IMAGE EVALUATION TEST TARGET (MT-3) V ^ A ^ a\ ^^^ f/. 4 1.0 I.I 1.25 2.2 m lAO 2.0 1.8 U ill 1.6 Photographic Sciences Corporation " bo«A on this subject.** \ !•«««-»• wx«- uA\^- ^^\^'h^' \ ^^ »• ^ "OJ". Brighton, Ont. "I have read Uiom through carefolly ana consiuer them the belt yorks onthe subject for the use of schools that I have ever seen. ^ 52 ?*!?*?^*'^y pleased with their simplicity of style and fair method of stating occurrences. I shall introduce them in the schoola li«^ and encourage their use in those of the vidnity.** « A 1 .1 * ^'^' ^^"H^* ^-^'^ I- ^' S.. Hcton. ^mti^l^S !F'«" my admiration of the very concise and systa- SSS^Xrwl?* ifi"?^«®^!°*.***'^P*®* ^^ **>«"' rendering them KSlnS* ^? Bnitable for students, and tending in no smaU degree to fftn'note a bettor system of pur suing this importaci study.** .. ,ST'a P'*"**^*^' ^^u' .^°ciP«l High School, Lindsay. «♦ Jj, r*^ *°»2®*' *^®"® ^"®f tsritOs authorised by the Minister of Education. They are clear ui# Bucdnct in statement MidSoVS «Wng the attention of the learner to definite periodr^an^ "« inabhng him to naaster difficult historical subjecUiwitlwut attamn^ ■•• tifeem in the bands of all my pupils." •"w««wi»yw Q. m J .i*: * ^J- «»,* ^mimmf TT !■• i/ i I MILLER'S SWp'rb^S LAHStTAfflS LlSiMS, rise new 4atSioHze4 Cii'aaiBiia% MILLER'S SWiHTOrS LAN&UAeE LESW BT J. A. McMillan, b. a. The only EdiUcn prmand a$ an Introductory Tmt BwOi to Mason'* Orammar. In Miller's Edition of Language Lesaoni IHe DeflBfttona oT the Pares of Speecb are uow made IdenUealwlOh miaeoii'e GraBimar. » « ««« wa»a The Olaaiafleatlon or Prononna, Verb*, Mooda. and Book *'«»»"»•"« ««8 t»»e aamo as In simmoES^SS ]II11]er>s Bdltlon is prei>ared as «n IntroduotoiT Text Book * ^S?"o iL***?**^*^' *^'* authorizod bool^ for adyanoo4 olasset for I^i^Wc Scho(rt8^ so that what is learned by a pupi^toam dlSS- tary text-book will not have to be unhwraed when the advaneed book to used, a aenous fault with many of the ^ed Public School BCoIdb. «.t?^^ Uj^ \ / t^ o _ ^ JT '^^ J '*■;{ /^ tf • * " " ' T~* ' l ^M. . -T t. i- i^^^^ = = /4 ■^ ' .' uf 1^ I •? IM -3 i1 ^ /^^ R E F A C E . r T^ / The importance of Arithmetic as a branch of in- struotion is universally admitted ; but, until a com- paratively recent period, the results of teaching it were very unsatisfactory, and not at all commensu- rate with the time usually devoted to it in our schools. This was not owing to any inherent difficulty in the subject itself, but to the method of teaching it. The rule was stated first, an example illustrating the rule followed, and the reason of it came last. Now exactly the reverse of this is adopted by all good ^ teachers. The examples and illustrations precede and lead up to the enunciation of the rule, whenever a rule fc IS considered necessary. But while the method of ^ -»— teaching Arithmetic has undergone a complete change ^ V ^ no corresponding change has taken place in our Mo- mentary text-books. To remedy this defect the fol- lowmg pages have been written. We would call attention to the general features of the work : 1. The Unitaky System — In aU our best schools this system has already superseded the cumbrous and illogical methods of our ordinary text-books Its advantages ai-e so great that it must soon become universal. It has been defined as a method of solving arithmetical problemsindependently of rules by reason- ing out each step of the solution from some previous one, until by a series of deductions, the result sought IS obtained. This system traius the pupils to habits ot neatness, exactness, and to logical habits of thought ^ but its chief advantage is its extreme sim- plicity, dispensmg with set rules, and enabhng the pupil to solve problems in Simple and Compound i-^l '7 k N>i 'f c/, ^\ V ■ ^ ^ ; V... N n PBSFAOK. ages, Profit and Loss, Partnership, Ac, by one uni- form, elegant, and simple process. 2, Abkanoement. — The different subjects have been arranged with reference to then- importance and their simphcity ; the less difficult and more practical first, and the more intricate and less important afterwards. Thus, problems in Canadian Money, Bills, &c., have been introduced immediately after Division, as being of greater importance than any other subject, within the range of the pupils' ability, at that stage of their progress. Each chapter is divided into sections with the same idea running through the section. This will oblige the pupil to confine his attention to one thing at a time, and thus, enable him the more easily to master it. 8. Oral Exercises. — Each subject has been elucid- ated by Oral Exercises leading up* to written work. This arrangement will assist the pupil in arriving at the reasons for the methods employed, and, to a certain extent, make him the author of his own definitions and rules. It has other advantages. Quickness in cal- culation is in itself an important object to attain. This can never be attained by mere slate-work in which the pupil is allowed to think at leisure at every step. 4. Rules. — The Rule is given as a convenient sum- mary of the methods employed in the solutions of the examples which precede it. The aim has been to lead the pupil to derive his own methods of operation. 5. Exercises. — Special pains has been taken in framing and selecting the exercises for the different sections in order to obtain such as will not only evolve thought on the part of the pupil, but more especially prepare him for the business relations of life. The authors will be pleased if these pages assist their fellow-teaahers in bringing this important sub- ject before their pupils in a manner more interesting and instructive than it has been hitherto. Toronto. May. 1878, ■\" CONTENTS. CHAP. I.— SIMPLE RULES. i Sbotxon I. — Definition, Notation and Numeration .... 1 Skotion n.— Addition 7 SscTioN m.— Subtraction X6 Skotion IV.— Multiplication 24 BsOTiOM V. — Division 34 CHAP. II.— CANADIAN MONEY. 65 CHAP, m.— MEASURES AND MULTIPLF3 Sbotiok I. — Prime Numbers, Prime Factors, &g 67 Sbotion II. — Cancellation 68 Section HI. —Highest Common Factor 69 Sbotion IV. — Least Common Multiple 72 CHAP. IV.— FRACTIONS. Bbotiom I. — Definitions 7g SacnoN II. — Reduction of Fractions 79 Bbotion 111. — Addition 8g Section IV. — Subtraction 83 Section V. — Multiplication 91 Section VL — Division 95 Section VII.— Complex Fractions 93 CHAP, v.— DECIMALS. Section I. — Definiticns 104 Section U. — Addition 1Q7 Section III. — Subtraction lOg Section IV.— Multiplication 109 Section V. — Division , ^ ^ ^ ^ 1 1 q Section VI. — Reduction of Deciles m Section VH.- Oiroiilating Decimals .,,,,,,__,' 119 VllL CONTENTf. CHAP. VI.- -OOMMERCIAL AlUTHMETIO. SxoTTOM I.— Tables and Reduction 117 Skotion n. — Compound Addition 124 Section III. — Compound Subtraction 126 Shotion IV.— Compound Multiplication 12f Section V. — Compdund Division 127 Sxci'ioM VI. — Denominate Fractions 128 Section Vn. — Practice 181 CHAP, vn.— AVERAGBB AND PERCENTAGES. Section I. — Averages IhS Section II. — Percentage 136 Section III.— Inauranco 136 Section IV.— Commission and Brokerage 138 Section V. — Interest 139 Section VI.— Present Worth and Discount 142 CHAP. Vni.— SQUARE ROOT. 145 CHAP. IX.— MEASUREMENT OP SURFACES AND SOLIDS. Section I. — Rectangles 147 Section II. — Carpeting Rooms 147 Section III. — Papering Rooms 148 Sbotion IV.— Measurement of Solidity 149 Miscellaneous Problems 160 Answers ]69 ELEMENTARY ARITHMETIC ON THa UNITARY SYSTEM. CHAPTEiv, I. Section l.~Deflnitions and Notation and Numeration. 1. Arithmetic is the science of numbers and the art of computing by them. 2. A Unit is a single thing ; as a man, a dog, a hall, &c. ' 3. A Number is a unit or collection of units ; one dollar is a unit ; Jive dollars is a collection of units. 4. In common arithmetic, all numbers are expressed by means of the significant figures, 1 q 3 4 5 6 7 .8 9 called one, two, three, four, Jive, six, seven, eight, nine, and the figure 0, which is called a cipher or zero, and which has no value in itself. 5. Numbers are considered as being either Ab- stract or Concrete. A Concrete Number is one applied to a particu- lar unit ; as 6 men, 6 horses, 9 dollars. An Abstract Number is one not applied to any particular unit ; as 8, 6, 8. t A. Si'TT'tlar Numbers are such as have the same mit; as 6 boys, 8 boys, 10 hoys. 1 ELBMENTABY AKITHMETIO. Exercise i. 1. How many units in 5 ? In 5 books ? In 9 pencils ? 2. What is the unit of 6 ? Of 5 books ? Of 3 balls ? 8. State which are ibstract and which concrete of the following numbers : 6, 7, 8 books, 9 men, 3, 4, 5 apple?, 2, 1 cont. 4. What is the unit of 8 miles ? 9 miles ? 7 ? 6 cents 1 6. Which are the similar numbers in the followin«T rr«nr.L will vmi lin.vA vfimaininfir ? 6. How many are 6 apples less 8 apples, 7. How many are 6 less 8 ? 6 leas 4 ? • liV TT 16 ■LEMENTABT ARITHMETIC. SUBTRACTION TABLE. f 1 2 8 4 5 6 7 8 9 10 1^ 1 1 1 1 1 1 1 1 1 1 1 2 8 4 5 G < 8 9 f 2 3 4 5 6 7 8 9 10 11 2 2 2 2 2 2 2 2 2 2 2 ^mtmmmtm 1 2 3 4 5 6 7 8 9 f 3 4 5 6 7 8 9 10 11 12 8^ 3 3 8 3 8 8 8 3 8 8 1 2 3 4 5 6 • 8 9 r 4 5 6 7 8 9 10 11 12 13 4h • 4 4 4 4 4 4 4 6 4 7 4 8 4 9 1 2 ■ » 3 4 5 f S 6 7 8 9 10 11 12 13 14 6 5 5 6 5 5 5 5 5 7 5 8 5 9 1 2 3 4 5 6 f 6 7 8 9 10 11 12 13 14 15 6- 6 6 6 6 6 6 6 6 6 6 1 2 3 4 6 6 7 8 9 f 7 8 9 10 11 12 13 14 15 16 7- 7 7 7 7 7 7 7 7 7 7 1 2 3 4 5 6 7 8 9 r 8 9 10 11 12 13 14 15 16 17 8- 8 8 8 8 8 8 8 8 8 8 8 9 « 1 2 3 4 4 6 7 f ^ 10 11 12 13 14 15 16 17 18 9 9 9 9 9 3 9 _4_ 9 ^6 9 6 9 9 9 1 2 __7_ 8 9 Oral Exercises. 1. Suhtract by 2*8 from 100 to 2 ; thus, 2 from lOOIeavM 08, 2 from 98 leaves 96, and so on. 2. Subtract by S's from 100 to 1 ; by 4*8 from 100 to 0, 8. Subtract by 4"'8 from 95 to 3 ; by 5'8 from lUO to 0. 4. Subtract by 6'b from 100 to 4 ; by T's from 100 to 2, 6. Subtract by 7'8 from 99 to 1 ; by S's from 100 to 4, SUBTRACTION 17 6. Subtract by 9'i from 100 to 1 ; by 9*s from 99 to 0. 7. Couui by 4'8 from 8 to 89, and back agam to 19. 8. Count by 5'8 from 6 to 66, and back again to 26. 9. Count by 7'8 from 18 to 68, and back agam to U. , 10. Count by 8's from 25 to 65, and back again to 1. 11. Jane is 11 years old, and Mary is 7 years younger; wliat is Mary's age ? , ., • j o 12. A grocer sold tea for 10 dollars, and thus gam«d B dollars ; what did the tea cost him ? 13. If I buy cloth for 7 dollars, »t what pnoe must 1 sell in order to lose 4 dollars ? „ - , i 14. John has 11 dollars ; he pays 2 dollars for books, and 3 dollars for a hat ; how much money has he left ? 15. Mary has 9 dollars ; she pays 7 dollars for a dress, and then earns 3 dollars more ; how much has she now ? 16. A boy having 12 apples, bought 6 more, and then sold 8 ; how many had he left ? , , ,, 17. James had 6 dollars, he earned 6 dollars more, and then spent 6 doUarB ; how much did he then have ? 18. A merchant gave 8 dollars for a certidn article, and paid 4 dollars for carriage ; at what price must he sell to gain 8 dollars ? •^5. Finding the difference between two numbers is called Subtraction. *^0. The number found by taking one number from another is called the Difference or Remainder. ^ '^1. The number from which the other is taken is called the Minuend. •^8. That which is taken fi'om the Minuend is caUed the Subtrahend. , '40. The sign of subtraction, --, is called Mtntu, and when i^laced between two numbers shows that the one on the right of the sign is to be taken from the one on tlie left of it. Thus, 6-2 is read 6 mmus 2, and means that 2 is to he taken from 6. ;iO Principle.— Only similar numbers can be subtracted; thus, 4 boys from 7 boys; 6 cents from 8 cents, &c. ;^l. Subtraction may bo divided into two cases : 1. When nojigure of the subtrahend is yreaier ihan its corresfmndixKj fi()ure of the minuend, t. When a figure of the subtrahend is greater than its corresponding finurt' of the minuend. 18 ELEMFNTAET AKITHMBTICt Case I. 32, To subtract when no figure of the sub- trahend is greater than its corresponding figure of the minuend. Ex. I. A grocer bought 678 oranges, and sold 835 of them ; how many had he left ? 678 oranga 835 " 843 «« Here we are required to find the diffirenct between 678 and 835. We write the less number under the greater, placing units under units and tens under tens. Beginning with the units we say : 5 units from 8 units leave 8 units, and we set the 3 in the units' column below. Then 3 tens from 7 tens leave 4 tens, and we set the 4 in the tens' column. Lastly, 3 hundreds from 6 hundreds leave 8 hundreds, and we set the three in the hundreds' column. Hence we have as the whole remainder 3 hundreds 4 tens »nd 3 anitb, or 343. Exercise x. (1) 625 812 (2) 456 215 (3) 763 512 (4) 617 215 (5) 767 123 (6) 896 432 (7) 279 136 (8) 807 502 (9) 706 462 (10) 736 432 ai) 967 234 (12) 875 345 — — — ■ — — (18) 8763 4321 (14) 9076 4054 (15) 8769 1546 (16) 6076 8075 (17) 4872 2342 (18) 7659 8237 (19) 8769 ^257 (20) 4876 2142 (21) 8275 3251 (22) 8799 2542 (23) 8591 7230 (24) 6857 lii34 (25) 784 861 (26) 82845 22121 (27) 57596 21321 (28) 72678 41362 (29) 27397 22315 (30) 67385 24128 BUBTHAOTIOM w (81) (32) (33) (84) (85; 67897 67858 87678 96754 81296 21472 32721 21335 (88) '. 1428 20135 (86) (37) (89) (40) 263786 472589 87695 66728 98786 218123 212324 23542 21306 21342 (41) (42) (48) (44) (45) 873967 873972 72587 95837 89976 212851 132421 51234 51321 82742 46. 814 from 678. 51. 1235 from 8768. 47. 425 from 668. 52. 3726 from 4969. 48. 561 from 789. 53. 2532 from 8748. 49. 254 from 576. 54. 4720 from 87856. 50. 437 from 869. 55. 12345 from 68799. Exerc dse xi. Practical Problems. 1. In a school of 74 pupils, 31 are boys; how many girls are there ? 2. A girl had 75 cents and paid 81 cents Tor a slate ; how many cents had she left ? 3. A man bought a horse for 98 dollars, and sold it for 82 dollars ; what did he lose ? 4. Two parties played a game of base ball and made 87 runs. One party made 68 runs ; how many did the other party make? 6. Jane and Susan together answered 87 questions in geography. Jane answered 43 of them ; how many did Susan answer? 6. A gentleman bought a buggy for 225 dollarg, and sold it for 268 dollars ; what was his i^rdff? 7. A man bought a horse for 265 dollars, and sold it for 282 dollars ; how much did he lose ? 8. A man deposited 5237 dollars in the bank ; h« after- wards drew out 8125 dollars ; how much remained ? i~hmmr\^^ M "d. A. man uymg leii zioa'k aoiiars w ms son ana ma daughter. The share of the son was 13452 dollars ; what was the daughter's share ? ELEMENTARY ARITHMETIC. Case II. 33. To subtract when a figure in the Sub- trahend is greater than its corresponding figure in the Minuend. Ex. 2. From 522 dollars subtr-act 286 dollars. 622 dollars. 285 ** 237 41 We begin at the right, but as we cannot take 6 units from 2 units, we borrow 1 ten from the 2 tens, and adding the 1 ten, = 10 units, to the 2 units, we have 12 vm,its. Then 6 units from 12 units leave 7 units, which we write under the units' column. Now, as we borrowed 1 tsn from the 2 tens, we left only 1 ten. As we cannot take 8 tens from 1 ten, we borrow 1 hundred from the 5 hundreds^ and considering the 1 }mndred borrowed as 10 tens, we add it to the 1 ten, making it 11 tens; then 8 tens from 11 tens leave '6 tens, which we write in the tens' column. Now, as we borrowed 1 hundred from 6 hun- dreds, we left only 4 hundreds : hence we say, 2 hundreds from 4 hundreds leave 2 hundreds, which we write in the hundreds' column, making the remainder 2 hundreds 8 tens and 7 units, or 237. There is another method of performing subtraction, which depends on the following principle : The difference between two numbers renuiim the same when each of them is increased bxj the same number. For example, 5-2=8. Now, if we add 10 to each, we have 16-12=8, as before. In Ex. 2, if we ad^^ units to 2 units we have 12 units. Then 6 units from 12 units leave 7 units, which we write m the place of units. Now, as we added 10 units to the mmuend, if we add an equal number to the subtrahend the difference will remain the same. But 10 units = 1 ten. Adding 1 ten to 8 tens wo have 9 tens : and as we cannot take 9 tens from 2 tens, we add 10 tens, thereby makmg 12 tens ; then 9 tens from 12 tens leave 8 tens, which ws write in the place of tens. Since we added 10 SUBTRACTION. dl g tenfl to the minuend, we must add an equal number to the Bubtrahend, in order that the difference may remain the same. But 10 tens =» 1 hundred. Adding 1 hundred to 2 hundreds we get 8 hundreds ; and taking 8 hundreds from 5 hundreds we get 2 hundreds, which we write in the hundreds' place. This is the method usually employed. 34. PROOF.— Add the remainder to the subtrahend ; the sum will equal the minuend if the work is correct. Exercise xii. (1) 578 248 (2) 748 376 (8) 885 573 (4) 968 676 (5) 839 584 (6) 638 394 (7) 659 475 (8) 839 688 (9) 547 284 (10) 658 872 (11) 735 878 (12) 848 589 (18) 524 856 (14) 762 887 (15) 845 579 (16) 807 138 (17) 456 887 (18) 450 382 (19) (20) (21) (22) (23) (24) 854 948 607 500 704 408 896 766 809 825 507 286 (25) (26) (27) (28) (29)- (80) 726 857 785 792 807 650 887 389 558 295 828 357 (31) 8876 2379 (82) (33) (34)^ (35) (86) 6385 6735 407f|P 4070 4185 3627 2547 8128 2137 1216 (88) (39) (40) (41) 6283 8175 2534 6735 2426 2836 1235 6376 ELEMENTARY ARITHMETIC. li ,' (48) (44) (45) 8522 7135 6347 6243 1872 2563 (49) 85672 23828 (50) (61) 43763 87253 24235 34365 (46) 8185 2453 (62) 73875 38376 (47) 7345 2876 (58) C3527 14238 (48) 4372 2583 (54) 53418 28401 Exercise xiii Practical Problems. 1. A horse was bought for 125 dollars, and sold for 117 dollars ; how much was lost by the sale ? 2. A roll of carpet contained 156 yards, but 79 yards were sold from it ; how much remains ? 8. A house cost 5440 dollars, and was sold for 6000 dollars ; how much was the gain ? ^' 1,^ ^^"^ o^'®^ '"^ ^^'^^ ^^ *^® age of 76 years ; when was he born ? 6. A town which ten years ago had a population of 3746. has now a population of 6996 ; what is the gain ? f went to a store and bought a knife for 66 cents, and gave the storekeeper a four dollar bill (400 cents) to pav tor It ; how much change did he give me back '> 7. Two little girls picked 74 quarts of blackberries one ot^^'Tck ? ''''^ ^ ^'^ *^'''^^*^* ^""^ "^^""^ ^^^''*^ ^^^ *^« 8. Mr. Kobinson's horse and carriage cost four hundred dollars; what did the horse cost, if the carriage cost two hundred and twenty-five dollars ? 9. Suppose a man's income is 2453 doUars a year, and his expenses are 1947 dollars, how much can he save in a year / 10. Smith bought of Brown 875 acres of land for 23400 dollars. For 500 acma of the tract he paid 11379 dollars • how many acres wei!»i the remainder of the tract ? And tor what sum was it purchased ? Addition and Subtraction. Exercise xiv. Find the result of 1. 768+276-369+284-782. 2. 869+784+468-266-368-248. SUBTRACTION. 28 L7 )0 in 5, d y le d o d a [) » i 6. 4. 5. 6. 7. 1764-8394-7864-724- 868-256. 136 - 769 - 284 + 968 + 268 + 372. 269-1846+368-2744-2976 + 769. 769 + 785 + 368 - 784 - 369 - 249. .. 1869-2846 + 362-4894-3007+249. 8. 2845+3624-78695+784+93768. 9. 7369-245-12456+85769-2572. 10. 3004 + 2006-5008-3604 + 7200. Exercise, xv. Practical Problems. 1. A man owing 1369 dollars, paid at one time 264 dol- lars, and at another 748 dollars ; how much did he still owe ? 2. A man bought a farm for 6780 dollars; he spent 1876 dollars for improvements and 977 dollars for stock. He then sold the whole for 9000 dollars ; did he gain or loB©, and how much ? , - . . 3. The sum of fcur numbers is 986287 ; the tot w 23789, the second is 11892 less than the first, the third is 85416 more than the second ; what is the fourth ? 4. What number increased bv the difference between 1458 and 2862 will make the sum of 8641, 789 and 7008 1 6. A collector received 1200 dollars from four men ; from the first he got 852 dollars, from the second 67 dollars more than this, and from the third 94 dollars less; how much did ho receive from the fourth ? 6. At an election, in which there were two candidates, the whole number of votes was 3694 ; the defeated candi- date received 1369 votes ; what was the majority ? 7. A boy shot an arrow up the road 173 feet and an- other down the road 234 feet ; his little brother brought them to iiim ; how far did he walk to get them ? 8. John and James play marbles, John has 24 at the beginning and James 36. The first game John wins 4, the next he winy 6, the next he los|| 5, the next he loses 3, the next he wins 2 ; how many marbles has each now ? 9. Find the final remainder in subtracting 64868 as many times as possible from 476209. 10. From the difference between 576 and 7862, take the i1itf/^..AV^/>a Kaf-nraon 1 Q1 01 n.nri 184.58, VliiitJi OiiV- 11. The sum of two numbers is 8764 ; the difference of the same two numbers is 1668 ;' what are the nnmbers ? i « BLEMENTART ARITHBfETIO. Section IV.-Multiplication. 1. Ihere are 6 oranges in each of three dishes • how many are there altogether ? 6 and 6 and 5 are how many? Three 5's or three times 5 are how •7 * 2. If there are 8 herries in one cluster, how many berries are there in 6 clusters ? 8 + 8 + 8 + 8 -<.S are how many ? t -r -r o -r o 8. There are 8 feet in one yard, how many feet are therem2yaa:ds? In 4 yards? In 6 yards ? 4. ihere are 6 workmg days in 1 week, how many a^alllfla^^ ^t'^ ^^ 2 ^^«^«? ^ ^ weeks ? o + o + b-+b + b are how many ? 5. What will 8 hats cost at 2 dollars each ? Since 1 hat costs 2 dollars, 8 hats will cost 2 + 2 + 2 ^ToreLiir;. ' '""''^''^ °^ ' ^°"^- ^-- ' ^^ts 6. If John walks 8 miles an hour, how far will he go m 4 hours ? 7. If a First Book costs 8 cents, what wiU 6 First Uooks cost ? 8. What will 4 buns cost at 2 cents each 9. If httle James takes 2 steps in a yard, how many steps wiU he take in going 6 yards ? 10. John bought 4 tops at 8 cents a piece, how much money did he spend ? ^'5. When any number is to be added to itself two or more times the work may be shortened by a pro- cess called Multiplication. ^ i. ?a?i'7.u^''^"'^i'" ''^^'^^ ^ ^^^^ *^^ Multiphcation IS called the Product. ,?T- The number to be added several times to it- self is called the OTiltiplicand. 38. The number denoting how many times the Mul^phcand 18 to be taken is called the Multiplier. rf» ihe Sujn of Multiplication is formed by two flUort hues crossing each other slantingly; thus, x . X. saowsthat the second of the two numbers be- tween which it is placed is to be multiplied by the nret, thus 4 times 8 is written 4x8. MULTIPLICATION. 25 ^ 40. Principles. — 1. Tlic Multiplicaud may be either an abstract or a coucreto number. The multiplier must always be regarded as an abstract number. 2. The Product is always of the same kind as the Multiplicand. Thus 8x8 cents are 9 cents ; 2 X 5 boys are 10 boys. MULTIPLICATION TABLE. n^^TinA Tliroe F our Five Six 1 Seven. ' , times. times. timos. times. times. 1 is ?. i 1 is 8 1 is 4 1 is 6 1 is 6 1 is 7 & ... 4' 2 ... 6 2 .. 8 2 .. 10 2 ... 12 2 ... 14 «... 6i 3 ... 9 8 ... 12 8 ... 16 3 ... 18 8 ... 21 ♦ .. 8i 4 ... 12' 4,.. 16: 4 ... 20 4 ... 24 4 ... 28 5 ... 10 ; 5 ... 15 5 ... 20 1 5 ... 25 5 ... 30 6 . 36 6 ... 12 6 ... 181 6 ... 24 6 ... 30 6 ... 36 6 ... 42 ? ... 14 7 ... 21 ! 7 ... 28 7 ... 85 7 ... 42 7 ... 49 8 ... 16 8 ... 24 8 ... 32 8 ... 40 8 ...48 8 ... 66 n ... 18 9 ... 27 9. ..36 9 ... 45 9 ... 54 9 ... 63 10 ... 20 10 .. so! 10... 40 10 ... 60 10 ... 60 10 ... 70 11 ... 22 11 ... 33' 11 ... 44 11 ... 65 11 ... 66 11 ... 77 12 ... 24; 12... 86! 12 ... 48 12 ... 60 12 ... 72 12 ... 84 Eight 1 Nine Ten Eleven Twelve times. 1 timos. timos. times. times. 1 i» 8j 1 is 9 1 ia 10 1 is 11 1 is 12 2 ... 16! 2 ... 18 2 ... 20 2... 22 2 ... 24 S ... 24 1 B ... 27 8 ... 80 8 ... 88 8 ... 36 J 4 ... 32 i 4 ... 36 4 . . 40 4... 44 4 ... 48 5 ...40 6 ... 45! 5 ... 60 6 ... 55 5 ... 60 6 ... 48! 6 ... 54 6 ... 60 6 ... 66 6 ... 72 7 ... 56 1 7 ... 63 7 ... 70 7 ... 77 7 ... 84 8 ... 64 8 ... 72 8 ... 80 8 ... 88 8... 96 9 ... 72 9 ... 81 9 ... 90 9 .. 99 9 ... 108 10 ... 80 10 ... 90 10 ... 100 10 ... 110 10... 120 11 ... 88 11 ... 99 11 ... 110 if... 121 11 ... 132 12 ... 96 12 ... 108 12 ... 120 12 ... 182 12 ... 144 Oral Exercise. 1. Multiply b\ 2 from 1 to 12 ; by 3 from 1 to 6. 2. Multiply by 4 from 3 to 9 ; by"5 from 12 to 4. 8. Multiply by 6 from 3 to 10 ; by 7 from 12 to 6. 4. Mul til )lv 8 frc >m '. L2to2 : bv< 9 fro EU 1 to 11. 26 ELBM£NTABT ARITHMKTIO. ! 6. What will 5 hat? cost at 7 dollaro eachf If 1 hat costs 7 dolJais, 5 Lata will cost 5 timof 7 dollarB, or 35 dollarH. 6. What will 4 pairs of boots ooct at 6 dollars a pair f 7. A sheep costs 7 dollars ; how much will 6 sheep cost at the same rate ? 8. How much will 12 tons of coal cost at 6 dollare a ton? 9. A man can earn 4 dollars a day ; how much can ho earn in 9 days ? 10. At the rate of 8 marbles for a cent, how many can be bought for 12 cents ? 11. What will 7 lead pencils cost at 7 cents apiece ? 12. If a man cuts 8 cords of wood in a day, how many cords can he cut in 12 days ? 