 3/4, 4/5 to their least common denominator. {=30/60} {=40/60} {=45/60} {=48/60}
q: Reduce 2/3, 2/5, 3/7, 5/8 to their least common denominator. {=560/840} {=336/840} {=360/840} {=525/840}
q: Reduce 2/7, 5/14, 9/21, 11/28 to their least common denominator. {=8/28} {=10/28} {=12/28} {=11/28}
q: Reduce 2/5, 3/4, 6/9, 15/18 to their least common denominator. {=24/60} {=45/60} {=40/60} {=50/60}
q: Reduce 2, 3/4, 5/9, 7/12 to their least common denominator. {=72/36} {=27/36} {=20/36} {=21/36}
q: Reduce 2 2/3, 3/5, 4, 5 5/6 to their least common denominator. {=80/30} {=18/30} {=120/30} {=175/30}
q: Reduce 2 1/2, 3 1/8, 4 1/4, 5 to their least common denominator. {=20/8} {=25/8} {=34/8} {=40/8}
q: Reduce 7/16, 11/18, 17/24, 19/36, 25/48 to their least common denominator. {=63/144} {=88/144} {=102/144} {=76/144} {=75/144}
q: Reduce 4/7, 3/10, 5/12, 17/35, 4/63, 15/28 to their least common denominator. {=720/1260} {=378/1260} {=525/1260} {=612/1260} {=80/1260} {=675/1260}
q: Reduce 3/5, 7/10, 6/25, 11/30, 13/45, 23/60 to their least common denominator. {=540/900} {=630/900} {=216/900} {=330/900} {=260/900} {=345/900}
== 32 | Fractions | Adding and subtracting fractions
note: Reduce the fractions to a common denominator; then add or subtract the numerators, and write the common denominator under the result. Give the answer in its lowest terms.
q: Add 1/5, 2/5, and 3/5. {~6/5}
sol: The sum of 1 fifth, 2 fifths, and 3 fifths is 6 fifths; 6/5 are equal to 1 1/5.
q: Add 1/4, 2/4, 3/4. {~1/4 + 2/4 + 3/4}
q: Add 1/5, 2/5, 3/5, 4/5. {~1/5 + 2/5 + 3/5 + 4/5}
q: Add 1/7, 2/7, 3/7, 6/7. {~1/7 + 2/7 + 3/7 + 6/7}
q: Add 4/9, 5/9, 7/9, 8/9. {~4/9 + 5/9 + 7/9 + 8/9}
q: Add 3/11, 7/11, 8/11, 10/11. {~3/11 + 7/11 + 8/11 + 10/11}
q: Add 5/13, 8/13, 9/13, 11/13. {~5/13 + 8/13 + 9/13 + 11/13}
q: Add 7/15, 8/15, 11/15, 13/15. {~7/15 + 8/15 + 11/15 + 13/15}
q: Add 9/20, 11/20, 13/20, 17/20. {~9/20 + 11/20 + 13/20 + 17/20}
q: Add 12/25, 16/25, 18/25, 24/25. {~12/25 + 16/25 + 18/25 + 24/25}
q: Add 5/6, 8/9, and 11/12. {~5/6 + 8/9 + 11/12}
sol: Reducing to a common denominator, 5/6 = 30/36, 8/9 = 32/36, and 11/12 = 33/36; the sum is 95/36, equal to 2 23/36.
q: Add 1/2 and 1/3. {~1/2 + 1/3}
q: Add 1/3 and 1/4. {~1/3 + 1/4}
q: Add 1/2 and 3/5. {~1/2 + 3/5}
q: Add 5/6 and 1/9. {~5/6 + 1/9}
q: Add 3/4 and 5/6. {~3/4 + 5/6}
q: Add 7/8 and 11/12. {~7/8 + 11/12}
q: Add 2 1/2 and 3 2/3. {~2_1/2 + 3_2/3}
sol: For 2 1/2 and 3 2/3: the sum of 1/2 and 2/3 is 7/6 = 1 1/6; write the 1/6 and carry the 1 to the integers: 1, 3 and 2 are 6. The sum is 6 1/6.
q: Add 2/3, 3/4, 5/6. {~2/3 + 3/4 + 5/6}
q: Add 1/4, 7/8, 11/12. {~1/4 + 7/8 + 11/12}
q: Add 1/8, 1/9, 2/11. {~1/8 + 1/9 + 2/11}
q: Add 4/5, 7 1/2, 8 3/4. {~4/5 + 7_1/2 + 8_3/4}
q: Add 1/12, 1/13, 1/14, 1/15. {~1/12 + 1/13 + 1/14 + 1/15}
q: Add 13/18, 8/15, 11/20, 13/30. {~13/18 + 8/15 + 11/20 + 13/30}
q: Add 7/12, 2 5/6, 3 3/8, 3 4/9. {~7/12 + 2_5/6 + 3_3/8 + 3_4/9}
q: Add 16 2/3, 12 3/4, 8 3/5, 2 1/4. {~16_2/3 + 12_3/4 + 8_3/5 + 2_1/4}
q: Add 1/2, 1/3, 1/4, 1/5, 1/6. {~1/2 + 1/3 + 1/4 + 1/5 + 1/6}
q: Add 2/5, 7/16, 7/50, 3/140, 3/2800. {~2/5 + 7/16 + 7/50 + 3/140 + 3/2800}
q: Add 1/20, 7/16, 11/12, 1 2/15, 2 11/18. {~1/20 + 7/16 + 11/12 + 1_2/15 + 2_11/18}
q: Add 2/3, 2 1/2, 4 1/5, 6 1/3, 8 1/4. {~2/3 + 2_1/2 + 4_1/5 + 6_1/3 + 8_1/4}
q: Add 1 1/3, 4 2/7, 2 1/5, 2 1/21. {~1_1/3 + 4_2/7 + 2_1/5 + 2_1/21}
q: From 5/7 subtract 2/7. {~5/7 - 2/7}
sol: 2 sevenths from 5 sevenths leaves 3 sevenths.
q: From 3/4 subtract 1/4. {~3/4 - 1/4}
q: From 7/8 subtract 5/8. {~7/8 - 5/8}
q: From 5/9 subtract 2/9. {~5/9 - 2/9}
q: From 8/10 subtract 3/10. {~8/10 - 3/10}
q: From 3 1/8 subtract 1 3/8. {~3_1/8 - 1_3/8}
sol: For 3 1/8 less 1 3/8: 3/8 can not be taken from 1/8, so borrow 1 from 3; 1 equals 8/8, and 8/8 and 1/8 are 9/8; 3/8 from 9/8 leaves 6/8 = 3/4. 2 from 3 leaves 1: 1 3/4.
q: From 4 1/4 subtract 2 3/4. {~4_1/4 - 2_3/4}
q: From 8 1/3 subtract 3 2/3. {~8_1/3 - 3_2/3}
q: From 23 7/20 subtract 17 11/20. {~23_7/20 - 17_11/20}
q: From 9/10 subtract 5/6. {~9/10 - 5/6}
sol: Reducing to a common denominator, 5/6 = 25/30 and 9/10 = 27/30; 25/30 from 27/30 leaves 2/30 = 1/15.
q: From 1/2 subtract 1/3. {~1/2 - 1/3}
q: From 1/3 subtract 1/4. {~1/3 - 1/4}
q: From 3/4 subtract 2/3. {~3/4 - 2/3}
q: From 4/5 subtract 1/2. {~4/5 - 1/2}
q: From 5/6 subtract 3/10. {~5/6 - 3/10}
q: From 5/6 subtract 3/8. {~5/6 - 3/8}
q: From 5/9 subtract 1/6. {~5/9 - 1/6}
q: From 4/15 subtract 1/10. {~4/15 - 1/10}
q: From 16/21 subtract 5/14. {~16/21 - 5/14}
q: From 3 1/2 subtract 1 2/3. {~3_1/2 - 1_2/3}
q: From 5 subtract 3/8. {~5 - 3/8}
q: From 5 2/3 subtract 4 1/2. {~5_2/3 - 4_1/2}
q: From 7 2/3 subtract 4 3/4. {~7_2/3 - 4_3/4}
q: From 14 1/4 subtract 12 2/3. {~14_1/4 - 12_2/3}
q: From 5 3/14 subtract 2 10/21. {~5_3/14 - 2_10/21}
q: From 4 1/24 subtract 3 1/16. {~4_1/24 - 3_1/16}
q: From 56 1/3 subtract 42 1/4. {~56_1/3 - 42_1/4}
q: From 60 4/5 subtract 41 3/10. {~60_4/5 - 41_3/10}
q: From 97 1/2 subtract 48 5/6. {~97_1/2 - 48_5/6}
== 33 | Fractions | Multiplying fractions
note: Multiply the numerators together for the numerator, and the denominators for the denominator. Make mixed numbers improper first, and cancel common factors where you can.
q: If 1 apple cost 4/5 of a cent, what will 3 apples cost? Cents? {~3 * 4/5}
sol: They will cost 3 times 4/5 of a cent = 12/5 of a cent, which equals 2 2/5.
q: At 12 ct. a yard, what will 2/5 of a yard of ribbon cost? Cents? {~12 * 2/5}
q: What will 4/7 of a yard of cloth cost, at 3/5 of a dollar per yard? Dollars? {~4/7 * 3/5}
sol: 1/7 of a yard will cost 1/7 of 3/5 = 3/35 of a dollar; then, 4/7 of a yard will cost 4 times 3/35 = 12/35 of a dollar.
q: Multiply 2/3 by 4/5. {~2/3 * 4/5}
q: Multiply 3/4 by 3. {~3/4 * 3}
q: Multiply 8 by 2/3. {~8 * 2/3}
q: Multiply 3/4 by 5/7. {~3/4 * 5/7}
q: Multiply 2/3 by 4. {~2/3 * 4}
q: Multiply 5 by 3/4. {~5 * 3/4}
q: Multiply 8/9 by 3/4. {~8/9 * 3/4}
q: Multiply 2/3 by 6. {~2/3 * 6}
q: Multiply 20 by 3/4. {~20 * 3/4}
q: Multiply 8/13 by 11/16. {~8/13 * 11/16}
q: Multiply 3/5 by 10. {~3/5 * 10}
q: Multiply 12 by 2/3. {~12 * 2/3}
q: Multiply 9/13 by 3/7. {~9/13 * 3/7}
q: Multiply 3/7 by 6. {~3/7 * 6}
q: Multiply 7 by 2/3. {~7 * 2/3}
q: Multiply 2 1/4 by 3 1/2. {~2_1/4 * 3_1/2}
q: Multiply 18 3/4 by 8. {~18_3/4 * 8}
q: Multiply 8 by 3 2/3. {~8 * 3_2/3}
q: Multiply 2 1/2 by 2 1/2. {~2_1/2 * 2_1/2}
q: Multiply 10 7/9 by 7. {~10_7/9 * 7}
q: Multiply 25 by 8 3/5. {~25 * 8_3/5}
q: Multiply 9/10 by 17 3/11. {~9/10 * 17_3/11}
q: Multiply 10 5/6 by 9. {~10_5/6 * 9}
q: Multiply 64 by 8 7/8. {~64 * 8_7/8}
q: Multiply 8 3/4 by 3/7. {~8_3/4 * 3/7}
q: Multiply together 5/12, 9/16, and 2 2/11. {~5/12 * 9/16 * 2_2/11}
q: Multiply together 2 1/16, 8/11, and 1 7/9. {~2_1/16 * 8/11 * 1_7/9}
q: Multiply together 6 3/4, 2 8/9, and 21. {~6_3/4 * 2_8/9 * 21}
q: Multiply together 2 1/2, 3 2/3, 4 3/4, and 1 1/7. {~2_1/2 * 3_2/3 * 4_3/4 * 1_1/7}
q: Multiply together 2 1/5, 2 3/26, 3 1/4, and 1 5/11. {~2_1/5 * 2_3/26 * 3_1/4 * 1_5/11}
q: Multiply together 7/8, 3/10, 8/9, 5/6, 2/3, and 6/7. {~7/8 * 3/10 * 8/9 * 5/6 * 2/3 * 6/7}
q: Multiply together 1/4, 9/7, 4/5, 7/9, 5/4, 2/3, and 6. {~1/4 * 9/7 * 4/5 * 7/9 * 5/4 * 2/3 * 6}
q: Multiply together 6/7, 4/9, 1 3/4, 1/6, 3/4, 5/6, 2/5, and 20. {~6/7 * 4/9 * 1_3/4 * 1/6 * 3/4 * 5/6 * 2/5 * 20}
q: Multiply together 2 1/2, 6 2/5, 3 1/4, 7/13, 2, and 3/7. {~2_1/2 * 6_2/5 * 3_1/4 * 7/13 * 2 * 3/7}
q: What is 2/3 of 2? {~2/3 * 2}
sol: 1/3 of 2 is 2/3; then, 2/3 of 2 is 2 times 2/3 = 4/3 = 1 1/3.
q: What is 3/4 of 5? {~3/4 * 5}
q: What is 2/5 of 7? {~2/5 * 7}
q: What is 4/5 of 10? {~4/5 * 10}
q: What is 5/6 of 12? {~5/6 * 12}
q: What is 5/6 of 15? {~5/6 * 15}
q: What is 8/9 of 21? {~8/9 * 21}
q: What is 7/10 of 25? {~7/10 * 25}
q: What is 5/12 of 27? {~5/12 * 27}
q: What is 7/12 of 28? {~7/12 * 28}
q: Reduce 2/3 of 4/5 to a simple fraction. {~2/3 * 4/5}
sol: Multiplying 2/3 by 4/5, the result is 8/15.
q: Reduce 2/3 of 5/7 to a simple fraction. {~2/3 * 5/7}
q: Reduce 3/5 of 7/8 to a simple fraction. {~3/5 * 7/8}
q: Reduce 1/2 of 3/5 of 2 3/4 to a simple fraction. {~1/2 * 3/5 * 2_3/4}
q: Reduce 7/11 of 2/3 to a simple fraction. {~7/11 * 2/3}
q: Reduce 3/4 of 5/8 to a simple fraction. {~3/4 * 5/8}
q: Reduce 2/3 of 5/7 of 1 4/9 to a simple fraction. {~2/3 * 5/7 * 1_4/9}
q: Reduce 2/3 of 3/4 of 4/5 to a simple fraction. {~2/3 * 3/4 * 4/5}
q: Reduce 1/3 of 3/4 of 5/6 to a simple fraction. {~1/3 * 3/4 * 5/6}
q: Reduce 3/5 of 5/7 of 7/8 to a simple fraction. {~3/5 * 5/7 * 7/8}
q: Reduce 3/5 of 4/9 of 7/12 of 18/35 to a simple fraction. {~3/5 * 4/9 * 7/12 * 18/35}
q: Reduce 1/3 of 3/4 of 4/9 to a simple fraction. {~1/3 * 3/4 * 4/9}
q: Reduce 1/9 of 3/4 of 1 1/3 to a simple fraction. {~1/9 * 3/4 * 1_1/3}
q: Reduce 3/5 of 6/7 of 1 17/18 to an integer. {~3/5 * 6/7 * 1_17/18}
q: Reduce 3/7 of 2 2/3 of 1 3/4 to an integer. {~3/7 * 2_2/3 * 1_3/4}
q: Reduce 9/13 of 7/18 of 1 6/7 to a simple fraction. {~9/13 * 7/18 * 1_6/7}
q: Reduce 1/2 of 4/5 of 1/8 of 5 to a simple fraction. {~1/2 * 4/5 * 1/8 * 5}
q: Reduce 1/2 of 2/3 of 3/4 of 4/5 of 5/8 of 5/9 of 9/10 to a simple fraction. {~1/2 * 2/3 * 3/4 * 4/5 * 5/8 * 5/9 * 9/10}
note: What will be the cost? Answer in the money named in the question.
q: What is the cost of 2 1/3 lb. of meat, at 13 1/5 ct. a lb.? Cents? {~2_1/3 * 13_1/5}
q: What is the cost of 3 yd. linen, at $2/3 a yd.? {~3 * 2/3} Of 5 yd.? {~5 * 2/3} Of 7 yd.? {~7 * 2/3} Of 6 1/2 yd.? {~6_1/2 * 2/3} Of 5 3/4 yd.? {~5_3/4 * 2/3}
q: What is the cost of 3 1/3 lb. of rice, at 4 4/5 ct. a lb.? Cents? {~3_1/3 * 4_4/5}
q: What is the cost of 3 1/5 tons of iron, at $18 3/4 per T.? {~3_1/5 * 18_3/4}
q: What is the cost of 1 2/3 yd. of muslin, at $3/20 per yd.? {~1_2/3 * 3/20}
q: What is the cost of 2 1/2 lb. of tea, at $4/5 per lb.? {~2_1/2 * 4/5}
q: What is the cost of 5 5/9 cords of wood, at $1 1/5 per C.? {~5_5/9 * 1_1/5}
q: At the rate of 5 1/2 miles an hour, how far will a man travel in 7 3/4 hours? Miles? {~5_1/2 * 7_3/4}
q: I own 2/3 of a steamboat, and sell 3/5 of my share: what part of the boat do I sell? {~2/3 * 3/5}
q: At $6 3/4 per yard, what cost 2/9 of a piece of cloth containing 5 1/2 yards? {~6_3/4 * 2/9 * 5_1/2}
q: 3/7 of 5/9 of 16 1/2 x 2/3 of 7/8 of 15 = what? {~3/7 * 5/9 * 16_1/2 * 2/3 * 7/8 * 15}
q: What is the sum of 2/3 + 3/4 and 2/3 x 3/4? {~2/3 + 3/4 + 2/3 * 3/4}
== 34 | Fractions | Dividing fractions
note: Multiply the dividend by the divisor with its terms inverted. Make integers and mixed numbers into fractions first.
q: If 3 yards of ribbon cost 6/7 of a dollar, what will 1 yard cost? {~6/7 / 3}
sol: 1 yard will cost 1/3 of 6/7 = 2/7 of a dollar.
q: At 2 dollars a yard, what part of a yard of cloth can be bought for 3/5 of a dollar? {~3/5 / 2}
q: At 2/3 of a cent for 1 apple, how many can be bought for 4 cents? {~4 / (2/3)}
sol: For 1/3 of a cent 1/2 an apple can be bought, and for 1 cent 3 times 1/2 = 3/2 of an apple; then, for 4 cents, 4 times 3/2 = 6 apples.
q: At 2/3 of a cent for 1 apple, how many apples can be bought for 3/4 of a cent? {~3/4 / (2/3)}
q: Divide 3/4 by 4/5. {~3/4 / (4/5)}
sol: Multiply the dividend by the divisor with its terms inverted: 3/4 x 5/4 = 15/16.
q: If 4 yards of muslin cost 8/9 of a dollar, what will 1 yard cost? {~8/9 / 4}
q: At 1/2 a cent each, how many apples can be bought for 3 cents? {6}
q: At 1/5 of a dollar per yard, how many yards of muslin can be bought for $9/10? {~9/10 / (1/5)}
q: If 1 orange cost 3 cents, what part of an orange could be purchased for 1/2 a cent? {~1/2 / 3}
q: At 3/4 of a dollar per yard, how many yards of cloth can you buy for 6 dollars? {8}
q: At 1/5 of a dollar per yard, how many yards of ribbon can be purchased for 3/4 of a dollar? {~3/4 / (1/5)}
q: If 7 pounds of rice cost 14/25 of a dollar, what will 1 pound cost? {~14/25 / 7}
q: Divide 4 1/2 by 1 1/3. {~4_1/2 / 1_1/3}
sol: Reducing 4 1/2 and 1 1/3 to improper fractions, we have 9/2 and 4/3. Dividing 9/2 by 4/3, that is 9/2 x 3/4, the result is 27/8 = 3 3/8.
q: Divide 2 2/5 by 6. {~2_2/5 / (6)}
q: Divide 22 by 5 1/2. {~22 / (5_1/2)}
q: Divide 2 1/2 by 1/16. {~2_1/2 / (1/16)}
q: Divide 4 4/5 by 8. {~4_4/5 / (8)}
q: Divide 6 by 2 2/5. {~6 / (2_2/5)}
q: Divide 4 3/4 by 5 1/8. {~4_3/4 / (5_1/8)}
q: Divide 12 4/7 by 11. {~12_4/7 / (11)}
q: Divide 30 by 3 3/4. {~30 / (3_3/4)}
q: Divide 2 1/4 by 7 1/2. {~2_1/4 / (7_1/2)}
q: Divide 3 2/3 by 7. {~3_2/3 / (7)}
q: Divide 50 by 4 3/7. {~50 / (4_3/7)}
q: Divide 1/2 by 1/50. {~1/2 / (1/50)}
q: Divide 47 2/5 by 15. {~47_2/5 / (15)}
q: Divide 56 by 5 4/9. {~56 / (5_4/9)}
q: Divide 14/15 by 21. {~14/15 / (21)}
q: Divide 130 2/3 by 18. {~130_2/3 / (18)}
q: Divide 1/2 of 2/3 by 3/4 of 4/5. {~(1/2 * 2/3) / (3/4 * 4/5)}
q: Divide 3/5 of 8/9 by 6/7 of 3/4. {~(3/5 * 8/9) / (6/7 * 3/4)}
q: Divide 1/3 of 5 1/8 by 3/4 of 17 1/2. {~(1/3 * 5_1/8) / (3/4 * 17_1/2)}
q: Divide 5/18 of 2/5 of 12 3/10 by 1/5 of 8 1/5. {~(5/18 * 2/5 * 12_3/10) / (1/5 * 8_1/5)}
q: Divide 2/7 of 7/8 by 3/4 of 1/3 of 5. {~(2/7 * 7/8) / (3/4 * 1/3 * 5)}
q: Divide 5/18 of 2/5 of 12 3/10 by 1/5 of 4 1/10 of 20. {~(5/18 * 2/5 * 12_3/10) / (1/5 * 4_1/10 * 20)}
sol: For a compound divisor, invert the terms of each of its fractions, as in the case of the divisor being a simple fraction.
q: 1 is what part of 2? {~1/2}
sol: 1 is 1/2 of 2; for 1/2 of 2 is 2/2, or 1.
q: 2 is what part of 3? {~2/3}
q: 1/2 is what part of 3? {~1/2 / 3}
sol: 1 is 1/3 of 3; then, 1/2 is 1/2 of 1/3 = 1/6 of 3.
q: 2/3 is what part of 3/4? {~2/3 / (3/4)}
q: 3 is what part of 4? {~3 / (4)}
q: 3/4 is what part of 5? {~3/4 / (5)}
q: 1/4 is what part of 1/2? {~1/4 / (1/2)}
q: 2/3 is what part of 5/6? {~2/3 / (5/6)}
q: 3 3/4 is what part of 5? {~3_3/4 / (5)}
q: 5/6 is what part of 8/9? {~5/6 / (8/9)}
q: 8 5/9 is what part of 11? {~8_5/9 / (11)}
q: 21/32 is what part of 35/48? {~21/32 / (35/48)}
note: A complex fraction is read as its top divided by its bottom: reduce both to improper fractions and divide.
q: Reduce 1 1/4 over 2 1/3 (1 1/4 divided by 2 1/3) to a simple fraction. {~1_1/4 / (2_1/3)}
sol: Reducing 1 1/4 and 2 1/3 to improper fractions, we have 5/4 and 7/3. Dividing 5/4 by 7/3, the result is 15/28.
q: Reduce 6/7 over 11/5 to a simple fraction. {~6/7 / (11/5)}
q: Reduce 2/3 over 5 to a simple fraction. {~2/3 / (5)}
q: Reduce 2 over 3 2/3 to a simple fraction. {~2 / (3_2/3)}
q: Reduce 3 1/8 over 4 5/7 to a simple fraction. {~3_1/8 / (4_5/7)}
q: Reduce 2 1/3 over 4 1/2 to a simple fraction. {~2_1/3 / (4_1/2)}
q: Reduce 3 3/4 over 5 5/8 to a simple fraction. {~3_3/4 / (5_5/8)}
q: Reduce 9 7/9 over 2 1/27 to a mixed number. {~9_7/9 / (2_1/27)}
q: Reduce 8 3/4 over 5 5/8 to a mixed number. {~8_3/4 / (5_5/8)}
q: Reduce 7 5/6 over 8 9/11 to a simple fraction. {~7_5/6 / (8_9/11)}
== 35 | Fractions | Examples in fractions
q: At 1/2 a dollar per yard, how many yards of silk can be bought for $3 1/4? {~3_1/4 / (1/2)}
q: At 3/5 of a dollar per pound, how many pounds of tea can be purchased for $2 3/10? {~2_3/10 / (3/5)}
q: At 3 3/4 dollars per yard for cloth, how many yards can be purchased with $42 1/2? {~42_1/2 / (3_3/4)}
q: By what must 3/8 be multiplied that the product may be 10? {~10 / (3/8)}
q: Divide 3 3/7 by 3/7 of 1 1/2. {~3_3/7 / (3/7 * 1_1/2)}
q: Divide 4/11 of 27 1/2 by 3/10 of 21 1/4. {~4/11 * 27_1/2 / (3/10 * 21_1/4)}
q: Multiply 1 1/2 over 2 1/3 by 3/7 over 4 1/2. {~1_1/2 / (2_1/3) * (3/7 / (4_1/2))}
q: Multiply 7 8/15 over 9 5/12, of 2 1/9 over 3 2/15, by 1 1/4 over 5/6. {~(7_8/15 / (9_5/12)) * (2_1/9 / (3_2/15)) * (1_1/4 / (5/6))}
q: Divide 1 1/2 over 2/3 by 2 2/5 over 2 1/6. {~(1_1/2 / (2/3)) / (2_2/5 / (2_1/6))}
q: Divide 1 2/3 over 2 1/2 by 5 1/7 over 84 6/7. {~(1_2/3 / (2_1/2)) / (5_1/7 / (84_6/7))}
note: Money with a fraction of a cent: $32.31 1/4 is 32 dollars 31 1/4 cents. Type it as a decimal (32.3125) or as dollars and a fraction.
