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Las diagrammes suivants iliustrent la m^thode. 1 2 3 1 2 3 4 5 6 r Sanabian gcvies oi §chool 1300I10, ELEMENTARY ARITHMETIC, FOR CANADIAN SCHOOLS REV. BARNARD SMITH, M. A., St. Peter's College, Cambridge. AND ARCHIBALD MacMURCHY, M.A., University College, Toronto, THIKD EDITION, REVISED. T O R O N T*0 : THE COPP, CLARK COMPANY (Limited), 9 FRONT STREET WEST, 1890, Entered ««ord,„g to Act ^, ^ fwf Af fu ^' ^*^^^^^« respectively ; the numbers between any ?horU?;"e"2oVnd^^^^^ '"^^ '^^"^^ ^^ ^^^ ^^"^ -^^ - ««yiw"?v.''''"'® "t^ l^""^*^ *^ ^^ (m;i€^^.7im^), or nine tern t^S'ei ^ """"^ ^^'""^ ''^'' be expressed by Ex. I. Write the following numbers in figures. (1) Three, four, two, seven, nine, six, eight (2) Ten, one, twelve, nineteen, five, eleven, sixteen ,^ir:t a-K^^^if.'^'J^^^'^^^ twenty-seven, thirty-three, forty nine, sixty, fifty-five, seventeen, thirty-six. (4) Eighty-eight, thirty-five, sixty-three, twenty-nine 8eventy-six, eighty, ninety-four, thirteen, fifty-two. ' NOTATION. 11 (5) Write clown in figures all the numbers between eight and eighteen, between forty-five and fi^'ty-oue, and between eighty-seven and ninety-nine. The next number after 99 is one hundred, written in fig- ures thus, 100 ; the 1 in 100, standing in the tliird place Irom the right hand, now expreysing not one unit, nor one ten units, but one hundred units. All numbers from 100 to 200 {two hundred) arc formed ex- actly in the same ,way, as we formed those from to 100 ; thus we go on 101 {one hundred and one), 102, &c., up to 110 {one hundred and ten), then 111 {one hundred and eleven)^ 112, &c., up to 120 {one himdred and twenty), them 121 (one hundred and twenty-one), 122, &c., up to 130 {one hundred and thirty), and so on up to 200; then 201, 202, &c., up to 300 {three hundred), and so on up to 400 {four hundred), 500 {Jive hun '"erf), GOO {six hundred), 700 {seven hundred), 800 {eight hundred), 900 {;iiine hundred), 999 {nine hundred and ninety- nine), or- nine Jmndreds, nine tens, and nine, the greatest num- ber which can be expressed by three figures. Ex. II. Write down the following numbers in figures. (1) One hundred and six, one hundred and fifty, two hun« dred, two hundred and eighty-seven, three hundred and ten, four hundred and thirty-one, five hundred and fifty -five, nine hundred and nineteen, eight hundred and sixty-seven. (2) Write all the numbers in figures from one hundred and ninety-five to two hundred and fourteen, from yix hun- dred and eleven to six hundred and twenty, and from nine hundred and forty-seven to nine hundred ana seventy. I '1 The next number after 999 is one thousand, written in figures thus, 1000 ; the 1 in 1000, standing in the fourth place from the right hand, now expressing one thousand units. All numbers from 1000 up to 9999 {nine thousand nine hundred and ninety-nine) are formed thus, 1001 {one thou- sand and one), 1002, &c., up to 2000 {two thousand), up to 3000 {three thousand), and so on. The next number after 9999 is ten tTiousand, written in figures thus, 10000; the 1 in 10000, standing in the fifth 13 MUTIIMETIG, place from the right liaud, now expressing one ten tlmisand a mis. All numbers from 10000 up to 99999 iiiinety-nine thousand nine hundred ami mnrff/-nine), are formed thus, 10001 {ten thousand and one), 10002, kc, up to 20000 {twenty thousand), then 20001 {twenty thousand ami one), 20003, &c., up to 30000 {thirty thousand), and so on. Ex. III. Write the following numbers in figures. (1) Four thousand five hundred and eighty-five, seven thousand three hundred and twenty-one, niiie thousand seven hundred and ninety-eight, seven tliousand and six, (3) Five thousand and four, five tliousa..d four hundred, five thousand and forty, eight thousand and thirty-six, ei^'-ht thousand three hundred and six, eight thousand thn^e hun- dred and sixty, nine thousand nine hundred and nine. '^^3) Seventy-five thousand six hundred and thirty-five ninety thousand nine hundred and nine, ten tliousand and four, eighty-seven thousand and fifty, ninety thousand and one, sixty-four thousand and sixty-four, eighty-three thousand. The next number after 99999 is one hundred tJmisand written in figures thus, 100000, the 1 in 100000, standing in the sixth place from the right hand, now expressing one hundred tJwusand units, and so on up to 999999 {nine hun- dred arod. ninety-nine thousand nine hundred and ninety-nine) then we come to one w.Ulhn, written in figures thus, lOOOOOo' the 1 expressing one million units, and so on up to te,ns oi millioris (10000000), humlreds of millions (100000000). billiom (1000000000), and so on. ^' Thus one is written ten one hundred ' * ' one thousand. \\ ten thousniid \. . .'. one hundred tliousand one million ] ten million '...'.'.'. V^.'.\'. l6,o66,o6o one hiinared millions 100,000,000 one billion 1,000,000,000 6. From tlie above table, we see that dividing any uum- 1 10 100 1,000 10,000 100,000 1,000,000 NOTATIOJS la i I i! *er Into periods of three figures each, begiuning at the right ii y 1 1 ^^ ARITHMETIC. 1 m (8) (9) 276608567 306738673 897203685 76293568 68345658 28678326 683937285 928337368 206738638 «38668589 9283678 728397328 211839297 238906594 563435639 36302562 93567836 912368834 397G12397 207867398 6383563 583967323 30673613 83297609 960039368 928327563 603536239 543832586 568302126 736397564 782395678 203386517 933506593 (10) (") (12) 72867863 868936709 878684976 97605812 76385673 79683886 7638516 467808758 468976896 316527808 900009900 786847512 275607836 90909999 9276U7038 97673904 938568378 90809008 1 268937318 712050760 768385006 718768926 77807689 708209600 203685738 234593368 87967839 96359568 99213567 862006764 397569387 837346395 993387686 113) Add toEjether nine mUlions four hundred and sitty. SIX thousand four hundred and ninety-five, three hundred and seventy -five millions five hiindred and seventy -three thou- sand seven hundred and thirty-five, seven hundred and fifty-four thousand five hundred and forty-seven, three mil- lions seven hundred and eighty-nine thousand two hundred and eighty-four, twenty-nine millions eight hundred and eiffhty-six thousand seven hundred and ninety-nine, nine hundred and ninety-two thousand and eighty-four, two hun- dred and nmety-three thousand six hundred and nlnety-five. two millions six hundred and eighty-four thousand four hundred and eighty -seven. ree millions five hundred and nin<4y-two thousand eight hundred and seventy-three, seven millions eight hundred and forty-nine thousand three hun- nred and lortv six. (14) A farmer had forty-four sheep, thirty-five head of cattle, fifteen pigs, six horses. How many animals had be {altogether I SIMPLE aUBTBACTlON. 28 (15) In one year a farmer's crop was as follows : Five hundred and twenty-three bushels of wheat, a hundred ana twenty bushels of oats, sixty-four bushels of peas, two hun- dred and thirty-seven bushels of potatoes, thirty-eight bush- els ot turnips. Hov, many bushels had he ? (16) A man bought a farm for sixteen hundred and fitly dollars, he spen a hundred und sixty in putting on it now fences, live hundred and seventy -five in building a new house, in repairing the barn and sheds two hundred; lie then sold it and made a protit of six hundred dollars. How much did he get for the farm 'i (17) In 1861 the population of the counties on Lake Erie was: Essex, tweuty-flve thousand two hundred and eleven; Elgin, thirty-two thousand and fifty \ Kent, thirty-one thou- sand one hundred and eighty-three; Norfolk, twenty-eight thousand five hundred and ninety , Haldimand, twenty-tiiree thousand seven hundred and eighty ; Welland, twenty-tour thousand nine hundred and eighty-eight. What was the total population of the six counties on Lake Erie? SIMPLE SUBTRACTIOiN, 13. Simple Subtraction is the method of finding what number remains, when a smaller number is taken from a greater number of the same kind. The number so found is called the Remainder, or Uip- FERENCE. The number subtracted from, is called the Minuend ; the number subtracted, the Subtrahend. 13. The sign — called minus, placed between two num- bers, means that the second number is to be subtracted from the first number : thus 7 — 3. or 7 minus 3, means that o is to be subtracted from 7. ." 7 — 3 = 4. Rule for Simple Subtraction,, 14. Rule. Write down the less number under the freater number, so that units may come under units, tens un- er tens, hundreds under hundreds, and so on ; then draw a straight line under the lower number. Take, if you can, the number of units in each figure of the lower number from the number of units in each figure of the upper number which stands directly over it, and place the remainder under the line just drawn, units under units, tens under tens, and so on. Butv if the units in any figure fai the lower number int P m Ir'll Ml 24 ARITHMEriC. li It' greater than the number of nnita in the figure jufet above it, then add ten to tlie upper figure, and tlien subtract the num- ber of units in tlie lower figure from the number in the up- per figure thus increased, and write down the remainder a« before. Add one to the next number in the lower number, and then take tliis figure tlius increased from the figure just above it, by one of tlie methods already explained. Go on thus with all the figures. The whole difference, or remainder, so written down, will be the diff"c'reuce or remainder of the given numbers. Ex. 1. Subtract 547 from 859. By the Rule, g59 Metlwd. 7 from 9 leave 2, i. e. 7 units from 9 547 units leave 2 units; write down 2 in the units' ,.-, _ rr^ place. 4 from 5 leave 1 ; i. e. 4 tens from 5 tens dm.— dl^ leave 1 ten ; write down 1 in the tens' place. 5 from 8 leave 8, i. e. 5 hundreds from 8 hundreds leave 3 hundreds ; write down 3 in the hundreds' place. Ex. 2. Find the difference between seven hundred and forty-two, and two hundred and sixty-eight. By the Rule, »j'42 I cannot take 8 from 2, i. e. 8 units from 2 268 ^^^i^s-. •"• I ^^^ 10 to 2, which makes 12, 8 from _ j-j 12 leave 4; write 4 in the units' place, dm — 474 J XwwQ added 10 to the upper number 742, 1 must .•. add 10 to the lower number 2C8 (so as not to alter the difference between 742 and 268), i. e. 268 must be made 278, or 1 must be added to the 6. Then I cannot take 7 from 4, i. e. 7 tens from 4 tens, .*. 1 add 10 to the 4, really 10 tens or 1 hundred to the 4 tens, which makes it 14, really 14 tens, then 7 from 14 leave 7, really 7 tens; write 7 in the tens' place. I have just added 10 tens, or 1 hund'^'^i' ' * u e upper:? um- ber, I must .'. add 1 hundred to thf ' • '. amber, i. e. I must add 1 to the 2, really 1 hundred to 2 hundreds, making it 3, really 3 hundreds, then 3 from 7 leave 4, really 4 hun- dreds ; write 4 in the hundreds' place. Ex. 3. How much greater is eight thousand two hundred .\u;'a six thousand three hundred and nine ? 8200 ^ ^''^^^ ^ cannot, then 9 from 10 leave 1 ; 6309 write 1 in the units' place; carry 1, reaiiy 1 'fff — TsqT ^^°' *^^^^ ^ ivom I cannot, then 1 from 10 '.Km — ISVJI leaves 9, really 1 ten from 10 tens leaves 9 tens ; SIMPLE suirnucTioN 25 write 9 in llie teiiH' place; carry 1, really 1 hundred, then 4 I'roni 2 I cannot, then 4 Iroin 12 leave 8, reuliy 4 hundreds I'rom 13 hundreds leave 8 hundreds; write 8 in the hundreds' place, carr> 1, really 1 thouHand, then 7 iroin 8 leave 1, really 7 thousands from 8 thousands leave 1 tliousaud; write 1 in the thousands' place. Note^ The truth of all sums in subtraction may be proved hy addiniT the less number to the ditlerenceor remainder; if this sum ecpials the larger number, the sum will probably have been worked correctly. Thus, Proof of Ex. 53 Less number + remainder = 6309 + 1891 = 8200, the greater number. Ex. X. From 18 Take 14 (2) 27 15 39 11 (4) 55 5 (5) 86 60 (6) 538 23 (7) 759 603 (8) 24 18 (0) 51 49 (10) 64 6 (11) 83 47 (12) 98 89 (13) 70 54 (14) 64 29 (15) 200 16 (10) 547 880 (17) 896 708 (18) 702 504 (19) 800 199 (20) 650 56 (21) 912 707 (22) 563 476 (23) 309 120 (24) 608 499 (25) 48({ 307 (20) 843 745 (27) 900 791 (28) 505 107 (29) Subtract thirty-seven from tifty; twenty-nine from «eventy-one r sixty-six trom one hundred and four ninety- eeven from two hundred and eleven; one hundred and live from three hundred and three ; f«)ur nundred and seventy- five from six hundred and forty-nine. (30) A gentleman bought a horses and a carriage for five hundred and sixty dollars, the horse was valueu at three hundred dolbirs. How mi <;h was the carriage worth ^ and Low much was the horse worth more than the carriage? (31) In a school there are 75 ehildren. there are 28 girla ^Tow many more boys tliaa girls arc tiiere r (32) Charles had 167 marbles, he gave John 49, James 65, \^ I'll i 1 26 ARITIIMRTIC I fi. Thomas all the rest but 19 ; how many marbles had Thomas less than James? (33) By how much does the sum of 6 and 4 exceed theh difference ? (34) A boy's father gave him 40 cents to pay 10 cents for a skte, 3 cents for pencils, 8 cents for a copy-book, 5 cents for ink, 3 cents for a postage stamp ; after paying for the above he lost all but 4 cents through a hole in his pocket; how much did he lose ? Ex. XI. (1) From 5467 (3) 7601 (3) 3000 (4) 4536 (5) 5480 Take 3546 3890 2001 2297 996 (8) 7009 (7) (8) 8052 5281 (9) 7210 (10) 8888 (11) 5600 5080 4847 597 3809 999 2575 (12) 14748 (13) 54833 (14) 80408 (15) 70007 (16) 43520 13942 29648 59385 69999 25347 (17) 445673 (18) 9200000 (19) 87125391 ( (20) 350030042 277594 560506 08050092 94090096 (21) What number taken from three thousand will leave one hundred and oneV What nmnber added to seventy- two thousand five himdred and seventy-six will make one million seventy thousand four hundred and nine ? (22) The sum ot three numbers is twenty three thousand two hundred and fifty-seven ; the first is 9277, and the sec- ond is twelve hundred and eighty-three less than the first; find the third number (23) What IS the difference between 23047 + 175 - 368 -f 495 - 132 and 10000 - «406 - 704 + 7305 ? (24) Wlien will the Prince of Wales, who was bom in the year 1841, be as old as the Queen now, in the year 1869, is who was born in the year 1819 ? How old will the Queen then be ? (25) John says to Henry, I have 97 marbles ; H^mu'v re SIMPLE SUBTRACTION 27 v)lics I have 29 less than you : Charhe adds, I have as many as both of you less 25. How many marbles had Henry, and how many had Charlie ? (26) A man whose yearly income is 1000 dollars, spends 84 dollars for house rent, 135 dollars for servants, 39 dollars in travellinff, 58 dollars in clothing, as much on his garden as in travelling and clothing, 804 dollars in household bills. Will he have saved anything or be in debt at the end of the year, «\nd to what amount ? (27) Harry goes up sixteen steps ot a ladder, which has 45 steps, then down 7 steps, then up 10, then down 2 then down 4 then up 11, then down 9, then up 7, then up 5, then down 8, what step from the top and bottom will he then be standinp- upon ? (28) In a union workhouse there are 133 inmates. The number is made up thus : infirm and able-bodied 70 ; able- bodied and children 105; children and officers 63; officers 5. Find the number of each class. (29) A basket contained oranges, nuts, and eggs ; in all 1769 , there were 1696 oranges and nuts, and 1262 nuts and eggs.' How many more nuts were there than oranges? (30) The population of the counties on the river St. Law- rence in 1861, was one hundred and seventeen thousand nine hundred and eighty-six, that of those on the Ottawa river was seventy-two thousand two hundred and sixty-eight, i md the difference between the population of these counties.^ (31) What is the difference between thirty-seven millions nine hundred and six thousand seven hundred and three, and forty- five millions three thousand and eight ? (32) The subtrahend is fifty-si xmillions two hundred and twelve thousand three hundred, the remainder seventy-seven thousand three hundred and thirteen. What is the minuend if (33) The minuend is sixty-six millions three hundred and four thousand, the difference twelve thousand five hundred and eighty-six. Find the subtrahend. (34, A man bought 305 sheep for 8 dollars a head, and after spending 45 dollars on them lor food, sold them tor 4 dollars a head, how many dollars did he gain by his bargain ? (35) For the year 1861 the Imports into Canada were torty-three millions fifty-foui thousand eight hundiea and iiiirty-six dollars, and the Exi)orts were thirty four millioua -■.'-.jn w ']■ f 28 ARITHMETIC seven linndrecl and seventeen thousand two hundred and forty-eight' dollars. Find by how much the Imports exceed ed the Exports for the year 1801. 15. Roman Notation. I, denotes one; V, five; X, ten; L, fifty C, one hundred ; D, five hundred , M, one thousand. Rule. Where any one of the alx)ve letters is aftei\ or to the right hand of, one of equal or greater value, it is tb be added to it, but when put bejore one of greater value, it is tc be fiubtracted from it. Thus II = 1 + 1 == 2, III = 1 + 1 -+ 1 = 3, IV= 5 less 1 = 4, VI rr: 5 + 1 = 6, VIII = 5 + 1 + 1 + 1 =: 8, IX=10 less 1 = 9, XIII = 10 + 1 + 1 + 1 = 13, XIV= 10 plus 5 less 1 = 10 + 4 = 14, LXXIX = 50 + 10 + 10 + 10 less 1 = 70 + 9 = 79, XC = 100 less 10 ^^ 90. Note. A line over a letteiv^r letters, increases their value a thousandfold : thus V= 5, V= 5000 ; C = 100, "C = lOOOOO- Ex. XIL 1. Express in the Roman Notation, three; seven; eleven; nine; twelve; sixteen; 18; 25; 28; 37; 40; 53; 59; 62; 77; 84; 103; 157; 190; 200; 051; 783; 1204; 1527, 1805. 2. Express m words, and also in Arabic; figures, III ; VI - VIII; XIII; XV; XVII. XX; LIV; LXXXI; CXI; DCV VII; MC; MM; DCCXLIX; MDCCCLXV SIMPLE MULTIPLICATION. 16. Simple Multiplication is a short method of re' peated addition; thus, when 2 is multiplied by 3, tne num- ber obtained is the sum of 2 repeated tlu-ee times, which sum = 2 + 2 + 2 = 6. The number, which is to be repeated or added to itself, h called the IMultiplicand : thus, in the above example, 2 i« the multiplicand. The number, which shews how often the multiplicand i» to be repeated, is called the Multiplier thus, in the above example, 3 is the mullii)lier. The number found by multiplication, for instance m the above example, is called the Product. The multiplier and multiplicand are ^^otik ihu. s called Fac- tors, because they are factors, or makers, of the product The sign X, called into, or multm'i.ied ry, placed l)c SIMPLE MULTIPLICATION. 29 tween two numbers, means that tb.e numbers are to be multiplied together. , The following Table, called the Multiplication Table, ought to be learned correctly : r^4 Twice 3 times 4 times 5 times 6 times 7 times 1 makes 2 1 makes 3 1 iiiakes4 1 makes 5 1 makes 6 1 makes 7 2 .. 4 2 .. 6 2 .. 8 2 .. 10 2 .. 12 2 .. 14 3 .. 6 3 .. 9 3 .. 12 3 .. 15 3 .. 18 3 .. 21 4 .. 8 4 .. 12 4 .. 16 4 .. 20 4 .. 24 4 .. 28 5 .. 10 5 .. 15 5 .. 20 5 .. 25 5 .. 30 5 .. 35 6 .. 12 6 .. 18 6 . 24 6 .. 30 6 .. 36 6 .. 42 7 .. 14 7 .. 21 7 .. 28 7 .. 35 7 .. 42 7 .. 49 8 .. 16 8 . 24 8 .. 32 8 .. 40 8 *. 48 8 .. 56 9 .. 18 9 .. 27 9 .. 36 9 .. 45 9 .. 54 9 .. 63 10 ..201 10 .. 30 10 .. 40 10 .. 50 10 .. 60 10 .. 70 11 ..22] LI .. 33 11 .. 44 11 .. 55 11 .. 66 11 .. 77 12 ..24] L2 .. 36 12 .. 48 12 .. 60 12 .. 72 12 .. 84 8 times 9 times 10 times 11 times 12 times 1 makes i ) 1 makes 9 1 makes 10 1 makes 11 1 makes 12 2 .. 1( > 2 .. 18 2 .. 20 2 .. 22 2 .. 24 3 .. 2^ t 3 .. 27 3 .. 30 3 .. 33 3 . . 36 4 .. 3^ I 4 .. 36 4 .. 40 4 .. 44 4 .. 48 5 .. 4( ) 5 .. 45 5 .. 50 5 .. 55 5 . . 60 6 .. 4^ ^ 6 ,. 54 6 .. 60 6 .. 66 6 . . T 7 .. 5( 3 7 .. 63 7 .. 70 7 .. 77 7 . 84 8 .. 6- t 8 .. 72 8 .. 80 8 .. 88 8 . . 96 9 .. 7$ I 9 .. 81 9 .. 90 9 .. 99 9 . . 108 10 .. 8( 310 .. 90 10 .. 10010 .. 110 10 . . 120 11 .. 8( m .. 99 11 .. llOlll .. 121 11 . . 132 (12 V — .. 9( 312 .. 108 12 .. 120|12 .. 133 12 . . 144 17. Rule for Simple Multiplkation^ when tlie midtiplier is f, number not larger than 12, Rule. Place the multiplier under the multiplicand, units under units, and (if the multipher be 10, 11, or 12) tens under tens ; then draw a line under the multiplier. Multiply each figure of the multiplicand, beginning with the units, oy the figure, or figures of the multiplier (by means of the Mul^plication Table). Write down and carry as in Simple Addition m iV Ii l 'I l\ n I 80 ARtTHMETIO. II I* • i! 1 ■ nr Ex. 1. Multiply 531 by 3. By the Rule. 531 Twice 1 unit makes 2 units ; write 2 in the 2 units' place of the product Twice 3 tens of units make 6 tens of units ; write 6 m the tens' place of the product Twice 5 hundreds of units make 1062 r»T' 10 hundreds of units, or 1 thousand hundred ; write in the hundreds' place, and 1 in tlie thousands' place. Ex. 2. Find the product of 5063 and 6. By the Rule, 5003 ^ times 3 units = 18 units = 1 ten and 8 units ; g write 8 units, carry 1 ten. Next n times 6 ~ tens = 36 tens, which added to the 1 ten carried 30378 _ 37 tens _ 3 hundreds and 7 tens ; write 7 tens and catry 3 hundreds. Next. 6 times hundreds = 0, which added to the 3 hun- dreds carried = 300 hundreds, write 3 in the hundreds' place. Next, 6 times 5 thousands = 30 thousands = 3 tens of thou- sands and thousands ; write in the thousands' place, and 3 in the tens of thousands' place. Note. It will be seen from the Multiplication Table, that to multiply any number by 10, we have only to write to the right hand of the number, thus, 3 x 1 = 3, G x 10 = 30 ; also 5893 x 10 - 58930, and 58930 x 10 = 589300. Similarly 3 x 100 = 300, 3 x 1000 = 3000: and so on. Also if any number be multiplied by 20, the result is the same as if the number were multiplied by 2, and written on the right hand of the product; thus, 6 x 20 — • 6 x 2 x 10 = 12 X 10 = 120; also 60 x 20 = 1200, for 60 x 20 - 60 x 2 x 10 = 120 X 10 = 1200 ; and so of any other number. Similarly 00 x 200 = 12000, 60 x 2000 = 120000, and so on. ^H (1) Multiply 53 By 2 (10) (11) Ul Oo 4 4 (2) 47 2 (12) 90 5 Ex. XIII. (3) (4) (5) (6) (7) (8) 88 56 48 60 29 75 2 2 3 3 3 3 (13) (14) (15) (16) (17) 67 43 30 99 78 5 5 6 6 6 (9) 27 4 (181 ■ ^ 27 7 SIMPLE MUL TIPLICA TION 31 ^19) (20) (21) ('23) (23) (24) (25) (26) (27) 53 45 77 69 54 20 99 53 87 7 8 8 9 9 10 10 11 11 i^i 'hm (28) (29) (30) (31) (32) (33) (34) (35) (36) 91 60 49 687 800 • 697 276 777 497 11 12 12 2 3 3 4 5 6 (37) (38) (39) (40) (41) (42) (43) (44) 479 905 835 487 560 538 888 704 7 7 8 9 10 11 12 12 (45) Supposing an acre of land to produce 39 bushels of wheat, how many bushels will 11 of such acres produce, and what will be their value at 6 shillings a bushel ? (46) There are 21 shilUngs in 1 guinea, and 12 pence in 1 shilling ; how many pence are there in 3, 7, 12 guineas ? (47) Charlie bought of Quintm 11 rabbits at 23 cents each, and Quintin bought of Charhe 9 hens at 33 cents each, how many cents had Qumtin to give to Charlie ? (48) What is the difference between 12 dozen and 8, and 8 dozen and 12 ? [Note, 1 dozen — 12 ] (49) ^ has seven thousand four hundred and one pota- toes ; he sells B fifty-seven dozen and five ; C one hundred and twelve dozen and eleven , D two hundred and fifty-nine dozen and nine ; and E the remainder, How many more did ^ buy than C? Ex. XIV. (1) Multiply 9048 By 2 (2) (B) (4) (5) (6) 5849 9873 38076 6057 97068 2 3 3 4 5 I ji i (7) (8) (9) (10) (11) (12) 69360 80965 439090 48508 33069 38476 6 5 7 8 7 9 49216 12 (14) 69432 12 (15) 21357 11 (16) 91537 13 8d ARITHMETIC. (17) Multiply (1) 3870493, (3) 4609758, (3) 85973864, (4\ 9090853, (5) 55880093, (6) 987654331, by each of the follow- ing, 2, 5, 3, 7, 4, 9, 6, 8, 11, and 13. (18) Two persons start from the same place, and travel in the same direc.'lion, one at the rate of 93 miles a day, the other at the rate of 79 miles a day ; how far apart will they be at the end of a week V (19) If the second person at the end of two days turn back, and travel each day in the opi)08ite direction the same number of miles as before ; how far will they be apart at the end of a week ? ! !■ • 1' ilii 18. Rule for Simple Multiplication^ ^chen the Multiplier is a number larger than 12. Rule. Place the multiplier under the multiplicand, units under units, tens under tens, and so on ; then draw a line under the multiplier. Multiply each figure of the multiplicand, beginning with the units, by the figure in the units' place of the multiplier (by means of the table given for Multiplication) ; write down and carry as in Addition. Then multiply each figure of the multiplicand, beginning with the units, by the figure in the tens' place of the multi- pliei, placing the first figure so obtained under the tens of the line above, the next figure under the hundreds, and so on. Proceed in the same way with each succeeding figure of the multiplier. Then add up all the results thus obtained by the rt^le of Simple Addition. Ex. 1. Multii)ly 2307 by 358. By the Rule, 3307 358 18456 11535 6931 since 358 = 300 -f 50 -f 8, when we mul- tiply by tlie 5, we in fact multiply by 50, and 3307 x 50=115350 ; again, when we multiply by the 3, we in fact multiply by 300, and 2307x300 = 692100; hence product=825906 {! \®, "^"'^^ f^^l^ that we may multiply ^ , by the snnple figures 5 and 3, if we only take care to place tlie first figure in the second line under tlie tens place of the first line, and the first figure of tiie thu-d line under the hundreds' place. i ,11 iS SIMPLE MULTIPLICATION. 3d 7.)8 _609 6823 4548 _ 461()32 £x. 2. Find the procUici of 758 and 609. Since 758, or any other number, multiplied by gives as a product, .'. in this case -we multiply by 9 and then by 6, writing the first figure of the second line luider the hundreds' phice, and not under the tens' liluce of the lino above, lor 609 = 600 -h 9. Note 1 If the Multiplikr or Multiplicand, or both, end with cyphers, we may omit them in the working ; tak- ing care to place on the right hand of the product jis many cyi)hers as we have omitted from the end of the multiplier or multiplicand, or both. Thus, if 270 be multiplied by 507, and 2700 be multiplied by 50700, we have In the first case, when we multiply 7 by 7, in fact we multiply 70 by 7, and 70 x 7 = 490. In the second case, when w^e multiply 7 by 7, in fact S70 507 189 135 i;]6890 270 50700 189 135 13689000 we multiply 70 by 700, and 70 x 700 = 49000 Note 2. 2 X 3 = 2 + 2 + 2 := 6, and 3 X 2 = 3 + 3 = 6. 2x3 = 3x2; and this is true of all numbers Note S. If more than two lactors have to be multiplied together, as 2 x 4 X 9, it is termed continued multiplica- tion, and since 2 x 4 =- 8, and 8 x 9 = 72, and .2x4x9 = 72, we shall ot course obtain the same result, whether we multiply any number l)y 72, or by its factors 2, 4, and 9, by continued multiplication , and so of any other number. 35x72 = 2520, and 35x2x4x9 = 70x4x9 = 280x9 = 2520. 