18. If 8 men can do a piece of work in 7 days, how many days will it take one man to do it 1 14. How many dollars will buy 9 tons of hay at 12 del- lars a ton ? 16. In an orchard there are 12 rows of trees, and 11 trees in each row ; how many trees arc .there in the orchard ? 16. If a quantity of provisions will last 8 men 7 days, how many days will the same provisions last one man ? 17. If 12 bushels of apples are picked from each of 8 trees, how many bushels are picked from all ? 41 . Multiplication may be divided into two case : 1. When the Multiplier does not exceed twelve. 2. When the Multiplier exceeds twelve. Case I. 43. When the Multiplier does not exceed twelve. Ex. I. — How many are 4 times 87 boys ? ist Operation. 2nd Operation. 87 boys. 87 boys. 87 *' 4 87 ** . 87 " 848 boys. f^ni ^AQ. k/^1TC In the first operation we find the result by Additimi. In the second operation, which is much shorter, we m MULTIPLICATION. 8V write down 87 once^ and we put 4, tlio number of timeii it IB to be taken, under the units' figure of the Multiplicand. We then begin at the right hand aide to multiply by 4; 4 times 7 units are 28 units, or 2 tens and 8 units. We write the 8 unit« under the units and add the 2 tens to the product of the tens. Wo next take 4 times 8 teni. 4 UraoB 8 tens are 32 tens and 2 tens make 84 tens, or 8 hundreds and 4 tens. Then we write down 4 in the t^ne' plaoo and 8 in the hundreds" place. Exercise xvi. (1) (2) (8) ' Multiply 7482 8432 72812 By 2 2 8 89 boyg 6 47 oents. 6 (7)' 187 7 fm. ' _ .—.I (8) 186 applet. 8 (4) 92128 4 (9) 284 girlg. 26. 6742 by^S. 26. m4yi^ 9# 2t. 61 Multiply 10. 316 by 6. 11. 480 by 7. 12. 614 by 6. 13. 7842 by 8. 14. 6848 by 7. 16. 8742 by 6. 16. 9764 by 8. 17. 8978 by 6. 18. 14068 by 6. 19. 18007 by 4. 20. 82709 by 8. 21. 21876 by 7. 22. 70095 by 9. 23. 68799 by 6. 24. 71878 by 9. Exercise xvii. ^ Practical Problems. 1. What wiU 4070 lemons cost at 4 cents each ? 2. What will 87086 oranges cost at 5 cents each ? 8. A man paid 887 dollars for a house ; how muish should he give for 7 such houses ? 4. What will 8048 pairs of boots cost mi 6 dollars » pair? 28 ir.EMENTART ARITHMETIC. 6. There are 56 sheep in one flock ; how itiftny sheep are there in 6 tuch flocka ? What if the value of each tlook at 7 dollars a head ? 6. A man bought 884 poundg of cngar ; he sold 290 pounds ; how much had ho left ] How nriuch did he re- ceive for what he sold, at 9 ccntu a pound ? What i« the remainder worth at 8 cents a pound ? At 7 cunts a pound ? 7. A merchant sold 878 kegs of nails at 9 dollars a keg ; 82 hundred woiglit of iron at 7 dollars a hundred weight ; what did each of the articles come to ? What did both come to ? Ho paid away 1889 dollars ; how much money had ho left ? 8. I have a book with 220 pageo, there are 6 paragraphs jn each page ; there are 9 linfjH in each paragraph ; t*li*ro are 8 words in each line ; there are, on an average, 5 let- ters in oach word ; how many paragraphs are there in the book ? How many lines ? How many words ? How many letters ? 9. A grocer sold 37 pounds of rice at 8 cents a pound ; 46 pounds of sugar at 9 cents a pound ; what did the rice come 10 *? What did the sugar come to ? What did both come to ? What did one cost more than the other? 10. A man bought 187 pints of chcstnute at 8 cents a pint ; 246 pint& of peanuts at 9 cents a pint ; what did each cost ? What did both cost ? How much did one cost more thp^n the other ? 43. To multiply by the factors of a number, 44. The Factors of a number are those niimbera which multiplied together will produce it. Thus, 8 and 5 are the factors of 16. Ex. 2. Multiply 742 by 86. 86 = 6x6, or 9x4, or 12x3. 742 742 742 742 86 6 9 12 4452 2226 4452 6 26712 6678 4 8904 8 26712 26712 26712 26712 It is thus seen that the Mnitiplicaud multiplied by tue Multiplier, or by any set of factors into which it can be B».parated, gives the same product. MULTIPUOATION. »i Multiply 1. 478 by 25. 2. 976 by 42. «. 187U by 08. 4. 1862 by 49. 5. 8936 by 54. 6. 4729 by 72. Exercise xviil. 7. 2345 hy 81. 8. 8764 by 04. 9. 2978 by 45. 10. 8475 bv '18. 11. 7649 by 24. 12. 9365 by 144. 18. In one wile there are 1760 yards, how many vardi aro there in 56 miles ? y y '^' 14. If Bound travels 1142 feet in one second how far will it movp in one minute or 6U seconds. 15. Wiuit will 72 bushels of wheat cost at 118 cents for one bushel I 16. If 27 men can do a piece of work in 17 days, how lonj; will it take one man to do the Hamo work ? 17. What is the cost of 24 horses at the rate of 125 dollars each ? 18. If a yoke of oxen costs 186 dollars, what will 68 yoke cost ? 19. If a man spends 945 dollars in a year, how much will he spend at the same rate in 21 years ? 20. There are 1440 minutes in a day ; how many min utes are there in 28 days ? Case II. 45. When the Multiplier exceeds twelve. Ex. 3. Multiply 479 by 67. 479 57 J isf partial product 8358 = 7 times the Multiplicand. 2nd " " 2895 =50 (( i( (( ti it n Entire «« 27808 =57 Since 57 is composed of 7 units and 5 tens or 50, 57 times the numJ)er must be equal to 7 times the number, plus 50 times the number. 7 times 479 ib 3853, the first times 479 and then multiplying this result by 10. 5 timei 479 IS 2395 and 10 times 2395 is 23950, the second partial product. We write thia ur.der the tirst pioduot so that 11 80 CLBMENTART ABITHMETIO. units may come under units, tens under tens, &o., anci then we add the two partial products together. In actual practice we always omit the and rite the second partial product as above. 46. P2200i^^.— MuUiply the Multiplier by the Multiplicand. If the product is the same as before the work is likely to be correct. Exercise xix. Multiply 1. 744 by 636. 2. 895 by 336. 3. 972 by 243. 4. 825 by 682. 6. 973 by 745. 6. 84G2 by 781. 7. 9643 by 683. 8. 8532 by 763. 9. 8984 by 133. 10. 4659 by 886. 11. 28352 by 845. 12. 41678 by 287. 13. 34073 by 435. 14. 40735 by 628. 15. 29304 by 789. 16. 90705 by 897. 17. 43445 by 678. 18. 37436 by 835. 19. 88888 by 789. 20. 23567 by 597. 21. 6484 by 6372. 22. 7856 by 3375. 23. 6748 by 6334. 24. 4878 by 3437. 25. 8547 by 7733. 26. 85474 by 2547. 27. 46887 by 3489. 28. 56184 by 5474. 29. 56664 by 4871. 30. 25473 bv 4487. 31. 73519 by 4735. 32. 81897 by 8456, 33. 21346 by 31452. 34. 47309 by 45233. 85. 25737 by 63252. 36. 43629 by 28516. 37. 10786 by 31672. 88. 47396 by 73462. 89. 76448 by 54178. 40. 28354 by 31867. 47. To multiply when the Multiplicand, the Multiplier, or both, contain ciphers. Ex.4. Multiply 2479 by 4006. 2479 _4006 14874 9916__ '9960874 4006 times 2479 equals 4000 times 2479 plus 6 times 2479. 6 times 2479 is 14874 ; 4000 times 2479 is 9910000. These partial products are written one being omitted. Multiply 1. 41o by 2. 7004 by KUT.T1FLI0AT10N. Exercise xx. 81 807. 902. b. 1'.64 by 6004. 4. 2709 by 708. 6. 9006 by 7036. 0. 1684 bv 4008. 7. 2002 by 4103. 8. 3678 by 7003. 9. 9999 by 8008. 10. 3674 by 200901. Ex 5. Multiply 614000 by 700. This result is the same as that ob- 614000 tained by miiltiplyiug 614 by 7, and 700 thon annexing to the right Jive noughts, which is the sum of the 429800000 number of noughts to the right of both the multiplicand, 614, and the multiplier, 7. Exercise Find tlie value ' xxi 1. 0; 743x600. . 2. Of 847X700. 8. Of 9642x6300. I 4. 0/1875x6340. | 6. Of 27X9000. i 6. Of 6000X13. 7. Of 18000x623. 8. Of 6400x640. 9. Of 650x650. 10. Of 83600x7500. 11. Of 9230X7000. 12. Of 8000X61000. Exercise xxii. Practical Problems. ; 1. In 1 rpam 01 paper there are 480 sheets. How many nhe4>trt are there in 947 reams ? 3. If ^ «otiou mill manufactures 637 yards oi clotb in •. day, how many yards will it make in 307 days ? o. At 12!) dollars each what will 49 horses cost ? 4. A merchant bought 29 pieces of cloth ; in each piece therft wer« 57 yards ; how many yards were there in the whole ? 6. If 19008 pounds of hay are required for the horses of a cavalry regiment for one day, how many pounds will be needed for 200 days ? 1^j6. What would be the cost of constructing 809 miles TTm plank road, at 3975 dollars a mile ? ■ 7. How many apples will an orchard containing 208 trees produce, if the average yield is 1269 apples for each iff 89 ELEMENTARY ARITHMETIC. / V / I < 8. In 8 editions of 750 books each, liow many pages are there, if each book contains 407 pages ? 9. How many yards of sheeting are there in 57 bales, each bale containing 25 pieces and each piece 43 yards i 10. In a cotton mill there are 29 looms ; each loom can weave 42 yards daily. At this rate, how many yards can he wov(?n in 159 days ? 11. A lot cost 420 dollars ; how much will 105 lots cost at the same rate ? 12. A drover has 406 cows worth 80 dollars oach ; how much are they all worth ? 13. How much will it cost to build 307 miles of railroad at 4060 dollars a mile ? 14. A eouiractor built G04 miles of railroad at 6500 dol- lars a mile ; how mtich did he get for it ? 15. If it requires 720 barrels of provisionc to eupply an army for one day, how many ba' rels will bo required for 365 days? " 1(). If one acre of laud costs 9620 dollars, how much will 736 acres cost ? 17. If it costs 98650 dollar-, to build one mile of railroad, how much will it cost to build 2809 miles ? 18. There are 15 fiolds of ocrii ; in each field there are 97 rows, and 2ou liills in each row ; how many hills are there in the 15 fiold-i i 19. How many yiird-, o2 oloth are there in 43 bales, each bale contaiiiing 72 pieces, and each piece 29 yards ? Sj^ 20. If a railway train ?< es 18 miles an hour, how far 'will it go in 17 day- of 24 hour> each ? Exercise i^xiii. Practical Problems involving the t'reviouo Rules. Jl &^iTrnwfrriirHoliso for 2900 dollars, and gavo for it 98 cows at 24 doUarf, each, and the rest in money ; how much money did he pay? 2. One army contains 4575 mon, and n.nother 30 times aB many, lacking 1930 men ; how nuiuy nicxi are there in the second army ? 3. Mr. Peters has 24t>l gallons o( coal oil, Mr. Martin has 1140 gallons, and Mr, Bcusou has 147 times as much as both ; how much has Mr. Deason 1 4. A farmer sold 129 cows a,t 37 dollars each, and re- V. IZ: u < A V Z.-d Oft^-H' LTIPLICATION. ^^^-m lived in payment 2000 dollars; liow much yet remains i t V [16 7 5. B isold 76 hens at 73 cents each, 96 turkeys at 324 cents each, and received in payment 24000 cents; how much remains due ? 6. A's barn cost 2485 dollars, his house cost 3 times as much, and his farm cost as much as both ; what was the cost of the house ? what was the cost of the farm ? —.7. A drover b(>nj]^ht 86 horses at .145 dollars^ head, and 96 cows at 28 ^:0tlaT'S~Trima;d^ which cost the most, and how much ? /^'^^ -.8'--A'sHt>ook contains 248 pages,-with 2850 letter^ gb*-«, V^ page, and B's contains 825 pages, with 3465 letters on a page ; how many letters in A's book ?x how many ill B's ? 9. A man has 75 bags of apples, each bag containing 2 bushels; howm;uch will he receive for them, at 125 cents a bushel,? 10- A farmer sold 25 firkins of butter, each firkin con- taining 126 pouncls, and received for each pound 87 cents ; how much did he receive for it all ? II. Find the product ot the sum and difference of 784 and 397. ^ 12. 11 472 men cut 800 cords of wood in two days, how long would it take one man to do it ? 13. A farmer sold 129 cows at 29 dollars each, and re- ceived in payment 2300 dollars; how much yet remains due? 14. A's bam cost 175 dollars ; his house cost 16 times as much, and his farm cost as much as both ; what was the cost of the house ? what was the cost of the farm ? 15. A man bought 56 acres of land at 46 cV^Hars an acre, and 78 acres at 62 dollars an acre, andr sold the whole at 53 dollars an acre. Did he gain or lose, and how much ? 16. A merch nt bought 1600 barrels of flour at 7 dollars a barrel ; he sold 900 barrels at 12 dollars a barrel ; and the remainder at 5 dollars a barrel. Did he gain or lose, and how much ? 17. If a house is worth 3250 dollars, and the farm on which it stands 3 times as much and 450 dollars more, and the stock on the farm twice as much as the house \\ lacking 2368 dollars; what is tlie value of the whole ? 18. A has 4278 dollnrs more than B, and 1225 dollars less than C, w'ao has 78G4 dollars ; and D has as muohaa A and B together. How much has D ? (^ ad I m .^.A.,..:^.:».u^tSi.^i^aajfcff.i^:-ii^^ jji.^ -..5W.»^.-— -. 84 ELEMENTARY ABITHMETIO. r I 19. A man invests in trade 450 dollars at one time, at another 840 dollars, at another 1125 dollars, and at an- other 1640 dollars ; how much must be added to thepe sums that the amount invested by him shall be increased three fold ? 20. A man sold his house for 4500 dollars, and 250 acres of land at 75 dollars m acre ; he got in payment 6000 dollars in cash, 239 cattle at 25 dollars each, and 317 sheep at 6 dollars each ; how much was still due him? Section IV. -Division. 1. John has 9 ap;oles wliicli he wishes to divide equally among his 3 brothers ; how many apples can he give to each ? Hore we are reqnirr^d to divide 9 9 apples, apples into 3 eqaal parts. If John 8 gives each brother one apple, it will require 8 apples, and 6 apples would be left. If, now, he j^ives each of thorn another apple, it will require 3 more apples, and 3 apples would be left. If he gives theia one apiece a third time there would be none left. Hence, it — is plain that he can give each of his 3rd remainder, brothers 8 apples. In this example we see that 9 contains 3 three timey, for if we subtract 3 from 9 three times, nothing is left. A number, therefore, may be divided into equal parts by sub- traction. Hence, we see that division is simply a short method of performing several successive subtractions of the same number. "We might have obtained the result in a shorter way, as follows : Sinco 3 times 3 is 9, wo see that 3 is contained in 9 three times. Hence, to find how many times one num- ber is contained in a second, we have merely to find what number multiplied by the first will produce the second. 2. How many times 2 liorses are 6 horses ? 8. How many times 3 cents are 12 cents ? 4. How many times is 5 contained in 15? Since 3 times 5 is 15, 5 is contained 3 times in 16. . xiOW Ullliij SjiLU^D IS U \iWJ-iiiaiii U iU t/v7 i 6. If a boy earns 24 dollars, how many times 4 dollare does he earn ? 6 1st remainder. 3 3 2nd remainder, B ( DIVISION. 85 7 I 7. How many times 6 boys are 30 boys ? 8. Three dogs have 12 feet ; how many feet has 1 dog ? 9. How many times 4 feet are 12 feet ? 10. A bush has 8 roses ; how mp.ny times 2 rosea has it ? How many times 4 roses ? 11. How many times 9 boys are 27 boys ? 12. A house has 12 doors; how many times 8 doors has it ? 13. How many times 7 horses are 21 horses ? 14. How many times is 7 contained in 28 ? 15. How many times is 4 contained in 20 ? 16. How many times is 5 contained in 30 ? 48. Wlien it is required to find how many times one number contains another the process is called Division. 49. The number to be divided is called the Divi- dend. 50. The number by which we divide is called the Divisor. 51. The number of times the Divisor is contained in the Dividend is called the Quotient. 53. AVhen the Divisor does not go an e,xact num- ber of times into the Dividend, the excess is called the Remainder, 53. The remainder, being part of the Dividend, will always be of the same kind or denomination as the Dividend. 54. The Sign of Division is a short horizontal line, with a dot above it and another below it ; thu>i, -r . It shows that the number hefora it is to be divided by the number after it. Thus 8 -r 2 ^ 4 is read, 8 divided by 2 is equal to 4. 55. Division is frequently indicated by a line, with the dividend above it and the divisor below i* • thus, f signifies that 9 is to be divided by 8. Isis. Division mav be divided into two cases; 1. Wht\i the I'ivisor does not exceed twelve. 2. When the divisor exrmds twflir. )l 86 i •I h' ELEMENTARY ARITHMRTIO, DIVISION TABLE. 9 in 9 1 time 18 2 times 27 3 86 4 45 5 54 6 63 7 72 8 81 9 90 10 99 11 "1 r\m ■• i-ii ^ 10 in 10 1 time 20 2 times 30 3 « 40 4 " 50 6 •' 60 6 " 70 7 «• 80 8 90 9 11 in 11 1 time 22 2 times 33 3 " 100 10 |lJO 11 '120 12 <( (( (< t( << (( it (( (( , 12 in 12 1 time I 24 2 times j 36 3 « i 48 4 " , 60 5 ** i 72 6 ♦• ' 84 7 " 96 8 ♦' ;108 9 " !120 10 " 1 q.> 11 < I 144 12 «' ^ t* \\l: \ I r %i 1 V DIVISION. Oral Exercise. 87 1. 86 is how manv times 4'? How many times 12? 2. How many times 7 is 28? Is 42? Is 84? Ib 35 ? 8. How many times 9 in 27? In 45 ? In 63 ? In 99 ? 4. A farmer recoived 8 dollars for 2 sheep ; what was the price of each ? Since he received 8 dollars for 2 sheep, for 1 sheep he must get as many dollars as the number o\ times 2 is contained in 8. 2 is contained 4 times in 8, because 4 times 2 is 8 ; hence 4 dol- lars was the price of each sheep. 6. If a man walks 24 miles in 6 hours, how far will he walk in 1 hour ? 6. If 1 man can do a piece of work in 82 days, how long will it take 8 men to do it ? 7. If 7 yards of silk can be got for 21 dollars, how much will 1 yard cost ? 8. If 27 yards of cloth can bo bought for 8 dollars, how many yards can be got for 1 dollar. 9. If 3 hats cost 9 dollars, how much will 1 hat DOst ? How much will 7 cost ? How much will 12 cost? 10. How many times 5 oranges are 50 oranges ? Is the result a concrete number, or an abstract number ? 11. If you can buy a lead pencil foir 3 cents, how many can you buy for 24 cents ? 12. How many barrels of apples, at 2 dollars a barrel, can be bought for 24 dollars ? 13. If a man walks 3 miles an hour, bow many houvv m]\ it take him to walk 18 miles ? 14. A farmer divides 84 bushels of apples equally among 12 men ; how ma'iy bushels does each receive ? 16. 72 cents are paid for 12 eggs ; how much will 1 cost at the same rate ? 16. How long will it take 12 men to perform a piece ol work that 1 man can do in 60 days ? 17. A man planted an orchard of 120 trees anl put 10 in each row ; how many row;> are there in the orchard ? 18. How many uien at 9 dollars a month can be hired 1 iU 20. If 6 barrels of flour coBt 64 dollars, how much wiU 1 barrel cost? 38 ELEMENTARY ARITHMETIC. Case I. 5T. When the divisor does not exceed Twelve. Ex. I. How many times is 7 coDtained in 952 ? Divisor. T>ividend. Qtiotient. 7 ) 952 ( im 952 7_ 25 21 42 42 We write the divisor at the left, and the Quotient at the right of the Divi- dend, and begin at the left to divide. 7 is con- tained in 9 hundreds 1 hundred times and a remainder. We write the 1 hundred m the Quotient, and multiply the Divisor 7 by the 1 hun- u 1' ^^^^ ^^^^^ "^ "^ hundreds, which we write under the hundreds of the Dividend. We then subtract the 7 hundreds from the 9 hundreds and the remainder is 2 hundreds, or 20 tens. We add the 5 tens of the dividend to these 20 tens and set down the 25 tens. 7 is contained m 25 tens 3 tens times, and a remainder. We write the 3 tens in the Quotient and multiply the Divisor by the 8 tens of the Quotient. This gives 21 tens, which we write under the ;)ar , ?; ^ ^"^rPf ^ ^as 24 cows and 93 sheep, worth 1521 *>« dollars ; it the sheep are worth 6 dollars each, how much is each cow worth ? ^' H?^ '^^'^ ''^^^ ^^^^ ^®^^*^ ^" a day, and 25 boys earn 6450 cents m a day, how much more does one man earn m a day than one boy ? 7. How many barrels of flour at 6 dollars a barrel are equaljn value to 1100 tons of coal at 9 dollars a ton ? 8. if a mechanic earns 52 dollars a, month, and his ex- penses are 34 dollars a month, how long will it take him to pa^ for ft farm of 36 acres, wonh ]2 dollars an acr> ^ Ih'>jh^^ "/^ ^ V i mvisK^. 4t 9. A clerk's salary is 1200 dollars a yen-r ; he pays 5 dollars a week for board, 2 dollars a month for car fare, and bis other expenses amount to about 1 dollar a day ; how much can he save in a year ? 10. Mr. Jones bought a farm of 110 acres at 75 dollars an acre, 2200 dollars to be paid down, and the remainder in five equal yearly instalments ; what must he pay each year ? 11. A man has 18 piles of wood, each containing 26 V\j cords, and each cord 128 cubic feet ; how many cubic feet of wood has he ? 12. A man exchanged 169 cords of wood at 6 dollars a cord, for a horse valued at 144 dollars, and the balance in sheep at 8 dollars each ; how many sheep did he receive 1 13. A merchant balancing his accounts ' found that he had on hand merchandize worth 476 dollars, and cash amounting to 2570 dollars ; he had 16st by bad debts " dollars, and owed 625 dollars ; if his original capital was 2000 doUatb, what had he gained ? 14. A cistern containing 18500 gallons is filled by two pipes, one discharging 250 gallons an hour, and the other 300 gallons, but, by a leak in one of the pipes, 100 gal- lons are lost in an hour ; if the cistern is empty, how long will it take to fill it ? Ex. 2, If 8 pounds of coffee cost 80 cents, what will 8 pounds cost ? ' The cost of 3 pounds of coffee = 30 cents ; « 1 pound 8 pounds it <( 3 = -^ = 10 cents ; ( =8x10 cents = 80 cents. ^ 16. What will 15 slates cost, if 5 slates cost 80 cents ? 16. If 4 trees cost 72 cents, what will 3 trees cost ? 17. If 6 barrels of flour cost 48 dollars, what will 7 barrels cost ? r 18. What will be the cost of 16 cords of wood, if 4 cords cost 24 dollars ? 19. If 15 yards of cloth cost 75 dollars, what will 20 yards cost ? 20. If 7 pounds of beef cost 56 cents, what will 6 pounds cost? 11 xz znBu can 6m n «~)o uoiiars in a a ay, now Diiiea Can 4 men earn in the same time ? 22. If 28 acres of land cost 4480 dollar"?, how mnch will 48 acres cost at the. same rate ? |.^ ^-•^ 50 lif i l|!l| IH M ft ELEMENTAKY ARITHMETIO. hnw'J^^''"*'^'' ^^^"^-'^ •^^^'^s o^ oloth for 61 dollars %^°Vi7 yards canyon get for 876 dollars? ' ^b. ItdS acres ot land coat 11172 dollars hnw »>,««« ^'?.' Ti'.^^^^^^-^^^f^'l^^SlO dollars? ' ^ ^^''^ cost f ''''* ^^'^^^ ^°"^'^«» ^^a* ^i» 25 housee horst cost'? ""''''' '''' ^''^ ^°"^^«' ^^- '^^^eh will fl hnw ' J^'^'^r ?? ^^'^ ^'^"^ 1^95 ^^shels of corn in a dav • now many bushels can 27 busk ? ^ ' Ex.3. If 7 men do a piece of work in 86 davs in how many days can 28 men do it ? ^' Time for 7 men to do the work = 86 days- Iman " " =7x86 days; 28 men «* ** ~ 7X36 - , 28~ ~ ^^''* <« 81. If 16 workmen can do a piece of work in 26 r^axro ^a'J^^* *i^! °^° 2^ ^^^ d'^ «^« same ? ^^ ^^y'' Oii. A held can be mowed bv 40 mpn ir. o a«.^^ • i. many days would it be finished ty 80 m"'''^'' " '°" Ju1d{t^'^k:^?srn't:itL'?t^^^^" '' ^^^^'^- ^-^ JutL\%Tl'nfA"''?i^^^'^^°^^^ clays, how long aV tT"? o 7 *° d° *^® same work ? ^ 00. 11 la horses can cart away the earth from a cellar i>. 76 days, ,n how many days would 27 horses do tL work ? 8of;h! 1°''''!°^^"*^*'^^^^^ ^«^se m 63 dayTbut fotuKTolsfr ^^^'' '^^ ^^"^ -^" '' '^^^ '^' -t loiflw^,?/^/^^"*^""' °*° ^""d ^ ^««SQ in 72 days, how 88 H i f* ""''"^If.^ carpenters to build the sameV 88. How long will it take 40 men to build a wall if 19 men can do it in 20 days ? * ^* ^^ 3f w^r^tTat e""^!'" ^* *f ' ^ r^^ *" '^^ *^« "^'^^ a°^°^°t >in IT , 6 n^en can do m Xii days ? ^Lth'^Z>%"'l'A»,'?„^",'-''«*<'.do a piece of work .-_...-. .,.^ ^j jxicu i.oo Hays to do '? Ex.. 4. If 80 men build a wall in 18 davs lio* many men will be required to do it in 12 dayY? EXAMINATION PAPERS. ei Ken required to build the wall in 18 days » 80 men ; 1 day =18X80 men; «t (I tt *i 4< II M ♦* 12days- 18X80 12 =45 men. 41. If 4 men can dig a garden in 7 days, how man^ taen would be required to dig it in 1 day ? 42. If 28 mon can mow a field of grass in 12 dayi, ho\( oaany men will be required to mow it in 4 days ? 43. If 7 men can reap a field of wheat in 18 days, ho'w many men would be required to do the same wtn-k in 6 days? 44. A piece of work was to have been performed by 144 tnen in 36 days, but a number of them having been dis- uharged, the work was performed in 48 days ; how many men worked ? 45. If 20 men can perform a piece of work in 16 days, How many men will it take to do it in 12 days ? 46. How many men in 26 days can perform th« same amount of work that 89 men can do in 76 days ? 47. A drain is dug by 49 men in 96 days ; how many men would have been required to dig it in 84 days ? 48. If 8 workmen can build a wall in 27 days, how many workmen would be required to build it in 8 days ? 49. If 100 workmen can perform a piece of work in 12 days, how many men are sufficient to perform the work in 8 days? 50. A gentleman met a number of beggars, and relieved 9 of them by giving 25 cents to each one ; how many «vould he have reheved for the same sum had he given them only 15 cents apiece ? EXAMINATION PAPERS. I. 1. Define the following tei-ms : Unit, Number; Notation^ and Numeration. 2. Add together four millions twenty thousand and sev- Buty-niue, twelve millions two thousand and seven, aad one million and five thousand, and subtract 16688107 from the sum. o ■n'.'—j ii-^ •— 1 ri. 1-t J^i;i- -It-- —--;--, -^t- -— r, t;. iiuu uiiv miitiinutsr aiuBr BUODroiClilug JiUB uuixaucxo 44444, 9099, 666, 77, 1, ^n succession from 1000000. 4. Add together the sum, difference, product, and quo* tient of the two numbers 826 and 9818876. 