q: Add $16 1/16; $9 1/8; $5 7/16; $2 13/16. {~16_1/16 + 9_1/8 + 5_7/16 + 2_13/16}
q: I paid for books $9 1/8; for paper, $4 7/16; for a slate, $3/8; for pens, $1 5/8: what amount did I expend? {~9_1/8 + 4_7/16 + 3/8 + 1_5/8}
q: Having $50 1/4, I paid a bill of $27 3/16: how much had I left? {~50_1/4 - 27_3/16}
q: From $32.31 1/4 take $15.12 1/2. {~32.3125 - 15.125}
q: From $5.81 1/4 take $1.18 3/4. {~5.8125 - 1.1875}
q: Find the cost of 9 yd. of muslin, at 12 1/2 ct. a yd. Dollars? {~9 * .125}
q: Find the cost of 21 lb. of sugar, at 6 1/4 ct. a lb. Dollars? {~21 * .0625}
q: Find the cost of 15 yd. of cloth, at $3.18 3/4 per yd. {~15 * 3.1875}
q: Find the cost of 5 1/2 yd. of linen, at $0.62 1/2 per yd. {~5_1/2 * .625}
q: Find the cost of 12 1/2 yd. of ribbon, at 18 3/4 ct. per yd. Dollars? {~12_1/2 * .1875}
q: Find the cost of 13 1/2 yd. of calico, at 16 2/3 ct. per yd. Dollars? {~13_1/2 * 16_2/3 / 100}
q: Find the cost of 10 1/4 yd. of cloth, at $3.37 1/2 a yard. {~10_1/4 * 3.375}
q: Find the cost of 17 2/3 dozen books, at $3.75 per dozen. {~17_2/3 * 3.75}
q: At 18 3/4 ct. per yard, how many yards of muslin can be purchased for $2.25? {~2.25 / .1875}
q: At 37 1/2 ct. per bushel, how many bushels of barley can you buy for $5.81 1/4? {~5.8125 / .375}
q: If 5 yards of cloth cost $11.56 1/4, what cost one yard? {~11.5625 / 5}
q: Seven men share $31.06 1/4 equally: what is the share of each man? {~31.0625 / 7}
== 36 | Fractions | Fractional compound numbers
note: A rod is 5 1/2 yards, or 16 1/2 feet. To reduce a fraction to a lower denomination, multiply; to a higher, divide.
q: Reduce 5 mi. to inches. {316800}
q: Reduce 2 mi. 2 rd. 2 ft. to feet. {10595}
q: Reduce 20 yd. to rods. Rods? {3} Yards? {~7/2}
q: Reduce 15875 ft. to miles. Mi.? {3} Rd.? {2} Ft.? {2}
q: Reduce 142634 in. to miles. Mi.? {2} Rd.? {80} Yd.? {2} Ft.? {0} In.? {2}
q: How many steps, of 2 ft. 8 in. each, will a man take in walking 2 miles? {3960 @ 126720 / 32}
q: How many revolutions will a wheel, of 9 ft. 2 in. circumference, make in running 65 miles? {37440 @ 65 * 63360 / 110}
q: Reduce 1 A. 136 sq. rd. 25 sq. yd. to square yards. {8979 @ 296 * 30_1/4 + 25}
q: Reduce 7506 sq. yd. to acres. A.? {1} Sq. rd.? {88} Sq. yd.? {4}
q: Reduce 5 chains 15 links to inches. (A link is 7 92/100 inches.) {~515 * 7_92/100}
q: How many acres in a field 40 1/2 rd. long and 32 rd. wide? A.? {8} Sq. rd.? {16}
q: Reduce 4 years to hours (a year being 365 1/4 days). {35064}
q: Reduce 914092 hr. to centuries (a year being 365 1/4 days). Cen.? {1} Yr.? {4} Da.? {101} Hr.? {4}
q: In what time will a body move from the earth to the moon, at the rate of 31 miles per day, the distance being 238545 miles? (A year is 365 1/4 days.) Years? {21} Days? {~99/4}
q: Reduce 1/24 of a peck to the fraction of a pint. {~1/24 * 8 * 2}
sol: To reduce 1/24 of a peck to the fraction of a pint, multiply by 8 and by 2. The result is 2/3 of a pint.
q: Reduce 1/40 bu. to the fraction of a quart. {~32/40}
q: Reduce 1/28 lb. to the fraction of an ounce. {~16/28}
q: Reduce 1/16 lb. Troy to the fraction of an ounce. {~12/16}
q: Reduce 1/20 rd. to the fraction of a foot. {~16_1/2 / 20}
q: Reduce 7/1280 A. to the fraction of a square rod. {~7 * 160 / 1280}
q: Reduce $3/350 to the fraction of a cent. {~300/350}
q: Reduce 1/1584 da. to the fraction of a minute. {~1440/1584}
q: Reduce 3/320 bu. to the fraction of a pint. {~3 * 64 / 320}
q: Find the value of 2/5 of a day in integers. Hr.? {9} Min.? {36}
sol: To reduce 2/5 of a day to hours, multiply by 24; the result is 9 3/5 hr. To reduce 3/5 of an hour to minutes, multiply by 60; the result is 36 min.
q: Find the value of 4/5 mi. in integers. Rods? {256}
q: Find the value of $3/5 in integers. Cents? {60}
q: Find the value of 2/5 mi. in integers. Rods? {128}
q: Find the value of 4/5 lb. Troy in integers. Oz.? {9} Pwt.? {12}
q: Find the value of 7/16 T. in integers. Cwt.? {8} Lb.? {75}
q: Find the value of 5/8 A. in integers. Sq. rd.? {100}
q: Find the value of 7/8 of 63 gallons of wine in integers. Gal.? {55} Qt.? {0} Pt.? {1}
q: Reduce 2/3 of a pint to the fraction of a peck. {~2/3 / 2 / 8}
sol: To reduce 2/3 of a pint to the fraction of a peck, divide by 2 and by 8. The result is 1/24 of a peck.
q: Reduce 4/5 qt. to the fraction of a bushel. {~4/5 / 32}
q: Reduce 4/5 ft. to the fraction of a rod. {~4/5 / (16_1/2)}
q: Reduce 3/80 oz. to the fraction of a pound. {~3/80 / 16}
q: Reduce 4/9 lb. to the fraction of a ton. {~4/9 / 2000}
q: Reduce 3/5 pt. to the fraction of a bushel. {~3/5 / 64}
q: Reduce 4/7 oz. to the fraction of a hundred-weight. {~4/7 / 1600}
q: Reduce 3/4 in. to the fraction of a rod. {~3/4 / 198}
q: Reduce 8/9 min. to the fraction of a day. {~8/9 / 1440}
q: Reduce 5/112 oz. to the fraction of a hundred-weight. {~5/112 / 1600}
q: 2 ft. 3 in. is what part of a yard? {~27/36}
sol: 2 ft. 3 in. equals 27 in.; 1 yd. equals 36 in. 27 in. are 27/36 of 36 in. 27/36 equals 3/4.
q: 2 ft. 6 in. is what part of 6 ft. 8 in.? {~30/80}
q: 2 pk. 4 qt. is what part of a bushel? {~20/32}
q: What part is 2 yd. 9 in. of 8 yd. 2 ft. 3 in.? {~81/315}
q: What part of a day is 13 hr. 30 min.? {~810/1440}
q: What part of a mile is 145 rd.? {~145/320}
q: What part of a yard is 2 ft. 8 in.? {~32/36}
q: 15 mi. 123 rd. is what part of 35 mi. 287 rd.? {~4923/11487}
q: A man has a farm of 168 A. 28 sq. rd.; if he sell 37 A. 94 sq. rd., what part of his farm will he dispose of? {~6014/26908}
q: What part of a pound is 7 1/9 oz.? {~7_1/9 / 16}
q: 2 qt. 1 1/3 pt. is what part of 1 bu. 1 qt. 1 2/3 pt.? {~5_1/3 / (67_2/3)}
q: 1 yd. 1 ft. 1 9/11 in. is what part of 3 yd. 2 ft. 8 6/7 in.? {~49_9/11 / (140_6/7)}
note: To add or subtract fractional compound numbers, find the value of each fraction in integers, then add or subtract.
q: Add 3/4 yd. and 5/6 ft. Ft.? {3} In.? {1}
sol: 3/4 yd. equals 2 ft. 3 in.; 5/6 ft. equals 10 in.; the sum is 3 ft. 1 in.
q: From 2/9 da. subtract 5/6 hr. Hr.? {4} Min.? {30}
q: Add 2/3 da. and 3/4 hr. Hr.? {16} Min.? {45}
q: Add 1/4 wk. 1/4 da. and 1/4 hr. Da.? {2} Hr.? {0} Min.? {15}
q: Add 2/3 wk. 5/9 da. 2/3 hr. and 2/3 min. Da.? {5} Hr.? {6} Min.? {0} Sec.? {40}
q: Add 11/12 gal. and 1/12 qt. Qt.? {3} Pt.? {1} Gi.? {2}
q: From 7/9 da. subtract 1/18 hr. Hr.? {18} Min.? {36} Sec.? {40}
q: From $5/8 subtract $3/40. Cents? {55}
q: From 3/8 lb. subtract 7/8 oz. Ounces? {~6 - 7/8}
q: From 1/7 da. subtract 6/7 hr. Hr.? {2} Min.? {34} Sec.? {~120/7}
== 37 | Fractions | Promiscuous examples
q: Reduce 32989/56981 to its lowest terms. {~32989/56981}
q: Add 5/14, 8/21, 2 1/2, 3 2/3. {~5/14 + 8/21 + 2_1/2 + 3_2/3}
q: From 3 4/7 subtract 1 4/5. {~3_4/7 - 1_4/5}
q: From 3 5/8 subtract 1/3 of 3 1/2. {~3_5/8 - 1/3 * 3_1/2}
q: Add 5/9 of 7/10 and 2/5 of 7/12. {~5/9 * 7/10 + 2/5 * 7/12}
q: Add 1 3/4 divided by 2 1/2, and 5 1/2 divided by 3 1/8. {~1_3/4 / (2_1/2) + 5_1/2 / (3_1/8)}
q: What number divided by 3/5 will give 10 for a quotient? {6}
q: What number multiplied by 3/5 will give 10 for a product? {~10 / (3/5)}
q: What number is that, from which if you take 3/7 of itself, the remainder will be 16? {28}
sol: If 3/7 be taken away, 4/7 remain; 4/7 of the number is 16; 1/7 is 4; the number is 28.
q: What number is that, to which if you add 3/7 of itself, the sum will be 20? {14}
q: A boat is worth $900; a merchant owns 5/8 of it, and sells 1/3 of his share: what part has he left? {~5/8 * 2/3} What is it worth? {~900 * 5/8 * 2/3}
q: I own 7/12 of a ship, and sell 1/3 of my share for $1944 4/9: what is the whole ship worth? {~1944_4/9 / (7/12 * 1/3)}
q: What part of 3 cents is 2/3 of 2 cents? {~2/3 * 2 / 3}
q: What part of 368 is 176? {~176/368}
q: From 25/37 subtract the sum of 1/3, 1/18, and 13/111. {~25/37 - (1/3 + 1/18 + 13/111)}
q: From 1 subtract 3/10 of 7/12 of 4 9/14. {~1 - 3/10 * 7/12 * 4_9/14}
q: From 2/3 divided by 5/7 subtract 5/8 divided by 10/11. {~2/3 / (5/7) - 5/8 / (10/11)}
q: If I ride 2044 rods in 7/15 of an hour, at that rate how far will I ride in 1 14/15 hr.? Rods? {~2044 / (7/15) * 1_14/15}
q: What part of 1 1/4 feet are 3 1/3 inches? {~3_1/3 / 15}
q: Two men bought a barrel of flour; one paid $3 1/5, and the other $3 2/3: what part of it should each have? First? {~3_1/5 / (3_1/5 + 3_2/3)} Second? {~3_2/3 / (3_1/5 + 3_2/3)}
q: A has $2400; 5/8 of his money, + $500, is 5/4 of B's: what sum has B? {1600 @ (2400 * 5/8 + 500) / (5/4)}
q: John Jones divided his estate among 2 sons and 3 daughters, the latter sharing equally with each other. The younger son received $2200, which was 5/12 of the share of the elder, whose share was 16/35 of the whole estate: find the share of each daughter. {~(2200 * 12/5 * 35/16 - 2200 * 12/5 - 2200) / 3}
== 38 | Practice | Aliquot parts of a dollar
note: An aliquot part is an exact divisor: 12 1/2 ct. is 1/8 of a dollar; 16 2/3 ct., 1/6; 25 ct., 1/4; 33 1/3 ct., 1/3; 37 1/2 ct., 3/8; 62 1/2 ct., 5/8; 75 ct., 3/4. Money with part of a cent may be typed as a decimal ($2.34 3/8 = 2.34375).
q: What is 12 1/2 ct. part of a dollar? {=1/8} 16 2/3 ct.? {=1/6} 33 1/3 ct.? {=1/3} 6 1/4 ct.? {=1/16} 18 3/4 ct.? {=3/16} 87 1/2 ct.? {=7/8}
q: What will 24 yd. of muslin cost at 25 ct. a yd.? {6}
sol: Since 25 ct. is 1/4 of a dollar, the cost will be 1/4 as many dollars as there are yards. 1/4 of $24 is $6.
q: I spent $1.12 1/2 for muslin at 12 1/2 ct. a yd.: how many yd. did I buy? {9}
sol: Since 12 1/2 ct. is 1/8 of a dollar, there will be 8 times as many yards as there are dollars. 8 times 1 1/8 = 9 yd.
q: What cost 12 1/2 yd. of ribbon at 18 3/4 ct. a yd? {~12_1/2 * .1875}
q: Paid $2.25 for muslin at 18 3/4 ct. a yd. how many yd. did I buy? {12 @ 2.25 / .1875}
q: What will 5 1/2 yd. of linen cost at $0.62 1/2 a yd.? {~5_1/2 * .625}
q: Paid $66.25 for books at $3.75 a dozen: how many doz. books did I buy? {~66.25 / 3.75}
q: What will 80 gal. of wine cost at $2.37 1/2 a gal.? {190 @ 80 * 2.375}
q: A number of men divide $39 so that each one receives $4.87 1/2: how many men are there? {8 @ 39 / 4.875}
q: What will 36 barrels of flour cost at $8.33 1/3 a barrel? {300 @ 36 * 8_1/3}
q: How many yd. of cloth at $1.33 1/3 a yd. can be bought for $246.66 2/3? {185 @ 246_2/3 / (1_1/3)}
q: What will 4 A. 60 sq. rd. of land cost at $16.50 an acre? {~4_60/160 * 16.5}
sol: 4 A. cost $16.50 x 4 = $66. 40 sq. rd. = 1/4 A.: 1/4 of $16.50 = $4.12 1/2. 20 sq. rd. cost 1/2 of that, $2.06 1/4. The total is $72.18 3/4.
q: At $18.33 1/3 per acre, how much land can be bought for $229.16 2/3? Acres? {12} Sq. rd.? {80}
q: What will 11 A. 120 sq. rd. of land cost, at $125.60 per acre? {1475.80 @ 11_3/4 * 125.6}
q: At $250 a lot, containing 50 x 150 ft., how much land can be bought for $10000? Lots? {40} Square feet? {300000}
q: What will 83 bu. 3 pk. 2 qt. of grass seed cost, at $6.20 a bu.? {~(83 + 3/4 + 2/32) * 6.2}
q: At $0.75 a bushel, how many bushels can be bought for $167.50? {~167.5 / .75}
q: What will 3 1/2 yd. cost, at $1.75 a yard? {~3_1/2 * 1.75}
q: At $1.50 a yard, how much cloth can be bought for $7.12 1/2? Yards? {~7.125 / 1.5}
q: What will 45 lb. 12 oz. of butter cost, at $0.37 1/2 per pound? {~45_3/4 * .375}
q: At $0.12 1/2 per pound, how much sugar can be bought for $2.93 3/4? Pounds? {~2.9375 / .125}
q: What is the cost of 2 T. 9 cwt. of wool at 37 1/2 ct. a pound? {1837.50 @ 4900 * .375}
q: What is the cost of 100 readers at $3.90 a dozen? {32.50 @ 100 / 12 * 3.9}
q: What is the cost of 3 3/4 dozen knives at $5.40 a dozen? {20.25 @ 3_3/4 * 5.4}
q: A farmer sold 6 1/2 doz. chickens, at $0.33 1/3 apiece, and 37 1/2 lb. butter, at $0.37 1/2 per pound: he received $36 in money, and the remainder in sugar, at $0.12 1/2 per pound: how many pounds of sugar did he get? {~(78 * 1/3 + 37_1/2 * .375 - 36) / .125}
== 39 | Decimals | Writing and reading decimals
note: Tenths stand in the first order right of the point; hundredths, the second; thousandths, the third. Type the point with the . key: two hundred and sixty-five thousandths is .265.
q: Write 1/10 as a decimal. {.1} 3/10? {.3} 1/100? {.01} 7/100? {.07} 1/1000? {.001} 5/1000? {.005}
sol: The number of orders in the decimal is always the same as the number of ciphers in the denominator of the common fraction.
q: Write two hundred and sixty-five thousandths. {.265}
sol: Write the numerator, 265; the figure 5 must stand in the order thousandths; the decimal point is placed before the figure 2.
q: Write two hundred and sixty-five millionths. {.000265}
sol: The figure 5 must stand in the order millionths; ciphers must be written in the orders thousandths, hundredths, and tenths.
q: Write two hundred and sixty-five hundredths. {2.65}
q: Write four hundred and ninety-eight and two hundred and sixty-five millionths. {498.000265}
q: Write: Twenty-six hundredths. {.26}
q: Write: Thirty-five hundredths. {.35}
q: Write: Eighty-seven hundredths. {.87}
q: Write: Four hundred and nineteen hundredths. {4.19}
q: Write: Five thousandths. {.005}
q: Write: Fifty-four thousandths. {.054}
q: Write: Three hundred and four thousandths. {.304}
q: Write: Seven thousand two hundred and ninety-three thousandths. {7.293}
q: Write: Twenty-five and forty-seven thousandths. {25.047}
q: Write: Two hundred and five ten-thousandths. {.0205}
q: Write: Four thousand one hundred and twenty-five ten-thousandths. {.4125}
q: Write: Nine hundred-thousandths. {.00009}
q: Write: Nine hundred thousandths. {.900}
q: Write: Six hundred and five hundred-thousandths. {.00605}
q: Write: Twenty thousand three hundred and four hundred-thousandths. {.20304}
q: Write: Seven millionths. {.000007}
q: Write: Two hundred and three millionths. {.000203}
q: Write: Three hundred thousand and four millionths. {.300004}
q: Write: Twenty-four ten-millionths. {.0000024}
q: Write: Eighty thousand and six ten-millionths. {.0080006}
q: Write: Two hundred millionths. {.000200}
q: Write: Two hundred-millionths. {.00000002}
q: Write: Nine hundred and seven hundred-millionths. {.00000907}
q: Write: Twenty million twenty thousand and three hundred-millionths. {.20020003}
q: Write: One million ten thousand and one hundred millionths. {1.0101}
q: Write: One million ten thousand and one hundred-millionths. {.01010001}
q: Write: One hundred and six and thirty-seven thousandths. {106.037}
q: Write: One thousand and one thousandth. {1000.001}
q: Write: Two hundred and twenty-five thousandths. {.225}
q: Write: Two hundred units and twenty-five thousandths. {200.025}
q: Write: Two thousand nine hundred and twenty-nine millionths. {.002929}
q: Write: Two thousand nine hundred units and twenty-nine millionths. {2900.000029}
q: Write: One million and five billionths. {.001000005}
q: Write: Two hundred and two ten-billionths. {.0000000202}
q: Write: Two hundred units and two ten-billionths. {200.0000000002}
q: Write: Sixty-five and six thousand and five millionths. {65.006005}
q: Change to decimals: 3/10? {.3} 7/10? {.7} 9/100? {.09} 17/100? {.17} 28/100? {.28} 41/100? {.41} 53/100? {.53}
q: Change to decimals: 87/100? {.87} 97/100? {.97} 123/1000? {.123} 289/1000? {.289} 487/1000? {.487} 733/1000? {.733}
q: Change to decimals: 3/1000? {.003} 101/10000? {.0101} 53/1000000? {.000053} 503/10000000? {.0000503}
q: Read out loud: .265. {open}
sol: Two hundred and sixty-five thousandths. Disregarding the decimal point, the number is 265; the right hand order is thousandths.
q: Read out loud: .028; .341; 2.327; 50.005; 184.173. {open}
sol: Twenty-eight thousandths; three hundred and forty-one thousandths; two and three hundred and twenty-seven thousandths; fifty and five thousandths; one hundred and eighty-four and one hundred and seventy-three thousandths.
q: Read out loud: .0003; .0625; .2374; .2006; .0104. {open}
sol: Three ten-thousandths; six hundred and twenty-five ten-thousandths; two thousand three hundred and seventy-four ten-thousandths; two thousand six ten-thousandths; one hundred and four ten-thousandths.
q: Read out loud: 3.0205; 810.2406; 10720.0905. {open}
sol: Three and two hundred and five ten-thousandths; eight hundred and ten and two thousand four hundred and six ten-thousandths; ten thousand seven hundred and twenty and nine hundred and five ten-thousandths.
q: Read out loud: .00004; .00137; .02376; .01007. {open}
sol: Four hundred-thousandths; one hundred and thirty-seven hundred-thousandths; two thousand three hundred and seventy-six hundred-thousandths; one thousand seven hundred-thousandths.
q: Read out loud: .001768; .040035; 70.360004. {open}
sol: One thousand seven hundred and sixty-eight millionths; forty thousand thirty-five millionths; seventy and three hundred and sixty thousand four millionths.
q: Read out loud: .1010101; .00040005; .00100304. {open}
sol: One million ten thousand one hundred and one ten-millionths; forty thousand five hundred-millionths; one hundred thousand three hundred and four hundred-millionths.
q: Read out loud: .31456; .000133; 60.04; 45.1003. {open}
sol: Thirty-one thousand four hundred and fifty-six hundred-thousandths; one hundred and thirty-three millionths; sixty and four hundredths; forty-five and one thousand three ten-thousandths.
q: Read out loud: 357.75; .4928; 5.945; 681.0002. {open}
sol: Three hundred and fifty-seven and seventy-five hundredths; four thousand nine hundred and twenty-eight ten-thousandths; five and nine hundred and forty-five thousandths; six hundred and eighty-one and two ten-thousandths.
q: Read out loud: 70.1200764; 954.203; 38.027. {open}
sol: Seventy and one million two hundred thousand seven hundred and sixty-four ten-millionths; nine hundred and fifty-four and two hundred and three thousandths; thirty-eight and twenty-seven thousandths.
q: Read out loud: 1007.3154; 7496.35491768. {open}
sol: One thousand seven and three thousand one hundred and fifty-four ten-thousandths; seven thousand four hundred and ninety-six and thirty-five million four hundred and ninety-one thousand seven hundred and sixty-eight hundred-millionths.
q: Read out loud: .00715; 3.00005; 28.10065701. {open}
sol: Seven hundred and fifteen hundred-thousandths; three and five hundred-thousandths; twenty-eight and ten million sixty-five thousand seven hundred and one hundred-millionths.
q: Read out loud: 13.0008241094710947. {open}
sol: Thirteen and eight trillion two hundred and forty-one billion ninety-four million seven hundred and ten thousand nine hundred and forty-seven ten-quadrillionths.
q: Reduce .75 to a common fraction. {=3/4}
sol: 75 hundredths written as a common fraction is 75/100; 75/100 reduced to its lowest terms is 3/4.
q: Change to common fractions in lowest terms: .9? {=9/10} .13? {=13/100} .19? {=19/100} .29? {=29/100} .37? {=37/100} .73? {=73/100}
q: Change to common fractions in lowest terms: .91? {=91/100} .347? {=347/1000} .513? {=513/1000} .691? {=691/1000} .851? {=851/1000} .917? {=917/1000}
q: Change to common fractions in lowest terms: .007? {=7/1000} .0207? {=207/10000} .00079? {=79/100000} .001007? {=1007/1000000}
q: Change to common fractions in lowest terms: .3421? {=3421/10000} .03401? {=3401/100000} .0900? {=9/100}
q: Change to common fractions in lowest terms: .001? {=1/1000} .5302? {=2651/5000} .000053? {=53/1000000}
q: Change to common fractions in lowest terms: .6? {=3/5} .25? {=1/4} .375? {=3/8} .035? {=7/200} .5625? {=9/16} .34375? {=11/32} .1484375? {=19/128}
q: Change 1.36 to a mixed number. {=1 9/25} 8.01? {=8 1/100}
q: Express 4.02 as an integer and common fraction. {=4 1/50} 8.415? {=8 83/200}
q: Reduce 3/4 to a decimal. {.75}
sol: Annexing a decimal cipher to 3, it is 3.0; 30 tenths divided by 4 is 7 tenths, and 2 tenths remaining. Annexing a cipher, 20 hundredths divided by 4 is 5 hundredths. The result is .75.
q: Reduce 4/5 to a decimal. {.8}
q: Reduce 5/8 to a decimal. {.625}
q: Reduce 7/25 to a decimal. {.28}
q: Reduce 3/40 to a decimal. {.075}
q: Reduce 15/16 to a decimal. {.9375}
q: Reduce 1/1250 to a decimal. {.0008}
q: Reduce 9/400 to a decimal. {.0225}
q: Reduce 1/256 to a decimal. {.00390625}
q: Reduce 5/6 to a decimal, to four places. {.8333}
q: Reduce 1/11 to a decimal, to four places. {.0909}
q: Reduce 4/33 to a decimal, to four places. {.1212}
note: These three never come out even: the figures repeat without end.