19, Numbers which are produced by multiplying togethe r two or more numbers respectively greater than unity, are callsd Composite Numbers. Thus 4 = 2x2, 36 = 6x6, or = 2x3x2x3. and such like, are Composite Numbeus. Numbers wli'rch cannot be broken up into factors, as 3, 5, 7, 11, and such like, are Prime Numbers. Note If-. The truth of all results in Multiplication may be proved by using the multiplicand as multiplier, and the mul- tiplier as multiplicand ; if the product thus obtained be the same as the product found at first, ihe results are in all prob- fi-nlitv true. V H-* 'M .1 li 'il If '' 84 >;t (1) Multiply 463 By ^ (7) (8) (9) 798 407 869 80 55 89 (15) 1263 36 ARITHMETIC. Ex. XV. (2) (3) (4) 678 276 601 27 33 54 (5) (W 946 837 61 89 (10) 917 46 (16) 5613 54 (11) 692 73 (17) 96732 72 (12) 909 88 (13) (14) 305 463 715 608 (19) (20) (21) 495 690 417 370 480 739 (22) 278 900 (26) 50738 9706 (32) 92035 8007 (27) 86370 90900 (33) 84009 7898 (28) 47672 5126 (23) 904 803 (29) 68109 2065 (18) 67628 6£ (24) 3259 497 (30) 45094 7838 (34) 678000 876000 (35) 90058 90009 (25) 15900 3300 (31) 56888 6049 (36) 80108 7770 (37) Find the product of seven thousand and thirty-nine by four thousand seven hundred and nine ; three thousand nine hundred and ten by three hundred and fifty thousand ; •ighty-seven thousand nine hundred .by nine thousand and six ; seven millions eight thousand and five by four hundred thousand seven hundred and three. (38) Find the product of the sum and difference of four hundred and ninety-six, and three hundred and twelve. (39) Multiply (1) 973 by 63, and also by its factors 3, 3, and 7, and (2) 33000 by 1560, and also by its fafCtors 13, 5, 4, and 6. (40) As in (39) do also, (15), (16), (17), (18) (1) muiiiuiy r will be found by multiplying the second remainder by the- first divisor, and adding to the product the first re- mainder. E.\. 4. Divide 56732 by 45. ^ 9 ' 5(57^3, i e.. 56733 units, 45 j 5 I 6303 rem. 5, i. e. 6303 nines and rem. 5 units, 12()0 rem. 3, t. e. 1200 forty-fives, and rem. 3 nines, .'. the true rem. = 9 x 3 units -f 5 units = 27 + 5, or 33 units. SiMPLE DtVISiOHf. 41 Therefore the quotient arising from the division of 56732 by 45 is 1200, with a remainder 32 over. viW %■ f- Ex. XVIII. Divide (1) 192 by IG ; 720 by 18 ; 795 by 15 ; 1786 by 19. (2) 1035 by 23 ; 1073 by 37 ; 2730 by 42 ; 5432 by 56. (3) 45G0 by 80 ; 3871 by 49 ; 7744 by 88 ; 6935 by 95. (4) 5375 by 25 ; 29526 by 37 ; 25605 by 29 ; 4590 by 45. (5) 09230 by 86; 37510 by 55; 10287 by 81; 23919 by 67 ; 25760 by 56 ; 538840 by 76. (6) 35626 by 94; 31339 by 77; 80840 by 86; 28782 by 39; 9009196416 by 96; 41765256 by 72. (7) 88832 by 256 ; 175252 by 308 ; 321776 by 104. (8) 653723 by 329 ; 3577926 By 506 ; 542100 by 834. (9) 8189181 by 900 ; 4049820 by 745 ; 342604 by 883. (10) 7848600 by 365 ; 2339100 by 678 ; 90625 by 727. (11) 27291888 by 478; 30387310 by 397 ; 3273068 by 703. (12) 87624792 by 843; 90273189 by 513; 53006751 by 609 ; 30073074 by 358 ; 630762540981 by 652. (13) 519387042 by 2731 ; 10101255 by 2185 ; 154725876 by 3076 ; 632798014 by 7243. (14) 2015029 by 1004; 131686100 by 6487; 395494875 by 6007; 50696184 by 1617. (15) 4519559744 by 5008 ; 16322853 by 9306 ; 23617103000 by 1579 ; 2106144185 by 2735. (16) 142997420 by 3782; 19554707200 by 6016; 2828882701578 by 38706. (17) What number multiplied by 79 will give the same product as 257 multiplied by 553 ? (18) How many pairs of stockings, at 66 cents a pair, should be given for 9 dozen pairs of gloves, at 110 cents a pair? (19) What number must be added to tliirty millions nine hundred and eighty-four thousand and fifty-one, that the ouni mav be exactlv divisible bv two hundred, and ciffhtv- »ight? (20) If the sum of 274 and 108 be multiplied by tneu* I'll ^;tj li H-i lt#f i% AIllTHMEfia 1. .1' 11 1 '^t :i! ,.n I . i •in < 'I 1 i; tliflference, and the product be divided by 17(>, what Will be the quotient ? (31) A farmer bought 75 sheep at 4 dollars each ; 94 sheep at 3 doHare each ; and 106 slieep at 2 dollars each ; at what price per head must he sell the sheep, so as to gain 147 dol- lars ' y his bargain ? (32) A hatter sold 267 hats for 1068 dollars, gainmg there- by 1 dollar on each hat, what did each hat cost him ? (23) If the sum of 103, 29, and 267 be divided by 19, and the quotient be multiplied by 57, and the product be dimin- ishea by 197, what will the remainder be? % (24) 8 lambs are worth 16 doLa«, and 15 sheep are worth 60 dollaw ; how many of such sheep ought to be given in 3xchang« for 840 of such lambs? (25) The sum of the product of two numbers and 355 is ei_;iity-8even thousand four hundred and three ; one of the numbers is 21G; find the other camber. (26) What number must 416 be multipned by to produce 154979552? (27) What number subtracted 28 times from 479632 will leave 20 as a remainder ? (28) A farmer bought 29 bullocks for 1885 dollars, and after keeping them for 3 months, and spending on each 5 dollars per month, he sold all the bullocks for 2610 dollars ; what was his gain on each bullock ? 24. If the Divisor terminate with a cyph&r or cyphers^ the process ofBimmn can he shortened by thefoUowing Mule. Rule. Cut off the cypher or cyphers from the divisor, and as many figures from the right-hand of the dividend, as there are cyphers so cut off" at the right-hand end of the divisor ; then proceed with the remaining figures according to the Rule, Art 21 or Art. 22, as the case maybe; and to the last remainder affix the flgm'es cut off from the divi- dend for the true remainder. Ex. 1. Divide 57 by 20. 2 J5 7 57 = 50 -f 7 ; now 20 goes 2 in 50 with rem. 4 rem. 1. ^ jg j-eaiiy i ten, or 10, and the true rem. = lO-H 7 or 17. Ill simple: mvistoN. 43 i Ex. 2. Bivide 46431 by 500. 46431 = 46400 -f- 31, and 46400 divided 5,00) 464,31 by 500 = 92 with rem. 400, .'. when the 92 rem. 4. ^^^ is divided by the 5, the rem. 4 i8 really 400, and the true rem. is 431. Ex. 3. Divide 375340 by 5900. 59,00)3753,40(63 354 .-. quotient = 63, and rem. 213 = 3640 177 36 £x. 4. Divide 563854 by 10, by 1000, ana by 100000. We may write down the quotient and remainder for each ques- tion at once. Thus • 1st quotient = 56385, and rem. = 4. 2d . . . . = 563, ... = 854. 3rd ... = 5, ... =63854. Ex. XIX. (1) Divide 34, 43, 56, 80, 135, 260, 1504, by 10, 20, and 30. (2) Divide 237, 840, 673, 291, 6019, 7820, 81229, 327800, by 40, 60, 70, 100, and 200. (3) Divide 79048, 6870, 890061, by 240, 1000, 1500, and 2600 ; and 830678103490 by 100000000. (4) 806753246-7-9067. (5) 612709066-70602. (6) 60005836 -f89C. (7) 70867509-r9986. (8) 8673466964-r868. (9) 200000783-93256. (10) Multiply ue09by7t»anddiTidetheproductby80«7. (11) How many regiments of 1000 men, and also of 1 200 ^en can be formed out of one million one hundred thou- 8ana men f (13) Add together twenty-flTe millions seven hundred and sixty thousand and thirty-four, 7621 1879 and 4637862 • •>y-^^.avv icii uiMiiuiiH ana seveniynve From the sum- dmde the remainder by 100000. \> ^ . ft'-' Hl^l 44 ARlTHMhSTiU. SECTION II 1*1 m MONEY TABLES. CANAn^AN CCRRENCY 25 Tbe Sliver coiiisi are : a 5 cent piece a 10 •♦ ■ " a 20 •' a 25 •• *• a 50 •' 100 cents make oiid dollar, or $1. Note I. Tlie cent, which is made of bronze, is onf inril ia diametei, and lOu cents weigh one pound avoirdupoi* HAI TFAX OK OLD CANADIAN CURRENCY. 26 2 Farthings make 1 Half-penny, or |d. 2 Half-pence 1 Penny Id. 12 Pence 1 Shilling Is. 5 Shillings 1 Dollar $1. 4 Dollars 1 P" und £1. Note ^ The farthing is written thus, |d ; and three f«t ♦Hings thus, ^d. ENGLISH OR STERUNG CURRENCY. 27. 2 Farthings make 1 Half-penny, or ^d. 2 Half-pence 1 Penny Id. 12 Pence 1 Shilling Is. 20 Shillings 1 Pound £1. The sovereign, a gold coin = 20 shillings. The guinea, a gold coin not now in use = 21 shilhngs. Notts. The sterling pound = B-86§ Canadian Qurreartv 28. UNITED STATES CURRENCY. JO Mills make 1 Cent. 10 Cents 1 Dime 10 Dir?K'?. , .,,... 1 Dolla! 10 Dollars 1 Eagle. TABLES— WEIGHTS ANI> MEASURES. 45 WEIGHTS AND MEASURES. TABLE OP TROY WEIGHT. 29. Trot Weight is used in weighing gold, silver, dia- monds, and other articles of a costly nature ; and also in determining specific gravities. 24 Grains, gr make 1 Pennyweight 1 dwt. 20 Pennyweights 1 Ounce 1 oz. 13 Ounce 1 Pound ...... .1 lb. or 1 lb. TABLE OF AVOIRDUPOIS WEIGHT. 30. Avoirdupois Weight is used in weighing all heavy articles, which are coarse and drossy, or subject to waste, as butter, meat, and the like, and all objects of commerce, with the exception of medicines, gold, silver, and some pre- cious stones. 16 Drams, dr make 1 Ounce 1 oz. 16 Ounces 1 Pound 1 lb. 25 Pounds 1 Quarter, 1 qr. 4 Quartei-s, or 100 lbs. . . 1 Hundredweight ... 1 cwt 20 Hundredweights 1 Ton 1 ton. Note. 1 lb. Avoirdupois weighs 7000 grs. Troy. TABLE OF apothecaries' WEIGHT. 31 . Apothecaries' Weight is used in mi xing medicines. 20 Grains, gr make 1 Scruple 1 sc. or 1 3 3 Scruples 1 Dram 1 dr. or 1 3 8 Drams 1 Ounce loz. orl | 12 Ounces 1 Pound 1 lb. or 1 lb TABLE OP LINEAL MEASURE. 32. In this measure, which is used to measure distances, lengths, breadths, heights, depths, and the like, of places or things: 12 Lines make 1 Inch 11. 12 Inches 1 Foot 1ft. 3 Feet, or 36 in 1 Yard 1yd. 6 Feet 1 Fathom .... 1 fth. 5i Yards, meaning 5 yards and [ 1 Rod, Pole, ) . a half yard ] or Perch \ ^ P"* *IU X V71CC, \jt. Ai;.\j y lia i x" uliuilg .... 1 luT. 8 Furlongs, or 1760 yds 1 Mile 1 mi. 3 Miles 1 League 1 lea. M! 46 ABITHMETta The following measurements may be added, as usefttk id certain cases : 4 Inches make 1 Hand (used in measuring horses). 22 Yards 1 Chain ) , y . • i i\ 100 Links 1 Chain \ ^"'^'^ ^^ measurmg land). A degree is equal to 60 geographical, or nearly 60^ En- glish miles. TABLE >TH MEASURE. 33. In this measure, \ drapers : 2i 4 4 5 6 ;n is used by linen and woollen Inches make 1 Nail 1 nl. Nails 1 Quarter 1 qr. Quarters 1 Yard 1 yd. Quarters. ... 1 Ell (English). Quarters 1 Ell (French). TABLE OF SQUARE MEASURE. 34. This measure is used to measure all kinds of surface or superficies, such as land, paving, flooring, in fact every- thing in which length and breadth are to be taken into ac- count A Square is a four-sided figure, whose sides are equal, each side being perpendicular to the adjacent sides. See figure below. A square inch is a square, each of whose sides is an inch in length; a square yard is a square, each of whose sides is a yard in length. 144 Square Inches mak( 1 Square Foot.. .1 eq. ft. or 1 ft. 9 Square Feet 1 Square Yard. . 1 sq. yd. or 1 yd. 30i Square Yards 1 Square Pole. . . 1 sq. po. or 1 po 40 Square Poles 1 Square Rood. .1 ro. 4 Roods 1 Acre 1 ac. 25000 Square Links = 1 Rood. 100000 =lAcre. 10 Chains = 1 Acre. 4840 Yards = 1 Acre. 640 Acres = 1 Square Mile. Note. This table is formed from the table for lineal mea- sure, by multiplying each lineal dimension by itself. Tlie truth of the above table will appea/rfrom the foUowma considerations. E F B 1 2 3 4 5 6 7 8 9 TABLES- WEmilTS AND MEA8UME8. 4? Suppose AB and ^C to be lineal yards placed perpendicu- larly to each other. Then ABOD is a square yard. liAE, EF, FB, AG, OH, ^(7, each O = 1 lineal foot, it appears from the figure that there are 9 squares in the ff square yard, and that each square is 1 square foot. The same explanation holds good ^ of the other dimensions. TABLE OP SOLID OR CUBIC MEASURE. 35. This measure is used to measure all kinds of solids or figures which consist of three dimensions, length, breadth' and depth or thickness. * A CUBE is a solid figure contained by six equal squares; lor mstance, a die is a cube. A cubic inch is a cube whose side is a square mch. A cubic yard is a cube whose side is a square yard. 1728 Cubic Inches make 1 Cubic Foot, or 1 c. ft 27 Cubic Feet l Cubic Yard, or 1 c. yd. 40 Cubic Feet of Rough Timber or 50 Cubic Feet of Hewn Timber 1 Load. V^CubicFeet i Ton of Shipping. 128 Cubic Feet of Fire-wood. ... 1 Cord. 16 Cubic Feet of Fire-wood 1 Cord-foot. The truth of the first part of above table wiU appear from thefolUming considerations. ViAB, AG, and AD be per- pendicular to each other, and each of them a lineal yard in length, then the figure DE ia a cubic yard. Suppose DH a lineal foot, and IIKLM a plane drawn parallel to side DC. By the table Art. 34, there are 9 square feet in side DG. There will tlierefore be 9 cu- bic feet in the solid figure DL. Similarly if another lineal riffif m % ^''' ;''^-»; ^^ I ■ f "■■' i ( 48 ARITHMETIC. foot EN were taken, and a plane NO were drawn para\ lei to IIL, there would be 9 cubic feet contained in the solid figure IIO. Similarly, there would be 9 cubic feet in the solid figure Therefore, there are 27 cubic feet in the soUd figure DE, or in 1 cubic yard. Note. A pile of wood 4 feet high, 4 feet wide, and 8 feet long, makes a cord. 4 il MEASURES OF CAPACITY. TABLE OF WINE MEASURE. 36. In this measure, by which wines and all liquids, with the exception of malt liquors and water, are measured 4 Gills make 1 Pint 1 pt. 2 Pints 1 Quart 1 qt. 4 Quarts 1 Uallon 1 gal. 63 Gallons 1 Hogshead . 1 hhd. 2 Hogsheads 1 Pipe 1 pipe. 2 Pipes 1 Tun 1 tun. TABLE OF ALE AND BEER MEASURE. 37. In this measure, by which all malt liquors and vralei are measured : 2 Pints make 1 Quart 1 qt. 4 Quarts 1 Gallon 1 gal. 9 GaUons 1 Firkin ... .1 fir. 18 Gallons 1 Kilderkin. 1 kil. 36 Gallons 1 Barrel 1 bar. 1^ Barrels, or 54 Gallons 1 Hogshead. 1 hlul 2 Hogsheads 1 Butt 1 butt 2 Butts 1 Tun 1 tun. TABLE OP DRY MEASURE. 38. 3 Pint* make 1 Quart 1 qt. 4 Quai'ts 1 Gallon ... 1 gaL 2 Gallons 1 Peck 1 pk. 4 Pecks ,.,,__..,,,,, 1 Buahel. . . 1 bu. 36 Bushels 1 Chaldron. 1 ch. ii a- m TABLES— MEASUREti OF TIME. 4^ i II; 39. 34 Pounds make 1 Bushel of Oats. * 48 Pounds 1 Bushel of B'kwheat, Barley or Timothv 60 Pounds 1 Bushel of Flax Seed. 56 Pounds 1 Bushel of Rye or Indian Corn. 60 Pounds 1 Bushel of Wheat, Potatoes, Peas, Beans, Onions, or Red Clover Seed. Note 1. Grains are sold by the cental (100 Ibs.l. or bv parts thereof. ^ MEi^SURES OF TIME. TABLE OF TIME. . • 40. i Seeonfl is written thus 1". * 60 Seconds make 1 Minute 1' 60 Minutes 1 Hour \\\' i hr 24 Hours l Day " i dav 7 Days i Week i wk ' 4 Weeks, or 28 days . . 1 Lunar month 1 mo. 365 Days i Civil or common year. . 1 yr. Note 2. 60 minutes make 1 degree, or 60' make 1°. A degree is the 360th part of the circumference of a circle, A year is divided into 12 months, called Calendar Months, the number of days in each of which may be easily remem* bered by means of the following lines : Thirty days hath September, April, June and November : February has twenty-eight alone, And all the rest have thirty-one : But leap-year coming once in four, February- then has one day more. Notts A civil or common, year = 52 wks, 1 day. A leap year = 366 days. Every year which is divisible by 4 without a remainder is a Leaf on Bj.ssextilf Year ; except those years which complete » century, (i. e. a hundred years), the numbers ex- pressing which century, are iwl. divisible by 4 ; thus 1600 and 2000 are leap years, because 16 and 20 are exactly divisible by 4 ; but 1700, 1800 and 1900 are not leap years, because Hj \6, and 19 are cot exactly divisible by 4. ' ' i \l/M I 80 ARITHMETia MISCELLANEOUS TABLE. 41. 12 Unita make 1 Dozen. 12 Dozen 1 Gross. 12 Gross 1 Great Gross. 20 Units 1 Score. 24 Sheets of Paper ... 1 Quire. 20 Quires 1 Ream. 100 Pounds 1 Quintal. 196 Pounds 1 Barrel of Flour. 200 Pounds 1 Barrel of Pork or Beef Note. A sheet folded into two leaves is called a folio, mto 4 leaves a quarto, into 8 leaves an octavo, into 16 leftvicis J . J mo, into 18 leaves an 18 mo, &c. REDUCTION. 42. When a number is expressed in one or more denom' inations, the method of finding its value in one or more othei denominations is called Reduction. Thus, £1 is of the samr value as 240d!., and 7a. 1^. is of the same value as 342 far things, and conversely : the method or process by which we find this to be so, is Reduction. 43. First. To express a number of a higher denanmiation lyr of higher denominations in units of a lower denomination. Rule. Multiply the number of the highest denomination in the proposed quantity by the number of units of the next lower denomination contained in one unit ot" the highest and to the product add the number of that lower denominalion, if there be any in the proposed quantity, Repeat, this process for each succeeding denomination, till the required one is arrived at. Ex. 1. How many cents in $75.65 cents? JKv thft Rnlft By the Rule, $75.65 100 7500 + 65 = 7565 ^^nts. Berwn. Siaace 100 cents ma>;e one dollar ; $75=(75 x 100 cts.) = 7500 cts., .-. $75.65 = 7500 CRT nftaK. $75.65 = 7586 cents. illli REDUCTION. Ex. 2. Reduce £2 to farthings. By the Rule, £ Reason f 07' tlie Rule. £1 = 20«., .-. £2 = (2 X 20>s.= 40«. ls.= 12c?., .-. 40«.= (40 X 12)d. = 480f/. ld= 4y., .-. 4806?.= (480 X 4)(/.=. 1920^. .-. £2 = 40s.= mid.- 1920«?. 51 2 20 40s. 12 480c?; 4 19203-. Reduce (1) £709. 16«., M, to farthings. (2) 17 mis., 1 fur., 2 ft., 6 in. to inches. (3) 8 tons, 2 cwts., 3 qrs., 5 lbs. to drams. (4) 612 ac, 2 r., 27^ yds. to square inches. (5) 10 mis., 5 fur., 5 po , 5 yds., ft., 5 in., 5 Is. to lines, (6) 5 ac, 3 per., 29 yds. to square inches. (7) 17 days to minutes. (8) 2 lbs., 11 oz., 20 grs. to grains. (9) 2 lea., 2 mis., 7 fur. to yards. (10) 23 cub. yds., 1000 in. to cubic inchei. (11) 13 galls., 3 qtri. to gills. (12) 220 bushels to quarts. (13) 3 yrs., 315 days to minutes. (14) 27 lbs., 5 oz., 16 dwts. to grains. (15) 47 lbs., 11 oz., 6 drs., 2 sc. to grainy (16) £200. 17s., 8K to halfpence. (17) 219 ac, 2 r., 16 per. to square yardi. (18) 218 yds., 2 qrs., 3 nls. to nails. (19) £2376. 19s., 8ic?. to farthings. (20) 216 cwt., 2 qrs., 17 lbs. to poundg. (21) 25° 36' to seconds. (22) 8 mis., 3 fur., 4 yds. to inches. (23) £312. 17s., 6^6?. to fiirthinga (24) 1C5 lbs. Troy to grains, 4 * * I I 'h; , n ,1 \yy.. m -V W *.a ARITFMETIC. (25) 26 English ella to nail8. (28) 37 Fri'uch ells to nails. (27) £567. Os. 6^(1. to farthin,ij;s. (28) 287 lbs., 6 oz. to scruples. (29) 3 pipes to gallons. (30) £200. 19s. Oid. to liirthings. M 44. Secondly. To erprem a rinmher of lower denominatian #r demminations in units of a higher denomination. Rule. Divide the given number by the number cvf unit*? which connect that denomination with the next higher, and the remainder, if any, will be the number of surplus uints ol the lower denomination. Carry on this process, till you arrive at the denommation required. Ex. 1. How many tons, cwts.,&c., are there in 27658 drams? By the Rule, 16 2 ^8 ( 2 1728 1M8 25 I ?; I 27658 Reason for the Rule. TYsaq 10 dr^ 16 drs.= loz.,.-. 27658-^16 = 1728 LJ8^J-10 (irs. oz.-f-lOdrs. 16 oz.=l lb., . . 1728 oz.-^-16 = 108 864-0 oz. lbs. + oz. 25 lbs. = l (ir., .-. 108 lbs. 4-25 == 4 qi's.+81bs. "^ Iw iv^ 4 qrs. = l cwt., .*. 4 qrs.-^4=l -iP^"^' cwt.+0qr3. Icwt, qrs., 8 lbs., oz., lOtbs. .•. 27658 drams = 1 cwt., qrs., 8 lbs., oz., 10 drs. Ex. 2. In 17392 cents, how many dollars and cents? By the Rule, ( 10' 17392 Reason for tlie Rule. lOOi .J 17^Q o 100 cents = $1, .-. 17392 cts.-r- 106 ( 1U| ii6J-. ^ ^^^g ^ yo ^^g^ . -^^jjy^ cents 108 21 4 |173-92cts. =: $173.92 cts. Note. From the above example, we see that by cutting off the last 2 flffures on the right of any number of cents, (fives the dollars^ and the figures so cut off will be the ceuts. \m I COMPOUND ADDITION, S3 Ex. XXI. neaucft (1) 123390 farthings to pounds. (3) 13173 grs. to lbs. Troy. \ (3) 18191 pts. to gallons. f (4) How many leagut's in 70787.108 inches? * * (5) How many tons, &c., in 2007008 drams? (6) How many acres in 93827 perches ? (7) In 167813 grs., how many lbs. Troy V (8) In 8756765037 lines, how many miles, &c ? (9) In 7678678956 drs., how many tons, &c. ? (10) In 121605 in., how many miles, &c. ? (11) In 98006 grs., how many lbs. Troy, &c. ? (12) In 2022753 drs., how many tons, «&c. ? (13) How many lbs., ozs., drs., &c., m 702917 grs. ? (14) How many years (365 ds.), &c., in 1737893 seconds ? (15) How many acres, &c., in 172435 yards ? (16) How many yards in 13850832 cubic inches? (17) How many acres in 1244160000 sq. inches? (18) How many yards, &c., m 500 nails ? (19) In 131075 seconds, how many degrees, &c. ? (20) III 31557600 seconds, how many days, «&c. ? (31) In 219612 pts., how many hogsheads of beer ? (33) In 300738 pts., how many hogsheads of wine ? (23) In 912715 lbs., how many bushels of wheat ? (24) In 1000000 lbs. of oats, how many bushels? (35) In 7303 lbs. of timothy seed, how many bushels? (2fc) In 30747 cents, how many dollars? ^37) Howmanypounds,&c., in 973647 farthmgs? COMPOUND ADDITION. 45. Compound Addition is the method of collectinir severaa numbers of the same kind, but containing different denominations of that kind, into one sum. Rule. Arrange the numbers, so that those of the same .V«^"V V' '"">,^^ "xjvtci cai;ii uLuur m ihe same coiuuin, and draw a hne below them. 54 ARITHMETIC. 11 ■ , 1 1''' i:'; a. I i Add the numbers of the lowest denommation together, and find by Reduction how many units of the nex.t higher de- nomination are contained in this sum. Write the remainder, if any, under the column just added, and carry the quotient to the next column. Proceed thus with all the columns. Ex. I. Add together $31.97, $28.76, $38.39. By the Rule, $21.97 $28.76 $38.39 12 way as reason. was The sum of the right hand column is 23 i write 2 under that column, and carry 2 to the next : the sum of the next column together with the 2 carried is 21 ; write 1 under that column and carry 2 to the next, and so on ; the same done in the Simple Rules, and for the same Ex. 2 Find the sum of £6. 6s.. £3. 13«. 0|fl.. £35. 15s. Hid, and £43. Os. 8id \q. + 2g. + 3^. = 6g. = \\d. • write down ^d.y and carry Id. Then Id. + Sd.+ lld.= 20d.= Is. Sd. • write down Sd., and carry Is. Then Is.-f- 15s. 4- 13s. -[- 6s. = 35s. = £1. 15s. ; write down 15s., and carry £1. Then £1+ £43 4- £35 + £3 + £6 = £88 ; write down £88. Note. The method of proof in the Compoui^d Rules is th^ •ame as in the Simple Rules. £ 6 8 35 43 ». 6 13 15 £88 . 15 d. 01 IH 8i Ex. XXII. Add together, £ s. d. qrs. lbs. o«. (l) $26.79 (2) 6 . 9 . 8 (3) 2. 17 . 12 $39.17 8 . 10 . 4 6 . 24 . 13 $28.68 _5 . 12 . 3 1 . 6 . _8 lbs. oz. dwt. gr. lbs. oz. dr. BC. gr. (4) 35 . 3 . 4 . 12 <5) 17 .8.2 . 1 . 5 27 . 8 . 14 . 22 12 . 10 . 6 . . 19 41 . 9 . 17 . 10 6 .6.4 . 2 . 18 2 . 3 . 13 . 21 17 . 11 . 7 . 2 . 10 COMPOUND SUBrRAGTION. 65 ^ (6) $ 230.97 6120.35 517.08 9012.07 712.15 yds. qrs. nls. (8) 27 . 2 . 3 35 . 3 217 . 1 89 . 2 207 . 3 2 3 2 2 £ (10) 38 29 . 16 39 . 17 21 . 18 8. a. 6 . 7i ' 8i 6f 7 15 . 17 . 8 (7) luuH cwt. qrs. lbs. 21 . IQ . 2 . 24 . oz. 10 26 . 5 . 1 . 22 . 9 1 . 17 . 3 . 19 . 12 19 . 12 . . 18 . 9 218 . 10 . 1 . 12 . 8 mis. ftir. per. yds . ft. 9) 2 . 3 . 8 . 2- . 2 25 . 7 . 21 . 4 . 1 3 . 6 . 23 . 2 . 17 . 4 . 19 . 3 . 2 29 . 5 . 16 . 1 . 1 dys. hrs. min. sec. 11) 2 . 16 . 16 . 17 27 . 22 . 22 . 33 19 . 21 . 80 . 37 28 . 23 . 39 . 50 36 . 20 . 45 . 55 /12) $2219.64 (13) 3812.75 7 913.25 • 9 837.19 19 687.^ .^^v ^^^ ^^^' vs. lbs. oz. drs. (14) 23 . 15 . 2 . 20 . 5 . 21 . 17 . . 24 . 1 . 13 43 . 19 . 3 . 24 . 15 . 15 ac. ro. 5 . 3 1 2 3 per. yds. ft. 7 9 16 13 22 29 27 28 2 8 2 6 3 in. 5 107 96 108 12 3 . 9 . 2 . 17 . 13 . 11 6 . 6 . 1 . . 7 . 8 (15) $5617.28 208.09 516.99 3712.89 984.75 1 f I , COMPOUND SUBTRACTION. 46. Compound Subtraction is the method of flndinff the (litterence between two numbers of the same kind, but con- taining different denominations of that kind. Rule. Place the less number below the greater, so that t.ie numbers of the same denomination may be under eacfc other in the same column, and thaw a line below them 56 ARITHMETIC. Begin at the right hand, and subtract if possiMe each number of the lower hne from that which stands above it, and set tlie remainder underneath. But wlien any number in the lower line is greater than i\\f* s^ number above it, add to the upper one as many units of th^ \A same denomination as make one unit of the next highejr ^e- \i^\nomination ; subtract as before, and carry one to the numbei of the next higher denomination in the lower line. Proceed thus throughout the columns. tv* Ex. 1. From £51. By the Rule, £ s. d. 51 . 0.8^ 47 . 18 . 7f Os. Bid, take £47. 18.«. 7fd £3 3 MetJiod of working. I cannot take ^q. from 2q., so I add Irf., or 4^., to the 25'., making it 6g. ; then, 3g. from %q. Of leaves Zq.; write down the Sg.; m order to increase the lower number equally with the upper, I add Id to the 7d, making it 8d ; then M. from 8d leaves Od; write down Od. I work the remaining columns in the same way, and find the required answer. Ex. 2. From $978.29 take $678.93. $978.29 $678.93 $299.36 This example is worked in the same way as Simple Subtraction. Ex. XXIII. £ 8. d. (1) 33 . 17 . 4 18 . 8 . 10 Ibi. 02. dwt. (3) 12 . 6 . 3 9 . 7 . 16 yds. qrs. nls. In. (5) 106 . 1 . 2 . 1 92 . 3 . 3 . U lbs oz, drs. sc. grs. (2) 27 . 8 . 6 . 2 . 15 17 . 9 . 3 . 1 . 19 mis. ftir. per. yds. ft. (4) 25 . 6 32 . 4 . 2 22 . 7 . 37 . 3 . 2 c. yds. c. ft. c. in. (6) 325 . 22 . 101 296 . 35 . 386 ac. lo. per. yds. ft. In. wks, dyii. hre. mip. bm (7) 39 . 2 . 27 . 29 . 2 . 6 (8) 7.5. 6 . 36 . IT 27 . 3 . 29 . 37 . 8 . 8 6 6 . 20 . 46 . 2/> COMPOUND MULTIPLICATION ^1 £ s. d. (9) 129 . 10 . 8i 75 . 18 . 9* (11) $3967.78 1898.89 ac. Gwt. qrs. lbs. oz. drp. (10) 7 . 2 . 15 . 6 . 12 6 . 3 . 24 . 10 . 14 cords, c. ft. (12) 193 . 107 97 . 125 (13) $325.68 297.99 MA^ cu^^ ^?- ?f^ ^^^' ^^- ^"- c. yds. c. ft. c. in. (14) 297 . 1 . 23 . 2 . 1 . 101 , (15) 278 . 3 . 1127 189 . 2 . 28 . 2i . 2 . 127 198 . 8 . 147H i :?>.> \i. :--.||' , » mli^. fur. per. yds, ft. in, (16) 117 . . 27 . 5 . 1 . 9 89 . 7 . 38 . 4 . 2 . 11 degs. min. 8eo. (17) 29 . 29 . 38 22 . 49 . 5:) , V. ! -I tons cwl. qrs. lbs. (18) 293 . 16 . 1 . 21 287 . 19 . 2 . 22 "^'- (^l»'»' yds. qrs. nls. in. 6 . 15 (19) 1209 .1.1.1 11 • 14 1198 . 2 . 2 . Ij- m bu. pk. gal. Qt. (20) 268 . 2 . 1 .1 197 . 3 . 1 . 3 bu. pk. gal. ' t. (21) 19672 .0.1.1 18998 .3.1.3 COMPOUND MULTIPLICATION 47. Compound Multiplication is the method of finding the amount of any proposed compovmd number, that is. of any number composed of ditterent denominations, but all of the same kind, when it is repeated a given number of times Rule. Place the mult1i)lier under the lowest denomina- tion of the multiplicand. Multiply the number of the lowest denomination by the multiplier, and find the number of miits of the next denom- matioii cont'iined in this first product ; if tliere be a re- mainder, write it down ; for the second product, multiply the number of the ne.xt denomination in the mutiplicand by tlio multiplier, and aft^r adding to it the above-mentioned num- ber of Liuite, proceed with the result as witli the first product Carry tliifl o;H;ration through with all the different denom- inations of th . multiplicand. Mnltipliei not yreater than 12. \ ■ > • r-:n 58 ARITHMETIC. v. I' i! '1^ n t ijii s. 14 d. 11 Ex. 1. Multiply £1. 14s, 9fr?. by 11. 3?. X 11 = 33(?.= 8id ; write down id. , then 9d X 11+ 8d= 99d+ 8d= 107d= 8*.. lid ; write down lid ; then 14s. x 11 + 8rf. = 154s. + 8s. = 162s.= £8. 2s. ; write down £19 . 2 . Hi 2s. ; then £1 x 11 +£8 = £19 ; write down £19. Ex. 2. Multiply $27.78 by 9. $27.78 I^ ^^^^ example we do the same aa in Simple Multiplication, observing to place the point separating the doUais and cents in its proper place. Ex. XXIV. 9 $250.02 (1) £ s. 12 . 9 . yds. qrs. 27 . 3 . vvt. qrs. Ibg 16 . . 17 d. 6 2 nls. 3 5 . oz. .0. (2) (5) drs. 15 8 lbs. 17 mli 21 oz. drs. sc. .5.6.2 3 5. far. per. yds. ' . 7 . 26 . 4 (3) ft. . 2 6 lbs. oz. dwt. grs. 18 . 6 . 5 . 10 4 (4) (6) $237.19 7 c (7) mis. fur. per. yds. (8) 6 . 4 . 6 . 2 ft. . 1 . in. 9 (9) $609.93 9 10 (10) wks. 7 dys. . 5 . hrs. 18 . min. 16 11 ac. ro (11) 7 . 3 per. yds. ft. In. . 29 . 20 . 1 . 108 12 (12) £ 20 . 17 d. . n 7 (13) lbs. oz. drs. sc. 74 . 11 . 5 . 2 12 bu. pk. qt. (14) 7.3.1 3 US) 2 . 3 . 59 (16) 1554 . 3 . 1 (17) 365 . 5 . 48 . 57 10 11 12 (18) 73 . 17 ..8| 11 uc. ro. per. (19) 14 . 3 . 39 9 20) $297.68 12 COMPOUND MULTIPLICATION. 59 237.19 7 309.93 10 In. . 108 12 [>k. qt. 3 . 1 3 1. sec. 1 . 57 12 bu. pk. gal. £ «. d. lbs. ob. dwt.grs. (31) 2782 .2.1 (22) 70 . . IH (23) 18 . 3 . 14 . 5 9 13 12 (24) $917.75 8 (35) £ B. 17 . 15 d. .01 9 mis. (27) 54 ftir. per. . 3 . 18 yds. . 