52 ELEMENTARY ARITHMETIC. lor wiiat It cost ; how much did I gain by the bargain ? II. 1. Explain the meaning of the followinff t&rma or^A 2. Inud the sum of the followino' numbers And n^r^v««c, t'?.e\rs?^rtL: nSe/r "^^ '^""""^"' --^ - - 4. Express MMDGXCIX. and CCCXXIX in thi, nrfli nary numerioal characters ; find their ptoduo, and expr^s^ the result in Roman characters. express pi,f; 1??'' ""*,"y ''"*«l8 of wheat, at 125 cents per bnshel t^^tZT"^"''^'"' ^^"'"'^ of Bngar7ar8 'It , r. III. "li'^ ■" ''^'J^ '"""^ ^"' ""'l '^■'Plain the process of ^ borro^ng and carrying » in the common rule of Sac GiveSrrxam;^*"' P"""""^' °' ™''*''^«''" »>« -^rifiec" eo uhp^"Msr?rtradSnr«- 6. Bought a farm for 35380 dollars, and ha W made Z rl 5 "^'■'' ^.* ^^ ^^"*^« »" acre ; how many acres did I purchase, and at what price per acre ? ^ IV. 1. What is the object of division ? Show that if mo^^ be considered a shortened subtraction. '* ""^^ Q* ); V • '^**'^ *^® •^"<'^^* o^ a number ? BuLLVvrZ;^^?!???^^^^^^^ performed by remamd^:mayl>e"l>Sir " Ex:lSi^^??^^ril^^^ ' much can a man earn Bam 48 dollars in 24 days ? 114 days, if he can EXAMINATION PAPEBS. se 6. A. man bought a number of sheep at the rate of B for 18 dollars ; how many did he buy for 8648 dollars ? V. 1. Wliat is multiplication ? Show by an example that 11 IS a Bliort method of performing addition ? 2. Show by an example that two or more factors wiU give the same product in whatever order they are multi* pliecL ,>. ilow many times must 1874 be added to itself to make a total oflG3038 ? 4. The product of 75 by 43 is 3225 ; how much must be added to it to obtain the product of 77 by 48 ? 6. A drover ])ought 79 oxen at 42 dollars each ; he sold 25 at 40 dollars each ; for how much per head must he sell tlie rost so as to gain 544 dollars on the whole tran- gaction f VI. 1. Given the divisor, quotient and remainder, how is the dividend found ? 2. I bought a farm of 150 acres for 12000 dollars ; I sold 29 acres at 95 dollars an acre, 76 at 112 dollars an acre, and the rest at 96 dollars an acre ; what did I gain by this transaction ? 8. What number is that, which being multiplied by 15, tho product divided by 16, the quotient multiplied by 7, 85 subtracted from the product, the remainder divided by ten, and 52 subtracted from the quotient, the remainder is 18 ? 4. I bought a farm for 6480 dollars, and after spending 890 dollars on improvements on it, I sold one half of it for 4050 dollars at 45 dollars an acre ; how many acres did E buy, and at what price per acre ? 5. If 16 men can perform a piece of work in 36 days, in how many days can they do it with the aid of 8 more men? -^ VII, 1. Explain why in addition of numbers the operation is begun at the units' place. Is this necessary ? Illustrate by an example. {% >k«t/*k«^^««4-w « j a n' '• fmmummi(mmmi i m¥ctAr\r\rt j._ 840001 cents. 10000091 cents. \l- 8B ■LBME.NTPARY ABITHMXTZO. iM i ^ M ill it ■ .5 i I Addition, Oral Exercises. vhu H""^ «o«* ^1-25, aad a slate 50 cents ; how roudh Old they both cost ? » ^^viuu 2. A pound of toa cost $1, a pound of coffee 25 cents and a ham 75 cents, what w&s the total cost ? * 3. If I pay $1.20 for a turkey, $1.15 for a gooeo and b;/ cents for butter, how much do I pay for all ? 4. Bou^^ht a loi for $6, a bag of fl.ur for $4, and a cord of wood lor $7.50 ; how much did t pay for all ? 5. Paid 90 cents for paper, 10 cents for pins, and «1.25 for a book ; how much did I pay for all ? 6. A book costs 90 cents, a pen-holder 10 cents, and a slate 65 cents ; how much do they all cost ? anflsO^O ^^^ ^'*^®^'^^'* $'^•^7, $20.78, $0.29 $187.04 * ol'nl ^^^ ^® f^"^* '^^^"^ ^^^'°^s of the same kind. n li ^® ^^^^^^ '^"^'^"^^ ^^"^^^ dollare and cents under ifirj'f? cents, letting the points ra,nge in a straight Iflnn ^'''''- ^^'''' regarding the dollars and cants as 500^0 go many cents, we add at in simple numbr^rs iT^ZH T T.^^*® ^^^ P^^"* ^" ^¥ ^"^^ two places » 1^4. 4b from the right to reduco the cents to dollars (1) #71.iJ6 169.08 208.72 714.89 Exercise xxxviii. , (2) (3) $184.36 $1843.21 769.28 978.89 41.07 36.07 669.36 802.48 1105.20 110.00 409.05 1000.65 5. A farmer receives $15.87 for a cow, $75 for a horse, $3.13 for B(-me potatoes, and $5.55 lor some poultry how much does he receive in all ? x- j » 6. Sold some velvet for $3.33, broadcloth for $18.75, siik lor $1-2.50, muslin for $5.40, carpeting for $30.05, a sbtiwl lor $12.25 ; what is the amount of the biU ? <^.l'^ }L^ K^^^^ °^^^ ^3487.75; repairs, $53.37 ; painting, $119.23; furniture, $1563.39; moving, $9; what was the whole cost? 8. A lady gives 25 cents for needles, $17.50 for a drpss 12.68 for trimmings, $1.50 for a cap,"aud 12 cents for thread; how muoh does she lay out ? SUBTBAOTION. S9 Subtraction. Oral Exercises 1. John bouj?ht a book for $1.50 and sold it for $1.75 ; how much did he gain ? 2. A merchant bou<^ht goods for $1.75 and sold them for $6 ; how much did he gain ? 3. John had $10; he paid $2.50 for Rome hooka, and $1.50 lor a satchel ; how much money has he loft ? 4. Mary had $1.25 ; she paid 75 cents for some ribbons, and 25 cents for car tickets ; how much has 8he loft ? 5. Bought some rice for 60 cents, some sugar for 45 /cents and some tea for $1 ; how much change should I get from a five dollar bill 1 > 6. Bought a hoxse for $li^O, a saddle for $15, and sold doth for $150 ; what was my gain ? V 7. I bought a pound of rice for 8 cents, crackers for 15 cents, raisins for 18 cents, candy for 10 cents ; how much change should I get back if I gavo the clerk $1.00. . Ex. 4. John owes $137.35 and pays $29.17; how much does he »till owe ? $137.35 Flacing dollars under dollars and 29.17 cents under cents, we regard the dollars — and cents as so many cents and sub- $108.18 tract as in simple numbers. We then write the point two places irom the right of the remainder to reduce the cents to dollars. (1) $104.36 9.78 Exercise xxxix. , (2) (3) $76.14 $200.00 17.39 166.81 $782.36 189.76 6. A man has $10000 ; he buys a house worth $4829.36; how much money has he remaining ? 6. John has $17.21, James has $41.00; how much has James more than John ? 7. My salary is $1000 a year; I pay for rent $150, for groceries $325.40, tor gutter $(50.30, for dry goods $127.68, and for other expenses $75.60 ; how much do I savo ? / 8. A man worth $10000 g»ve away $956.38, and losi 4 127.82: what was he thonVorth? • i^ $1127.82; what was h6 thenVorth ? 9. If a lady gives 12 cents ink, 63 cents for pens, r 1- i] ■ 1 60 ELEMENTARY ARITHMETIO. $13.80 for books, and $1.'87 for paper ; howmnah oh&n((d must she got tor a twenty- dollar bill? /' 10. Bmirriit ^76 worth o1 bay, and $25.2S worth of corn ; paid $49.88 ; bow much is still due ? 11. I paid $-1037.25 for a iarm, $3075.25 for build- >^ ing a bouse, and $2890.87 for building a barn ; I Bold iny property lor $13000 ; bow much did I gain ? 12. I paid $240.75 lor a borse, $325.45 for ft mule, $42.25 for an ox, $37.50 for a cow ; I sold tbem all for $603.50 ; wbat was tbe loss ? Multiplication. Oral Exercises. 1. Wbat will 10 pounds of fish cost at 12 conts a pound ? 2. Wbat will 3 pairs of boots cost at $5.25 a pair ? 8. If I earn $10.50 in 1 weeii, bow much can I earn in 2 weeks ? 4. Bought 2 bats at $1.25 each, and 8 collars at 25 cents each ; bow much did I pay for them ? 5. Tbomas earns 75 cents a day; his oxpenses are 62 cents a day ; bow much does be save in 7 dayp ? 6. A man bought 4 bushels of wheat at $1.12, and sold the flour for $5 ; how much did he gain ? 7. Bought 5 barrels of flour at $8.60 a barrel, and 6 bushels of wheat at $1.25 e» bushel; what was the cost oi botli ? 8. What is tbe cost of 2 pairs of chickens at 76 cents a pair, and 5 pairs of ducks at 60 ccnt«i a pair f 9. Bought 5 pounds of cofl'ce ai 35 cents a pound, and 12 pounds of hftm at 22 cent? a pound how much chango did I get from a five-dollar bill / Ex. 5. Multiply §78.89 by 8. $78.89 We regard tho dollars and >^ent8 a& 8 80 many ceni=«i, anl multiply as in Simple Multiplication, and then we $027.12 place the point two places from the right of tho product, to roduce the cents to doUars. Exercise xl. Multiply $78.87 By 6 J2)_ sp2'iv.lG 6 J3)_ $48.75 19 . (4) 1781.86 125 DIVISION. «1 5. A farmer sold 176 acres of land at $37.50 an acre : how much did he get for the land ? 6. A miller sold 626 barrels of flour at $6.71 a barrel • how much did he receive for all of it ? 7. What will 42 calves cost at $3.75 tpiece f 8. At 37 cents apiece, what will 75 geese cost ? ?n ^m K^^'Vu^ .f'"'^" "^ ^^^^ ^°«* at ^3. 78 a cord ? ftl 20 r anlT '"''"^ ''^' ^^ ^^''^' °*' ''^^^^ '^^^ ** yC 11. If a boy's wages are $4.76 a week, how much will ' he earn in a year, or 52 weeks ? yC 12. If a clerk earns $8 a week, and spends $4 76 a week / ho^ much will he lay by in a year ? ' 13. What will it cost six persons to board for a vear ftt the rate of $5.75 apiece each week ? " '°^ «- y^ar 14. What is the value of 17 chests of tea, each weighing 69 pounds, at $0.72 a pound ? weiguiug onn^* ^ T'^'i^fn^ ^""^^ ^^ ^^*"^^^« '^^ PO^k, each weighing 200 pounds at 12 cents a pound ; what did he receive P^ lb. A lady goes to market with 10 dollars ; she buvs 6 tT'^y!'T^^^ V T*«' 7 pounds of meat kt 16 cents" hL^L'TemitT'^^"^^ ^^ ^'-'"'^ ^^^ --^ --^y t,.int5/q ^'j;'^ ^""^l"* 15 hogsheads of molasses, con. tainmg 63 ga ons each, at 65 cents a gallon, and sild It at $1.10 a gallon ; what was his gain ? Division.^ Oral Exercises. 1. If 7 hens cost $3.57, what will 1 cost f payaw^e'u^ ' ' '" ' '''^^^''' ^^^^^ ' bow mlh did I $3/72t* ^ ''^''*^ ^^''^' ^'^'^ '"^"^ ^^'^''''^ ^^^ ^ ^^y f<>r 6. If 4 hats cost $5, what will 7 such hats cost ? b. A yard of calico IS worth 12 cents; if I buv 15 varda , 7. If a barrel of flour costs $6.25. how mflnxr l...vv^i„ De bought for $50 ? ' "' —'^^^ "«-" 8 At the rate of 16 cents a dozen, how many doaen buttone can b© bought for $3 ? ^ w 62 ELEMENTARY ARTTHMETIO, I I -! t 9. If I buy 17 pounds of sugar at 10 cents a pound, how naany oranges at 5 cents each can I got for the change due me from a five -dollar bill 1 10. A yard of calico is worth 9 cents ; how many yards Cf>.u I get for 10 dozen of eggs, worth 18 cents iv dozen ? 11. If I trade 6 pounds of butter at 20 cents a pound, and 10 pounds of lard at 12 cents a pound, for sugar at 12 cents a pound, how many pounds of sugar do I get»? Ex. 6. Divide .^563975 by 5. We regard the dollars and cents as so 6)039.75 many cents, and divido as in simple division. Then wo place the point in ^127.95 the quotient to separate the dollars from the cents. Ex. 7. When potatoes are worth $1.25 a bag, how many bagfuls can be bought for $46.25 ? 125(4025(37 Wo are required to find how often 375 !$1.25 is contained in $46.25. Wo re- gard $1.25 as 125 cent^J and $40.25 ao 875 4625 cents and then w© divido in tho 875 usual way. r . i i ! Exercise xli. ^^^(1) (2) (3) (4) 6)|76^ 7 )$149. 59 8 )$145.3 6 9) $237.0e 6. If a person spends $410.28 in a year, how much la that a week, allowing 52 weeks to a year ? 6. Divide $2117.71 equally among 35 families, and find tlie share of each. 7. A man pays for some land $400 cash and $192.80 in produce. If there were 57 acres, how much does the land cost him per acre ? 8. How many sheep can be bought for $302.95 at $4.15 each. 9. If 93 oranges cost $5.58, what will 37 cost ? 10. I bought a house for $3453, and paid for it in instalments of $575.50 each; how many payments did I have to make ? >^ 11. William earned $3.25 a day, and paid 75 cents for board : in how manv davs would he s.ave ft91'2.l>0 ? 12. A merchant received $853.25 for a case of silk, including $1.25 cost of box. IIow many pieces of silk were in the ca»e, if he received $53.25 apiece ? KLLLS AND A0OOUNT8. 68 BILLS. 70. A Bill of goods is a written statement of the goods field, giving the quantity and price of each article and total cost, also the date of tiio sale, with the names of the buyer and seller. 71. The party who owes is called a Debtor, and the party to whom a debL is owed is called a Creditor, SPECIMEN OF A BILL, Toronto, February 23, 1878. James Brown, Esq., Bought of C. Meredith. 1878. Jan. (I Feb. 19 15 1b. Coffee at 32o... 23 16 " Lard at 15c . . 2 I 25 *' Sugar at 13c ... 20 I 16 *• Ham at 16c..., $ 4 2 3 2 $13 c. 80 40 ••• ••••■• 25 56 01 / / John Smith, Dr. SPfiOIMEN QH^A RECEIPTED BILL. Toronto, March 1, 1878. To George Brown. 1878, Jan. Feb. Jan. Feb. 1 2 7 2 To 75 lbs. of sugar at $0. 12, *• 47 yds. of cloth " 3.25, Or. By 75 bu. of corn, at $0.78, " 48 bu. of apples" 1.25. Balance due, $9; 00 162 i6 $58 63 50 75 161 112 c. 75 25 $491 5G A\jiO, i«i.aiCn idbB. Received Payment, George Brown. u mt I ■LBMBNTARY ARITHMBTIO. Exercise xlii. Make out bills for the foUowing accounts, supplying dates : 1. Mr. J. Jones bought of R. Walker 10 yards silk, at $2.60; 12 yards of flannel, at 40 cents; 16 yards calico, at 15 cents. 2. Mr. Brown bought of McChing & Bros. 10 pounds tea, at 75 cents ; 8 lbs. raisins, at 18 cents ; 5 pounds rice, at 10 cents ; 12 lbs. butter, at 21 cents. 8. James Taylor bouf'ht of Thomas Yellowlees 6 quires foolscap, at 25 cents ; 1 Hamblin Smith's Arithmetic, at 75 cents ; 3 rolls wall paper, at 45 cents ; 4 dolls, at 26 cents. 4. Darid Montgomery bought of F. F. McArthur 20 yards cotton, at 11 cents ; 15 yards print, at 16 cents ; 12 yards braid, at 6 cents; 3 pairs gloves, at 27 cents; 26 yards dress goods, at 63 cents ; 1 hat, at $5.25. 5. Robert Davey bought of Murdoch Bros. 18 bags salt, at 75 cents ; 4 barrels plaster, at 98 cents ; 10 pounds coflFee, at 35 cents ; 1 chest tea, 18 pounds, at 65 cents ; 48 grain bags, at $3.60 a doz. 6. Levi Van Camp sold VVm. Burns A Co. 257 bushels wheat, at 81.12 ; 475 bushels oats, at 86 cents ; 45 bushels corn, at 76 cents; 175 bushels pease, at 82 cents ; 867 bushels barley, at 69 cents. 7. A. Thompson bought of A. Harrison 82 pounds sugar, at 12 cents; U pounds coffee, at 35 cents ; 26 pounds soap, at 8 cents; 14 pounds rice, at 9 cents ; 7 pounds fish at 15 cents ; 18 pounds crackers, at 12 cents. 8. W. West bought of T. Brown 27 pairs calfskm boots, at $4.50 ; 96 pairs gaiters, at $3.25 ; 126 pairs overshoes, at 91 cents ; 18 pairs slippers, at 95 cents; 75 pairs heavy boots, at $2.75. ^ 9. Mrs. Jones bought of R. Walker & Co. 25 yards i'rtlico, at 12 cents; 12 spools cotton, at 5 cents; 16 yards 'Ipaca, at 75 cents ; 17 yards muslin, at 18 cents ; 6 .>Keins tape, at 2 cents. 10. Murdoch Bros, sold to A. Preston the following: z7 yards calico at 18 cents ; 45 yards muslin at 1^ cents ; 16 yards linen at 45 cents ; 17 yards cambric at 15 cents; and 9 handkerchiefs at 45 cents ; and took in exchange 12 ousbels potatoes at 65 ceum ; 3 barrels apples at $3.25, 18 lbs. butter at 85 cents, ^nd the remainder in cash. How much cash was paid ? Make out a r«ceipt«d bill. ! -A l^A EXAMINATION PAPBBST EXAMINATION PAPERS. I. 66 1 A farmer gave $43.60 for sheep, at the rate of $7.23 fol 3 shf^p ; xiow many did he buy ? No. of sheep bought for $7.25 «» 8 sheep ; 8 M M «< €1 Hh 60- «< 7.26 48.50 X 8 sheep; 7.25 = 18 sheep. 2. If 18 chickens cost $4.20, how much will 8 chickens eoBif 8. A merchant bought 9 pieces of cloth, each contain- in« 60 yards, for which ho paid $2317.60 ; what was the ccst of a single yard ? 4. \ banker has $20000 in cash ; he pays for 60 shares 01 stock, at $97.50 a share ; and 100 shnree, at $110 a share ; how many shares, at $41.25 each, can he buy with the remainder of his money ? 5. I owed $276 and paid $17.25 on it ; how many times must I pay such a sum to cancel the debt ? II, 1. I retail envelopes at 12 cents a pack, gaining 8 cents on each pack of 24 ; what did they cost me per 1000. Cost of 24 envelopes == 9 cents. "1 •• __ , t« ** 1000 " — •— " «• ^^ = $8.75. 2. A grocer sold 9760 pounds of flour, at $4.26 per 100 lbs. ; what was the amount of the sale ? 8. Messrs. Smith & Co., burn in their store, in a year, 62560 cubic feet of gas, at $4.50 per 1000 fe^t ; what is their gas bill ibr a year ? 4. A man bought a quantity of coal for $250,' and by retailing it at $5.76 a ton, he gained $87.60 ; how many tons did he buy ? 6. The charge of sending a telegram to a certain place is 40 cents tor ten words, and 5 cents for each additional word ; what would a despatch of 24 words cost me ? III. 1. A horse worth $160, and 7 cows at $25 each, were exchanged for 67 sheep and $25.75 in money ; what were the sheep valued at per head ? ELEMENTARY ABITHMSTIO. i' ! ill Valne of horse and cows = Value of sheep = Hence " 67 sheep = Therefore " 1 sheep = $160 + 7X$26=.$825. $825 -$25.75==92d9.25. $299.25 ; $299.25 67 = $6.g6. 2. A merchant bought 6 pieces of cloth of equal lengths, at $3.26 a yard ; he gained $18.75 on the whole cost by selling 4 of the pieces for $750 ; how many yards were there in each piece ? 3. At an election there were three candidates A, B, and C ; the total number of votes polled was 7734. The suc- cessful candidate, A, got 208 votes more than C, who got 107 votes less than one-third of the total vote polled ,*;, what was A's majority over B ? 4. A father divided his property worth $4767 among his three sons A, B, and C, in such a way that A got at much as B and C together^ and B and C shared alike ; what was C's share ? 6. If the continued product of 275, 376, 484 and 196 be divided by 77x28x47X55, what will be the quotient? IV. 1. A merchant expended $547.40 for cloth. He aold t certain number of yards for $522, at $1.45 per yard, and gained on what he sold $108. How many yards did h< buy and how much did he gain yer yard on the cloth h< sold? 2. A farmer exchanged 390 bushels of wheat worth $1.20 a bushel, for an equal number of bushels of barlej at 75 cents a bushel, an T oats at 42 cents a bushel ; how many bushels f. ' each did he receive ? 8. John Turner has manufactured in 4 years 7740 paire of shoes, making each successive year 250 pairs more than the year before; how many pairs did he manufacture the first year ? 4. If 80 men have sufficient provisions for 75 days, and 20 men go a way, how long will they last the rest ? 6. The product of 275 and 86 is 23650 ; how much must be taken from the product to give the product of 276 and 82 ; and to give the product of 270 and 86 ? CHAPTER in. I' MEASURES AND MULTIPLES. Section 1.— Prime Numbers, Prime Factors, &c. 72' In the series of numbers 1, 2, 3, 4, &c., a dis- tinction may be observed of odd and even numbers. An Odd number is one which cannot be divided into two equal whole numbers, as 1, 3, 5,&c. An Even number is one which can be divided into two equal whole numbers, as 2, 4, 6, &c. 73. There is another, and a more important divi- sion of numbers into two classes, one class consisting of numbers, each of which is divisible only by 1 and a number equal to itself, as 2, 8, 5, &c.; and the other class consisting of numbers which admit of other (livisors, as 4, 6, 8, &c. The numbers in the former class are called prime numbers ; and those in the latter class composite numbers. (Art. 61). 74' A Prime Number is one which can be ex- actly divided only by unity and a number equal to itself. 15. The Prime Factors of a number are the prime numbers, which when multiplied together will produce it ; thus, 2, 2 and 3 are the prime factors of 12. Oral Exercises. 1. What are the prime factors of 30 ? The prime factors of 30 are 3, 2 and o, since these are the only prime numbers which multiplied together will produce 30. 2. Name the prime numbers from 16 to 63 ? from 53 to 101 ? 3. What are the prime factors of 12 ? 16 ? 16 ? 18 ? A XKTlyai- atftk +V1/1 nvi*viA -f-'n /if nwa ^-P 01 O OK O Or? O OO O OO O Is TTXXC.VB uii.\j viAV |_j-iijLii--' i.anjv\j±a •■ji. iii s Ml! i ^t i ij^ : tfiJ *' 34? 5. What prime factor is found in both and 9 ? 6. What prime factor is found in both 20 and 26 ? 67 lis >l 68 ■LKMINTABY ABITHMXIXa !■;; ilif 7. What prime faotor is oommon to 12 and 80 ? 21 and 28? 8. What prime factor is common to 85 and 60 ? 14 and 70 ? ya und 99? 42 and 48 ? 26 and 39 ? YG. To resolve a number into its Prime Factors. Ex. I. Find the prime factors of 105. 8)106 Pi'^iding 106 by 8, a prime factor, we have '66 ; dividing 35 by 6, a p»ime factor, 5)85 we have 7, a prime number, therefore the 9 prime factors of 105 are 3, 5, 7. Exercise xliii. Find the prime factors of I. 2. 8. 4. 48. 72. 81. 108. 6. 6. 7. 8. 175. 270. 160. 325. 9. 10. 11. 12. 429. 276. 800. 180. 18. li. 15. 16. 818. 836. 855. 1165. What prime factors are common to 17. 50 and 70? 18. 81 and 96? 19. 63 and 147 ? 20. 120 and 600 ? Section II.— Cancellation. 77' Cancellation is tlie process of shor.^ning operations in division by rejecting or cancelling equ^ factors common to both dividend and divisor. Ex. I. Divide 28 by 8. /--8^. ^8^_4X7 _ 8""" 4x2 ' 2 Write the divisor 8, under the dividend 28. Kesolve 28 into 4X7, and 8 into 4x2. Cancel the oommon factor 4 in dividend and divisor, and we have 7 divided by 2 or 8^. The same result will be obtained by dividing both dividend and divisor by 4. Hence cancelling a common factor frmn both dividend and divis&r does not chanye the quotient* Exercise xliv. 1. c\ . . t r\ UxVUXG io X ^ X u uy o X i: X au. 2. Divide 7 x 16 x 6 by 14 x 3 x 8. ftiid and me we stor, the 5. ling [ual olve ctor I or )oth iend ' THB HIGHEST COMMON FAOTOB. 8. Divide 9x7x16x16 by 21x82X2. 4. Divide 27 X 12 x 14 by 9 x 4 X 7. 6. Divide 72 x 46 x 140 by 18 x 24 x 36. 6. Divide 24 x 82 x 86 x 144 by 64 x 108 x 8. 7. How many yards of muslin, worth 12 cents a yard may be bought for 16 pounds of butter, worth 16 cents a pound ? 8. How many bushels of potatoes at 75 cents a bushel must a farmer give for 86 yards of carpet worth $1.60 a >yard ? ' 9. A tailor bought 12 pieces of cloth, each containing 22 yards, wortb ^i.iiS a yard; he made 27 suits oi clothes ; how much must he get per suit so as not to lose ? 10. If a farmer exchange 26 bushels of wheat at $1.20 a bushel for cloth at 40 cents a yard, how many yards does he get ? / 11. Three pieces of cloth containing 80 yards each, ^orth $5 a yard^wereexchanged for 5 pieces of cloth con- taining 46 yards each ; what was the second kind worth per yard ? 12. Divide the continued product of 16, 18, 24, 26, 86 and 46 by the continued product of 27, 72 and 100. Section IIL—The Highest Common Factor, Oral Exercises. Name a common faotor 1. Of 6 and 9. I 4. Of 16 and 20. 2. Of 12 and 10. 6. Of 12 and 18. 8. Of 27 and 24. j 6. Of 16 and 40. What is the highest common factor J 7. Of 12 and 10 ? 10. Of 24 and 72 ? 8. Of20andl6? , 11. Of24andl2? 9. Of 26 and 60? \ 12. Of 72 and 144? 78. A Common Factor of two or more num- bers is a number that will exactly divide each of the given numbers. 10. The Highest Common Factor, called oJ&A the (rTBfitist Gonitiion A1£o.sutb- of two or more numbers is the largest number that will exactly divide each of the given numbers. If! HI 70 BLEMENTABY ABITHMBTIO. f Ex. I. Fiud the highest common factor of 18, 86 and 72. 6) 18, 86, 72 We pikoe the numbers as in the margin. 8[^ 6^12 By dividing each number by 6, we tak» 1, 2, 4. out the common factor 6 ; by dividing each of the quotients by 3 we take out the common factor 8; since the quotients 1, 2, 4 have no factor common to all of them 6 and 3 are all the common factors ol the given numbers, hence 6x3, or 18 is their H. C. P. Hence to fuid the H. C. i^ of two or more numbers, iv$ divide hij anij common factor of all the numbers: we then divide the quotients iWthe same manner, and thus continue until the quotients have no common factor ; the product oj all the divisors will be the highest common / actor. Exercise xlv. Find the H. 0. F. 1. Of 15, 20, 30. 2. Of 16, 20, 24. 3. Of 24, 96, 80. 4. Of 28, 56, 42. 6. Of 80, 60, 60. 6. Of 84, 126, 210. 7. Of 120, 240, 72. 8. Of 44, 110, 77. 9. Of 75, 800, 450. 10. Of 144, 576, 720. 11. A man has two logs which he wishes to cut into boards of equal length ; one is 24 feet, and the otjier 16 feet long ; what is the greatest length into wTiich the boards can be cut ? 12. Wliat is the greatest equal lengths into which two trees can be cut, one being 105 feet in length and th« other 84 feet? 13. Three pieces of carpet, of 48, 64 and 80 yards, ii cut into the longest possiMp equal lengths, wiU'exactly cover a parlor floor, each piece being the length O) the parlor ; how long is the parlor ? 14. A grocer has 136 quarts of strawberries, ana 152 quarts of plums, which he wishes to put into boxes, each box to hold the same number ot quarts, and the largest number possible ; how many quarts may he put into each box? 15. What is the greatest number of pears you could buy with 180 cents, or 225 cents, or 315 cents, so as to get the same number each time ? 16. A certain sclioul coupi^ts of 1B2 pupils in the lower BcUuol, and 99 in the upper school ; how might each ol THE HIOHEST COMMON FAC5T0B. 71 thnae be divided so that the whole school ibould be dis- tributed into equal aeofcions ? 