== 40 | Decimals | Adding decimals
note: Write the numbers so that the decimal points stand in a column; add as in simple numbers, and place the point in the sum under the column of points.
q: Add 375.83; 49.627; 5842.1963; 813.9762. [[+ 375.83 49.627 5842.1963 813.9762]] {7081.6295}
q: Add 37.1065; 432.07; 4.20733; 11.706. [[+ 37.1065 432.07 4.20733 11.706]] {?}
q: Add 4 and 4 ten-thousandths; 28 and 35 thousandths; 8 and 7 hundredths; and 9404 hundred-thousandths. [[+ 4.0004 28.035 8.07 .09404]] {?}
q: Find the sum of 3 units and 25 hundredths; 6 units and 4 tenths; and 35 hundredths. [[+ 3.25 6.4 .35]] {?}
q: Add 21.611; 6888.32; 3.4167. [[+ 21.611 6888.32 3.4167]] {?}
q: Add 6.61; 636.1; 6516.14; 67.1234; and 5.1233. [[+ 6.61 636.1 6516.14 67.1234 5.1233]] {?}
q: Add 4 and 8 tenths; 43 and 31 hundredths; 74 and 19 thousandths; 11 and 204 thousandths. [[+ 4.8 43.31 74.019 11.204]] {?}
q: Add 45 and 19 thousandths; 7 and 71 hundred-thousandths; 93 and 4327 ten-thousandths; 6 and 401 ten-thousandths. [[+ 45.019 7.00071 93.4327 6.0401]] {?}
q: Add 432 and 432 thousandths; 61 and 793 ten-thousandths; 100 and 7794 hundred-thousandths; 6.009; 1000 and 1001 ten-thousandths. [[+ 432.432 61.0793 100.07794 6.009 1000.1001]] {?}
q: Add 16 and 41 thousandths; 9 and 94 millionths; 33 and 27 hundredths; 8 and 969 thousandths; 32 and 719906 millionths. [[+ 16.041 9.000094 33.27 8.969 32.719906]] {?}
q: Add 204 and 9 ten-thousandths; 103 and 9 hundred-millionths; 42 and 9099 millionths; 430 and 99 hundredths; 220.0000009. [[+ 204.0009 103.00000009 42.009099 430.99 220.0000009]] {?}
q: Add 35 ten-thousandths; .00035; 35 millionths, and 35 ten-millionths. [[+ .0035 .00035 .000035 .0000035]] {?}
== 41 | Decimals | Subtracting, multiplying and dividing decimals
note: Subtract with the points in a column. In multiplying, point off as many decimal orders as in both factors. In dividing, the quotient has as many orders as the dividend has more than the divisor; annex ciphers to the dividend when needed.
q: From 729.835 subtract 461.5738. [[- 729.835 461.5738]] {268.2612}
sol: The ten-thousandth place in the minuend may be regarded as occupied by a cipher; subtract as in simple numbers, and place the point under the column of points.
q: From 97.5168 subtract 38.25942. [[- 97.5168 38.25942]] {?}
q: From 20.014 subtract 7.0021. [[- 20.014 7.0021]] {?}
q: From 5.03 subtract 2.115. [[- 5.03 2.115]] {?}
q: From 24.0042 subtract 13.7013. [[- 24.0042 13.7013]] {?}
q: From 170.0035 subtract 68.00181. [[- 170.0035 68.00181]] {?}
q: From .0142 subtract .005. [[- .0142 .005]] {?}
q: From .05 subtract .0024. [[- .05 .0024]] {?}
q: From 13.5 subtract 8.037. [[- 13.5 8.037]] {?}
q: From 3 subtract .00003. [[- 3 .00003]] {?}
q: From 29.0029 subtract 19.003. [[- 29.0029 19.003]] {?}
q: From 5 subtract .125. [[- 5 .125]] {?}
q: From 1 thousand subtract 1 ten-thousandth. {999.9999}
q: From 1 subtract 1 millionth. {.999999}
q: From 25 thousandths take 25 millionths. {.024975}
q: Multiply 2.149 by 6.34. [[x 2.149 6.34]] {13.62466}
sol: Multiply as in simple numbers, 2149 by 634. There are three decimal orders in 2.149, and two in 6.34; hence there must be five decimal orders in the product: 13.62466.
q: Multiply .0276 by .035. {.000966}
sol: 276 times 35 is 9660. There must be seven decimal orders in the product; three ciphers must be prefixed: .0009660, which is .000966.
q: Multiply 2.075 by 100. {207.5}
sol: Move the decimal point two places to the right.
q: Multiply 33.21 by 4.41. {146.4561}
q: Multiply 32.16 by 22.5. {723.6}
q: Multiply .125 by 9. {1.125}
q: Multiply .35 by 7. {2.45}
q: Multiply .2 by .8. {.16}
q: Multiply .02 by .4. {.008}
q: Multiply .15 by .7. {.105}
q: Multiply 125.015 by .001. {.125015}
q: Multiply .135 by .005. {.000675}
q: Multiply 1.035 by 17. {17.595}
q: Multiply 19 by .125. {2.375}
q: Multiply 4.5 by 4. {18}
q: Multiply .625 by 64. {40}
q: Multiply 61.76 by .0071. {.438496}
q: Multiply 1.325 by .0716. {.09487}
q: Multiply 4.87 by 10. {48.7}
q: Multiply 5.3 by 100. {530}
q: Multiply 17.62 by 100. {1762}
q: Multiply 1.01 by 10. {10.1}
q: Multiply .0001 by 100. {.01}
q: Multiply .09375 by 1.064. {.09975}
q: Multiply 1 tenth by 1 hundredth. {.001} 1 hundred by 1 ten-thousandth? {.01} 43 thousandths by 21 ten-thousandths? {.0000903} 40000 by 1 millionth? {.04}
q: Divide 2.125 by .5. {4.25}
sol: Divide 2125 by 5. There are three decimal orders in 2.125, and one in .5; hence there must be two decimal orders in the quotient: 4.25.
q: Divide .048 by .006. {8}
q: Divide .3 by .004. {75}
sol: Annex two ciphers to .3, making .300; then 300 divided by 4 is 75, with no decimal orders.
q: Divide 83.1 by 4. {20.775}
q: Divide 2.11 by 3, to three places. {.703}
q: Divide 475.6 by 100. {4.756}
q: Divide 1.125 by .03. {37.5}
q: Divide 86.075 by 27.5. {3.13}
q: Divide 24.73704 by 3.44. {7.191}
q: Divide 206.166492 by 4.123. {50.004}
q: Divide .96 by .24. {4}
q: Divide .0425 by .0085. {5}
q: Divide 21 by .5. {42}
q: Divide 2 by .008. {250}
q: Divide 37.2 by 5. {7.44}
q: Divide 100.8788 by 454. {.2222}
q: Divide .000343 by 3.43. {.0001}
q: Divide 9811.0047 by .108649. {90300}
q: Divide .21318 by .19. {1.122}
q: Divide 102048 by .3189. {320000}
q: Divide .102048 by 3189. {.000032}
q: Divide 9.9 by .0225. {440}
q: Divide 872.6 by 100. {8.726}
q: Divide 4.5 by 1000. {.0045}
q: Divide 400 by 10000. {.04}
q: Divide 1.7 by 64. {.0265625}
q: Divide 1 tenth by 10. {.01} 1 by 1 tenth? {10} 10 by 1 hundredth? {1000}
q: Divide .08 by 80. {.001} 1.5 by 7, to four places? {.2143} 11.1 by 32.76, to four places? {.3388} .0123 by 3.21, to five places? {.00383}
== 42 | Decimals | Decimal compound numbers
note: A decimal is reduced to a lower denomination by multiplying, and to a higher by dividing, just as with common fractions.
q: Reduce .05 gal. to the decimal of a pint. {.4}
sol: To reduce .05 gal. to the decimal of a pint, multiply by 4 and by 2. The result is .4 pint.
q: Reduce .035 pk. to the decimal of a pint. {.56}
q: Reduce .0075 bu. to the decimal of a quart. {.24}
q: Reduce .005 yd. to the decimal of an inch. {.18}
q: Reduce .00546875 A. to the decimal of a square rod. {.875}
q: Find the value of .3125 bu. in integers. Pk.? {1} Qt.? {2}
sol: To reduce .3125 bu. to pecks, multiply by 4; the result is 1.25 pk. To reduce .25 pk. to quarts, multiply by 8; the result is 2 qt.
q: Find the value of .75 yd. in integers. Ft.? {2} In.? {3}
q: Find the value of .3375 A. in integers. Sq. rd.? {54}
q: Find the value of .7 lb. Troy in integers. Oz.? {8} Pwt.? {8}
q: Find the value of .8125 bu. in integers. Pk.? {3} Qt.? {2}
q: Find the value of .44 mi. in integers. Rd.? {140} Yd.? {4} Ft.? {1} In.? {2.4}
q: Find the value of .33625 cwt. in integers. Lb.? {33} Oz.? {10}
q: Reduce .64 pt. to the decimal of a gallon. {.08}
sol: To reduce .64 pt. to the decimal of a gallon, divide by 2 and by 4. The result is .08 gal.
q: Reduce .72 qt. to the decimal of a bushel. {.0225}
q: Reduce .77 yd. to the decimal of a mile. {.0004375}
q: Reduce .25 pt. to the decimal of a gallon. {.03125}
q: Reduce .6 pt. to the decimal of a bushel. {.009375}
q: Reduce .7 rd. to the decimal of a mile. {.0021875}
== 43 | Decimals | Examples in decimals
q: What is the cost of 9 yd. flannel, at $0.40 per yard, and 12 yd., at $0.75 per yard? {12.60 @ 9 * .4 + 12 * .75}
q: What is the cost of 2.3 yd. of ribbon, at $0.45 per yard, and 1.5 yd., at $0.375 per yard? {1.5975 @ 2.3 * .45 + 1.5 * .375}
q: What is the cost of 16.25 yd. of cloth, at $2.6875 per yard? {43.671875 @ 16.25 * 2.6875}
q: At $0.75 per bushel, how much wheat can be bought for $35.25? Bushels? {47 @ 35.25 / .75}
q: At $2.5625 per yard, how much cloth can be bought for $98.40? Yards? {38.4 @ 98.4 / 2.5625}
q: What will 6 cwt. 50 lb. of hops cost at $3.25 per hundred-weight? {21.125 @ 6.5 * 3.25}
q: What will 14 bu. 3 pk. 4 qt. of corn cost, at $0.625 per bushel? {9.296875 @ 14.875 * .625}
q: What will 13 A. 115 sq. rd. of land cost, at $17.28 per acre? {237.06 @ 13.71875 * 17.28}
q: At $0.3125 per bushel, how much corn can be bought for $9.296875? Bu.? {29} Pk.? {3}
q: At $4.32 per acre, how much land can be bought for $59.265? A.? {13} Sq. rd.? {115}
q: If 63 gal. of wine cost $49, what will 464 gal. cost? {~464 * 49 / 63}
q: Add .34 yd., 1.07 ft. and 8.92 in. Ft.? {2} In.? {10}
sol: .34 yd. = 1.02 ft. = 1 ft. .24 in.; 1.07 ft. = 1 ft. .84 in.; adding these and 8.92 in. gives 2 ft. 10 in.
q: Add .625 gal. and .75 qt. Qt.? {3} Pt.? {.5}
q: From 1.53 yd. subtract 2 ft. 3.08 in. Ft.? {2} In.? {4}
q: From .05 yr. subtract .5 hr. (A year is 365.25 days.) Da.? {18} Hr.? {5} Min.? {48}
q: From .41 da. subtract .16 hr. Hr.? {9} Min.? {40} Sec.? {48}
q: Find the value of .3 yr. in integers. (A year is 365.25 days.) Da.? {109} Hr.? {13} Min.? {48}
q: What is the cost of 343 yd. 2 ft. 3 in. of tubing, at $0.16 per yard? {55 @ 343.75 * .16}
q: At $690.35 per mile, what is the cost of a road 17 mi. 135 rd. long? {~(17 + 135/320) * 690.35}
== 44 | Metric System | The metric system
note: Milli is .001; centi, .01; deci, .1; deka, 10; hekto, 100; kilo, 1000; myria, 10000. To reduce to a lower denomination move the point to the right; to a higher, to the left. A meter is 39.37 inches; a kilometer .62137 mile; a liter 1.0567 quarts; a gram 15.432 grains; a kilogram 2.2046 pounds.
q: Reduce 5.638 m. to centimeters. {563.8}
sol: To reduce meters to centimeters, multiply by 100: move the point two orders to the right.
q: Reduce 3642.9 m. to kilometers. {3.6429}
q: Reduce 4.27 Dm. to centimeters. {4270}
q: Reduce 5.6 dm. to hektometers. {.0056}
q: Reduce 30.75 m. to centimeters. {3075} 4.5 Km. to meters? {4500} 75 mm. to meters? {.075}
q: Reduce .025 Dm. to decimeters. {2.5} 36.5 dm. to dekameters? {.365} .4875 Km. to centimeters? {48750}
q: Reduce 2.625 a. (ars) to centars. {262.5} 397.8 a. to hektars? {3.978} 2500 ca. to hektars? {.25} 3.8 a. to square meters? {380}
q: Reduce 2.456 l. to centiliters. {245.6} 873.5 l. to hektoliters? {8.735} 1.83 Hl. to deciliters? {1830} 2400 cl. to dekaliters? {2.4} 1400 l. to cubic meters? {1.4}
q: Reduce 1428.06 g. to kilograms. {1.42806} .28 Kg. to grams? {280} 1713.5 Kg. to metric tons? {1.7135} .00654 Hg. to centigrams? {65.4} 192.7 dg. to dekagrams? {1.927}
q: How many yards, feet, etc., in 4 m.? Yd.? {4} Ft.? {1} In.? {1.48}
sol: In 4 meters there are 4 times 39.37 in., which are 157.48 in.; reduced to higher denominations, 4 yd. 1 ft. 1.48 in.
q: What is the value of 36 lb. in kilograms, to three places? {16.329}
sol: In 36 pounds there are as many kilograms as 2.2046 is contained times in 36, which are 16.329+.
q: What is the value of 20 Km. in miles? {12.4274}
q: How many hektars in 160 acres, to two places? {64.75}
q: What is the value of 49 m. in inches? {1929.13} In rods, yards and inches: Rd.? {9} Yd.? {4} In.? {3.13}
q: What is the value of 15 g. in grains Troy? {231.48} In pennyweights and grains: Pwt.? {9} Gr.? {15.48}
q: How many hektoliters in 42 bu., to one place? {14.8}
q: How many cords in 500 sters? {137.95}
q: How many square yards in a roll of paper 9 m. long and .5 m. wide? {5.382}
q: 32 l. are how many gallons? {8.4536}
q: What is the sum of 127 dl., 4.87 l., 1563 cl., and 234.5 dl.? Liters? {56.65}
q: What will be the cost of 45 Ha. of land, at $3.32 an ar? {14940}
q: A merchant paid $457.92 for cloth, at $3 a meter: how many meters did he buy? {152.64}
q: A block of marble .72 m. long, .48 m. wide, and .5 m. thick cost $.864: what is the cost of the marble per cubic meter? {5}
q: A manufacturer bought 380 sters of wood for $454.10: how much was that a ster? {1.195}
q: How many hektoliters of oats in 4685 sacks, each containing 1.6 Hl.? {7496}
q: I bought 346.75 Kg. of coffee for $194.18: what did I pay per kilogram? {.56}
q: The nickel 5-cent coin weighs 5 g. and is 2 cm. in diameter: what would be the weight of enough of these coins laid in a row, to make a meter in length? Grams? {250}
q: How much lining 1.85 m. wide will it take for a garment made of 6.5 m. of cloth 1.25 m. wide? Meters, to two places? {4.39}
q: How many kilometers from Cincinnati to Dayton, the distance being 60 miles? To two places? {96.56}
q: A map is 29 mm. long and 22.4 mm. wide: what space does it cover? Square millimeters? {649.6}
q: The distance between two towns is 13.24037 Km.: how many steps of .715 m. each, must I take to walk that distance? {18518}
sol: 13240.37 m. divided by .715 m. is 18517 and a little more, which would make 18518 steps.
== 45 | Percentage | The four cases of percentage
note: Any per cent of a number is so many hundredths of it: 5 per cent is 5/100 = 1/20 = .05. The base times the rate is the percentage; the percentage divided by the base is the rate; the percentage divided by the rate is the base.
q: Express 10 per cent as a common fraction. {~10 / 100} As a decimal? {~d 10 / 100}
q: Express 15 per cent as a common fraction. {~15 / 100} As a decimal? {~d 15 / 100}
q: Express 20 per cent as a common fraction. {~20 / 100} As a decimal? {~d 20 / 100}
q: Express 30 per cent as a common fraction. {~30 / 100} As a decimal? {~d 30 / 100}
q: Express 50 per cent as a common fraction. {~50 / 100} As a decimal? {~d 50 / 100}
q: Express 2 1/2 per cent as a common fraction. {~2_1/2 / 100} As a decimal? {~d 2_1/2 / 100}
q: Express 6 1/4 per cent as a common fraction. {~6_1/4 / 100} As a decimal? {~d 6_1/4 / 100}
q: Express 12 1/2 per cent as a common fraction. {~12_1/2 / 100} As a decimal? {~d 12_1/2 / 100}
q: Express 18 3/4 per cent as a common fraction. {~18_3/4 / 100} As a decimal? {~d 18_3/4 / 100}
q: Express 37 1/2 per cent as a common fraction. {~37_1/2 / 100} As a decimal? {~d 37_1/2 / 100}
q: Express 56 1/4 per cent as a common fraction. {~56_1/4 / 100} As a decimal? {~d 56_1/4 / 100}
q: Express 87 1/2 per cent as a common fraction. {~87_1/2 / 100} As a decimal? {~d 87_1/2 / 100}
q: How many per cent are equal to 2/25? {~2/25 * 100}
q: How many per cent are equal to 3/25? {~3/25 * 100}
q: How many per cent are equal to 4/25? {~4/25 * 100}
q: How many per cent are equal to 1/80? {~1/80 * 100}
q: How many per cent are equal to 1/12? {~1/12 * 100}
q: How many per cent are equal to 1/4? {~1/4 * 100}
q: How many per cent are equal to 2/5? {~2/5 * 100}
q: How many per cent are equal to 3/4? {~3/4 * 100}
q: How many per cent are equal to 1/8? {~1/8 * 100}
q: How many per cent are equal to 7/16? {~7/16 * 100}
sol: Common fractions may be reduced to hundredths, and then read as per cent. Thus, 1/6 = 16 2/3 hundredths, or 16 2/3 per cent.
q: What is 25 per cent of 32? {8}
sol: 25 per cent is 1/4. 1/4 of 32 is 8.
q: What is 7 per cent of 162? {11.34}
sol: 7 per cent is .07. Multiplying 162 by .07, the result is 11.34.
q: What is 1 per cent of 278? {~d 1 / 100 * 278}
q: What is 2 per cent of 180? {~d 2 / 100 * 180}
q: What is 3 per cent of 97? {~d 3 / 100 * 97}
q: What is 3 1/3 per cent of 165? {~d 3_1/3 / 100 * 165}
q: What is 3 3/4 per cent of 240? {~d 3_3/4 / 100 * 240}
q: What is 4 per cent of 140? {~d 4 / 100 * 140}
q: What is 5 per cent of 118? {~d 5 / 100 * 118}
q: What is 5 1/3 per cent of 150? {~d 5_1/3 / 100 * 150}
q: What is 6 per cent of 250? {~d 6 / 100 * 250}
q: What is 6 2/3 per cent of 450? {~d 6_2/3 / 100 * 450}
q: What is 8 per cent of 11? {~d 8 / 100 * 11}
q: What is 8 1/3 per cent of 384? {~d 8_1/3 / 100 * 384}
q: What is 10 per cent of 57? {~d 10 / 100 * 57}
q: What is 12 1/2 per cent of 292? {~d 12_1/2 / 100 * 292}
q: What is 15 per cent of 95? {~d 15 / 100 * 95}
q: What is 17 per cent of 53.4? {~d 17 / 100 * 53.4}
q: What is 18 3/4 per cent of 11.2? {~d 18_3/4 / 100 * 11.2}
q: What is 20 per cent of 9.85? {~d 20 / 100 * 9.85}
q: What is 25 per cent of 43? {~d 25 / 100 * 43}
q: What is 33 1/3 per cent of 6.93? {~d 33_1/3 / 100 * 6.93}
q: What is 45 per cent of 5.7? {~d 45 / 100 * 5.7}
q: What is 50 per cent of 38.75? {~d 50 / 100 * 38.75}
q: What is 1/2 per cent of 456? {~d 1/2 / 100 * 456}
q: What is 3/8 per cent of 464? {~d 3/8 / 100 * 464}
q: What is 7/16 per cent of 144? {~d 7/16 / 100 * 144}
q: What is 125 per cent of 36? {~d 125 / 100 * 36}
q: What is 208 per cent of 650? {~d 208 / 100 * 650}
q: What is 450 per cent of 12? {~d 450 / 100 * 12}
q: What is 1000 per cent of 24.75? {~d 1000 / 100 * 24.75}
q: What per cent of 8 is 2? {25}
sol: 2 is 1/4 of 8. 1/4 is 25 per cent.
q: What per cent of 56 is 3.5? {~3.5 / 56 * 100}
sol: Dividing 3.5 by 56, the result is .0625, which is 6 1/4 per cent.
q: What per cent of 15 is 3? {~(3) / (15) * 100}
q: What per cent of 50 is 6? {~(6) / (50) * 100}
q: What per cent of 75 is 4.5? {~(4.5) / (75) * 100}
q: What per cent of 9 is 3? {~(3) / (9) * 100}
q: What per cent of 25 is .25? {~(.25) / (25) * 100}
q: What per cent of 142.6 is 7.13? {~(7.13) / (142.6) * 100}
q: What per cent of 9 is 9? {~(9) / (9) * 100}
q: What per cent of 9 is 13.5? {~(13.5) / (9) * 100}
q: What per cent of 243 is 8.505? {~(8.505) / (243) * 100}
q: What per cent of 2 is .002? {~(.002) / (2) * 100}
q: What per cent of 3532 is 13.245? {~(13.245) / (3532) * 100}
q: What per cent of 4/5 is 3/25? {~(3/25) / (4/5) * 100}
q: What per cent of 2/3 is 2/15? {~(2/15) / (2/3) * 100}
q: What per cent of 16/21 is 2/7? {~(2/7) / (16/21) * 100}
q: What per cent of 11 2/3 is 5 1/4? {~(5_1/4) / (11_2/3) * 100}
q: What per cent of 57 7/9 is 10 5/6? {~(10_5/6) / (57_7/9) * 100}
q: 15 is 25 per cent of what number? {60}
sol: 25 per cent is 1/4. Since 15 is 1/4 of some number, the number is 4 x 15 = 60.
q: 4.93 is 17 per cent of what number? {29}
q: 60 is 20 per cent of what number? {~(60) / (20) * 100}
q: 90 is 75 per cent of what number? {~(90) / (75) * 100}
q: 85 is 125 per cent of what number? {~(85) / (125) * 100}
q: 7.13 is 23 per cent of what number? {~(7.13) / (23) * 100}
q: 20.23 is 34 per cent of what number? {~(20.23) / (34) * 100}
q: 23.5 is 47 per cent of what number? {~(23.5) / (47) * 100}
q: 45 is 1 1/2 per cent of what number? {~(45) / (1_1/2) * 100}
q: 2.25 is 12 1/2 per cent of what number? {~(2.25) / (12_1/2) * 100}
q: 3/4 is 250 per cent of what number? {~(3/4) / (250) * 100}
q: 14 2/7 is 16 2/3 per cent of what number? {~(14_2/7) / (16_2/3) * 100}
q: A number, plus 35 per cent of itself, equals 675: what is the number? {500}
sol: The number plus .35 of itself equals 1.35 of it; then the number is 675 divided by 1.35, or 500.
q: A number, minus 5 per cent of itself, equals 57: what is the number? {60}
sol: 5 per cent is 1/20. The number minus 1/20 of itself equals 19/20 of it; 19/20 of the number is 57, 1/20 of it is 3, and the number is 60.
q: 721 is 3 per cent greater than a certain number; what is the number? {700 @ 721 / 1.03}
q: 68 is 66 per cent less than what number? {200 @ 68 / .34}
q: What number, increased by 25 per cent of itself, amounts to 2125? {1700 @ 2125 / 1.25}
q: What number, diminished by 6 per cent of itself, is equal to 7.52? {8 @ 7.52 / .94}
q: 8250 is 37 1/2 per cent greater than what number? {6000 @ 8250 / 1.375}
q: What fraction, less 10 per cent of itself, equals 3/8? {~3/8 / (9/10)}
q: 6.6 is 20 per cent more than what number? {5.5 @ 6.6 / 1.2}
== 46 | Percentage | Examples in percentage
q: I had $800 in bank and drew out 36 per cent of it: how much had I left? {512 @ 800 * .64}
q: A man had $300; after he had spent $225, what per cent did he have left? {25}
q: A merchant withdrew 40 per cent of his deposits, leaving $3000 remaining in the bank: what amount did he withdraw? {2000 @ 3000 / .6 * .4}
q: A grain dealer sold corn for 56 ct. a bushel, which was 40 per cent more than it cost him: what was the cost per bushel? Cents? {40 @ 56 / 1.4}
q: A man sold a horse for $175, which was 12 1/2 per cent less than the horse cost: what did the horse cost? {200 @ 175 / .875}
q: A grocer bought 4 sacks of coffee of 75 pounds each; 12 1/2 per cent was lost by waste: what was the remainder worth at 35 cents per pound? {~300 * 7/8 * .35}
q: A man owed $500; he paid $425: what per cent of the debt remains unpaid? {15}
q: A speculator invested 75 per cent of his estate in bonds, and the remainder of it, amounting to $5000, in real estate: how much did he invest in bonds? {15000}
q: A farmer owned a farm of 250 A. 86 sq. rd., which was 12 1/2 per cent more than his neighbor owned: how much land did his neighbor own? A.? {222} Sq. rd.? {112}
q: A flock of 160 sheep increased 35 per cent in one year: how many were then in the flock? {216}
q: A miller takes for toll 6 qt. from every 5 bu. of wheat ground: what per cent does he take? {~6 / 160 * 100}
q: A farmer owning 45 per cent of a tract of land, sold 540 acres, which was 60 per cent of what he owned: how many acres were there in the tract? {2000 @ 540 / .6 / .45}
q: When the gold dollar is worth 7 per cent more than the greenback dollar, how much in gold are $371.29 in greenbacks worth? {347 @ 371.29 / 1.07}
q: A's salary is $800 a year; he spends 18 per cent of it for rent, 15 per cent for clothing, 23 per cent for provisions, and 12 per cent for sundries: how much does he save annually? {256 @ 800 * .32}
q: A pupil at an examination answered 17 of the 20 questions correctly: what per cent did he make? {85}
q: 2 bu. 3 pk. are 33 1/3 per cent of what number? Bu.? {8} Pk.? {1}
q: The number of pupils attending school on a certain day was 37; this was 7 1/2 per cent less than the number enrolled: how many were enrolled? {40 @ 37 / .925}
q: A gold dollar weighs 25.8 grains Troy; 10 per cent of it is alloy: how many grains of pure gold does it contain? {23.22}
q: The five-cent piece weighs 5 grams, of which 1.25 G. are nickel and the remainder copper: what is the per cent of copper? {75}
q: A man sold a horse for $150, which was 25 per cent more than it cost him: if he had sold the horse for $200, how many per cent would it have been more than it cost him? {~(200 - 120) / 120 * 100}
== 47 | Percentage | Commission and trade discount
note: The value of the goods is the base; the commission is the percentage; the net proceeds is the base less the percentage. In trade discount, "20 and 5 off" means 20 per cent off the price, then 5 per cent off the remainder.