5 7 (26) $1875.25 12 If the Multiplier he a Com/podte number, each of wlwm f(W- tors is less than i^, multiply by one of them, and the resulting product by another, and so on. The last product so obtained^ is the required product. Find the product of 2 cwt, 3 qr., 17 lbs. by 63. ewt. qrs. lbs. 2 . 3 . 17 The factors of 63 are 9 and 7. First, we 9 multiply by 9 and the product we get by 7 ; Qfi 1 q which clearly is the same as multiplying 3 7 cwt., 3 qr., 17 lbs. by 63. Note. The same result is obtained, by 183 . 3 . 31 taking the factor 7 first, and then the 9. ac. ro. per. (1) 56 . 3 . 9 38 lbs. dwt. grs. (4) 21 . 13 . 17 77 Ex. XXV. mis. ftir. per. (3) 37 . 6 . 9 54 £ s. d. (5) 17 . 11 . 8i 30 £ 8. d. (3) 19 . 11 . 4 144 yds. qr. nls. in. (6) 37 . 1 . 3 . 3 54 cwt. qrs. lbs. oz. drs. £ s. d. (7) 3 . 3 . 23 . 13 . 6 (8) 72 . 19 . 9| (9) $209.18 63 81 35 u. %. yds. c. ft. c. in. Ibb. oa. dwt. grs. £ <«. nO) 17 . 21 . 57 (11) 8 . 8 . 15 . 13 (12) 42 . 10 . 9i 84 49 88 ■1 M 'J 11! •i; : ■t«ji •ill ■: S |b». ■■■f:j ■! i .1 [.J li i; '^t- 60 ARtTllMETia dys. hrs. min. eec. ^ (13) 5 . 17 . 39 . 20 120 lbs. oz. dwt, ffrs, (15) « . 2 . 3 . l7 5382 (17) 20 . 2 . 17 . 15 . 3 . 3 64 £ 5. c?. (19) 2 . 6 . 8i 900 lbs. or. drs. bc. '^ 2 84 (14) 74 . 11 . £ 8. d. (IC) 13 . 7 . 4| 275 mis. fur. p«r. yds. ft. in. (18) 2.6.2.3.0 5 375 (20) $237.15 500 bu. pk. gal. (21) 10 . 2 . 1 800 ijl ■ 1 m ,*■;. fhl^l^ f/ ^^^]mer is not a Compodte number and larger trsand P- ^^ ^^^ ^^^^ *^ ^^ *^^*'^ *^^^ number iitofac- Thus,29 = 4x7 + 1; 19=:6x3 + l; 39 =r 12x3 + 3. oofi""' h.^'^^^'^^y ^^^^9. Os. OK by 2331. 2331 = 2000 + 300 + 30+1 = 1000x2 + 100x3 + 10x3 + 1 = 10x10x10x2 + 10x10x3 + 10x3 + 1. 2579 . . Of for 1 10 25790 . Tn for 10 10 "257900T 6 . 3 for 10 x 10, or 100 10 2579003 . 2 . 6 for 100 x 10, or 1000 2 .rW ^l^X 'A- ^ ^^^ 1000x2, or 2000. add S • \^ • 1^. I''' f i^^™- «^- ^^- X 3' «r for 300. add '2I79 : I : 'ol lol r'"'- ''' '^'' ^ '' ^^^ ^^^- ^'^^• "8i for 2000 + 300 + 30+1, or 2:]::i. 6011656 . 5 dOMPOUND DIVISION. 61 cwt. qrp. lbs. oz. (1) 3 . 3 . 21 . 5 89 i:x. XXVI. lbs. oz. dwt. grrs. (2) G . 2 . 3 . 17 54G3 cwt. qrs. lbs. oz. drs. <4) 2 . 3 . 23 . 6 . 7 627 £ s. d. (3) 2 . 6 . 9* 938 £ 8. d. (5) 4 . 18 . n 561 1 ' ' lbs, oz. drs. sc. grs. (6) 15 . 2 . 3 . 2 . 7 712 (7) If a man gets $3.25 a day, how mncli will that be in 209 days? (8) When wheat is selling for $1.27 a bushel, how many lollars will a farmer get for a load of 52 bushels of wheat? (9) A butcher buys an ox weighing 1625 lbs. , live weight, %t 6 centsa pound, howmuch will he hav«» to payaltogetlier ■ (10) A boiler-builder bought 29 boiler plates, each weigh- !ng 1 qr., 17 lbs., 8oz., what was the weight of the whole of them? (11) If the Governm of Ontario sells one hundred thou^ sand acres of wild land for forty cents an acre, how many dollars will it obtain for the whole? COMPOUND DIVISION. 48. Compound Division is the method jf dividing a com- pound number, that is, a number composed of several denom- inations, but all of the same kind, into as many equal parts as the divisor contains units ; and also of finding how often one compound number is contained in another of the same kind. WJien tlie Didisor zs a member eitJier larqer, or not laraer than 12. ^ Rule. Place the numbers as in Simple Division : then find how often the divisor is contained in the highest denomi- nation of the dividend ; put this number down in the quo- tient ; multiply as in Simi)le Division and subtract. If ther«i be a remainder, reduce that remainder to the next i IJ: Ill if-' 1 Ulj 1-1 6d ARITBMETta inferior denomination, adding to it the number of that (It* nomination in tlie dividend, and repeat the division. Carry on this process through the wliole dividend. WJien tJte Divisor is less tJuin 12. Ex. 1. Divide £G76. 19s. 9id by 11. 11 1676 . 19 . 9^ £676 11 gives £61 as a quotient and £5 over ; £5+19s.=119«., 119s. bl . lU . 10irem.8g. ^ ^^ ^j^^^ ^^^ as a quotient and 9s over; 9s.4- M.= 117d, 117d-4- 11 gives lOd as a quotient and Id. over; 7dH- 2y.= 30^., 30?. -i- 11 gives 2 as a quotient and rem. ^q. When the Divisor is greater than 12 and not a Composite number^ the work may stand thus : Ex. 2. Divide £297. 4s. M. by 73. By the Rule, "We first subtract £4 taken 73 times, i. e. £292 from £297. 4s. 8d, there remains £5. 4s. %d. Now £5. 4s. 8d = 104s. 8^., from this we subtract Is. taken 73 times, i. e. 73s. from 104s., there remains 31s., .-. there is Is. in quotient. 31s. M. - 380d, from this we subtract bd. taken 73 times, i. e. 365d, there remains \M. over .'. £4. Is. M. goes 73 times in £297. 4s. M., and \M. over. £. 8. d. 73)297 . 4 . 8(£4 292 5 20 [add the 4s.] 73) 104 (Is. 73 31 _12_[add the 8d] 73')380(5d 365_ 15" .*. the Quotient is £4. Is. 5d and 15d over. Whm the Divisor is a Composite number greater than .12, we may divide as in Ex. 1, successively ii each factor, and the last quovent so obtained wUl be the requirei quotient. Ex. 3. Divide 975 mis., 3 fur., 24 per. by 56. Since 56 = 8 x 7, the work may stand thus : mlp. ftir, per. 8 975 . 3 . 24 121 . 7 . 18 17 . 3 . A4 Note. The same result tained by dividing first by 7 and then by 8. iiil Compound division. 63 Ex. XXVII. d., £S. 15.S. 7d. = 2\07d: AM. subtracted 49 times from 2107f/. leaves no remainder. .'. 49 times is the answer. Ex. 2. I employ twice as many men as women, the wages of the former are Ss. Qd. each, and of the latter U. lOd. each per day. The weekly wages amount to £23. 17«. How many men, and how many women do I employ ? £'^3. 17«.-f- G = £3. 19.-*. 6^. = 954d= am^ of daily wage- Daily wages of 2 men and 1 woman = Hs. Gd.x2+ 1.^. 10a. = 8s.lOd.= 10M. 106)954(9 954 .-. there are 18 men and 9 women Ex. XXVHI. Divide, (1) £684. 7.V. iUI. by £76. Os. lOd. '(2) £171. l.s. 10.i -^ 12 = 190 -j- 12 = 15 \ ^ $291.95^* Therefore the required answer is $291.95^§. Ex. XXIX. How many dollars and cents in (1) £25. Gs. 3d (2) £57. 19.9. Sd. (3) £207. 17.9. Sd. (4) £153. 18.S. nd. (5) £217. 17.9. Od. (6) £319. 15.9, 7hd. (7) £612. 19s. Hid. (8) £63. 9.9. OJd (9) £912. 12.9. 6cZ. (10) £711. .5.9. 5M. (11) £1117. Os. 7K (13) £47. 7.9. 9d (13) £2017. Ck9. Sd. (14) £7.5. 9s. 8M (15) £37. 18s. 7M. (16) £87. 13s. dd. 50. To reduce dollars and cents to Halifax or old Canadian Cvrrency. Rule. Divide the dollars by 4, the quotient is pounds. If there is any remainder bring it to cents and add the given cents if any ; then divide by 20, the quotient is shillings. If any cents are left, multiply them by 3 and divide by 5 ; the quotient is pence. By arranging these several quotients properly, the required answer is obtained. How many pounds, shillings and pence in .$1279. t2i^ ? 4 I P79 12* P + l!^i cts. = 300 cts. + 12i\"cts. = 45Ta^~V^o 813icts.; 313icts. -T-30 = 15.9. and £319 and %6 over. ^2^ cts. over ; 12.^ cts. x 3 ^ 5 = 7id Therftfovo the answer is £310, 1.5.s'. 7M The above is evi- Iciitly wrrect; because $4 = £1, 20 cts.=: 13d, 5 cts.=: 3d ■ f\U\ I f I If'il I: I, : I i no AiiiTHMirrw. Ex. XXX. flow niftn} pounds, shilliu^ and pence in (1) $217.35 (3) $327.55 (3) $17.3A (i) 184.50 (5) $75.95 (0) $125.37^ (7) $867.87i (8) $1162.40 (9) $1393.02^ (10) $1937.20 (11) $2220.39 (12) $3785.48 Ex. XXXI. MISCELLANEOUS EXAMPLES. PAPER I. (1) The population of tlie counties on the river St. Law- rence in 1801 was as follows : Leeds, thirty-live thousand seven hundred; Grenville, twenty-four thousand one hun- dred and ninety-one; Dundas, eighteen thousand seven hundred and seventy-seven; Stormont, eighteen thousand one hundred and twenty-nine ; Glengarry, twenty-one thou- sand one hundred and eighty-seven. Find the total popula- tion of these five counties. (2) By the census of 1848, the population of Montreal was fifty-five thousand one hundred and forty-six ; of To- ronto, twenty-three thousand five hundred and three; of HamJlton, nine thousand dght hundred and eighty-nine ; of Ottawa, six thousand two hundred and seventy-five; of Kingston, eight thousand three hundred and sixty-nine ; of London, four thousand five hundred and eighty-four. Find the whole population of those cities. (3) Add, one hundred thousand, two hundred and twenty- nine thousand seven hundred and thirteen, fifty-eight thou- sand seven hundred and five, six hundred and twelve thousand five hundred and seventeen, nine hundred and ninety-nine thousand nine hundred and ninety-nine, eight hundred and thirty-three thousand seven hundred and nine- teen, seven hundred and sixty eight thousand three hundred and nine, 'fifty thousand and fifty. (4) Add, five thousand and five, seven thousand and eight- een, seventeen thousand nine hundred and fifteen, twenty- eight thousand seven hundred and nineteen, nine thousand and twelve, eight hundred and seven thousand five hundred and twelve, seven hundred and seventeen thousand and seventeen, ninety-three thousand five hundred and t^io, tvvo lumdred and twelve thousand six huudi'ed and seveu MISCELLANEOUS EXAMPLES, 87 n • . I itm B I (5) How many miles in 178006 inches? (0) In 1848 the value of the imports into Canada was 18375180.20; in 1801,thevalue of thcimports was $43054836; the i>opiilation at the former date was 1493332, at the latter 25067r)5. Find 1st., the value of the imi)orts for each person In 1848 and in 1861 , and 2nd., the diflFerence between these values. PAPER II. (1) What is the price of 818 bushels of wheat at 8.^. lOid, per bushel ? (2) A farmer sold 67 bushels of wheat at $1.62 a bushel ; bought a suit of clothes for $18, 82 yards cotton at 13^ cents a yard, a stove for $16. How much was left of the price of the wheat ? (3) If a Government was to divide 72812 acres equally amon^ 397 discharged soldiers, how much would each re- :jeive f (4) A farmer brought 160 bushels of wheat to mill when wheat was worth $1.60 per bushel, and in exchange got 27 barrels of tlour. How much was he charged for the floui* per barrel ? (5) A merchant has a piece of cloth containing 42^ yards, worth 6s. Q\d. a yard. How many dresses of 8^ yards each can be made out of it, and what will each cost ?' (6) A farmer sold in the Toronto market 618 barrels of flour for £1. 13'?. M. per barrel; and bought 84 yards of cotton at 17 cents a yard, 5 lbs. tea at 3-9. M. a 'b., 2 tons of coal at £1. 15.S. per ton, 8 sheep at £2, \\s. 9d. each, 15 head of cattle at £12. 19.9. 9d each. How much can he deposit in a bank, allowing that he takes $50 home with him ? PAPER III. (1) In one year there were coined in the British mint 203761 pounds of gold, value £9520732. 14». 6d Required the value of each pound ? (2) Three persons bought a ship for $63000 ; the first taking one share, the second three, and the third five. How much do they severally pay ? (3) If a contribution of £354. ll*. 6d. is made up in «qual shares by 26 men, how much must each give ? (4) What is the 29th part of XO bc. , 2 xo. , 7 1: I'll i |ii;i i| i' ( V ■ I f I 68 ABITIIMETia fJ Ui} (5) Divide 300 tons, 15 cwt., 3 qrs., equally among 347 men. How much will each get ? (6) Sokliei-s marching in quick time, make 110 steps in a minute, each step 3 ft. 6 in. long. In what time would a company of soldiers march 30 miles in quick time, allowing half an hour for rest? PAPER IV. (1) Add together £6. 17.s. 6d, $30.37, S^. 13s. 9d., $75.83; giving your answ^er in decimal currency'. > (3) Three boys went out together to fish, the first caught eight, the second as many and three nSte, the third as many as his two comrades all b^t one. Hotf luimy did each of the last two boys catch ? • (3) Three boys, Thomas, William, and Alexander, had between them 6 cents ; Ttfemas had 9nt?; William two, and Alexander three ; they bought fifty-l«p[Hjiarbles with their money. How many o.u^t each boy *9^t ? (4) Four men went.6ut^ne night to fish, borrowing both boat and nets. A man waA to have 4 shares of the catch as often as the owner of the net was to have one ; but, a man was to have omy two shares as often as the owner of the boat had one.* The catch was four barrels^^f herrings. What w^as each party's share in dozens ; each barrel contain- ing 38 dozens of herrings ? (5) It is fotmd by observation that m each square inch of the human skin there are about 1000 pores ; and the surface of the body (J •ft middle sized man contains about 3304 square inches, or l^Aquare feet. Required, the number of pores in the surface drsuch a body, 999 being supposed to be con- tained in eadSTsquare inch ? y. (6) Thejuiiifcof two numbers is 84889 ; the difference be- tween thei#{s 889. What are the numbers ? PAPER V. (I) Find the product of 73678397 and 86073 ? (3) Tlie quotient is 73697; the remainder 3087 ; the di Visor 11689. Find the dividend? (3) The minuend is twenty-seven thousand eight hundred and twelve ; the difference, fifteen thousand nine hundred and eight. Find tlie subtrahend? (4) Tliere are seven addends all equal ; their sum is eii.;hly-nine thousand two hundrtnl and iiixty-four. Find Diif; of Djeml^ y^ MISCELLANEOUS EXAMPLES. 69 347 Vi (5) In the census of 1861, Rutland contained twenty-two thousand nine hundred and eighty-three inhabitants ; North- amptonshire, ni«iety-six thousand eight hundred and one; Huntingdonshire, sixty-four thousand one hundred and eighty-three ; Leieestersliire, ninety-one thousand three hun- dred and eight ; Nottingliamshire, one hundred and ninety thousand and sixty. What was the sum of the population of the above 5 counties in 1861 ? (6) During the Crimean war, out of the French army there were killed in action or missing ten thousand two hundred and forty; drowned hi a wreck, seven hundred and four; died of various diseases before the battle of Alma, eight thousand and eighty-four ; died of disease before Sebastopol, four thousand three hundred and twelve ; died in hospitals, &c., seventy-two thousand two hundred and forty-seven. How many were lost altogether ? PAPER VI. (1) In 1861 the population of Edinburgh was 160302 ; of Glasgow, 168795 more than that of Edinburgh ; of Aberdeen, 71973; of Inverness, 24537 more than that of Aberdeen What was the total population of all these places in 1861 ? (2) The paid up capital of each of the following Banks doing business in Ontario, is: of the Bank of Montreal, 16000000; of Bank British North America, $4866666; of Quebec Bank, $1467750 ; of Bank of Toronto, $800000 ; of Ontario Bank, $1909640 ; of Royal Canadian Bank, $590382 ; of Merchants' Bank, $862033. Find the total amount of the paid up capital of the above named Banks ? (3) The amount of revenue, fi-om the named sources du- ring 1866, was as follows: Customs, $7328146.68; Excise, $1888576.76; Postage, $621936.42; Public-works, $417474 1 Education, $66554; Common School Fund, $122142.77. Find the whole revenue from these sources ? (4) A person has $975. He buys a team for $375, a wagon for $82, a plough for $16, a stove $16, a reaping machinetor $153, 12 sheep for $8 each, 2 cows $25 each, 3 pigs $6 a piece, pays his servantman 3 months' wages at $20 a month, and the re^it he lays out in flour at $l!75 per 100 pounds HoVv" many pounds of flour will he have ? (o) Among 635 men dividt! equally 86895 acres. (6) How many kic,U'?s In IC i^ls., 3- |:er», 4 yds. ? i, ' " :: •■^; '• H ' . i'i ..i;'; ''M\\ lii^ 1 fhft, H i! n W'h ! ^1' M " .11 70 ARITHMETIO. I- If '•' .- m 1 1-^' "\'i. I 1 '!,■>?: h ^ii'is: SECTION III. GREATEST COMMON MEASURE. 51. A MEASURE of any given number is a number which will divide the given number exactly, i. e. without a remain- der. Thus, 3 is a measure of 6, because 2 is contained 3 times exactly in 6. 52. A MULTIPLE of any given number is a number which contains it an exact number of times. Thus, 6 is a multiple of 2. 53. A COMMON MEASURE of two or morc given Humbers is a number which will divide each of the given numbers exactly. Thus, 3 is a common measure of 18, 27, and 36. The GREATEST COMMON MEASURE (g. c. m.) of two or more given numbers, is the greatest number which will divide eaca of the given numbers exactly. Thus, 9 is the greatest com- mon measure of 18, 27, and 36. 54. To find the greatest common measure of two numbers. Rule. Divide the greater number by the less. If there be a remainder, divide the first divisor by it. If there be still a remainder, divide the second divisor br this remainder, and so on ; always dividing the last preced- ing divisor by the last remainder, till nothing remains. The last divisor will be the greatest common measure re- quired. Ex. Find the g. c. m. of 144 and 240. By the Rule, 144)240(1 144 96 ) 144 ( 1 bringing down last diviscr 144 for a dividend 96 48)96(3 96.. .. .. g6_ .% 48 is Q. c. M. required lMAST COMMON' MlfLTiPLE. 1\ Ex. XXXII. Find the Q. c. m. of (1) 8 and 18. (2) (4) 16 and 28. (5) (7) 28 and 44. (8) (10) 46 and 116. (11) (13) 366 and 128. (14) (16) 1216 and 424. (17) (19) (21) (28) (25) 3042 and 3094. 1441 and 1572. 21168 and 204624. 828597 and 738140. 6 and 15. 20 and 32. 30 and 42. 58 and 174. 180 and 210. 127 and 445 (20) (22) (24) (26) (3) 4 and 22. (6) 24 and 39. (9) 36 and 56. (12) 315 and 378. (15) 310 and 030. . (18) 6408 and 7264. 7040 and 7392. 46436 and 23025. 97482 and 29579. 326337 and 737800. LEAST COMMON MULTIPLE. 55. A COMMON MULTIPLE of two or more given numbers is a number which will contain each of the given numbers an exact number of times. Thus, 144 is a common multiple of3, 9, 18, and24. The LEAST COMMON MULTIPLE (l. c. m.) of two or morc given numbers is the least number which will contain each of the given numbei-s an exact number of times. Thus, 72 is tlie least common multiple of 3, 9, 18, and 24 56. When tTie least common multiple of several numbers is required^ (lie mx>si convenient practical metliod is that given hy the following Rule. Rule. Arrange the numbers in a line from left to right, with a comma placed between every two. Divide those numbers which have a common measure by that common measure, and place the quotients so obtained and the undivided numbei*s in a line beneath, separated as before. Proceed in the same way with the second line, and so on with those which follow, until a row of numbers is obtained in which there are no two numbers which have any common measure greater than unity. Thti the continued product of all the divisor:: and the numbers in the last line "will be the least common multiple required. Note. It will in general be found advantageous to begin li \n . 'hi rfjiij ! . i H 73 ARITHMETIC. 2 1 10, 12, 16 2 1 5, 6 , 8 3, 4 A. 5, with the lowest prime number 2 as a divisor, and to repeat this as often as can be done ; and then to proceed with the prime numbers 3, 5, &c., in the same way. Ex. 1. Find the l. c. m. of 10, 12, and 16. Uy the Rule, 10 = 2 X 5, 12=2 X 2 X 3, 16=2 x 3 x 3 x 2. .•. L. c. M. must clearly contain as factors 2 X 5 for 10. 2 x 5 X 2 X 3 for 10 and 12. 2 X 5 X 2 X 3 X 2 X 2 for 10, 12, and 16. .-. L. c. M.= 2 X 2 X 5 X 3 X 4 = 240. Note. The process of finding the L. c. M. may often be shortened by striking out in the same line e\ ery number which exactly measures any other number in that line. Ex. 2. Find the l. c. m. of 9, 14, 16, 18, 24, 36, and 38. ^, 14, 16, ;^, 24, 36,38 Every multiple of 36 must be 7 8 12 18 19 ^ multiple of 9 and of 18; .*. ;^' — j' "r~ ^'"[tr strike out 9 and 18: for the ' 1 0, t^jLJ same reason strike out 3 in the 7, 2, ^, 9,19 4th line. L. c. M.= 2 X 2 X 2 X 7 X 2 X 9 X 19 = 19152. 2 2 o Ex. XXXIII. (2) 8, 9, and 12. (4) 20, 28, and 36. (6) 24, 56, and 84. (8) 6, 33, 24, and 33. (10) 7, 8, 9, 10, and 13. Find the l. c. m. of . (1) 2, 4, and 10. (3) 12, 16, and 18. (5) 16, 24, and 30. (7) 15, 25, and 105. (9) 7, 21, 6, 14, and 25. (11) 24, 28, 36, 22, and 16. (12) 2, 5, 45, 15 and 25. (13) 9, 4, 8, 15, and 27. (14) 15, 20, 24, 21, and 35. (15) 4, 5, 7, 8, 15, 21, and 30. (16) 2, 7, 9, 13, 15, 52, and 63. (17) 3, 7, 21, 11, 77, and 198. (18) 100, 56, 35, 125, and 150. (19) 22, 55, 19, 15, 95, and 133. (2e^ 4a Gi 37, 33, 110 and 165. .1 -■4 FUAGTlom. n SECTION IV. i FRACTIONS. 57 II. Let unity be represented by the line ABy which we will consider to be 1 yard in lengtli. Suppose AB to be divided into 3 equal parts AD^ DE, EB ; then one of such parts AD A D E B F O C is a foot or one-third part of the \ ;— 7 \ \ yard, and it is dcjoted thus ^ (read one-third) \ two of them AE, or two feet, thus f (read two-tMrds) ; three of them AB^ or tliree feet, or the whole yard, thus | or ^.. If another equal portion /?^^ of a second yard JBC, divided in the same manner as tlie fust, be added, then AF^ or four feet, is denoted thus ;^ ; and so on. Such expressions, representing any number of the equal parts of a unit, i. e. of a quantity which is denotect by 1» are called Broken Numbers or Fractions. 58. A Fraction denotes one or more of che equal parts of a unit; it is expressed by two numbers piaced one above the other with a line between them ; tLe lower number is called the Denominator (Den^), and snews into how many equal parts the unit is divided ; the upper is called the Nu- merator (Num^), and shews how many of such parts are taken to form tlie fraction. 59. A Fraction also represents the quotient of the num'; by the den'. o Thus '^ = 3 -f- 3 : for we obtain the same result, whether 3 we divide one unit, AB or 1 yard, into 3 equal parts AD, DE^ EB, eacn = 1 ft. or 12 in., and talic 2 of such parts AE (represented by f),= 12 in. x 2 = 24 in., or divitle 2 units, AC or 2 yards, into 3 equal parts, AE, EF, FC, each = 2 ft. or 24 in., and take 1 ot such parts AE; which m equal to ^rd part of AC or 2 units, or = 2-^-3. Hence f and 2 -r- 3 have the same meaning. 00. When fractions are denoted in the manner above ex- Jilained, they ai'c called Vulgar Fractions. ■I ,t It i f. I u ARITHMETIC. 61. Fractions, whose den", are composed of 10, or of 10 multiplied by itself any number of times, are called Deci- mal Fractions, or Decimals. IS I ' ' T ' VULGAR FRACTIONS. 62. In treating of the subject of Vulgar Fractions, it is usual to make the following distinctions : (1) A PROPER fraction is one whose num'. is less than the den'. ; thus, f , f , f , are proper fractions. (2) An IMPROPER FRACTION is One whose num'. is equal to or greater than the den^ ; thus, f , f , J are improper frac- tions. (3) A SIMPLE FRACTION Is One whose num'. and den^ are simple integer numbers ; thus, i, f are simple fractious. (I) A MIXED NUMBER is composcd of a whole number and#a fraction ; thus, 5^, 7| are mixed numbers, representing respectively 5 units, together with ^th of a unit ; and 7 units, together with f ths of a unit. (5) A COMPOUND FRACTION is a fraction of a fraction ; thus i of f, f of ^ of 1^0, are compound fractions. (6) A COMPLEX FRACTION is ouc which has either a frac- tion or a mixed number in one or both terms of the fraction ; ^ 2^ 3 21 fofi thus, t ^ __ f 3' 4f' 5^' 2i are complex fractions. 63. It is clear from what has been said, that every whole number or integer may be considered as a fraction whose den', is 1 ; feus, 5 = f , for the unit is divided into 1 part comprising the whole unit, and 5 of such parts, that is 5 miits, are taken. 64. To multiply a fraction by a wlwle number. Rule. Multiply the numerator by the whole number. 2 2x24 -^^^ ^^ ^ ^^^ 6 ' ^^^ ^^i^ is divided into ^ X 2 = — ir- = -• 5 equal parts, and twice as many part3 are 5 taken in ^ as are taken in f . 5 Ex. XXXIV. Multiply (1) f and H each separately by 2, 3, 5, 7, 9, and 12; and (2) Sf and -,2,&- each sepurately by 6, 8, 11, 10(J and 157. VULGAR mACTlOM. ?d 65. To divide a fraction by a whole number. Rule. Multiply the denominator by the whole number. 2 2 2 '^^^ value of each part in ^ is twice 10' = — . as large as the value of each part in i(t 1 but the same number of parts are 5 • ^~5x2 taken in each, . . ^ is twice as large as i\-, or ^ -?- 2 = /(j. Ex. XXXV. Divide (1) f and \ each separately by 2, 3, 5, 6, 9, and 12 ; and (2) ^f and ij each separately by 3, 5, 11, 56, and 100. 66. If the numerator and denominator of a fraction be both multiplied, or both divided, by the same number, the value of the fraction will not be altered. Since 8 = 4x2, two of the parts in f are equivalent to ons of the parts in f ; but since 6 = 3x2, there are twice as many 3x2 4x2 6 8 parts taken in | as there ai*e in f , therefore f = f. In figure, Art 57, ^^ represents either ^rd or gths (^ AG. 67. Hence it follows that a whole number may be con- verted into a vulgar fraction with any required den'., by multiplying the number by the required den', for the num'. of the fraction, and placing the required den', underneath. For 5 = -, and to convert it into a fraction with a den'. 6 30 or 17,wehave5 = 5 = — 1 1x6 — IT » 6 also5 = 5 = 5iii7 = §5. 1 1 X 17 17 Ex. XXXVI. Reduce (1) 3, 5, 8, 15, to fractions with den'". 2, 9, and 13; and (2) 9, 12, 17, 37, to fractions with den'«. 8, 10, and 57. 68. To represent an improper fraction as a whole or mixed number. Rule. Divide the numerator by the denominator. If there T e no remainder, the quotient will be a whole number. If there be a remainder, put down the quotient as the in- tegral part, and the remainder as the num'. of the fractional part, and the given den', as the den', of the fractional part 24 24 Ex. Reduce — and -=- to whole or mixed numbers. 4 5 1;' .1 'I'M i ;, 76 ARITHMETIC, By the rule, 5 ^^• -, 24 4x0 ^ , K . nn\ a For — = -—-= - (Art, 00) — 6. 4 4x1 1 ^ 24 20+4 For — - — 5 20 4 . 4 ,, 5 +5 = 4-^5- '^^ li f l^;i 6 • ilia 1 .J J • Ex. XXXVII. Express the following improper fractions as miAed or whole numbersi : (1) h (2) I. (3) ^h '4) \\ (5^ (6) ¥. (7) h\ (8) n. (0) H. (10) (11) W. (12) ^A^. (13) V^,a (14) ^|^^J.(i5) 69. To reduce a mixed mimhev to an improper fraction. Rule. Multiply the whole number or integer by the flenominator of tlie fraction, and to the product add tlie nu- merator of the fractional part. Tlie result will be tlie required uum'., and the den', of the fractional part the required den^ Ex. Convert 3f into an improper fraction. By the Rule, „„ 3x4 + 3 124 3 15 3| = For3|=:| + ? Ex. XXXVIII. lEtoduce the following mixed numbers to improper frac- tions : (1) H. (2) 2,V. (3) lf5-. (4) 17f. (5) 12f (6) 203H. (7) 2H. (8) 29^. (9) 704-,^,^. aO) 900-?,ff. (11) S^^TT. (12) 53^^. (13) 21tAtf. (14) 148IH. (15) 13^?i{}. (10) 25/A%. (17) l^U^h- 70. To reduce a compound fraction to its equivalent simple fra/ition. Rule. Multiply the several numerators together for the numerator of the simple fraction, and the several denomina' ^rs together ibr its denommator. 4 4 - 4* 3x4 ~lx4 3 12 ~ 4 3 + 4 = 12 + 3 4 " 15 "" 4* VULGAR FRACTIONS. Ex. 1. Convert | of ^ into a simple fraction. By the Rule, 77 2 5 _ 2^x5 3 6 ~ 3 X 6 For I of?: 5 twice H ^^ Q~ ^^^^^ g 10 18* 5 S = iwice ^(Art.65) 5x2,, ,.,, 10 = -3^(Art.()4)=.-. Mte 1. Before applying the above Rule mixed numbers must be reduced to improper fractions. Note 2. In reducing compound fractions to simple ones, we may strike out from any num^ and any den^ such fac- tors as ai-e common to both ; for tliis is in fact simply di- viding the num^ and den^ of a fraction by the same number. (Art. 66.) Ex. 2. Reduce f of 2 1^ of 1 h to a simple fraction. 3 ^„i ^.1 3 „25 „16 3x(5x5)x(4x4) -of2,i, of 1,^ = 5 of- of---^^^^3^^-^^^^^ = 3x^x^x |x4 ^ 4 ^-^j^j^^. ^^^^, ^n,^ ,^e,,r. i,y 3^ 5, 5, 4, ^xpx^x3xp 6 factors common to both. Ex. XXXIX. . Reduce the following compound fractions to simple ones.- (1) fof^. (2) fofii (3) ^ofH. (4) A of A. (5) tof2f. (6) tofli. (7) 18|of5fejofl0. (8) mofSf. (9) fof2iof9. (10) 1^of3fof3i (U) iofi^rofllofif. (12) Aof4tofAof6-,arof-«V (13) -,^of2^offofl0i. (14) 5- of 12 i of ^ off off of 9. (15) -h of \ of \% of \ of -h of 2 of i,', (16) f of f of f of 701 of -h of 1-,^ of 147. 