80. To find the H. C. F. when the num. i(^rs are large. Ex. 2. Find the H. C. F. of 91 and 143. n ) 143 ( 1 91 Tiuit which we are seeking to find is the largest number "~~ ^ tbat will divide both num- 02 ) 91 ( 1 bera. Now any number thai 52 will divide two other Humbert —- will also divide their difference 89 ) 52 (1 or their sum, and as we can 89 floe the factors of a small — number more easily than oi 18 ) 39 ( 8 a large one, we divide the 39 • greater of the two numbers . , ■— by the less ; then we divide ''iie less number by the remainder, and each former re- mainder by the new remainder, till we find a number that will divide the last remainder exactly. This will be the H. C. F. of the two numbers. To find the H. C. F. of more than two numbers, first find the H. C. F. of two of thorn ; then find the H. 0. F. of the common factor thus found and a third numher ; and so on through all the numbers. The last common factor found will be the H. 0. F. of all the numbers. Exercise xlvi. Fmd the H. C. F. of 1. 115 and 161. 2. 833 and 592. 8. 697 and 820. 4. 392 and 672. 5. 405 and 900. 6. 1220 and 2013. 7. 6006 and 8318. 8. 2871 and 4213. 9. 43902 and 49593. 10. 23940 and 28360. 11. 1435, 1084 and 2135. „ , , - 12. 14385, 20391 and 49287. 16. A grocer has two hogsheads of sugar, one containing 1104 pounds, an^ the other 1288 pounds. He wishes to put this sugar mto barrels, each barrel to contain the same number of pounds, and this the greatest number ~'.^ ■■■"■" i^«i^j yx.'xiii\.ir^ uiiisv eucu oarrei consist 7 nnssihiA c 14. A and B purchased horses at the same rate per head ; the value of A's horses was $623; and of B's $1068; what was the mimber purchased by each ? •'w? -I 72 ELEBIENTABT ARITHMITIO, .. Section IV.—Least Common Multiple. , Oral Exercises. 1. What number is three times 6 ? four times 7 f A number which is one or more times anotre »"«^ber is called a mwifip^c of that number. 2. \V?iat number is a multiple of 3? of 6 ? of 9 ? 8. Name two multiples oi 8 ; three multiples of 7. 4. What number is a multiple oi boih 4 and 6 ? 3 and 6 ? 6. What multiple is commor to both 8 and 4 ? 4 and 7 ? 6. Name all the multiples of. 4 from 8 to 30. 7. What is the least number of which 3 and 6 ar€ factors. 8. What is the least number exactly divisible by 8, 4. and 8. ' 9. What is the least number .exactly divisible by 10 and 12? by 8 and 12 ? by 6 and 10 ? by 12 and 18 ? 10. James has just enough money to buy oranges at 6 cents each, pears at 4 cents each, or tops at 6 cents each ; how much money has he ? 81. A Multiple of a numbor is a number that is exactly divisible by that number. 8^. A Common Multiple of two or more num- bers is a number that is exactly divisible by each oi the given numbers. Thus, 24 is a common multiple of 4 and 6, because it is exactly divisible by each of them. 83. The Least Common Multiple (L. C. M.), ol two or more numbers is the least number that ig exactly divisible by each of them. Ex. I. Find the least common multiple of 24, 20. and 88. 24 = 2x2x2x8 20 = 2x2x5 83 =r 3X11 L.C.M. = 2X'2x2x3X5xll = 1820. The L. 0. M. of the q:iven numbers nfust contain the factors 2, "v., 2 and fi to be divisible by 24; it must contain the factors 2, 2 and 5, to be divisible'by 20 ; it must con- tain the factors 8 and 11 to be divisible bv 88. S^Anoe. the- number 1320 contains all those lactors arid no others, it if the leait common multiple of 24, 20. and 88. LBA9T OOMMON MULTIPLE. 78 S«M$ to find ths L. C. M. of two or more immhers we find (he prime factors of the numbers^ and take the product of ihese facforsj using each the greatest numher of times it occurs in any of tht given numbers. 84. \VheD the peveral numbere are not large, the pro- C8B8 may be shortened by Buccessive divisions of tbe given fiumbeis, by pnme factors which are common to two or more of the given numberg. By this means, all the divisors will consist of the prime factors common to two or more of the numbers, and the numbers left after the divisijns will be the factors which are not common to any two of Ihe numbers. Then the product of these common prime fac':nrs, and the lactors which are not common, will be the least common multiple ol th.^ given numbers. Ex. 2. Find the L. C. M. of 15, 24, 8G, and 42. 2) 15, 24, 36, 42 Here 2, 2, 3 are the prime factors common to two or more of the numbers, and 5, 2, 3, 7 are the factors not common. L. C. M. = 2X2X3X5X2X8X7 - 2520. Exercise xlvii. Find the L. C. M. of 2) 15, 12, 18, 21 8) 15," 6, 9," 21 6," "2, 8," 7 6. 5, 9,. 12 and 15. 7. 12,15, 18 and 24. 8. 22, 55, 77 and 110. 9. 15, 30, 42 and 72. 10. 21, 54, 56 and 84. 1. 15, 10 and 6. 2. 20, 10 and 30. 8. 9, 12 and 18. 4. 10, 25 and 30. 6. 24, 30 and 30. 11. 6, 7, 16, 28, 48 and 21. 12. 16, 12, 14, 32, 50 and 75. 13. 15, 18, 24, 40, 50, 60 and 90. 14. The even numbers from 14 +o 28 inclu8rvq9voQ^- -J ^ by 482X25X8X30? ^*^ ^^X 15 Xd2x 23 divided 4. What is the least number of mnrhloB fUo* ^ i. d.vidoa equally among 10, 21, 24 » sTboys.*** """ ^' 5. A can dig 24 post holes in a day; Beau die 25 Rr,^ C SO,n the same time. Vih.t is ^he sr,ae7 '^^0' fhioh will fnmic!!-. ri— r^ ^--t ii J ^11 C^CiC mg Alone or for all working together? ^ u**j,ft moor eituer lor each work- ammttaimmm EXAMINATION PAPEBB. 75 III. 1. The product of four conseoutive mimberfl is 73440 ; find tlie numbers. 2. What is the least number of acres in a farm that can bo exactly divided into lots oi 12 acres, 16 acres, 18 acres or 25 acres each ? 8. A farmer sold 4 loads of apples, each containing 16 barrels, and each barrel 8 bushels at 60 cents a bushel. He received as payment 6 barrels of pork, each weighing 200 pounds ; what was the pork worth a pound ? 4. The product of two numbers is 152368,. and 7 times one of them is 2996 ; what is the other one ? 6. How many rails will enclose a field 7163 feet longbv 8816 feet wide, provided the fence is straight, 6 rails high, the rails of equal length, and the longe«fc that can be used? IV. 1. A farmer exchanp^ed 9 tubs of butter, each contain- ing 56 pounds, worth 25 cents per pound, for 4 chests of tea, each containing 42 pounds : what was the tea worth per pound ? 2. What is the smallest sum of money with which I can buy sheep at $5 each, cows at $24 each, oxen at $64 each, or horses at $135 each ? 3. Divide the continued product jL 61, 72, 144, 972, and 750 by the continued product of 9, 17, 18, 24, 36 and 45. 4. Find the least number which divided by 1595, 2530, and 3168, will leave the same remainder, 71P 6. The following are the prime factors of a number ; 2, 2, 3, 5, 6. 7, 11, 11, 13, 19, 89, and 227 ; find the number. V. 1. State and provo the rule for findini,^ th^ H. C. F. of tT^o numbers, and find the H. C. F. of 12&7000 ^nd 504504. 2. Find the L. 0. M. of 16, 24, ,md 80 d explain the method. 3. A school was found to contain such a number of boys, that when arranged in sixes, stv -lis, nines or twelves, there wero always live over ; how many children, %t least, did the school contain ? 4. The fore and hind wheels of a carriage are 12 and 16 feet in circumferen^'e ; find the lea?fc n ; .aber of revolutions of each that will give the same ,lon<»tii. n. Exnlain the terrns i^fciHu.Ti*. '^.xa o-o*^.v^. y^ji-k Aia Mrt a. . .^.A --«,1 prove, by means of an example measure of the dividend and divisi;? reiJLiainder. lat i.^ a every common measure of the iri CHAPTER IV. Ir I 111 [t FRACTIONS. Section I. Definitions* Oral Exercises. 1. If an apple is divided into two epaal parts, what in one of these equal parts called ? 2. How many halves are there in anything? Wxlt« down one-half. (See example 7, page 42). 3. When I divide an orange into three equal paj-tt, what is one of these equal parts called ? What are two of them called ? 4. How many thirds are there in anything? How many fourths are there in anything ? 5. How would you get fourths ? fifths ? sixths ? 6. How many thirds make a ?/hole? How many fourths ? sevenths ? tenths ? 7. Into how many equal parts mast a thing be divided to get halves? fifths? sevenths? eighths'/ 8. Two halves of an apj^e are equal to how many whole apples ? 9. What are four fourths of a pear equal to ? 10. Which are the smaller, halves or thirds ? Halves or fourths ? tliirds or fourths ? The value of the part varies according to the number ci equal parts into which the whcle is divided. The more parts it is divided into, the smaller they must be. Halt I Half Third Third { Third Fotirfch Fourth Fonrth Fourth One half of a thing is greater than one third ; one third is greater than one fourth. 85. A Fraction is an expression representing on© or more of the equal parts of ft \mit. VBAOnONS. 77 86. Fractions are divided into two classes, Com mon, or Vulgar Fractions, and Decimal Frac- tions. 81. A Common Fraction is one which is expressed by two numbers one placed above the other with a line between them ; thus four-fifths is written T » nine-elevenths, y\ ; ten thirty-fifths, ^f . 88. One of these equal parts is called the Frac- tional Unit and instead of the name of this unit being written after the number of such units as in whole numbers, it is placed under it. Thus, three apples is written 3 apples and 3 fourths, ^. 89. The number written below the line is called the Denominator or *' name-giver'' because it indi- cates the name of the fractional unit, i. e. it shows into how many equal parts the whole is divided. !>0. The number written above the line is called the Numerator, i.e. the ''numbered' or "counter," be- cause it indicates how many of the parts named bv the denominator are to be taken. 91. The Terms of a fraction are the numerator and denominator. Thus, | is a fraction— 5 and 8 are its terms. 0^. A Proper Fraction is one whose numerator is less than its denominator. Thus, ^, §, -J are proper fractions. 03. If we cut an apple into tivo equal parts, one-half will be represented by i. If we cut an apple into four equal parts, one-half will be represented by f . If we cut an apple into eight equal parts, one-haK will be represented by |. .-. i = I == |. Similarly, If we cut an apple into three equal parts, one-third will be represented by ^. If we out an apple into nins equal parts, one third will be represented by .|. .'. f (auds for the word tli^sfor: 78 ■LBMENTARY ARITHMETIC. )f I' If we cut an apple into eighteen o(jual paii*8, onc-tiiird m]\ be rejji-eseutecl by y^^. 1 7 3 a Hence, we conclude the value of a fraction is not altered by multiplying or dividing both its numerator and its denominator by the same number. The following is another proof cf this important proposition : B E Mill I I ! II ) II ! I F D and but CD = j% of EF, AB = -I of EF ; AB = CD; I 15- 94. By the help of the following proposition, whicJi is best explained by an example, we shall b© able to obtain another definition of a fraction. Ex. I. Prove that | of 1 = -J- of 2. Since 1 = j^w-fifths of a unit, 2 = ten-MtliH of a unit ; ,*. -J^ oi 2 :^ ^ of len-Mths of a unit = ttvo-Mths of a unit = *otl; .-. i of 2 = 2. of 1, Hencej we may define a fraction as a simple manner of indicating tlmt its numerator is to be divided by its de- nominator. 95. Since 8 apples multiplied by 2 = 6 apples, BO 3 eighths (f ) " 2 = 6 eighths (|) ; .. iS A -g- -35-. Ilertce to multiply a fraction by a whoU number^ ws dm/fly muUlply the numerator by iht whole number^ and retain tie denominator. rRAOTKmS. 79 Sinae 8 marbles divided by 2 = 4 marbles, •o 8 ninths (|) " 2 = 4 ninths (J) \ .. -g- -r ^ «= -jj-. Hence a fraction is divided by any numher by dividing th" numerator by tJie number and retaining the denomin- ator. 96. From the preceding article it appears that fractions may, in general, be treated as whole num- bers. They are in all respects exactly like other do- nominations, such as a cent, which is -j-^ of a dollar, a shilling, which is ^^j- of a pound, or a penny, which is y*5- of a shilling, only it is impossible to find a name for every kind of fractional unit which wc can employ. It is enough to indicate the unit, as jJy, 6 of them being -j^y. Section II— Reduction of Fractions. J Case I. 97. To reduce whole or mixed numbers to improper Fractions. 9^. A mixed Number consists of a whole num- bar and a fraction ; as 3^, 4|, &c. 99. An Improper Fraction is one whose nu- merator is not less than its denominator. Oral Exercises. J 1. How many halves Id 5 apples ? In 1 apple there are 2 halves, and in 5 apples there are 6 times 2 halves, or 10 halves. 2. How many halves in 6 ? In 10 ? In 13 ? In 40 ? 3. How many fourths in 4 ? In 6 ? In 9 ? In 12 ? 4. How many fifths in 4j ? In 1 there are 5 fifths, and in 4 there are 4 times 6 fifths or 20 fifths, which added to 3 fifth*, make 23 fifths ; tiioreforo 4^ = y " - 6. How many fourths in 7 J ? In 5f ? In 9| ? In 125? «. How many aixths in 4f ? In 8| ? In 9j f In llj:-* 80 BLBMENTARY ARITHMBTIO* Ex. I. Beduce 27i to fourths. 27i = 27 4- i ; Now, 1 = . .-. 27 •.-. 27 • «-^« 4 . 37X4 108 + i = ~' 4 + j = LU 37x4 + 8, 4 ' * — 4 Hence, to reduce improper fractions to mixed iivmbtrs^ wt multiply the whole number by the denominator of the fracf tio7h, add the numerator to the product, and write th deuominator under the sum. Exercise xlviii. Keduce to improj^er fractions 7. 85]^. 1. 2. 8. 4. 6. 6. 8^. Vtfm 12^. 8. 9. 10. 11. 12. 82^. 99yy. 18. 14. 15. 16. 17. 18. 237A^- 5¥* 304^VV. 1806^9^. 2500|f4. 1001,Vt. 2897Jif. Case II. 100. To reduce improper fractions to whole or mixed numbers. Ex. 2. Beduce -fr to a mixed number. Since dividing both terms of a fraction by the same number does not change its value (Aifc. 98), 87 _ 87 -=-ll TT - ri=n 710. -=-¥=m- Hence to reduce an improper fraction to a whole or mixed number we simply divide the numerator hy tlie denominator. Exercise xlix. Beduce the following fractions to whole or mixed numbers : 1. 2. 8. 4. 5. 19 81 Y • 40 T • 7a 99 T- 193 7. 8. w 983 9. 736 10. 330 11. 476 XT 12. S82 IT 18. 14. 16. 16. 17. 18. 4407 TSIS ' 6919 69 17007 32460 " • 48 i 8 6 6_0 ^ 6 9 4 139 frao e tht 3 6 r^' 9 PTT' 84 rrsr* 1 1 ' 16 iiole jT the *le or i tlw ixed 101. terms. FRAOnOMg. 61 Case IIL To reduce a fraction to its lowest 10^. A fraction is in its Lowest Terms when the numerator and denominator have no common fac tor. Ex. 3. Reduce jY? *o its lowest terms. 4 8 _ 12 _ 4 TTTir — TT — TT* Dividing both terms of ^ by the common factor 4, reduces it to ^; dividing both terms of this fraction by 8, reduces it to J. Since $ has its numerator and denominator prime to each other it is in its lowest terms. We might have found the H. C. F. of the numerator and denominator and divided both terms by it at once. tlence to reduce a fraction to its lowest terms tve divids both terms by a common factor, and the result again by a common factor, and so on till the terms have no common factor. Or we may divide both terms of the fraction by their Highest Common Factor, Exercise 1. Reduce the following fractions to their lowest terms. 1. 2. 8. 4. 6. 6. 7. 8. 13 inr* 18 **• 94 14 TTT' 80 TT7* 2 1 9. 10. 11. 12. 18. 14. 15. 16. 288 792 840 1176 T7T2f eiBl TTTT' 5400 I 17. 18. 19. 20. 21. 22. 28. 24 9 41» 304 660 §72 lg84 ♦135 7315* 276 TsM' Case IV. ii9«i. i u icuuvic i* v^^umpound "r* ractlon to a Simple one. !t •82 ■LEMKNTABY ARITHMBTIO. 104. A Compound Fraction is a liaction of « fraction ; as J of f ; f ot |, Ac. 105. A Simple Fraction is one in which both numerator and denominator are whole numbers ; M t, I, &o. Oral Exercises. 1. What is i oi 6 apples ? Of 1 boys ? Oi 16 oents ? 2. What is ^ oi 6 ninths? ^f 10 elevenths ? Of 16 twentieths ? 3. What is i of I? Since ^ of 6 apples = 8 apples, i of 6 ninths («) - 8 ninths (J). 4. Whatis 1 otf2? ot|J? of J-f ? of^Sg? 6. What is 1-01 «? |of«? |oi|f? J ol §J. ? 6. What is i of i ? i = 3 7. What is ^ ot ^ ? ^ oi ^ ? i. of 4 ? 8. A boy had -^ of a dollar, and lost ^ of it ; what part of a dollar did he lose ? 9. A man owned \ of a farm, and sold J of his ehpre ; how much did he seU ? 10. I had I a ton ot coal, and gave my neighbour -J of it ; he gave his brother ^ ot his share ; how much did his brother get ? Ex. 4. Keduce | of J to a simple fraction. 7 44 y 01 18 = i 01 ^^ = :j^ , I of ^ = 3 X .. , 7 21 F — *^'^TT7 — TV — the product of the nntnorators the product of the denominators * 3X7 « X 8 Hence to reduce a compound fraction to a simple one^ multiply the numerators together for a new numerator j and. the denominators together for a new denominator. Note. — Before performing the multiplication, mixed «%•« w* r\rw«ei n l^ /Nt^ 1 rM w\r\ inr\i\-\^ r^j^rX 4-«-\ v-v»«/~kv^r««« *w .;i ^. fftolor common to a numerator and denominator cuaoelled iHAOTIONS. 88 * of ^j. 2. J of /^. 8. 4. 5. 6. 7. 8. 9. i of 4i. 2 J of |- . of $of $ of ^, of |. of I of 2 J. ^ V,. f of 1 of ^. * of 3 3 TT 4ol 01 ¥ 9 Tiff 7' Of A. Exercise U. Simplify the following fractions 10. i 11. ^ 12. 18. 14. 16. 16. 17. 18. of f of ^ of ^. of^of Aof 3 TT ^* TIT "* ¥• of I of ^V of 7. of j\ of IJ of 5|. T-Vofifof9t. 1 3 I* of ^ of 8| of i of 2|. y*ff of I of i of 4i. f of f of I of 9. i of ^ of 3|f of 6. Exercise lii. 1. Some boys owned f of a boat ; they sold f of their share ; what part of the boat did they sell ? 2. Having ^ of a bushel of potatoes, I gave away i oi what I had ; what part of a bushel did I give away ? 8. A boy had ^^ of a dollar, and spent f of it ; how much did he spend ? 4. A gentleman owning f of a factory gave f of what he owned to his son ; what part of the whole factory was the son's share. 5. A has ;^ of a ton of hay, which is | as much as B has ; how much has B ? i of what B has = ^ of a ton ; .'. i " = i of y of a ton = t^ of a ton; .*. i ** = 4 X A of a toil = if of a ton; .:. B has ^f of a ton. 6. A owns Y of a railroad, and f of this is 8y times what B owns ; how much does B own t -> 7. How many acres of land has B, if ^^ of 18 is -^j of his number ? 8. A's money equals y^ of $8760, and A*8 is fj of B'a money ; how much money has B ? Case V. 100. To reduce Fractions to equivalent ones having the least common denominator. r ^j. #r- iACiiV'fniVJi/UiiO'ili be the fractions in their simpl^t form. prvnie to eacfi o titer atui I f I I .1 4i I M I A#o. IMAGE EVALUATION TEST TARGET (MT-3) 1.0 I.I 2.2 2.0 18 'yi i u 1 1.6 Sdences Corporation 23 WEST MAIN STREET WEBSTER. N.Y. 14580 (716) 873-4503 ■1>^ iV 5v 'v^ 4^^^. % r-u^ M ELEMENTARY ARITHMSTIG. Ex. 6. Reduce }, |, and ^ to their least ooannoD demcmmator. Since multiplyiDg the terms of a fraction by the same niunber does not change its value, we have, 3 -if 4X8X8 4X« X3 - 40 - JTCff 3 -_ 3X5x3 45 * 8 X a X 4 XT' 3. 4. 5. 3 3 ff 3» ir» T 6 v. 4 T» 3"' If 3 2 3 «» 3> ¥ 7 3 13 1 '• 6» ^> T» ¥• ft » 3 4 11 *^' 7» ¥' 7> TTT* Q 3 4 8 ^' 7' If' lEr» TT* ^. Let 'he denominators be not prime to each other, Ex. 6. Reduce f , f , «, and J to their least com- mon denominator. L. C. M. ofS, 4, 6, 8=24. 2 _ 8X3 ^ 8X3 — 1« — ^-4- 3 ^sm — 18 *^ ~ 6 X 4 ~ ITf 4X5 — 20 We find the L. C. M. of the denomi- nators to be 24, henoe 24 is the least common denominator. Dividing 24 by 3, the denominator of f , we find we must multiply 8 by 8 to produce 24, similarly with the other fractions. 7 —JLll — 21 ff ~~ 3 X 8 — 5f T" to reduce fractions to a commmi denominator ^ lie least common multiple of the denominators^ divide this by each denominator, and multiply both terms of the fraction by the quui,vent. r V ' FBAOTIONB. 66 Exercise liv. Beduoe the following fractions to their least oom- mon denominator. 1. 2. 8. 4. 6. 6. 8. 9 3 7j "B"' igr« 13 7 3' T» -g-' 7 9 7 ¥» TTT» a7» « 7 11 IF' T!r» Tff' 4 7 7» XT' 3 7 6 T» TTT' ff* 7 17 11 6 4 2 ¥» Tr» 7* 7 3 4 ¥» ¥» Z' 10. 2, T»„ 4. 11. 2f, 8^, f. 12. h H, S- 18. 8, H, tV- 7 6 6 4 6 9 T» Tr» TT» 4 4.1 4 O 7 17 11 6 6 Tir» "ST' T8» TT* 18. 2i, 7i, 8|, 4tV i. 16. 16. 17. if. 6. lOT. Comparison of Fractions with respect to Magnitude. To compare fractions we must express them in terms of the same fractional unit, that is we must hring them to a common denominator. "When they are so expressed they are compared as other numbers are. Ex. 7. Arrange the fractions $, $, y^j, in order of magnitude. Eeducins; to equivalent fractions having a common denominator we have JJ, ^, §^ ; hence the order o^ magnitude is $, ^y f . li two fractions happen to have the same numerator, that which has the smaller denominator is the greater; for its units are greater, and there are the same num- ber of them in each. Exercise Iv. Find which is the greater, or 1 * 2. f _^ 3. tV or i?. or||. 4. ^V or 1 ' TTT 21 27' 6 7* or /t- 7. _ 8. :^ or li. 9. A or iJ. 6. ^or 6. if or if Which is the greatest and which the least of the following : 10. xVi?.^^ fi ig " 11 12. TT» Tf7' ^TJ I. Tff» ff» 77 18. A» hh U ? 14. 9 13 , TT» TJf' FT 15. f , ^, « ? ^^ ■LBMHNTABY ABITHMETia Arrange in ascending order of magnitude ^*' ¥» if it ii i^* 16 Tr» iV> ¥7' Ti nF» ?• I Section III. Addition. Case I. 108. To add proper fractions. Oral Exercises. 1. What is the sum of 2 apples, 3 apples and 5 apples f 2. What is the sum of 2 elevenths, 3 elevenths and 5 elevenths ? ' 8. How many ninths are |, ♦, ^, and J? 4. James paid ^ for a slate, f f for a reader, and $♦ for an arithmetic ; how much did he pay f^r all ? 6. Mary paid $* for some ribbon and $| for a pair of gloves ; how much did she pay for both ? 6. Jane bought | of a yard of ribbon at one time and J of a yard at another time ; how much did she buy at both times ? f = I, and 1+7 ^ i^a ^ xf . She, therefore, bought 1 » yards. 7. A farmer sold ^ of his grain to one man, and ^ of it to another ; how much did he sell altogether ? 8. If I pay J of a dollar for butter, ^ of a dollar for eggs, and i a dollar for cheese ; how much do I pay for all? 9. What is the sum of ^ and J- ? of i and J ? of i and 8 8 4 4 10. What is the sum of i- and i ? of i and # ? of 1 and I? » 8 7^ 7 Ex. I. Find the sum of |, |, and ^. 4.4-» J- 3 12 I 25 19 46 23 1 8 In this example we are required to add fifths, sixths^ and tenths together. As the addends have not the same name, we cannot add them till they are changed into others having the same fractional unit. Wc, iberelore, change the fractions into others having a VBAOTiONf. 87 common denominator, we then add the numerators together for a new numerator, and call the sum SOths. We reduce the improper fraction to its lowest terms, and then to the mixed number, 1^^. Hence, to idd fractions, we reduce them to others hav- ing a common denominator ; we then add the numeraton together for a new numerator and place the sum over the common denominator. Exercise Ivi. Add together the following fractions : 1. 2. 3. 4. 5. 6. 1 and |. ^ and J. A and tV I, ^ and |. h i and 4. 1^, ^z and axP Cas( 7. -|,f,|andT-V 8. h tV' f and ^%. q 3 4 13 ar\(\ 7 10. i,i,|andf 11. i^.UandJf. 12. ^,i,TVandyV- J II. 109. To add mixed numbers. Ex. 2. Add together 2|-, 8|, 7t^ and ^ 2. Add together 2|-, 8f , 7t% and 2i + 8J + 7t% + tV = 2 4-8 + 7 + 1: + ^ + t\ + ^^^ IQ4. 15 4. 48 I 26 _L 43 = 12+ VTr^ = 12 + 21 = la. 7 ITT Note. — ^When there are mixed numbers in the example we add the sum of the whole numbers to the sam of the fractions. Exercise Ivii. Find the sum of the following fractions : 1. 2. 3. 4. 5. 6. 8^- 91 24,84 1« 1 and 4J. and 7|^. and 4^5. and T^^. i and 24. Tir» ^3^ 80|, 4x»v and 10}. 7. 2J, 4J, 7tV and 8^^. 8. 7f, 10T^„4Jand7/^ 9. H, 8J, 2f and 5^. I 10. 41, 5i, 8Jand9^V 11. 4^, 3^, 4f and ^\, 12. 6|, 6J, 8f and |i. HV " 88 ELEMENTARY ARITHMETIC. Section IV. Subtraction. Case I. 110. To subtract one fraction from another. Oral Exercisee. 1. John has 7 marbles, James has 4 ; how many mar- bles has John more than James ? •" 2. John has yj of an apple, James has -j^y of an apple ; how much has John more than James, ? 8. How much less is f than ^ ? | than f ? f than -f ? 4. John has ^ of an apple, James has J of an apple ; how much has John more than James ? I 7 4 and J 3 . 1 7 1 _ 4 6. A boy spent ^ of his money for a coat and i of it for a hat ; how much- had he left ? 6. What is the difference between ^ and ^ ? f and ^ ? 7. What is the difference between | and t^ ? between iandi? Ex. I. From ^ take ^. 7 _ 7 S 6 2 I . ,_ 14 T TJ ?Cr — Tff ^ir — ^TF - - TTT In this example we are required to take twentieths from twelfths. As we can only subtract numbers that have the same name, we must change the frac- tions into equivalent ones, having a common denomi- nator. ^ becomes t#, and i^ becomes ^y We be -j^ becomes f f , and ^ now find the difference between |^ and |^ + which reduced to its lowest terms is ^ 14 ^TT* Hence to subtract one fraction from another we re- duce the fractions to otiiera Iiaving a common denomina- tor; we then subtract the numerator of tJie mjihtraJtend from the numerator of the minuend ; and place the difference over the common denominator. Exercise Iviii Find the difference between 1. ^ and %. i 6. a. |andf. 6. 8. I and |i. 7. 4. ll and |. t 8. 2¥ ^^^ tI* landy*,. I and i%. 9. 10. 11. 12. and ^. 2 1 'It and 1 ^ -»- and ^ 7T ^¥* FKAOnONS. 