q: An agent whose commission is 5 per cent, receives how much upon a sale of goods amounting to $240? {12}
q: An auctioneer received $11.50 for selling a lot of goods amounting to $460: what per cent commission did he receive? {~d 11.5 / 460 * 100}
q: At a commission of 2 1/2 per cent a commission merchant receives $8.12 1/2 for selling 25 barrels of molasses: for how much per barrel did he sell the molasses? {13 @ 8.125 / .025 / 25}
q: An agent receives $210 with which to buy goods: after deducting his commission of 5 per cent what sum must he expend? {200 @ 210 / 1.05}
q: What are the net proceeds on a sale of goods amounting to $180, at 4 per cent commission? {172.80 @ 180 * .96}
q: A lawyer received $11.25 for collecting a debt: his commission being 5 per cent, what was the amount of the debt? {225 @ 11.25 / .05}
q: An agent receives $1323.54 to cover cost of goods and commission at 8 per cent: what is his commission? {98.04 @ 1323.54 - 1323.54 / 1.08}
q: A commission merchant sells 250 bbl. pork, at $15 per bbl.; 175 bbl. flour, at $7 per bbl.; and 1456 lb. feathers, at 25 ct. per lb.; his commission is 3 per cent: what sum does he remit the owner? {5178.83 @ (3750 + 1225 + 364) * .97}
q: A bill of goods amounted to $225.50 list; 20 per cent off being allowed, what was paid for the goods? {180.40 @ 225.5 * .8}
q: A bill of articles amounted to $725.16, the purchaser being allowed 1/3 and 5 off, what did he pay? To the nearest cent? {459.27}
q: I paid $1430.75 for a lot of groceries, which was 3 per cent discount from the face of the bill: what was the amount of the bill? {1475 @ 1430.75 / .97}
q: A bill of goods cost $390.45 at 25 and 5 off: what was the list price? {548 @ 390.45 / .75 / .95}
q: Sold 20 doz. feather dusters, giving the purchaser a discount of 10, 10, and 10 per cent: his discounts amounting to $325.20, how much was my price per dozen? {60 @ 325.2 / (1 - .9 * .9 * .9) / 20}
q: Bought 100 doz. stay bindings, at 60 ct. per dozen, for 40, 10, and 7 1/2 per cent off: what did I pay for them? {29.97 @ 60 * .6 * .9 * .925}
q: A retail dealer buys a case of slates containing 10 dozen for $50 list, and gets off 50, 10, and 10 per cent; paying for them in the usual time, he gets an additional 2 per cent: what did he pay per dozen for the slates? To the nearest cent? {1.98}
== 48 | Percentage | Profit and loss
note: The cost is the base; the profit or the loss is the percentage; and the selling price is the cost plus the profit, or less the loss.
q: A merchant's profit on a piece of cloth which cost $40 is 10 per cent: for how much does he sell it? {44}
q: Prints that cost 6 ct. a yard are sold for 5 ct. a yard: what is the per cent of loss? {~100/6}
q: A grocer, by retailing coffee at 27 cents per pound, gains 12 1/2 per cent: what did it cost per pound? Cents? {24 @ 27 / 1.125}
q: Selling a lot of goods at a loss of 4 per cent, the loss on the entire lot was $15.30: what did the goods cost? {382.50 @ 15.3 / .04}
q: To make a profit of 37 1/2 per cent, at what price must a dry-goods merchant sell shawls that cost $8? {11 @ 8 * 1.375}
q: A bookseller sells a grammar for 90 ct. which cost 75 ct.: what is his gain per cent? {20}
q: What is the cost of tea, which, when sold at 6 1/4 per cent profit yields a profit of 5 ct. per pound? Cents? {80 @ 5 / .0625}
q: A grocer sells apples at $4.75 per barrel, making a profit of 18 3/4 per cent: what was the cost? {4 @ 4.75 / 1.1875}
q: Sold silk at $1.35 per yard, and lost 10 per cent: at what price per yard would I have sold it to make a profit of 16 2/3 per cent? {1.75 @ 1.35 / .9 * 7/6}
q: A peddler bought a stock of goods for $874, and disposed of them at a profit of 25 per cent: how much money did he make? {218.50 @ 874 * .25}
q: If a bookseller makes 25 ct. on an atlas, which he sells for $1.75: what is his per cent of profit? {~25 / 150 * 100}
q: A dealer sold two horses for $150 each; on one he gained 25 per cent, and on the other he lost 25 per cent: how much did he lose by the transaction? {20 @ 120 + 200 - 300}
q: A merchant reduced the price of a certain piece of cloth 5 ct. per yard, and thereby reduced his profit on the cloth from 10 per cent to 8 per cent: what was the cost of the cloth per yard? {2.50 @ .05 / .02}
q: A speculator bought 10000 bushels of corn, at 60 ct. per bushel; he sold 7000 bushels, at 65 ct. per bushel; corn then falling in price, he disposed of the remainder at 55 ct. per bushel: what per cent profit did he get out of the transaction? {~(7000 * 65 + 3000 * 55 - 600000) / 600000 * 100}
q: A speculator in real estate sold a house and lot for $12000, which sale afforded him a profit of 33 1/3 per cent on the cost; he then invested the $12000 in city lots, which he was obliged to sell at a loss of 33 1/3 per cent: how much did he lose by the two transactions? {1000 @ 4000 - 3000}
q: A bookseller purchases books from the publisher at 20 per cent off the list price; if he retail them at the list price, what will be his per cent of profit? {25}
q: A grocer bought 5 half chests of tea of 74 lb. each, at 45 ct. per lb., at 2 per cent off for cash: if he retail it at 12 1/3 per cent advance, what will be his profit? To the nearest cent? {20.12}
q: Bought 5 assorted cases of men's boots, containing 12 pairs each, for $45 per case, 5 per cent off for cash; I retail them at $4.25 a pair: what is my profit? {41.25 @ 60 * 4.25 - 225 * .95}
q: Sold a case of hats containing 3 dozen, on which I had received a discount of 10 per cent and made a profit of 12 1/2 per cent or 37 1/2 ct. on each hat: what was the wholesale merchant's price per case? {120 @ .375 / .125 / .9 * 36}
q: A merchant bought 100 packs of pins, of 12 papers each, for $1.00 per pack, 60, 5 and 5 per cent off; if he retail them so as to make a profit of $23.90, for how much a paper will he sell them? Cents? {5 @ (100 * .4 * .95 * .95 + 23.9) / 1200 * 100}
q: I sent a car-load of flour, 100 bbl., to a commission merchant in New York; he disposed of the flour at $9.50 per barrel, his commission was 2 1/2 per cent with charges of $17.25: if the flour cost me $7.50 per barrel, how much did I make? {159 @ 950 * .975 - 17.25 - 750}
q: A contractor bought 80 horses for government, at $125 apiece; the freight was $200, and the agent's commission was such that the horses cost the government $10450: what per cent was the commission? {~d 250 / 10000 * 100}
q: A commission merchant sells a consignment of 50 hhd. of sugar, 1500 lb. each net, at 10 1/2 ct. per pound; his commission is 2 per cent and charges $22.50; the consignor clears 14 per cent by the transaction: what did he pay per pound for the sugar? Cents? {9 @ (7875 * .98 - 22.5) / 1.14 / 75000 * 100}
q: A dealer in notions buys 60 gross shoe-strings, at 70 ct. per gross, list, for 50, 10 and 5 per cent off; if he sell them at 20, 10 and 5 per cent off list, what will be his profit? To the nearest cent? {10.77}
q: Bought 50 gross of rubber buttons for 25, 10 and 5 per cent off; disposed of the lot for $35.91, at a profit of 12 per cent: what was the list price of the buttons per gross? {1 @ 35.91 / 1.12 / (.75 * .9 * .95) / 50}
== 49 | Percentage | Stocks, bonds and gold
note: A share is $100 unless named. Brokerage is reckoned on the par value. "Harlem, 139 3/4" means the stock sells for 139 3/4 per cent of its par value. Income is reckoned on the par value of the bonds.
q: A broker bought for me 75 shares New York Central and Hudson River stock: required the brokerage at 1/4 per cent. {18.75 @ 7500 * .0025}
q: The brokerage for buying 50 shares of Chicago and Rock Island stock was $6.25: what was the per cent? {~6.25 / 5000 * 100}
q: At 1/4 per cent brokerage a broker received $10 for making an investment in bank stock: how many shares did he buy? {40}
q: A broker buys 17 shares Milwaukee and St. Paul preferred stock: what is his brokerage, at 1/4 per cent? {4.25 @ 1700 * .0025}
q: The brokerage on 95 shares of Vermont Central stock is $11.87 1/2: what is the per cent? {~11.875 / 9500 * 100}
q: A broker received $9.50, or a brokerage of 1/4 per cent, for buying Union Pacific stock: how many shares did he purchase? {38}
q: I own 35 shares of bank stock; if the bank declare a dividend of 4 per cent, what will I receive? {140}
q: A man pays an assessment of 7 1/2 per cent, or $300, on his insurance stock: how many shares does he own? {40}
q: A mining company declares a dividend of 15 per cent: what does Mr. Jones receive who owns 80 shares of stock? {1200}
q: A man owns 60 shares of railroad stock: if the company declare a dividend of 5 per cent payable in stock, how much stock will he then own? Shares? {63}
q: A gas company has a capital stock of $160000; its gross earnings are $15700, and its expenses $4500 annually: what per cent does it pay the stockholders? {7}
q: What will be the cost of 150 shares ($50 each) of Harlem, at 139 3/4, brokerage 1/4 per cent? {10500 @ 7500 * 1.4}
q: Bought $8000 in gold at 110, brokerage 1/8 per cent: what did I pay for the gold in currency? {8810 @ 8000 * 1.10125}
q: My broker sells 50 shares of Chicago and Northwestern, brokerage 1/4 per cent; he remits me $2475: at what per cent did the stock sell? {~(2475 + 12.5) / 5000 * 100}
q: What will be the cost of 25 1000-dollar 5-20 U. S. Bonds of 1867, at 114 1/4, brokerage 1/8 per cent? {28593.75 @ 25000 * 1.14375}
q: I paid $1560 for Milwaukee and St. Paul, at 19 1/4, brokerage 1/4 per cent: how many shares did I buy? {80 @ 1560 / 19.5}
q: When gold is at 105, what is the value in gold of a dollar in currency? Cents? {~100 / 1.05}
q: When gold was at 112 1/2, what was the value of a dollar in currency? Cents in gold? {~100 / 1.125}
q: In 1864, the "greenback" dollar was worth only 35 5/7 ct. in gold: what was the price of gold? {280}
q: A merchant paid $8946.25 for gold, at 105, brokerage 1/4 per cent: how much gold did he buy? {8500 @ 8946.25 / 1.0525}
q: My broker sells a certain amount of gold, and remits me $25734.37 1/2; his brokerage, at 1/16 per cent, was $15.62 1/2: what was the price of the gold? {103 @ (25734.375 + 15.625) / 25000 * 100}
q: If I invest $39900 in 6 per cent bonds, at par, what will be my income? {2394 @ 39900 * .06}
q: If I invest $39900 in 6 per cent bonds, at 105, what will be my income? {2280 @ 39900 / 1.05 * .06}
q: If I invest $39900 in 6 per cent bonds, at 95, what will be my income? {2520 @ 39900 / .95 * .06}
q: What is a man's income who owns 20 1000-dollar U. S. 6 per cent bonds, when gold is 107? In currency? {1284 @ 20000 * .06 * 1.07}
q: What income in currency would a man receive by investing $5220 in U. S. 5-20 6 per cent bonds, at 116, when gold is 105? {283.50 @ 5220 / 1.16 * .06 * 1.05}
q: What per cent of income do U. S. 4 1/2 per cents, at 108, yield when gold is 105? {~4.5 * 1.05 / 1.08}
q: If I receive an annual dividend of 6 per cent on Michigan Central stock, which cost me but 37 1/2, what per cent of income do I receive on my investment? {16}
q: What sum invested in U. S. 5's of 1881, at 118, yielded an annual income of $1921 in currency, when gold was at 113? {40120 @ 1921 / 1.13 / .05 * 1.18}
q: How many shares of stock bought at 95 1/4, and sold at 105, brokerage 1/4 per cent on each transaction, will yield a profit of $925? {100 @ 925 / 9.25}
q: What must be paid for 6 per cent bonds to realize an income of 8 per cent? {75}
q: When U. S. 4 per cent bonds are quoted at 106, what yearly income will be received in gold from the bonds that can be bought for $4982? {188 @ 4982 / 1.06 * .04}
q: If I pay 87 1/2 for railroad bonds that yield an annual income of 7 per cent, what per cent do I get on my investment? {8}
q: What could I afford to pay for bonds yielding an annual income of 7 per cent to invest my money so as to realize 6 per cent? {~700/6}
== 50 | Interest | Simple interest
note: Interest for 1 year is the principal times the rate; for years, months and days, multiply by the time in years (30 days to the month, 12 months to the year). The amount is the principal and interest. Money answers: to the nearest cent.
q: Find the interest of $25 for 1 yr., at 6 per cent. {1.50}
sol: 6 per cent is .06; the interest for 1 yr. is $25 x .06 = $1.50.
q: Find the interest of $18.75 for 1 yr., at 6 2/3 per cent. {1.25}
sol: 6 2/3 per cent is 1/15. The interest for 1 yr. is $18.75 divided by 15 = $1.25.
q: Find the amount of $215 for 1 yr., at 6 per cent. {227.90}
q: Find the interest of $200 for 1 yr., at 8 per cent. {16}
q: Find the interest of $150 for 1 yr., at 5 per cent. {7.5}
q: Find the interest of $85 for 1 yr., at 7 per cent. {5.95}
q: Find the interest of $7200 for 1 yr., at 6 1/4 per cent. {450}
q: Find the amount of $28.20 for 1 yr., at 8 1/3 per cent. {30.55}
q: Find the amount of $45.50 for 1 yr., at 10 per cent. {50.05}
q: Find the amount of $420 for 1 yr., at 5 1/3 per cent. {442.4}
q: Find the amount of $857 for 1 yr., at 9 per cent. {934.13}
q: Find the amount of $96 for 1 yr., at 8 1/2 per cent. {104.16}
q: Find the amount of $2000 for 1 yr., at 4 1/2 per cent. {2090}
q: Find the amount of $164 for 1 yr., at 12 1/2 per cent. {184.5}
q: Find the interest of $50 for 3 yr., at 7 per cent. {10.50}
sol: The interest of $50 for 1 yr., at 7 per cent, is $3.50; then, the interest for 3 yr. is $3.50 x 3 = $10.50.
q: Find the amount of $225.18 for 3 yr., at 4 1/2 per cent. {255.58|255.5793}
sol: The interest for 1 yr. is $10.1331; for 3 yr., $30.3993; the amount is $255.5793, which is $255.58 to the nearest cent.
q: Find the interest of $65 for 4 yr., at 5 per cent. {13}
q: Find the interest of $300 for 2 yr., at 6 per cent. {36}
q: Find the interest of $275 for 3 yr., at 6 per cent. {49.5}
q: Find the interest of $187.50 for 4 yr., at 5 per cent. {37.5}
q: Find the interest of $233.80 for 10 yr., at 6 per cent. {140.28}
q: Find the amount of $45 for 2 yr., at 8 per cent. {52.2}
q: Find the amount of $80 for 4 yr., at 7 per cent. {102.4}
q: Find the amount of $237.16 for 2 yr., at 3 3/4 per cent. {254.95|254.947}
q: Find the amount of $74.75 for 5 yr., at 4 per cent. {89.7}
q: Find the amount of $85.45 for 4 yr., at 6 per cent. {105.96|105.958}
q: Find the amount of $325 for 3 yr., at 5 2/5 per cent. {377.65}
q: Find the amount of $129.36 for 4 yr., at 4 3/8 per cent. {152|151.998}
q: Find the amount of $8745 for 2 yr., at 8 per cent. {10144.2}
q: Find the interest of $24 for 9 mo., at 6 per cent. {1.08}
sol: 9 mo. are 3/4 of a year. The interest of $24 for 1 yr., at 6 per cent, is $1.44; then, the interest for 9 mo. is 3/4 of $1.44, which is $1.08.
q: Find the interest of $300 for 1 mo., at 6 per cent. {1.5}
q: Find the interest of $240 for 2 mo., at 8 per cent. {3.2}
q: Find the interest of $50 for 5 mo., at 6 per cent. {1.25}
q: Find the interest of $86 for 3 mo., at 6 per cent. {1.29}
q: Find the interest of $50 for 4 mo., at 8 per cent. {1.33|1.333}
q: Find the amount of $150.25 for 6 mo., at 8 per cent. {156.26}
q: Find the amount of $360 for 7 mo., at 5 per cent. {370.5}
q: Find the amount of $204 for 11 mo., at 7 per cent. {217.09}
q: Find the amount of $228 for 9 mo., at 6 per cent. {238.26}
q: Find the amount of $137.50 for 8 mo., at 6 per cent. {143}
q: Find the amount of $7596 for 10 mo., at 8 per cent. {8102.4}
q: Find the interest of $288 for 24 da., at 5 per cent. {.96}
sol: 24 da. are 4/5 of a month. The interest of $288 for 1 mo., at 5 per cent, is $1.20; then, the interest for 24 da. is 4/5 of $1.20, which is 96 ct.
q: Find the interest of $360 for 20 da., at 6 per cent. {1.2}
q: Find the interest of $726 for 10 da., at 6 per cent. {1.21}
q: Find the interest of $1200 for 15 da., at 6 per cent. {3}
q: Find the interest of $180 for 19 da., at 8 per cent. {.76}
q: Find the interest of $240 for 27 da., at 7 per cent. {1.26}
q: Find the interest of $320 for 21 da., at 5 per cent. {.93|.933}
q: Find the interest of $450 for 25 da., at 10 per cent. {3.13|3.125}
q: Find the amount of $100.80 for 28 da., at 5 per cent. {101.19|101.192}
q: Find the amount of $150 for 18 da., at 5 per cent. {150.38|150.375}
q: Find the amount of $360 for 11 da., at 6 per cent. {360.66}
q: Find the amount of $264 for 9 da., at 6 per cent. {264.4|264.396}
q: Find the amount of $900 for 14 da., at 7 per cent. {902.45}
q: Find the amount of $430 for 19 da., at 4 1/2 per cent. {431.02|431.021}
== 51 | Interest | Years, months and days
note: Find the interest for each period and add; or reduce the months and days to the fraction of a year and multiply. The twelve per cent method: the interest of $1 at 12 per cent is as many cents as there are months, and 1/3 as many mills as there are days.
q: Find the interest of $360 for 2 yr. 7 mo. 25 da., at 8 per cent. {76.40}
sol: The interest for 1 yr. is $28.80; for 2 yr., $57.60; for 7 mo., $16.80; for 25 da., $2. In all, $76.40.
q: Find the interest of $120 for 4 yr. 6 mo. 20 da., at 6 per cent. {32.80}
sol: 20 da. are 2/3 of a mo.; 6 2/3 mo. are 5/9 of a year. Then, the interest will be $120 x .06 x 4 5/9 = $32.80.
q: Find the interest of $150 for 4 yr. 2 mo., at 6 per cent. {37.5}
q: Find the interest of $375.40 for 1 yr. 8 mo., at 6 per cent. {37.54}
q: Find the interest of $92.75 for 3 yr. 5 mo., at 6 per cent. {19.01|19.014|19.01375}
q: Find the interest of $500 for 1 yr. 1 mo. 18 da., at 6 per cent. {34}
q: Find the interest of $560 for 2 yr. 4 mo. 15 da., at 8 per cent. {106.4}
q: Find the interest of $750 for 4 yr. 3 mo. 6 da., at 6 per cent. {192}
q: Find the interest of $456 for 3 yr. 5 mo. 18 da., at 5 per cent. {79.04}
q: Find the interest of $216 for 5 yr. 7 mo. 27 da., at 10 per cent. {122.22}
q: Find the interest of $380 for 3 yr. 9 mo. 9 da., at 15 per cent. {215.18|215.175}
q: Find the amount of $300 for 3 yr. 8 mo., at 6 per cent. {366}
q: Find the amount of $250 for 1 yr. 7 mo., at 6 per cent. {273.75}
q: Find the amount of $205.25 for 2 yr. 8 mo. 15 da., at 6 per cent. {238.6|238.603|238.603125}
q: Find the amount of $150.62 for 3 yr. 5 mo. 12 da., at 5 per cent. {176.6|176.602|176.60195}
q: Find the amount of $210.25 for 2 yr. 7 mo. 20 da., at 7 per cent. {249.09|249.088}
q: Find the amount of $57.85 for 2 yr. 3 mo. 23 da., at 5 per cent. {64.54|64.543}
q: Find the interest of $150, from January 9, 1847, to April 19, 1849, at 6 per cent. {20.5}
q: Find the interest of $240, from February 15, 1848, to April 27, 1849, at 8 per cent. {23.04}
q: Find the interest of $180, from May 14, 1843, to August 28, 1845, at 7 per cent. {28.84}
q: Find the interest of $137.50, from July 3, to November 27, at 9 per cent. {4.95}
q: Find the amount of $125.40, from March 1, to August 28, at 8 1/2 per cent. {130.64|130.641|130.640675}
q: Find the amount of $234.60, from August 2, 1847, to March 9, 1848, at 5 1/4 per cent. {242.02|242.024|242.0241125}
q: Find the amount of $153.80, from October 25, 1846, to July 24, 1847, at 5 per cent. {159.55|159.546}
sol: To find the time between two dates, subtract the earlier date from the later, counting 30 days to a month.
q: Find the interest of $1, at 12 per cent, for 9 mo. 12 da.. {.09|.094}
q: Find the interest of $1, at 12 per cent, for 4 mo. 18 da.. {.05|.046}
q: Find the interest of $1, at 12 per cent, for 7 mo. 12 da.. {.07|.074}
q: Find the interest of $1, at 12 per cent, for 9 mo. 3 da.. {.09|.091}
q: Find the interest of $1, at 12 per cent, for 1 yr. 4 mo.. {.16}
q: Find the interest of $1, at 12 per cent, for 1 yr. 5 mo. 27 da.. {.18|.179}
q: Find the interest of $1, at 12 per cent, for 2 yr. 3 mo. 21 da.. {.28|.277}
q: Find the interest of $1, at 12 per cent, for 3 yr. 7 mo. 12 da.. {.43|.434}
q: Find the interest of $1, at 12 per cent, for 4 yr. 2 mo. 15 da.. {.51|.505}
q: Find the interest of $1, at 12 per cent, for 2 mo. 1 da.. {.02}
q: Find the interest of $1, at 12 per cent, for 5 mo. 17 da.. {.06|.056}
q: Find the interest of $1, at 12 per cent, for 10 mo. 13 da.. {.1|.104}
q: Find the interest of $1, at 12 per cent, for 1 yr. 2 mo. 4 da.. {.14|.141}
q: Find the interest of $1, at 12 per cent, for 2 yr. 9 mo. 20 da.. {.34|.337}
q: Find the interest of $1, at 12 per cent, for 3 yr. 5 mo. 29 da.. {.42}
sol: Call the months cents, and one-third of the days mills.
q: Find the interest of $1, for 2 yr. 5 mo. 18 da., at 6 per cent. {.148}
sol: 6 per cent is 1/2 of 12 per cent. The interest at 12 per cent is $0.296; at 6 per cent, half of that, $0.148.
q: Find the interest of $1, for 3 yr. 7 mo. 20 da., at 8 per cent. {.291|.29|.2911}
q: Find the interest of $1, for 7 mo. 24 da., at 6 per cent. {.04|.039}
q: Find the interest of $1, for 10 mo. 15 da., at 5 per cent. {.04|.044|.04375}
q: Find the interest of $1, for 11 mo. 18 da., at 9 per cent. {.09|.087}
q: Find the interest of $1, for 1 yr. 2 mo. 9 da., at 6 per cent. {.07|.072|.0715}
q: Find the interest of $1, for 2 yr. 5 mo. 12 da., at 8 per cent. {.2|.196}
q: Find the interest of $1, for 3 yr. 10 mo. 17 da., at 10 per cent. {.39|.388}
q: Find the interest of $1, for 4 yr. 3 mo. 11 da., at 7 per cent. {.3}
q: Find the interest of $1, for 5 yr. 7 mo. 24 da., at 4 per cent. {.23|.226}
q: Find the interest of $25, for 1 yr. 5 mo. 18 da., at 6 per cent. {2.20}
q: Find the interest of $134.45, for 1 yr. 7 mo. 15 da., at 8 per cent. {17.48|17.479|17.4785}
q: Find the interest of $40, for 6 mo. 21 da., at 6 per cent. {1.34}
q: Find the interest of $50, for 8 mo. 24 da., at 9 per cent. {3.3}
q: Find the interest of $120, for 10 mo. 12 da., at 7 per cent. {7.28}
q: Find the interest of $200, for 11 mo. 15 da., at 6 per cent. {11.5}
q: Find the interest of $500, for 1 yr. 3 mo. 6 da., at 3 per cent. {19}
q: Find the interest of $750, for 1 yr. 5 mo. 27 da., at 8 per cent. {89.5}
q: Find the interest of $48.75, for 1 yr. 9 mo. 3 da., at 6 per cent. {5.14|5.143|5.143125}
q: Find the interest of $76.32, for 1 yr. 10 mo. 25 da., at 4 per cent. {5.81|5.809}
q: Find the amount of $600, for 2 yr. 1 mo. 9 da., at 5 per cent. {663.25}
q: Find the amount of $900, for 2 yr. 4 mo. 10 da., at 6 per cent. {1027.5}
q: Find the amount of $86.25, for 2 yr. 7 mo. 17 da., at 9 per cent. {106.67|106.6696875}
q: Find the amount of $450, for 3 yr. 2 mo. 13 da., at 8 per cent. {565.3}
q: Find the amount of $534.78, for 3 yr. 5 mo. 22 da., at 4 per cent. {609.17|609.174}
q: Find the amount of $1200, for 3 yr. 11 mo. 15 da., at 10 per cent. {1675}
== 52 | Interest | Finding the time, rate and principal
note: Time: divide the interest by the interest for one year. Rate: find the interest for one year, and what per cent it is of the principal. Principal: multiply the rate by the time, and divide the interest by the product.