71. A fraction is in its t.owrst terms, wlien its numer- »*or and denominator are prime to each other. !?2. To reduce a fraction to its lowest terms. i 1 i 1 i. ■ "• i I! is ARifHMliftO. 1 .1- KuLE. Divide the numerator and denominator by thelj greatest common measure. Ex. Reduce i|f to its lowest terms. By the J .iile, find the g. c. m. of 176 and 484. 170)484(3 352 133)176(1 132 44)1f>3(3 132 44)176(4 44)484(11 176 44 „ 176 44x4 4,, ^ .-. fraction in its lowest terms = r^ I'l ■ ' M • I" Ex. XL. Reduce each of the following fractions to its lowest terms: 2. 4" mi 2 2tj8' (2) (6) (10) (14) (18) (32) ii If Tt;64' mi (3) (7) (11) (15) (19) (23) A. 1 0T7. (4) (8) (12) (16) (20) (24) h (I) (5) (9) (13) (17) (21) denmiin^t&r^^^ ^^^' ^ ^ ^^uivalent ones with a common Rule. Find the least common multiple of the denomina- tors; this will be the common denominator. Ihen divide the common multipl ^— — ^' ^— r— r; or 3x5x7 5x3x7 7x3x5' 105 105 105" !, Ex. XLI. Reduce the fractions in each ol the following sets to equiv- alent fractions, having the least common den^ : (1) fandf. (2) | and f . (3) f and f (4) fandf. (5) H and fi (6) H and H. (7) T^andH§. (8) H^andM?^. (9) i U, and ji^o. (10) A', A, and ^. (11) fs, H, audi (12) |,A,-i?,andiV (13) If, ||, U, anu a. (14) -h, i^, \l H, and ,V (15) fi, f J, h. and ii (16) A, If, /g, ^1, II, and f i (17) f,itandH. m f.iiand.V (19) h i h I, and ,%. (20) h I, i and h. 74. Whenever a cmnparison has to be made between frac- tions, in respect of their magnitudes, they mmi be reduced to equivalent ones with a common den'. ; because then we shall have the unit divided, in the case of each fraction so ob- tamed, into the same number of equal parta ; and iiie 1 1- flpectivt num". wiU ahew us how manv of «ich parts are ARTTUMETIG. So nei" ZLT^. "^ "^^^^ '^ ^^« ^-^^-^ fr-tion. which ^''' ^Tl'x 4 r>'^S ^/^ftest and which the lea^t of the fractions ~i ^1^, ^^1±, 11+4 5xy 4x10 0x8 "5 + ir^ The fractn*. in theh- lowest terms are ^, 1, ??, and 1? L. c. M. of tlie den".= 2520. ^^ ^^ ^"^ .-. the fractions become ^i^ or ^^?^, ^ ^ ^'^^ 2268 45 X 56 "^ 2520 10^7252 ""' 2520' 25x105 2625 15x180 2700 24 X 105 ' * 2520' lllPm """^ 2520 11 +4 . , 12 y ^ IS the greatest, and j^ the least. 5 + 9 4x10 Ex. XLII. Compare the values of (1) f and f . (2) I and -,V (4) (7) (9) (10) (3) H and i§. ofw '*' ^ "' ^' '"' "' '^ "' ''' ^"^ ^ ^' ^^ -' ^ m of -e^-, 14 of 6i of H of 1,V and If of 1* of 5,1, S i 01 l.f„. Which is the greater, (11) f of a yd. or f of a yd. (12) iofayd. orf ofayd (13) II of -a, of H of Hof aloaf, orf of yf^ of 5i loaves? ADDITION OF VULGAR FRACTIONS ^eJ^L^^^:^^^^^ ^« equivalent ones witb ^rtftif^^ciii^r^ zz£r^^^'^^^' ^^^ "-^- ^^^^^ -- B^tLii;;!:;;^'^^""^^^'*'-^^- VULGAR FRACTIONS. 81 TllO I4 C. M . ft*act"». bc'coine Their sum of the dnn". is 24. 1x13 13 1x8 3x13 ^^ 24' 3x8 13 + 84-15_35_ «»• -' uTTu- or ;rj. 34 5x3 15 24 34 R£ason for tJw Rule. In each of tlio equivalent fractions unity is divided into 34 equal parts, and 13, 8, and 15, of such parts are taken, tlierefore their sum must be 13 4- 8 4- 15, or 35 of such parts, and will be represented by the fraction Jf , or by lU- Note 1. If the sum of the fractions be a fraction which is not in its lowest terms, reduce it to its lowest terms ; and if the result be an improper fraction, then reduce it to a whole 147 49 or mixed number : thus . _— = ;t^ = Uf *• the same remarlc lOo 35 * applies to all results in Vulgar Fractions. Note 2. Before applying the Rule, reduce all fractious to their lowest terms, improper fractions to whole or mixed numbers, and compound fractions to simple ones. Note 3. If any of the given numbers be whole or mixed numbers ; the whole numbers may be added together as in simple addition, and the fractional parts by the Rule given above. Ex. 2. Find the sum of 3-,^, 3^, 2 fg, and f of 3f . f of3f = f ofV- = V = 3f; :. aum of fractions = 3 + 3 + 3 + 3 + 1^ + ^ + -i\- 4- f , ,_ 5x4 1x8 7x3 3x13,.. = 1" + 1 s — 1 + ?r— Q + ^n — 3 + -A — ^ o (since L. c. M of den". = 48) 13x46x8 16x34x13 ' 76 The sign () or { }, called bracket enclosing numbers within It, and the sign called a vinculum, placed over two or more numbers, denotes that all the numbers within the bracket or under the vinculum are equally affected by anything outside the bracket or vinculum, thus (3 + 3) apples or 2 + 3 apples would mean 3 apples + 3 apples, or 5 apples; whereas 2 + 3 apples would mean 3 units +3 apples. 'An-A 1 M.; i .;! 83 AniTEMETia hi I 1^' I Again i+i of (34-i)=i+ ' ^ff=i+&=Uf=f =^=H. (i+i) of (2+i)=(f4-t) of (J+i)==f oi-f=ff=2,J,. (i+i) of 2+i=(f +f) of 2+i=f of 2 M=Y+|:=i^^2i Ex. 3. Find the valueof|+iof(2+i)+^of 2i+iofr^+4^). value=Hi o^M of f+J of (f-} t)=HI+ A+AT 11 5 1 44 + 15 + 13_71__ " ~" 36 "" -^^^^ 9'^12"*"3~ 36 Find the sum of, (1) iandf (4) landi (7) land,^,. m Handli (13) 4 4 nnri 1 Ex. XLIII f, I, and rt. (3) 3andi (6) I and h. (9) hh^indh. (13) liof2iand6i. (2) fandf. (5) h and h (8) t and ,i,. (11) 7tand8. ,... ., (14) 2|, ,^, and 31^2- (15) 6,\,ioflf,and2|. (16) 9i of 2i, ||, and gS. (17) f,iandiof(l4-li). Find the value of, (18) i+f-M+f, (19) 2i+3i4-4i+5i (20) 5^,-f-13^+f|+2M. (21) 4f+,i«+16^+25H. (22) 3|+16^+7,^-hfof3f. (23) (2f+3^)of2A+3iof(16|+3i)+lfofllof2^. ni?3^l 4 ^^'i^^S^n*^ ^^^® ^^^^ t<> ^» ^H to 5, £3A to C £4H to A and m to Jg^. How much did he give away ? (25) A man ate t,% of a 4 lb, loef on Mon., ^^ of a similar loat on Tus., f, on Wed., ,% on Thurs., ^ on Frid., and on ba.. and Sun as much as on Mon., Tues., and Wed. How many lbs. of bread did he eat during the week? SUBTRACTION. . "^V ?^^^- I^e^^uce the fractions to equivalent ones hav- mg the least common denominator. Take the difference of the new numerators, and place the common denominator underneath. Ex. 1. Subtract i from |. By the Rule, The fract". become i^ or -, 2x4 8' their difference = 6-4 8 1 a* A 5 andg, I VULQAH FRACTIONS. 89 Kemonforthe Rule. In each of the equivalent fractions onity IS divided into 8 equal parts, and there are 5 and 4 parts respectively taken, .-. the diiierence must be 5 - 4 or 1 ot such parts, which is represented by i. ' Ifote 1. Before applying the Rule, reduce fractions to tiieu- lowest terms, improper fractions to whole or mixed numbers, and compound fractions to simple ones. Note 2, If either of the given fractions be a whole or mixed number, it is most convenient to take separatelv the ditference of the integral parts and that of the fractional parts, and then add the two results together, as in the fol- lowing examples. ^^'^' ,?^o«i 4f take 2i, or from (4 + f) take (3 + +). Diff— (4 + f) - (2 + i) = 4 + f - 2 - i (Art. VeT = (4-2) + (t-i)=:2 + (t-t)l2 + i = 2i Ex. 3. Find the difference between 2f and 4^ i is greater than i, and .-. cannot be taken from it .-. we write 4J thus (4 + 1 + l\ or (3 + 4^ ' Fi/id the diff^. between (1) iand Ex. XLIV. 5'« (2) iardi (3) fandA. i (4) it and -U. (5) 31 and 2, t. (6) 7 and 2, V (7) lOiVandS^. (8) 17f and 13&. (9) 1/, and f. 110) 4f and 2^7. (U) I5f and 7$. (12) 20,^ and 8,^ than h ^^^^ ate f of a cake, how much less did he leave tol^ wiirmake'™?"" ""^'^^^ ^^^ ^"^ '' "^"^ "'^'^^ ^^- ^^^ <^^ (15) I copied down by mis->ke H instead of %d what amount of error did I make? , vvuat 78. mampU% imolmig both Adaition and Subtraction of vulgar Fractions. '' * Ex. 1. Find the value of 54- - 21 f 1 + 2^ - A TMue = (5-24-2)4-(i-UiVi-.,^)* ''* _^. 4-8 + 24-4-1 ^ , «i :i; MP B4 ARITHMETIC. i ' !• Ex. 2. Find the value of % -+- \ of (3 - ^) -- \ of 2^ -h ^ iofa-i). Value = | + iof(?^-^)-iof| + |-iof(-a-i) =H4 of f-^ of i+i-i of §=^+a— ,^,+i-,^ 6-6 =^.^^^.1 12 Find the vahie of 1. Ex. XLV. (1) i + 2| + 13,%-3,=\y. (3) i,-5, + -^_J.a. (3) 12H-§i + 7l'?-iof|f + i|of3i. (4) (16|-3i)of3^— 16| + 3iof3i (5) 6i + f,ofAof3i-^§-5f. (6) 6i + f,ofAof(3f, ^^5.)_5|. (7) Wliat number must be added to the sum of |, i, and f i, to make S-rg-o V (8) A bought f of a cheese, and sold -j of liis ])urchase to 5, \ of what then remained to C, \ of what tlieii reinahuul to I>\ what part of tlie cheese had B^ C\ and Z>, and what part had Ay after the sales V I! ' I • h ' I MULTIPLICATION. 79. Rule. Multiply all the numei-ators together fo** a new numerator, and all the denominators together for a ntsw denominator. Becusonfar the Bale. f multiplied by 5, gives h^ (Art. 64). But J;j^ must tie 7 times Ex. 1. Multiply f by f By the Rule, 3 "^"Itiphed by ^ = g--- = -. too large, since ^ is one-tr ti e Joi-^n one fraction the sign of x being phiced between ea^h of t he factors in the num^ and den^ to cancel those faetors wh h are common to both, before carrying into t-lfect the t • multiplication. Thus, in the following examples : Ex. 1. Multiply - and ~ together. r> Ai 3x4 3 J^'^S' <^^ivif^ing nu"!""- and den^ by 4. Ex. 2. Multiply I i^, |, and | together. Prod* =-'' 1^^^J^4_? 9x24x30x60 = (2x^x^) x(2xgx ^_x2) xj_^_x ;^ x 3) X (3 ^ .3 X ^') (3 X ^) X (/ X 2 X ;^ X 3) X ('fx ^ x]J) X (~2 x>J x ^ ±>;) 2 =g-r-ing bv 2 X 2 X 3 x 2 X 2 X 2 X 3 x 3 X 3 X 3 X 3 X 5. Ex. 3. Multiply 3i, 3|, 10^ 20^, and 5 A together. D ,, -1 27 81 184 124 P^"^-^-^¥^T^ir^y3- _ 5x(^x3)x (9 x9)x(8 x 2g ) x (4 x 31 ) 2x(2x:l)x^xyx23 5x3x9x9x31 3T()()5 ' = 2xT— ^-4~='^^l^i- Ex. 4. Simplify (f of U of |j +3* of 2i^-2|) x 3f. Vaine =1 - ot 7 of -- -f - of 1 x — V7 4 15 3 31 3/7 =( =( 3x2x5x3x7 7x2x3x3x5 ''--"^ 2x3x7 ~3/ 8\ 27 20 8\ 27 3+30-8 27 31 ^ + ¥-3^-7=^" •> 27 07 i 86 ARTTHMETTa f"*'i Find the value of Ex. XLVI. (1) (5) (8) (10) ^x|. (2) 5xi (3) T^xi (4) Axi 7i X 3i (6) f of i X 17i. (7) -h of U x 3^^ x i ^ . t X 3A X 19i X ^i (9) f« of 1 h of lif X 2i X 3f . Ili«f3fx4f of 2^^x13. (11) 2H of(4i-t-3A) X Uof2f9xUV (12) (3f-l-iV+n-2H)x38iof ,V (13) fof(^ + i--.^^ + i)xf of (2,^6 + 1). (14) |(i 4- i) of ( H + 2f )} X {(2i^4 - H ) of ( 3^ - ?)}. (15) {1? of 26i of (1 - §)| X [2t of (4i - 3|) of -i^4'). DIVISION. 80. Rule. Invert the divisor, i. e. take its numerator as a denominator and its denominator as a numerator, and pro eeed as in Multiplication. Ex.1. Divide f by |. By theRule, f -5-f = f xf = iV Reason for the Rule. If f be divided by 2, the result is -,^4 (Art. 65). This quotient is only < ne-third part of the required quo- tient, since the divisor is one-third part of 2 ; hence ,*t mupt be multiplied by 3, in order to give the true quotient, and -,\x3 = -,V (Art. 64). Note. Before applying this Rule, mixed numbers must be re'\iced to improper fractions, and compound fractions to simple ones. Ex. 2. Find the quotient of 3 2^ by 4|. ,,78 22 78 5 ;2x39x^ _39 55' «i26 -5- ^t = " 35 ■ "^ 5 ~'25'^22~^x5x;ixll Ex. XLVII. Divide (1) i^byi (2) Ibyi (3) (4) 4^by6|. (5) 56by5i {^ 7fby4^V. (7) ^of20|byl0|. If by f . (8) f of 5^ by -A of 9. (9) (i of 7i--,8,) by 1|. (10) Divide i -f- 1 - ^ by the sam of i and i VULOAR FHACTtONS. 8^ (11) What number multiplied by 216 will produee 6| ? (12) What must f be divided by in order to produce 2 ? (13) What is the least fraction which must be added to the sum of 4 and i divided by their difference to make the result a whole number ? Nol^. Complex Fractions may by this Rule be re- duced to simple ones. (1) (2) (3) ^ = | = i-f(Art.59) = ixf = -,V 41 a — — — — 9. ^ aa — 9 V -J, — ^i- 4-.^-+2f ^ 6+fif+f _ 6+-,^+A 13t^^-3i 10+ ,^ -i ~ 10+ ,^ -tV 6+-.^i 10+1^ Simplify, (1) 6f (2) ^ 3f 2i- Ex.^XLVIII. (3) 2i 6* S¥- (4) 6A m' (5) 6^ 2f 1 1 (6) ^ (7) I fofH (8) H4 (9) 5i+6» (10) ir+~+~ ^h' ISof.V 2r 6f-5y =f-?-4* (11) (3^ 2 5 4) (12) /5f .,\ /3f' '\ (13) 5f-^7| ^ 2ix8^ (14) of 13 n •i-.v 2|-n"^ 44i(i-i)- '.1^) 2f+i""3l 81. To find the valite of a fraction in terms of t?ie same or lower denomination. Rule. Divide (if possible) the numerator by the denomi- nator; if there be a remainder, reduce it to the next lower *i ITU', and divide the product by the denominator; repeat Iht latter operation as often as necessary. 1' ind the value of f of £15. By the Rule, 2x15 „.S0 „.„ „. 2x20 40 ? of £15 = £---- £y- £4^ ; £?="-y^-«. = ~8. =: 5^- ii I m '■■ *'" ! 8c AUimMETtC. fs.= 5xl!3 ,,..*. d. = ~d. =8|d ; H = -«- \d. ^., ood. 'lotJiet lie re- 3r th« vhiclr 72 in 13) 1 lb. Troy to Iho fr. of 1 11). Avoirdupois. ) '^h of 10 cwt., 3 qrs., to the fr. of 1 ton. (()) 3^ of 3 ac, 3 ro. to the fr. of 3 ro., 3^ per. (7) I lb. ^^i-oy to the fr. of a lb. Av. (8) 1t\ of £3. 4.V 1[\d. to the fr. of 5.*?. (9) h of 3| mis. to the fr. or'| of I mis. (10) 6^ of 3 cords to the fr. of 5 cord ft. (11) 8^ of C lbs., 3 sc. to the fr. of a lb. (12) \ of t of $31 to the fr. of $r. (13) ^\ of 8 yds., 3 nls. to the fr. of 2^ ells (English). (14) 3| of 10 hrs. to the fr. of 1 dny. 84. Mimellaneous Examples in Vulgar Fractions worked out. Ex. 1. At the 'call over' at a certain school, f of the children on the reuist^'r answered to their names; the rest, 18 in nnnilnr, were absent. How many children were there ou the reiTister V •r * ■ ! 4' . \ .J f, v([ 00 ARITHMETIC. iff t I § of the no. were present, .*. ^ of no. were ?i.Ksest By the question, ^ of no.= 18. .•.no.= 18x6 = 108. 4 ^^\^' ^ P^^^ woman lost through a hole in lier pocket A of her money ; only 3s. Ofrf. was left. How much money had she at first, and how much did she lose ? After losing A of her money, f,- of it was left, .*. i^f of her money = 3s. Ofrf. .*. A of her money = 3s. Of d -^ 7 = hid. .: her money = h\d. x 11 = 4s. 9fd She lost -h of 4s. nd.=: ^^^ = Is. M. Ex. 3. A, B, (7, D run a race over 1 mile. First A and B race, when A wins by 20 yds. ; then (7 and I) race, when 6 wins by 60 yds. ; then A and Crace, which will win, and by how much, supposing that if B and D had run against each other, B would have won by 40 yds *>> 17^5^^'"? ^n""^ ^Ifj^b ^ ^""« ^^^^ y^^«- ? while C runs 1760 yds., D runs 1700 yds., or while D runs 1 yd., C runs i^BS yds. ; while B runs 1760 yds., D would have run l'?20 yds. or while 5 runs 1 yd., D would have nm i^|§ yds. While ^ runs 1760 yds., B runs 1740 yds. "^ " " 2> runs ( 1740 X HI) yds. " G runs (1740 x Hf x Hg), or 1760^ yds .• (7 will win by -,«/ yds. Ex. 4. Divide 15s. M. between A and J5, so that ^'s share may be less than ^'s share by % of ^'s share. To represent ^'s share fix on some number which is ex- actly divisible by 5 ; let 5 represent ^'s share. Then 5's share r:- 5 - f of 5, or 5 - 3, or 3. . . 15s 6^. has to be divided into 5 + 3, or 8 shares, of which A is to have 5, and ^ 3 ; .*. value of each share = 1?^ = \s. Wid. :. A's share = Is. Hid. x 5 = 9s. 8K, 7?'s = Is. 11^. x 3 = ^s. 930- how much had she in her purse a I first V ' onofti ""^^ fi^^""^^'-s"»'i(:k being worth $90, find the value (8) X person after paying an income-tax of 5 cents in the dollar, has a net income of $855; tind his gross income. (9) If, when the income tax was 6 cents in the dollar a person paid $54; how much less will he now pay the tix being reduced 4 cents in the $? ^ •^' "^^ ^"^^ wn!.?h i^^ I ''^'•'' '"''^y'\ .^^w^^^th f.v., and 3 of a rabbit be woi th 2^s of a pig ; what is the value oi' 100 pigs ? (11) If, in practising, 7 riflemen shoot 2G rounds in 1 lir II S^lr^ll!^ ^"""'^ ''''' '' ^"^'""^'^ '''^'' "^ ^ ^- ^J^^\ ^.1^""^ ^^ l^^'''^^^' ^'^ divided into 4 parts, which are to each other as the numbers 1, 2, 8, 4; and a person who receives I of each share, obtains altogether $12 GO- fin d t I sum of the several shares ? ^ ' ^ (13) If 15 cows or 28 sheep can graze a field of 5 ac in 11 days, how many days ought a similar field of 18 ac' to serve 33 cows and 20 sheepv'' '^ '^'- ^^ VVLQAr FnACTTONS 98 (14) Divide $91.r)0 between A ami B; (1) <,nviug A half /IS much a^ain as B; (2) giving A'n share loss half A's share to 5. (15) V bankrupt owes to one creditor 500 d llar^ fo each of two others $250, to each of tliree others ., 75: 1' prop- erty \N Tth $025. How much ( 'in he pay in the dollar, and iiv,w much will the first credit* -r received (10) A mine is worth $10000 ; a p* '•son for -?^ of his share receives $750. What part of tlie mine did he possess? (17) A scliool is composed of three divisions; there are i^ths of the whole number of boys in the first, .l,th in the second, and the r<'st, 80 in number, in the third : how many boys are there altogether? (18) A can do a i^iece of w^ork in 10 days, which B could do in in what time would they do it together ? (19) A father left U^ the elder of his two sons || of his es- tate, and \% of the rem ' der to the younger, and tlie residue to the widow; fiii 1 their respective shares, it being found that the elder son uKteived $1090 more than the younger. (20) Divid« -^5 ac. 2 ro. of land between /I, />*, and C, so that B''8 share = -^x of yl'.v share, and that 6".s siiare shall be 9 ac. more than the united shares of A and B. (21) A fine of $14.40 had to be raised among a number of boys ; one-third paid 18 cents ea(;h, as many more 30 cents eacli, and tlie remainder 42 cents each. How many boys tvere there ? (22) A. cistern has 3 pipes in it, by one of which it could be fiUci^ in 3 minutes, and by the other two it could be emptied in and 7 minutes respectively ; in what time will it be filled, if they are all open together ? (23) A and B together can do a piece of work in 30 days, B by hims(!lf can do it in 70 days; (I) in what time could A ilo It by himself ? (2) how much more of the work does A do than B^ when they work together ? (24) A and B can do a piece of work in C| days, A and G in 5i days, and A, B, and G in 3f days. In how many days £&n A do it alone ? (25) There arc 4 casks of different sizes. The 1st is filled with liquid, the rest are empty. The 2nd cask is filled from the 1st, and fths of the original li(iuid in the 1st remains. The 3rd is then tilled from the 2nd, aad ^th of the liquid in ii.::-ii IMAGE EVALUATION TEST TARGET (MT-3) k A {./ < % /- M/.A « ?. ^ 1.0 I.I ^ I4£ 12.0 25 2.2 |25 I U 1.6 V] <^ /a /

> ft# ,^-^v /> # riL _x. rnumgiapmu Sciences Corporation 23 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) 872-4503 4>^ iV iV « #v^ ^v ^/^. ^1>^ :^ A ^ ^^ ^#^^ m^ ^9^ A. n? Z % "^ 94 ARITBUSTIG. K^Q) ^ ^ m 2 days can do as much work as B osm dn {« q days ; together they take 12 days to dTa certain work In what time would A alone have done it ? WW It" '• t yi 'M : r "'^ DECIMALS. ♦h??' .^T^^^ ^^ *^® "^its' place of any number exnrp«,« their ^^^;, values, while those to the ^^/^ Vthe unit^' K mcrease m value te;./^;d! at each step f^m the un?te' D?ace fro^ ?h!' ^Tr^i?^ to the same notation, as we proceed from the units' place to the right every succeXe Xm. would decrease in value tenfoll Wo £« ♦? "'^"**^ whole numbers or inte^rstKrtal'fSionsZS:?;^^^^^ ten at length would stand thus : "umoer writ« 5x1000 + 6x100 + 7x10 + 3+ A . _1 10^100 1000* The dot is termed the decimal nm'nt onri oii « ^ ., right of it are called DECiMl^orXcZi FrI^^^^^^^^ t^ riOx%'^x?o'ir«??r^'^^ '""'"'y 10,100or^^0xlTi000 or lu X lu X 10, &c., as their respective denominators The .^^.7^^ iyr^^milib^ 71/k will be represented thus : §" 7 6 5 4 3 2 1 c a. cc O tr o g g' p CO 1^ Cr o o p CI. CD 09 2 3 4 5 6 I CO a a; en a a O 00 o o 'S o W g 4 ;r:i DECIMALS. 95 86. 10, called i\ie first Power of 10, is written thus, 10'. 10 X 10, or 100, called the secoTid Power of 10, is written thus, 101 10 X 10 X 10, or 1000, called the third Power of 10, is writ- ten thus, 10*, and so on ; similarly of other numbers : thus the fifth power of4 is 4x4x4x4x4, and is written thus, 4^ The small figures 1, 3, 3, &c., at the right of the number, a little above the line, are called Indices. 87. •306=^- 6 8 X 100 X 10 100 300 1000 4- 1000 4- + 6 10 X 100 100 X 10 6 308 1000 1000 3 Again, -0306.^ j^+^^^ 1000 ~ 1000' _0_ 6 ■^ 1000 "^ 10000 0x1000 3x100 + 0x10 100x100 Again, 80-306 = 80 + + 6 10 X 1000 + 300 + + 6 306 1000x10 ' 10000 10000 ~ 10000' 306 80000 + 306 80306 1000 . 1000 ~ 1000 Hence to '>onvert decimals to vulgar fra/Mons: from the above examples wo deduce the following : 88. Rule. Write the fl inures which compose the deci- mal as numerator, and for ^mominator 1, followed by aa many cyphers as there are figures after the decimal point Ex. LIII. Express as vulgar fractions, (1) -3; 13; 19; '301; 270; 5653. (3) -504; -73201; '791003; 03; 0045. (3) -300; 18-741; 21; -000001; 50007. (4) 34702007; 500005; 5-60746805; -0000500. (5) 290050; 20-607; 500038. 89. Any fraction, having 10, or any power of 10, for its denominator, as Wcnnf . may be expressed thus, 800036. ForVoWo^ = 80+To%n + TTrfRT5 = 80 + fu + Tl^O+T-(?«-Ti+T4M = 800036 (by the Notation we have assumed). 90 -241 — -,2^*iL. '0241 — -r 3^4.1-^ -941 — -ma. — .aii, MU. ,«/tA 1(M)0» '^-i/tl — T0 0T5> 'ii^-l-U — 100(10 — 10 off* ■^'^ see that -341, 0341, and 3410 are respectively equiva- .' .. » /fractions which have the same nvuneratoi:, i^o \^i- ' ;i iSJ n I 96 ABITHMETIC, first and third of which have also the same denoi iAHjs. while the denominator of the second is greater Heace-^l IS equal to -2410, but 0241 m less than either ^^ 1 he vjilue ot a decimal is therefore not affected bv aifb^nn cyphers to the right of it; but it. value is decrease^W /UY//./ cyphers : which effect is exactly ()pposite to that wldrh IS prodijced by affixing aud preffxingVphertfiS^e^^^^^^^^ .n. A decunal is maUiphed by 10, it the decimal nnint im pwits, Ijy 1000 It ttresplncos; anc soon: and converselv aSdsOon ' ''>'l"»''"""M>laces; l)y 1000. it7Am places ; 5S'o-''*«''i'l= '''£.''* =i^'^^ ". >»«»=^S X 1000 = 5600, = 6056" "•>< '^ = * = -5*'i 5«-s-1000 = HxT-A, = T#,„ ™. .. , , Ex. LIV. Write down "a decimala, (1) A;H:W;t3,i iVA-iTfc. ? !t!o^ ^f^- -"'Wo-.; m,; rom^. ^ '"""i, iP'^"'' ^'''''""'^■«' ^''='«>^''i WoW; TooaoiioooTTo- (4) Seven-tenths; thirty thousandths. ^^^ ^sandth'""^""'^ '''''^ ""''''^' thousandths; one ten thou. ^'^ "^'te^^'imhs'""'"' ^"' '^"' ™^"^^^^^^^ --«-*^ Express in words the meaning of b;l^otis«lVJeVnt^/^"•*•-^-p^^^^^ b/Vo! lO^oilHl 'by'&i""'"'' """ ''^' «-" -P'-'-^y (12) What is the quotient of203 by a million? ^ ADDITION OF DECIMALS. 92. Rule. Place the numbers under each othpr nnif* under units tens under tens, &c., tenths under tenths Z so that the decimals be all und^r each Xr. Add' ^^ to' DECIMALS. 9"^ tvhole numbers, and place the decimal point in the sum un- tler the decimal point above. Ex. Add together 23, 050, 37, and 3-60015. By the Rule. 2*3^ ^ By fractions, jj^:^^^ 3-3 + -050 + 37 + 3-()00ir) = 1-?,+ ii^{io4-¥ + ??rSm O OUUl«J I OOOOO T^ I (MIOoTlT^ MKKHH) +l(MIO(M» 43^5615 =-■ ^i^M^^^^ = 42-1)5015 (Art. 89). Ex. LV. Add (I) 1-035 (2) 24- (3) lH()-8 (4) 94-25 •00043 185-3009 35-2779 008 27- •98795 • 9000- 18796009 2-2140 3-098 9 201 57-3910 53009 •70000 83005764 5-998347 Add together, and verif} c;ach result by fractions ; * (5) 12-5, 20-043, 7-63201, and 0561. (0) 0573, 15, 204, and 567-98075. (7) 505-0003, 13-98, 5853097, and 960. (8) 600734, 54, 157008701 2, 800003, and 9-987789. . (9) Find the sum c T thirteen hundredths, seven and thre« ten-thousandths, four hundred and eight and five tenths, nine hundred and seventy-eight, and eight hundred anil eight ten-thousandths. SUBTRACTION OF DECIMALS. 93. Rule. Place the less number under the greater, units under units, tens under tens, &c., tenths under tenths, «&c.; suppose cyphers to be su])plied if necessary in the upper line to the right of the decimal. Then subtract as in whole numbers, and place the decimal point in the remainder under the decimal point above. Ex. Subtract 3084 from 5-7. By the Rule, ^''^ By fractions, ?i084 5.7 _ 3-084 = ^-h - nu = nn - nu §•616 -fgi^ = 2-61Q. Il't w M l» ¥ •! tif -t-"*i 98 (1) From 5 345 Take 3087 ARITHMETIC. Ex. LVI. (2) 26002 (3) 15-67 18-9564 9-7003 (4)21 19.9009 (5) Find the difference between, verifvinfr each result hv fractions, 1) 13 and 13; 207 and 207 2) 763 and 7-fi^^. 67-3 and 67-5803. (3) 501 and 428-90456 -53 24 and 5S24' (4) 4-42 and -00042; -0000007 and 'OOT. ' 2-3^and%^?'''^ "'"''^ ^""^^ ^^ ^''''^^^ ^^ difference between (7) Find the difference (1) between one-tenth and five IweS^l^l'tfol^Sr ^"^^^^ ^^^ ^^^ ^^--^^^^' -" se4?n ttoSZn?rC^^^^^^^^^^ ^^^^^^" l-2^-12^-m2 l^nl^fn ^'mi"^"; ^¥^^ ^^^^^^^ t« the sum of 1 ^, 1^, 012, and'210, will make the result a whole number. (10) Find the value of (1) 3125 — 305Q +-9^^fi7f^ft 184-00t)3; (2) 215-263- (7-0004 -05r-(45-08V8(^^^^^^^^ MULTIPLICATION OF DECIMALS. 94. Rule. Multiply the numbers together as if thev SnJl^nW ""'^r' ^"^ Pr .^ «ff i" theVoduct as man? fsMH^o l^^^S^fu^ ^^'f ? ^y^ decimal places in both the nml- tiphcand and the multiplier ; if there are not figures enough supply the deficiency by prefixing cyphers. «'^««ffii. •0002.' ^^^ ^*'^^"^' ^^ ^^^ '^'^^ ^^ ■^^' ^^^ ^t' ^'^^ "^y ^y the Rule, (1) 7-35 By fractions, 7-35 X -23 = m X 1^ = \mh = 1-6905. •23 2205 1470 1-6905 (2) 8-27 •0002 •001054 8-27 X -0002 = m X Ti^hn^ = ^U^ ='001654. Multiply %^'ll ^^^ ^"^P? (^^ ^'^^4 (4) 680-35 (5) 20607 «y j5 _J± 203 0049 -20607 DECIMALS. 99 fllultiply, and verify each result by fractions : (6) (10-71 by 11; 57068 by 2004; 536 by 700; 7-01509 by 50805. (7) 48067 by 00037 ; 54-3047 by 900005 ; 2-568 by -00025. (8) Find tlie continued product (1) of 5-5, -055. 550. and •0055 ; (2) of 1-75, OS, 85, and -0004. , . , ^ (9) How many yds. of cloth are there in 735 pieces of cloth, each of which contains 3785 yds. ? (10) A man eats -95 of a loaf daily ; how many loaves will he eat in the year 1866 ? DIVISION OF DECIMALS. 95. First. WJien the number ofdedrrMl places in the divi- dend exceeds tlie number of decimal places in the dimsar. Rule. Divide as in whole numbers, and mark off in the quotient a number of decimal places equal to the excess of the number of decimal places in the dividend over the num- Der of decimal places in the divisor; if there are not figures mfflcient, prefix cyphers as m Multiplication. Ex. 1. Divide (1) 21125 by 845, (2) 0021125 by 84-5. By the Rule, (1) 8-45)2-1125(25 By fractions, 1690 2-1125 -5- 8-45 = HiM -*- m = ^U x • 4225 m = HiF X ^mj5 = ¥ X ik = m 4235 = .25. No. of dec . piaees in quotient = 4~2 = 2, .*. quotient =-25. (2) 84-5)0021125(25 By fractions. 1 690 "4225 4225 •0021125 -5- 84-5 = -rU.^lho -^ W = roW(f(Ajo X -Ih = Hit ^ X TUU^^oTTTT = ¥ X Tinrku = To-o^^oF = -000025. No. of dec^ places in quotient=7— 1=6, .-. quotient =000025. 96. Secondly. When the number of decimal places in the dividend is less than the number of deciinal places in the dimor. Rule. Affix cyphers to the dividend until the number of decimal places in the dividend equals the number of deci- mal places in the divisor ; the quotient up to this point of wUe division will be a whole number. If tUea:e be a remainder, and the division be carried ^m 'Mr ifc I'"!: .'I 100 ARITlIMETia |M ^1 ttl further, the figures in the quotieut after this point will be uecimals. Ex. 2. Divide 2112-5 by '845. By the Rule, •845)2112-500(2500 1690 ]iy fractions, 2112-5 ^ -845 = ^V.P -*- T'n^nl. = ^-^iV* X ^i-^^^ =Hif ^ X J-^S^ = 25 X 100 = 2500. 4225 4225 00 Ex. LVIII. Divide and verify each result by fractions. (I) 33-372 by 2-7. (2) 33372 by -27. (3) '33372 by 27. (4) 33372 by -27. (5) 33372 by '00027. (6) 561-0833 by -323. (7) 5610833 by 323. (8) 66108-33 by 3-23. (9) 5610833 by 0000323. (10) 552-5325 by 3-25, and also by -00325. (II) 2'419003 by 4643, and also by 004643. (12) '000081 by 27, by 0027, and also by 27000. (18) 218051-081884 by 200099, and by 200099. (14) '121 by 11, by 1100, and also by 0011. (15) 393'72 by 000193, by 193 and also by 193000. (16) 590-4825 by 03275, and also by 327500. (17) 213-419596 by 1 00103, and also by 100103. (18) Divide the sum of twenty-four ten thousandths and twenty-four hundredths by twenty-four. (19) Two ten thousandths by twenty-five hundredths. (20) If a man mow 175 ac. of grass In one day, how long will it take him to mow 21 875 ac. ? (21) How often is "75 min. contained in 64- 125 min. (22) The product of two numbers is seventy thousand two hundred and forty-two hundred millionths ; one of the numbers is twenty-three thousandths ; find the other num- ber. Ex. 3. Divide 240-13 by 73'4 to three places of decimals. Before dividing, aflUx two cyphers to the dividend, so as to make the number of decimal places in the divide'^d excee^^ DECIMALS. 