89 Case II. 111. To subtract when one or both the fractions are mixed numbers. Ex. a. From 16^ take 9f . m = 16A ; 16 A + « = 16f J. 9i = 9A + 1 = 10^ TV 6^ Tar* We first reduce the fractional parts of the SP^vt>n numbers to their least common denominator, 12. Then, since -^ cannot be taken from y*^, we add 1 or If to both minuend and subtrahend, (Art. 88). This gives 16||- for minuend and lOyH,^ for subtrahend. Then subtracting the fractions and integers sepa- rately we have 16|J — lO^^^j = 6^^, the required result. Note. — The fractional part of the result can be obtained much more readily by subtracting the numerator of the fraction in the subtrahend from the common denomina- tor and adding the remainder to the numerator of the fraction in the minuend. Thus 12 — 9 = 3 ; 3 + 4 = 7. This is the numerator of the fractional part of the result. The integral part is obtained as above. Exercise lix. Find the value of 1. 8|-1|. 2. 21-1^1^. 8. 6^-2^. 4. 8f -6-}. 5. 6. 7. 8. .9 * . 18 9«— 1 7 K 3 O20 47—1 1 3 9. 10. 11. 12. 28^1 43 6* 8A XT' QK 7 \Alo Exercise Ix. Practical Problems. 1. The sum of two numbers is 26^ and the less is 7f ; what is the greater ? 2. From a barrel of vinegar containing 81| galloiif, 1444 gallons were drawn ; how much was there left ? 3. To what fraction must the sum of ^ and y^ be added, that the sum may be 44 ^ 4. From a piece of silk containing 36^ yards, I4f yardi «vere sold ; how much remained in the piece f 1 1 I 1 90 ELEMINTARY ABITHBiKTIQ. b From no, $21 were given to James, $81 to Jane. Sli w Emily, and the remainder to Mary ; what did she reoeire ? 6. A has two farms, one of 70 J acres and the other of 118^ acres; if he sells 87if acres, how much land has he leit ? 7. How much paper has a printer left if he had on hand 30^^ reams, and has used 7j reams for one job. and 84 reams for another ? ^ » 4 8. A grocer, having mixed 16^ pounds cf tea with WjAl pounds of a different kind, sold all the mixture but isi pounds ; how much did he sell ? y. B started on a journey of 100 miles ; the first day he travelled 30x\ miles, the second day 86| miles ; how far has he yet to go ? 10. Henry had $47i, and James as much lacking $9 A- ; how many dollars .had James ? » i o » 11. The selling price of a horse was $126|; the gain was $i6f ; what was the cost price ? 12. It^nd the sum of the greatest and least of the frac- tions I, 3^, ^, ^, the sum of the other two, and the dif- ference of these sums. Exercise Ixi. Practical Problems. 1. $249| is $134 J less than the value of my horse anfl carriage ; what are they worth ? 2. A boy paid ^ for a ball, ^ for a slate, ^ for a knife and $^ for a book ; how much did he spend ? 8. What is the entire weight of 4 crocks of butter weighing as follows : the first lOj pounds ; the second, llf pounds ; the third, ISy^ pounds, and the fourth 14* ponndi) f 4. Three men bought a horse ; the first paid $414, the second paid $53 j^, and the third as much as the other two ; what was the cost of the horse ? 5. A grocer has 3 barrels of molasses ; the first contains 271 gallons, the second, 42^^, and the third, 36| gallons; how many gallons are there in the three barrels ? 6. What number is that from which if 6-[^ is taken the remainder will De 2j| ? 7. A merchant sold 84f yards of cloth for $94/^, 39|4 FRACTIONS. 91 yards for 1124$, and 70f yards for 9184-| ; how many yards of oloth did he sell and how much did he receive for the whole ? 8. Four geese weigh respectively 9f , lOf, 12^ and llj^ pounds ; what is their entire weight ? 9. A lady hired a gardener at 15 cents an hoar for 8 days ; how much did she pay hina if he worked 6^ hotirs the first day, 7% the second, and 6j the third ? 10. If 6$ gallons of brandy are mixed with 1-^ gallonb of water and S^j gallons of whiskey, how many gallons are there in the mixture ? 11. A paid $46^ for an ox, and $57^7 more than this for a horse ; for how much must he sell them to gain $26}- ? 12. A owns 7 If acres of land, B owns I12y\ acres ; owns 217iJ acres, and D owns 872j[-J acres; how many acres do they together own ? Section V. Multiplication of Fractions. Case I. 11^. To multiply a fraction by a whole number. Ex. I. Multiply A ^y ^• Since 8 apples multiplied by 6 = 16 apples, so 8 tenths {^) multiplied hy 5 = 16 tenths (■^) ; • K v^ — -14.— -?-><•.. But »-^ = |(Art.77) = j^^- Hence, to multiply a fraction hy any number we eithei multiply the numerator or divide the denominator hy it. Oral Exercises. 1. What is the cost of 6 yards of cloth at f of a dollar a. yard ? Cost of 1 yard = $| ; " 6 yards = 6 X$i = $ V = $4^. 2. If a man earns f of a dollar in 1 day, how mnoh can he earn at the same ratis in 12 days? 3. If a yard of cloth costs y of a dollar, how much will 9 yards cost ? 4. If a bushel of potatoes costs I of a dollar, what will 5 bushels cost ? 92 BLEMBNTABY ABITHMETIO. {J If » yard of ribbon costs f of a dollar, how much mil 10 yards cost ? 6^ If a basket holds $ of a bushel of apples, how manv bushels will 9 baskets hold ? 7. If a barrel of flour costs $8^, what will 6 barrels BOSt? Multiply the fractional and integral parts separately and add the products. 8. If a yard of cloth costs »2|, what will 7 yards cost? 9. How much is 6 times | ? ^^ir ? 44 ? J ? 10. How much ie 8 times -ff ? y^vj- ? f ? « 9 11. How much is 10 times l,^^ ? 2/^ ? 'g'i ? g^*^ ? ^ 1. 2. 8. Multiply H by 9. n by 8. A by 7. Exercise Ixii. 4. 5. 6. ^1 by 10. 7. 8. 9. n by 21. A by 24. iH^ by 86. f?- by 49. ri by 91. 10. At $1^ a day, how much doeg a man earn in 4 weeks, of 6 days each? 11. What is the cost of 86 dozen eggs, at 86^ cents a 12. At U6i a month, what wiU a boy earn in 12 months ? "^ 13. What is the cost of 12 pounds of beef at 14i cents a pound ? * 14. What is the cost of 14 bushels of oats at 62* cents a bushel ? * Case II. 113. To multiply a whole number or a fraction by a fraction. The definition of multiplication given in Art. 85, has no meaning when applied to fractions. Since the multiplier must always be an abstract number (Art. 40) we can no more multiply by | than by 2 apples.' We must, therefore, extend the meaning of the sign x when we apply it to fractions. ' ' It w found convmient to agree that f x J shall mean I 9f h IBAOnONS. Ex. a. Multiply | by f . i = ll. iof J = iof|J,orA; that is, f X J = T^ ax4 3X« __ th« product of the numetatura the product of tb« denominator* Hence, the product of two fractions is found by multi- plying the two numerators together for the numerator ^ and the two denominators together for the denominator of the product. In a similar way it may be shown that the product of more fractions than two is found by multiplying all the numerators together for the numerator, and all the denominators together for the denominator of the product. Oral Exercises. 1. "What is the cost of f of a yard of cloth, at $8 a yard ? 2. What is the cost of ^ of an acre of laud, at $240 an acre? 8. How much is i of ^1,^? ^X^^ fof^? fX^? 4. i of $60 is 8 times what a shawl cost ; what did the shawl cost ? 5. A horse cost $120, and a wagon -^ as much ; what did the wagon cost ? 6. If a ton of hay costs $25, how much will | of a ton cost ? 7. If ^ of 14 yards of calico cost 60 cents, what is 1 yard worth ? 8. What will J of 70 pounds of coffee cost, at 85 cents a pound ? 9. A father is 48 years old ; his son is f as old ; how much older is the father than the son ? 10. If ^ of 68 yards of caUco are worth $8.24 ; what are f of 16 yards worth ? Exercise Ixiii. Note. — Cancel the factors common to the numerator! ftnd the denominators. i 94 ■LSMBNTABY ABITHMBTIO. Find the value of 1. 2. 8. 4. A S 8 TF li ir 7 X 18. X 46. X 48. X 124. 6. n X A. "• TO" ^ UT X 75* 7 1 X .^ X 1* ft I V '^ V 7 '^* IT X TIT X Tff* 9- JSxilxlJ. 10. ifx^xj^. 11. A X if X ti- 1Q 319^20 V^ 18. What should be paid for i of | ot a pound of tea, at the rate of ^J of a dollar per pound ? 14. What should be paid for f of a barrel of apples, ii the whole barrel is worth ^^ of a dollar ? 16. A has i of $876, B has $ as much, and C { as much as both ; how many dollars has each, and how many have they all ? Ex. 3. Multiply 61 by 7$. 7^ = V, and 6f = V*. 7f X = V X V= ¥ X ^=52. Exercise IzIt. Find the value vu 1. 8i 2. 6^ 8. X 6^. V 17f X 16f. 4. 89| 6. 6^ 0. 8 X 7i X 88i. x77 xUx X4A X 4^, 7. If a cord of wood costs $4^, what will 8^ cords cost? Oo*tofloord =$4^; " Si cords = 3i X $4i = jx$v=^4-'=«i4;. 8. Tf » pound of sugar is worth 9^ cents, what will 4^ pounds cost ? 9. If a man reaps 8^ acres of wheat in a day, how many acres could he reap in 2f days ? 10. What would be the cost of 18| acres of land at $18f per acre ? 11. If a ton of coal costs 6^ dollars, what will be the cost of 9i tons at the same rate ? 12. Mr. Jones rented a house at $42f a month, taking I, lease for 6 years ; but disposed ol the lease at the end of 8i years ; how much rent did he pay ? 13. A bill of books at retail amounts to $376f, but I got a reduction of ^ for wholesale and -^^ for cash ; what was the exact amount of the hill ? FMAOnONS. 1 Seotir::! VI.— Division of Fractions Case I. 114. To divide a fraction by an integer. Oral Exercises. 1. If 8 applea are divided by 4, what i i the quotient f 2. If 8 ninths are divided by 4, what is the quotient ? 8. Divide 4 by 2 ; tt by 8 ; | J by 8. I. If 3 ducki cost xiT of a dollar, how much will 1 duck eost? Gostof 8duoks = $/{f; " 1 duck = i of %^% = $,2y = $1 - 20 cents. 5. If 8 caps cost jjf of a dollar, how much will 1 cap cost ? 6. William had yy of an cran»e, and divided it eqi^ally among 3 of his schoolmates ; what part of an orange did he give to each? 7. A man shares | of a ton of coal among 6 persons, how much does each get ? Share of 6 persons = f of a ton ; ** 1 person = ^ of f of a ton ; = Y^y of a ton. 8. If 8 men can do § of a piece of work, how much unit 1 3. If 1 tenth of An apple is divided into ten equal parts what part of the whole apple is 1 part 1 3 parts ? 9 parts ? 4. How are hundredths got ? How are they got from tenths 7 5. What part of 1 tenth is 1 hundredth ? How many 1 hundredths in 1 unit ? in 1 tenth 1 6. If 1 hundredth of an apple is divided into ten equal I parts, what is the fractional unit called ? jT 7. How many thousandths are equal to 1 hundredth f ^ To 1 tenth ? To 1 unit ? 8. What is ^ of ^j^ ? J-of -1^ ofyV m "^ TUT) ' 121. A Decimal Fraction is one which has for its denominator 10, 100, 1000, or some power of ten. 12/8. The Power of a number is the product ob- tained by multiplying the number by itself one, or more times. Thus 9 is the second power of 3, for 9 =8x3. 27 " third «' 3, for 27 = 8 x 3 X 3. 81 " fourth " 8, for 81 = 3 X 3 X 3 X 8. 123. The Denominator of a decimal fraction is never expressed, but is always miderstood. For brevity decimal fractions are usually called Decimalt, A decimal fractiuii is expressed by writing the Numerator with a poiut (.) before it. DBOIMALflU 106 Thus, -fif is wiitten •!. •8. •01. •79. •001. •189. 3 Tff ToTF 79 1 T6WV . t3» TTTDUr it « «« l< <« 121. The Point placed before decimals is called the decimal point. It separates the fractional part from whole numbers. 125. The ^first place to the right of the decimal pomt is that of tenths ; the second place is that of hun- dredths ; the third, that of thousandths ; the fourth, that of ten-thousandths ; the Jifth, that of hundred-thousandths ; &c. Thus, 23-046 = 2 X 10 -|- 3 + A + T^ + tAif- Hence it appears that decimals are simply an ex- tension of the ordinary system of notation and nu- meration. 120. Zeros affixed to a decimal have no effect on its value ; that is -9, -90, -900 are all equal ; for '9 = A .qn 9 9 127. To convert a decimal to a vulgar frac- tion. Since 'SIS means 8 tenths, 7 hundredths, and 8 thousandths ; ,*. '878=^ + jjpf + looff ^ 300+70+8 1000 378 Similarly "00807 means 3 thousandths and 7 hundred- tiiousandths ; .-. -00807 = Tinny + Trnnrmr 3 00+7 — lOOOOU" — = 3 7 ^ — i u « u wTT * HmM to express a decimal as a vulgar fraction wfiu the given decimal & whole number for the numerator ^ ' i }• i . I ■ t 106 BLSmNTABT ARITHmnO, ths vulgar fraetion^ and for the denominator writs 1 followed by as vtany ciylier* as there are decimal places in the given decimal. Conversely a fract'on having 10, 100, 1000, &o., for denominator may be expressed as a decimal by writing the yiwruirator and counting off from the right at many figures as there are ciphers in the denominator. Thus Sj\\% = 3-175, and yj^ff - -076. Exercise Izx. Express the, following decimals as common frao- tions : 1. -7. 6. -4128. 2. -86. 7. -0614. 8. -08. 8. -0078. 4. .-784. 9. -7614. 6. -709. 10. -3005. 11. -00427. 12. -00036. 18. -02007. 14. -712465. 16. -000006. Express the following fractions as decimals : 24. 126tVA. 25. 26. 27. 16x^^. 16. 17. 18. 19. TtF* 27 7 TJSlf' 20. 21. 22. 28. 1 3 6 A 1 6 1 8496 Q 7 'lOOOOO* Exercise Ixxi. Write the following decimals in words : 1. 2. 8. 4. •9. •27. •368. •064. 6. 6. 7. 8. 4-81. 7-216. 8-814. 6-8167. 9. 10. 11. 12. 21-8601. 17-0064. 18-00081. 20-01458. Express in figures the following : 18. Eight tenths ; two, and seven hundredths ; nina thousandths. 14. Eight hundred and seven, and ninety-four thou- sandths ; three thousand and seventeen, and seven hun- dred and nine ten-thousandths ; three, and one thousand _ ■» •_! 1 -ii-- -■•■'- *■' 16. Six, and four ten-thousandths ; eighty, and six hun- dred and nine ten-millionths ; one hundred and one, and one thoaiaad and one hundred- thoaiandtha. f DBOIMALf. 107 Section II. Addition. l!8g. To add Decimals. Ex. I. What is the sum of 8-7, 14-086, 81-64 and •7166 ? 8-7000 8'7 140860 14-085 81-6400 or 81-64 •7166 '7165 ^ 100-0916 100-0916 Since we can add figures of the same name only we write the addends so that units will be under units, tenths under tenths, &c. This is always the case when the points range in the same straight line. Then beginning at the lowest order we add as if the figures were integers and place the decimal point in the sum before the tenths. Exercise Ixxii. (1) 42-8 18-06 8-049 1-6 •037 (2) 12-82tt 204-00 8-8024 52-007 824-1 (8) 4031-06 108-304 9-001345 76-739 260-0007 (4) •608242 •0816044 •8084 -086 •9106 > n« m- Qd in- nd Find the sum of 5. 4-6 4- 70-63 4" 1*079 + 25. b. -126 + 8-06 + -07 + -628 + 7-098. 7. 111-306 + -03174- 2-793 -|- -007. 8. 470-05 -f 72-701 + 8-0316 + 413-2658. 9. 12-3987 4- 4-1462 + -02063 + 18 + 10 962 10. 210-7+14563-21+ -0173 + 382-74156. 11. 9-127 + 17-72 + -004] f 2-31 + 170-96. 12. -101285 + 17-061 + 3-2001 + 5-88706. 13. 2-325 + -0012 + 5-086 + 219-6832 + -407. 14. A merchant has 4 pieces of calico measuring res- poctively, 25-5 yards, 29 126 yards, 34-25 yards and 88-76 yards ; how many yards are there in the 4 pieces ? •'% '^,y>,f.»{.. 4'rtl 1 /-kWO 1 K»?17K orrAa. 12-6125 acres, 14-005 acres, 16-6 acres ; how many aoree do the four fields contain 9 r |; ' 106 ELEMENTARY ARITHME-nO. \ Section m. Subtraction. ' 1!^. To subtract decimals. Ex. I. From 17-018 take 1-90764. 17-01800 1-90764 17-018 or 1*90764 16(^10686 16-10680 "We write the subtrahend tinder the minnond, plaemf tenths under tonths, hundredths under hundredths, &o. Then, as there are more figures in the suhtrahend than in the minuend, we may annex as many ciphers a-, will ren- der the number of df^cimal placep in » aeh the same. This will not affect the vahie of the minuend (Art. 1261 We then subtract as in whole numbers and place the aeoimal point in tho remainder immediately to the left ol th» tenths. (1) From 18-6 Take 2-8476 Exercise Ixxiii. (2) (8) 2-8706 •60876 •49 -006 (4 •86 •12704 From 6. 1-869 take 6. -0061 " 7. 6-723 " 8. 9-306 «• •0874. •00089. 2-7981 7-9 9. 2041 10. 1000 11. 2 12. 17-36 take 36-002. " 999-99. " 1-8678. •• 9-0184. li I Find the value of 18. (7-2 --2-75) - (1-9- -0027). 14. 36 + 7-07 - 24-896 -(S-164- -789). 16. (278-29-41 -802) - (7*162 + 61-386- "'^^ 16. The length of a' seconds pendulum is 89-1892 inches, and that of a French metre 39-371 inches. Find the difference in length between them. 17. A sovereign weighs 123-274 grains, and a shilling 87*272 (grains ; find their difference in weight. 18. '^i/ke eleven thousandths from eleven hundredths. 19. Afid togeth*?r the sum and difference of seventy- go* From a piece of muslin containing 27*6 yards, a merchant sold 13-76 yards ; how much was left ? Aur^. ^. Section IV. Multii lication. 150. To multiply decimals. Ex. I. Multiplv -7 by '9. Since -7 = TV »n^ "^ = A J .-. -9 X -7= rV X tV = tW =*60. Ex. 2. Multiply -781 by OG. Sinf e -781 == iVA and 06 = jStt '» .'. -OG X'781 = xJir X roVo = iSoSo-o == '04380. Ex. 3. Multiply 8'76 by 2- 1. Since 8-7C = \i% and 2-4 = f J ; .-. 2-4XB-70 = ? J X fJ5 = ?-g J-J = 9-024. Hence to viultiply decimals, rmiUiply an in the ease of integers and w ark off from the right of the prodi(£t as many decimal places as there ore d£cimah in tJie factors. ^^ ^ 'Viultiply t (3) 68-062 -1346 4-53 -208 •008 MiJtiply t. 718 ty 8-47. C. 8-96 by -008. 7. 9 07 by 1-06. 8. -008 by -009. 9. 1314 by 0236. 10. 714-6 by 1124. 11. 9-006 by -0045. 12. 1-001 by 1-009. X i8- A square link contains 62-720 square inches ; what »ii the area in incLei of r)827 square links ? 14 A pint of water weighs 1*26 pounds avoirdupois ; what is the weight of 7-8 pints ? ^ >L 16. Gold is 19-26 times heavier than water; what c. eight of gold is of the same bulk as 17-842 pounds of water? Q^ * ' • > 16. The circumference of a circle measures 3-14159 iKmes its diameter ; what will be the length of the cir- cumference of a circle whose diameter measures 87268 miles? 17 i?inf? fh- tifridnftt nf the sum and difference of -27 What is the weight of five cubic feet of water, if a and eubio foot weighs 62*455 pounds avohdupois 7 yi, 11 1 I T 110 ELEMENTARY ARITHMETIC. 131. To multiply by i followed by ciphers. Ex. 4. Multiply 71-181; by 10; by 100; by 100000. 71-134 71-134 71-134 10 100 100000 711-340 7113-400 7113400-000 From tlie^e examplGLS it will be seen that tlie deci- mal point has been moved to the right in the product as many places as there are ciphers in the multipliti-i'. Hence to mulnj'hf hy 1 followed by ciphers, rrwve the decimal point as mam/ places to the right in the muUipli- cand as there are ciphers in the multiplier, and the result will be the p>roduct. Section V.— Division. 11X%. To divide one decimal by another. Ex. I, Divide 9 by -3. In this example we multiply both divisor and dividena by 1 0. This makes the divisor a whole number. We then proceed as in ordinary division. Ex. 2. Divide 97-92 by 9. 9 ) 97-92 As the divisor is already a whole number T(>88 we proceed to divide as usual. Ex. 3. Divide 3-24 by -00081. Multiplying both divisor and dividend by 100000 we get 324000 -7- 81, which can easily be worked by ordinary division. Ex. 4. Divide -736644 by 234-6. "We multiply the divisor and dividend by 10 ; the divisor is now a whole number. The operation will then stand as follows : 2346) 7-36644 {-00814. 7 038 8284 2346 9384 9884 We first bring down 3 tenths and put the decimal point in the quotient. The divisor is not contained in 73 tenths ; we therefore put a in the - J " X — -_ J3 ln««C «^ «^ A r\-%Try\ r? hundredths. Since the divisor is not contained in 786 bun dredths, we put another in DECIMALS. Ill the quotient and bring down 6 thousandths. The divisor is now contained in 7366 thousandths. The rest of the work proceeds as in ordinary division. Hence, if the divisor is a decivial, ive muUiphj both divisor and dividmd by such a power of 10 as will make the divisor a whole number, and then we divide as in simple division placing the deci7nal point in the quotient as soon as the tenths figure of the dividend is brought down. Exercise Ixxv. Divide 1. 16-578 by 5-4. 2. 48-591 by 96. 8. 2.50 by '0032.. 4. 1-126 by 640. 6. ^'1 by -0025. 6. -0012 by 1-6. 7. -0774 by 480. 8. 21-3 by 37-5. 9. 202 by -01. 10. 406-8 by -018. 11. 1-006 by 13. 12. 15-77 by -19. 13;{. To divide by i followed by ciphers. Ex. 4. Divide 186-15 by 10; by 100 ; by 10000. 10 I 136-15 100 I 13605 10000 i-'3615 136-16 13-615 1-3615 -013615 From these examples, it Vvill be seen that the deci- mal point has been moved to the left in the quotient as many places as there are ciphers in the divisor. He7ice to divide by 1 followed by ciphers, move the decimal point as miiny places to the left in the dividend as there are ciphers in the divisor, and the residt will be th" qm- tient. Section VI —Reduction of Decimals. 134. To reduce^ Vulgar Fraction to a De- cimal. Ex. I. Reduce -^ to a decimal. 40 ) 300 ( -075 280 200 200 thonsandths, and ^^ bonce /^ =» '070. A equals ^\y of 3 (Art. 94). 8 eqnala 80 tenths, and ^ of 30 tenths is tenths. 30 tenths equals 300 hun- dredthB, and ^^ of 300 hundredths is 7 hundredths, and 20 hundredths re- maiuini?. 20 hundredths equals 200 of 200 thousandths is 5 thousandths ; JJ2 ELEMENTARY ABITHMETIO. Hence to reduce a vulgar fraction to a dediml, annex dphers to the numerator and divide by the demmmator oj the fraction, and place the decimal point m the quotient as soon as the first cipher is annexed. Exercise Ixxvi. Bediice the following to decimals 6. 7. 8. 9. 10. • 1. 2. 3. 4. 6. A. 3 6 9 5 Try 1 ru' 3 7 IS' 5 11. 12. 7 TTff* 6.^ 13. 24t^^. U. 15. Q2 1 46 » Tff* Section VII.— Circulating Decimals. 135. To reduce a circulating decimal to a vulgar fraction. In reducing vulgar fractions to decimals we find that sometimes the division wiU not termmate. but the same figure or figures will be repeated over again continually. Ex. I. Eeduce ^ to a decimal. ^ = -3333, &c. Ex. 2. Reduce ^^ to a decimal. ^6j.= -4645, &c. 136 Decimals of this kind are called Repeating or Circulating Decimals. The part repeated is .railed the Period or Repetend. 131 It is usual to express the repetend by writmg it down and placing a dot oyer the first and last figures of the part repeated. When there is only one figure repeated the dot is placed over it. Thus, -SSSS, &o., is indicated "S. t( 45. (( Ex. 3 •4545, &c., •2838, &c., , Reduce ^ to a decimal 23. ^ = 1868686, &c., :=- 186. DJlClMAIiB. 118 138. A pure circulating decimal ia one in which the figures that repeat begin immediately after the decimal point. . , . 1:59. A mixed circulating decimal is one m which the figures that repeat do not be^ mime- 4iately after the decimal point. ^I40. Since t = -11111... Also A = ^ - 11 = -fn-oioi.-. = -050505... -•171717... Similarly, i^w = * "^ ^^^ == -001001... ^5 = 125125... From the preceding examples it is evident that a Purt Circulating Decimal may be expressed as a fraction by writing the figures that repeat as numerator, and as many nines as there are figures in tlie repetendfor denominator of the fraction, Thus,-65 ==-- /ff. '^78 =g^-|. ^= -11111... J == -22222... 56565. 17 ■54 = U- 8-4 = 34. 5;48^ = 5|4. -0378 = -Qvs^' ^ 141 Mixed Circulating Decimals may bo re- fWluced to vulgar fractions in the following manner • Ex. 4. 3* •03i = -03|-=^Q^Q 3 1 _34-3, 9^0 ff 9 e43 8 4 3—6 Ex. 5. -0548 =*-05ff = fob"^*^^^^ »»oo * 18f , _ 136-13 Ex. 6. -0186 -= •018|=]^QQ = wdiF— Vooo • V From these examples it is evident that & Mixed Cir- A ^dating Decimal may be expressed as a fraction by subtracting the part of the decimal which does not repeat from the whole decinml and placing tlui remainder, as nuTTwrator, and as many nines as there are figures in the repHend, folbtved by as many ciphers as there are figur^ in Liu part which does not repeal, ai denummator Oj ■^.-•c fraction. H 114 XLEMSNTABY ABITHMETKJ, Exercise Ixxvil. Bednoe to vulgar fractions : 1. -8. 2. -64. 3. -729. 4. -829. 6. -024. 6. -314. 7. -00676. 8. -0443. 9. 4-0581. 10. 11-287. 11. 8-4*18. 12. 2-846. 142. The Addition and Subtraction of Circu- lating Decimals is generally performed by repeat- ing the period as many times as seems sufficient to insure the required degree of accuracy, and then add- ing or subtracting. 143. Multiplication and Division of Circulat- ing Decimals may also be performed by carrying out the repetend, but these operations are more usually performed by reducing the decimals to vulgar frac- tions, then multiplying or dividing these fractions, and reducing the results onco more to decimals. Ex. 7. Multiply -28 by -86. •30 X -23=11 X f i- -rV\ = -084. Ex. 8. Divide -16 by -0027. •16 -r- '0027 = iTT -^ W9iru f ^J- * Exercise Ixxviii. Find the value of 1. -31007 + 21-008 + 41-607342. 2. -8 — -09 and -04 - -00769238. 8. 87-23 X "26 and 7.72 X -297. 4. -3 -T- 09 and -042 ^ -036. EXAMINATION PAPERS. 1. What are decimal fractions ? How does the ubo of fchem facilitate oalcuiatiou ? S Represent as vulgar fractions 1-26, -0004 How does it affect the value of a decimal to place ciphers (I) of EXAMINATION PAPEBS. 116 after the decimal places, (2) between the decimal places and the decimal point. Decimals may be multiplied and divided by 10, 100, 1000, &c., merely by ahiftmg the decimal point ; show this. Divide -000121 by 11. 8. What are the advantages of decimal fractions ? Ex- press as a decimal, 17859 divided by one miUion. Divide •00126 by 2'5. If the number of decimal places in the divisor exceed the number in the dividend, how do you •proceed ? Explain this, by making 2-6 the dividend and •00125 the divisor. .