q: The interest of $225 for a certain time, at 4 per cent, was $66: what was the time? Years? {7} Months? {4}
sol: The interest of $225 for 1 yr., at 4 per cent, is $9; then $66 is the interest for as many years as 9 is contained times in 66, which is 7 1/3, or 7 yr. 4 mo.
q: In what time, at 10 per cent, will $500 amount to $800? Years? {6}
q: In what time, at 8 per cent, will any principal double itself? Years? {12} Months? {6}
sol: A principal has doubled itself when the interest becomes 100 per cent: 100 divided by 8 is 12 1/2 years.
q: I lent $200, at 6 per cent, and received $36 interest: how long was the money lent? Years? {3}
q: In what time, at 5 per cent, will $60 amount to $72? Years? {4}
q: In what time, at 6 per cent, will any principal be doubled? Years? {16} Months? {8} Days? {0}
q: A man lent $375, at 8 per cent, and received $90 interest: how long was it lent? Years? {3}
q: In what time, at 9 per cent, will $600 amount to $798? Years? {3} Months? {8}
q: In what time, at 10 per cent, will any principal double itself? Years? {10}
q: How long will it take $250, at 6 per cent, to yield $34.50 interest? Years? {2} Months? {3} Days? {18}
q: How long will it take $60, at 6 per cent, to amount to $73.77? Years? {3} Months? {9} Days? {27}
q: How long will it take any principal to treble itself, at 6 per cent? Years? {33} Months? {4}
q: The interest on $400, at 7 per cent, was $68.60: how long was it loaned? Years? {2} Months? {5} Days? {12}
q: In what time, at 9 per cent, will $700 amount to $924.70? Years? {3} Months? {6} Days? {24}
q: How long will it take any principal to increase one-half, at 8 per cent? Years? {6} Months? {3}
q: In what time, at 10 per cent, will $1200 amount to $1675? Years? {3} Months? {11} Days? {15}
q: A merchant paid $30 interest for the use of $300, for 1 yr. 8 mo.: what was the per cent? {6}
sol: 1 yr. 8 mo. are 5/3 yr. The interest for 1 yr. is $18; $18 is 3/50 of $300; 3/50 are 6 per cent.
q: At what per cent will any principal double itself in 20 yr? {5}
q: I borrowed $600 for 2 years and paid $48 interest: what per cent did I pay? {~48 / 600 / 2 * 100}
q: A broker paid $200 interest for the use of $1000 for 2 yr. 6 mo.: what was the per cent? {~200 / 1000 / 2_1/2 * 100}
q: The amount of $250 for 2 yr. 4 mo. 24 da. was $310: what was the per cent? {~60 / 250 / (2 + 4/12 + 24/360) * 100}
q: $23.40 interest was paid for the use of $260 for 2 yr.: what was the per cent? {~23.4 / 260 / 2 * 100}
q: At what per cent will any principal double itself in 12 yr. 6 mo.? {~100 / 12_1/2}
q: The amount of $175 for 3 yr. 7 mo. was $250.25: what was the per cent? {~75.25 / 175 / (3 + 7/12) * 100}
q: The interest of $450 for 1 yr. 8 mo. 12 da. is $61.20: what is the per cent? {~61.2 / 450 / (1 + 8/12 + 12/360) * 100}
q: At what per cent will any principal double itself in 11 yr. 1 mo. 10 da.? {~100 / (11 + 1/12 + 10/360)}
q: The amount of $650 for 2 yr. 5 mo. 18 da. is $746.20: what is the per cent? {~96.2 / 650 / (2 + 5/12 + 18/360) * 100}
q: The interest of $640 for 6 yr. was $110.40: what was the per cent? {~110.4 / 640 / 6 * 100}
q: The interest for 2 yr., at 6 per cent, is $27: what is the principal? {225}
sol: 6 per cent is 3/50; for 2 yr. the interest is 3/25 of the principal. Then, 3/25 of the principal are $27, 1/25 is $9, and the principal is $225.
q: The interest for 3 yr., at 9 per cent, is $21.60: what is the principal? {80}
q: The interest for 3 yr., at 5 per cent, is $8.25: what is the principal? {55}
q: The interest for 3 yr., at 5 per cent, is $341.25: what is the principal? {2275}
q: The interest for 1 yr. 4 mo., at 6 per cent, is $2.26: what is the principal? {28.25}
q: What principal, at 5 per cent, will produce a yearly interest of $1023.75? {20475}
q: What principal, at 8 per cent, will produce $30.24 interest in 1 yr. 6 mo. 27 da.? {240}
q: What principal, at 9 per cent, will produce $525.40 interest in 12 yr. 3 mo. 20 da.? To the nearest cent? {474.40|474.4}
q: The interest for 2 yr. 7 mo. 11 da., at 4 per cent, is $9.41: what is the principal? {90}
q: The interest for 5 yr. 8 mo. 24 da., at 6 per cent, is $28.38: what is the principal? {82.50|82.5}
q: What principal in 5 yr., at 6 per cent, will amount to $650? {500}
sol: The interest for 5 yr. is 3/10 of the principal, and the amount is 13/10 of it. Then, 13/10 of the principal are $650, 1/10 is $50, and the principal is $500.
q: What principal in 9 yr., at 5 per cent, will amount to $435? {300}
q: The amount for 4 yr., at 5 per cent, is $571.20: what is the interest? {95.20|95.2}
q: The amount for 6 yr., at 7 per cent, is $532.50: what is the interest? {157.50|157.5}
q: The amount for 2 yr. 9 mo., at 8 per cent, is $285.48: what is the principal? {234}
q: The amount for 2 yr. 6 mo., at 6 per cent, is $690: what is the interest? {90}
q: The amount for 3 yr. 4 mo. 24 da., at 7 per cent, is $643.76: what is the principal? {520}
q: The amount for 4 yr. 3 mo. 27 da., at 4 per cent, is $914.94: what is the interest? {134.94}
== 53 | Interest | Compound and annual interest
note: In compound interest the interest is added to the principal at the end of each year (or half-year, or quarter); each new amount is the principal for the next period. Work to the nearest cent at each step, as the book does.
q: Find the compound interest of $300 for 3 yr., at 6 per cent. {57.30|57.3}
sol: The amount for the 1st yr. is $318; for the 2d, $337.08; for the 3d, $357.3048. The compound interest is $357.3048 - $300 = $57.30.
q: Find the amount, at 6 per cent compound interest, of $500, for 3 years. {595.51}
q: Find the amount, at 6 per cent compound interest, of $800, for 4 years. {1009.98}
q: Find the compound interest of $250, for 3 yr., at 6 per cent. {47.75}
q: Find the compound interest of $300, for 4 yr., at 5 per cent. {64.65}
q: Find the compound interest of $200, for 2 yr., at 6 per cent, payable semi-annually. {25.10|25.1}
q: Find the amount of $500, for 2 yr., at 20 per cent compound interest, payable quarterly. {738.73}
q: What is the compound interest of $300, for 2 yr. 6 mo., at 6 per cent? {47.19}
q: What is the compound interest of $1000, for 2 yr. 8 mo. 15 da., at 6 per cent? {171.35}
q: What is the amount of $620 at compound interest semi-annually, for 3 yr. 6 mo., at 6 per cent? {762.52}
q: What is the difference between simple interest and compound interest on $500, for 4 yr. 8 mo., at 6 per cent? {16.49}
q: No interest having been paid, find the amount due in 4 yr. 8 mo. 24 da., on a note for $400, with interest at 6 per cent, payable annually. {526.46}
sol: The interest of $400 for 4 yr. 8 mo. 24 da. is $113.60. Each annual interest of $24 remains unpaid 3 yr. 8 mo. 24 da., 2 yr. 8 mo. 24 da., 1 yr. 8 mo. 24 da., and 8 mo. 24 da.: in all 8 yr. 11 mo. 6 da., and the interest of $24 for that time is $12.864. The amount is $12.864 + $113.60 + $400 = $526.46.
q: No interest having been paid, find the amount due in 3 yr. on a note for $800, with interest at 8 per cent, payable annually. {1007.36}
q: The interest having been paid for 2 yr., find the amount due in 5 yr. on a note for $750, with interest at 10 per cent, payable annually. {997.50|997.5}
q: No interest having been paid for 4 yr., find the interest due on a bond for $10000, with interest at 5 per cent, payable annually. {2150}
q: No interest having been paid, find the amount due Sept. 1, 1877, on a note for $500, dated June 1, 1875, with interest at 6 per cent, payable semi-annually. {571.10|571.1}
q: A note for $1200 dated May 12, 1873, with interest at 6 per cent, payable annually. No interest having been paid, what was the amount due on this note, September 20, 1877? {1545.66}
q: A note for $1500 dated October 10, 1872, payable May 1, 1877, with interest payable annually at 5 per cent. No interest having been paid, what amount was due at maturity? {1872.75}
q: What is the difference between simple and annual interest on $1000 for 5 yr., at 6 per cent? {36}
q: What will be due on six 500-dollar city bonds running 3 yr., with interest at 6 per cent, payable semi-annually, if the interest should not be paid? {3580.50|3580.5}
q: The interest on U. S. 4 per cent bonds is payable quarterly in gold; granting that the income from them might be immediately invested, at 6 per cent, payable in gold, what would the income on 20 1000-dollar bonds amount to in 5 yr., with gold at 105? In currency? {4798.50|4798.5}
== 54 | Interest | Partial payments
note: Apply each payment first to the interest then due; any surplus goes toward the principal. If a payment is less than the interest, keep reckoning interest on the same principal until the payments together exceed the interest. Count 30 days to the month.
q: A note for $1000, dated May 1, 1875, with interest at 6 per cent, has these payments indorsed: November 25, 1875, $134; March 7, 1876, $315.30; August 13, 1876, $15.60; June 1, 1877, $25; April 25, 1878, $236.20. What was the amount due on settlement, September 10, 1878? {409}
sol: To Nov. 25, 1875, interest $34: new principal $900. To March 7, 1876, interest $15.30: principal $600. To Aug. 13, 1876, interest $15.60 equals the payment: principal $600. To June 1, 1877, interest $28.80 is more than the $25 paid; to April 25, 1878, the interest is $61.20 and the payments $261.20: principal $400. To Sept. 10, 1878, interest $9: $409.
q: A note for $350, dated July 1, 1875, with interest at 6 per cent. Indorsements: March 1, 1876, $44; October 1, 1876, $10; January 1, 1877, $26; December 1, 1877, $15. What was the amount due on settlement, March 16, 1878? {306.75}
q: A note of $200 is dated January 1, 1875. Indorsement: January 1, 1876, $70. What was the amount due January 1, 1877, interest at 6 per cent? {150.52}
q: A note of $300 is dated July 1, 1873. Indorsements: January 1, 1874, $109; July 1, 1874, $100. What was the amount due January 1, 1875, interest at 6 per cent? {109.18}
q: A note of $150 is dated May 10, 1870. Indorsements: September 10, 1871, $32; September 10, 1872, $6.80. What was the amount due November 10, 1872, interest at 6 per cent? {132.30|132.3}
q: A note of $200 is dated March 5, 1871. Indorsements: June 5, 1872, $20; December 5, 1872, $50.50. What was the amount due June 5, 1874, interest at 10 per cent? {189.18}
q: A note of $250 is dated January 1, 1875. Indorsements: June 1, 1875, $6; January 1, 1876, $21.50. What was the amount due July 1, 1876, interest at 7 per cent? {248.40|248.4}
q: A note of $180 is dated August 1, 1874. Indorsements: February 1, 1875, $25.40; August 1, 1875, $4.30; January 1, 1876, $30. What was the amount due July 1, 1876, interest at 6 per cent? {138.54}
q: A note of $400 is dated March 1, 1875. Indorsements: September 1, 1875, $10; January 1, 1876, $30; July 1, 1876, $11; September 1, 1876, $80. What was the amount due March 1, 1877, interest at 6 per cent? {313.33}
q: A note of $450 is dated April 16, 1876. Indorsements: January 1, 1877, $20; April 1, 1877, $14; July 16, 1877, $31; December 25, 1877, $10; July 4, 1878, $18. What was the amount due June 1, 1879, interest at 8 per cent? {466.50|466.5}
q: A note of $1000 is dated January 1, 1870. Indorsements: May 1, 1870, $18; September 4, 1870, $20; December 16, 1870, $15; April 10, 1871, $21; July 13, 1871, $118; December 23, 1871, $324. What was the amount due October 1, 1873, interest at 6 per cent? {663.80|663.8}
note: Mercantile rule, for notes and accounts running a year or less: find the amount of the principal to the day of settlement, and of each payment from its date; subtract the sum of the payments' amounts.
q: A note of $320 is dated Jan. 1, 1876. Indorsements: May 1, 1876, $50; Nov. 16, 1876, $100. What was the amount due Jan. 1, 1877, by the mercantile rule, interest at 6 per cent? {186.45}
q: An account of $540 was due March 1, 1877. Credits: May 1, 1877, $90; July 1, 1877, $100; Aug. 1, 1877, $150; Oct. 11, 1877, $180. What was the amount due on settlement Jan. 1, 1878, by the mercantile rule, interest at 8 per cent? {39}
== 55 | Discount | Bank discount
note: Bank discount is simple interest on the face of the note, taken in advance, for the time it runs and 3 days of grace. The proceeds is the face less the discount. Months here are counted as 30 days.
q: A note of $700, dated June 25, 1877, payable in sixty days, is discounted at 6 per cent: find the bank discount. {7.35} The proceeds? {692.65}
sol: The note is due August 24, with grace August 27, 1877. The interest of $1 for 63 days, at 6 per cent, is $0.0105, and of $700 it is $7.35; this is the discount. $700 - $7.35 = $692.65, the proceeds.
q: A note of $100, dated June 20, payable in 60 days, discounted at 6 per cent. Bank discount? {1.05} Proceeds? {98.95}
sol: Due Aug. 19/22.
q: A note of $120, dated October 12, payable in 30 days, discounted at 8 per cent. Bank discount? {.88} Proceeds? {119.12}
sol: Due Nov. 11/14.
q: A note of $140, dated January 15, payable in 4 months, discounted at 6 per cent. Bank discount? {2.87} Proceeds? {137.13}
sol: Due May 15/18.
q: A note of $180, dated April 10, payable in 6 months, discounted at 4 per cent. Bank discount? {3.66} Proceeds? {176.34}
sol: Due Oct. 10/13.
q: A note of $250, dated December 1, payable in 5 months, discounted at 8 per cent. Bank discount? {8.50|8.5} Proceeds? {241.50|241.5}
sol: Due May 1/4.
q: A note of $375, dated August 4, payable in 30 days, discounted at 6 per cent. Bank discount? {2.06} Proceeds? {372.94}
sol: Due Sept. 3/6.
q: A note of $600, dated February 12, 1876, payable in 60 days, discounted at 9 per cent. Bank discount? {9.45} Proceeds? {590.55}
sol: Due April 12/15, 1876.
q: A note of $1200, dated February 20, 1877, payable in 90 days, discounted at 10 per cent. Bank discount? {31} Proceeds? {1169}
sol: Due May 21/24, 1877.
q: A note of $1780, dated January 11, 1872, payable in 90 days, discounted at 6 per cent. Bank discount? {27.59} Proceeds? {1752.41}
sol: 1872 is a leap year; due April 10/13, 1872.
q: A note of $600, dated Sept. 15, 1876, payable one year after date, is discounted May 21, 1877, at 10 per cent. Days of discount? {120} Bank discount? {20} Proceeds? {580}
sol: Due Sept. 15/18, 1877; from May 21 to Sept. 18 is 120 days.
q: A note of $1000, dated May 8, 1877, payable ninety days after date, is discounted June 22, 1877, at 6 per cent. Days of discount? {48} Bank discount? {8} Proceeds? {992}
q: A note of $1500, dated July 10, 1877, payable six months after date, is discounted October 25, 1877, at 6 per cent. Days of discount? {81} Bank discount? {20.25} Proceeds? {1479.75}
note: When the note bears interest, find the amount of the note for its full time, then the bank discount and proceeds of this amount.
q: A note of $800, dated January 5, 1877, payable six months after date with interest at 6 per cent, is discounted April 15, 1877, at 8 per cent. Days of discount? {84} Bank discount? {15.39} Proceeds? {809.01}
sol: The note is due July 5/8, 1877. The amount of $800 for 6 mo. 3 da., at 6 per cent, is $824.40. The bank discount of $824.40 for 84 days, at 8 per cent, is $15.39. The proceeds are $809.01.
q: A note of $150, dated May 20, 1875, payable in 6 months, with interest at 6 per cent, discounted September 9, 1875, at 8 per cent. Days of discount? {75} Bank discount? {2.58} Proceeds? {152}
q: A note of $300, dated August 5, 1876, payable in 1 year, with interest at 8 per cent, discounted April 16, 1877, at 6 per cent. Days of discount? {114} Bank discount? {6.16} Proceeds? {318.04}
q: A note of $450, dated March 4, 1877, due January 1, 1878, with interest at 6 per cent, discounted August 13, 1877, at 10 per cent. Days of discount? {144} Bank discount? {18.90|18.9} Proceeds? {453.60|453.6}
q: A note of $650, dated May 16, 1876, due Sept. 1, 1878, with interest at 9 per cent, discounted April 25, 1878, at 6 per cent. Days of discount? {132} Bank discount? {17.26} Proceeds? {767.29}
q: A note of $840, dated September 1, 1875, payable in 6 months, with interest at 10 per cent, discounted December 20, 1875, at 8 per cent. Days of discount? {75} Bank discount? {14.71} Proceeds? {867.99}
q: A note of $1400, dated July 19, 1875, due May 1, 1876, with interest at 6 per cent, discounted Jan. 17, 1876, at 10 per cent. Days of discount? {108} Bank discount? {44} Proceeds? {1422.50|1422.5}
q: A note of $2400, dated Oct. 16, 1876, due Jan. 1, 1878, with interest at 8 per cent, discounted July 26, 1877, at 10 per cent. Days of discount? {162} Bank discount? {118.51} Proceeds? {2515.09}
q: A note of $3500, dated October 15, 1877, payable one year after date with interest at 6 per cent, is discounted May 15, 1878, at 9 per cent. Days of discount? {156} Bank discount? {144.76} Proceeds? {3566.99}
q: A note of $6000, dated May 10, 1875, payable one year after date with interest at 8 per cent, is discounted November 21, 1875, at 10 per cent. Days of discount? {174} Bank discount? {313.39} Proceeds? {6170.61}
q: For what sum due 90 days hence, must I give a note to a bank, that, when discounted at 6 per cent, the proceeds will be $177.21? {180}
sol: The bank discount of $1 for 93 days, at 6 per cent, is $0.0155, and the proceeds $0.9845. Then $177.21 is the proceeds of 177.21 divided by .9845 = $180.
q: The proceeds of a note discounted at a bank for 60 days, at 6 per cent, were $197.90: what was the face of the note? {200}
q: For what sum must a note be made, so that when discounted at a bank, for 90 days, at 6 per cent, the proceeds will be $393.80? {400}
q: What must be the face of a note, that when discounted at a bank for 5 months, at 8 per cent, the proceeds may be $217.35? {225}
q: The proceeds of a note are $352.62, the time 4 months, and the discount at 6 per cent: what is the face? {360}
q: I wish to borrow $400 from a bank for 30 days: what must be the face of my note, that, when discounted at 6 per cent, I may receive this amount? {402.21}
q: I wish to obtain from a bank $500 for 60 days: for what sum must I give my note, at 8 per cent discount? {507.10|507.1}
q: I wish to use $1500 for 6 months; if I can obtain money from a bank, at a discount of 10 per cent, for what sum must I give my note to realize this amount? {1580.33}
q: A note dated February 19, 1876, payable January 1, 1877, and bearing 8 per cent interest, was discounted October 12, 1876, at 6 per cent; the proceeds were $1055.02: what was the face of the note? {1000}
sol: The note is due Jan. 1/4, 1877. Discount on $1 for 2 mo. 24 da. at 6 per cent is $0.014; $1055.02 divided by .986 is $1070, the amount due at maturity; the interest for 10 mo. 15 da. at 8 per cent is 7 per cent, so the face is $1070 divided by 1.07 = $1000.
== 56 | Discount | True discount and exchange
note: The present worth is the sum which, put at interest for the time, would amount to the debt: divide the debt by the amount of $1 for the time. The true discount is the debt less the present worth.
q: Find the present worth, at 6 per cent, of a note of $430.50, due in 2 yr. 5 mo. 18 da. {375} The discount? {55.50|55.5}
sol: The amount of $1 for 2 yr. 5 mo. 18 da., at 6 per cent, is $1.148. Then, the present worth of $430.50 is 430.50 divided by 1.148 = $375; and the discount is $430.50 - $375 = $55.50.
q: Find the present worth, at 8 per cent, of a note of $500, due in 3 yr., and bearing interest at 6 per cent. {475.81} The discount? {114.19}
sol: The amount of the note for 3 yr. at 6 per cent is $590. The amount of $1 for 3 yr. at 8 per cent is $1.24. Then the present worth is 590 divided by 1.24 = $475.81; the discount is $114.19.
q: Find the present worth, at 6 per cent, of a note of $224, due in 2 yr. {200} The discount? {24}
q: Find the present worth, at 6 per cent, of a note of $300, due in 2 yr., and bearing interest at 8 per cent. {310.71} The discount? {37.29}
q: Find the present worth, at 6 per cent, of a debt of $675, due in 5 yr. 10 mo. {500} The discount? {175}
q: Find the present worth for 5 mo., at 10 per cent, of an account of $368.75. {354} The discount? {14.75}
q: A note of $800, dated September 10, 1876, due January 1, 1878, and bearing interest at 6 per cent, was disposed of for the present worth, at 10 per cent, July 19, 1877: what was the present worth at this date? {825.65} The discount? {37.15}
q: A merchant bought a bill of goods amounting to $775, on 4 months' credit: if money is worth 10 per cent to him, what might he pay for the goods in cash? {750}
q: Bought a bill of goods, amounting to $260, on 8 months' credit: if money is worth 6 per cent, what sum will pay the debt in cash? {250}
q: A merchant buys a bill of goods amounting to $2480; he can have 4 months' credit, or 5 per cent off, for cash: if money is worth only 10 per cent to him, what is his gain if he pays cash? {45.47}
q: Find the present worth, at 5 per cent, of a debt of $956.34, one-third to be paid in 1 yr., one-third in 2 yr., and one-third in 3 yr. {870.60|870.6}
q: Omitting the three days of grace, what is the difference between the true discount and the bank discount of $535, for 1 yr., at 7 per cent? {2.45}
q: A man was offered $1122 for a house, in cash, or $1221, payable in 10 mo., without interest. He chose the latter: what is his present loss, if money is worth 12 per cent to him? {12}
q: A man offers to sell his farm for $8000 in cash, or for $10296, payable in three equal installments at the end of 1, 2, and 3 years, without interest: considering money to be worth 10 per cent, what will be the gain to the buyer by paying cash? {620}
q: A note of $2000, dated July 4, 1876, due May 1, 1878, and bearing interest at 8 per cent, was cancelled October 25, 1877, by payment of the present worth at 6 per cent: what was the present worth, at this date? {2223.08} The discount? {68.92}
note: Exchange: premium and discount are reckoned on the face of the draft. A pound sterling is 20 shillings; a shilling, 12 pence. A franc is 100 centimes.
q: What is the cost of a sight draft on New York for $1400, at 1/2 per cent premium? {1407}
q: What is the cost of a sight draft on Boston, for $2580, at 1/2 per cent discount? {2567.10|2567.1}
q: What is the face of a sight draft on Wheeling, which cost $375.87, at 1/8 per cent premium? {375.40|375.4}
q: What is the cost of a sight draft on Chicago, for $2785, at 1/4 per cent discount? {2778.04}
q: What is the face of a sight draft, which cost $1852.55, at 1 1/4 per cent discount? {1876}
q: What is the cost of a draft on New Orleans for $5680, payable in 60 days, exchange being at 1/2 per cent premium, and interest 6 per cent? {5648.76}
sol: The interest of $5680 for 63 days at 6 per cent is $59.64; the premium is $28.40; $59.64 - $28.40 = $31.24 less than the face: $5648.76.
q: What is the cost of a draft on Boston for $1575, payable in 30 days, exchange being at 3/4 per cent premium, and interest 6 per cent? {1578.15}
q: The face of a draft, payable in 60 days, is $2625; exchange being at 1 1/2 per cent premium, and interest 6 per cent, what is the cost of the draft? {2636.81}
q: What will a sight bill on London, for 500 pounds 10 shillings, cost in New York, exchange being at $4.87? {2437.44|2437.435}
sol: Since 20 s. = 1 pound, 10 s. = .5 pound. If 1 pound is worth $4.87, 500.5 pounds are worth $4.87 x 500.5 = $2437.44.
q: How large a bill on London can be bought for $1808.04, exchange being at $4.88? Pounds? {370} Shillings? {10}
q: What will a bill on London for 890 pounds 8 shillings cost, exchange being at $4.86? {4327.34}
q: How large a bill on London can be bought for $2130.12, exchange being at $4.88? Pounds? {436} Shillings? {10}
q: What will a bill on Paris cost for 1290 francs, exchange being 5 fr. 15 centimes to $1? {250.49}
q: How large a bill on Paris can be bought for $1657.60, exchange being at 5 fr. 16 centimes? Francs? {8553.22|8553.216}
q: What will a bill on Berlin cost for 12680 reichsmarks, exchange being $.97 per 4 reichsmarks? {3074.90|3074.9}
q: How large a bill on Frankfort can be bought for $1470, exchange being at $.98 for 4 marks? Marks? {6000}
== 57 | Percentage | Insurance
note: The premium in fire and marine insurance is a per cent of the amount insured; the cost of the policy is added. In life insurance the annual premium is so much on each $1000.
q: What is the cost of insuring a house worth $3375, at 2/3 of its value, the premium being 1 1/2 per cent and the policy costing $1? {34.75}
sol: 2/3 of the value of the house is $2250. The premium is 1 1/2 per cent of $2250, which is $33.75; adding $1, the cost of the policy, the sum is $34.75.