101 the number of decimal places in the divisor by 3 ; if we di- vide up to this point, the quotient will contain 3 decimal r'aces by Rule 1. 73-4) 2401300 (3-271 220 1993 1468 5250 5138 1120 734 386 By fractions, 240-13 -r- 73-4 = ^%h^ -f- W = HSi^ X -,m (we multiply num'. and den', by 10, to make den'. 1000, since the quotient is to contain three decimal places) = ^^iF^ X -nfrj=^^H=3-271. u li Ex. LIX. Divide to three places of decimals, and verify each result by fractions, (1) 1-9 by -3, by -03, and by 300. (2) 4-937 by 159, by 159, and by 1590. (3) 329744 by 53, by 0053, and by 5300. 97. Certain Vulgar Fractions can be expressed accuratelp 08 Decimals. Rule. Reduce the fraction to its lowest terms; then place a dot after the numerator and affix cyphers for deci- mals ; divide by the denominator, as in division of decimals, and the quotient will be the decimal required. - Ex. 1. Convert f , -4% j^ji, each mto a decimal. 4 )3'00 No. of dec', places in quotient=no. of dec*, places •75 in dividend —no. of dec', places in divisor =2 — 0=2. •i\5=f-*-10=-75'^10=075; :r^TJ=f-5-100=-75-i-100=-0075. Ex. 2. Reduce 77; to a decimal. 16) ^-0000 (-8.^2.5 48_ '20 16_ 40 32_ 80 80 or thus, 16- 4 4 «-00 1-2500 •3125 16 = -3135 «--vi iiH ti. : is 'I! ^:| I,. ?| 'Ilk ll J- 103 ABITIIMETia Ex. 8. Convert 5it^t) + '75 of - of 7i into a decimal. o (J40j 8 8 8 10 5000 •625000 •078125 •0078125 .6 •75of?of7i = 75of§ofi^ •75 X 9 = 675 .'. 5^5 +-75 of g of ^ = 50078125 + 675 = 117578125. Ex. LX. It ' \ MS <5„ 1 •»! Reduce to decimals, (1) (3) (4) (5) 98. As 8if ; ill; H; ni i; t; f; 6i; V; f; Si^,. (2) 1^ ; hh ; B i^-ij ; -«^^- ; 84t-oV4-- fofH; 3iof2i; 3iot4iof5i If - if 4- 'S-is ; Hi +-75 of n of 6t. To convert a vulgar fraction into a decimal, we have in fact, after reducing the fraction to its lowest terms, and affixing cyphers to the num-., to divide 10, or some multiple of 10 or of its powers, by the den^ ; now 10=2 x 5, and these are the only factors into which 10 can be broken up ; therefore, when the fraction is in its lowest terms, if the den', be not composed solely of the factors 2 and 5, or one of them, or of powers of 2 and 5, or one of them, then the division of the num'. by the den^ will never terminate. Decimals of this kind are called indeterminate decimals, and they are also called Circulating, Repeating, or Re- curring Decimals, from the fact that when a decimal does not terminate, the same figures must come round again, or r^cur, or be repeated : for since we always affix a cypher to the dividend, whenever any former remahader recurs, the quotient will also recur. 99. Pure Circulating Decimals are those which re- cur from the beginnmg: thus, •333.., -2727.., are pure cu-culafg. decK ^ f Mixed Circulating Decimals are those which do not ''^gin to recur, till after a certain number of figures : thus, m *' '^113636. ., are mixed circulate. dec'». The circulating part is called tlie Period or Repetend. Pure and mixed circulating decimals are generally written h DECIMALS. 103 down only to the end of the first period, a dot being placed over tho first and last figures of that period. Thus JJ represents the pure circulate, dec'. '883. . •36 -3686.. •639 '. -639639. . •138 mixed 1388. . •01136 0113036.. 100. Pure Circulating Decimah may he converted into their equivalent Vulgar Frmtionn by tMfollawing Rule. Rule. Make the period or repetend the numerator of the fraction, and for the denominator put down as many nines as there are figures in the period or repetend. This fraction, reduced to its lowest terms, will be the fra• I DECIMALS. 105 JVote. If inc quantity, tlu! viiluo of whose decimal part is tc, 1)0 found, be a (toni|M>und (luantity, it nniHt be redutea to one denomination hvAora tlie rule is applied. Ex. iJ. Find the value of 7405 of 15 mi., 5 fur., 31 po. po. 15 mi., 5 fur., :M po - 50:U 7-405 25155 20124 35217 37254-555 po. r>i 2-775 •2775 30525 yds. 36 3150 1575 1-8900 in Ex. 3. Fmd the value of -5410 of ^ cwt 1st method. po. yds. In. .•.vaK= 37254 3 1-89 mi. I'ur.po. ydfl. in. or = 116 3 14 3 189 cwt. lbs. 4* = 450 -541606 450 27083300 2166664 of;; )cwt 243 749700 lbs. 16 2d. method. /5416-541 •5416of4i=(-^^ = r^.|)cwt.=(l|4x.00>. D V9000 -n«200oz. ^ ='^V^'.''w .•.value=243 lbs., 12 oz. nearly. The 2nd method is the bet ter one in most cases. Ex. LXII. Find the value of, (I) •75of$l. (2) -875 of $5. (3) 625 of $1. (4) -625 of 1 cwt. (5) -375 of a mi. (6) -175 of a ton. *7) -46875 of £2. 10.^. (8) •06?,5 of 7s. 6(?, (9) 8175 of lib. Troy. (10) 4-65of4iac (II) 1004of2iro. (12) 256 of 10«. llM (13) 5-00875 of3wks. 4 dys. (14) 16 504 days. 1. \« 1 s 1 t: r i 100 AttlTnMETxi^' rii (15) 305of5 1bs. 23. (17) 7034oflaa, 3ro., 5po. (18) 5005 of 16 lbs., 1 oz., 6 grs. Troy. (19) -3 of $2. (20^ -54 of 16s. 6d (16) 30085ofjC-i.lA. (21) -243 0^^10^ (22) 6-83 of £5. (23) 2383 of 2i lbs. T'y. (24) 62 of a c. yd (25) 18-72 of an ac. (26) 2-063 of 1000 guineas. (27) £-634375 + -025of25s. + 316of30.?. (28) -6 of an ac. + 625 of a ro. — A- po. (S9) 6-714285 of Is. M. - 0833 of £7. ^s. H- 251190476 of 6s. 8dC 104, To reduce a nurnber or fraction of one or more denomt nations to tJw decimal of another denomination of the same kind. Rule. Reduce the ^iven number or fraction to a fraction! of the proposed denomination ; and then reduce this fraction to its equivalent decimal. Ex. 1. Reduce f of£l to the decimal of a guinea. I of £1 = ias.= 8s. 1 guin*.= 21s., .-. fraction req'*.= -j^f. Now 8 -^ 21 = -38095238. ..,.-. dec'. req'».= -380952. Ex. 2. What decimal of £2 is lis. 9^d. V lis. nd-miq. ; £2 := 1920g. 'rSh=lH, .-.dec'. reqd.=189-H640:=-2953152; .*. fract". req<^ or thus, 4 300 12 9-75 4,0 11-8125 •2953125 We first reduce f ^. to the decJ. of Id., by div». 3d by 4, which = -75^, next 9'75d. to the dec', of is., by div». by 12, which = -81255., then 11-81258. to the dec', of £S, by div^. by 40, which = £-2953125. Ex. LXIII. Reduce, . (1) 1 qr., 5 lbs. to the dec', of a cwt. (2) $2.50 to the dec', of | (3) (4) (5) (6) C) (8) 3 hrs., 30' to the dec', of a day. 3 ro., 11 per. to the dec', of an acre. Q^d. to the dec', of a shilling. 3^ in. to the dee', of 2 furlongs. 2 oz., 13 dwt. to the dec', of a 11). 4 lbs., 2 sc. to the dec', of an oz. * ' tonv yd. ame tion tioa 152; liv». lec«. lien 40, 135. MISCELLANEOUS EXAMPLES. 107 (9) 2 sq. ft. 73 in. to the dec>. of a aq. yd. (10) 1 lb. Troy to the dec', of a lb. Avoir. (11) 10a. 9(i. to the dec>. of £1. (12) 1?». Id. to the dec». of £1. (13) 2 wks. 6i dys. to the dec', of 4 dya., 3 hrs. (14) 2 lbs. 14 oz. to the dec', of 18 lbs. Ex. LXIV. MISCELLANEOUS EXAMPLES. PAPER I. (1) Define a unit; a number. Into what classes are num- bers divided ? Explain the difference between them. Define Notation and Numeration. (2^ Write down in words the following numbers : 70340 ; 125004321 ; 5607605213403 ; and express by numbers eight hundred and ten thousand four hundred and one; sixty-four billions two millions six hundred and forty-six thousand and two. (3) (1) Add together one million eighteen thousand two hundred and sixty-nine; twenty thousand nine hundred and seventy-nine ; one hundred millions one thousand and htty ; fiftv-four billions three thousand; four hundred millions and six • nine hundred and ninety-nine thousand nine hundred and ninety. (2) Subtract 300725 from 400001. Explain clearly why you carry 1 when you borrow 10. (4) (1) Multiply 268936785 by 5689, and verify by division. (2) Divide 27027027027 by 6974, and verify by multiplica- (5) The product is 99626417315464, the multiplier 73568 ; what is the multiplicand ? (6) In 12 mi., 2 fur., 6 per., how many inches ? Shew that your result is correct. PAPER II. (1) When is a number said to be a multiple of another number V What is a common wultiple? What is the leoAt common multiple oi two or more numbers ? Find l. c m. ot 27, 36, 42, 48. (2) Explain the meaning of the signs, +,-, =. Wlien cavi questions in Additior, be performed by Multiplwatwn. ••r 108 AniTIIMETW. *■!•' If ^ . I ^■1 )•! (3) A cask is rcqnirocl to ho oxacjtly filled by any one ot the IblUnvin^ measures: 1 ])int, 3 i>inlH, 5} pints, 5 pints, 6 pints, or 1) pints; lind tlie smallest cask lor tlie purpose. (4) The torewheel of a \va.sj;on is S leet, round, and the hind-wheel rourteen ; how many feet will the wai^on travel over before each wheel shall iiavc; madj; a number of com- plete turns? IIow often will this happen in 1000 feetV (5) The lens^lh and cost of building the undernamed Canadian Canals, were as follows: The Uideau Canal, I'JfJf miles, $4880000; the St. Ltiwrnice Canal, 40,^ miles, $8550- 000; the Ottawa Canal, lO.V miles, $1500000; th(^ Chambly and St. Ours Lo(!k, i\h miles, $550000; the Welland Canal and feeder, 50.^ miles, $7000000; the Burlington and Des- jardins Bridge cost $5((0000. Find (1) the total length of the above canals, (2) their total cost, and (; J) the average cost per mile, excluding the Burlingt(m and Desjardin* Bridge (G) Define a vulgar fraction? Distinguish between a vulgar and decimal fraction ? Give an example of the different kinds of vulgar fractions ? PAPER III. (X) Simplify (1)2HH§) + HI -I). (2) A person who owns half of a steam-vessel, sells \ of his share for $15000 ; what is the remaining pari of hia share worth ? (3) Simplify (l)-^(8i-2i)- HI -1^.) (2) 12 |(8|-2^)-i(S-A)}. (4) A clerk coi)ied 55 of £5 instead of 5o of £5, what was the amount of th(3 error? (5) It takes 87 yds. of carpet, 1-25 yd. wide, to cover a room, how many more yds. will it take, if the width be •7.'» yd? (6) A gave 5 of an orange to B\ "3 of what rem.'iined to G\ how much of the orange had A left for himself? PAPER IV. (1) A drover sold I of his Hock to A,^, of tin; remainder to B, and the rest to C. llow many had he at first, suppos ingCgot32? (2) Add together 13^, 5Cf, and 14^ by vulgar and dec- inal fractions, and shew that the results coincide. MI8CEL NEOCfS EXAMPLES. t09 (3) The product < vo deciniiils is (KitJUT'^ ; one of tliem Is 2-7 : find the other. ^ ^ ^ (4) Add together £27. 6s. «.J(?., $17.22, £10. 5.s. 8(Z., |li)8.05, £3. 12«. 7(/. Tlie answer to he in dee', currency. (5) At a foothall match there were 875 as many on one !ide as on tlie other, and the players on hoth sides \v(Te £qual in numher to 025 ol' the l()t)kers on : if there were 21 on the smaller side, how many were playinj^ on tlie other eide, and how many were looking onV (6) If in a cricket match one side scores 014 of 1^ ^^^'gi of ^ of 4-5 of ?— !- of 71^ of the score made by the other side ; which side wins ? PAPER V. (1) (7 owes B -6 of what B owes A, B gives C5.9. to put the accounts between them all straight. What is 5's debt (2) Out of a bag of silver, I take 50. former. When the terms have been properly reduced in i- tiply the second and tlurd together, and 'divide by the firs ta-eatmg all three as abstract numbers. The nuoti4t n-l! I- the answer to the question, in the denomination to 'which the third term was reduced. in i RULE OF THREE. 113 If W bushels of potatoes cost $15.20, how many bushels can be bought for $83.30 ? Since 10 bus. is of the same kind as the req**. term, viz., bus., we make 19 bus. the 8'**. term ; (Since $83.20 can buy more bus. than $15.20, we make $83.2(1 the 2"*^. term, and $15.20 the \^\ term : $ c. $ c. bus. 15.20 : 83.20:: 19 : na of bus. req*. or 1520 cts. : 8320 cts. :: 19 bus. : no. of bus. req*. ., . 8320x19 ... .-. no. of bus. req*. = — ■■,,., = 104. 1520 Ex. 2. A gentleman hired a servant for the year 1865 for £32. 13s. 11:^., the man left his service on the evening of the last day of June: what amount of wages ought to be paid to him? From Jan. 1 to June 30, both included, there are (31 -+- 28 + 31 + 30 + 31 + 30) days = 181 days ; We place £32. ISs. lli(^., the given quantity of the req*. kind, in the ^'^. term ; wages for 181 days will be less than wages for 365 days, ..place 181 days in the 2"*. term, and 865 days in the \^\ term. days. days. ^ s. d. :. 365 : 181 :: 32 13 Hi : req*^. am*, of wages. or 365 days : 181 days :: 31390^. .: in ?. /. req** am*, of wages = 31390 X 181 365 <7.= £16. 4s. 3K Ex 3 A bankrupt can pay 9s. 014. in the £, and his assets amount to £1069. 3s. (\^d. ; find the amount of his debts. ^^ , n^ . .tu For every asset of 9s 0\d. he owes £1, .'. place £1 m thr 9s. o'id : £1069. 3s. ^d. :: £1 : am*, of debts in £'s,_ cr 217 half-pence : 513205 half-pence :: £1 : am*, of debts m £ s, .-. am*, of debts in £'s = — ^pp = ^365. Ex. 4. If 0625 of 1 lb. cost 4585. ; what will '075 of a ton |«()St ? lb. ton. f. J . . t Mv •0625 • 075 :: -458 : req*. price m shillmgs, lb. lbs. s. ^ , . , .1,. or 0625 • 075 x 20 x 112 :: "458 : req*. price m shillings^ ^rice - : 458x07 5_x_20xllg^^^6i n,. 1.248^ 1- I'' I ! ■ f" 114 ahithmetig. \m\'. Ex. 6. A owned -^^ths of a ship, and sold -^ of f of hi ghare for £V^h ; what was the value of 7I of f ths of the ves- 4i Bel? A of I of i^' : ^ of t :: £,\%h '- req*. value m £'8, \ 2x4 ? 4 2 .400- 11x3x17 ? i. ^^17 oFti — i^ — 7^ 5 7 ^ 7^ ^ i •• £ 1J5" • f^^**- value in £'s ; 33 . , . -, 400x2 11x3x17 ^^^ .*. req*. value m £ s = ^.^ „„ x — ^^—-j — = 100. 33x17 2x4 Note 1. There are certain examples in which, at first sight more than three terms appear to be given, but they, in certain cases, come under this Rule, as in the following in- stances. Ex. 6. If the carriage of 5 cwt., 7 lbs., for 84 miles cost £3. 18«. 4d., what will it cost to have 21 cwt., 1 qr., 14 lbs. carried the same distance ? 84 miles may be left out of consideration, the distance in both cases being the same. .-. 5 cwt, 7 lbs. : 21 cwt., 1 qr., 14 lbs. :: £3. 18«. Ad. : req**. cost; whence, req"*. cost = £16. 10«. Sid. ^q. Ex. 7. If 12 men can reap a field in 4 days, in what time can the same work be performed by 32 men ? 32 men require less than 4 days to perform the work ; .-. 32 : 12 :: 4 days : req<*. time in days ; • /. req*. time = days = li days. Note S. Examples such as the following are easily worked by Rule of Three. Ex. 8. A gentlemen after paying an income-tax of 7d. in the £, has £248. 10s. 8d. ; what was his gross annual income ? After paying inc«. tax on £1, he had £1 less 7d, or Ids. 5d. .-. 19s. 5d : £248. 10». Sd. ::£!.: req*. income; whence, req*^. income = £256. Ex. 9. A hare, pursued by a greyhound, was 130 yards before him at starting; whilst the hare ran 5 yards the dog ran 7 yards ; how far had the hare gone when she was caught by the greyhound ? Since the dog gams 2 yds. on every 5 yds. which the hare RULE OF THREE. 115 Ains, "we require to find how many yards the hare mus^ i un for tlie dog to gain liJO yds. .'. 3 yds. : 130 yds. :: 5 yds. : no. of yds. the hare must run: i- , . 130x5 „_ .'. no. ot yds. re(i'>. = — - — = 335. (Swp, 159,) Ex. LXVI. (I) If 8 bushels of wheat cost $10, what will 34 bushels cost at the same rate V (3) If 3 bushels of oats cost $1.10, how much will 33 bushels cost ? (3) If 9 bushels clover seed cost $36, how much will 4 bus., 30 lbs. cost? (4) When oats are seiMng at 55 cents a bushel; how many bushels can be bought for $31.35 V (5) The price of a bushel of pease being 84 cents ; how many bushels can be bought for $17.30? (6) Find the value of a silver salver, weighing 31 lbs., 4 oz. at 6s. 5d. an oz. (7) How mucli cheese at 16 cts. per lb. can be bought for $463.36? (8) A bankrupt, who owes $33856, can pay |10496.64; ;vhat will be the dividend in the $? • (9) A pensioner received $106.14 for the year 1864; find the amount of his daily pension. (10) 1 mile of road cost $393.75 • what will 30 mi., 5 fur., 33 yds. of the same kind of road cost? (II) What weight of sugar may be bought for $449.38, when the cost of 6 cwt., 3 qrs. is $133.13. (13) The taxes on a house rated at $188.75 amount to $33.15; the taxes on another house in tne same village amount to $386.66^^; find the rateab'e value of the 3nd house. (13) A bankrupt's debts amount to $10000, and his prop- erty to $3875, what will each of his creditors lose in the $ ? (14) A ship was provisioned for a crew of 84 men for 5 months ; how much longer would the pi^ovisions last, if a crew of only 60 men were taken on board ? (l&i) A merchant exchanged U34yds, of velvet foru313 8 I I 116 A HITHMETia n ' tf I ' ! yds. of silk at 3s. 4id a yd.^, find the value of the velvet a yd.? (16) What are the effects of a bankrupt worth, whose debts amount to £3057. VZs., and who can ixiy 17s. M. in the£? (17) A man on the averajuje walks ovvr 10 ft., Sin. in 4 steps, what number of steps will he take between two places, a distance of 1 mi., 12S0 yds. apart V (18) If 31 ac, 3 ro., 9 po., 21 yds. of i^round cost £3025 12«. ^d.y what will be the price of 41) ac, 3 ro., 38 i)o., 2$ yds. of ground of the same quality V (19) A bankrupt pays 59 cts. in the $ ; what will bt; lost on a debt of $13675. (30) How many minutes must a boy, who runs 6 mi. ai» hour, start before another boy, who runs 7^ ml. an hour, in order that they may be together at the end of 10 mi. V (21) Two boats start in a race, and one of them gains 5 ft upon the other in every 55 yds.; how much will it have gained at the end of half a mile V (22) How many pairs of mits at 45 cts. a pair should bo exchanged for 36 (lozen pairs of stockings at 55 cts. a pairV (23) How many men would perform in 168 days a piece of work, which 108 men can do in 266 days? (24) If an incorporated village be rated at $12571.871 and a rate be granted of $419.061 ; how much is the rate in tho $? How much will be paid by a house rated at $1734.37^. (25) A gentleman's income in 1863 was $2500, out of which he saved $994.37i ; find his average daily expenditure. (26) If 100 men can finish a piece of work in 27 days, how many men \7ill finish it in 20 days ? (27) A special train on the Grand Trunk Railway, which travels at the uniform rate of 44 ft. in a second, leaves Belle- ville for Toronto, a distance of 109 miles, at 8 o'clock a. m. ; at what time will the train reach Toronto. (28^; A bankrupt owes to one creditor a certain sum, t<» each of two others $1250, to each of three others $816: his property is worth $1718.75, and he can pay 22 cts. in the $. riow much will the first creditor lose ? ^S59) If. when '^heat is 42.«« i qr. (8 bus.), the 4 lb. loaf ^i!f RULE OF TUBES. 117 costs 4i(t., what ought the 4 lb. loaf to cost when wheat is 70«, a qr. ? (30) In what time ought 10 men to perform the same work, which 5 men and 5 boys can jx-rforni in 15 days, it being given that 3 mta can perform the same amount of work as 5 boys V (31) Find a 4th proportional to lib., 10 oz., lOdvvts. ; 1 oz. ; and £6. 3ai. M. (33) How much might a person liave spent in Jan., 18154, who wished to save in that year $250 out of an income of 13034.50? (33) A person, after paying an income-tax of iSd. in the £, has £877. 10s. let;t, find his" original income. (34) Find (1) the inconje which pays £29. 3«. 4rl tax at the rate of Id. in the £ ; (3) the income from which, after paying tax at the same rate, the remainder is £932. (35) A piece of gold at £3. lis. IQ^d. per oz. is worth £150; what will be the worth of a piece of silver of equaU weight at 548. M. per lb. (36) A certain piece of work was to be done by 25 men in 16 days; after 4 days 15 men go away ; hov/ long will it take the rest of the men to finish the work ? (37) A person after paymg tor the 1st half of a year an income-tax of 1 ct. in the $, and for the 2nd hiilf (fiie of 1^ <^ts. in the $ on his income, has $1855 left ; what was the income on yr^iich he paid ? (38) If f of a qr. of wheat cost 54s., what will be the price of ^ of a bus. ? (39) If If of a cwt. cost £7. 3s., what will -h of a ton cost ? (40) If ti? of f of 3i of 40 lbs. of beef cost X^d., how many lbs. can be bought for £1. 6«. Qd. ? (41) A dock marks the true time on Sunday morning at 6 o'clock, and on Tuesday at noon it has gained 24 minutes, what will be the true time when it shews 1 o'clock on Sat- iirday afternoon ? (42) The hour and minute hands of a watch are together at 13 o'clock, when will they next be together? (43) If 5 lbs. of sugar cost '0703125 of $4, what will 0625 ,, 4. -.^fUfi n».-.^y^ ^—.n.n-.' r^^n*-^ Cwl. Oi. LUC sauic su^ai \;usi. : (4^) ^ certain piece of work can be «k)ne in 18 days by u m 11^ ' M 1U' !ri 118 AlilTIIMETia n; ' 4 men, 7 women, or 9 boys ; how long will the same vfdk » occupy 5 men, 4 women, and 2 boysV (45) If after selling Ktlis of an estate, I sell ^ of I of tie remainder for l|j of § of £G0O§, what is the value of ^rds of ity (4G) What will be the value of a gold cup weighl^.g 3«83 lbs.; when 1 oz. of it is worth £401)? (47) 4 men and 5 boys earn $33.13 in 7 days, and 3 mc-n and 8 boys earn $38.1)8 In i) days; in what time will 12 men and 13 boys earn $18(>.48V (48) A can do a pi(?(!e of work in 5 hours, J? in 9 hours, and in 15 hours. How long will it take C to flnish the work, after A has worked at it for 40 minutes, and B for H hours V (49) If a garrison of 1500 men have provisions for 13 mo., how long wdl their provisions last, if at the end of 2 mo. they be reinforced by 700 men ? (50) Two men start at S.oO a. m., one from Toronto and the other from Whitby, a distance of 30 miles, and they ap- l)roach each other at the rat(!s of 4^ and 3 miles an hour ; at what time will they meet, and at what distance from a place, which is 2 miles nearer to Toronto than Whitby is? (51) Tw*) trains respectively 210 feet and 180 feet in length are going in opposite directions, the first at the rate of 24 miles per hour, and the other at the rate of 27 miles per iour ; find how long they will take to pass each other. DOUBLE RULE OF THREE. 117. The Double Rule op Three is a shorter method of working out such ciuestions as would require two or more applications of the Rule of Three. 118. For the sake of convenience, we m? ■ •'^ide each question in the Double Rule of Tliree into two parts, the suppositwn and the demand : the supposition being the part whi{;h expresses the conditions of the question, and the de- i21, wiKr* 'A'oujff tlie carriage the WJ-xi; i> -iJC cariiagu \jI la Cwt. lOi x* uiiica v;v7afc 4?*!, form liic supposition ; and the words " what would the car- riage ot ?1 cwt. for 16 miles cost?" forin the deinw4 VOCBLE UULE OF TltliEE 110 Adopting this distinction we muy p^ive tlie follow ln<; Uiilo for working out exiiinplcs in the Double Uule of Three. 119. KuLE. Take from the supposition thiit (piantity which coiTcsponds to tlie quantity sought in the demand ; and write it down as a third term. Then tjike one of the other quantities in the suppositicm and the corresponding quantity in the demand, and consider them with reference to the third term only (regarding each other quantity in tlie supposition and its corresponding quantity in the demand as bein^ equal to each other); when the two quantities arc so considered, if from the nature of the cas?, the fom-th term would be greater than the third, then, as in the Rule of Three, put the larger of the two quantities in the second term, and the smaller in the first term ; but if less, put the smaller in the second term, and the larger in the tirst term. ^ Again, take anotlker of the quantities given in the suppo- sition, and the corresponding quantity in the demand ; and retaining the same third term, proceed in the same way to make one of those quantities a first term and the other a second term. If there be other quantities in the supposition and demand, proceed in like manner with them. In each of these statings reduce the first and second terms to the same denomination. Let the common third term be also reduced to a single denomination if it be not already in that state. The terms may then be treated as abstract num- ber. Multiply all the first terms together for a final first term, and all the second terms together for a final second term, and retain the former third term. In this final stating mul- tiply tne second and third tcnis together and divide the product by the first. The quotient will be the answer to the question in the denomination to which the third term was reduced. Ex. 1. If 5 men earn £18. 15s. in 12 weeks, how much will 16 men earn in 20 weeks ? 16 men will earn more mon- ey than 5 men in a given time ; in 20 wks. 7nore mon- ey will be earned than in 12 wks. by a given no. of m^n. 16x20:: 375s. : no. of shillings req".; 16 X '■'0 X 3J3 ..no. of shillings req'«.= — flyj-^ = 2000s. = £100. By the Rule, 5 men : 16 men 12 wks. : 20 wks. 5x12 [== £18. 15s. 120 ARITHMETIC. ,)'.! Ex. 2. If 10 liorscs eat 56 bus. of corn in 32 days, in liov» many days Avill 8 liorscs eat 84 bus. ? A given no. of bus. will last 10 liorses ^ .. 3^ ^^^^y^ g fiorses more days than 10 8 liorses : 50 bus. : 84 bus. S ' horses; 84 bus. will last a 10 X 84 X 32 - _ gweii no. of horses irnore .•.no.daysreq-i.^-^^g^r— =90.t]ays than 56 bus. Ex. 3. If 15 inimps, working 8 hours a day, can raise 1200 tons of water in 7 days; how many pumps, working 12 hours a day, will be required to raise 7500 tons of water in 14 days? Fewer pumps work^. 12 hrs. 12 hrs. : 8 hrs. ) a day are req**. to raise a 1200 tons : 7500 tons [- :: 15 pu'"?^ giwn weight of water in a given no. of days than if they worked 8 hrs. a day; niore pumps are req"*. to = 30. raise 7500 tons than to raise 1200 tons in a given no. of 14 days : 7 days ) .•. no. of pumps req'. _ 8 X 7500 X 7 X 15 ~T^^a200xi4 days, works, a given no. of hrs. each day ; fewer pumps are req'^, work*?, for 14 days a given no of hrs. eacli day, to raise a given iceight of water, than if they worked only for 7 days. Ex. 4. If 25 men can perform a piece of work in 16 davs working 12 hours a day, in what time will 20 men perform a similar piece of work 4 times as large, when they wots only 8 hours a day ? Call the Isic piece of work 1, then the 2nd piece will = 4 20 men : 25 men ) .'. no. of days req**. 1:4 [-irlOdays. 25x4x12x10 12 hrs. 8 hrs. 20x8 = 120. Ex. 5. A contractor engages to make a road 5| mi. lonL» in 100 days; but after employing 135 men upon it for 100 days, he tinds that only 3 mi., 700 yards are completed ; how many extra men must he employ in order to complete Uib contract ? 5^ mi.-3 mi., 700 yds.=9080 yds.-5980 yds.=3700 yds. • no. of men req"*. 5980vds. : 3700 yds. ) .,.0^, ,^^„ 3700x100x135 00 days : 100 days, f 5980 x 00 =130^ 2»yi 140 men umst be employed, or additional men. DuCliLE RULE OF THliEk m Ex. LXVIl. ^1) If 10 sacks of oats supply 12 horses for 4 weeks, how IDiig ^'ill 15 sacks supply 9 horses V (2) If 42 men finish a work in 36 days, how many will finish twice as great a work in 27 days ? (3) If 60 men in 36 days finish a work, in how many days will 135 men finish four times as great a work ? (4) If 104 tons carried 34 miles cost $87.36, what will 103 t(ms carried 122 miles cost? (5) If a man with a capital of $100000 gain $2500 in 3 (nonths, what sum will he gain with a capital of $1500000 in 7 months ? (6) If 21 cwt. be carried 40 miles for $2.80, how far ought 7 cwt. to be carried for $4.06 ? (7) If 7 horses be kept 20 days for $70, what will it cost to keep 45 horses for 9 days ? (8) If 140 horses eat 560 bus. of oats in 1 6 days, how many horses may be kept for 24 days on 1200 bus. of oats? (9) If with a capital of $5000 a person gains by trade $250 in 16 months, in how many months will he gain $625 with a capital of $2000. (10) If a regiment of 1878 soldiers consume 702 qrs. of wheat in 336 days, how many qrs. will an army of 22536 men consume in 112 days? (1 1) If 6 horses can plough 17^ acres in 4 days, how much land can 24 horses plough in 2i days ? (12) If £240 be paid for bread for 49 persons for 20 mo., when wheat is 48.s. a qr. ; how long will £234 find bread for 91 persons, when wheat is £2. 