«„ , 4. Multiply 2-664 by -047, and divide -00169 by -018. Verify the result by putting the decimals in the form of vulgar fractions. . , ,, 6. Wliat aro recurrmg decimals ? Fmd the recurrmg decimal equivalent to f, and find the vulgar fraction equivalent to the recurring decimal -81246246 IL 1. Explain the notation of decimal fractions, and show how the value of a decimal is affected by moving the decimal point two places to the right or left. Write yVxrV as a decimal, and express the one-millionth part of the name fraction as a decimal. Multiply 86*345 by 4-176. Divide 25-6 by -00016. 2. Divide -365 by 20. If in obtaining the quotient you cut off the cipher from the divisor and actually divide by 2, what corresponding change should be made in the dividend ? 8. Prove that -3333 X -212121 = -070707. 4. Prove the rule for fixing the position of the decimal point, when one decimal fraction is multiplied by an- other. , . , Express as vulgar fractions in their lowest terms : (1) -0625 X 0032; (2) -016 -4- '64 ; (3) -46 -'46. 6. Simplify il^ X -f-^, and divide the result by •162 •00126. 2-95 * III. 1. Prove the rule for dividing on© decimal fraction by •06 X '05 X -05 + 1 another, and find the value of j/65~ * State and for WA^iiAinn ^. Dtate auu U2.pia'u fraction to ft decimal fraction. Find the value of i -4- 'OlOOl and of 10-01 ulgar zV jl L16 BLBMENTABT ARITHMSTIO. 8, Reduce -064 and 16-626 to vulgar fractions ; multi- plv them together in that form, and then reduce the re- mit to decimals. Prove by multiplying the deomials aa ^^7. Which is llie greater, ^ |X2t. or -018X216 ? 6. Suppose unity represents -0012, what number rep- resents -0001 ? X V • 1 Whether is 1-121472663 more accurately represented- by i-1214726 or 1-1214727, and why ? ^ 2. Express in decimal notation the value of 8-0626 - 6ifg -•00375 + 1-09236-^1^. ^. , , 8 A bought a house with -25 of his money ; he spent 576 5 it in buying a farm and had ^2100 left; find the cost of the house and farm respectively. 4 What is the smallest number that can be exactly divi'ded by the nine signiiioant figures? Simphfy * 5 What number is that, firom which if there be taken -j ot -375, and to the remainder -63 of -3125 be added, the sum ia 10 1 V. 1. Find the value of A of (4 + If), and prove it equal ^""2 Pr^ov^e^e rule for finding the value of a circulating decimal ; and reduce 1 -^ 99999 and 1 -^ 10001 to cir- culating decimals. 8 Prove that 46-2 -^ 92-4 = -76 X -6. 4 Prove that •02X-02X-005X-005 = -0001X-0001. 6. Divide i-hi+xV+TV+gV by i+ro+Tra+TnT. »nd reduce the result to a decimaL i 'iin f\ '35 CHAPTEE VI. COMMERCIAL ARITHMETIC. Section 1.— Tables and Reduction. 144. ▼ ENGLISH OR STERLING MONEY. ' 4 farthings (far.) = 1 penny, or 3 d. 12 penco = 1 shilling, " Is. 20 shillings = 1 pound, »* £1. Note 1. — Farthings are usually written as a fraction of Id. Thus 1 far. is written ^d.; 2 far., ^.; 3 far., |d. -£1 sterling = $4. 86f, and Is. = 24i cents. Note 2. Oral Exercises. m Repeat the tahle of Englioh money. How many far. in 2d. ? in 3d. ? in Gd. ? in 8d. ? How many pence in 12 far. ? in 16 far. ? in 20 far. f How many pence in 2s. ? in 8s. ? in 5s. ? in 6s. ? How many far. in Is. ? in 2s. ? in 3s. ? in 5b. ? How many shillings in £1 12s. ? in £2 16s. ? 145. There are two kinds of Reduction, Re- duction Descending and Reduction Ascend- ing. 146. Reduction Descending in the process of changing a number from a higher to a lower denomi- nation. 141. Reduction Ascending is the process of changing a number from a hiver to a higher denomi- nation, Ex. I. Reduce £6 5s. S^d. to farthings. uv If , M it 118 ELEMENTARY ARITHMETIC, £6 6s. 8id. In 1 pound there are 20 shil- lings, and in ^G there are C times 20s., or 120b.; 120s. plus 5s. are 125s.; in 1 shillmg there are 12 pence, and in 1258. there are 125 times 12d., or 1500d.; 1500d. plus 8d. are 150od.; in Id. there arc 4 farthings, and in 1603d. there are 1603 times 4 far., or G012 far.; 6012 far. plus 1 far. are 6013 far. 20 126s. ^ 12 1608d. 4 6013 far. Ex 2. How many £ s. d. in 3679 farthings ? far. 4 I 8679 12 J^ 919 3 far. 20 I 76 7d. £d 16s. Ans. £6 IGs. 7fd. Thorc are 4 far. in Id., hsnct in 3679 far. there are as many pence as the number of times 4 is contained in 3C79 ; 8G79 -^ 4 = 919 and 8 over. This o is 3 far. Tiiero are 12d. in Is., hence in 919d. there are as many shillings as the number of times 12 is contained in 919 ; 919 -i- 12 = 76 and 7 over. This 7 is 7d. There are 20s. in £1, faence in 76s. there are as many pounds as the number of times 20 is contained in 76 ; 76 -i- 20 == 3 and 16 over. This 10 is 16 shillings. lH' Exercise Ixxix. Beduce 1. 7s. 8d. to pence. 2. £1 38. to farthings. 3. 7145d. to £, &o. 4. 6185s. to £, &c. 5. jfilO Os. 6d. to pence. 6. £2 6s. 8d. to pence. 7. 3910 far. to £, &o. 8. 71C3d. to £, &o. 9. £191 9s. ll^d. to far. 10. £3 Cs, lO^d. to far. 11. 78916d. to £, &c. 12. £100 7d. to far. 148. UNITED STATES MONE^^ 10 mills (m.) . . . . = 1 cent, 10 cents .... — 1 dime, 10 dimes ...-. = 1 dollar, 10 dollars .... =^ 1 eagle, 14«>. AVOIRDUPOIS WEIGHT. 16 drams (dr.) = 1 ounce, 16 ounces . . — 1 pound, 26 pounds . . = i quarter, 4 quarters . . =1 hundred-\ 20 hundred-weight =»» 1 ton. or 1 c. " 1 d. " 1$. *' 1 0. or a 1 1 1 OZ. lb. qr. 1 owt 1 f . COMMERCIAL ARITHMETIC. 119 Note 1.— Avoirdnpoio Weight Ib ased for weighing evorytliing except jewels, prooionB metals, and medicines, when dispensed. Note 2.-28 pounds equal 1 quarter in Great Britain. Oral Exercises. Repeat the table of Avoirdupois Weight. II ow many ouncos in 2 lb. ; in 3 lb. 4 oz. ; in 4 lb. 7 How many quarters in 28 lb. ; in 49 lb. : in 100 lb. ? How many drams in 2 oz. 6 dr. ; in 8 oz. 4 dr ? How many tons in 68 ewt. ? in 112 cwt.? in 200 cwt. ? Ex 3. Bcducc 2 cwt. Ex. 4. Keduce 147668 4 oz. 11 dr. to drams. jibs, to tons, etc. cwt. qra. 2 4 8qr. 2d 200 lb. le 8204 oz 16 51275 dr. Reduce lb. oz. 4 dr. 11 26- lbs. 6|147658 5,29531 . . . 8 ^ 415906 . .81b. 2011470 . . . 2 qr. 73 tons 16 cwt. lAns. 78 t. lOjaFt- 2 qr. 8 lb. w Exercise Ixxx. 1. 2t. 8 qr. 6 lb. to drams. 12. 76385 qrs. to tons, &c. 8. Sib. 6 oz. 14 dr. to drams. 14. 3 cwt. 8 lb. 6 oz. to ounces. 6. 21645 oz. to cwt., &c. lO. 61649 lb. to tons, &c. 150. TEOY WEIGHT. 24 grains (gr.) . . . = 1 pennyweight, . . 1 dwt. 20 pennyweights . . == 1 ounce, . . . . 1 oz. 12 ounces . . . ,• =1 pound, .... 1 lb. Note 1.— This is chiefly used for weighing gold, silver and jewels. SWvn Note 2.— 1 lb. avoirdupois =^ 7000 grains, v^^ ^ 1 lb. troy . . = 57^ gr%ins^y;^t/\. Oral Exercises: * How many oz. in 2 lb.? in, 3 lb.? in 6 lb.?- 120 ■LEMENTART ARITHHETIO. 1 How many lb. in 86 oz. ? in 48 oz. ? in 60 oz.? in 44 oz. ? in 78 oz. ? How many dwt. in 2 oz. ? in 8 oz 1 in 4 oz. ? in 48 gr. ? 151. APOTHECARIES' WEIGHT. 20 grains (gr.) . . = 1 scruple, . , or 1 so. or 1 9. 8 scruples, . . = 1 dram, . . . or 1 dr. or 1 3- 8 drama, . . . = 1 ounce, . • . or 1 oz. or 1 5- 12 ounces, . . . = 1 pound, . . or 1 lb. or 1 lt>. Note 1. — The ounce and pound of Apothecaries' Weight are the same as in Troy Weight. Note 2. — Apothecaries' Weight is used in mixing medi- cines. Those are bought and sold by Avoirdupois Weight. How many 1. Grains in 79? 113? 2. Scruples in 9 3 ? 16 3? a. Drams in 24 9? 96 3? 4. Drams in 5 § ? 7 § ? 5. Ounces in 88 3? 963? 6. Pounds in 1085? 1685? Exercise Ixxxi. Beduce 1. 1 lb. 4 oz. to ounces. 2. 7163 so. to lb. &o. 8. 7685 dwt. to lb. &c. 4. 11 oz. 8 drs. to grains. 6. 3 oz. 6 dwt. to grains. 6. 73564 grains to lb. (Troy) 153. LONG MEASURE. or 1 ft. 1 yd. 1 rd. 1 fur. 1 mi. 1 1. Cloth is 12 inches (in.) = 1 foot, 8 feet = 1 yard, 5i yards = 1 rod, 40 rods = 1 furlong, 8 furlongs = 1 mile, 8 miles ....... = 1 league, Note 1. — Cloth Measure is not now used, bought by the yard, half-yard, quarter-yard, etc. Note 2. — Gunter*s Chain is used in measuring land. It is 22 yards long and is divided into 100 links. Note 8. — Mariners use the following : 6 feet =1 fathom. 120 fathoms =1 cable length. 880 fathoms =1 mile, Bepeat the table of Lineal Measure. How many feet in 4 yd. ? in 6 yd. 1 ft. f COMMERCIAL ARITHMETin. 121 How many miles in 17 fur, How many leet in 9 fatb. 7 Ex. 5. How many feet inl2rd. 8yd. 2 ft.? rd. yd. ft. 12 8 2 \ ? in 820 rods ? in 69 far. ? in 2 rd. ? in 12 yd. ? Ex. 6. How many rode in 209 ft. ? feet. 8)209 68 09 yd. 8 209 ft. Ans. Note. — We multiply by 6, and add to the product the 8 yds., and then multi- plying by i, we have 69 yd. 5i ) 69 yd. 2 2 2 ft. >V' 1I]18S 12 = 6halfya. = 8yd. Ana. 12 rd. 8 yd. 2 ft. Note. — To divide by 5^, we reduce both to halros, then the remainder is halves^ which we reduce to whol4%^ by dividing by 2. Exercise Ixxxii. Eeduce 1. 1 mi. 1 fur. 1 rd. to inches. 4. 2 rd. 1 y 2. 764,52 in. to mi., etc. 6. 7 chains feet to mi., etc. 6. 16752 in rd^^ feet. )et. fathoms. IStJ. SURFACE OB SQUARE MEASURE. t( it " DtoauB .601!). Gatff...V.....84 lb. Peas.'.: !.:.". 60 lb. Backwheat48 lb. ilUMifM?/ Mllf'I J4t')<| I No*iB. t.' — A barre*^ of beer contains 86 gals. nbttowm A hog^h^aci of beer tt citlpt. or 1 qt. «rlgal. *.^ <•'< . ""•'[^ : A^hogshead of wine " 68 gals.'- -^^•''- - N6Tii 2. — The mne gallon contains 281 cubic inehes ; the beer gallop ogntains 282 cubic inches, and the Imperial or standkfd^feiitllon, 277'274o^Wc inches, nj >:^c; Note 8.-:ii wine g^^i., 5== ,0 standard gak^^ ^^^m MM^i^qmur* .p . MEA8UBE OP TIM^MPIv OtM ' i *0 iieeoiid4 >(0do.) 1 v ' • v ^ iv « v ' ;== 1 minute^ / 1 , • ; 1 miii.i' 60minute8»fvi « ;;^r- .wmI. wu == Ikoxxri, n^iU'rl h. jj., 24hpws, , ^,^,^ . ..i , , =14>y, . . Ida,, w".flftF^»,.ts h ..%,!• »• , • !• 1=1 week, . . 1 wk. 12 calendar months or 866days = 1 year, . . ' * ' 1 w,' ^^days, . .^. , * •»• =1 leap year. ' "^ ' NoTB 1.— Tlie numbed of ^ays in each month i£ay be remembered by means of the' following lines : Thirty days has September, April, Juno, and November; Pjibruary has twenty-eioht alone- All the reat have thirty-one ; But leap year comins once in four, . , Febraary then has one day more. Note 2. — The leap years are those that can be divided by 4 without a remainder : as, 1864* 1863, 1872, etc. 3»t of the even hundreds, only thqse that can be divided hiy.AflO Are IsBD vearS;. TH* -uan^r l^nn rnU] n/>i \%k ^ i^s^ .^mhxL% gC^oVili be,, r- it V V - rt^.T 33'»-|ius. '>'ur,i!l- i i iH 124 158. 12 units 12 dozen 12 gross 20 units BLBMBNTABT AKITHMBTIO. 1 dozen. 1 gross. 1 great gross. 1 score. MISCELLANEOUS TABLE. 24 sheets . . = 1 quire. 20 quires . . = 1 ream. 196 lb. flour . = 1 barrel. 200 lb. pork . = 1 barrel. Oral Exercises. Repeat Time Measure. How many lays in 3 weeks ? in 6 weeks and 8 days? How many dozen in 84 ? in 182 ? in 160 ? Was 1600 a Leap year ? 1876 ? 1854 ? . ' ,^ . , How many hours in 860 min.? in 788 mm.? 600 mm.? Exercise Ixxxiv. Reduce 1. 7 da. 16 hr. to seconds. 2. 7684 pints to bushels, etc. 3. 84 gal. 8 gills to gills. 4. 86 bu. 8 qt. 1 pt. to pints. 6. 2686 gills to gallons. 6. 17 qr. 8 bu. to pecks. 7. 8685 lb. of wheat to bu. 8. 785693 sec. to weeks, etc. 9. 8586 lbs. Timothy seed to bu., etc. 10. 78 da. 9 min. to seconds. 11. 1676 cu. ft. to oordi. Section II— Compound Addition. 159. Toiiidd compound numbers. 100 A compound number is one composed of 2 or more numbers of different denominations which can be reduced to the same denomination. The sum of £6 and £4 is tound by simpU addition. The sum of ^6 12s. and £4 98. is found by compound addition. _ Ex. I. Find the sum of £7 6s. 8d., £5 9s. 8cl., £8 9s'. 7d.,and £9 7s. 9d. £ *. d. £ 8 7 9 d 7 6 8 9 6 9 9 7 B. 6 9 9 7 d. 8 8 7 9 29 81 27 80 13 8 ^ We write the numbers so that units oi liic same uaiiio wiU be^ the same column. Then we add the pence rolumn^s in shnple addition and find the sum to be 27. SiSly^th the other columns. Hence the correct I J J5 C t COMPOUND 8UBTEA0TI0N. 126 eum is £29 Sle. 27d. But it is usual in writing de- nominate numbers not to have more units of any denomi- nation than 1 less than the number required to make 1 of the next higher denomination ; thus, a rod 12 in. long is said to be 1 ft. in length. We do not say 20 owt. of hay, but 1 ton, &c. We therefore change the 27d. to 28. 8d' We set down the 3d. under the pence' column and add the 2s. to 81s.; 31s. + 2s. = 83s.; 83fl. = £1 18s. We set lown the ISs. under the shilhngB' column and add the £1 ^ #9 ; £29 + £1 = £S0. y ^ 4 ^ lb. 17 25 72 67 (1) oz. 9 6 n 10 dwt. 16 12 18 19 Exercise Ixxzv. (2) owt. qr. lb. o«. '-^0 ^ 12 11 16 2 16 12 17 22 T5 19 1 18 13 rd. 17 21 28 26 (8) yd. ft. 4 2 8 6 2 1 2 in. 6 7 8 9 M 6 8 2 18 6 (4) 8. 6 1 6 d. 6 n m 6 bn. 10 2 5 8 15 (6) pk. qt. 1 3 2 3 2 1 6 3 1 4 Pt. 1 1 1 rd. 87 80 1 25 (6) yd. ft. 4 1 6 2 3 2 2 1 1 in. 9 2 7 10 11 7. Find the sum of 1 wk. 2 da. 13 h. 40 min. 30 sec; 2 wk. 6 da. 10 h. 8 min. 3 sec; 6 da. 22 h. 55 min. 46 sec. ; 4 h. 1 min. 15 sec; and 1 wk. 2 da. 4 h. 5 min 8. Add together 10 rd. 4 yd. 2 ft. 8 ia ; 1 rd. 8 yd. 6 in.; 8 rd. 2 yd. 1 ft. 6 in. ; 1 rd. 4 in. ; and 2 yd. 1 ft. 9 in. Section III. Compound Subtraction. 101. To subtract Compound Numbers. Ex. I. From 16 lb. 8 oz. 6 dwt. take 7 lb. 4 ox. 12 dwt. dwt. We write the subtrahend under the 6 minuend, so that units of the same 12 name will be in the same column, and begin at the right to subtract. . ^ 8 14 Since we cannot take 12 dwt. from 6 dwt., we take 1 oz. or 20 dwt. from the 8 oz. and add it lb. 16 7 oz. 8 4 4-.". s-iiw u uwif., lUH/iiuLu^ zo aws. ZU dwi. — lis dwt. ^ l4 dwt. Since we took 1 o». from 8 oe., we left only 7 oi. - 4 oz. = 8 o». 16 lb. - 7 lb. s= 9 lb. Toil 120 n HLEMENTARY ARITHMETIC. Exercise IxxxvL (1) (2) <'^ . Q> § 3 9 gr. mi. far. rd. a. r. sq.rd. 24 7 2 1 16 60 69 8 25 16 10 S 2 17 40 7 89 10 88 (4) (6) ^^^ . fur. rd. yd. ft. in. £ s. d. r. p. yd. 7 31 1 1 8 43 11 10 8 17 18 1 39 1 2 7 16 14 6 2 18 30 7. A farmer had 200 bu. of wheat, and sold 28 bu. 2 pk. 6 qt. 1 pt. to one man, and as much to another ; how much remained ? 8. A miner having 112 lb. of gold sent his mother 17 lb. 10 oz. 16 dwt. 20 gr., and 8 lb. 16 dwt. less +o his father ; how much did he retain ? 9. lYom a barrel of beer containing 54 gallons, a per- son drew 12 gal. 3 qt. one day, and 9 gal. 2 qt. 1 pt. an- other ; how much was left ? 10. From 89 sq. rd. 29 sq. yd. 128 sq. in., subtract 17 sq. rd. 16 sq. yd. 5 sq. ft. , i « 11. A grocer has 1 cwt. 18 lb. of sugar in one barrel, 8 or. 21 lb. in another, and 1 cwt. 2 qr. 11 lb. in a third. After selling 1 cwt. 8 qr. 15 lb., how much will he h»ve left? Section IV. Compound Multiplication. 10/J. To multiply a Compound Number. Ex. I. Multiply 3 da. 19 hrs. 69 min. by 97. da. hrs. min. da. hrs. min. 8 19 69 97 19 69 97 291 1843 6728 871 18 23 We multiply each denomination separately, as in simple multiplication, and obtain as product, 293 da. 1843 hrs. 6723 min. But as 6723 min. = 96 hrs. 23 min., we write down 23 min., and add the 96 hrs. to 1848 hrs. ; 1848 hrs. + 96 hrs. = 1988 hrs. = 80 da. lo hrs. dto. Note.— The usual method of working this example is to multiply first by 10, this product by 9, then to multiply the 8 da. 19 hrs. 59 min. by 7, and add the result to ♦ha OOMPOTTND DTVTSION. 127 yd. 18 30 last product. We recommend the method in the ex- ample as being on the whole easier and more convenient. Exercise Ixxxvii. ,. (2) (8) lb. oz. dwt. gr. da. h. min. sec. 16 8 15 17 10 20 80 40 3 7 a) CWt. lb. OB. 18 16 9 5 4. What IS the value of 39 oxen at ^15 76. ll^d. each ? 5. What IS the weight of 846 hogsheads of sugar, each weighing 14 cwt. 1 qr. 20 lb. ? 6. What is the weight of 1 doz. spoons, each weighing 1 oz. 2 dwt. 16 gr. ? , ft f» 7. If a man owning 6 farms, of 120 a. 1 r. 12 sq. rd. each, sells 460 a. 8 r. 26 sq. rd., how much land has he left? 8. If 2 gal. 2 qt. 1 pt. 1 gi. leak out of a water pipe in 1 hour, what will be the waste in 1 leap year ? 9. Suppose a person to walk, on an average, 3 mi. 2 fur every morning, and 8 mi. 20 rd. 1 yd. every afternoon : how far will he walk in 2 weeks ? 10. If from 2 lb. of silver enough is taken to make a dozen spoons, weighing 1 oz. 10 dwt. 2 gr. each, how much will be left ? 11. What cost 97 tons of lead at ^2 17s. #|d. per ton ? 12. If a man travel 17 mi. 8 fur. 19 rd. 3 yd. 2 ft. 7 in. in one day, how far would he travel in 88 days f 18. If 1 acre will produce 27 bu. 3 pk. 6 qt. 1 pt. of corn wnat will 98 acres produce f * Section V, Compound Division. 163. To divide a Compound Number. Ex, I. Divide 80 da. 6 h. 40 min. bv 17 17>>80 «S m. d*. 40r4 b. 17 m. fo. iSda. 94 394h. 17 134 Il» 5 h. «0 340 S40 We write the divisor at the left of the dividend. 17 is contained 4 times in 80 da. and 12 da. over ; 12 da. = 288 h. ; 288 h. + 6 h. - 294 h. 17 is oontwjQed 17 times in 294 h. and 6 "• °*«^ ; h. = 800 min. ; 800 min. 4-40 mm. - 840 min. 17 is contained 20 times in 840 min. I 128 ELEMENTABT ABITHMETIO. Ex. 2. Divide £12 Is. 6d. by £1 6s. lOd. £12 Is. 6d. ^ 2898d. ^ £1 63. lOd. " 322d. Ex. 3. A divided a field of 11a. into lots of 1 r. 4 per. each ; how many lots were there ? 11a. __ 1760 per. 44 per. = 40. Ir. 4 per. When we divide one compound number by another, we reduce each to the lowest denomination named in either, and divide as in simple division. Exercise Ixxxviii. (2) (^) iu lb. oz. dwt. gr. t. cwt. qr. lb. 6)76 10 14 12 7)112 16 2 16 £ B. d. 4)61 18 4 4. Divide 4 gal. 2 qt. by 144. 6. Divide 40 cu. yd. 10 cu. ft. by 18. 6. Divide £48 7s. 4d. by £6 Hd. , , . ^, 7. Divide 69 bu. 3 pk. 6 qt. by 6 bu. 3 pk. 6 qt. 8. Divide 697 lb. 7 oz. 5 dr. by 60 lb. 10 oz. 6 dr. 9. Divide 80 bu. 2 pk. 4 qt. by 13 bu. 3 pk. 5 qt. 10 A farmer put up 1000 bushels of apples m 350 bar- rels of uniform size ; how many bushels, Ac, did each barrel contain ? . , • • o --«i q «♦ 11. How many demijohns, each oontwmng 2 gal. 3 qt. I pt., can be filled from a tank holding 71 gal. 3 qt. 1 pt '^t'^A drove of cattle ate 6 T. 19 cwt 87 lb. of hay in a v^eek ; how long will 34 T. 19 cwt. 85 lb. last them ? Section VI.— Denominate Fractions. 104. To find the value of a Fraction of a Denominate Number. Ex. I. How maaiy shillings, &c., ore there in | of a £ 8. d. Smce -tg = i, "^ -'-- v— - -- - — ks in compound division. 8)8 vide ^8 by 7 6 Ex. 2. Find the value of 8^ of A oi 2 1. 8 cwt. of a MINOMINATE PRAOT10N8. H of T% of a t. 8cwt. = V of A of 2 t. Scwt. = i of 2 1. 8 cwt. «= li-LEI!*! a. It. 12owt. Iqjr. Exercise ly-gyiv. What is the valuo 1. Of I of a bushel ? 2. Of J of a mile ? 8. Of J of a rod? 1^ 4. Of A of ft mile f 6. Of I of an ton ? 6. Of f of an acre ? 7. Of f of £3 16s. 8id. ; of £18 16s. 7 J». may be written in any of the following* ways : 1. Beduce 4 lb. to the fraotion of 8 lb. 2. What fraction of 8 lb. is 4 lb. ? 8. What part of 8 lb. is 41b. ? 4. If 81b. is the unit, what is the measure of 4 lb. ? Exercise zo. 1. What part of an ounce is -^ of a scruple I 2. What part of a ton is t of an ounce f 8. What part of a mile is {^ of a rod ? 4. What part of an acre is | of a square foot 7 6. Reduce f of a giU to the fraotion of a gallon. 6. Beduce y of an inch to the fraction of a rod. 7. Beduce f of a lb. to the fraction of a ton. 8. What fraotion of ^8 Ss. 6id. is 14s. lO^d. f 9. Express IBs. 10l|d. as a fraction of ^£2 Os. 7d. 10. Express 2 a. 31 per. as a fraction of 4 a. 2 r. 17 per. 11. Beduce gfllet of a ton to the fraction of an ounce. 12. Beduce 1-9770 of a mile to the fraotion of an inch. . j; : H^ m ■UEinENTARY ABITHMETIO. 106. To find the value of a Decimal of t Denominate Number. Ex. 4. What is the value of -7875 of £1 ? £•7876 -7876 of £1 == -7876 of 208. 20 = 16-758. P. 16-7600 12 •76ofl8.^-76ofl2d. = 9d. d.9-0000 Hence '7876 of £1 = 158. »d. Ex. 5. Find the value of 2-16 of 1 yd. 2-16of lyd. = 2|«oflyd.= V of 1 yd. = 2 yd. C in. Exercise xci. Find the value of 1. -94376 of 1 acre. 2. -816626 of 1 lb. Troy. 8. -876 of 1b. 4. •786oflhr. 6. -497 of 1 day. 6. -4876 of £1. 7. -966626 of 1 mil«. 8. -778126 of 1 ton. 9. -628125 of £1. 10. 8-4688 of Is. 11. 2-6884876 of 1 day. 12. -002083 of £1. 167. To Express a Compound Number as a Decimal of a Higher Denomination. Ex. 6. Reduce 8 r. 16 per. to the decimal of 1 a. ; and express 6 a. 8r. 16 per. in acres only. 16 40 ) 16 per. 16 per. 40 r.— •4r. 4)3-4r. .*. 8 r. 16 per. t- 8.4 r. -86 a. 8-4 r. 8-4 a. -86 a. Henoe 6 a. 8 r. 16 per. -• 6*85 a. Exeroise xoii. Beduee 1. IOb. 6d. to the decimal of £1. 2. 6 cwt. 2 qr. 14 lb. to the decimal of 1 ton. 8. 16 dwt- 16 gr, to the deciinal 1 o^, trov. 4. 6 fur. 8 rd. to the decimal of 1 mile. 6. 2 qt. 1 pt. to the decimal of 1 peck. 0. Express ^09 6«. 4^ in pounds oofy. of « Gin. r as a.; r. ft. a. FRAOnOB. idi 7. Express 17 owt. 8 qrs. 14 lbs. 8 oz. in owt. only. 8. Express 7 bu. 8 pk. 1 gal. in bushels only. 9. Express 8} ft. as the decimal of 1 fathom. 10. What decimal of 4 os. is 2 oz. 16 dwt. 19 '2 gr. 11. Express 5 da. 9 hr. 46 min. 48 sec. in hours only. 12. Express f of | of 22f lb. as the decimal of 1 ton. Section VII. Practice. 108. Practice is a convenient method of solving many examples in Multiplication of Compound Num- bers. Ex. I. Find the oostof 864 articles at 88| cents each. B8io-$i $864 m, cost at $1 each. $12188^ - " 88io. each. Ex. a. Find the price of 2 a. 8 r. 14 per. of land at $160 per acre. 4 40 2 X $160 = $320 = price of 2 a. 8r. 8 X $40 = 120 = «« 14 X $1 = 14 = " 14 per. $354 =r entire cost. Ex. 3. Find the cost of 7 t. 6 cwt. 8 qr. 6 lb. of iron at $60 per ton. 20' 4 25 6X$3 = 7 X $60 ~ $420 = cost of 7 t. 18 = ** 6 owt 2.26 = •• n qr. .16 = ••6 lb. 8 X $.76 = 5 X $.08 = $440.40 = entire eo«t. Exercise xciil. Find the price of 8. 297 8. 864 4. 291 K 60c. $1.20. $1.88i. •jTvt-- ;:x SIV19S ess ^pj-w. 6. 828 7. 147 8. 264 $1.87i. $8.87i. $1,164. \\ f jl 1,1 182 RLIMKNTART AKITHMETia 9. 15 a. 8 r. 35 per. of land at $24 per a&r«. 10. 9 gal. 8 qt. 1 pt. of wine at $3.60 per gallon 11. 84 bu. 3 pk. 1 gal. of wheat at $1.20 per bnshel. 12. 7 0%. 16 dwt. 6 gr. of gold at $16 per ounce. 1 8. 29 a. 8 r. 17 per, of land at $80 per acre. 14. 8 t. 13 owt. 1 qr. 16 lb. of hay at $12 per ton. 16. What is the cost of oonstmcting a road 17 mi. 8 for. 15 rd. long at $1880 per mile. Exercise xciv. Problems' Involving the Previous Rules. 1. What is the value of a silver pitcher weighing S lb. 10 oz. avoirdupois, at $2.24 per ounce Troy ? 1 oz. Troy. =«= 480 gr. 1 lb. Avoird. « 7000 gr. 2 lb. 10 oz. " =« 2 1 X 7000 gr. — V x VW> oz. Troy. Price 1 oz. Troy = $2.24. Price of ¥ X '^TWU oz- Troy 2. How many pounds of gold are actually as heavy as 10 1b. of iron? 8. K a druggist buys 25 lb. avoirdupois of drugs at $8^ a pound, and sells them in prescriptions at 76 cents an ounce apothecaries' weight, what is the gain ? 4. How many sovereigns will weigh one ounce avoirdu- pois, if 1869 we'igh 40 pounds troy ? 6. If I of an inch on a map corresponds to seven mile* of a country, what distance on the map represents 20 miles ? ~ V X Vnf X $2.24 = $86.75. 6. The value of 1 lb. troy of standard gold is ^646 148 6d. ; calculate the value of a vase of the same material whose weight is 39 oz. 18 dwt. lib. a« 240 dwt. ; 39 oz. 18 dwt. =» 798 dwt. £46 14s. 6d. ^ 11214d. Cost of 240 dwt. = I1214d. ; •• 1 dwt. «« 798 dwt. 940 130 ' 7«8X5«0 7a ^ 183X880 73 130 R79ftfi44d. 20 i>15.5 7s, 2lid- 7. If 81 owt. of cheese cost ^669 4s. 8d., what wifl 16 cwt. 2 qr. cost ? 8. Bought 2 oz. of tea for 7id., what is that per lb. T PBOBLBMB nrVOLVINO THE PBEVIOUS* BULXl. 136 9. n 8 qr. 24 ib. eoet £4 16g. 8iO°,68, May «r«».68, June 61*».84, July 6r.48, Aug. 66».8fJ, S«pt. 69*.10, Oct. 46«.74, Nov. 86<>.08, Dec. 26°.66. What was th« aver- age yearly temperature during that period t Section n. Percentage. 1 70. The term, ptr emt. means ky or on a hundred ; thnn. ft -nfir nflnf. f\-n n -ITT cr maona fw\ y\«Ty%. *i-a-w-crz^C! hJ w'JLi UT^TX xiuu- dred of it. Hence 1 per cent, of a number ig j^ of it; 2 per cent, is ^J^ of it ; 7 per cent, is ^ of it, &c, 186 ■LSmNTABT ABITHMETIO. ITl. The sign, %, is generally used to represent the words per cent. Thus, 8% is read 8 per cent. Ex. I. Find 6 per cent of $360. Since $100 yields $5 ; $1 $360 (( • 3 6 O X s 100 Exercise xcvi. or $18. 4. 5^ percent, of $200. . 6. 2^ " of 600 men. 6. 7| " of 680. Find 1. 16 per cent, of 450. 2. 20 •* of $75. 3. 88 J " of 69 sheep. Ex. 2. A merchant sold 80 yd. of cloth from a web containing 260 yd.; what per cent of the web did he sell ? From 250 yd. he sold 80 yd ; • 1yd. " ir%yd; 100 yd. '• lAoJlAo yd. or 82 yd. •^ 960 ^ •' .-. he sold 82%. 7. A farmer who had 800 bu. of wheat sold 820 bu.; what per cent, of his wheat did he sell ? 8. A fourth of a field has been ploughed ; what per cent, of the field remains to be ploughed ? 9. 780 is what per cent, of 1300? of 2145 f Ex. 3. Of what number is 60, 8 % ? Sinoe 8 = 8 % of 100 ; 1 = 8 % of i4-« ; 60 = 8% of ^V^ 760. 10. Find the number of which 276 is 26 %. 11. How much must be a clerk's salary in order that 17 % of it may be $204? Section III.— Insurance. 17^. Insurance is security guaranteed by one party on being paid a certain sum, to another against any loss. 11^. xne jrrciiiiUiii la lue uuiu. yum loi l«u iiusui- ance. It ia always a certain per cent, of the sum insured. INIUBANOV. 