q: What is the cost of insuring a house worth $5000, at 3/4 of its value, the premium being 1/2 per cent, and the policy costing $1.50? {20.25}
q: A store is valued at $12600, and the goods at $14400; 2/3 of the value of the store is insured at 3/4 per cent and 1/2 the value of the goods at 2 per cent; the cost of the two policies is $1.25 apiece: what was the total cost of insurance? {209.50|209.5}
q: A man owns a manufactory valued at $21000, and a dwelling-house worth $7200: what will it cost to insure the manufactory, at 4/7 of its value, at 1 1/2 per cent, and the house, at its full value, at 3/4 per cent, the two policies costing $1.25 each? {236.50|236.5}
q: A man's dwelling, valued at $5600, was burned; it had been insured, in a certain company, 20 years, for 3/4 of its value, at 1 1/2 per cent: how much did he receive from the company more than the sum total of the annual premiums? {2940}
q: A man secures a policy of insurance, on his house, for $3600, furniture for $1600, and library $800; the premium is 7/8 per cent, and cost of policy $1.25: what is the cost of the insurance? {53.75}
q: A hotel is insured, for 2/3 of its value, at 1 1/2 per cent; the policy costs $1.25 and the total cost of insurance is $151.25: at what sum is the hotel valued? {15000}
q: The cost of insuring a house worth $4500, for 4/5 of its value, was $32.75; the cost of the policy was $1.25: what was the per cent of insurance? {~31.5 / 3600 * 100}
q: A farmer, with an insurance of $1000 on his house, and $1500 on his barn, in the Vermont Mutual, pays an annual assessment of $3.50: what is the per cent of the premium? {~3.5 / 2500 * 100}
q: A man at the age of 40 insures his life for $5000; the company's annual premium on $1000, for a life policy at this age, is $31.30; if he dies at the age of 70, how much money will he have paid the company? {4695}
sol: Since the annual premium on $1000 is $31.30, on $5000 it is $156.50; then, the amount paid, in 30 yr., will be $156.50 x 30 = $4695.
q: Mr. Harris, aged 35, takes out an endowment policy for $10000, payable in 10 years; the annual premium on $1000, at his age, is $105.53: if he lives to receive the endowment, what will be the cost of the paid-up policy, without interest? {10553}
q: At the age of 50, the cost of a life policy, payable annually, is $47.18 on $1000; the cost of an endowment policy, payable in 20 years, is $60.45 on $1000; at the end of 20 years, how much more will have been paid on a policy of $8000 by the endowment plan than by the life plan? {2123.20|2123.2}
q: At the age of 44, a man insures his life to the amount of $12000 in favor of his wife; the company's annual premium at this age, for a life policy, is $36.46 on $1000: if the man die after the payment of 5 premiums, how much more than he paid out, will his widow receive? {9812.40|9812.4}
q: At the age of 21, a young man takes out a life policy for $5000, upon which the annual premium is $19.89 on $1000: if he lives to the age of 75, how much will it cost him to keep up his insurance? {5370.30|5370.3}
q: At the age of 30, to secure an endowment policy for $1000, payable in 10 years, costs an annual premium of $104.58; what will be the amount of the ten payments at the end of the time, allowing interest at 6 per cent? {1390.91}
sol: The ten payments are $1045.80; the first draws interest 10 years, the second 9, and so on: 55 years in all, and the interest of $104.58 for 55 years at 6 per cent is $345.11.
q: At the age of 38, a gentleman took out a policy for $6000, on the life plan, paying annually $29.15 on $1000. After keeping up his premiums for 15 years, he suffered his policy to lapse: how much money had he paid out, allowing interest at 6 per cent? {3882.78}
== 58 | Percentage | Taxes and duties
note: Subtract the poll-tax from the whole tax; divide the rest by the valuation for the rate (so many mills on each dollar). A man's tax is his property times the rate, and his poll-tax. Duties: specific, so much a pound; ad valorem, a per cent of the cost.
q: The property of a certain town is valued at $1049905; there are 483 persons subject to poll-tax. The total taxes of the town are $13323.36; the poll-tax being $1.50 for each person, what is the rate of taxation upon the property? Mills on $1? {12}
sol: The poll-tax is $1.50 x 483 = $724.50; the property tax is $13323.36 - $724.50 = $12598.86. The tax on $1 is $12598.86 divided by 1049905 = $0.012, 12 mills, or 1 1/5 per cent.
q: A tax of $2500 is assessed upon a certain district to build a school-house. The property of the district is valued at $618000, and there are 28 persons subject to poll-tax: if the poll-tax is $1, what will be the rate of taxation? Mills on $1? {4}
q: Upon a valuation of $2876475 the tax is $18409.44: there being no poll-tax, what is the rate? Mills on $1? {6.4}
q: The total valuation of property in the State of Wisconsin, for 1874, was $421285359; the tax levied upon this valuation was $656491.61: what was the rate to the hundredth of a mill? {1.56}
q: A tax of $1373.64 is assessed upon a village, the property of which is valued at $748500; 57 persons pay a poll-tax of $1.25 each: find the rate of taxation. Mills on $1? {1.74}
q: At this rate, James Turner's property is valued at $7851, and he pays poll-tax for 2 persons: what is his tax? {16.16}
sol: The tax on $7851 at 1.74 mills is $13.66; the poll-tax is $1.25 x 2 = $2.50. His tax is $16.16.
q: John Brown's property is valued at $2576, and he pays poll-tax for 1 person: what is his tax? {5.73}
q: Henry Adams' property is valued at $9265, and he pays poll-tax for 3 persons: what is his tax? {19.87}
q: Amos Clarke's property is valued at $4759, and he pays poll-tax for 1 person: what is his tax? {9.53}
q: Emily Wood's property is valued at $8367: what is her tax? {14.56}
q: The tax to be raised in a city is $64375; its taxable property is valued at $16869758; find the rate of taxation to thousandths of a mill. {3.816}
q: At this rate, William Mill's property is valued at $56875: what is his tax? {217.04}
q: Samuel Young's property is valued at $27543: what is his tax? {105.10|105.1}
q: Charles O'Neil's property is valued at $83612: what is his tax? {319.06}
q: Adolph Meyer's property is valued at $72968: what is his tax? {278.45}
q: Louis Ganot's property is valued at $69547: what is his tax? {265.39}
q: The public lands are disposed of at $1.25 per acre: what will the government receive for a township containing 36 sq. miles? {28800}
q: Letter postage is 2 ct. for each ounce, or fraction thereof: what is the necessary postage on a letter weighing 1 1/4 oz.? Cents? {4}
q: The postage on books is 1 ct. for each 2 oz., or fraction thereof: what is the postage on a book weighing 1 lb. 5 oz.? Cents? {11}
q: The tax on proof spirits is 70 ct. per gallon: what is the tax on a barrel of 40 gallons? {28}
q: The tax on cigars per 1000 is $5: how much does this enhance the price of a single cigar? Cents? {~1/2}
q: The tax on beer is $1 per barrel of 31 gal. Each wholesale dealer in malt liquors pays a special tax of $50, and each retail dealer a special tax of $20; in a certain city there are 12 wholesale dealers, 250 retail dealers, and the annual manufacture of beer is 30000 bbl.: what is the revenue to government? {35600}
q: The gross weight of a hogshead of imported sugar is 1760 lb.; allowing 12 1/2 per cent tare, what is the duty at 1 3/4 ct. per pound? {26.95}
q: A manufacturer imported from Spain 40 bales of wool, of 400 lb. each, tare 5 per cent; the cost was 45 ct. per pound: what was the duty, at 9 ct. per pound and 10 per cent ad valorem? {2052}
q: A merchant imported a case of glassware; the cost of the ware in France was 365.15 francs, the case and charges were 57.15 francs, and the commission 5 per cent: what was the duty at 40 per cent in U. S. money, reckoning the franc at 19 3/10 ct.? {34.40|34.4}
sol: The total cost being $85.58, the dutiable value is $86.00; 40 per cent of $86 is $34.40.
q: A book-seller imports a case of books; their cost in Germany was 1317.04 marks, case and charges 34.36 marks, and commission 6 per cent: what was the duty at 25 per cent in U. S. money, the mark being estimated at 23.8 ct.? {85.25}
q: A merchant imports six cases of woolen cloth, net weight 1500 lb.; the cost in England was 500 pounds, cases and charges 8 pounds 4s. 6d., commission 2 1/2 per cent: what was the duty, at 50 ct. per lb. and 35 per cent ad valorem in U. S. money, estimating the pound at $4.8665? {1637.25}
== 59 | Ratio | Ratio
note: The ratio of two numbers is the first (the antecedent) divided by the second (the consequent). Terms of different denominations must first be reduced to the same denomination.
q: What is the ratio of 6 to 3? {2}
sol: The ratio of 6 to 3 is 6 divided by 3, equal to 2.
q: What is the ratio of 2/3 to 3/4? {~2/3 / (3/4)}
sol: The ratio of 2/3 to 3/4 is 2/3 divided by 3/4, or 2/3 multiplied by 4/3, equal to 8/9.
q: What is the ratio of 12 to 3? {4} 30 to 5? {6} 35 to 7? {5} 56 to 8? {7}
q: What is the ratio of 5 to 10? {~5/10} 7 to 21? {~7/21} 12 to 18? {~12/18} 15 to 20? {~15/20}
q: What is the ratio of 15 to 25? {~15/25} 25 to 15? {~25/15} 36 to 28? {~36/28} 49 to 35? {~49/35}
q: What is the ratio of 1/2 to 2/3? {~1/2 / (2/3)} 1/3 to 3/4? {~1/3 / (3/4)} 1/3 to 1/2? {~1/3 / (1/2)} 1/5 to 1/3? {~1/5 / (1/3)}
q: What is the ratio of 1 1/2 to 3/4? {~1_1/2 / (3/4)} 3 1/2 to 2 1/4? {~3_1/2 / (2_1/4)} 5 5/6 to 2 1/3? {~5_5/6 / (2_1/3)} 6 9/10 to 4 3/5? {~6_9/10 / (4_3/5)}
q: What is the ratio of $18 to $6? {3} 54 days to 9 days? {6} 96 men to 12 men? {8} 221 bu. to 17 bu.? {13}
q: What is the ratio of 1 ft. 9 in. to 3 in.? {7} 5 yd. 1 ft. to 5 ft. 4 in.? {3}
q: 7 is the ratio of what number to 4? {28}
sol: The number is 4 multiplied by 7, equal to 28.
q: 4 is the ratio of what number to 13? {52} 5/9 is the ratio of what number to 27? {15} 7/13, of what number to 52? {28}
q: 2 5/8 is the ratio of what number to 24? {63} 4 2/5 is the ratio of what number to 1 2/3? {~4_2/5 * 1_2/3}
q: 3 is the ratio of what to 75 ct.? Cents? {225} 7/8 is the ratio of what to 4 lb. 8 oz.? Ounces? {63} 2.6 is the ratio of what to $4? {10.40|10.4}
q: 5 is the ratio of 45 to what number? {9}
sol: The number is 45 divided by 5, equal to 9.
q: 4 is the ratio of 56 to what number? {14} 7/10 is the ratio of 42 to what number? {60} 2 3/4 is the ratio of 23 3/8 to what number? {~23_3/8 / (2_3/4)}
q: 7 5/9 is the ratio of $27.20 to what? {3.60|3.6}
q: Find the value of the compound ratio 6 : 2 and 9 : 3. {9}
sol: The product of the antecedents 6 and 9 is 54, the product of the consequents 2 and 3 is 6; then, the value of the compound ratio is 54 divided by 6, equal to 9.
q: Find the value of the compound ratio 10 : 5 and 9 : 6. {3}
q: Find the value of the compound ratio 12 1/2 : 6 1/4 and 33 1/3 : 8 1/3. {8}
q: Find the value of the compound ratio 2/3 : 1/2 and 3/4 : 1 1/4. {~4/5}
q: Find the value of the compound ratio 8 men : 2 men and 12 da. : 24 da. {2}
q: Find the value of the compound ratio $6.75 : $2.25 and 6 bu. : 3 bu. {6}
q: Find the value of the compound ratio 5 : 2, 7 : 3 and 9 : 5. {~21/2}
q: Reduce 16 : 24 to its lowest terms. First term? {2} Second? {3}
sol: The G. C. D. of 16 and 24 is 8; dividing both terms by 8, it becomes 2 : 3.
q: Reduce 20 : 25 to its lowest terms. First? {4} Second? {5}
q: Reduce 10 : 30 to its lowest terms. First? {1} Second? {3}
q: Reduce 34 : 51 to its lowest terms. First? {2} Second? {3}
q: Reduce 95 : 133 to its lowest terms. First? {5} Second? {7}
q: Reduce 75 : 125 to its lowest terms. First? {3} Second? {5}
q: Reduce 217 : 279 to its lowest terms. First? {7} Second? {9}
q: Clear 1 1/2 : 2 1/3 of fractions. First? {9} Second? {14}
sol: The L. C. M. of the denominators 2 and 3 is 6; multiplying both terms by 6, it becomes 9 : 14.
q: Clear 3 3/4 : 4 2/5 of fractions. First? {75} Second? {88}
q: Clear 7 1/2 : 10 2/3 of fractions. First? {45} Second? {64}
q: Clear 5/6 : 7/9 of fractions. First? {15} Second? {14}
q: Clear 6 3/10 : 9 7/15 of fractions. First? {189} Second? {284}
== 60 | Proportion | Simple proportion
note: In every proportion the product of the extremes equals the product of the means. For the third term write the number of the same kind as the answer; for the second, the greater of the other two when the answer is to be greater, the less when it is to be less; then multiply the second and third, and divide by the first.
q: What : 6 :: 4 : 8? {3}
sol: The product of the means 6 and 4 is 24; then, 24 divided by 8, one of the extremes, equals 3, the other extreme.
q: 4 : what :: 3 x 10 : 5 x 12? {8}
sol: The product of the extremes, 4 x 5 x 12, divided by 3 x 10, one of the means, equals 8, the other mean.
q: 2 : 8 :: 6 : what? {24}
q: 5 : 7 :: 10 : what? {14}
q: What : 8 :: 6 : 16? {3}
q: 5 : what :: 6 : 12? {10}
q: 3 : 7 :: what : 14? {6}
q: 7 : 14 :: 9 : what? {18}
q: The compound ratio 2 : 4 and 8 : 9 equals what : 45? {20}
q: The compound ratio 5 : 8 and 4 : 10 equals the compound ratio 3 : 4 and 7 : what? {21}
q: The compound ratio 10 : what and 14 : 21 equals the compound ratio 22 : 33 and 26 : 39? {15}
q: 2/3 : 3/4 :: 4/5 : what? {~3/4 * 4/5 / (2/3)}
q: 3/5 : what :: 9/10 : 1 1/4? {~3/5 * 1_1/4 / (9/10)}
q: What : 4 2/3 :: 7 1/2 : 10 1/2? {~4_2/3 * 7_1/2 / (10_1/2)}
q: 4 : 6 :: 6 : what? {9}
q: If 2 yd. of cloth cost $4, what will 6 yd. cost? {12}
sol: The answer is dollars, so the third term is $4. Since 6 yd. cost more than 2 yd., 6 yd. is the second term and 2 yd. the first: 2 : 6 :: 4 : what? 6 x 4 divided by 2 is $12.
q: If 3 men can dig a cellar in 10 days, in how many days can 5 men dig it? {6}
sol: 5 men dig it in fewer days than 3 men, so 3 is the second term and 5 the first: 5 : 3 :: 10 : what? 3 x 10 divided by 5 is 6 days.
q: If 3 men can dig a cellar in 12 days, how many men will dig it in 6 days? {6}
q: If 3 yd. cloth cost $8, what cost 6 yd.? {16}
q: If 5 bl. flour cost $30, what cost 3 bl.? {18}
q: If 3 lb. 12 oz. tea cost $3.50, what cost 11 lb. 4 oz.? {10.50|10.5}
q: If 2 lb. 8 oz. of tea cost $2, what quantity can you buy for $5? Lb.? {6} Oz.? {4}
q: If 4 hats cost $14, what cost 10 hats? {35}
q: If 3 caps cost 69 cents, what cost 11 caps? Dollars? {2.53}
q: If 4 yd. cloth cost $7, what cost 9 yd.? {15.75}
q: If 8 yd. cloth cost $32, what cost 12 yd.? {48}
q: If 12 yd. cloth cost $48, what cost 8 yd.? {32}
q: If $32 purchase 8 yd. of cloth, how many yards will $48 buy? {12}
q: If $48 purchase 12 yd. of cloth, how many yards can be bought for $32? {8}
q: A man receives $152 for 19 months' work: how much should he have for 4 months' work? {32}
q: If 8 men perform a piece of work in 24 days, in what time can 12 men perform it? {16}
q: If 60 men perform a piece of work in 8 da., how many men will perform it in 2 days? {240}
q: If 15 oz. of pepper cost 25 ct., what cost 6 lb.? {1.60|1.6}
q: If 6 gal. of molasses cost $2.70, what cost 26 gal.? {11.70|11.7}
q: If 5 cwt. 85 lb. of sugar cost $42.12, what will 35 cwt. 25 lb. cost? {253.80|253.8}
q: If 1 1/2 yd. of cloth cost $2.50, what will be the cost of 1 1/8 yd.? {1.875|1.88}
q: If 90 bu. of oats supply 40 horses 6 da., how long will 450 bu. supply them? {30}
q: If 6 men build a wall in 15 da., how many men can build it in 5 da.? {18}
q: If 15 bu. of corn pay for 30 bu. of potatoes, how much corn can be had for 140 bu. potatoes? {70}
q: If 3 cwt. 25 lb. of sugar cost $22.60, what will be the cost of 16 cwt. 25 lb.? {113}
q: If a perpendicular staff, 3 ft. long, cast a shadow 4 ft. 6 in., what is the height of a steeple whose shadow measures 180 ft.? {120}
q: If a man perform a journey in 60 da., traveling 9 hr. each day, in how many days can he perform it by traveling 12 hr. a day? {45}
q: A merchant, failing, paid 60 ct. on each dollar of his debts. He owed A $2200, and B $1800: what did each receive? A? {1320} B? {1080}
q: A merchant, having failed, owes A $800.30; B $250; C $375.10; D $500; F $115; his property, worth $612.12, goes to his creditors: how much will this pay on the dollar? {.30|.3}
q: If the 4-cent loaf weigh 9 oz. when flour is $8 a bl., what will it weigh when flour is $6 a bl.? Ounces? {12}
q: I borrowed $250 for 6 mo.: how long should I lend $300 to compensate the favor? Months? {5}
q: A starts on a journey, and travels 27 mi. a day; 7 da. after, B starts and travels the same road 36 mi. a day: in how many days will B overtake A? {21}
q: If William's services are worth $15 2/3 a mo., when he labors 9 hr. a day, what ought he to receive for 4 2/5 mo., when he labors 12 hr. a day? {91.91|91.911|91 41/45}
q: If 5 lb. of butter cost $5/8, what cost 3/4 lb.? {~5/8 * 3/4 / 5}
q: If 6 yd. cloth cost $5 3/5, what cost 7 3/8 yd.? {~5_3/5 * 7_3/8 / 6}
q: If 1/3 bu. wheat cost $3/8, what cost 1/2 bu.? {~3/8 * 1/2 / (1/3)}
q: If 1 3/4 yd. cloth cost $7/24, what cost 2 yd.? {~7/24 * 2 / (1_3/4)}
q: If $29 3/4 buy 59 1/2 yd. of cloth, how much will $31 1/4 buy? Yards? {~59_1/2 * 31_1/4 / (29_3/4)}
q: If .85 of a gallon of wine cost $1.36, what will be the cost of .25 of a gallon? {.40|.4}
q: If 61.3 lb. of tea cost $44.9942, what will be the cost of 1.08 lb.? To the nearest cent? {.79|.793}
q: If 5/7 of a yard of cloth cost $3/5, what will 9/11 of a yard cost? {~3/5 * 9/11 / (5/7)}
q: If 3/7 of a yard of velvet cost $4 3/5, what cost 17 3/8 yd.? {~4_3/5 * 17_3/8 / (3/7)}
q: A wheel has 35 cogs; a smaller wheel working in it, 26 cogs: in how many revolutions of the larger wheel will the smaller gain 10 revolutions? {~260/9}
q: If a grocer, instead of a true gallon, use a measure deficient by 1 gill, what will be the true measure of 100 of these false gallons? Gallons? {~100 * 31/32}
q: If the velocity of sound be 1142 feet per sec., and the number of pulsations in a person 70 per min., what is the distance of a cloud, if 20 pulsations are counted between the time of seeing a flash of lightning and hearing the thunder? Feet? {~1142 * 20 * 60 / 70}
q: The length of a wall, by a measuring line, was 643 ft. 8 in., but the line was found to be 25 ft. 5.25 in. long, instead of 25 feet, its supposed length: what was the true length of the wall? Feet? {654} Inches? {11.17}
== 61 | Proportion | Compound proportion
note: For the third term write the number of the same kind as the answer. Arrange each pair of the other numbers as if, with the third term, they made a simple proportion. Divide the product of the second and third terms by the product of the first terms.
q: If 2 men earn $20 in 5 da., what sum can 6 men earn in 10 da.? {120}
sol: 6 men earn more than 2 men, and in 10 da. more than in 5 da.: 2 x 5 : 6 x 10 :: 20 : what? 6 x 10 x 20 divided by 2 x 5 is $120.
q: If 6 men, in 10 da., build a wall 20 ft. long, 3 ft. high, and 2 ft. thick, in how many days could 15 men build a wall 80 ft. long, 2 ft. high, and 3 ft. thick? {16}
sol: Dividing the product of 6, 80, 2, 3, and 10 by the product of 15, 20, 3, and 2, the required term is 16 da.
q: If a man travel 24 mi. in 2 da., by walking 4 hr. a day: at the same rate, how far will he travel in 10 da., walking 8 hr. a day? {240}
q: If 16 men build 18 rods of fence in 12 days, how many men can build 72 rd. in 8 da.? {96}
q: If 6 men spend $150 in 8 mo., how much will 15 men spend in 20 mo.? {937.50|937.5}
q: I travel 217 mi. in 7 days of 6 hr. each: how far can I travel in 9 days of 11 hr. each? Miles? {~217 * 9 * 11 / (7 * 6)}
q: If $100 gain $6 in 12 mo., what sum will $75 gain in 9 mo.? {3.375|3.38}
q: If 100 lb. be carried 20 mi. for 20 ct., how far will 10100 lb. be carried for $60.60? Miles? {60}
q: To carry 12 cwt. 75 lb. 400 mi., costs $57.12: what will it cost to carry 10 tons 75 mi.? {168}
q: If 18 men, in 15 da., build a wall 40 rd. long, 5 ft. high, 4 ft. thick, in what time could 20 men build a wall 87 rd. long, 8 ft. high, and 5 ft. thick? Days? {~15 * 18 * 87 * 8 * 5 / (20 * 40 * 5 * 4)}
q: If 180 men, in 6 days, of 10 hr. each, dig a trench 200 yd. long, 3 yd. wide, 2 yd. deep, in how many days can 100 men, working 8 hr. a day, dig a trench 180 yd. long, 4 yd. wide, and 3 yd. deep? {24.3}
== 62 | Partnership | Partnership and bankruptcy
note: Take such part of the whole gain or loss as each partner's stock is of the whole stock. With time: multiply each stock by the months it was used, and divide by these products. Money answers to the nearest cent.
q: A and B engaged in trade; A's capital was $200; B's, $300; they gained $100: find each partner's share. A? {40} B? {60}
sol: The whole capital is $500; A owns 2/5 and B owns 3/5 of it; A's gain is 2/5 of $100 = $40, and B's is 3/5 of $100 = $60.
q: A and B form a partnership, with a capital of $800: A's part is $300; B's, $500; they gain $232: what is the share of each? A? {87} B? {145}
q: A's stock was $70; B's, $150; C's, $80; they gained $120: what was each man's share? A? {28} B? {60} C? {32}
q: A, B, and C traded together: A put in $200; B, $400; C, $600: they gained $427.26: find each man's share. A? {71.21} B? {142.42} C? {213.63}
q: Divide $90 among 3 persons, so that the parts shall be to each other as 1, 3, and 5. First? {10} Second? {30} Third? {50}
q: Divide $735.93 among 4 men, in the ratio of 2, 3, 5, and 7. First? {86.58} Second? {129.87} Third? {216.45} Fourth? {303.03}
q: A person left an estate of $22361 to be divided among 6 children, in the ratio of their ages, which are 3, 6, 9, 11, 13, and 17 yr.: what are the shares? Age 3? {1137} Age 6? {2274} Age 9? {3411} Age 11? {4169} Age 13? {4927} Age 17? {6443}
q: Divide $692.23 into 3 parts, that shall be to each other as 1/3, 3/5, and 7/8. First? {127.60|127.6} Second? {229.68} Third? {334.95}
q: A man, failing, owes A $175; B, $500; C, $600; D, $210; E, $42.50; F, $20; G, $10; his property is worth $934.50: what part of each dollar can he pay? Cents? {60} What is A's share? {105} C's? {360}
q: A man owes A $234; B, $175; C, $326: his property is worth $492.45: what can he pay on $1? Cents? {67} What will A get? {156.78} B? {117.25} C? {218.42}
q: Mr. Smith failed in business, owing $37000. His assignee sold the stock for $25000, and charged $4650 for expenses: how much did he pay on the dollar? Cents? {55}
q: A, B, and C freighted a ship with 108 tuns of wine. A owned 48, B 36, and C 24 tuns; they were obliged to cast 45 tuns overboard: how much of the loss must each sustain? A, tuns? {20} B? {15} C? {10}
q: From a ship valued at $10000, with a cargo valued at $15000, there was thrown overboard goods valued at $1125: what per cent was the general average? {4.5} What was the loss of A, whose goods were valued at $2150? {96.75}
q: A and B built a wall for $82; A had 4 men at work 5 days, and B 3 men 7 days: how should they divide the money? A? {40} B? {42}
sol: The work of 4 men 5 da. equals 20 men 1 da.; of 3 men 7 da., 21 men 1 da. Divide $82 as 20 to 21: A has 20/41 of $82 = $40; B, 21/41 of $82 = $42.