16.s. a qr. ? (13) If 100-8 lbs. of flour support 20 men for 3 days, how many men will 46305 cwt. support for 735 weeks ? (14) If 26 men can reap a field of 85 ac. in 12 days, how many men will reap another similar field one-half the size of the 1st field in one-seventh part of the time ? (15) 3 men can do a piece of work in 6 days, if they work 10 hours a day ; how long will it take 16 men to do twice the amount of work, when they are working at it 9 hours a day ? (16) If the wages of 25 men amount to £76. 13s. ^. in 16 days, how many men must work 24 days to receive I ;i( . r: 122 AMITBMETIC. fe'i w |4 y^ fi £108, lO.s., the diiily wages of each of the hitter being oue- half that of eacli of those of the former ? (17) If 00G4 men, on half rations, consume 857 qra. of wheat in 57 days, how many qrs. of wheat will 798 men, on full rations, cor.oame in 119 days? (18) If the 16 cts. loaf weighs 3-35 lbs., when wheat is $1.14 a bus., what ought to be the price of wheat per bus., when 47'5 lbs. of bread cost $8.20. (19) If when wheat is $14.40 a qr., the 12 cts. loaf weighs 4 lbs., what should be the price of wheat per qr., when 25 lbs. of bread cost 37^ cts. ? (20) If 4 men, each working 8 hrs. a day, take 11 days to pave a road 220 yds. long, and 85 ft. broad ; how many days will 6 men, each working 12 hrs. a day, take to pave a road 175 yds. long, and 30 ft. broad ? (21) If 100 horses consume a stack of hay 20 ft. long, 11 ft., 3 in. broad, and 31 ft., 6 in. high, in 9 days, how long will a stack 18 ft. long, 5 ft. broad, and 14 ft. high supply 80 horses ? (22) If 3 men can dig a ditch 105 yds. long, 4 ft. deep, and 5 ft. wide in 10 days, how long will it take 5 men to dig a ditch 175 yds. long, 4^ ft deep, find 6 ft. wide. (23) If the 8 cts. loaf weighs 1 lb., 11 oz., 12drs. when wheat is $1.80 per bu., what ought the 12 cts. loaf to weigh when wheat is $1.26 per bus. ? (24) If 5 horses require as much corn as 8 ponies, and 15 qrs. last 12 ponies for 64 days, how many horses may be kept 48 days for £41. 5«. when corn is 22s. a qr. ? (25) A. contractor agrees to execute a certain piece of work in a certain time. He employs 55 men who work 9 hrs. daily. When f ths of the time is expired, he finds that only ^ths of the work is done. How many men must he employ during the remaining part of the time, working 11 hrs. daily, in order that he may fulfil his contract ? (26) If 5 pumps, each having a length of stroke of 3 feet, working 15 hours a day for 5 days, empty the water out of a mine ; what must be the length of stroke of each of 15 pumps which, working 10 hours a day for 12 days, would empty the same mine, tb.e strokes of tlie form^^r set of pumps being per formed four times as fast as those of the latter ? %A I. PBACTICM m to PRACTICE. 1 20 An Aliquot part of a number is such a part as, when t^kena certain number of times, will exactly make up that number. „.. . „.o Thus, 4 is an aUquot part of 13 ; 6s. ot 18«. TABLES OP ALIQUOT PARTS. Parts of a ctot. {100 lbs.) \ Parts of a cwt. {in lbs.) 50 lbs. or 2 qrs. = i cwt. 25 lbs. or 1 qr. = i " 20 lbs. = i 10 lbs. =-ijo ^, 5 lbs. = ro Note. The parts of a $ the nme as of the cwt. (100 lbs). (( Parts of a £1. 10s. 6s. Sd. 5s. 4s. 3s. 4d 2s. M. 2s. Is. 8d Is. Ad. Is. dd. Is. = i£l = i _ 1 — 8 1, — I — I i i. — 1 6 = -h (( (( (( u u u 56 lbs. or 2 qrs. = i cw\ 28 lbs. or 1 qr. 16 lbs. 14 lbs. 7 lbs. 4 lbs. 2 lbs. = i = I = i u (i t( (t (i (t Parts of a shUHng. 4d. 3d. 2d. i\d. Id. id I of Is. i — 1, " u Note. In workmg examples in Practice the above tables tvili often have to be varied; the knowledge which the To olar now has, will render him expert in taking such ah- qiiot parts as he may require in any particular example. 121 Practice is a short method of finding the value of Jv number of artU^^^^ by means of ^^^g^^^^ Parts, when the vafue o7a unit of any denomination is given. Practice may ibc divided into Simple and Compound. SIMPLE PRACTICE. .--, T *v.?= ra"- ♦^^ trUrpn number is expressed in the Sace, 27 bushels at |1.30 per bushel. ii :'^ ^i' lilM 124 AUITIIMETia t'l- ■\* The Rule lor Simi)lc Practice will be easily shewn by tho following examples. Ex. 1. Find the value of 1296 things at 16«. lO^fZ. each. The method of working such an example is the following . If the cost of the things be £1 each ; then the total cost = £1296 : .*. cost at £. 10s. 0^. each = i of the above sum = 648 5s. Od each = | the cost at 10.9. each . . = 324 Is. 3d each =: | the cost at 5s. each . . . = 81 Os. 7|deach = i the cost at Is. M. each = 40 .-. by adding up the vertical columns, cost at 16s. lOid each = £1093 The operation is usually written thus: £. s. d. 1206 . 0.0 = cost at £1 each. cost at 10s. each. s. d. . . . 10 . 10 . 10s.= iof£l. 5s. = ^ of 10s. Is. 3d =i of 5s. 7^d=iof Is. 3d 048 324 81 40 10 — cost at 5s each. =: cost at Is. 3d each. = cost at 7^d each. £1093 . 10 . = c( )st at 1 6s. lOM each Note. The student must use his own judgment in select- ing the most convenient 'aliquot' parts; taking care that the sum of those taken make up the gicen prke of the unit. Ex. 2. Fmd the value of 825 bushels of wheat at $1.30 per bus. If 1 bus. cost $1, cost of 825 bus.= $825 at $1 each. ^825.00 = \ralue at $1. each 165.00 = value at 20 cts. each. 82.50 = value at 10 cts. each. 20cts.= ^of$l. 10cts.= iof20cts. Find the value of, (1) 75 at $2.25. (3) 910 at $1.75. (5) 1075 at $3.25. (7) 397 at £1. Is. (9) 1324 at $3.75. (11) 973 at 16s. ^\d. $1072.50 = value at $1.30 each. Ex. LXVIII. (2) 105 at $1.50. (4) 870 at $2.20. (6) 1278 at $1.87i (8) 250 at £2. 8s. (10) 2078 at £2. 7s. (Sd. (12) 230 at £7. 5s. 11 JO PRACTICE. 125 (13) 9978 at £8. 13s. Sid (14) 15739 at £9. 175. 9H (15) 27H3r> at $9.63^. (IG) 37833 at $18.90. (17) A bankrupt whose debts amount to $250215 pays 89 cts. in the dollar ; what are his effects worth ? (18) A gentleman's gross income is $12815, his rates and taxes .1 mount to 25 cts. in the $, find his net income. (19) vVhat will be the loss on a debt of £4970, if a divi- dend of 8s. 10i<^. in the £ be paid ? (20) What will be the total cost of 83i yds. of calico @ \\\d, per yd., of 57f yds. of flannel @ Is. lOcZ. a yd., and of tl8 yds. of ribbon @ 9f(i. a yd. COMPOUND PRACTICE. 123. In this case the given number is not wholly expressed In the same denomination as the unit whose value is giveu^ as for instance, 1 cwt. 2 qrs., 14 lbs. at $10.24 per cwt. The Rule for Compound Practice will be easily shewn from \he following examples. Ex. 1. Find the value of 60 cwt., 3 qrs., 5 lbs. of sugar @ 18.50 per cwt. t • .i ^ n • The method of working such an example is the tollowmg : The value of 1 cwt. of sugar being $8.50; .-. value of 6(^cwt. = ($8.50 x 60) = $510 00 2 qrs.=: \ (value of 1 cwt.) = i ($8.50) 1 qr.= i (value of 2 cjrs.) = i($4.25) 5 lbs.= ^ (valu • of qr.) = \ ($2.12i^ Therefore adding up the vertical columns, value of 60 cwt 3 qrs., C ibs The operation is usually written thus ; = $4.25 = |2.12i = $0.42i = $516.80 2 qrs. = i cwt $8.50 10 Iqr. =:iof2qrs 51b3.= iof lor 85.00 6 = value of 1 cwt = value of 10 cwt 5i0.oo = value of 60 cwt 4.25 = value of 2 qrs. 2.12i =: value of 1 qr. .42-^ -: value of 5 ll)s. I ' ii $516.80 = value of 60 cwt, 3 qrs., 5 lbs. 12b ARITHMETIC, >r - r ' 1'' ^i ii 2 qr8.= i cwt subtracting 1 qr. = ^ of 2 qrs. 141bs.= iofl qr. 21bs.= |ofl41bs o Ex. 2. Find the value of 319 cwt, 3 qrs., 16 lbs. a- 12«. 6d per cwt. s. d. 12 . (j = value of 1 cwt 10 5 . = vaiue ot 10 cwt 4 = value of 40 cwt S = value of 320 cwt 6 = value of 1 cwt 6 = value ot 319 cwt 3 = value ot 2 qrs. £2 105 . 840 2 12 837 1 7 6 13 . 1^=: value of 1 qr. 6 . 61= value of 14 lbs. . lli=valueof21bs. £839 . 14 . 4J = value of 319 cwt. 3 qrs., 16 lbs. Ex. LXIX. Find the value of (1) 55 bus., 25 lbs. wheat @ $1.20 per bushel. (2) 16 cwt, 2 qrs., 20 lbs. of sugar @ 10 cts. per lb. (3) 96 ac, 2 ro., 10 pei'. at $15.50 per ac. (4) 2 lbs., 8 oz., 13 dwt at 7s. Id. per oz. (5) 15 yds., 2 ft., 7 in. at 128. M. per yd. (6) 28 sq. yds., 7 ft., 110 in. at £1. 7s. per sq. ft (7) 11 mis., 3 fur., 55 yds. at $11000 per mile. (8) What is the value of 5 tubs of butter, each of 2 of them containing 57^ lbs., and each of the rest 73f lbs., at $25 per cwt? (9) What will 3460 ft. of timber cost at $8 per 100 ft. ? (10) What will 24650 bricks cost at $4 per 1000. ? (11) What will 46590 ft. lumber cost at $10.25 per 1000 ft.? Find the amount of each of the following bills : (12) 17| yds. calico at 19^ cts. a yd., 35 .^^^ yds. flannel at nni cti^.. a yd., 96,^- yds. sheeting at 70i cts. a yd., 104^ yds. of Holland at 32^ cts. a yd., 12f yds. of ribbon at 17i cts. a yd. (13) 25|f lbs. of beef at 12i cts. a lb., 19H veal at 11 cts. . tt £2 i rt 7t vt. b. 1 of 2 of Mbs., at )Oft.? 1000 ft? annel at Lvnfj J VIE). cts. a yd. It 11 cts. SIMPLE INTEREST. 121 a lb., 35^ lbs. of pork at 8^ cts. a lb., 17i lbs. lamb at 6i cts. alb. (14) 17t lbs. crushed sugar at 12)^ cts. a lb., 18f lbs. cheese at 17i cts. a 1 »., 5^ lbs. of tea at 75 cts a lb., 10> \bs. coffee at 40 cts. a lb , 7f lbs. of honey at 25 cts. a lb. Note 1. The scholar should bring the last tliree questions in the form of a bill, to the master. INTEREST. 124. Interest (Int.) is the sum of money paid for the loan or use of some other sum of money, lent for a certain time at a fixed rate ; generally at so much for each $100 tor one year. The money lent is called the Principal. The int. of $100 for a year is called the Rate per Cent. The principal + the interest is called the Amount. " Interest is divided into Simple and Compound. When ui- terest is reckoned only on the principal or sum lent, it is Simple Interest . , . . , When the interest at the end of the first period, instead of being paid by the borrower, is retained by him and added as principal to the former principal, interest being calculated on the new principal for the next period, and this interest again, instead of being paid, is retained and added on to the last principal for anew principal, and so on ; it is Compound Interest. SIMPLE INTEREST. 125. To find iTie interest of a given sum of 'nwney at a given rate per cent for a year. Rule. Multiply the principal by the rate per cent., and divide the product by 100. Note 2. The int. for any given number of years will be found by multiplying the int. for 1 year, by the number ot years ; and the int. for any part of a year may be toimd trom the int. for 1 year, by Practice, or by tlie Rule of Three. Note 3. If the interest has to be calcnilated from one given day to another, as for instance from the 30th of Jan. to the 7th of Peb., the 30th of Jan. must oe left out in the ^alcu :i- tion ana the 7th of Feb. must be taken into account, for the borrower will not have had the use oi ihe money tor one day till the 31st of Jan. Note Jf. ' tf the amount be recjuired, the int. has firs^ \o ^§ 1.' ii- I- Mil t I I ' '1 ''» ! I ill"' 1 *, 128 ARITHMETIC. fbiincl for the given time, and the principal has then to b* added to it. Ex. 1. Find the simple int. of $250 for one year, at 9 pel cent. pe. annum. By nc Rule, or by the Rule of Three. 1250 $100 : $250 :: $9 : Int. req-*., ...I„.S1.50 ••• In.. ro...= $?^ =1.2.50. Ex. 2. Fnid the amount of £1370. lis. J3d at 4i per cent from Apr. 6 to Aug. oO. £. 8. d. £. s. d. 1376 . 11 . 3 *137G . 11 . 8 4|- 3 5506 .5.0 4)41 29 . 13 . 9 1033 • 8 . 5i * ■ £1032T~8 . 5i £65-38 . 13 . 5i 20 s. 7-73 .-. Int. for 1 yr.= £65. Is. 8-8125d 13 _ d. 8-8125 since 5id = 5-25rf. No. of days from Apr. 6 to Aug. 30=24 + 31 f 30 + 31 + 3l = 146; .-. 365 days : 146 days:: £65. Is. 88125^. : int. req**. or 5 : 2 :: £65. Is. 88125^ : int. req-*. .-. intreq^J.^I of £65. 7s. 8-8125f«.=£26. 3s. l*125d; .-, Am».= £1376. lis. 3fl?. + £26. 3s. l-125d=£1402. 14s. 4-125d Note. Since £1376. lis. 3d = £13765625, and 4| = 475 . , , . . . . ^/1376-5625x4-75\ we might have found the mt. thus : mt.=£l r^ 1 = £65-38671875. Ex. LXX. Find the Simple Int. and also the Ami of (i) $217.25 for 1 year at 8 per cent, per ann". (2) $217.25 for 2 yrs. at 8 per cent (3) $527.37i for 8 yrs. at 7 (4) $93.50 for 2 yrs. at 6 (5) $75.75 for 2^ yrs. at 7 (6) £62. 18s. 9H for 3i yrs. at 8 , • COMPOUND INTEREST. r?» (7> $1075.75 for 4i yrs. at 8 per cent, per anit (8) $084 for 5 yrs., 8 nio. at 8 (9) £7500 from May 5 to Oct. 20, at 3^ (10) £4865. lis. M. from Jan. 1 to An^. 38, 18(58, at 5f .. (11) In what time will $073 at 8 per cent. simp. int. amount to $1)1)4.50 V (12) At what rate per cent., simp, int., will the mt. on $810 amount to $840.80 in 5 yrs. V (i;j) What sum of money will amount to £138. 2«. 0(/. m 15 mo. at 5 per cent, per ann""., simp. int.V (U) If £1 = 10 florins = 100 cents = 1000 mills, find the Bimp. int. on £578. 3 fl. 1 c 2im. for 2i yrs. at 2^ per cent. ^ (15) At what rate per cent., simp, int., will $2293.75 double itself in 25 yrs. V COMPOUND INTEllEST. 126 To find the Comjwund IntercM of a given mm of money at a given rate j)er cent, for any number of years. Rule. At the end of each year add the interest of that veai found by (Art. 110), to the princ^ipal at tlie be^nnnm- f,f • t is will be the princii.al for the next year; proceed n the same way as far'as may be requ red by the question ?dd together the interests so arisin.i,^ in the several years, P thi'rcsult will be the compound interest tor the given ^""ex^. Find the Comp. Int. and Am*, of $600 for 3 yrs. at 8 per cent, per ann. $600 8_ $48.00 .-.$648 8 $51.84 /. $699.84 8 Int. for 1st JT. Prin' for 2nd yr. Int. for 2nd yr. Prin'. for 3rd yr. $55-9872 Int. for 3rd yr. ... Comp^ int.= $55-9872 +^l^ + ^48 = $155-8273. Am^ $600 + $155-8273 = $7.')a-83,3. ^ Ex. 2. Find, working with decimals, the comp. mt. and ■!. \\\V of £690 for 3 yrs. ^X ^ per cent, per auu. U«\ .30 AIUTHMETW. £ 690 4i= 45 3450 2760_ £31 050 = Int. for 1st yr. £690^ _ £721 050 = Prin'. for 2nd yr. 45 ^ 360525" 288420 £32-44725 = Int. for 2ncl yr. £7 2105 £753-49725 = Prin'. for 3rcl yr. or amount req-i. 20 ^ 9-94500*. _12 _ ll-346rf. 4 1 36g. .-. am*.= £753. 9.s. \\\d. nearly, and Int.= £753. 9s. Hid., nearly - £690 = £63. 9«. n^d. nearly. Note 1. It is customary, if the comp^. int. be required for any number of entire yrs. and a part of a vr. (for instance for 5| yrs.) to find the comp-i. int. for the 6th yr., and then take fths of the last int. for the fths of the 6th yr. Note. 2. If the int. be payable half-yearly, or quarterly, it is clear that the comp**. int. of a given sum foj* a given time will be greater as the length of each given period is less ; the simp. int. will not be affected by the length of each period. Ex. LXXI. Find the Compound Int. and Am*, of (1) $800 for 2 yrs. at 7 per cent, per annum. (2) $742 for 3 yrs. at 8 (3) $560 for 5 yrs. at 10 (4) $308 for \\ yrs. at 6 .paid quarterly. (5) $610 for 2 yrs. at 8 . . .paid half-yearly $1009 for 3 yrs. at 7 paid half-yearly PRESENT WOUTIl. iiil ''7) Find the diltcroiu'c Ixitwccn the Amounts at simp. <'um( coiTii*. int. of (1) i:HS() for 2 yrs. ut ;Ji per cent. (3) j:x4l3l. 5«. for tlnee yrs. tit 4 per eent. PUESEN'l' WOUTir AND DISCOUNT. 13T A owes 7i St^^OO, whleh is to he paid ut tlie end of 9 iHoullis from tlie present time: it is clcMr thai, if the debt l)e paid ill onee (int. bem<«^ reeiioned, we will siipjjosc, at H per eent. per ammm), li oui-lit to receive a less sum of money than $.■)()(); in liiet such a sum of money as will, heinjjj now put out at 8 per eent. int., amonnt to ^oOO at the end of 9 moiiths. The sum whieh B oui^ht to receive voir is called the I'rcscnt Worth of the 15(10, due 9 months hence, and tl.^e Bum to be dedm^ted from the $500, in eonseiiue- ee of inunedi- ute payment, whieh is in fact the int. of the Pres(!nt Worth, is calU'd the Discount of the $500 paid 9 months before it is due; hence, Present Worth is the actual worth at the present time '.)f a sum of money due some time hence, at a given rate of mterest. Discount of a sum of money is the interest of the Present Worth of that sum, cidculatcd from the ju'csent time to the time when the sum would be properly payable. •. Disc».= given smn km its P. Worth, and P. Worth = given sum fes.9 its Disc^ PRESENT WORTH. 128. Rule. Find the interest of $100 fot the given time a« the given rate per cent., and state thus: $100 + its interest for the given ti ue at the given rate per cent. : given sum :: $100 : present worth required. i;:v. 1. Find the present worth of $010, due months hence, at 8 per cent, per annum. By the Rule, Int. on $100 for 6 mo. at 8 per cent. = $4. $104 : $()TG:: $100 : P. Worth req^. , , ()7ft X 100 ^^^^ hence P. Worth rccj'^ =- $ - ^^^ = $050. Ecason $100 is the P. Worth of $104, du'e, 6 mo. hence, • we have tlie above statement by the Rule of Three. Ex- 2. Find tlic present worth of £275. Gs. 8d due 15 rpowtJis hence dt 4 per cent, per annum. ;ii 132 ARlTllMF/na m IT * i It" • i I \ '1 "% •: i! r ' ;•■ ' 11 fi:.': 'I-.. of e's^sii;., 15 Int of £100 for 15 mo. at 4 per cent. = r', «t'£4 = £5. .-. £105 : £375^:: £100 : P. Worth mi''. ... P. Worth veq<».= £''^5— = S^' 4... 5^ nearly DISCOUNT. 129. Rule. Find the interest of $100 for the given time at the given rate per cent., and state tlius : $100 + its interest fen* the given time at t^AC given rale per cent. : given sum :: interest of .$100 lor th^ given tune at the given rate per cent. : discount reciuired. Ex. 1. Find the discount of $250. «'5 due 17 montlife liencf. at 8 per cent, per annum, simple interest By the Rule, 17 Int of $100 for 17 mo. at 8 per cent. = ^^ /. $11 IJ : $250| : : $11^ • disc*, req**. .-. disc*, req*. =$ — sr\ — =$^0.41)2^. Baosoii. $lli is the interest on $100 or the discount 01: $111J for 17 mo. at 8 per cent., ;. we have the above statement by the Rule of Three. 1 30 In the discharge of a tradesman's bill before it has be- come due, it is usual to deduct interest instead of discount S if 2 contracts with A a debt ot $100, A giving 12 months' credit, it is usual, if the interest of money be reckoned at « nor rent Tier annum, and the bill be discharged at oiur, t fto ?hn w off $8, or Ibr A to receive $02 instead of $ 1 00; but if i were to put out the $02 at 8 per cent, interest lor 12 monthf it will not amount to $100; tli(>r(;fore such a pro- ceed g s to the advantage of />': the sum ot money wind S s "Itness ought to have been deducted, was not $8, he iXS on the^vhole debt, but $T.;K;, the interest on the present worth of the debt, i. e. the discount. 131 Bankers and lyierchants 111 discounting bills calculate i^t'i H"tP'Kl of (ijcrunint m\ the sum drawn for in the bill, KteUm'oflheir discounting it t,> tl>c .i.ne ,vi..jn it Ik;- comes am HuAm? thukk ^avs of ■^'«\'''^:^I''."' ''' ^ ^^ •jsuaiiy ».ii»wea atter .m. time iv oi.. .s ,vOMiNA„L> >.uc, ik I' ' I- ! f DISCOUNT. isa fore it Ifl LEGALLY duc. Wlu'ix a bill is payable on demand, llie days of grat^e arc not allowed. rure^admng^nllie days of grace, on the lird of March. HillB which fall due on a Sunday, are paid on the previous hatur- ^^£x A bill of £1000 is drawn on Feb. Ifith 18(54 at 7 months' cUte : it is discounted on the Hlh day ot July at o per cent. What does the banker gain by the transaction i The bill is legally due on Sept. 19; trom July 8 to Sept. 19 are 73 days. o^A t-»- t i?o la i « Int. of £1000 for 73 days = £10_^ Disc».= £9. ISji^T^., •. banker's gain = £10 — £9. IStot-"*-— li oi^-» Ex. LXXII. Find the present worth of (1) $216 due 1 yr. hence at 8 per ct. per aim. simp. int. (2) $968 . . . .3 yr 7 (3) $1236 ... .6 mo 6 (4) $225.25 ... .9 mo 10 (5) $1057.50 . . . .2i yrs 7 (6) £161. 13«. 5if?. 71 yrs 3^ (7) £193. 17.S. 4ir/. 19 mo 5 (8) £458. 8s. 9if/. 31 days 5 Find the Discount on ^ . • ♦ (9) $217 due 3 yrs. hence at 8 per ct. per aifc. simp. int. (10) $22100 ....Uyrs 7 (11) $2000 ....6 mo 10 (12) $1750 9mo 8 (13) £345. 16«. 3d . .86 days 4 • (U\ What is the difference between the true and mer- canme discount on £549 for 32 days at 5 per cent, per an- num? ^ J ^ • J',. (15> A bill for £450 drawn March 3, at 9 mo date is dis- counted by a banker on Oct. 22 at 5 per cent. Fmd h,3 profit. II i\ 134 ARITHMETIC (16) From a bill of £3 Us M. due 18 mo. hence, a tradesman deducts S*. ; which is the rate per cent, at which the true discount is calculated ? STOCKS 133 If the 6 per cent. •' Dominion of Canada" stock be quoted in the money market ut 105^ iLe meiininu; is, that for $l05i ol money a man can ]nu'cli;«se $100 ot such stock. iov which he will receive a document which will entitle him to half-vearly paymentsof Interest or Dividends, as they are called, from the Government of the country, at the rate ol 6 \^^iv cent, per annum on the stock held by him, until the Government choose to pay off the debt. Similarly, ii shares in any trading company, whi«h were oriftinally fixed at any given amount, say $100 each, be ad- vertised in the share-market at 86, the meaning is, that lor $86 of money om share can be obtained, and the holder of such share will receive a dividend at the end ot each half- year upcm the $100 share according tx) the stiite of the finan- ces ol tfie company. Stock may therefore be defined to be the capital of trading companies ; or to bt the money borrowed by our or any other Government, a1 so much per cent, to defray the ex- pp. ',es ot the nation. , , ^ ^ - ^^ a The amount oi debt owing by the Government is called the N\TioNAL Debt, oi the Funds. The Government re- serves tr itsch the option of paying ofi the principal or debt at anv fu^-ire time, pledging itself however to pay the in- terest on )t regulaily at fixed periods in the mean time. From i Vanety of causes the price of stock is continually varying \ f urid'holdei can at any time sell his stock, and sc convert iT into money, and it will depend upon the price at which he disposes of it as compared with the price at which he bought it, whether he will gain or lose by the transaction. Note 1. Purchases oi sales ot stock are made through Brokers, who generally charge $i,or 12icts. per cent, upon the stock bought or sold : so that, when stock is bought by auy party, every $100 stock costs that party %\ more than tlie market-price of the stock : and when stock is sold, the o«uo» .rote ^i 1ps« fni evcrv sftlOC stock sold than the market- price. Thus, the actual cost oi $10C stock in thu 3 per cents at M0CK8. 135 i»4|, is $(94i + i), or $94i The actual sum received for {^100 stock ill tlie 3 per cents, at 94i, is $(94i - i), or |94. Unless tlie brolierage is mentioned, it need not be noticed iji working examples in stocks. NoU. When $100 stock costs $100 in money, the stock is said to be at par; wlien $100 stock cost more than $100 in money, the stock is said to be -dX ix. premium; wlien $100 stock costs less than $100 money, the stock is said to be at a discount. All Examples in Stocks depend on the principles of Propor- tion, aiid may tlierefore he worked by the Ride of Thiee. Ex. 1. What sum of money will purchase $3000 6 per cent, stock at 93 ? $100 stock (st.) costs $93 in money ; .-. $100 St. : $3000 St. :: $93 : req^. sum ; 2G00x93 _.,Q .-. req. cost ; .'. ann». cost = $1.50 x 16 = $24. 134. All questions which relate to gain or loss in mer- cantile transactions fall under the head of Profit and Loss. Tradesmen measure their Profit or Loss by the actual amount gained or lost, or by the amount gained or lost on every $100 of the capital they invest. Ex 4. If tea be bought at 84 cts. per lb., and sold at 93 cts. per lb., find the gdn per cent. (93 cts.— 84 cts.)= 9 cts. ; .'. gain on 84 cts.= 9 cts. .'. 84 cts. : $100 :: 9 cts. : gain per cent. ; .-. gain per cent.= — -^ cts.= $10.71 f. Ex. 5. If tea be bought at 93 cts. per lb. and sold at 84 cts. per lb., find the loss per cent. In this case 9 cts. is lost on 93 cts., .'. 93 cts. : $100 :: 9 cts. : loss per cent. • whence loss per cent. = $9,671?. Ex. 6. By selling cheese at £3. 13s. M. a cwt. a grocer realizf d a profit of 22 J^ per cent, what did it cost him per cwt? He sells cheese for which he gave £100 for £122i. ; £122i : £3. 13s. 6d or £3H - £100 = prime cost per cwt. ; , o3Hxl00 „^ .-. prime cost per cwt.— £ — ^^ = £d. Ex. 7. By selling cheese at £3. 13s. 6d. a cwt. a grocer lost 22^ per cent., find the prime cost of the cheese per cwt. In this case he sells cheese, for which he gave £100, for (£100-£22i), or for £77i .'. £77i : £3|^ :: £100 : prime cost of cheese per cwt. ; 32.1 X 100 .-. prime cost per cwt.= £ * ' . — = £4. 14s. 10 ^d. Ex. 8. By selling sheep for $19 the seller loses 5 per cent, on liis.ontlav: what would have been his loss or gain per cent, if he had sold the «Ueep for $23.75 ? iifli 4 I II 140 ARITHMETIC. m H*k, 1st. $95 : $19 :: $100 : prime cost of sheep, .*. prime cost of sheep = $30. 2nd. $20 : $100:: $8.75 : i?ain per cent, if the sheep be sold for $23.75 ; ' .-. gain per cent. = $^^|^ = $18.75. This sum might have been worked thus, $19 : $28f :: $95, 1 e. w'> :* 'i^lOO v/ill realize if the slieep be sold for $19 : what $100 ; ; . . alize if the sheep be sokl for $23|. .-. $100, if sheep sold for $23|, will realize $--^i??*, or $118 J : 19 ' ^ ' /. gam per cent.= $118f -$100 = $18f = $18.75. 135. Tables respecting the increase or decrease of Popu- lation, &c., are constructed with reference to the increase or decrease on every 100 of such population ; Education returns are constructed in the same way ; and so are other Statistical Tables. Ex. 9. In 1852 the population of the County of Wellington was 2G79G, in 1801 it was 49200; find the increase per cent 49200-26796=^22404; .-. 26796 : 100::22404 : incr". per cent 2240400 /. mcrease per cent.=: -^^^-^ = 83609. . .per cent ^5^- }^' . P^tween the years 1841 and 1851 the population of England mcreased 14-2 per cent. In 1851 it was 21121290. what was it in 1841 ? For every 100 persons in l«n there were 114*2 in 1851 ; .-. 114-2 : 21121290 .. 100 : population in 1841 ; .-. population in 1841 = ?l™^iii22 .^ 18495000. 1 ^\ WJ^ ^^ ^ regiment of 750 men, 26 per cent, are in hospital, 32 per cent, in trenches, and the rest in camp, how many are in hospital, trenches, and camp, respectively ? 100 : 750 :: 26 : no. in hosp». ; .-. no. in hosp'.= '^5^^ - io-, ^ 100 ~ 100 : 750 :: 32 : no. in tren^^ ; .-. no. in tren^^ =: I52i!-?? _ 040 100 ~ .-. number in camp = 750-(195 + 240) == 315. Ex. 12. The percentage of children who are learning to ,:ij APPLICATIONS OF TERM " PER CENTr 141 write is 65 in a scliool of 60 children, and 78 in anotlier school of 70, what is the percentage in tlie two schools together i In the 1st school, ^^ ^^ 00x65 100 : 60 :: 65 : no. who write ; .'. no. who write = ^...^ -^^^ In the 2nd school, 100 : 70 :: 78 : no. who write; .*. no. yf\\o write= 100 70x78 100 =54i •. in a school of 130, there are 93| who write ; .-. 130 : 100::93f : percent, req'^. ; .*. percent, req*. 10 X 93^ 130 = 72. Ex. LXXIV. (1) Wliat will be the broker's commission on the purchase of $4300 6 per cents, at 90^, at i per cent. ? (2) What is the premium on a policy of insurance for $9626.55 at $2.60 per cent. ? (3) The commission on the purchase of $1560 Dominioi) stock at 104 amounted to $4.60, what was the rate per cent, i (4) For what sum would the life of a person aged 23 be insured by the annual payment of $45.60, the premium tor that age being $2.40 per cent. ? (5) A draper at Hamilton buys 25 pieces of calico, each containing 36 yds., for £32. 