187 174. The Policy is the written contr»ot of insur- ance. Ex. I. What is the premium for insuring a house valued at $6000 at 1^ per cent. Premium on $100 == $li ; $6000 =$^«l^ii=^ 162.60. Exercise xcvii. Find the premium'on 1. $600 at 8 y.. 2. $840 at li%. 8. $760 at 2 %. 4. $876 at 8 %. 6. $8000 at li %. 6. $7860 at 1^ %. 7. $9600 at IJ %. 8. $4890 at li %. Ex. 2. For what sum should goods worth $i.4900 be insured at 2% so that, in case of loss, the owner may recover both the value of the goods and premium paid ? Premium on |100 at 2 % is $2. Insurance on goods worth $98 = $100 ; i« «< i* $79.14 ; what is the value oi the property insured ? 188 ELSMENTABY ARITHMETIC. Section IV. Commission and Brokerage. i75. Commission is the charge made by aa agent for buying or belling goods, and is ge»erAUji.ft percentage on the money emphyed in thstraiuMtion. lit; Brokerage is the charge made by a broker for buying or selling stocks, bills of exchange, etc. Ex I. My agent has boug}a»t t^a, on my account, to the amount of $750. What is his commiBWon at 2%? The oommission on $100 = $2 ; ,. ii«> '^1 ,±i « 5«i"k- '"" ' i( l.< Hi •«'• $760 = $1»P^ =i $15. K tK'^'CK* h Exercise xcviii. Find the commission on , > 6. $7600 at &i %v,, ,,^,. 6. $4800 at 2i %. \, i:>3j6at4%. I. $790 at 2 %. " 8. 1^800 at li %. "• ; : '- V uuij Ex. 2. I send my agent $1470 with mstructionb to deduct his commission at 5 % and inve^ the ^if- ance in wheat. How much does he mvest ? •; '7^'Sfetlt$2«00 to my agent to invest after deduiatin^ his «n'nimM8loii at 41. What sum did he invest ? n sen? my agent $9180 with instructions t6 deduct How much wheat did he purchase at $1.20 per btishi^ I ft An aiient receites $31.56 as Ms compensation for purihtLgToods at 4 % fommission. What 1. the^alfie of goodri|)Jir^Hi*ed?^,;^^;^,„ ^^ ,^..u«^«k^ V ^ai «70D. What is his brokerage at i per cent ? n. If a commission of $106.47 is paid for selhng $8276 wortii of goodi, what is the rate per cent ? INTBBE8T. 189 Section V.— Interest. 1. If I lend you $600 and you have to pay me $1 for the use of each $100 per year, how much will I receive for 1 year ? 2. How much must you pay for the use of $600 for 1 year, if you have to pay $2 for the use of each $100 per year, or 2 cents for each dollar ? If you have to pay $3 ? $4? $8? ^^ 177. The sum paid for the use of money is called Interest. 1 78. The money on which the ini^reat is paid is called the Principal. 179. The number of dollars paid for the use of $100 for one year is called the Rate per cent. Note 1. — When the rate per cent, is stated without the mention of any length of time, the time is understood to be one year. Ex. I. What is the interest on 12760 for 1 year at 8 per cent. ? Interest on $100 for 1 year » $8 ; «« $1 ** $2760 •• = $TTnr : = $?-L*.?JH loo « $220. Exercise zcix. 1. What is the interest on $600 for 1 year at 8 %? 2. What \B the interest on $560 for 1 year at 7 % f 8. What is the interest on $3152.16 for 1 year at 7^ % ? 4. A man borrowed $7200 for 1 year, viz., $1260 at 77 • $1340 ai, 7^/. ; $2360 at 8% ; and the remainder at 8^%! How much interest has he to pay at the end of the year"? 6. Four brothers have to divide equally the interest of $26800 at 7%. How much does each receive each year ? Ex. a. What is the interest on $575 for 5 years at 7 % ? Interest on $100 for 1 year ac $7 ; $1 " =$Thrl ... 6x7 $1 for years = $, 00 ; $675 ♦* = $ M U . 100 = $201.25. 140 ■LBMENTAST ARITHMBWO. . 7. What is the interest on $986 for 4 years at 6 % ? 8. What is the interest on ^1573 for 4 years at 8 % ? 9. What is the interest on S500 for 2 years at 8^ ^ ? 10. What is the intei-est on $2245.85 for 5 years at 7^ % 1 Ex. 3. What is the interest on $672 for 4 yr. 8 mo. lit 9 % ? 4 yr. 8 mo. = 4t^^ yr. = 4^ years. Interest on $100 for 1 year = $9 ; -* $1 " =$tStf; • $1 for 41 years = «-^/ ; j: 100 = $282.24. 11. What is the interest on $924 for 3 yr. 7 mo. at 6 % ? 12, What is the interest on $954 for 4 yr. 8 mo. at 7 % ? 18. What is the interest on $604.72 for 8 yr. 10 mo.°at 8%? 14. What is the interest on $640.76 for 8 yr. 4 mo. at9%? 180. From the preceding examples we have the fol- lowing rule for finding the interest on a given sum of money at a given rate per cent, for any number of years. Multiply the Principal by the Rate per cent., the product by the number of years, and di- vide this result by loo. 181. The Amount is the name given to the prin- cipal and interest together. Ex. 4. K a man borrows $480 for 8 months at 8%, what amount should he return at the end of that period ? ^ Interest on $480 for 12 months = ox** it «« it «( 1 month = Interest — $ 25.60 jfrinoipal — $480.00 100 » I 4 8 0X8 13X100 » j. 8 X480xi ^ 12 X 100 = $86.60. Amount — $505.60 B%? at 7i % 1 4yr.8 at6%? at7%? mo. at 4 mo. thefol- mm of iber of cent., Id di- B prin- ths at 3f that 16. What is the INTEREST. $840 141 ? Wh * • .u *°^°»^^ 0* «840 for 10 months .o. What 18 the amount of $1673 for 4 years at 8% ? " 17. To what sum will $784 amount in 2^. 9mos at 7% 1 t«nJ?fi^ ^ ^""^f^'?-^ ^""^/"P^®" ^® *^»^« expressed the months as a fraction of a year, but in actual practice more accuracy is generally required, and we mus? express the given parts bf a year in days, men interest is required from one date to another lof/^^'J^^^^*^® interest on $1200 from March 1 1876, to May 81, 1878, at 7 per cent. ' Time from March 1, 1876. to May 81, 1878 = 8 yr. 91 davs Interest on $100 for 866 days = $7 ; <« M (t i( $1 $1200 91days=$r-^^-^^^. •^ ^100X365 » 1900X 9 1x7 *< = $ 1 00X380 ; ^100X730 ^ $20 942 ... Interest on $1200 for 8 yr. at 8% = $288. 00. to ^ '' for 8 yr. 91 days at 8% =$308.94. 1Q w'°;? .*?® •''^''^l* ^° 15^^ f°^ 1^6 days at 7 Vo. at 8 % "'^'* ""^ ^^^° ^^^ ^*y ^ *« Oit. 27, 20. Find the interest on $8000 from Jan 2fi ift7A *« March 81, 1878, at 7^ %. ' ^^^^' *° Ex. 6. At what rate per cent, must $766 be put at mterest for 4 years to yield $241.92 ? ^^^P"^^^ Interest on $766 for 1 year = $?il:»i» = |60.48 • i% •1 $100 « •( C 60. 4i = $l£0_x«o^ 01 A * — $8, or 8 per cent. «u„* • "xiT"" ,i'""^" *<»"- ^or luo use of $900 tor 1 vaar what 18 the rate per cent ? ^ *^» 22. A man lent $484 for 6 years, and reoeivad llfti sn for the mterest; what was the rate p^oeT? ^^^'^ 142 BLBMBNTABY ABITHMETIO. I il I :i I M M M 23. If $108.68 interest is received on a principal of $482 for 4 years, what is the rate per cent. ? Ex. 7. What principal will bring $200 interegt in 146 days at 5 per cent. 7 Principal to give $5 in 865 days = $100 ; 31 u =3^|-^-$20; $200 " = 200 X $20 =$4000; " inl day =^865 X $4000; " 146 days = $?i±2Lt?±l = $10000. 24. A man borrowed money at 7 per cent, and paid $245 interest a year ; how much money did he borrow ? 25. A man bequeathes his wife $876 a year, his daugh- ter $770 a year, and his son $630 a year ; what sum must be invested at 7 per cent, to produce these amounts ? 26. Suppose a gentleman's interest on money, at 6 per cent., is $45 per month. How much is he worth ? Ex. 8. In what time will $800 amonnt to $880 at 8 per cent. Interest = $880 — $800 = $80. The interest of $800 for 1 yoar at 8 per cent, is $64. Time to produce $64 = 1 year ; «l = i^year; •• •• $80 = »o = 1^ years = 1 yr. 3 mo. 27. BTow long a time would be required for $525 to gain $110.25 at 7 per cent. ? 28. How long a time would it require for $625 to amount to $756.25 at 7 per cent. ? 29. A principal of $600 was loaned May 20th, 1878, at 7J per cent. At what date did it amount to $796.87^ ? 30. A note given for $273.25 at 7 per cent, remained unpaid until the interest equalled the principal ? How long did it run f Section VI.— Present Worth and Discount John Smith owes me a debt of $108 to be paid at the end of a year, without interest ; how much ia th^ rlollf tXmffVl of. rkVAar^nf a-nA Vi/^tET «Miiy>1-i nUrx.'.i;] be allowed for the immediate payment of the debt, money being worth 8 per cent ? WKMBssT mmtm amd msoouinr. 148 If I receive f 100, and pnt it out to interest at 8 % for one year, it will amount to $108 ; hence, the pres- ent worth of the debt is $100. Evidently $8 should be allowed for immediate payment. 18^. The Present Worth of a note or debt, payable at some future time, without interest, is such a sum as, being put out to interest, will amount to the given debt when it becomes due. 183. The allowance or deduction made for the payment of the debt before it becomes due is called Discount. Ex. I. What is the present worth of $685, pay- able in 1 year, the rate of interest being 7 per cent ? Amount of $100 in 1 yr. at 7 % = $107. Prrscnt worth of $107 = $100 ; $1 = (( M nop . Tttt» »8 36X 100 $500. 107 Exerclfle o. 1. What iB the present worth of $1250.609, payable in 1 year, the rate of interest being 7 % ? 2. What is the present worth of $612.40, payable in 1 year, when money is worth 12 % ? Ex. 2. What is the present worth of $787.75 due in 2 yr. 6 mo., when money is worth 6 % ? Amount of $100 for 2 yr. 6 mo. at 6 % =°$115. Present worth of ^116 = $100 : « — JTOO . $787.76 = $0^1«i> 1. 8 ft. by 12 ft. 2. 6i ft. by 14 ft. 8. 21 ft. by 25 ft. 4. 2 yd. 2 ft. by 7 yd. 6. 17 yd. by 20 yd. 2 ft. 6. 19 ft. 7 in. by 24 ft. Section II.— Carpeting Rooms. 18d. Carpets are sold in strips, and when the width 01 a stnp IS known, we can ascertain what Imath of carpet wiU be required to cover a given surface. ILX. I. How many yards of carpet 2 ft. 8 in. wide will be required for a room 21 ft. by 18 ft. ? _ Area of surface to be covered = 18 v 91 -«* f*- iiength oi carpet, 1 ft. wide, required " ' "'^^ '"' to cover given area « 18 x 21 feet ; U8 ■LBMKNTABY JJUTHMBTK). Length of carpet, 3| ft. wide, required to cover given area itxli ft. = 66 yards. Exercise cUi. How many yards of carpet 27 in. wide will be re- quired for rooms whose dimensions are 1. 27 ft. by. 21 ft. ? 2. 15 ft. by 12 ft. f 8. 18 ft. by 24 ft. ? 4. 26 ft. by 86 ft. ? Find the cost of carpeting rooms whose dimensions are: 5. 18 ft. by 20 ft. with carpet 8 ft. wide, at $1.20 a yd. 6. 20 ft. by 24 ft., with carpet 30 in. wide, at 90 cts. a yd. 7. 16 ft. by 17^ ft., with carpet 3 ft. wide, at $1 a yd. 8. The cost ol carpeting a room 18 ft. long by 16 ft. wide, with carpet worth $1.20 a yd., is $61.20 ; how wide is the carpet ? Section III.— Papering a Room. 190. Boom papers, like carpets, are sold in strips and we ascertain the quantity that will cover a wall in the same manner as we ascertained the quantity of carpet required to cover a floor. Ex. I. How many yards of paper 16 in. wide will be required for a room 18 ft. long, 14 ft. wide, and 8 ft. high, which contains 1 door 7 ft. high by 8i ft. wide and 8 windows each 6 ft. high by 2^^ ft. wide. Length of surface to be cov- ered- (18-|-14+184-14) ft.-64ft. Area of entire walls = (8 x 64) eq* ft. = 612 sq. ft. Area of door = (8ix7) sq. ft. = 24^ sq. ft. Area of 8 windows = (8 X 2| x 5) sq. ft. = 37^ sq. ft. Area of door and windows = (24^-j-37i) sq* ft. = 62 sq. ft. Area to be papered = (612-62) sq. ft. = 450 sq. ft. 460 sq. ft. = 450x144 sq. in. .'.length of paper required = *^"^/** =4060 i«. = 112^ yards. Exercise civ. 1. How many yards of paper 20 in. wide will be reouired for a room 20 ft. loner. 15 ft. wide, and 9 ft. hiarh f 8. How many sq. ft, of paper will be required for a room 18 ft. 9 in. long, 15 ft. 8 in. wide and 8| ft. high ? lOABUaEMKNT OF 80LXDITT. 149 8. A room 24 ft. long, 20 ft. wide and 10 ft. high con- tains 2 doors each 7 ft. by 4 ft. and 6 windows each 5^ ft. by 4 ft. ; find how many yards of paper 2 ft. wide will be required to paper it T 4. How many yards of paper 80 in. wide will it require to cover the walls of a room 15 ft. long, 12ft. wide and 8 ft. high ] 5. William Benson has agreed to plaster the walls and ceiling of the room in the last example, at 10 cents per sq. yd. ; what will his biQ amount to ? , Section IV.— Measurement of Solidity. Ex. I. Find the number of cubic feet in a rect- angular piece of timber 24 ft. long, 8 ft. wide, and 2 ft. thick. If this piece of timber be out into blocks 1 ft. long there will be 24 such blocks. Number of cu. ft. in 1 block = 6 cu. ft. '* 24 blocks = 24 X 6 cu. ft. = 144 cu. ft. Hence to find the cubic content of a rectangular solid , we take the product of its Unfjth , breadth^ and ihichnesa. Exeroise CT. Find the cubic content of the rectangular solids whose dimensians are 1. 8 ft., 6 ft., 6 ft. 8. 8 ft., 7i ft., 8i ft. Ex. 2. How many bricks will be required to build a wall 20 ft. long, 15 ft. high, and 18 in. thick, each brick being 8 in. long, 4 in. wide, and 8 in. thick ? Cubic content of wall = (20 X 12 X 15 X 12 x 18) cu.in. " brick = (8X4X3) cu. m. ,, . , . _ •0X12X16X13X18 .*. Number of bricks required = sxTxs = 8100. 5. How many bricks will be required to build a wall 46 ft. long, 20 ft. high, and 16 in. thick, each brick being 9 in. long, 4^ in. wide, and 3 in. thick ? 6. What will it cost to put a stone foundation under a barn 8G ft. long hj 24t ft. wide at 25 cents a cubic y;^ui the wall being 7 feet high and 2 ft. thick ? 2. 2^ ft., 5J ft., 11 ft. 4. 2-6 ft., 8-5 ft., 6 ft. 160 ■LBMBNTART AJUTHMETIO. 'M \ Miscellaneous Problems. ]. A garrison of 800 men had proTisions to last for 60 days, but 15 days afterwards 80 men were killed ; how long will they last the remainder ? They would last 800 men 45 days. ♦• 1 man 800 X 45 days. «• 720 men 1«L<1><*» days = 60 d«ys. 3. 28 shanty men have proyisions for 20 days, but 7 men more arriyed ; how long will the proTitioas now last ? 8. A garrison of 1000 men was yiotnalled for 29 days ; after 11 days it was reinforced by 2400 men ; how long will the provisions last ? 4. A garrison of 450 men had prorisiorn for 5 months, but 200 men were sent away ; how long will the proyi< sions last the remainder ? 6. A garrison of 1000 men was vietnalled for 80 days ; after 10 days it was reinforced by 8000 men ; in what time would the provisions be exhausted ? 6. A can do a piece of work in 8 days, and B can do it in 9 days ; how long vrill it reqvir* A and B working together to do it ? The part A does daily = ^ •" " AAB do. " .*. th'ey do tV i^ iV ^^.y ; .*. they do the whole work in ■}■? days, or 4^ days. 7. A can do a piece of work In 12 hours, and B can do it in 16 hours ; in what time can both working together do the work ? 8. A can do a piece of work in 20 dayn ; B can do it in 24 days, and can do it in 80 days ; in wli it time will they all do it working together f 9. A can build a wall in 8 day:, B in 12 day ard in 15 days; in what time can they all build it working together ? 10- A nnikntitv of flonr lasti a man and wife davs. and file wife alone 27 days ; how long would it last the maa 4 + * = «; MlflOKLLAimOUfl fBOBUUII. 161 11. A eftn do a piece of work in 20 dayi ; after working At it for 8 tlays B oomea to help him and they finish the work in 5 days ; how long would it take B by himself to do the work ? 12. A can do f of a piece of work in 8 days ; B can do \ of the same work in 12 days ; in what time oould both working together do 2 such pieces of work ? 18. A and B can mow a field in 12 days ; A and G in 15 days ; B and in 20 days \va. what time could A mow it by himself 7 A and B can do -^ of work in 1 day ; «• u •< l+tV+A u It «t iof I (i>5-W) (t u A and JBandO .-. 2 A's and 2 B's and 2 C'i " .*. A and ^B and .-. ^ .*. J^ can do the work in 20 days. 14. A and B can do a piece of work in 8 days ; A and O can do it in 9 days, and B and in 10 daj s ; in what time can all three working together do it ? 16. A and C can dig a garden in 10 days ; B and O can dig \ of the same garden in 4 days, and B alone can dig it in 20 days ; in what time can A do it by himself ? 16. A piece of work has been half done hy A^ By and C working together, in 8 days ; if A and B together can finish it in 12 days, in what timd could G have finished it ? 17. A can do a piece of work in 6 days of 10 hours each, anH B can do it in 8 days of 9 hours each ; for how many hours a day should A and B be engaged together, that the work may be done in 4 days ? 18. If 6 men or 9 women can do a piece of work in 12 days, in what time will 4 men and 7 women do it ? 6 men do the work m 12 days .*. 1 man does ^ of it in 1 day. 9 women do the work in 12 days .*. I woman does yi-g of it in 1 day. .'.4 men and 7 women do -f^ + -i-J-y or -jS^ of it in 1 day. " ** tJt of it in t^ day. •• do it in ^y/ day, or 8-^^ days. .T9. If 7 boys or 4 men can do a piece of work in 9 20. If 8 men or 5 women do a piece of work in 12 dayi, in what time can 2 men and 1 woman do it f 152 XLEM3NTART ABITHME7I0. 1^ i h I f ' 21. If 1 man. uuJ 2 women can do a pieoe of work in 8 days, and 8 men and 4 women can do it in 3 days, in what time can 1 man or 1 woman do it ? Since 1 man and 2 women do ^ of >i in 1 day 2 men tind 4 women do | " But 8 men and 4 women do J •* 1 man does i - i» or tV " 1 man will do it in 12 days. Now 1 man and 2 women do | of it in 1 day .*. 2 women do ^ — A, ^^ A" " . •. 1 woman will do it in 48 days. 22. If 3 men and 2 boya do a pieoe of work in 8 days, and 3 men and 7 boys can do it in 6 days, in what time can 1 man or 1 boy do it ? 28. If 2 men and 5 boys can do a pieoe of work in 20 days, and 1 man and 8 boys can do it in 18 days, in what time can 1 man or 1 boy do it ? 24. If 7 men and 6 women can do a piece of work in 2| days, and 8 men and 8 women can do it in 3^ days ; in what time can 1 man or 1 woman do the work ? 26. 8 women and 2 boys can do a work in 6f di\ys, and 2 women and three boys can do it in 7yY ^*ys ; in what time can 1 woman or 1 boy do it ? 26. A cistern is filled by 2 pipes in 8 and 10 hours respectively ; in what time will they fill it when they both run at the same time ? They fill ^ -f ^^ of the yessel in 1 hr. « 40 1 or A :fj^ in I hr. .*. they fill the vessel in V o^ 4| hr. 27. A vessel is filled by 3 taps, running separately, in 60, 75, and 90 minutes respectively; in what time wiU they fill it when they all run at the same time ? 28. Two pipea running together can empty a cistern in 8 hours, and one by itself can do it in 12 hours ; in what time can the other empty it I 29. Two pipes running together can empty a vessel in 60 minutes ; one of them can empty |^ of the vessel in 40 minn - 1 ; in what time can the other empty | of it ? 80. A cistern is filled by two pipes, A and jO, iu 20 and 24 minutes respectively, and is emptied by a tap, 0, in 80 minutes; m what time will it be filled by all running together ? MISOELLANEOUS PSOBLEMg. 168 81. A bath is filled by a ynipe in 60 minutes; it is emptied by a waste pipe in 40 minutes ; in what time will the bath be emptied if both pipes are opened at once ? One pipe empties ^ of ressel in 1 minute. The other fills -^ of vessel in 1 minute. .-.when both are running d^--^), or ^ of the vessel is emptied in 1 minute. .'. the vessel U emptied in 120 minutes. 82. A vessel can be filled by 2 taps running separately iu 80 and 36 minutes respectively, and emptied by a third in IS min. ; if the vessel is full and all 3 taps running at once in what time will it be emptied ? o . » 83. A bath can be filled by two taps running separately in 20 and 80 minutes respectively, and emptied by two others in 24 and 18 min. respect ^ely ; if the bath is full and all four taps opened, in what time will the bath be emptied ? 84. A gave to James \ of his money and to John of it, and had |2.10 left. How much had he at first ? He gave away g^ + A or |J of his money. 6 He had left f | — |i, or ^ tt /y of his money, «< $2.10; A •- - 1¥^ ; If, or hiB money — $^^^~^^ $5.40. 85. A father willed to his eldest son f of his property to his second son y of it, and to his youngest son the rest amounting to $7238. What was the property worth? 86. A post is i iu the earth, ^ in the water, and 13 feet above the water. What is the length of the post ? 87. A man devotes -12 of his income to charity, -2.5 for educating his children, 45 for household expenses, and •aves the remainder, which is $284.76. What is hig in- oome ? 88. A ship whose cargo was worth $25000 being dis- abled, '46^ of the whole cargo was thrown overheard. What would a meruluuit lose who owned '26 of the cargo 7 i ; I 154 ELEMENTARY ARITHMBTIO. 89. A laborer in one week dug 5 rods more than i the length of a ditch, and the next week he dug the remain- ing 20 rods; how long was the ditch ? Length of ditoh dug first week =* i ditch -H 5 rods ; .•. Length remaining = \ ditch less 5 rods ; .-. A length of ditch less 6 rods = 20 rods ; .-.1 length of ditch = 20 rods -f 6 'o^s = 26 rods ; .'. length of ditch = 50 rods. 40. A man invested $800 more than | of liis money in a house and $600 more than $ of the remainder in a lot and had now #©00 left. How much was he worth ? 41 If 10 men can chop 90 cords of wood in 8 days, now many cords can be chopped by 20 men in 4 days ? Cords chopped by 10 men m 8 days = 90 cords ; a 1 ma»* ** 8 days = ^ = 9 cords ; m 1 man " 1 day = | c« rda ; . , J 20X9 ^- •• 20 men " 1 day = -^- — = Y cords ; «• somen •' 4 days = ^^— = 90 cords. Note —When the pupil has become familiar with the unitary sYBtem.'and thoroughly understands the reason of each step, the process may be abridged by leaving out the steps m itahon. 42. If 8 men build 88 ft. of wall in 11 days, in how many days will 12 men build 86 feet ? i •„ .« 43; If 36 men earn $324 in 18 days, how mueh will 42 men earn in 87 days ? , ,- . 1 om 44 How many days will it take 15 men to cut 810 cords of wood, working 9 hours a day, if 18 men can cut 364 cords in 14 days, working 12 hours a day? 45 It costs a family of 5 persons $135 tor 6 weeks board, how much will it cost a family of 7 persons at the same rate for 3 weeks ? , 1 . « ■. 46. If 12 men can dig a ditch 16 rods long m 8 days, in how many days can 24 men dig a ditch of the same depth and width, 32 rods in length T Time in which 12 men will dig 16 rods =^ 8 days ; 24 men 32 rods = - 18 -^ 8Sx 19xt 94xl« 8 dayi. dayi mSOELLANBOUS PROBLEMS. 106 tton'^'i ^^^^}' aj;e carried the distanee of 50 miles for Au jTJ^nn^ "^'^i^^ ^'^^ '^^^^ i^ °^'^^ 40 miles ? 48. If $500 gam $60 in 2 yr. at 6%, how much will 3800 gain m 3 yr. at 8% ? 49. If 20 men can perform a piece of work in 12 days required the number of men who could perform anothei piece of work 3 times as great in J of the time ? 60. If a 10-cent loaf weighs l§ oz. when flour is $8 a bar- rel, how much will a 6-cent loaf weigh when flour is worth ^o a barrel ? 61. If it costs $36 to carpet a room 18 ft. long and 16 ft wide, how much will it cost to carpet a room 16 ft. lone and 9 ft. wide ? ** ^\ ? ^\ ^^^^^ ^^^^^ *^ '^^S * °®^^^ 40 ft- long 80 ft. wide and 6 ft. deep, how much will it cost to dig a cellar 80 ft. long, 8 ft. wide, and 5i ft. deep f ^ . 63. If the rent of a house worth $8200 is $240 for 9 wo^th I'sMoT^^* ^^^ ^^^ ^^^ °^"** * ™*^ '^^^ * ^°"^® I 64. I bought a horse for $130 and sold him for $162.60 : what was my gain per cent ? On an outlay of $180 my gain is $82.60 ; *2 " A S 9.0 «« $100 <( <( $ V 1 30 1 1 00x3 2.5 1»0 or $26; .*. I gain 257^. ih^t F\^^7r.^^^J^^ ^"^""^^ ^^^ ^6 and afterward seU them for $7.60 ; what per cent, do I gain ? coffhit^^Tp^'^^vf t-^\"^^ °.^ ^^*°^«« ^^^ «7.60 which cost liim ^6.25 ; what is his gain per cent. ? ^'^'^^ercha.nt buys sugar at 6 cents* per pound and sella It at 8 cents ; what per cent, does he ^in ? 68. I bought caUco at 12 cents a yard ; for what must I sell It to gam 26 per cent. ? That for wliich I gave $100 I must sell for $125 ; *« $.12 ' 100 » laxi'SA 100 = 16 cents. 69. A merchant bought silks at $1.26 n^r vord • f-- ^ must he seii them to gain 20 per cent ? ^ "' ' " 60. A bought a house for $8500 and afterwards sold it i^i a loss of 16% ; what did he get for the houwT 156 BLEMENTABY ARITHMETIC. i i: I 61. A grocer bought a quantity of sflgar for $116 ; fof what must he sell it to gain 18 per cent. "^ ^ ^^^^ , 62. A grocer sells a quantity of sugar for $324, and thereby loses 10 per cent. ; what Aid the sugar coat? That which sold for $90 cost $100 ; $1 " $ ~90 » $824 -$»^*> = |(1.0Q)»+t5itX$(1.06)2=$(1.06)» $400 " = 400^ $(1.06) 3 = $476.4064. Amount == $476,406 - Pxincip«i= 400.00 Compound Interest = $76,406 82. What is the compound interest of $660 for 8 yean at 6 per cent. 83. Find ihe amount of $1000 for 4 years at 6 per cent. 84. Find the difference between the simple and com- pound interest of $350 for 3 years at 8 per cent. 85. A sum of money put out at simple interest for 2 years at 8 per cent, amounted to $464; to what sum would it have amounted had it been lent at compound interest ? 86. The true discount on a sum of money for 8 years at 8 per cent, is $120 ; what is the compound interest of the same for the same time. 87. A man deposits in the Savings Bank $500, on which the interest at 6 % is to be added to the principal every 6 months ; how much money has the man in the hank at the end of two years ? (( (( I AKSl^ERS. Exercise i.—Page i. 1. 6 ; 6 ; 9. 2. 