q: A put in trade $50 for 4 mo.; B, $60 for 5 mo.; they gained $24: what was each man's share? A? {9.60|9.6} B? {14.40|14.4}
q: A and B hire a pasture for $54: A pastures 23 horses 27 da.; B, 21 horses 39 da.: what will each pay? A? {23.29} B? {30.71}
q: A put in $300 for 5 mo.; B, $400 for 8 mo.; C, $500 for 3 mo.: they lost $100; find each one's loss. A? {24.19} B? {51.61} C? {24.19}
q: A, B, and C hire a pasture for $18.12: A pastures 6 cows 30 da.; B, 5 cows 40 da.; C, 8 cows 28 da.: what shall each pay? A? {5.40|5.4} B? {6} C? {6.72}
q: Two men formed a partnership for 16 mo.: A put in, at first, $300, and, at the end of 8 mo., $100 more; B put in, at first, $600, but, at the end of 10 mo., drew out $300; they gained $442.20: find each man's share. A? {184.80|184.8} B? {257.40|257.4}
q: A and B are partners: A put in $800 for 12 mo., and B, $500. What sum must B put in at the end of 7 mo. to entitle him to half the year's profits? {720}
== 63 | Partnership | Equation of payments and average
note: Multiply each payment by the time till it is due; divide the sum of the products by the sum of the payments: the quotient is the equated time. The average price of a mixture is the whole cost divided by the whole quantity.
q: A owes B $2, due in 3 mo., and $4, due in 6 mo.: at what period can both sums be paid so that neither party will be the loser? Months? {5}
sol: The interest on $2 for 3 mo. equals the interest on $1 for 6 mo.; on $4 for 6 mo., the interest on $1 for 24 mo.; then $6 must be on interest 30 divided by 6 = 5 mo.
q: A owes B $2, due in 4 mo., and $6, due in 8 mo.: find the average time of paying both sums. Months? {7}
q: A owes B $8, due in 5 mo., and $4, due in 8 mo.: find the mean time of payment. Months? {6}
q: A buys $1500 worth of goods; $250 are to be paid in 2 mo., $500 in 5 mo., $750 in 8 mo.: find the mean time of payment. Months? {6}
q: A owes B $300; 1 third due in 6 mo.; 1 fourth in 8 mo.; the remainder in 12 mo.: what is the average time of payment? Months? {9}
q: I buy $200 worth of goods; 1 fifth to be paid now; 2 fifths in 5 mo.; the rest in 10 mo.: what is the average time of paying all? Months? {6}
q: A merchant buys goods as follows, on 60 days credit: May 1st, 1848, $100; June 15th, $200: how many days after May 1 is the average time of payment? {90}
sol: Counting from May 1, it is 60 days to the first payment and 105 days to the second; $100 x 60 + $200 x 105 = 27000, divided by $300 is 90 days from May 1st, that is, July 30th.
q: I bought goods on 90 days credit, as follows: April 2d, 1853, $200; June 1st, $300: how many days after April 2 is the average time of payment? {126}
sol: The payments fall due 90 and 150 days after April 2; (200 x 90 + 300 x 150) divided by 500 is 126 days: August 6, 1853.
q: A merchant bought goods as follows: April 6, 1876, on 3 mo., $1250; May 17, 1876, on 4 mo., $4280; June 21, 1876, on 6 mo., $675: how many days after July 6, 1876, is the average time of payment? {69}
sol: The bills fall due July 6, Sept. 17, and Dec. 21, which are 0, 73, and 168 days after July 6; the equated time is 69 days after July 6: September 13, 1876.
q: I mix 4 pounds of tea, worth 40 ct. a lb., with 6 lb., worth 50 ct. a lb.: what is 1 lb. of the mixture worth? Cents? {46}
sol: 4 lb. at 40 ct. are worth $1.60, and 6 lb. at 50 ct. are worth $3.00; 10 lb. are worth $4.60; 1 lb. is worth 46 ct.
q: Mix 6 lb. of sugar, at 3 ct. a lb., with 4 lb., at 8 ct. a lb., what will 1 lb. of the mixture be worth? Cents? {5}
q: Mix 25 lb. sugar, at 12 ct. a lb., 25 lb., at 18 ct., and 40 lb., at 25 ct.: what is 1 lb. of the mixture worth? Cents? {~1750/90}
q: A mixes 3 gal. water, with 12 gal. wine, at 50 ct. a gal.: what is 1 gal. of the mixture worth? Cents? {40}
q: I have 30 sheep: 10 are worth $3 each; 12, $4 each; the rest, $9 each: find the average value. {5}
q: On a certain day the mercury in the thermometer stood as follows: from 6 till 10 A. M., at 63 degrees; from 10 A. M. till 1 P. M., 70; from 1 till 3 P. M., 75; from 3 till 7 P. M., 73; from 7 P. M. till 6 A. M. of the next day, 55: what was the mean temperature of the day, from sunrise to sunrise? Degrees? {~1509/24}
== 64 | Powers and Roots | Involution
note: A power is the product of a number taken as a factor so many times: 3 squared is 3 x 3 = 9; 5 cubed is 5 x 5 x 5 = 125.
q: Find the cube of 75. {421875}
sol: 75 multiplied by 75 is 5625, the square of 75; 5625 multiplied by 75 is 421875, the cube.
q: Find the square of 65. {4225}
q: Find the cube of 25. {15625}
q: Find the fourth power of 12. {20736}
q: Find the fifth power of 10. {100000}
q: Find the sixth power of 9. {531441}
q: Find the eighth power of 2. {256}
q: Find the square of 2/3. {~2/3 * 2/3}
q: Find the cube of 3/4. {~3/4 * 3/4 * 3/4}
q: Find the fourth power of 4/5. {~4/5 * 4/5 * 4/5 * 4/5}
q: Find the fifth power of 2/3. {~2/3 * 2/3 * 2/3 * 2/3 * 2/3}
q: Find the square of 16 1/2. {~16_1/2 * 16_1/2}
q: Find the cube of 12 1/2. {~12_1/2 * 12_1/2 * 12_1/2}
q: Find the fourth power of .25. {.00390625}
q: 14 cubed = what? {2744}
q: 19 to the fourth power = what? {130321}
q: 2 1/3 to the fifth power = what? {~7/3 * 7/3 * 7/3 * 7/3 * 7/3}
== 65 | 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; double the root for a trial divisor, and so on. A root that does not come out even is given to 3 decimal places.
q: Extract the square root of 256. {16}
sol: The largest square in 2 is 1; subtract, and bring down 56: 156. Double the root, 2, for a trial divisor; 2 in 15 goes 6 times; 26 x 6 = 156, leaving 0. The root is 16.
q: Extract the square root of 758.436, to two decimal places. {27.53}
q: Extract the square root of 256/625. {~16/25}
q: Extract the square root of 3/8. {.612|.6123|.61237}
q: Extract the square root of 529. {23}
q: Extract the square root of 625. {25}
q: Extract the square root of 6561. {81}
q: Extract the square root of 56644. {238}
q: Extract the square root of 390625. {625}
q: Extract the square root of 1679616. {1296}
q: Extract the square root of 5764801. {2401}
q: Extract the square root of 43046721. {6561}
q: Extract the square root of 987656329. {31427}
q: Extract the square root of 289442169. {17013}
q: Extract the square root of 234.09. {15.3}
q: Extract the square root of 145.2025. {12.05}
q: Extract the square root of 915.0625. {30.25}
q: Extract the square root of .0196. {.14}
q: Extract the square root of 1.008016. {1.004}
q: Extract the square root of .00822649. {.0907}
q: Extract the square root of 10. {3.162}
q: Extract the square root of 2. {1.414}
q: Extract the square root of 25/729. {~5/27}
q: Extract the square root of 847/1183. {~11/13}
q: Extract the square root of 30 1/4. {~11/2}
q: Extract the square root of 2/3. {.816}
q: Extract the square root of 6 2/5. {2.529|2.53}
q: Extract the square root of 384 4/7. {19.61}
q: Extract the square root of 441, by factoring. {21}
sol: 441 = 3 x 3 x 7 x 7; hence its square root is 3 x 7 = 21.
q: Extract the square root of 16 by factoring. {4}
q: Extract the square root of 36 by factoring. {6}
q: Extract the square root of 100 by factoring. {10}
q: Extract the square root of 225 by factoring. {15}
q: Extract the square root of 400 by factoring. {20}
q: Extract the square root of 1764 by factoring. {42}
q: Extract the square root of 5184 by factoring. {72}
q: Extract the square root of 3025 by factoring. {55}
note: In a right-angled triangle the square on the hypotenuse equals the sum of the squares on the other two sides.
q: The base and perpendicular of a right-angled triangle are 30 and 40: what is the hypotenuse? {50}
q: The hypotenuse of a right-angled triangle is 100, and the base 60: what is the perpendicular? {80}
q: A castle 45 yd. high is surrounded by a ditch 60 yd. wide: what length of rope will reach from the outside of the ditch to the top of the castle? Yards? {75}
q: A ladder 60 ft. long reaches a window 37 ft. from the ground on one side of the street, and, without moving it at the foot, will reach one 23 ft. high on the other side: find the width of the street. Feet, to two places? {102.65}
q: A tree 140 ft. high is in the center of a circular island 100 ft. in diameter; a line 600 ft. long reaches from the top of the tree to the further shore: what is the breadth of the stream, the land on each side being of the same level? Feet, to two places? {533.44}
q: A room is 20 ft. long, 16 ft. wide, and 12 ft. high: what is the distance from one of the lower corners to the opposite upper corner? Feet? {28.284}
q: The area of a square field is 6241 sq. rd.: what is the length of one side? Rods? {79}
q: The surface of a square table contains 8 sq. ft. 4 sq. in.: what is the length of one side? Ft.? {2} In.? {10}
q: The area of a circle is 4096 sq. yd.: what is the side of a square of equal area? Yards? {64}
q: A square field measures 4 rd. on each side: what is the length of the side of a square field which contains 9 times as many square rods? {12}
q: What is the length of one side of a square lot containing 1 acre? Rods? {12.649}
== 66 | Powers and Roots | Cube root
note: Point off into periods of three figures. Find the greatest cube in the left period; for the trial divisor square the root found and multiply by 300, and so on.
q: Extract the cube root of 13824. {24}
sol: The largest cube in 13 is 8; its root is 2; the remainder with the next period is 5824. 2 x 2 x 300 = 1200, contained 4 times; 1200 + 2 x 4 x 30 + 4 x 4 = 1456; 1456 x 4 = 5824, leaving 0. The cube root is 24.
q: Extract the cube root of 413.5147, to two decimal places. {7.45}
q: Extract the cube root of 2197/13824. {~13/24}
q: Extract the cube root of 4/5. {.928}
q: Extract the cube root of 91125. {45}
q: Extract the cube root of 195112. {58}
q: Extract the cube root of 912673. {97}
q: Extract the cube root of 1225043. {107}
q: Extract the cube root of 13312053. {237}
q: Extract the cube root of 102503232. {468}
q: Extract the cube root of 529475129. {809}
q: Extract the cube root of 958585256. {986}
q: Extract the cube root of 14760213677. {2453}
q: Extract the cube root of 128100283921. {5041}
q: Extract the cube root of 53.157376. {3.76}
q: Extract the cube root of .199176704. {.584}
q: Extract the cube root of 2. {1.259|1.26}
q: Extract the cube root of 9. {2.08}
q: Extract the cube root of 200. {5.848}
q: Extract the cube root of 216/343. {~6/7}
q: Extract the cube root of 2744/6859. {~14/19}
q: Extract the cube root of 48778/118638. {~29/39}
q: Extract the cube root of 5 104/125. {~9/5}
q: Extract the cube root of 9 1/6. {2.092|2.093}
q: The contents of a cubical cellar are 1953.125 cu. ft.: find the length of one side. Feet? {12.5}
q: Sixty-four 3-inch cubes are piled in the form of a cube: what is the length of each side? Inches? {12}
q: A cubical box contains 512 half-inch cubes: what are the dimensions of the box inside? Inches each way? {4}
q: A cubical excavation contains 450 cu. yd. 17 cu. ft.: what are its dimensions? Feet each way? {23}
q: Find the side of a cube equal to a mass 288 ft. long, 216 ft. broad, and 48 ft. high. Feet? {144}
q: The side of a cubical vessel is 1 foot: find the side of another cubical vessel that shall contain 3 times as much. Feet? {1.442}
== 67 | Mensuration | Surfaces
note: Parallelogram: base x altitude. Trapezoid: half the sum of the parallel sides x altitude. Triangle: half of base x altitude. Circle: circumference = diameter x 3.1416; area = radius squared x 3.1416. Where an answer does not come out even, give it to two decimal places.
q: How many square feet in a floor 17 ft. long and 15 ft. wide? {255}
q: How many acres of land in a parallelogram, the length of which is 120 rd., and the perpendicular breadth 84 rd.? {63}
q: How many acres in a square field, each side of which is 65 rd.? A.? {26} Sq. rd.? {65}
q: How many acres in a field in the form of a rhombus, each side measuring 35 rd., and the perpendicular distance between two sides being 16 rd.? A.? {3} Sq. rd.? {80}
q: Find the difference in area between a floor 30 ft. square, and two others each 15 ft. square. Square feet? {450}
q: A table is 3 ft. 4 in. long, and 2 ft. 10 in. wide: how many sq. ft. in its surface? {~10/3 * 17/6}
sol: 3 ft. 4 in. = 10/3 ft.; 2 ft. 10 in. = 17/6 ft.; the surface is 10/3 x 17/6 = 9 4/9 sq. ft.
q: How many square feet in a marble slab 5 ft. 6 in. long and 1 ft. 8 in. wide? {~5_1/2 * 1_2/3}
q: How many square yards in a ceiling 25 ft. 9 in. long, and 21 ft. 3 in. wide? Sq. yd.? {60} Sq. ft.? {7} Sq. in.? {27}
q: A room is 10 ft. long: how wide must it be to contain 80 sq. ft.? Feet? {8}
q: How many yards of carpet, 1 1/2 yd. wide, will cover a floor 18 ft. long and 15 ft. wide? {20}
q: How many yards of flannel, 3/4 yd. wide, will it take to line 3 yd. of cloth, 1 1/2 yd. wide? {6}
q: How many yards of carpet, 1 1/4 yd. wide, will it take to cover a floor 21 ft. 3 in. long and 13 ft. 6 in. wide? {25.5}
q: A rectangular field is 15 rd. long: what must be its width to contain 1 A.? Rods? {~160/15}
q: The parallel sides of a trapezoid are 2 ft. 2 in. and 2 ft. 11 in.; its altitude is 11 in.: what is its area? Square inches? {335.5}
q: A field is in the form of a trapezoid; one of the parallel sides is 25 rd., and the other 19 rd.; the width is 32 rd.: how many acres in the field? A.? {4} Sq. rd.? {64}
q: How many square yards in a piece of roof 10 ft. 8 in. wide on the lower side, and 6 ft. 2 in. wide on the upper side, the length being 12 ft.? Sq. yd.? {11} Sq. ft.? {2}
q: The base of a triangle is 15 ft. and its altitude 12 ft.: what is its area? Square feet? {90}
q: One side of a triangular lot is 44 rd., and the perpendicular distance from the angle opposite to this side is 18 rd.: how many acres in the lot? A.? {2} Sq. rd.? {76}
q: What is the area of a triangle, of which the base is 12 ft. 6 in. and the altitude 16 ft. 9 in.? Sq. ft.? {104} Sq. in.? {99}
q: Find the area of a triangle whose sides are 13, 14, and 15 ft. Square feet? {84}
sol: Half the sum of the sides is 21; 21 - 13 = 8, 21 - 14 = 7, 21 - 15 = 6; 21 x 8 x 7 x 6 = 7056, whose square root is 84.
q: The sides of a triangle are 30, 40, and 50 ft.: what is the area? Square feet? {600}
q: Find the area of a field in the form of a trapezium, of which a diagonal is 50 rd. and the perpendiculars to the diagonal from the opposite angles 30 rd. and 20 rd. A.? {7} Sq. rd.? {130}
q: The diameter of a circle is 48 ft.: what is the circumference? Feet? {150.7968|150.80|150.8}
q: The circumference of a circle is 15 ft.: what is the diameter? Feet, to two places? {4.77}
q: The diameter of a wheel is 4 ft.: what is its circumference? Feet? {12.5664|12.57}
q: If the girth of a tree is 12 ft. 5 in., what is its diameter? Ft.? {3} In., to two places? {11.43}
q: What is the circumference of the earth, the diameter being 7912 mi.? Miles, to the nearest mile? {24856}
q: Find the area of a circle whose radius is 21 ft. Sq. yd.? {153} Sq. ft.? {8} Sq. in.? {64}
q: The area of a circle is 6 sq. ft. 98.115 sq. in.: what is its diameter? Inches? {35} Its circumference, inches to two places? {109.96}
q: How long a rope will it take to fasten a horse to a post so that he may graze over 1 A. of grass, and no more? Rd.? {7} Ft.? {2} In.? {3}
q: Two circles, 10 and 16 ft. in diameter, have the same center: what is the area of the ring between their circumferences? Sq. ft.? {122} Sq. in.? {75}
q: The area of a circle is 1 square foot, what is its diameter? Inches, to two places? {13.54}
== 68 | Mensuration | Solids and their uses
note: Prism or cylinder: volume = area of base x altitude. Pyramid or cone: one-third of that. Sphere: surface = diameter squared x 3.1416; volume = diameter cubed x .5236. Board measure: square feet x thickness in inches. A perch of stone is 24.75 cu. ft.; a bushel, 2150.4 cu. in.; a gallon, 231 cu. in.
q: Find the surface of a cube, each side being 37 in. Square inches? {8214}
q: Find the surface of a right prism, with a triangular base, each side of which is 4 ft.; the altitude of the prism is 5 ft. Square feet, to two places? {73.86|73.85}
q: Find the surface of a box which is 3 ft. 6 in. long, 2 ft. 9 in. wide, and 1 ft. 10 in. high. Square feet? {~42_1/6}
q: Find the surface of a cylinder, its altitude being 5 ft. and the radius of the base, 2 ft. Square feet, to two places? {87.96}
q: Find the volume of a right parallelepipedon, of which the length is 12 ft., the width 3 ft. 3 in., and the height 4 ft. 4 in. Cubic feet? {169}
q: How many cubic yards in a room 24 ft. long, 18 ft. 6 in. wide, and 10 ft. 7 in. high? Cu. yd.? {174} Cu. ft.? {1}
q: Each side of the base of a triangular prism is 2 ft.; its altitude is 14 ft.: what is the volume of the prism? Cubic feet, to two places? {24.25|24.24}
q: Find the volume of a cylinder whose altitude is 12 ft. and the radius of the base 2 ft. Cubic feet, to one place? {150.8}
q: How many cubic inches in a peck measure, the diameter of the bottom being 9 1/4 in. and the depth 8 in.? To one place? {537.6}
q: Find the entire surface of a right pyramid, with a triangular base, each side of which is 5 ft. 4 in.; the slant height of the pyramid is 7 ft. 6 in. Square feet, to one place? {72.3}
q: What is the convex surface of a cone of which the slant height is 25 ft. and the diameter of the base 8 ft. 6 in.? Square feet, to two places? {333.79|333.80|333.8}
q: Find the entire surface of a cone, of which the slant height is 4 ft. 7 in. and the diameter of the base 2 ft. 11 in. Square feet, to two places? {27.68}
q: Find the volume of a square pyramid, of which each side of the base is 5 ft. and the altitude 21 ft. Cubic feet? {175}
q: Find the volume of a cone, of which the altitude is 15 ft. and the radius of the base 5 ft. Cubic feet, to one place? {392.7}
q: A square pyramid is 477 ft. high; each side of its base is 720 ft.: how many cubic yards in the pyramid? {3052800}
q: The diameter of the base of a conical glass house, is 37 ft. 8 in., and its altitude 79 ft. 9 in.: what is the space inclosed? Cubic feet, to the nearest foot? {29622}
q: What is the surface of a sphere, of which the diameter is 1 ft.? Square feet? {3.1416}
q: What is the surface of a sphere, of which the diameter is 4 ft. 6 in.? Square feet, to one place? {63.6}
q: What is the area of the earth's surface, on the supposition that it be a perfect sphere 7912 miles in diameter? Square miles, to the nearest mile? {196663356|196663355}
q: Find the volume of a sphere 13 ft. in diameter. Cubic feet, to one place? {1150.3|1150.4}
q: Find the volume of a sphere 2 ft. 6 in. in diameter. Cubic feet, to two places? {8.18}
q: The volume of a sphere is 1 cu. ft.: what is its diameter? Inches, to one place? {14.9}
q: A room is 20 ft. 6 in. long, 16 ft. 3 in. broad, 10 ft. 1 in. high: how many yards of plastering in it, walls and ceiling, deducting a fire-place 6 ft. 3 in. by 4 ft. 2 in.; a door 7 ft. by 4 ft. 2 in., and two windows, each 6 ft. by 3 ft. 3 in.? Square yards, to two places? {108.89}
q: A room is 20 ft. long, 14 ft. 6 in. broad, and 10 ft. 4 in. high: what will the papering of the walls cost, at 27 ct. per square yard, deducting a fire-place 4 ft. by 4 ft. 4 in., and two windows, each 6 ft. by 3 ft. 2 in.? {19.73}
q: What will it cost to pave a rectangular court, 21 yd. long and 15 yd. broad, in which a foot-path, 5 ft. wide, runs the whole length: the path paved with flags, at 36 ct. per square yard, and the rest with bricks, at 24 ct. per square yard? {79.80|79.8}
q: At 10 ct. a square yard, what will it cost to paint both sides of a partition 15 ft. 6 in. long, 12 ft. 6 in. high? {4.31}
q: A house has three tiers of windows, seven in a tier: the height of the first tier is 6 ft. 11 in.; of the second, 5 ft. 4 in.; of the third, 4 ft. 3 in.; each window is 3 ft. 6 in. wide: what will the glazing cost, at 16 ct. per square foot? {64.68}
q: A floor is 36 ft. 3 in. long, 16 ft. 6 in. wide: what will it cost to lay it, at $3 a square of 100 sq. ft.? {17.94}
q: At $3.50 per square of 100 sq. ft., what will be the cost of a roof 40 ft. long, the rafters on each side 18 ft. 6 in. long? {51.80|51.8}
q: How many feet, board measure, in an inch board 16 ft. long and 1 ft. 3 in. wide? {20}
q: How many feet in a two-inch plank 12 ft. 6 in. long and 2 ft. 3 in. wide? {56.25}
q: How many feet in a piece of scantling 15 ft. long, 4 in. wide, and 3 in. thick? {15}
q: How many feet of inch boards will a stick of timber 12 ft. long and 2 ft. square make? {576}
q: How many feet in an inch board, 12 ft. 6 in. long, 1 ft. 3 in. wide at one end, and 11 in. wide at the other end? {~13_13/24}
q: How many perches in a stone wall 97 ft. 5 in. long, 18 ft. 3 in. high, 2 ft. 3 in. thick? To one place? {161.6}
q: What is the cost of a stone wall 53 ft. 6 in. long, 12 ft. 6 in. high, 2 ft. thick, at $2.25 a perch? {121.59}
q: How many bricks in a wall 48 ft. 4 in. long, 16 ft. 6 in. high, 1 ft. 6 in. thick, allowing 20 bricks to the cubic ft.? {23925}
q: How many bricks, each 8 in. long, 4 in. wide, 2.25 in. thick, will be required for a wall 120 ft. long, 8 ft. high, and 1 ft. 6 in. thick? {34560}
q: Find the cost of building a wall 240 ft. long, 6 ft. high, 3 ft. thick, at $3.25 per 1000, each brick being 9 in. long, 4 in. wide, and 2 in. thick. {336.96}
q: How many bushels in a bin 15 ft. long, 5 ft. wide, and 4 ft. deep? To the nearest bushel? {241}
q: How many gallons in a trough 10 ft. long, 5 ft. wide, and 4 ft. deep? To the nearest gallon? {1496}
q: How many bushels in a cylindrical tub 6 ft. in diameter and 8 ft. deep? To two places? {181.76}
q: How many barrels, of 31 1/2 gal. each, in a cistern, in the form of a cylinder, of which the diameter is 4 ft. and the depth 6 ft.? To one place? {17.9}
== 69 | Progressions | Arithmetical and geometrical progression
note: Arithmetical: the last term is the first, with the common difference added (or taken away) one time less than the number of terms; the sum is the sum of the extremes times the number of terms, halved. Geometrical: the last term is the first times the ratio raised to one less than the number of terms; the sum is (the greatest term x the ratio, less the least term) divided by the ratio less 1.
q: I bought 10 yd. of muslin, at 3 ct. for the first yard, 7 ct. for the second, 11 ct. for the third, and so on: what did the last yard cost? Cents? {39}
sol: To the cost of the first yard add 4 ct. nine times: 9 x 4 = 36, and 3 + 36 = 39 ct.
q: The first term of a decreasing series is 39, the common difference 4, and the number of terms 10: find the last term. {3}
q: Find the last term of an increasing series in which the first term is 2, the common difference 3, and the number of terms 50. {149}
q: What is the 54th term of a decreasing series in which the first term is 140, and common difference 2? {34}
q: What is the 99th term of a decreasing series in which the first term is 329, and common difference 7/8? {~329 - 98 * 7/8}
q: The first term of a series is 2, the last 20, and the number of terms 7: what is the common difference? {3}
sol: The difference of the extremes is 18; 18 divided by 6, the number of terms less 1, is 3.
q: The extremes are 3 and 300; the number of terms 10: find the common difference. {33}
q: A travels from Boston to Bangor in 10 da.; he goes 5 mi. the first day, and increases the distance traveled each day by the same number of miles; on the last day he goes 50 mi.: find the daily increase. Miles? {5}
q: Find the sum of 6 terms of the series whose first term is 1, and last term 11. {36}
sol: 1, 3, 5, 7, 9, 11 and the same written backward make six sums of 12; 6 x 12 = 72 is twice the series, which is therefore 36.
q: The extremes are 2 and 50; the number of terms, 24: find the sum of the series. {624}
q: How many strokes does the hammer of a clock strike in 12 hours? {78}
q: Place 100 apples in a right line, 3 yd. from each other, the first, 3 yd. from a basket: what distance will a boy travel who gathers them singly and places them in the basket? Yards? {30300}
q: A body falling by its own weight would descend in the first second a space of 16 ft. 1 in.; the next second, 3 times that space; the third, 5 times; the fourth, 7 times, etc.: at that rate, through what space would it fall in 1 minute? Feet? {57900}
q: The first term of an increasing geometric series is 2; the ratio, 3: what is the fifth term? {162}
sol: The fifth term is 2 multiplied by 3 taken four times as a factor: 2 x 81 = 162.
q: The first term of a decreasing geometric series is 192; the ratio, 2: what is the fourth term? {24}
q: The first term of an increasing series is 2; the ratio, 2; the number of terms, 13: find the last term. {8192}
q: The first term of a decreasing series is 262144; the ratio, 4; number of terms, 9: find the last term. {4}
q: The first term of an increasing series is 10; the ratio, 3: what is the tenth term? {196830}
q: Find the sum of 5 terms of the geometric series, whose first term is 4, and ratio 3. {484}
sol: 324 x 3 = 972; 972 - 4 = 968; divided by 3 - 1 = 2, the sum is 484.
q: The first term is 10; the ratio, 3; the number of terms, 7: what is the sum of the series? {10930}
q: A father gave his daughter on New Year's day $1; he doubled it the first day of every month for a year: what sum did she receive? {4095}
q: I sold 1 lb. of gold (12 oz. Troy) at 1 ct. for the first oz.; 4 ct. for the second, 16 ct. for the third, etc.: what sum did I get? {55924.05}
q: Find the sum of an infinite series, of which the greatest term is .3 and the ratio, 10; that is, of 3/10 + 3/100 + 3/1000, etc. {~1/3}
q: Find the sum of the infinite series 1/3, 1/9, 1/27, etc. {~1/2}
q: Find the sum of the infinite series 1/2, 1/4, 1/8, etc. {1}
EOFmake-book.ps1: builds any book: -Book practical writes public\books\practical.json.
cat > /root/rays-rebuild/make-book.ps1 <<'EOF'
# Builds public/books/<id>.json from a hand-corrected book in source/<Book>.txt (format at the top of source/primary.txt).