16s. 3d; the carriage costs him 6s. M. ; (1) What will he gain by selling the calico at lO^rf. a yd.? (2) What will he gain per cent.? (6) A merchant bought 1280 bus. of wheat at $1.20 a bu., Uie expenses of carriage, &c., averaged 3| cts. a b".; he sold the wheat at $1.40 a bu. (1) What was his gain? (2) What was his gain per cent. ? (3) At what price a bu. should he have sold the wheat in order to gain $400? (7) (1) A man buys a pig for 6s. 8d, and sells it for 7s. Ad.\ and his gain per cent. (2) What would have been the loss per cent had he bought the pig at 1b. Ad. and sold it at bs. (8) Tea is bought at $96 per cwt., at what price per lb. must it be sold to gam 25 per cent. ? (9) Sugar is bought at $6 per cwt., what will be the gam per cent, if it be sold at 10 cts. per lb. ? (10) At what price must a yd. of cloth be sold, which cost i8.M., so as to gain 12i per cent. ? 142 ARiTHMETia ^M hi. '\: Pr! "I rr ■ 'ti (11) If a yd. of cloth, sold at 4*. 8rf., give a profit of \2\ per cent. ; find the prime cost. (12) A grocer buys 40 lbs. of tea at 84 cts., 44 lbs. at 93 cts., and 55 lbs. at $1.08; and sells the mixture for $188.16., what is his gain per cent. ? (13) A grocer mixes 20 lbs. of tea at 5s.- 3d, 32 lbs. at 5s. 7rf., and 36 lbs. at 6s. \d. ; at what rate per lb. must he sell tlie mixture in order to gain 40 per cent, on his outlay ? (14) If I sell for 15s. I lose 10 per cent., what must I sell at to gain 10 per cent. V (15) A person buys a certain number of eggs and sells them again at such a price, that 11 are sold for the money 18 cost him. Find his gain per cent. (16) A boy sells another boy a cricket-bat for $1.56, gain- ing thereby 30 per cent. ; what did it cost him '? APPLICATIONS OF THE TERM "AVERAGE." 136. Questions are often given, in which the term " Ave- rage" occurs; two such examples will be worked by way of illustration, and others subjoined for practice. Ex. 1. A gentleman in each of the following years ex- pended the following sums : in 1845 $650, in 1846 $675, in 1847 $680, in 1848 $690, in 1849 $700, in 1850 $715, m 185? $790. Find his average yearly expenditure. The object is to find that fixed sum which he might have, spent in each of the seven years, so that his total expendi. ture in that case might be the same as his total expenditure was in the above question. Adding the various sums together we find that the total expenditure amounted to $4900 ; this sum divided by 7 gives $700 as the average yearly expenditure. Ex. 2. In a school of 27 boys, 1 of the boys is of the age of 17 years, 2 of 16, 4 of 15i, 1 of 14f , 2 of 14i, 5 of 13f , 10 of 12i, and 2 of 10 ; find the average age of the boys. The object is to find, what must be the age of each boy, supposing all to be of the same age, that the sum of their ages may equal the sum of the ages in the question. Sum of ages = 17 + 32 + 62 -+-14I -i- 29 +68f -+- 122i + 20 = 366 ; .•. average age = 366 yrs.-v- 27 = 13f years. Ex. LXXV. {1) The highest temperature registered in the shade on DIVISION INTO PROPORTIONAL PARTS, 143 Monday 13th July, 1808, in the following towns, was :— Ot- tiivva, 104- Montreal, DO: Toronto, 02; New York, 90; Buf- falo, 83; New Orleans, 81. Find their average highest tenii»erature? (3) On Sunday I spent no money, on Mond. $4.25, on Tuca. $5.75, on Wed. $«.(iO, on Thurs. $7.80, on Frid. $3.50, on Sat. $5.58; find my average daily expenditure during the week. (3) The highest temperatuc registered in the shade in the week endingon Midsummer-day, 1865,in the following towns, was :— Birmingham, 878 ; Manchester, 87-7 ; London, 876 ; Bristol, 86-8; Leeds, 850; Salford, 845 ; Dublin, 838; Edin- burgh, 780; Liverpool, 770; Glasgow, 77*6. Find their average highest temperature. (4) In a school, 17 children average 6 yrs. ; 26, 7^ yrs. ; 35, Oi yrs. ; 30, 10 yrs. ; and 8, 13i yrs. Find the average dgc of all the children. (5) The average age of 37 men is 57 years, that of the first eleven is 53 years, and that of the last eight 59i years. Find the average age of the rest. (6) The populations of 3 towns in 1851 were 31336, 43334, and 6706 ; in 1861 the first two had increased 13, and 10 per. cent, respectively, and the last had decreased 18 per cent. ; find the average population of the 3 towns in 1861. (7) A tradesman's average annual gain from the year 1853 to 1863, both inclusive,' was £184. lis. 6d ; in 1853 he lost £76. 88. 4d, and in 1864 he gained £151. 9.s. lOd What was his average annual gain from 1854 to 1864, both inclu- sive? DIVISION INTO PROPORTIONAL PARTS. 137. To divide a given number into part,% which shall he proportional to certain otiier given numbers. This is an application of the Rule of Three; still it may be well to state a general Rule, by which such Ex^ may be worked. Rule. As the sum of the given parts : any one of them :: the entire quantity to be divided : the corresponding part of it. This statement must be repeated for each of the parts, or ftt all events for all but the last part, which may either b« 144 AIUTIIMKTW. ih f it 1 1 found by the Rule, or by subtructiiig the sum of the values of the other parts from the entire quantity to be divided. Ex. 1. Divide 40 dollars among A, B, C, so that theil shares may be "s 7, 11, and 14 respectively. By the Rui^. Sum of shares = 7 4- 1 1 4- 14 = 32. .-. 32 : 7 :: $40 : yl's sli". ; 32 : 11 :: |40 : /i's sh«. ; whence yl's sh".= $8.75, J5's sh«.= $13.75., O's sh«.= $40 - ($8.75 + $13.75) = $17.50. Ex. 2. Divide £45 among A, B, (7, and D, so that ^'s share : S's share :: 1 : 2, ^'s : C's :: 3 : 4, and C's : Z>'s :: 4 : 5. The L. c. M. of 1, 2, 3, 4, and 5, is 60, .-. if i> has 60 shares, Cwill have^of60, ()r48; i? will have | of 48, or 36 ; and ^ will have i of 36, or 18. .-. (18 + 36 + 48 + 60), or 162 : 18 :: £45 : ^'s sh«. ; whence .4's sh«.= £5. Similarly J5's = £10, (7's = £13, 6«. m.y and i>'s = £16. 13«. 4d FELLOWSHIP OR PARTNERSHIP 138. Fellowship or Partnership is a method by which the respective gains or losses of partners in any mercantile transactions are determined. Fellowship is divided into Simple and Compound Fel- lowship : in the former, the sums of money put in by the several partners continue in the business forUhe same time ; in the latter, for different periods of time. The Rule in the last Art. applies for Simple Fellowship. Ex. Two merchants, A and B, form a joint capital; A puts in $240, and B $360 ; they gain $80. How ought the gain to be divided between them "i $(240 + 360) : $240 :: $80 : .4's sli«. in $'s .-. ^'s sh<'.= $32, and 5's sh«.=^ $(80 - 32) = $48. COMPOUND FELLOT/SHIP. 139. Rule. Reduce all the times into the same denom- ination, and multiply each man's stock by the lime of iti continuance, and then state thus ; The sum i all the products : each particular product:: the whole quantity to be divided : the corresponding share. Ex. A and B trade together ; A p- cS in $300 for 9 mo., and B $240 for 6 mo. ; they gain $115. How ought they to divide it ? Bv the Rule, ' ; ■ KqUATWN OF PAFMENT 145 1(300 X 9 + 340 X (}) : |(300 x 9) :: $115 : ^'s sh"., 1(300 X 9 4- 240 x (}) : $(340 x G) :: $115 : /^'s sli'"., whence, ^I's sli«.= $75, and JJ'a = $40. Eeaaon. $300 for 9 mo. = 9 times $300 for 1 mo., and $240 for 6 mo.= 6 times $240 for 1 mo. ; tlie example then be- comes one of Simple Fellovvsliip. EQUATION OF PAYMENTS. 140. When a person owes anotlier several sums of money, due at different times, the Rule by whicli we determine tlie just time wlien the wliole debt may be discliarged at one payment, is culled the Equation of Payments. Mte. It is assumed in this Rule that the sum of the ia- terests of the several debts for their respective times equals the interest of the sum of the debts tor the equated time. Rule. Multiply each debt into the time which will elapse before it becomes due, and then divide the sum of the pro- ducts by the sum of the debts; the quotient will be the equated time required. Ex. 1. A owes B $100, whereof $40 is to be paid in a mo., and $60 in 5 mo.; find the equated time. By the Rule, „ , , ,. . 40 X 3 + 60 X 5 420 , , equated time m mo.= -^oT^q— "-- ^^ = 4h Ex. 2. A owed B $10, to be paid at the end of 9 mo.; he pays however $2 at the end of 3 mo., and $3 at the end of 8 mo. ; when ought the remainder to be paid V In this case, 2 x 3 + 3 x 8 + 5 x no. of mo. req'i.= 10 x 9, or 6 4- 24 + 5 X no. of mo. req**.— 90 ; or, 30 4- 5 X no. of mo. req'i.= 90, or 5 x no. of mo. req*!. =i 90 - 30, or GO, .-. no. of mo. req'i.= 12. Ex. LXXVI. (1) Divide (1) 1008 into 3 parts, whicli shall be to each other as the numbers 2, 3, 4, respectively. (2) $260 uito 3 parts, which shall be to each other as 5, 11, and 16. (3) 145 ac. 3 ro. 33 po. between two persons in the ratio of 5 : 6. (4) £110 between 4 persons, whose shares shall be as 4, i, i, and i. (2) {\)A,B, and contribute to a fund $320, $560, $720, respectively. How are they to divide a profit of $680 ? (2) A, who has £422. 10."*., owes B, £175 : C\ £210 ; and i>, £265 ; what sum ought G to receive *«* ^fi i M 14G ARITUMETia I H 1 'Pijiij ■ ' mi •i ■■> . L...>l J (3) Sugar being composed of 48*856 per cent, of oxygen, 43-265 per cent, of carbon, and the rest hydrogen ; how many Iba. of each of these materials are there in 1 ton of sugar ? (4) Archimedes discovered tliat tlic crown made for King Hiero consisted of gold and silver in llie ratio of 3 : 1. How much per cent, was gold, and how much per cent, was silver ? (5) Find the equated time of payment of $150 due in 3 mo., $310 due in 6 mo., and $130 due in 7 mo. (6) A owes B $1000 to be paid at the end of 6 mo. ; A pays $400 at the end of 3 mo. ; when ought he to pay the remainder ? (7) A, B, and G remained partners for 3 years ; A brought in $4000' which remained the whole time ; B began with $300 and 6 months after put in $300 more ; G began with $300, and one year after put in $500 more. The whole gam was $7960. Determine each partner's share. (8) ^ is a working, B a sleeping partner in a bookseller's business Their capital amounts to £0400 ; of which £3400 belongs to A, the rest to B. Then* profits, at the end of the first year, amounted to £1600. A receives 10 per cent, of the profits for managing the business. How ought the re- maining part of the profits to be divided? (9) ^ 5 and C rent a field for $60 ; A puts in 20 horses, B 15 oxen, and G 10 sheep ; supposing the keep of a horse, ox, and sheep to be in the ratio of 3, 2, and 1 ; shew how the rent should be divided. (10) Some broth was distributed among a certain number of old men, 9 widows, and 6 single women ; the men had twice as much broth given among them as was given among the women ; also an old man's share was to a widow'b share ••6-5 and a widow's share to a single woman's share : *: 10 : 9*. Each single woman received 1 i pints. How many old men were there? SQUARE ROOT. 141 The Square of a given number is the product of that number multiplied by itself. Thus 6 x 6 or 36 ia the square of 6, or 36 = 6'^ Art. 86. 143 The SqiTAP-E Root of a given number is a numoer, A\ SQUARE ROOT. 147 which when multiplied by itself, will produce the given number. Thus 6 is the square root of 36 ; for 6 x 6 = 3G. The square root of a number is sometimes denoted by placing the sign \/ before the number, or by placing the traction ^ ibove the number a little to the right. Thus y^36, or (36) )< denotes the square root of 30 ; so that y^36, or (36)^ = 6. 143. Rule for extracting the Square Root of a number. Place a point or dot over the units' place of the given number ; and thence over every second figure to the left of that place ; and thence also over every second figure to the right, when the number contains decimals, annexing a oy< pher when the numbar of decimal figures is odd ; thus di- viding the given number into periods. The number of points over the whole numbers and decimals respectively will shew the number of whole numbers and decimal re- spectively in the square root. Find the greatest number whose square is contained in the first period at the left ; this is the nrst figure in the root, which place in the form of a quotient to the right of the given number. Subtract its square from the first period, and to the remainder bring down, on the right, the second period. Divide the number thus formed, omitting the last figure by twice the part of the root already obtained, and annex the result to the root and also to the divisor. Then multiply the divisor, as it now stands, by the part of the root last obtained, and subtract the product from the number formed, as above mentioned, by the first remainder and second period. If there be more periods to be brought down, the operation must be repeated. Ex. 1. Find the square root of 1369. 1369(37 After pointing, according to the 9 Rule, we take the first period, or 13, and find the greatest number whose square is contained in it. Since the square of 3 is 9, and that of 4 is 16, it is clear that 3 is the greatest number whose square is contained in 13 ; therefore place 3 in the form of a quotient to the right of the given number. Square this num- ber, and put down the square under the 13 ; subtract it from the 13, and to the remainder 4 affix the next period 69, thus forming the number 469. Take 2x3, or 6, for a divisor. 4i* 3«= {2x3=6} 67 469 469 .- ' '''h |'H'r> 's\ 5 ' m M r : -I'r'iisiu. 14S ARITIIMETia vide the 469, omitting tlie last fir to have 13^3^6} 61 i Too 3 periods, and .-. 3 dee'. plae.;s in 56261 33756 I 33756 61 39 root; since there is no number in the units' jilace, the first dot will lie over the second cypher from the units' place, and since first period is 00 \Ye place '0 as the first figure in the root. Ex. 5. Find the square root of i^,h- 529(23 2401(49 431 139 139 16__ 891 801 801 2 1 sq. root-- i$ CUBE ROOT. Ex. 6. Find the square root of ^ to 3 places of dec^^ 149 H •714285...; •714285 (-845... 04 164 1G85 742 656 8085 8425 5 .•. sq. root of - = -845. . . 260 Si,i^i Ex. LXXVII. Find the square roots of (1) 190; 289; 625. (2) 841; 900 ; 1704. (3) 2401 ; 7509 ; 9004. (4) 12321 ; 40000 ; 388129. (5) 494209; 582109; 259081. (0) 1234321 ; 28547049. (7) 62504830; 33010510; 49112064. (8) 182493081 ; 47-61. (9) •008836; 445336609. (10) '000633679929; '0000000009. Find the square roots, each to four places of decimals, of (11) 51; '51. (12) 5'1 ; -051. (13) 80652; 90304993. Find the square roots, each to 3 places of decimals where the roo^, does not come out exactly, of (14) -3. (15) '027. 2304 441 (IB) 4J§. (17) 3,8j. (IB) ii\ (19) ^ A father left liis child a box, containing sovereigns, and shillings; the sovereigns were w^irtli as many times the shillings, as tlie shillings were worth the box; the value of the box was 2.s'. Od, aiid there were 5832 sovereigns in the box. How many shillings were there '? CUBE ROOT. 144. The Cube of a given numbef is the product which arises from multii)lying tlial number by itself, and then mul- tiplying the result again by the same number. Thus 0x0 X 0, or 216, is the cube of O"; or 210 = 0'. Art. 80. 145. The Cube Hoot of a given number is a number, which, when multiplied into itself, and the result again mul- tiplied by it, will produce the given number. Thus is the cube root of 216 ; for 6 x (} = 30, and 30 x = 210. The cube root of a uumber is sometimes denoted by plac- 'f; IP 150 jxRITHMETIC. i'l'jAi.. 1 Ing the sign y—before the number, or placing ttie^fraction i above the number, a little to the right, ^is ;216or(216)H denotes the cube root of 216 ; so that V216 or (216)^ = 6. 146 Rule for extracting the Cube Boot of a number. Place a point or dot over the units' place of the given number an? Whence over every third figure to the left ol fhaTDlace and thence also over every thu'd figure to the St whei the number contains decimals, afflxmg one or Iwo cvDhe?s when necessary, to make the number of deci- mal Xes a multiple of 3; thus dividing the given number Sto perTods The number of points over the whole nunv beS and decimals respectively will shew, the number of Se numbers and decimais respectively in the cube root. & S greatest number whose cube is contamed m the first per od It the left; this is the first figure mtiie root, wllh place t the form'of a quotient to the right of the given """MTtmct its cube from the first period, and to the remain- der brino- down, on the right, tlie second period. i)iv d? the number thfis formed, omitting the two last fiffurls by 3 times the square of the part of the root akeady obtained and affix the result to the root. x.»i « . Now calSilate the value of 3 times the square of the first ficrnrP^nUe root M^^^ of course has the value of so many S +Vtorth^ of the two figures in the root + thTL^are Sf the la^t figure in the root. TVIultiply the value Sus found by the second figure in the root, and subtract thPVpSdt from the number formed, as above mentioned, by the firs" eSder and the second period. If there be more period to be brought down the operation must be repeated. Ex 1 Find the cube root of 15625. lh625(25 23 = 8 After pomtmg we TfiOK take the first period, or 15, and find the great- est number whose cube is contained in it. Since the cube of 2 is 8- and that of 3 is 3x2^ = 12 ax (20)2 = 3x400 = 1200 3x20x5= 300 5« = _25 1525 Multiply by 5 7625 7625 27, it is clear that 2 is the greatest number ^ho^ cube 18 coft 'i:t' ' CUBE ROOT, 151 iaitied In 15 ; .'. place 3 in the form of a quotient to the right of the given number. Cube 3, and put down its cube, viz. 8, under the 15 ; sub- tract it from the 15, and to the rem'. 7 affix the next period 625, thus forming the number 7635. Take 3 x 3*, or 13, for a divisor ; divide 76 by 13, 13 is contained 6 times in 76 ; but when the otiier terms of the divisor are brought down 6 would be found too great, therefore try 5. Affix the 5 to the 3 before obtained ; and calculate the value of 3 x (30)^ -f- 3 X 30 X 5 + 5«, which is 1535 ; multiplying 1535 by 5 we ob- tain 7635, which being subtracted from 7635 before formed leaves no rem'. ; .*. 35 is the cube root req*. Ex. 3. Find the cube root of 319-365327791. Place the flrat dot over the 9 in the units' place. 2i9-36533*779i(6031 68 = 316 3x6« = 108 Sx(60)« = _10800 •ix(600)« =1080000 3 X 600 X 3 = 5400 3» = 9 1085409 3 3356337 3k(603)«= 1090837 :jx(6030)« =109083700 ;]x 6030x1= 18090 1«= 1 109100791 3365 33 is not divisible by 108 ; 3365337 luring down the next pe- riod and affix to the root ; the trial divisor will then be 3 X (60)2 = 10800, and 33653^10800 goes 3 times, try 3. 3256327 109100 109100791 109100791 bring down next pe- riod 1091007-^1090837 goes once, try 1. .-. 6031 is the cube root required. Ex. 3. Find the cube root of 000007 to three places ot decimals. -OOOOOfOOOlOlO . 3xl« = 3 3x(10)«= 300 8x10x9= 370 9» = _81 651 9 6859 6000 5859 141 '*i 1 ii V'-\ 1 \ iii ''ft MM ARITHMETIC 147. Higher roots than the square and cube can some- times be extracted by means ot tne Kuies lor square anu cube root; thus the 4th root is lounci by taking the square root of the square root; the 6th root by taking the square root of the cube root, and so on. Ex. LXXVIII. Fmd the cube roots v f (1) 1738; 8000; 5832. (3) 74088; 431875; 778688. (3) 912673; 1092737. (4) 134217738; 64-481201. (5) 444194-947; -000202363003. (6) 131-019108039; 408518488000. Find the cube roots, to three places of decimals in those cases where the root does not terminate, of (7) ti (8) -A: ' (9) 3^. (10) 1. ^ (11) -1. (12) 01 (13) 10. (14) -037 t* MISCELLANEOUS QUESTIONS. Ex. LXXIX. PAPER I. 1. Subtract 2057312 from 5387301, and 205n'Lii again from tlie remainder. Explain how this is the same as di- viding 5387301 by 3057313. 3. (1) Reduce 553553 oz. to tons, cwts., &c. (cwt.= 112 lbs.) (3) Find the proportions of the Avoir d. and Troy oz.^ when the respective lbs. are as 175 : 144. 3. Find, by Practice, the cost of 16 cwt., 3 qrs., 16 lbs. at f 3. 7 cents a cwt., (113 lbs.= cwt.) £1 being - 10 florins = 100 cents = 1000 mils. 4. Define (1) the G. c. M., (3) the t.. c. m., of two or more numbers, (3) a Vulgar Fraction. Find the G. c. m. of 30803 and 67373 ; and the l. c. m. of 8, 9, 10, 13, 15, 18, 35 and 84. 5. (1) Add togetlicr f-of /', of 99H, f of | of 69At, f off of 306i. (3) Express 13.ntaiuing 13 ac, 81 yds. at the rate of 3i miles an hour ? PAPER v. 1. Find the T-irodiict of the following numbers*— (1) 301G X 701). '(2) 1)8307 X 087G. (:])'g0700 x 701)5 (4; 908175 X 39078. (5) 94871>18 X 7y8'>. MISCELLANEOUS. 15& I 2. A merchant bought 974 yds. cloth, and sold It all fi.r ^847.38, gaining $301.94; what was the cost per yard I 3 ^ and J5 own together 120 acres, A having 24 acres .nore than B, A sells his share for $84 per acre B sells his share for the same amount as A ; how much does ti get per acre? 4 If potatoes be bought at $20.35 and sold at $21.32 per* load, how much will be made on a sale amountmg to S6332 04 \ • 5. A merchant sold 45980 bushels of grain that cost him 98 cents at a gain of 29 cents per bushel, and with thTmoney bought 2299 head of cattle ; how much did he pay for each 1 x • • no f 6 If a milkman use a false measure containing 36 ot a pint instead of a pint, out of how much will he have Sed his customed when he has really sold 23 gallons 2 pints ] PAPER VI. 1 Find the length of a street in which the wheel of a barrow revolves exactly 150 times, the diameter of the V heel being U ft., and the r-^i- of the circumference to the diameter, 314159:1. „ ,. ^ , ., o 9 France is 128 millions of Enghsh acres, and the Pyrenees Tread over it would cover it to the depth of 115 feet ; find the bulk of the Pyrenees m cubic feet. 3 What is the height of a closet 8 ft. 4 in. , by 6 ft. 8 in. , vv4h will exactly contain 12 boxes 4 ft. 6 in. long, 3 ft. 4 in. wide, 2 ft. 6 in. deep? ^ 4 What sum of money must be left, in order that after a leduction of ten per cent, has ^een made f e^^^^^^^^ being invested in the 5 per cents, at OlJ, may give a yearly income of $100 ] . , i j fiiQnnn 5 A ship worth $6000 is entirely wrecked, W)U helonid to!l $2000 to B, and the rest to Cf. What are Soctive 'losses to A, B and C, Buppo-g the ship to have been injured only to the amount of $4o00. ,^'l iv\ m G. A can do a piece of work in 27 days, and B in 15 days. I U: > '": II '^'f 156 ARITHMETIC. hi \W4 A works lit it alone for 12 days, B tliuii works 5 days, and afterwards (J finishes it in 4 days. Find the time in which C alone could do the whole work. PAPER VII. 1. Find the product of the following numbers : — (1) 78398G70X 90785. (2) 9703078x679458. (3) 96870 X 708963. (4) 897463287X30974. (5) 906870690x90087. 2 Two boys go fishing : one catches 40 chub, 30 perch, and 26 trout ; the other catches an ec^ual number of each, in all 90 fish. They sell them, a chub for 5c. , a perch 8c. , and a trout, 12c. ; how much does each receive \ 3. A case of strawberries contains 54 b(jxes, each 1 lb. in weight at 7c. a box. What will be the cost of canning 2 cases, allowing 1 lb. sugar at 10c. to every 2 lbs. berries \ 4 Each man in an ^v\i\y of 60000 men gets two pairs of socks per year. How many sheep, each fieece 6 lbs. , are necessary to supply wool for the socks, 1 lb. wool making 8 socks i 5. Jones and Smith are farmers. Jones sold last year 200 bush, oats at 38c., 73 bush, peas at 81c,, 580 bush, wheat at 98c. , 156 bush, potatoes at 29c. , 138 bush, barley at 87v,. Smith sold 45 sheep at $5, 60 lambs at $3.30, 18 young cattle at $15, 18 large cattle at $29, and 26 tons hay at $19. What sum did each receive ? b. A merchant sold a cargo of wheat valued at $40000 for \ less than this amount, thus making a profit of only ^j on cost. At what advance on cost was the wheat valued at in the first instance ? PAPER VIII. 1. Find the product of the following numbers: — (1) 987798640x10970. (2) 793289765x40097. (3) 7968 X 8679. (4) 874598X39076. 2. A shopkeeper bought $0. 60 worth of steel pens at 32 cents per box, each containing 12 dozen, and retaihnl them at 5 cents per dozen. How uiueii did he gain ou his outlay \ W iM\ MISCELLANEOVS. 167 3. A person distributes $22.08 aniongst six men, eight women and twelve boys. Each vvdiiian had three times as .niich as each boy, and each man half as nuicli again as .ach woman. Find what each received. 4. Goods were bought for 8C48 dollars; there was further paid for packing, 20 dollars; for lake carriage, 55 dollars ; for land carriage, 115 dollars ; and for other charges, 350 dollars. The goods were then sold for 10000 dollars. What was the profit made on the sale 1 5. Divide 1120 cents between three boys, Alfred, Ben- jamin and Charles, so that Alfred may have three times as much as Benjamin, and Charles as much as Alfred and Benjamin together. 6. In 1871 the population of England and Wales was 22704108; of Scotland, 3358613; of Ireland, 5402759; of islands in the British seas, 144430 ; and of the army and navy, &c. , 207198. Find the total population of the United Kingdom at that date. PAPER IX. 1. Divide (1) 6022808 by 769 ; (2) 1942944984 by 9876 ; (3) 55596055703076 by 15487 ; (4) 326789039400120 by 90087. 2. If a locomotive travelled from Toronto to Whitby at a uniform rate of 880 yards a minute, it could perform exactly the distance in 60 minutes ; find the distance be- tween the two places in yards. 3. Three men, A, B and C, start on a journey, each with 126 dollars in his pocket, and agree to divide their expenses equally. On their return home, A has 106 dollars, B has 56 dollars, and C has (iG dollars. What ought A to pay B and C to settle their accounts ? 4. A farmer bought two farms, each of 130 acres, for 19500 dollars. What is the value of an acre of each farm, If two acres of one be worth three acres of the other ? 5. A gentleman in Toronto remits $10696. 93| to a friend in London. How much does it amount to in London, exchange at 109^, commission § /„, extra I HI \l 168 ABITHMETIC, G. Brown, in London, has £715 stg. He sends it o a friend in Toronto. How much does the friend realb t, exchange at 109^, commission \ % extra ? PAPER X, (admission TO HIGH SCHOOLS.) 1877. 1, How often is 6 yds. 2 ft. contained in 26 furlong»i? 2. If I buy 3 bushels, paying 5 cents for every 3 quarts, and sell at a profit of 10 cents per gallon, find th') celling price of the whole. 3 Simplify 2HS°n2-| 11 18H+5A- 82M «"»P''^^3iX-0l+Sr^3lX.l.^^^^_^^,^ 4. Reduce 2 hrs. 20 min. to the decimal of 3^ week». 5. A sum of money was divided among A , B and C, A received g of the sum ; 5, $20 less than § of what was left; and the remainder, which was | of .4*8 share, was given to C Find the sum divided. 6. Trees are planted 12 feet apart around the sides of a rectangular field (40 rods long) containing two acres. Find the number of trees. 7. I buy a farm containing 80 acres, and sell | of it for f of the cost of the farm ; 1 then sell the remainder at $60 per acre, and neither gain nor lose by the whole transaction. Find the cost of the farm. 8. Find the amoun., of the following bill of goods: — 18| cords of wood, at $3.50 per cord. 16 yards of cloth, at $1,12§ per yard. 12 bush. 25 lbs. of wheat, at $1.20 per bush. 1,400 feet of lumber, at $12.60 per thousand. 65 tons 12 cwt. of coal, at 30 cents per cwt. PAPER XI. (admission TO HIGH SCHOOLS.) 1878. 1. Define prime number, multiple of a number, highest common factor of two or more numbers, ratio between numbers. Find the prime factors of 1260. 2. The quotient is equal to six times the divisor ; the divisor is equal to six times the remainder, and the threw together, plus 45, amount to 661. Find the dividend. I MISCELLANEOUS. 