# Generates every table exactly, works out {?} answers, and checks typed answers wherever the question is plain
# arithmetic ("How many are 3 and 5?", "One from 9?", "Nine and how many make 17?", "How many twos in 8?").
# Usage: powershell -ExecutionPolicy Bypass -File RaysArithmetic\tools\make-book.ps1 [-Book primary|intellectual|...]
param([string]$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"
EOFmake-audio.ps1: records every line of a book with Eloquence: -Id practical.
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"
EOFapp.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', ' '); 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', ' '); const w = el('div', 'word', ' ');
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 };
})();
EOFindex.html: the screens and the dark look.
cat > /root/rays-rebuild/index.html <<'EOF'
<!doctype html>
<html lang="en"><head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width,initial-scale=1,maximum-scale=1,user-scalable=no">
<meta name="theme-color" content="#121212">
<title>Ray's Arithmetic</title>
<link rel="icon" type="image/png" sizes="192x192" href="icon-192.png">
<link rel="manifest" href="manifest.webmanifest">
<link rel="apple-touch-icon" href="icon-192.png">
<meta name="apple-mobile-web-app-capable" content="yes">
<meta name="mobile-web-app-capable" content="yes">
<meta name="apple-mobile-web-app-status-bar-style" content="black">
<meta name="apple-mobile-web-app-title" content="Ray's Math">
<style>
:root{color-scheme:dark;--paper:#121212;--ink:#ececec;--soft:#a8a8a8;--accent:#2f8fbf;--warm:#d9733f;--good:#3fae6e;--bad:#ff6b5e;--board:#1d3527;--card:#1e1e1e;--line:#6a6a6a;--hi:#1f3340;--a:#f0a03c;--b:#5ab4f0}
*{box-sizing:border-box;-webkit-tap-highlight-color:transparent}
html,body{margin:0;height:100%;background:var(--paper);color:var(--ink);font-family:Georgia,"Times New Roman",serif;overflow:hidden;user-select:none;-webkit-user-select:none;-webkit-text-size-adjust:100%;text-size-adjust:100%}
.screen{position:absolute;inset:0;display:none;flex-direction:column;padding:10px 16px}
.screen.on{display:flex}
h1{margin:4px 0 10px;font-size:clamp(26px,6vmin,44px);text-align:center;letter-spacing:.04em}
h2{margin:4px 0;font-size:clamp(19px,4.2vmin,30px);text-align:center}
button{font:inherit;font-size:clamp(18px,4vmin,26px);border:3px solid var(--line);background:var(--card);color:var(--ink);border-radius:14px;padding:8px 16px;min-height:50px;cursor:pointer}
button:active{transform:scale(.96)}
button.big{background:var(--accent);color:#fff;border-color:#164a63;font-size:clamp(21px,4.6vmin,30px);padding:12px 26px}
button.ghost{border-color:transparent;background:transparent}
button:disabled{opacity:.35}
.row{display:flex;gap:10px;justify-content:center;flex-wrap:wrap;align-items:center}
.grow{flex:1;min-height:0;position:relative}
.kids button{min-width:30vmin;font-size:clamp(24px,6vmin,40px)}
.kids button.sel{background:var(--accent);border-color:var(--accent);color:#fff}
.kids button.add,.kids button.ghost,.kids button.big{min-width:0}
.kids input{font:inherit;font-size:clamp(22px,5vmin,34px);width:min(60vw,320px);padding:8px 14px;border-radius:14px;border:3px solid var(--line);background:var(--card);color:var(--ink)}
#home{overflow-y:auto}
.shelf{display:flex;gap:3vmin;justify-content:center;flex-wrap:wrap;margin-top:3vmin;padding:8px}
.book{width:min(34vmin,210px);aspect-ratio:3/4;border-radius:6px 14px 14px 6px;color:#fff;padding:14px;display:flex;flex-direction:column;justify-content:space-between;box-shadow:inset 10px 0 0 rgba(0,0,0,.18),0 4px 10px rgba(0,0,0,.25);border:none;text-align:left}
.book b{font-size:clamp(18px,4vmin,26px)}.book small{font-size:clamp(13px,2.6vmin,17px);opacity:.9}
.book .bp{font-size:clamp(14px,3vmin,19px);font-weight:bold;background:rgba(0,0,0,.35);border-radius:8px;padding:4px 8px;align-self:flex-start}
.book.soon{filter:grayscale(.6);opacity:.6}
.book.cur{outline:5px solid #ffd166;outline-offset:3px;animation:pulse 1.6s infinite}
.map{display:grid;grid-template-columns:repeat(auto-fill,minmax(min(22vmin,120px),1fr));grid-auto-rows:min(24vmin,120px);gap:12px;overflow-y:auto;padding:6px;align-content:start}
.tile{height:100%;min-height:0;border-radius:18px;display:flex;flex-direction:column;align-items:center;justify-content:center;font-size:clamp(22px,6vmin,38px);padding:4px;line-height:1.1}
.tile small{font-size:clamp(11px,2.4vmin,15px);color:var(--soft)}
.tile .st{font-size:clamp(14px,3vmin,20px);color:#e0a000;letter-spacing:2px}
.tile.next{background:var(--hi);border-color:var(--accent);animation:pulse 1.4s infinite}
.tile.locked{opacity:.35}
.jump{display:flex;align-items:center;gap:8px;flex-wrap:wrap;margin:6px 0;font-size:clamp(16px,3.4vmin,22px)}
.jump select{flex:1;min-width:0;max-width:100%;font:inherit;padding:8px;background:var(--card);color:var(--ink);border:2px solid var(--line);border-radius:12px}
.tile.rev{border-style:dashed}
@keyframes pulse{50%{transform:scale(1.05)}}
.top{display:flex;align-items:center;gap:8px}
.top .t{flex:1;text-align:center;font-size:clamp(17px,3.8vmin,26px)}
.steps{display:flex;gap:6px;justify-content:center;margin:4px 0}
.steps i{width:clamp(18px,4vmin,28px);height:clamp(18px,4vmin,28px);border-radius:50%;border:3px solid var(--soft);font-style:normal;font-size:12px;display:flex;align-items:center;justify-content:center}
.steps i.done{background:var(--good);border-color:var(--good)}.steps i.cur{border-color:var(--accent);background:var(--hi)}
.hint{font-size:clamp(15px,3.2vmin,20px);color:var(--soft);text-align:center;min-height:1.4em}
.toast{position:fixed;left:50%;top:40%;transform:translate(-50%,-50%);background:#2a2a2a;border:3px solid var(--accent);color:#fff;padding:18px 28px;border-radius:18px;font-size:clamp(24px,6vmin,40px);display:none;z-index:9;text-align:center}
.stars{font-size:clamp(40px,12vmin,90px);color:#e0a000;text-align:center;letter-spacing:6px}
.parent{font-size:15px;color:var(--soft)}
.col{display:flex;flex-direction:column;align-items:center;gap:1.2vmin;height:100%;overflow-y:auto;justify-content:center}
.big-n{font-size:clamp(70px,22vmin,170px);line-height:1;font-weight:bold}
.word{font-size:clamp(22px,5vmin,36px);color:var(--soft)}
.fact{font-size:clamp(30px,8vmin,64px);text-align:center;line-height:1.15}
.fact .sm{display:block;font-size:clamp(16px,3.4vmin,22px);color:var(--soft)}
.dots{display:flex;flex-wrap:wrap;gap:1.4vmin;justify-content:center;align-items:center;max-width:94vw}
.grp{display:grid;grid-template-columns:repeat(5,auto);gap:.8vmin;padding:1vmin;border-radius:10px;border:2px dashed transparent}
.grp.box{border-color:var(--line)}
.dot{width:clamp(14px,3.6vmin,26px);height:clamp(14px,3.6vmin,26px);border-radius:50%;background:var(--a)}
.dot.b{background:var(--b)}.dot.x{background:#444;position:relative}.dot.x::after{content:'';position:absolute;left:-2px;right:-2px;top:45%;height:3px;background:var(--bad);transform:rotate(-35deg)}
.op{font-size:clamp(26px,6vmin,44px);color:var(--soft)}
.chips{display:flex;flex-wrap:wrap;gap:6px;justify-content:center;max-height:24vh;overflow-y:auto}
.chip{font-size:clamp(14px,3vmin,19px);min-height:38px;padding:4px 10px;border-radius:10px}
.chip.cur{background:var(--hi);border-color:var(--accent)}.chip.seen{border-color:var(--good)}
.balls{display:flex;flex-wrap:wrap;gap:2vmin;justify-content:center;max-width:90vw}
.ball{width:clamp(44px,11vmin,80px);height:clamp(44px,11vmin,80px);border-radius:50%;border:none;background:radial-gradient(circle at 35% 30%,#ffd9a8,var(--warm) 60%,#7a3412);padding:0;min-height:0;font-size:clamp(18px,4vmin,28px);color:#fff;font-weight:bold}
.ball.on{background:radial-gradient(circle at 35% 30%,#b8f0c8,var(--good) 60%,#1b5a35)}
.rods{display:flex;gap:1vmin;align-items:flex-end;justify-content:center;flex-wrap:wrap}
.rod{display:grid;grid-template-rows:repeat(10,1fr);gap:2px;height:clamp(110px,26vmin,190px);width:clamp(14px,3.4vmin,22px)}
.rod i{background:var(--b);border-radius:3px}
.choices{display:grid;grid-template-columns:repeat(2,minmax(0,1fr));gap:2vmin;width:min(90vw,520px)}
.choices button{font-size:clamp(40px,12vmin,86px);min-height:clamp(80px,18vmin,140px)}
.choices button.ok{background:var(--good);border-color:var(--good);color:#fff}.choices button.no{animation:shake .4s}
@keyframes shake{25%{transform:translateX(-8px)}75%{transform:translateX(8px)}}
.grid{display:flex;flex-wrap:wrap;gap:8px;justify-content:center;overflow-y:auto;max-height:100%;align-content:flex-start;padding:4px}
.grid button{min-width:clamp(56px,14vmin,90px);font-size:clamp(24px,6vmin,40px)}
.grid button.seen{border-color:var(--good)}
/* question + number pad */
.study{gap:1vmin}.study .chips{max-height:15vh;flex-shrink:0}.art{flex:1;min-height:0;overflow-y:auto;width:min(94vw,760px);background:#fffdf6;border-radius:12px;padding:10px 14px;color:#222}
.art{display:flex;flex-direction:column;gap:10px}.art .figs{display:flex;flex-wrap:wrap;gap:8px;justify-content:center;align-items:center}
.art img.fig{display:block;max-width:100%;max-height:26vh;mix-blend-mode:multiply}
@media (min-aspect-ratio:5/4){.art{flex-direction:row;align-items:flex-start}.art .figs{flex:0 0 38%;flex-direction:column;position:sticky;top:0}.art img.fig{max-height:30vh}.art p{flex:1}}
.art p{margin:0;font-size:clamp(18px,3.8vmin,28px);line-height:1.45}
.mc{display:flex;flex-direction:column;gap:1.4vmin;width:min(92vw,640px)}
.mc button{font-size:clamp(18px,4vmin,28px);min-height:56px;padding:8px 14px;text-align:left;line-height:1.3}
.mc button.ok{background:var(--good);border-color:var(--good);color:#fff}.mc button.no{animation:shake .4s;opacity:.55}
.qa{display:flex;flex-direction:column;align-items:center;gap:1vmin;height:100%;min-height:0}
.q{font-size:clamp(20px,4.6vmin,34px);line-height:1.35;text-align:center;overflow-y:auto;max-height:42%;padding:4px 10px;border-radius:12px}
.q .pt{opacity:.55}.q .pt.cur{opacity:1;background:var(--hi);border-radius:8px;padding:0 4px}
.q .got{display:inline-block;background:var(--good);color:#fff;border-radius:8px;padding:0 8px;margin:0 4px;font-weight:bold}
.q .miss{background:var(--warm)}
.q.has-stack{max-height:60%}.q.has-stack .pt{font-size:.85em}
.stack{display:table;margin:.2em auto;font-family:'Courier New',monospace;font-weight:bold;text-align:right;line-height:1.05;font-size:1.05em;border-bottom:3px solid currentColor;padding:0 .3em}
.stack b{float:left;margin-right:.6em}.stack .sum{color:var(--good);border-top:3px solid currentColor;margin-bottom:-3px}
.pic{font-size:clamp(30px,8vmin,56px);line-height:1}
.expr{font-size:clamp(46px,13vmin,100px);font-weight:bold;line-height:1.05;text-align:center}
.micb{min-width:min(86vw,360px);font-size:clamp(20px,4.6vmin,30px)}.micb.live{background:var(--bad);border-color:var(--bad);color:#fff;animation:pulse 1s infinite}
.expr.long{font-size:clamp(28px,7vmin,52px)}
.entry{min-width:3.4em;min-height:1.3em;font-size:clamp(36px,9vmin,64px);text-align:center;border-bottom:4px solid var(--accent);font-weight:bold;letter-spacing:.05em}
.entry.long{font-size:clamp(24px,6vmin,44px)}.entry.no{animation:shake .4s;border-color:var(--bad)}.entry.ok{color:var(--good)}
.pad{display:grid;grid-template-columns:repeat(3,1fr);gap:1.2vmin;width:min(86vw,360px)}
.pad button{font-size:clamp(24px,6vmin,38px);min-height:clamp(46px,9.5vmin,70px);padding:0}
.pad button.okb{background:var(--good);border-color:var(--good);color:#fff}
.algmode .entry{max-width:92vw;overflow-wrap:anywhere;letter-spacing:0}.algmode .entry.long{font-size:clamp(18px,4.2vmin,32px)}
.pad.alg,.algmode .pad{grid-template-columns:repeat(6,1fr);width:min(92vw,480px)}.pad.alg button,.algmode .pad button{font-size:clamp(20px,5vmin,32px);min-height:clamp(40px,8vmin,60px)}.pad.alg button.let{font-style:italic;font-family:Georgia,serif}
.seq{display:flex;flex-wrap:wrap;gap:6px;justify-content:center;max-height:22vh;overflow-y:auto}
.seq span{min-width:2.2em;text-align:center;border:2px solid var(--line);border-radius:8px;padding:2px 6px;font-size:clamp(18px,4vmin,28px)}
.seq span.got{border-color:var(--good);color:var(--good)}.seq span.cur{border-color:var(--accent);background:var(--hi)}
.sol{font-size:clamp(16px,3.4vmin,22px);color:#ffd166;text-align:center;max-width:90vw}
.tray{position:fixed;left:50%;bottom:2vh;transform:translateX(-50%);background:#1a1a1a;border:3px solid var(--accent);border-radius:16px;padding:10px;z-index:20;display:none;flex-direction:column;align-items:center;gap:8px}
.tray.on{display:flex}
.frame{display:grid;grid-template-columns:repeat(10,1fr);gap:5px}
.frame button{width:clamp(26px,6.5vmin,44px);height:clamp(26px,6.5vmin,44px);min-height:0;border-radius:50%;padding:0;border-width:2px}
.frame button.on{background:var(--a);border-color:var(--a)}
.timer{font-size:clamp(16px,3.4vmin,22px);color:var(--soft)}
.stk{font-size:clamp(40px,10vmin,72px)}
.stickers{display:flex;flex-wrap:wrap;gap:10px;justify-content:center;overflow-y:auto;font-size:clamp(34px,9vmin,60px)}
.stickers span{width:1.5em;height:1.5em;display:flex;align-items:center;justify-content:center;border:2px dashed var(--line);border-radius:12px}
.stickers span.got{border:2px solid #e0a000;background:#2a2410}
canvas{position:absolute;inset:0;width:100%;height:100%;touch-action:none;border-radius:14px}
.paper{background:#1a1a1a;border:3px solid var(--line);border-radius:14px}
.note{font-size:clamp(13px,2.8vmin,17px);color:var(--soft);text-align:center;max-width:90vw}
/* show your work: scratch paper beside the question (landscape) or under the number pad (portrait), never over them */
#controls button.on{background:var(--hi);border-color:var(--accent)}
#stage.working{display:flex;flex-direction:column;gap:8px}
#stage.working .qa{flex:1 1 auto;height:auto;min-height:0;overflow-y:auto}
#stage.working .qa .pic{display:none}
#stage.working .pad button{min-height:clamp(42px,7vmin,60px)}#stage.working .micb{min-height:44px;padding:4px 16px;font-size:clamp(17px,3.6vmin,24px)}#stage.working .entry{font-size:clamp(28px,6.5vmin,50px)}#stage.working .expr{font-size:clamp(32px,7.5vmin,64px)}#stage.working .expr.long{font-size:clamp(24px,5.5vmin,44px)}#stage.working .qa{gap:.6vmin;overflow:hidden}#stage.working .qa>*{flex-shrink:0}#stage.working .qa>.q{flex:0 1 auto;min-height:2.8em;max-height:24vh}#stage.working .qa>.picbox{flex:0 1 auto;overflow-y:auto}
.scratch{flex:0 0 35%;display:flex;flex-direction:column;gap:6px;min-height:0;min-width:0}
.sarea{flex:1;position:relative;min-height:0;border:3px solid var(--line);border-radius:14px;overflow:hidden;background:#fffdf6}
.sarea canvas{border-radius:0}
.stools{display:flex;gap:6px;justify-content:center;flex-wrap:wrap}
.stools button{font-size:clamp(16px,3.2vmin,22px);min-height:clamp(46px,8vmin,60px);padding:4px 12px;flex:1 1 0;max-width:180px;white-space:nowrap}
.stools button.on{background:var(--accent);border-color:#164a63;color:#fff}
.erz{display:inline-block;width:1.1em;height:.65em;background:linear-gradient(90deg,#f4a3b4 60%,#5a8fd0 60%);border-radius:3px;transform:rotate(-20deg);vertical-align:middle}
@media (min-aspect-ratio:5/4){#stage.working{flex-direction:row}#stage.working .qa{flex:0 0 44%;height:100%}#stage.working .qa .pic{display:block}.scratch{flex:1 1 0}}
.worklist{display:flex;flex-wrap:wrap;gap:12px;justify-content:center;align-content:flex-start;overflow-y:auto;padding:6px}
.wles{flex-basis:100%;margin:10px 0 0;font-size:clamp(17px,3.6vmin,24px)}
.wpage{display:flex;flex-direction:column;gap:4px;padding:6px;text-align:left;font-size:15px;line-height:1.3;max-width:min(316px,46vw);border-width:2px}
.wpage canvas,.workbig canvas{position:static;border-radius:6px}
.wcap{font-size:clamp(14px,2.8vmin,18px);max-height:4.2em;overflow:hidden}
.workbig .wcap{max-height:none;text-align:center;max-width:94vw}
.wres{color:var(--soft);font-size:14px}.wres.ok{color:var(--good)}.wres.miss{color:var(--a)}
.workbig{position:fixed;inset:0;background:var(--paper);z-index:30;display:none;flex-direction:column;align-items:center;justify-content:center;gap:10px;padding:12px}
.workbig.on{display:flex}
</style>
</head><body>
<section id="home" class="screen on">
<h1>Ray's Arithmetic</h1>
<h2>Who is doing math today?</h2>
<div class="row kids" id="kids"></div>
<div class="hint" id="whoHint"></div>
<div class="shelf" id="shelf"></div>
<div class="row" style="margin-top:auto"><button class="ghost parent" id="speedBtn">Voice: normal</button><button class="ghost parent" id="toStickers">Sticker book</button><a class="parent" href="/account/">Grown-ups: account</a><a class="parent" href="/readers/">Readers</a><a class="parent" href="/">About</a></div>
</section>
<section id="map" class="screen">
<div class="top"><button onclick="Rays.go('home')">◀</button><div class="t" id="mapTitle"></div></div>
<div class="jump"><label for="jumpSel">Go to lesson</label><select id="jumpSel"></select><button id="jumpGo">Go</button><button id="mapWork" class="parent" title="Grown-ups: the scratch paper saved with each question">✏️ See the work</button></div>
<div class="map grow" id="tiles"></div>
</section>
<section id="lesson" class="screen">
<div class="top"><button onclick="Rays.leave()">◀</button><div class="t" id="lesTitle"></div><button id="sayBtn" title="Say it again">🔊</button></div>
<div class="steps" id="steps"></div>
<h2 id="stepTitle"></h2>
<div class="grow" id="stage"></div>
<div class="hint" id="hint"></div>
<div class="row" id="controls"></div>
</section>
<section id="stickers" class="screen">
<div class="top"><button onclick="Rays.go('home')">◀</button><div class="t" id="stkTitle">Sticker book</div></div>
<div class="stickers grow" id="stkList"></div>
<div class="row"><button id="stkWork" class="parent">✏️ See the work</button></div>
</section>
<section id="work" class="screen">
<div class="top"><button id="workBack">◀</button><div class="t" id="workTitle">Work</div></div>
<div class="worklist grow" id="workList"></div>
<div class="row"><button id="workDel" class="ghost parent">Delete all work in this book</button></div>
<div class="workbig" id="workBig"></div>
</section>
<div class="tray" id="tray"><div class="frame" id="frame"></div><div class="row"><b id="trayN">0</b><button id="trayClear">Clear</button><button id="trayClose">Close</button></div></div>
<div class="toast" id="toast"></div>
<script src="hershey.js"></script>
<script src="trace.js"></script>
<script src="/account/sync.js"></script>
<script src="scratch.js"></script>
<script src="alg.js"></script>
<script src="app.js"></script>
<script>if ('serviceWorker' in navigator) navigator.serviceWorker.register('sw.js').catch(() => {});</script>
</body></html>
EOFprompts.json: the spoken instructions.
cat > /root/rays-rebuild/prompts.json <<'EOF'
{
"home": "Who is doing math today? Tap your name. Then tap the book.",
"greet": "Hi, {kid}! Tap the book to start.",
"hello": "Hi! Tap the book to start.",
"greatJob": "Great job!",
"map": "Pick a lesson. The glowing one is next.",
"count": "Count the balls. Tap each ball, and count out loud.",
"numbers": "Learn the numbers. Listen, and say each number with me.",
"find": "Find the number. Listen, then tap the number you hear.",
"say": "Read the numbers. Say each number out loud to a grown-up. Tap a number to hear it.",
"write": "Write the numbers. Listen, then type the number you hear, and tap OK.",
"trace": "Time to trace the numbers. Start at the green dot, and follow the arrow.",
"learn": "Learn the table. Listen, and say it with me.",
"rules": "Learn the table. Listen, and say it with me.",
"drill": "Practice time. Type the answer, then tap OK. Try to beat your best time!",
"problems": "Story problems. Listen to each question. Type the answer, then tap OK. Tap the counters if you need help.",
"study": "Study the book. Listen to each part, and look at the figure. Tap a part to hear it again.",
"quiz": "Quiz time. Listen to each question, then tap the right answer.",
"yes": "Yes!",
"tryAgain": "Not quite. Try again.",
"done": "You finished the lesson! You earned a sticker."
}
EOFsw.js: offline support (now keeps Book 3 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)));
});
EOFRebuild, step by step
1. Build the book data
PC (PowerShell). It prints "69 lessons, 2430 questions, 3273 answers", or a list of problems:
powershell -ExecutionPolicy Bypass -File C:\RaysArithmetic\tools\make-book.ps1 -Book practical2. Record the voice
PC (PowerShell), about 15 minutes:
powershell -ExecutionPolicy Bypass -File C:\RaysArithmetic\tools\make-audio.ps1 -Id practical3. 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 -Audio4. Check it
Open https://abcbooks.fyi/math/ , tap a name, and tap the green Book 3. Lesson 1 asks you to write "twenty-three" in figures. Lessons open one after another.
Troubleshooting
- Book 3 says "Coming soon":
app.jsstill hassoon: trueon thepracticalline of the shelf, orbooks/practical.jsonis missing on the server. - Book 3 is silent: check that its voice list exists. Proxmox web UI, pve, Shell:
pct exec 306 -- ls -l /var/www/abcbooks/rays/audio/practical.jsonand count the MP3s (all three books together are about 12,000 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 typed answer and the working disagree; one of them has a misread digit. Look at the page again.
- A long number does not fit: numbers longer than 16 digits can't be typed; the book's longest answer has 15.
- An old version shows after an update: change
rays-shell-v7insw.jsto the next number, deploy, and reload the page twice.
Notes
- Stickers are kept per book and per reader, so Book 3 has its own sticker page.
- Next on the shelf: Book 4, Ray's New Higher Arithmetic (Grades 7-8).