159 3. I sell \2\ tons of coal for $80, which is one-seventh more than the cost. Find the gain per cwt. 4 OOlX C)01-r0001. 5. A cistern is two-thirds full ; one pipe runs out and two run in. The first pipe can empty it in eight hours, the second can fill it in twelve hours, and the third can fill it in sixteen hours. There is also a leak half as large as the second pipe. In how many hours will the cistern be half full? 6. Ten men can do a piece of work in twelve days. After they have worked four days, three boys join them in the work, by which means the whole is done in ten days. What part of the work is done by one boy in one day \ 7. I buy a number of boxes of oranges for $(iOO, of .which 12 boxes are unsaleable. I sell two-thirds of the remainder for $400, and gain on them |40. How many boxes did I buy] 8. Find the total cost of the following :— Cutting a pile of wood 80 ft. long, G ft. high, and 4 ft. wide, at 60c. per cord. Digging a cellar 44 ft. long, 30 ft. wide, and 8 ft. deep, at 18c. per cubic yard. Plastering a room 24 ft. long, 16 ft. wide and 10 ft. high, at 15c. per scjuare yard. Sawing 6800 shingles, at 40c. per 1000. The Independent Method, or the Method of Reduction to the Unit, introduced at page 80, may with advantage be employed to solve questions which can also readily be done by the Rule of Three. We subjoin a few more examples, showing how to apply the method referred to. 1. If 27 men build a house in 63 days, in how many days will 42 men do the same \ 27 men build a house in 63 days ; /. Iman " ** 63x27 days; 42 men " " —7^ days; K amber of days required = 42 63x27 40^. Jin hr: f4 K, '\, 160 ARITHMETIC. 2. A person rows down a stream in 20 minutes, but without the aid of the stream it woukl have taken him half an hour. What is the rate of the stream per hour and how long would it take him to row against it ? Ist. Moving with stream : In 20', distance rowed — 1^ miles ; .'. in 1', ** =5^j miles. 2nd. Moving in still water In 30^, distance rowed = lJ miles; .*. in 1', '* -=^^ miles; .". rate of stream = f^^ - ^q ^ ^^ miles : .'. rate of stream per hour, fVX^^ = H "^i^es. Rate of stream in 1'=^^ miles, in still water, distance rowed = j\y miles; .'. distance rowed against stream = (^ - 5J5) miles =^:^^ miles; .'. time required to row 1\ miles = f -^J^(J= — - — =* G0'=1 hour. 3. At what time between 1 and 2 are the hands o' '< clock opposite to each other i Let 0(1 be Jie position of the hr. hand. Let Oi> be the position of the min. hand. At 1 o'clock OC over- lapped O B^ and OD overlapped A. Then BC space jiassed over by hr. hand, and A D space passed over by min, hand. 12 times /iO=^Z>(l). But AD^AB + BG^ CI). = 5 min. +BC+30 min. MISCELLANEOUS. 161 ;. substituting this value of A D for A I) in (1), we have VI times liC='^h min. + B. C. .". 11 times BC—'6i^i min., or ii C= 35 min. -^ 1 1 = 3 j^^ min. /. AD^^b min. + 3,'^, min. .. time ru(i[uired is 38 ^^^ min. i){ist 1 o'clock. In connection with the above we give the following statement : Since tlie minute hand moves twelve times as fast as the hour hand, therefore in 12 minutes the minute hand gains 11 minute spaces on the hour luind. * 4. The hands of a clock are together at 12, when will tliey be together again? The time must be after one ; therefore the minuto hand has 6' to gain. 11 minute spaces gained in 12'; ' .'. 1 minute space gained in [f ; 12 X 5' .*. 5 minute space*i gained in — — — \ .'. time required is 5j\' past 1. 5. After paying an income tax of ^W on a ^100, a person has $270() a year. What was liis entire income I 10 on a 100 = j^y on a unit; .'. ^"ff of every urdt of income left ; .-. 1^ ^ $2700 ; .-. tV =1300; • .-.1, or whole income = |?300 X 10 = |3000. 6. A stock o^ provisions will serve 75 men ^or 30 days. How many men must leave in order that the stock may hold out 45 days for those left i PruT'iaions last 30 days for 75 men ; " 1 day for 75x30 men; u ,r 1 . 75x'60 " 45 days tor — -- — men, or 50 men. 45 Ha^xc^j the innnber of men who must leave = 75 - 50 = 25 Exjrjis?e LVI., Slc, furnish <*xan:!plei» (( It n (i's 162 ARITHMETIC. IW'- J nhw. 1 1 Exchange is the Rule by which we find how much monej of one country is equivalent to a given sum of another country, according to a given Course of Exchange. By the Course of Exchange is meant the variable sum of the money of any place which is given in exchange for a fixed sum of money of another place. By tht Par of Exchange is meant the intrinsic value of the coin of one country as compared with a given fixed sum of money of another. Arbitration, or Comparison of Exchanges, is the method of fixing upon the rate of exchange, called the Par of Arbitration, between the first and last of a given number of places, v/here the course of exchange between the first and second, second and third, &c. , of these places is known. It is called Simple or Compound Arbitration, as three or more places are concerned. (For fuller infor- mation on Exchange, see Advanced Arithmetic, p. 227, ^8^0 are requirv^d in Toronto to pay £1800 in London, :^K^land, find the rate of exchange between the two cities. 10 A traveller for Paris wishing to provide himself with Frtinch money, calls at a broker and is informed that the sfjvereign in Lcmdon is worth 25f. 25c. , rate of exchange on London, 8^ premium, and ^ per cent, connuission. Find the sum in French money he ought to receive for $500 of our money. PAPER XIII. 1. Divide (1) 36017070414410 by 160388. (2) 75732561476 by 9487018. 2. If a gallon contain 277 "27 4 cubic inches, and a cubic foot of water weigh 1000 ounces, wliat quantity in gallons and what weight of water in pounds will fill a rectangular cistern 5 feet long, 3^ feet wide, and 2 feet 9 inches deep ! 3. Find the depth ^ oz to i-k o '^ "^ "^ ''i C;i 4i> CO to I— 1 ^^ i^r^ ^ 1— 1 ( o: ^ t- r *>. Oi tc t— k h-l .•/m 00 •<{ OS Ol 4i- cc to H-k iC ^ cc Oil a 00 C5| ><^ cc . to o: c ' tc > oc ' to ) OS to 4^ to to to c; H-l QC HJ o: ^ H-l 4^ t H-k fc-J ■ to © oc © »^ |to cc i en 1 Ol 4^ oc c;i »4^ tc . CO cc OS CO cc CC o to to 4- to H- Is t H-l Ol H-l to cc © CO *^ g OS <1 to S 05 o Ol o: or to 4^ oc OS o 4^^ 4^ 4^ o CO cs CO to to oc to 4^ to © H-k © H-k to 00 4^ o? (— ' c ^ 00 o OI CO OS 00 00 oo o s OS 4^ OT OS 4^ 00 4^ © CO to to 4*^ H-*. © 00 «o 1-* to CO 1— ^ 1-^ to OS 1-^ (— k CO CO CO o oc H-k ;^ © CO Ol 4^ 4^ CO c;t © to H-k 00 CO t-l o to o lO o o 1— ^ 00 I—'- § OI o CO o to o h- k 00 © © § §^ CO to © © H-k- © 1— I 00 1— I t— »• o o? 1— k CO CO to t— k to H-k H-k H-k o CO 00 00 © © Ol Ol :^ CO CO to to H-L H-l to to •• I i. '♦ -f 149. Such questions as 7 + 8 + 3, «fec., are how many t anc> 29 less 7, less 0, &c., are how many? or questions m which addition and subtraction are combined, we omit; because, any teacher, by a little practice, can very easily give such exercises to the class, and, moreover, every practical teacher knows that much of the mine of this part of the Arithmetic depends on the pupil not having seen the questions betore the lesson begins. 150. To find the value of 12 tldngs, the value of one thing being given. Rule. Reckon each penny in the given value as a shil- ling, and each fartiiing as M. Ex. Find the value of 12 things at 15|tZ. each. By the Rule, The value req^.= Is. x 15 + 3d x 3 = 15s. M. Jieasonfor the i*>'ocess. 12 things at Id. each = Is.; .-. 12 at 15^. each = Is. x 15 = 15s. 12. . . .?. . . .id = 3(Z.; .-. 12at|(Z =Sd.x3 = 9d.; .-. 12 things at 15|tZ. each = 15s. M. 151. To find tlie value of S4. things, tU value of one thing being given. Rule. Reckon each penny in the given value as 2s., and each farthing as (id. 152. To find tJie value of 48 things, tlie value of one thing being given. Rule. Reduce the given value into farthings, the result reckoned as so many shillings will be the value required. Ex. Find the value of 48 things at 18fd. each. By the Rule, since 18|<^.= 75(7., the value req'^ = 75s. = £3. 15s. Reason for the Process. 48 things at Id. = 48g.= Is. : .-, 48 things at 75r/.=: U. x 75 = 75s. = £3. 15s. 153. To find the value of lU things, the mine of one thing being given. Rule. (1) Find the value of 12 things by Rule 150 : then consider tills nvMv' as the value of one thiuii;. and apply Rule 150 a second time. Ex. Find the value of 144 things at 13ia. each. MENTAL AlitTHMETtC. 169 Value of 12 things ■= 13s. + M.^ 13s. 6d Value of 144 things = 13s. x 12 + Ga-. = inOs.-f 6s. = £8. 2s. 154. The following general Rule may be given " for find- ing the value of any number of thinfjH, the mlue of one thinq being glmnr liuLE. Reckon liow many dozens are contained in the given number, and how many single things remain over. Then by Rule 150, find the value of one dozen, which value multiply by the number of dozens, and add to the result the price of the single things which remained over. Ex. Find the value of 38 things at 4s. 7d each. 38 = 3 X 12 + 2, value . .f 12 things = £2. 8s. + 7s. = £2. 15s. •• 12 X 3 = £2. 15s. X 3 = £8. 5s. •• 2 = 4s. 7(/. X 2 = 9s. 2d .' 38 = £8. 5s. + Us. 2d = £8. 14s. 2d Ex. LXXX. 1. Find the value of 12 articles at the following prices tor a single article. (1) fd (2) 2d (3) M. (4) 7d ^5) lid (G) lid (7) 2id (8) 3fd (9) Gid (10) Bid ai) lOM (12) Is. Old (13) Is. 4d (14) Is. 6id (15) U. 9K (IG) Is. 8d (17) Is. ll^d (18) Is. 2fd (19) 2s 7d (20) 3s. Oid (21) 4s. 4d (22) 6s. Ifd. (23) 7s. 9d (24) 8s. S^d (25) lis. 7fd (26) 13s. 2d (27) IGs. 3id (28) 18s. lid (29) 19s. 9d (30) 19s. 6|d 2. At the prices named as the value of a single article iu (1) to (12) inclusive in the last question find the value of 24 articles ; at the prices named in (13) to (20) inclusive find the value of 48 articles ; and at tlie prices named in (21) to (30) inclusive find the value of 144 articles. 3. At the prices named as the value of one article in quest". 1. (G) to (20) inclusive, find the value of (1) 13; (2)21; (3)28; (4)35; (5)41; (6)44; (7)57; (8)72; (9) 153 ; (10) 182 articles. 155. To find the value of 20 things, the value of one thing being given. Rule. Reckon each shilling in the given value as £1, ana if there be pence, reckon each penny as the twelfth of £1, thus Id as Is. ^d., and if there be farthings, each farthing as one-fourth the valuf. of each penny, or \q. as 5d »fec. Ex. Find the value of 20 thin^^s at 2s. 8id each. I 170 MtJNTAL AlilTHMETtG. !;!' By tlie Rule, The value required = £1x2 + (Is Sd.) x 8 -f- 5, 5, 11, 16. 3. 14, 20, 27, 33, 40, GO, 55, 17, 36. 4. 88, 35, 63, 29, 76, 80, 1)4, 13, 52. 5. 9. 10, 11, 12, 13, 14, 15, 16, 17; 46, 47, 48, 49. 50; 88, 89, 90, 91, 92, 93, 94, 95, 90 97, 98. Ex. 11. (p. 11.) 1. 106, 150,-200, 287, 310, 431, 555, 919, 867. 2. 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213; 612, 613, (;14, 615, 616, 617, 618, 619; 948, 949, 950, 951, 952, 95;i. 954, 955, 956, 957, 958, 959, 960. 961, 962, 963, 964, 965, 966, 967, 968, 969. Ex. III. (p. 12.) 1. 4585, 7321, 9798, 7006. 2. 5004, 5400, 5040, 8036, 8306, 8360, 9909. 3. 75635, 90909, 10004, 87050, 90001, 64064, 83000. Ex. IV. (p. 13.) 1. 105, 8790, 37071, 30402, 77700, 24817. 2. 105409, 8008013, 7650090, 64000400, 89044001, 504023024, 900300800, 5300050:.. 3. 6006070007, 83401001010, 7004089200, 990000000. Ex. V. (p. 14.) 1. 8even, thirteen, four, uine, eighteen, five, twenty, eleven, five, fifty, tliirty-four, twenty-nine, three, seventeen, fifty- three. 2. Nineteen, eight, forty-one, eighty-eight, twenty-seven, seventy-two, ninety-four, forty-nine, sixteen, sixty-one, nine- ty-eight, eighty, fifty-six, twenty-ciglit. 3. One hundred and seven, one hundred and seventy, seventeen, four hundred and tliirty, six hundred and ninety- one, eighty, eight luuidred, eight, nine Inindred and fifty-six, eigiit Imndred ;ui I ilu*ec, six hundrwl and eighty-four. 4. Four thousand five hundred and six, five thousand eight AmWERS. 173 hundred and seventy, five thousand and eighty-seven, six thousand nine hundred, six thousand and nine, two tliousana five hundred and eighty, seven thousand and forty-five, seven thousand five lumdred and ninety-one, six tliousand two hundred and seventy-five. r>. Twenty-four thousand S(!ven huiidered and fourteen, twelve thousand five hundred, ten tliousand and twenty-five, ten thousand two hundred and five, seventy thousand four hundred and fifty-seven, seventy-four thousand and seven, sevenUr-seven thousand. 6. Three hundred thousand eight hundred an. $151.72i le. $350.7a. Ex. XXX. (p. 66.) 1. £54. 6s. M. 2. £81. 17.s. M. 4. £21. 2s. 6d 5. £18. lU.v. 9rf. 7. £216. 198. Aid. 8. £290. l^s. 10 £484. 6s. 11. £555. Is. ^^d. 3. £4. 6.S. M. 6. £;n. (In. lOirf. 9. £348. 8-s. Uj'. 12. £946. 76. m- 4. 1898307. 2. $63.47. 6. $3227.42. Ex. XXXI. (p. 66.) PAPER I. 1 117984. 2. 107766. 3. 3653012. 5. 2 mis., 6 fur., 18 per., 5 yds., 1 ft., 10 in. 6. 1st, $5.60, $17.17 ; 2nd, $11.57. PAPER II. 1. £362. 19s. 9d. 3. 183 ac, 1 ro., 24 per., 26 yds., 7^ ft- ^ 4. $ 9.48ijV. 5. 5 dresses, £2. 15s. 7K each. PAPER III. 1 £46. 14s. Qd. 2. $7000, $21000, $35000. 3. £13. 12s. 9d 4". 1 ro., 18 po., 5 yds., 2 ft., and 16i f ct over 5. 17 cwt., 1 qr., 8 lbs., 10 oz., 5 drs., and 89 di-s. over. 6. 6 hours, 54 min. PAPER IV. 1 $148.15. 2. 11, 18. 3. 9, 18, 27. 4. owner of net, 8 dozen • owner of b(jat, 16 dozen ; each man, 32 dozen. 5. 230i696 pores. 6. 42000, 42889. PAPER V. 1 6255647664981. 2. 861447920. 3. 11904. 4. 12752. 5. 465335. 6. 95587. PAPER VI. 1 657872 2. $16496471. 3. $10444830.63. 4. 6228^ lbs. 6. 136 ac, 3 ro., 14 po., 24A¥(f yds. 6. 634338 Ex. XXXII. (p. 71.) 12 2 3. 3. 2. 4. 4. 5. 4. 6. 3. 7. 2. 8. 6. 9. 4. lb 2 11 58. 12. 63. 13. 2. 14 30. 15. 10. 16. 8. 17 none 18. 8. 19. 26. 20. 352. 21. 131. 22. non«. 28" 7056. 24. U. 25. 17. 2G. 31. :iif- fmim0K ANSWEHS. ifel Ex. XXXIII. (p72.) I 20. 3. 7S. 3 144 4. 1260. 5. 340. 6. 168. 7. 525„ 8. 1056. 9. 1050. 10. 3520. 11. 11088. 13. 450. 13. 1080. 14. 840. 15. 840. 16. 16380. 17. 1386. 18. 21000 19. 43890. 20. 95C40. 0> 7, 19' f2^ ^^ ( 370 2109 57' "^ ' 10' 57* 72 9' 136 '8 ' 85 19' 107' 7_ 45' 445' 104 o ♦ 13 170 10' 6 ' 10' Ex. XXXVII. (p. 76.) 1 8. 2. 3i. 3. 41 4. 4. 5. 3^ 6. 6f. 7. 5^. 8. 6}^. 9 7. 10. 8. 11. 8U. 12. inn. lU. 9,3.?-,. 14. 1.02AH,. 15. I2i||. '' ^■'" "■ '^ ''*l « f^B^^^^H 182 a ^- 65' 13. 3376 63 • 60380 ^6- 2400* ANSWERS. Ex. XXXVIII. (p. 76.) 35 „ 16 88 89 ^h 'iB- ^'T '-r 300931 ^"•"401 • 69057 465 • 6. 3874 239 ^ 88716 ^- T" 126 ° 13. 2(5253 14. 17. 1. 8. 3 5" 3363 35 • 2 1250* 608543 3084 • Ex. XXXIX. (p. 77.) 9 .12 - 35 19 • 11 3407 •680* 2 9160 ^^' 2160" 3- in '*f;r» 19 55- 5. 16- «• 6" 35 9. -^ 15 10. ^. 11. 5H- 12. r^ ^, 175 _ 14 .. 2 6399 22 36' ir 5945 ^' 6 13 ?^ 3 2.3. 3.; 2 Ex. XL. (p. 78.) 4 5. 9* 3 ^l 8. TS* "^^ 1 ' ir 13" 10. 11 13" 11. 8" ^•21- 13 20" 14. 4" 235 19 3^^^. 35 103 20. ^3g. 16. I 21. =. 17. 12. 191 7_ "^•ll' 13 2^ 1^- 84' 22. 279* 945 18. 1529" 827_ 7337* 20 23- 31- 24 23 33" 9 10 ^12' 12' > 110 m_ 120' 130- 55 9^ 60' CO" 161 «• "" 83 42 ^•63' 63' 48' 48* 2712 3689 iri. 12. 210 EX.XLI. (page 79) 9 8 „ 6 7 . 27 35 2-12' 12- 8' 8 „ 140 183 ^ ^•200' 200' 6720' 6720' 189 384 560 n ^ 112 1^- 1008' 1008' T008- "• 210' 210' 1170, 9^ ^60' 6545 0120 7293 4455 8415' 8415' 8415' 8415* Amwms. 18J^ 5945 6 375 • 44' ^' 84" 827 21* 9^ \m 1225 1176 800 15. 1260' 1360' 1260' 3744 6075 4200 6000 . . 105 102 1J]0 135 84 ■ 180' 180' 180* 180' 180" 7200' 7200' 7200' 7200' 149940 319464 340170 484155 621180' 621180' 621180' 621180* 16 ??®^^ 448630 • 621180' 621180* 80 96 45 112 120' 120* 63 40 48 72' 72' 72* . 10 12 ' 'is' 15' 396 18. 20. 90 100 105 108 2. 120' 120' 120' 120* 220 264 165 jO 330' 330' 830' 330* Ex. XLIL (p. 80.) 28 27 _ 221 228 17 -— - 19. 120' 120* 48 60 72' 72* 36' 36* Q ZZl "• Q10» 504* 40 5. 315 396 672 882' 882' ^82* 312' 312* 6. 36 1239 168' 168' 168* 1728 5445 103040 8. 220 255 252 300' 300' 300' 300 28 99 420 ^^- «o> co> 392 315 '504' 504* 3339 3528 3420 5040' 5040' 6040* 290 16704 9. ggg^ i » 6336' 6336' 6336 * ""• 63' 63' 63 :l. ^ yd. is greater by 3^ yd. 12 * yd. is greater by i yd. 13. U of -1^- ofH of H of a loaf is greater by ^,- of a loaf. Ex. XLIII. (p. 82.) ■jQ 1. ~. 2. 1-,V 3. 3i 4. If,-. 5J 60* 6. li 7. 1,^. 10. 2h 11. 15i. 12. 9f^. 13. 2A-. ^' 56* "• 90* 14. 5H. 15. 9m. 16. 22t^. 17. 2-,%. 18. 2^. 19. 15^ 20.21f4¥.T. 21. 46fgi 22. 29^. 23. IIGH- 24. £111^ 25. 13/Mbs. Ex. XHV. (p. 83.) . 5 1. 2Q. S?. g. 3. g. 41 72* 5. U. 6. 4-,V 7. IM. 8. 3H. 9. tttt:. 10. 1 ,Vo- 11. ni 12. 12Hi 13. i of 100 8 cake. 14.(1)-. ^)5fff. M 1 72 d. It i IS4 AKSWEBS. L II Ex. XLV. (p. 84.) 1. m\l 3. i 3. 20;V. 4. 36H. 5. 1. 6. HI 7. 3^. o 111 3 8. B, C, i>, and A had respectively ^, g, ^, and ^ of cheese. Ex. XLVI. (p. 86.) 1 . 35 _ 5 .1 - - -. ^ 13 ^' 12" 2. 72* 3. 26' ^* 12* 5. 25. 6. 2|. 7. 40* 17 8. 10. 9. 4i 10. 329,\. 11. 4^. 12. 6^. 13. ^, 14. 5;,\% 15. 7i. Ex. XLVII. (p. 86.) 1. 1^- 2. 8. m 2 10 3' 11' 9. 3iH. 10. 4. 116 105- ^' ^'• 27 1 — - 11 — 88" 32* 12. 6. nt 3 8* 13. ^ 2 13 15* ■ Ex. XLVIII. (p. 87.) 1. im- 76 28 2. 2h 3. ^. 4. U. 5. IH. 6. --. 7. 2i 8 8 8. 153' 9. 12H. 10. I-2W5-. 11. Q. 14. 3HI^. 725* 12. 813 102949' 13 ^ AAA' Ex. XLIX. (p. 88.) 1. 40 cents. 2. 3 fur. 3. 1 qr., 17 lbs., 13 oz., llf drs. 4 19 cwt., 1 qr., 10 lbs. 5. 4 fur., 35 per. 6. 2 ac, 1 ro., 25 per., 20 yds., 4 ft., 136* in. 7. 4 lbs., 2 oz., lOdwt., 20gi-s. 8. 59 yds., 2 qrs., If nls. 9. £7. As. M. 10. 109 lbs 8 oz., 5 drs , 8t grs. 11. 5 hrs., 36 min. 12. 7 lbs., 9 oz., 9f drs. 13. $24. 14. 7 hrs., 12 min. 15. 13 cords, 64 c. ft. 1. . 6. |25.20. 7. $110 8. $900. 9. $30. 10. £70. 11. 385^% rounds. 12. $33.60. 13. 131^ days. 14. (1) A has $50.70, B lias $37.80 ; (2) A has $63, B has $31.50. 15. $|i $255.10jri 16. i 17. 3 boys. 18. 5|\days. 19. Elder son received $3250, younger son $1560, and widow $1440. 20. .4 has 24 ac, 3 ro. ; B has 13 ac, 2 ro. ; and O has 47 ac, 1 ro. 21. 48 boys. 23. 42 min. 23. (1) 52i days ; (2) {. 24. 15H ilJiys- 25. 1st cask contains 140 gals. ; 2d, 60 gals. ; 3d, 45 gals. ; 4th, 80 gals. 26. 20 days. Ex. LIII. (p. 95.) £ 11. il. -??1 115- ^^ 10' 100' Too' 1000' 1000' 10000' 73201 791003 3 45 I. 2. 504 100000' 1000000' 100' 10000* 1 50007 . 34702007 1000000' 10000 500 4. 1000' 500005 10000000' 3900,50 10000 100000 ' 1000 20607 500038 loob ' iooooo' 1000' 18741 31 1000 ' 10 ' 560746805 100000000 ' 1. -4; 3-3; 23-5; 502; 00502. Ex. LIV. (p. 96.) •04; -147; 047. 3. 5001; 9-51; 00951; 3. 35-6; 1700701; 0050005; 0000003} 186 It' ill it!!., 'Um 4. -7 ; "030. 5. 300003 ; -0001, 20-76854 ; '0000053002. 6. 4 000504 ; 0000070. 7. Six tenths ; seventeen hundredths ; seven hundredtha. 8. Seven thousandths ; seven hundred thousandths, oi seven tenths ; six and three thousand and four ten thou- sandths. 9 Thirty-five and two hundred and five hundred thou- sandths ; four hundred and thirty -four thousand one hundred thousandths, or four hundred and thirty-four hundredths. 10. 3, 30, 3000, 3000000 ; 1-3, 13, 1300, 1300000 , 540-003. 540003. 540003, 540003000; 74201, 742010, 74201000. 74*?0lO00000. 11. -5302, '05362, 000005362 ; "03 "003, •0000003; 7 00107, -700107, 0000700117; 600, 50, '005. 12. 00000203. Ex. LV. (p. 97.) 1 560 34603. 2.21408691. 3- 10061 '33654. 4. 345'608037. 5. 40-23111. 6. 68607806. 7. 73320773. 8. 9309602912. 9. 1393-7U1. Ex. LVI. (p. 98.) 1. 2-258. 2. 7.0456. 3. 5-9697. 4. l•099^ 5. a) M7 ; 204-93. (2) 68-67 ; '2803. (3) 7209544 ; 527076. (4) 4'41958 ; '0069993. 6. 20-93. 7. 095 ; 19 98. 8. 613 of it left 9. m. 10. (1) 79-8665. (2) 82-9319. Ex. LVII. (p. 98.) 1 11375. 2. 16-2945. 3. 81 20812. 4. 3-333715. 5 42^6-48449. 6. 667 81 ; 1 14364272; 3752; 356-40161745. i 01778479; 488 '745015235 ; '000642. 8. (1) -^150625. (2) '3689. 9. 2781975 yds. 10. 346S loaves. Ex. LVIII. (p. 100.) 1 12'36. 2. 1-236. 3. -01236. 4. 123600. 6. 123600000 6 1737-1. 7. 17371000. 8. 17371. 9. 17.3710000 lo' 170-01 ; 170010. 11. '00521 ; 521. 12. '00003 ; 03 •000000003. 13. 108971-6; 1089716. 14. Oil; '00011 110 16.2040000; 204; 00204. 16.18030; '001803. 17. 213-2; -002132. 18. 0101. 19. 0008. 20. 12i days. 21. 86-6 times. 22. 03054. Ex. LIX. (p. 101.) 1. 6333 ; 63-333 ; 006. 2. 931 ; 3105 ; 003. 3. 6211 684 62215849 056; 62-215. ANSWERS. is: .. ) Ex. LX. (p. 102.) 1 o^. -ft. 1-5- 6-2 • 7 8- (JSo; 53. 2. -1875; 8-9375; 05; 4. •5078125; 8'75; 7( 234375. 5. 39125; 16 36. Bit. LXI. (p. 104.) 1 -6. •!• 857l4)i; -588; -73. 2. 6037; 7131 ; 1001590; 2-8823529;il7^706. 3 11-135^ ^^3^0^^^^^^ 2 r 2 31 1 2 s n. 368. 202.W 4- h Qv ' iT' 198' 37' r 30' 495' 99990' 19H; .u ey^^^^ ^•««'^^^^^^- L^Jo™ 9 1' 817686 10. 44-494301). 11. 40-8 ; -258723 ; •01185. 12. 2-5416; '136; '0743; 30833953. i^x. LXII. (p. 105.) 1. 75ccnt9. 2. $4.37ict8. 8 62Vcts. 4 2 ir.l21bs.,8oz. 5. 3fur. 6. 3cwt aqr ^-^^.^^^J' \t'6 a?. o Q IKa 9 n7 2 dwt 10. 2vi IC., o ro., AiO i'u. ^*_" . ' iro 4 po. 12 £l 8». 18- 1 ' ^"s.. 8 days, 5 h" 1« ma EX.LXIII. (p. 106.) 25 3. 14583. 4. '81875. 5. '5416. 7 -22083 8. 48-083. 9. ■2785493827100. 11. -5375. 12. -87916. 13. 4.90. 14. -15972. Ex. LXIV. (p. 107.) PAPER I. 3. Seventy tho^d S.-^rufttuTancr&ZnS dred ana twenty-five ^°^^J^,^X^iZA seven billions :rx'Sr^d°ri five S^s"^^^ »^-1'e<' -^ "^'^^ %SStrymmi 5- 1372»6^ ■■\. w ^9) V ^^S^^ .,.^ %*■' ? .^. > .h 4. £24. 15s. 5. 58 yards. 8. i of the orange. PAPER IV. 1. 60. 2. 84-875 or 84|. 3. •01230. , 4. $410.27^ 5. 21 on smaller side, 24 on larger side, and 72 lookers on. 6. One side scores 7 times as many runs as the other, and therefore that side wins. PAPER V. 2. 275.^. 3. 42 ,S-. 1. 12s. 6d. 6. $48.27i. 4. $19.90. 5. $5.92. 1. 27. 8. -36. 9. 1 6- 5. 1204. 0. 15. 7. I Ex. LXV. (p. 112.) 2. Of. 3. 15. 4. f. 10. 3-2. Ex. LXVI. (p. 115.) 1. $48. 2. $18.15. 3. $17.33i 4. 38 bus., 21^,- lbs. 5. 20 bus., 28f lbs. 0. £82. 2s. 8d. 7. 28 cwt, 3 qr., 14 lbs. 12 oz. 8. 44 cents. 9. 29 cents. 10. $812001 ,V 11. 21 cwt., 3 qr., 18 lbs., 12 oz. 12. $1638-40-i^/«^;. 13. 61^. 14. 2 mo. 15. 15s. 9f(Z. 10. £2075. 8s. 17. 3420 steps. 18. £4754. 10s. lO^d. 19. $5000.75. 20. 20 min. 21. 20 yds., 2 ft. 22. 528 pairs. 23. 171 men. 24. 3i cts., $57-812i. 25. $4.12^. 20. 135 men. 27. llhrs., 38 min. 28. $2234.31 29. 7id. 30. 12 days. 31. 5s. 6d. 32. $151.14Hi 33. £900. 34. (1) £1000 ; (2)£900. 35. £8. 14s. Hid. -^% q. 36. 30 days. 37. $1902.50^^ 38. 3s. 6d 39. £90. 40. 104 lbs., 2§ oz. 41. 11 hr., 52§'. 42. 5A' past 1 o'clock. 43. 3515625 cents. 44. 8tH days. 45. £4005. 46. £132. Os. 4f d i q. 47. 21 days. 48. 10^ hrs 49. 9i mo. 50. 12.30 P. M. 10 mi. from place. 51. 5AS''. Ex. LXVII. (p. 121.) 1. 8wks. 2. 112 men. 3. 04 days. 4. $307.44. 5. $87500. 6. 174 miles. 7. $202-50. 8. 200 horses. 9. 100 months 10. 2808 qrs. 11. 39ac.,l ro.,20po. 12. 9 mo. 13. 00 men. (cwt. =112 lbs.) 14. 91 men. 15. 2i days. 10. 45 men. 17. 178 qrs., 4 bus. 18. $1-008. 19. $7.20. 20. 4 days. 21. 2 days. 22. Vd^ days. 23. 3 lbs., 11 oz., 7^ dVs. 24. 25 horses. 25. 180 men. 20. 2.V ft. Amwms. 18y -^Z) 1. $168.75. .-). $849;175. J). $4965. 12. £n22.6s.2d. 15. $267911.871 18. $9611.25. Ex. LXVIII. (p. 124.) 2. $157.50. 3. $ir.92.50. 4. $1927.20. 6. $2396.35. 7. £416. 17«. 8. £600. 10. £0360. 5s. 11. £812. 17.9. 2^d. 13. £86663.1.9.9^. 14. £15560110.9. lUrf. 16. $715024.80. 17. $72562.35. 19. £2764. lis. 3d 20. £14. Is. UK Ex. LXIX. (p. 126.) 1. $66.50. 2. $167. 3. $149671875. 4. £11. lis. Sid. 5. £9. 18s. 3id 6. £350. 13s. 7id. 7. $125468.75. 8. $84.06i. 9. $173. 10. $98.60. 11. $477-5475. -^- 14. $15.69i^|. 12. $127.57-2^^. 13. $9.61-A. Ex. LXX. (p. 128.) 1. $17.38, $234.63. 2. $34.76, $252.01. 3. $11074875 $63812375. 4. $11.22, $104.72. 5. $13-25625, $8900625. 6. £17. 12s. 5H + , £80. lis. Sd. 7. $305-755, $1441-505. 8. $310.08, $994.08. 9. £111. 14s. 7-Arf, £7611. 14s. 7-Ad 10. £171. 9s. 9-94. . .d, £5037. Is. 2-94. . .d. 11. 6 years. 12. 8i. 13. £130. 14. £32; 5 fl., 3c., 0-078125 m. 15.4. Ex. LXXI. (p. 130.) 1. $115.92, $915.92. 2. $192.70, $934.70. 3. $341.88, $901.88. 4. $28.78, $336.78. 5. $103.61, $713.61. 6. $229.25, $1229.25. 7. (1) £1. Is. 6^^ •S8q., (2) £6. 19s. 2^d. ISQq, Ex. LXXII. (p. 133.) 1. $200. 2. $800. 3. $1200. 4. $209.53 + . 5. md. mg- _ . , 11. $95.23H. 13. £3. 4s. 6^. U^q. 14. 2isd. -^^h-q 16. 5 per cent. Ex. LXXIII. (p. 137.) 1. $416.79 + . 2. $780.48fi 3. $1524.88. 5. $15069. 6. $1391. 7. (1) £10. 16s. 4d; (2) £3. 4s". ll-^^/S 8. 6perct. per ann™. nearly. 9. Bank of Toronto. 10. £25. U. His income less by £64. \2s. 12. $1392^ 13. £240000 stock. 14. Loss of income = £45. 10s. 15. £52. 10s. 16. Increase of income = £135. 5s. W^d. t!ik. Ex. LXXIY. (p. 141.) 1. $5.37i. 2. $250-2903. 3. 29^^ cents. 4. $1900. C. (1) £6. 5s. ; (2) £18. 17.s. W. Ihq. 6. (1) $208 ; (2) $13.13 ; 6. £129. 6s. 9d 7. £179. 12s Ufd WM. 9. $42. 10. $2100. 8. £456. 9s. 12. $99.05fi 15. 4.j%m^d. 4. $37.15fi l9C)i AJ^swsns. {^) ^1.55. 7. (1) 10 per cent. ; (2) £9. U. 9K M. 8. $1.20. 9. ()(ij. 10. 5s. M. 11. 4s. l|a. i,q. 12. 40^^. 13. 78. Hid. i\q. 14. 18s. id. 15. £63. 12s. 8f f. i?(?. 16. $1.20. Ex. LXXV. (p. 142.) 1.90-83. 2.15.58. 3.83-67. 4. 8667.. .yrs. 5. 60i yrs. 6. 29046-813. 7. £191. 8«. Ex. LXXVI. (p. 145.) 1. (1) 224, 336, 448 ; (2) $40.62^, $89.37^, $130 ; (3) 66 ac. h^'k}5 P?;.'.^?.^^-' ^ ^^'^ 1^ P<^- 5 (4) £42. 17s. lief, fq., £38 P* ^fe/^i- ^*- ^*^- ^q-' ^l'''- 3«. lOK iq. 2. (1) i is tc o ^''^li% J?.^^?^' ^^^ ^ ^^^- (2) £136. 10s. hr, }S?J\F^J^^- ^^ oxygen, 939-136 lbs. of carbon, 176-4896 lbs. of hydrogen, (cwt.= 112 lbs.). 4. 33^ per ^7^"in.^l^^^^-.^- ^ ^^- ^- S "^<^' "^^ ^ ought to have $6400, B $840, C $720. 8. A ought to have received £700, and B £900. 9. A should pay $30, i? $18, and C $6. 10. 24 men. Ex. LXXVII. (p. 149.) . \\Y^'^ll'^ ?L ^- ^^5 ^^' 42- 3. 49; 87; 98. 4. HI; 200; 623. 5. 703; 703; 509. 6. 1111; 5343. 7. 7906; 5746; 7008. 8. 13509; OO. 9. -094; 21103 10. -025173; -00003. 11. 71414; 7141. 12. 2*2583; -2258. 13. 28-3992 ; 310-3304. 14. 577. 15. 166. 16. 2-175 17. U. 18. 2-625. 19. 540s. Ex. LXXVIII. (p. 152.) 2. 42; 75; 92. 3. 97; 103. 4.512:4-01. 6. 5079; 7420. 7. f. 8. 643. 9. 1-660. 12. -215. 13. 2-154. 14. '333. Ex. LXXIX p. 152.) PAPEB I. 1. ?; rem. 117257. 2. (1) 15tons,8 cwt.,3qrs.,171bs.,loz. (cwt. =112 lbs.) (2) 1 oz. Avoird.= j^f of 1 oz. Troy. 3. £34. 9 fl. 6 c. 8-2142857i m. 4. 1 ; 2520. 5. (1) 63. (2) I (3) 4 cwt., 3 qrs., 3 lbs. (cwt.=112 lbs.) 6. (1) 21060. (2) -00002106. ^ w PAPER II. 1. (1) $90. (2) £281. 7s. 5d. 2. Gain per cent.= $95 ; Loss per cent.= $20. 3. (1) 30502. (2) 266. (3) 67 yds., 4 in, 4. 118625 5. 2 x 2 x 3 x 3 y7 x 7 x 13. 6. 34'. 27;j\" past 6 o'clock p. m. 1. 12; 20; 18. 5. 76-3; -0587. 10. 1. 11. -464. AJfTSWEHS. i9) PAPER III. 1. 6. 2. (1) -10625 ; (2) '030416 ; (3) The first ; (■i) ij%. 3. $520. 4. 37f square yai'ds. 5. 13H 0. $22.37H- PAPER IV, 1. (1) Uf; H; (2) 123 times; '. 2. £139 155. 3. (1) £20 95. 6(/.. ; (2) 52|. 4. £82. 5. Turkey, 16s. 6d. i fowl, 2s. lOd. 6. lO^^V- PAPER V. 1. (1) 3011404; (2) 971472492; (3) 430709070, (4) 37834342650; (5) 75732561476. 2. 56c. 3. $126 4. $288.09. 5. $25.40. 6. 1 gal. 2 qts. 102 pts. PAPER VI. 1. 706-85775 feet. 2. 641203200000000 cubic feet 3. 8 ft. l^in 4. $2025. 5. ^,$750; £,$500 0, $250. 6. 18 days. Paper vii. 1. (1) 7117423255950; (2) 0593445483924; (3) 68677245810; (4) 27798027851538; (5) 81697259850030. 2. $7.50; $7.52. 3. $12.96. 4. 5000. 5. $868.57; $1709. 6. ^. PAPER VIII. 1. (1) 10836151080800; (2) 31808539707205; (3) 69154272; (4) 34175791448. 2. $8.40. 3. $1.62; $1.08; 60c. 4. $812. 5. 420 cents; 140 cents; 500 cents. 6. 31817108. PAPER IX. 1. (1) 7832 ; (2) 196734 ; (3) 3589853148 ; (4) 3627482760. 2. 52800 yds. 3. $20 to B; $10 to 0. 4. $600: $900. 5. £2190. 6. $3463.85. PAPER X. 1. 825. 2. $4.20. 3. 312^|. 4. -004125 '> $200. 6. 130. 7. $3000. 8. $509. 62^. 192 AmWEHS, PAPER XI. ^ ^« k' ^' ^' ^ , ^' ^^1^^- 3. 4 cents. 4. 01 0. 8 hrs. 6. T^. 7. 1120. 8. $101.85^. 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