1 ; 1 book ; 1 ball. B. 6, 7, 3, 4, 2 are abstract ; 8 books, t) men, 6 apples, l ceut are concrete 4. 1 mile : 1 ; 1 cent. 5. 3 apples, 7 apples, and 6 apples ; 4 boys' and 9 boys ; 7, 9 and 8 ; 2 cents niid 5 cents ; 4 gals and 6 girls. Exercise n.— Page 8. 1. 7 ; 9 ; 4 ; 2. 2. 3G ; 84 ; 20 ; 69. 8. 44; 70; 96; 16. 4. 14; 12; 39; 56. 6. 48; 97; 36; 60. 6. Seven; eleven; fifteen; nineteen; fifty-nine; eighty- four; ninety-six; ninety-eight. 7. Seventy-one; twelvo ; twenty-eight; ninety-one; forty- four; seventeen; twenty two; thirty-four. 8. Twenty; thirt v- seven ; forty-eight; serenty-gix ; ninety, nine ; sixty-nine ; seventy ; eighty seven. 9. Fourteen ; thirty-five ; eighty-nine ; seventy-eight ; fifty- four; forty-nine; fifty; thirteen. 10. Ninety; eighty; thirty-nine; twenty-eight; eleTen; nineteen ; twenty-sev n ; thirty-one. Exercise m.— Page 4. 1. 149 ; 308 ; 974. 2. 200 ; 420 ; 694. 3. 660 ; 908 ; 444. 4. 736 ; 960 ; 406. 6. 309 ; 687 ; 672. 6. Two hundred and seven ; three hundred and seventy-one ; one hundred and eighty-five ; one hundred and ninety ; three hundred and sixty-eight. 7. Five hundred and seventy ; four hundred and seventy-two ; eight hundred and seven ; nine hundred and nine ; nine hun- dred and ninety. 8. Three hundred and sixty-eight ; five hundred and eighty- four ; seven hundred and sixty ; three hundred and twenty- otxe ; nine hundred and ninety-nine. 9. Three hundred and ninety-four; seven hundred and eighty-six ; four hundred and seventy-five ; seven hundred and •ifihty-two : seven hundred, iO. Five hundred and six ; three hundred ; four hundred and seven; seven hundred and forty; three hundred and ninety-Mven. m 160 ULRMENTART ARlTHBfBTIO. 1^ ! Exercise IV.— Page 6. 1. 6006 ; 4300 ; 9080. 2. 3700 ; 7906 ; 3084. 3. 64009 ; 807068 ; 700316. 4. 4030097 ; 809007039 ; 686000007: 5. 8000000000; 64007000024; 4004000004. 6. 408003009; 74000074004; 5000«)000(t500. 7. 80070000000; 800000008; 300000300090. 8. 57700000080; 11000011; 19000014000. 9. 7000000000070; 400000001; 600600000000600. 10. 99000000000008; 700070007000; 16016000000016. 11. Seven thousand and seventy-seven ; eighty-five thousand and seventy-nine ; fifty-six thousand nine hundred and fifty ; four hundred and seventy-three thousand, six hundred and twenty-eight. 12. Fifty-six thousand four hundred and eighteen; seven hundred and eighty-four thousand and six ; four hundred thousand five hundred and seven; three hundred and sixty thousand and four. 13. Three hundred thousand and seventy-one; nine hun- dred and one thousand and seven ; seven hundred and twenty thousand and nine; one hundred and eighty-two thousand and ten. ■, a • 14. Three millions one hundred and forty thousand and six ; fifty millions and six hundred; three billions six hundred miiUons ten thousand and seventy. ^ ^ 16. Fifty-one billions six hundred and thirty-six millions two hundred and seven thousand six hundred and forty ; sev- enty billions and one hundred ; nine hundred and twerty bill- ions seventy millions seventy thousand and seventy. Exercise v.— Page 6. 1 XIX- XXIV; XLIX; LXXXIV ; XCIX. 2 CLXXXVII ; CGVni ; DCCI.X .XI ; OMLXII ; CMXCIX; 3* MCCCI; MCCCXG; MDCLXXXIV; MDCCCXV. 6. MOV.U1 , m^^ , MDCCCLXXVni. 4. 44; 69; 94; 71. 6. 99; 129; 177. 6. 555; 1604; 1819: 1090. Exercise vi.— Page lo. 1. 46 horses. 6. 956. 9. 979, 13. 898. 17. 9879. 21. 87988. 25. 768989. 2. 98 boys. 6. 898. 10. 697. 14. 879. 18. 8989. 22. 88998. 26. 789689. 3. 39 girls. 7. 889. 11. 798. 15. 889. 19. 9989. 23. 79988. 27. 988989. 4. 978. 8. 879. 12. 998. IG. 8o8y. 20. 98878. 24. 797898. f- ^m- AMSWEItS. 161 Exercise vii.— Pageii. md ty; tud ven red xty un- nty Eind iix; red ons ?ev- siU- IX; Ql. 1. 79 cents. 4. 796 dollars. 7. 878 bales. lO. 899898 persons. Exercise" vni.— Page is. 2. 88 trees. 6. 989 miles. 8. 8989 dollars. 8. 968 acres. 6. 969 yards. 9. 97989 dollars. 1. 113 dollars. 5. 146. 9. 217. 13. 1915. 17. 2704. 21. 1842. 25. 16954. 29. 24459. 83. 22825. 37. 166681. 41. 24692. 46. 23378. 49. 75958. 2. 78 cents. 6. 247. 10. 213. 14. 1954. 18. 1656. 22. 2141. 26. 14978. 80. 31405. 34. 29165. 88. 33«306. 42. 25879. 46. 288390. 50. 103618. 3. 152 boys. 7. 162. 11. 1861. 15. 1931. 19. 1951. 23 23878. 27. 15118. 81. 29377 35. 266648. 39. 2033781. 43. 27265. 47. 246818. 51. 41121. 4. 145 girls. 8. 161. 12. 1357. 16. 1759. 20. 1976. 24, 18294. 28. 16046. 32. 21282. 36. 226871. 40. 199859286. 44. 24447. 48. 81148. Exercise ix.— Page 14. 1. 222 dollars. 2. 1661 acres. 4. 1061 miles. 5. 036 pounds. 7. 3441 acres. 8. 633 dollars. 10. 2237 dollars. 11. 1173 dollars. 12. B, 601 dollars ; C, 1066 dollars; 2182 dollars. Exercise X.— Page 18. 8. 120 days. 6. 7428bu8helb 9. 2104 nages. 1. 813. 5. 644. 9. 344. 13. 4442. 17. 2530. 21. 5j24. 25. 423. 29. 6082. 83. 66243. 37. 260265. 41. 161116. 45. 57234. 49. 322. 53. 6216. 2. 241. 6. 464. 10. 304. 14. 5022. 18. 4422. 22. 6257. 26. 60224. 30. 43262. 34. 75331. 38. 64153. 42. 741651. 46. 364. 50. 432. 54. 83136. 3. 251. 7. 148. 11. 788., 15. 2228. 19. 5512. 23. 1361. 27. 36275. 81. 36425. 35. 61161. 39. 35422. 43. 21358. 47. 233. 51. 2533. 55. 56454. 4. 402. 8. 305. 12. 530. 16. 2001. 20. 2734. 24. 4628. 28. 31216. 82. 35137. 86. 40663. 40. 77443. 44. 445 J 6. 48. 228. 52. 1243. '8. i98. Exercise XI.— Page 19. 1. 43 girls. 2. 44 cents. 4. 34 runs. 5. 44 questions. 7. 83 dollars. 8. 2112 dollars. 3. 16 dollars. 6. 48 dollars. 9. 14442 dollars. ^■■-■wn:.*^i^nfeA^«a h J B latiBtea* 'l2 'I 102 FLEMENTABT ABITHMETIO. Exercise xn.— Page 21. 1. 825. 5. 255. 9. 263. 13. 168. 17. 69. 21. 298. 25. 339. 29. 479. 33. 4188. 37. 4944. 41. 1359. 45. 3784. 49. 11844. 63. 49289. N 2, 873. 6. 144. 10. 386. 11. 8br 18. 68. 22. 175. 26. 468. 30. 293. 84. 948. 88. 2857. 42. 5247. 46. 6682. 60. 19528. 64. 25012. 8. 262. 7. 184. 11. 362. 15. 266. 19. 458. 23. 197. 27. 177. 81. 1497. 86. 19.S3. H'J. 53d9. 43. 2279. 47. 4469. 61. 52888. 4. 298. 8. 2C6. 12. 309. 16. 169. 20. 178. 24. 118. 28. 497. 82. 2858. 36. 2919. 40. 1299. 44. 5263. 48. 1789. 52. 35499, Exercise xin.— Page 22. 2. 77 yards. 3. 560 dollars. 4. 1803. 1. 177. 2. 739. 6. 2262. 6. 620. 9. 77866. 10. 3598. 1. 8 dollars. 6.3261. 6. 344 cents. 7. 87 quarts. 8. 175 dollars, t. 606 dollars. 10. 376 acres ; 12021 dollars. Exercise xiv.— Page 22. 3. 1811. 4. 601. 7. 2152. 8. 22326. 3. 853288. 6. 956. 9. 41266. 4. 368402. 8. 1488 apples. 12. 3070. 16. 78112. 20. 661672. 24. 646857. 28. 364704. 32. 6044346. 36. 9162527. Exercise xv.— Page 23. 1. 867 dollars. 2. Lost 632 dollars. 4. 10534. 6. 171 dollars. 7. 814 feet. 8. John 28, James 32. 10. 6628. 11. 5211 and 3553. 1. 14864. 5. 195 boys. 9. 2106 girls. 13. 23526. 17. 53838. 21. 153132. 25. 63936. 29. 428215. 33. 6283784. 37. 10860916. Exercise XVI. 2. 16864. 3. 6. 282 cents. 7. 10. 1890. 11. 14. 47901. 15. 18. 70340. 19. 22. 630855. 23. 26. 54360. 27. 30. 1023024. 31. 34. 2217177. 35. 38. 9091656. 39. —Page 27. 216936 959 cows. 3360. 43710. 72028. 352794. 432481. 3417355. 7865490. 10743888. Exercise XVll.--Page 27. 1. 16280 cents. 2. 185430 cents. 3. 2709 dollars. 4. 15215 dollars. 6. 336 sheep ; 392 dollars. 6. 94 pounds ; 2610 cents ; 762 cents ; 658 cents. ANSWERS. 168 9. 6. s. 7. 8402 dollarB ; 224 dollars ; »626 dollars ; 2237 dollars. 8. 1320 paragraphs ; 11880 lines ; 95040 words ; 475200 ktters. 9. 296 cents ; 414 cents ; 710 cents ; 118 cents. 10. 1096 cents ; 2214 cents ; 3310 cents ; 1118 cents. Exercise xvni.— Page 29. 1. 11950. 4. 66738. 7. 189946. 10. 62560 13. 98560 yards. 16. 469 days. 19. 19846 dollars. 2. 40992. 6. 482644. 8. 240896. 11. 183576. 14. 68620 feet. 17. 3000 dollars. 20. 40320 min. 8. 118377. 6. 340488. 9. 134010. 12. 1348660. 16. 841)6 cents. 18. 8605 dollars. Exercise xnL— Page so. 1. 472440. 4. 662650. 7. 6686169. 10. 4127874. 13. 14821755. 16. 81362385. 19. 70132632. 22. 26514000. 25. 66093951. 28. 307551216. 31. 848112465. 84. 2139927997. 87. 341614192. 40. 903666918. 2. 300720. 6. 724885. 8. 6509916. 11. 9781440. 14. 25581580. 17. 29456710. 20. 14069499. 23. 42741832. 26. 217702278. 29. 276010344. 32. 283036032. 35. 1627916724. 38. 3481804962. 3. 236196. 6. 6608822. 9. 1194872. 12. 11961686. 15. 23120856. 18. 31259060. 21. 41316048. 24. 16765686. 27. 163588743. 30. 114297351.. 33. 671374392. 36. 1244124564. 39. 4141417604. Exercise XX.— Page 31. 1. 127406. 2. 6317608. 8. 6825466. 6. 63366216. 6. 6749472. 7. 8214206. 9. 80071992. 10. 738110274. Exercise xxi.— Page 31. 1. 445300. 6. 243000. 9. 422600. 2. 692900. 6. 258000. 10. 627000000. 8. 60744600. 7. 11214000. 11. 64610000. 4. 1960452. 8. 26996104. 4. 11887500. 8. 4096000. 12. 488000000. Exercise xxn — Page 31. 1. 454560 sheets. 4. 1663 yards. 2. 6. 8. 4 r\ XU. 7. 263952 apples. ivowa yaras. ix. 18. 1246420 dollars,^ 14. 16. 7080320 dollarSwN 17. 19. 89784 yards. 20 8. 6. 9. 195559 yards, 3915648 pounds. 915750 pages. 3926000 dollars. 16. 277107850doll*r8 18. 73^ miles. 6126 dollars. 1228276 dollars. efi27fe yards. ^/^■«r\/\ t f\ 262860 barrels. 87S480 hill*. i i 164 ■LBMBNTARY ABITHMETIO. Exercise, xxm — Page 32. 1. «08 dollars. 2. 162764 men. 3. 630229 gal. 4. 277S dollars. 6. 12652 oeuts. 6. 7466 dollars ; 994C dollars. 7. The horses ; 263li dollars. 8. A'h 706800 ; B's 1126125 letters. 9. 18760 cents. 10. 116660 cents. 11. 457047. 12. 944 dayH. 18. 1441 dollars. 14. House, 2800 dollars ; Farm, 2976 dollars. 16. Loss 254 dollars. 16. Gain 3100 dollars 17. 17582 dollars. 18. 9000 dollars. 19. eilOdollars 20. 10690 dollars. Exercise XXIV — Page 39. 1. 18. 2. 29. 3 . 27. 4. 46. 6. 48. 6. 192. 7. ^5. 8. 241. 9 291 10. 325, 11. 213. 12. 191. 13. 233. 14. 144. 16. 187. 16. K7. 17. 170. 18. 195. 19. 149. 20. 167. 21 122. 22. 141. 23. 154. 24. 162. 26. 139. 26. 112. 27 114. 28. 119. 29. 138. 30. 137. 31. 112. 32. 117. 83. 118. 34. 122. 36. 124. 36. 52. 37. 64. 88. 96. 39 . 82. 40. 74. Exercise xxv.- -Page 39. 1. 228. 2. 368. 3. 274. 4. 187. 6. 269. 6. 246. 7. 272. 8. 174. 9. 138. 10. 246. 11. 223. 12. 171. 13. 182. 14. 255. 15. 275. 16. 128. 17. 166. 18. 183. 19. 144, 20. 206. 21. 184. 22. 204. 23. 243. 24 152. 25. 109. 26. 123. 27. 147. 28. 129. 29. 157. 30. 168. 31. 163. 32. 187. 83. 156. 34. 153. 36. 176. 36. 139. 87. 108. 88. 109. 39. 129. 40. 144. 41. 246. 42. 656. 43. 419. 44. 609. 45. 1223. 46. 367. 47. 676. 48. 1208. 49. 1337. 60. 1410. 61. 907. 62. 457. 68. 947. 64. 3669. 66. 1-3879. 66. 988. 67. 442. 68. 285. 69. 7032. « 60. 7484. Exercise XXVI.— Page 40. 1. 69 oranges. 2. 173 days* work. 3. 918 pounds. 4. 231 yards. 6. 96 rods. 6. 91 cents. 7. 16 bushels. 8. 123. 9. 6062 bushels. 10. 8 cords. Exercise xxvn.— Page 4i. 1. 2177i. 4. 1640^. 7. 4200611. 10. 729684i<»^. 13. 823956. 16. 420560. 19 8470853|». 22. 2730956^^. 26. 4660387W- 2. 1248*. 6. 12317^. 8. 672004|. 11. 1398260. 14. 6273804^. 17. 20073842. 20. 7298426. 23. 8844826 li. 26. 37376008. 3. 9606. 6. 69049). 9. 7770701. 12. 7400061. 15. 4238753^^. 18. 37037048. 21. 7480093. 24. 92506026. I AN! WEBS. 165 1. 482 barreli. 4. Hi dollars. 8. 13()J pounds. 11. 252. 14. 548 brkilrs. Exercise XXvni.-Page 41. 8. 6226 dollan. 6. 2423 minuter. 10. 62 J. weeks. 18. 15745{ pounds. 2. 1256 pounds. . 5. 8 dollars. 9. 125 i dollars. 12. 669} aores. T^ ercise xxix.- 2. 8H. 6. 39?J. 8. i)4r)«|. 11. 231|e. 14 2259 J i,. 17. 6205 jVA- 20. 2025^VA- 23. 4321. 26. 3615. 29. '>567. -Page 44. 1. 24H. 4. 74*f. 7. 53111. 10. 49bVt. 13. 375|J|. 16. 2831. 19. 3746. 22. 9710jy,'V- 26. 3180. 28. 7277MVt. Exercise XXX. -Page 44. 1. 43 days. 4. 82 dollars. 7. 46 dollars. 10. 2076 barrels. Exercise XXXI.— Page 46. 2. 38 days. 5. 129 years. 8. 545 bnles. 8. 185|. 6. 688f J. 9 654 iV,. 12. 498i|S. 16. 6050V/,. 18. 62(iO^Vo- 21. 4998//V^. 24. 4671. 27. 1142. 30. 60444. 3. 1090 feet. 6. 123 doHars. 9. 343/r miles. 1. 1734j%. 6. 246925 J J. 9. 1082^. 2. 13G6U. 3. 1549|i. 6. 149U7i8. 7. 349JJ. 10. 4236*5. 11. 2570. Exercise xxxn.— Page 46. 4. 807 J|. 8. 800 J j. 12, 5599,Vr 1. 24/o. 6. 804/,. Q 30 7 ft 1 2. 12713. 6. 3. mn, 7 2AV„. 4. 826AV- 12. 632tVVV,. 3. 123 pounds. 6. 236 dollars. 9. 43 bushak. 1183IS. 7. 2AVi 10. 153^85- 11. 673iJJ5. Exercise xxxin.— Page 46 1. 108 yards. 2. 66 hours. 4. 30 pounds. 6. 42 bushels. 7. 1378 quarters. 8. 237 bushels. 10. 38 miles. Exercise XXXIV.— Page 47. 1. 8814. 2. 129 3. 233289. 4. 348. 6. 186. 6. 272. 7. 10005100. 8. 19062. 9. 194 and 86. 10. 784623. Exercise xxxv.— Page 48. 1. 367 aores. 2. 2310 dollars. 3. 846 dollars. 4. 16 weeks. 5. 44 dollars. 6. 66 cents. 7. 1660 barrels. 8. 24 months. 9. 661 dollars. 166 i § ji ft'* I ELBMXNTART ARITHBfBTIO. 10. 1210 dollars. 13. 520 dollars. 16. 54 cents. 19. 100 dollars. 22. 6880 dollars. 25 6040 yards. 29. 954 dollars. 32. 12 days. 85. 50 days. 38. 6 days. 41. 28 men. 44. 108 men. 47. 56 men. 50. 15 beggars. 11. 41600 0. ft. 14. 30 i'onrs. 17. 5Gdonara, 20. 40ceit8. 23. 23725 days. 26. 365 acres. 30. 1971 bushels. 33. 32 days. 36. 90 days. 89. 10 days. 42. 84 men. 45. 25 men. 48. 72 men. 12. 217 sheep. 16. 240 cents. 18. 96 dollars. 12 dollars. 7056 pounds. 28. 31260 dollars. 31. 15 days. 34. 361 days. 37. 48 days. 40. 119 days. 43. 21 men. 46. 114 men. 49. 150 men. 21. 24 Examination Papersi-^ag« 51. I. 2. 488979. 3. 944813. 4. 7706307420. 5. 1316 dollars. n. 2. 29900000. Twenty-nine milliouR, nine hundred thousand. 3. 846055. 4. 2699,329; DCCCLXXXVfiCMLXXI. 5. 16 bushels. in. 3. 64366636568. 5. 580 acres ; 61 dollars. IV. 4. 228 dollars. 5. 608 sheep. V. 3. 86. 4. 86. ^ 5. 63 dollars. VL 2. 3571 dollars. 8. 112. 4. 180 acres ; 36 dollars. 5. 24 days. vn. 2. 40831 doUars. 3. 663 miles. 4. 247. 5. 6626 dollars. vni. 1. 4700 dollars. 2. 973. 3. 31 dollars. 4. 201 cents. 6. 964 miles; 1181 miles. Exercise xxxvni.— Page 68. 1. ni63.56 6. 199.06 1. $94.58. 6. »5170.64. 9. 94.08. 2. S1864.07. 6. «&2.28. 3. ^3220.66. 7. 95232.74 4. S1624.90. o. 922. Exercise XJlXIX.— Page 69. 2. $58.75. 3. $43.19. 6. »23.79. 7. ^261.07. 10, 160.37. 11. «1790.M. 4.^1592.61. 8. 57916.80. 33. 148.46. 1. 0391.86. 6. «6662.60. 9. »3364.20. 13. »1794. 17. »261.25. 1. «12 72. 5. $7.89. 9. $2.22. ▲NBWmBS. Exercise XL.— Page 60. 2. $1482.96. 3. 5926.25. 167 6. $3622.75 10. $16.80, 14. $722.16. 18. H25.26. 7. $157.60. 11. $247. 15. $360. Exercise XLL— Page 62. 2. $21.37. 3. $18.17. 6. $60.50|}. 7. $10.40. 10. 6. 11. 365 days. 4. $97670. 8. $27.76. 12. $169. 16. $3.51. 4. $26.34. 8. 73 sheep. 12. 16 pieces. Exercise Xm.— Page 64. 1. $32.20. 2. 11196. 3. $4.35. 4. $27.76. 6. $47.02. 6.. 1889.77. 7. $14.24. 8. $771.51 9. $18.78. 10. $3.31. Examination Papers.— Page 66. I. 2. 70 cents. 3. $6.16. 4. 100. 6. 15 times more. n. 2. $414.80 3. $281.62. 4. 50 tons. 6. $1.10. m. 2. 46 yards. 8. 86 votes. 4. $1191.76. 6. 1760. IV. 1. 476 yards ; 30 cents. 2. 400 bushels. 8. 1560 pairs. 4. 100 days. 6. 1100 ; 430. Exercise XLin.— Page 68. 1. 2, 2, 2, 2, 3. 4. Zy J, o, ti) d. 7. 2, 2, 2, 2, 2, 6. 10. 2, 2, 3, 23. 12, 2, 2, 3, 3, 6. 16. 3, 3, 5, 19. 18. 3. 2. 2, 2, 2, 3, Ot 6. 6, 5, 7. 8. 6, 6, 13. 6. 2,3,3,3,6. 9. 3, 11, 13. 11. 2,2,2,2,2,5,6. 13. No prime factors. 14. 2, 2, 2, 2, 3, 7. 16. 3, 5, 7, 11. 17. 2 and 5. 19. 7 and 3. 20. 2, 2, 2, 3 and 6. Exercise XUV — Page 68. 2. 2. 3. 12. 6. 30. 6. 72. 8. 72 bushels. 9. $22. 11. $2. 12. 1440. Exercise XL v.— Page 70. 1. 6. 2. 4. 8. 8. 4. 14. 6. 10. 6. 42. 7. 24. a 11. 9. 76. 10. 144. 11. 8 feet. 12. 21 feet. 13. 16 feet. 14. 8 quarts. 16. 46 peart. 16. 8, 11, or 83 pupils in each section. 1. 2. 4. 18. 7. 20 yards. 10. 76 yards. ¥ 168 KLEMENTABY ARITHMETIC, Exercise XL VI.— Page 71. 1. 23. 2. 37. 3. 41. 4. 66. 6. 45. 6. 61. 7. 42. 8. 11. 9. 813. 10. 630. 11. Prime. 12. 21. 13. 184 lbs. 14. 7 and 12. Exercise XLVII.— Page 73. 1. 30. 2. 60. 3. 36. 4. 160. 5. 360. 6. 180. 7. 360. 8. 770. 9. 2520. 10. 1512. 11. 1680. 12. 16800. 13. 1800. 14. 720720. 15. 50702925. 16. 173. 17. »2100. 18. 360 bushels. 19. 240 oentB 20. 84 bushels • 21. 120 days. 11 (1 f fl Examination Papers.— Page 74. I. 1- ttll, 707, and 1089 are comp. ; 643, 757, and 991 are prime. 2. 8. 3. $3048. 4. 643. 6. 25 acres. n. 2. 26. 3. 46. 4. 1680 marbles. 6. 47400 holes. ni. 1. 15, 16, 17, and 18. 2. 900 acres. 8. 9 cents. 4. 356. 6. 9672 rails. IV. 1. 76 cents. 2. $1080. 3. 3600. 4. 10565999. 5. 1267994828100. V. 1. 10a«6. 2. 240. 3. 257. 4. 6 and 4. i. 5. 9. 13. 17. V. 1 1 go as' • 3 B oar Its * lOl I IB Iftl 1. 8|. 5. 33. 9. 16/,. 13. 32^^^. 17. 36jji. Exercise XLVm.— Page 80. 2. G. 10. 14. .18. '-' V. V 1 «•• - ' — ^ • 1 • 7 18 19^ 33 S804S 1 a 3. 7. 11. 15. V. 1 Q>0 H • «««4TT ITa • B»a Exercise XLIX. — Page 80. 3. 6|. 7. 12,^. 11. 28. 15. 515^1. 2. 6^. 6. 17. 10. 13. 14. lOOi-. 18. 522TfVV. 4. V 12. '-^^' ' a» 1ft %*.*-••»*• ^"* •ra 8. 65-i',. 12. 61|.. 16. 6764|. / ANSWEBS. 169 are i. L*. 1. i. 7. f. 13. H- 19. XT* 1. f^ 13. H. 1. i. 6. J. 2. S. 8. j%, 14. |. 20. ,♦,. Exercise L.— Page 81. 3. |. 16. HI- 21. ii. 4. §. 5. |. 10. ii. 11. H- 16. ||^ 17. i^ 22. |. 23. i|. 6. f. 12. J. 18. ♦. 24. IVS^ 2. A. 8. fi. 14 3. Exercise LI.— Page 83. 3. 3tV. 4. IH- 9. ^js- 10. i§. 16. 2i|. 16. H- 5. A. 11. J. 17. IJ. 6. A. 12. /,. 18. II. Exercise Ln.-Page 83. i. f. 3. /j. /, 12^ acres. 8. «8750. Exercise LIIL— Page 84. 4. A- 1. H. If. 4 7« |5§i ilS' ilo' ^20 9» ffiS> S0T§) is JU' IsTw 2. f8, n- as TO fl« T» '• T»K» T5I» TTJS- 5T 5 TO a* • T0»» TOI» 10] 3. $f » fl» ,. 6. n, th n- 8. Hf8»mi.«liMII!- Exercise LIV.— Page 86. 1. 58. s«. n- 4. 3^,41 -«*- ♦ a 1 S' w w 7. i^, 4«. !«• 10. }h t'tt. a- 13. H, ih tV. 15. n, »f» Ay sv- 17. h;, j5^i. Hh ns 6» §§> fflT' Is* 8. t§» l^i IS' 11. V. V. i' 3. 18, fS. H. 6. U. $?, fg. 9. IM8,H- 12. h V» V. 14. m. iVr. tVs. i*»v 16. j%h,Hl^^%%^Hi- 18. ^8g, Wo", nh m- Exercise LV.— Page 85. 1. f. 6. iJ. 9. U. 13- i'bf t f i 16. ^, /, I-* J. Tj|« 6. ijv'T' 2. If. 6, If 14. u ; «• 3. H. 7- A. Ul (I . • 15. S ; i 4. i*. 18, } ; A. If % 17. h iS. S' f . s- Exercise LYI.— Page 87. 2. 1}J. 3. H- r. * 'ni* 6. m- 10. lAV- 7- 2?. 11. 3AV. 4. lA. 8. 2/,. 12. H. Exercise. LYH -Pag« 87. 1. loa 5. ^kii. f. i2|. 2. 10^. 6. 46A. 10. 21iJ. 8. 10||. '. 22TVff. 11. i2f. 4. r>^. 7.^. 16}. Mf «.a* ia il i rii»i i ' >fi Mi i i i ir i l ii H i » ii'm il»» :r r' , M. •„-f[P^- f'™""-»«°i»M a 2. il. 8. m- 170 1. 8*.. 7. ^,j. 1. Iff. 5. ItV 9. 4|. 1. 18i|. 4. 20ii yards. 7. 14^5 reams. 10. ^38^,. ELEMENTARY AEITHMETIO. Exercise LVIII.— Page 88. 3. H. 9. tVt- 4. 5^- 10. x'.. 6. ^. 11. ^. Exercise LIX.— Page 89. 3. S^. 7. 2ii. 11. 24^. Exercise LX.— Page 89. 2. Iju" 6. IJ. 10. 12/,. 6, t;|i. 12. A. 4. 2Si. 8. 2ff. 12. 10|. 2. 16^ gallons. 5. ^3i 8. 34if pounds. 11. «98||. 3. n- 6. IOIt'j acres. 9. 33^ miles. 12. tl, IS. »Vift- Exercise LXI.— Page 90. 2. mi' 3- *^" pounds. ,. ...x„. 6. 106H gallons. 6. SU- 7. 145U yards; $403^1. 8. 44/^ pounds. 9. %i. 10. lOirgallons. U. ^177,V 12. 774^ acres. 1. »383t^. 4. $191t>,. 1. 7*. 5. 37i. 9. 4t|. 13. »1.77. 1. 15. 6. A. Exercise Lxn.— Page 92. 2. 6|. 3. 2i. 6. 71i. 7. 26J. 10. ^30. 11. $13.14. 14. «8.75. Exercise LXIII.— Page 98. 2. 40. 3. 35i. 4. 54i^. 11 t\. 7. |. 12. n. 8. /j. 13. tj%»,. 9. 1. 14. 4. 9i. 8. 10t\. 12. »201. 6. 41- 10. n- 15. A, »281J; B, #225 ; C, $303| ; «810. Exercise LXIV.— Page 94. 1 17 tj, 2. 49^. 3. 290. 4. 1320. 5. 8789. 6. 6l|r 8*. 42f cents. 9. 10 acres. 10. »351t>». H. *67|. 12*. «1667i. 13. »227|ii. Exercise LXV.— Page 96. 2. i. 3. A. 4. .f}. el t' ?: f. 8. 8,v 11. 2i cords.l2. 3| miles. 13. «8§. 6. IJ. 9. 23ii. 10. 8Jaore». iA Exercise LXVI.— Pago 97. 9. 9A. 3. 341. 4. 11?. 5. 3. 7. Ifl. 8. if. 6 S?- l.oi. 10, ^M- ii- -^i*- ^'^- ^'''^' \ M. 9i* iS. ads. res. V 281i; ). ioitn. *? ANSWERS, 171 Exercise LXVII.— Page 97. a. 24 bushels. 3. 14 tons. 4. 27 bushels 5. 248 weeks. 6. 11 persons. 7. ^«J. 8. 2f weeks. 9. 11| bushels. 10. 190]^ days. 11. »67J. 1. 16. 6. 4). II. 7i. 16. 2J. 1. |. 5. B.Vo- 9. U- 13. 4»J. 12. lOil Exercise LXVllI.- Page 99. 2 |. 3. i»T. 4. jV' 6. 1| 7. 2. 8. IJ. 9. 2. 10. 2|. 12. 3^ 13. tn- 14. ItVj. 16. -A- 17. 38t\. 18 Hi. 19. in. 20. iJ Exercise LXix.~i>age lOO. 2. 3i?. 3. 1. 4. 22^1, 6. U||. 7. 4. 8. 1. 10. 1|. 14. 1. 11 ,», 12. 1( 2. §of A. f. «./«'*!• 6. $33075. 16. ,V3. 16. 3,\V. Examination. Papers. Page loi. 11. 4. 1|; iJ- ^' 24 days, m. 4. $21900. IV. s8. »5.60. 3. /,. 2. |. 8. 18 bags. &. 1^15.85. 4. Too large by iJ. V. 2. $359.45. 3. $660.80. 4. %. 5. 9 acres. VI. 2. 2. 3. $1840. 4. 49 acres. 6. *6187.50. Exercise LXX.--P^ge 106. t'o'V _4 I 8« T0T)(5TJ- 8 n oi 7 12 4 «*. TTSaOOf -f" •27. 416. 3-00007. 8. 7. 11. 15. 19. 23. 27. •07. 16126. 1600163. 4. 8. 12. 16. 20. 24. TS4_ TOOO* ToSaffw •8. .136. 126-36^ »1 i7i%' 1. /„. 2. 5. iVA- 6. 9. tVoVtj. 10. 13. jiUh- 1*- 17. -71. 18. 21. 207. 22. 26. -18406. 26. Exercise LXXi.-Page 106. 1 Nine-teuths. 2. Twenty-seven hnndredthp. 8. Three hundred and Mxty-eight tbousandths. 4 Sixty-four thou- sandths. 6. Four, and. thirty one hundredths. 6. beyen, and two hundred and sixteen thousandths. 7. Three, and thre« lir^AT^d and fourteen thousandths. 8. Five, and eight thorn- 172 KLEMRNTAKT \11ITHMSTI0. I sand one hundred and «*i^''y-««veuteu.tbou8andthB. 9. Twenty, one. and three thonaand six hundred and one te^-t^05;«^^J^^^^- 10. Seventeen, and sixty-four ten-thousandths. 11. liighteen, and eighty-one hundrodthousandthn. 12. Twenty, and one thousand four hundred and fifty^^eight l^^^^^^^J^ thousa^^^^^ 13 -S- 2-07; -009. U. 807 094; 8017-0709; 3-001008. is! 6-0004 ; 800000609 ; 10101001. 1. 65046. 4. 2-4397464. 7. 1141377. 10. 16166-66886. 13. 227-5024 Exercise Lxxn.— Page 107. 2. 600-7364. B. 101-209. 6. 959-0483. 9. 200-1211. 12. 122-625 yds. 15. 5. 8. 11. 14. 4475-106045 10-876. 40-52753. 25-749445. 58-4905 acres. 1. 4. 7. 10. 13. 16. 19. 16-1524. •23296. S-9249. •01. 2-5527. •2318 inches. •146. Exercise LXXiu.-Page 108. 2. 2-3806. 8. -43876. 6. 1-8316. 6. -00521. 8. 1-405. . 9. 168-098. 11. -6322. 12. 8-3416. 14. 15.799. 15. 173-03863. 17. S6-()02 grains. 18. -099. 20. 13-75 yard' . Exercise LXXiv.-Page 109. 1. 4. 7. 10. 240-37086. 2474-11. 15-544. 2. 5-4008. 5. 9-6142. 8. -000072. x^. 803-2104. 11. -040527. 13. 334141-402 sq. in. 14. 975 pounds. 16 117-04936022 mi.l7. 728-9271. 3. -0273238 6. -26928. 9. -310104. ' 12. 1-010009. 15. 334-00692 pounds. 1. 5. 9. 1. 5. 9. 13. 3-07. 1240. 20200. •1875. •15625. •06876. 24-008. 18. 312-275 pounds Exercise Lxxv.— Page ill. 2. 50-615625. 3. 800. 4. -006446876. 6 -00075. 7. -00016125. 8. -568. lo! 22600. 11. -082. 12. 83. Exercise LXXYI.-Page 112. 2 .75 - 3. -626. 4. '225. 6 -026. 7. -0376. 8. -875. 10. -078125. 11. 056. 12. 6 6. 14. 3-525. 15. 46.3125. Exercise LXXVn.-Page 114. 1. i 7. xi»- 2. 8. T*r- 3. If. 4. |?|. 5. tIj. 9. l,tU- 10. lU!- 11- S||. 12. 2i|. Exercise LXXViil.-Page 114. 1, 62-9204133494430524. ». 9-928 : 2 297. 2. -24; -03271165. 4. 3-6 ; I-IU. ANSWEBS. 178 Examination Papers.— Page 114. I. «. li; 11*00 ; -oooou. 4. -120508 ; 13. n. 8. -017359; 0005. 6. -714286; iUh- 1. -375 ; -000000375 ; 866-315375 ; 160000. 1. -9525. 4. If X2t. 1. 1-1214727. 4. 2520 ; 3|. 1. |. 2. 24'975024; 500-5. 6. Vi- IV. 2. -54321. 6. 95|. V. 2. -OOOOl ; -00009999. 2. -01825. 6. 8; 6400. 3. tS, ; 15f ; 1» 3. 83000, $6900. 5. 1.60546875. 1. 92d. 4. £309 58. 7. £4 Is. 5id. 10. 3209 far. Exercise LXXix.—Page 118. 2. 1104 far. 3. £29 158. 5d. 6. 2406d. 6. 560d. 8. £29 168. lid. 9. 183839 far. 11. £328 16s. 4d. 12. 96028 far. Exercise LXXX.— Page 119. 1. 1044736 dr. 2. 1390 dr. 3. 13cwt. 2qr. 21b 13 oz. 4. 954 1. 16 cwt. 1 qr. 5. 4933 oz. 6. 25 1. 16 cwt. 1 qr. 24 lb. Exercise Lxxxi.— Page 120. 2. 24 lb. 10 oz. 3 dr. 1 scr. 3. 32 lb. 5 dwt. 5. 1684 gr. 6. 12 lb. 9oz. 5 dwt. 4gr. Exercise LXXXII.— Page 121. 1. 71478 in. 3. 1 mi. 3 fur. 18 per. 3 yd. 2 ft. 5. 462 ft. 1. 16 oz. 4. 5460 gr. 2. 1 mi. 1 fur. 26 per. 2 ft. 4. 86 ft. 6. 232fath. 4 ft. Exercise Lxxxm.— Page 122. 1. 12 a. 1 r. 37 rd. 2. 117900 in. 3. 4 cu. ft. 1557 in. 4. 60 c. 9 ft. 6. 75506904 cu. in. 6. 136424 cu. in- Exercise LXXXIV.— Page 124. 1. 662400 sec. 2. 120 bu. 2 qt. 8. 2691 gi. 4. 2311 pt. 6. 83 gal. 3 qt. 1 pt. 1 gi. 6. 566 pk. 7. 61 bu. 25 lb. 8. 1 wk. 2 da. 2 hr. 14 min. 63 sec. Q 74 ba. 84 lb- 10.6739740 860. 11. 12 e. 40 It 3is»*f ! 1 174 ELEMENTARY ARITHMETIC. Exercise LXXXV.— Page 126. 8. 88 rd. 5 yds. 1 ft. 6 in. 4. £34 Us. 8d. 6. 43 bu. 1 pk. 1 pt. 6. 95 rd. 5 yd. 2 ft. 3 in 7. 6 wk. 3 da. 6 h. 50 min. 33 see. 8. 22 rd. 2 yd. 8 in. Exercise LXXXVI.— Page 126. 1. 7 lb. 8oz. 6 dr. 1 scr. 19 gr. 2. 19 mi. 1 rd 3, 69 a. 2 r. 27 rd. 5. £27 17s. 4d. 7. 142 bu. 2 pk. 6 qt. 9. 31 gal. 2 qt. 1 pt. 11. 1 cwt, 8 qr. 10 lb 4. 5 fur. 31 rd. 5 yd. 2 in. 6. 38 per. 18 yd. 2 ft. 36 in. 8. 79 lb. 3 oz. 5 dwt. 4 gr. 10. 22sq.rd.r2yd.4lt.l28m. Exercise Lxxxvn.— Page 127. 1. 90 cwt. 3 qr. 7 lb. 13 oz. 3. 70 da. 23 h. 34 min. 40 sec. 5. 4985 cwt. 1 qr. 7. 150 a. 2 r. 35 sq. rd. 9. 88 mi. 3 far. 2 rd. 3 yd. I2: f62 mi''4 f m^28 x d. 3 yd. 2 ft. 2 in. 13. 2739 bu. 1 pk. 5 qt. 2. 50 lb. 2 oz. 7 dwt. 3 gr. 4. £600 9s. 6|d. 6. 1 lb. 1 oz 12 dwt. 8. 23332 gal. 2 qt. 10. 5 oz. 19 dwt. Exercise LXXXVIII.—Page 128. 1. £15 9s. 7d. 3. 16 t. 2 cwt. 1 qr. 13 lb. 6. 2 cu. yd. 6 ft. 960 in. 7. lOjS. 11. 25 dimijohns. 2. 12 lb. 9 oz. 15 dwt. 18 gr, 4. 1 gi. 6. 8. 8. nnn- , „ , 10. 2 bu. 3 pk. 3| qt. 12. 5 weeks. Exercise LXXXix.— Page 129. 2. 5fur.l3rd. 1yd. 2ft.6in. 4. 2 fur. 16 rd. 6. 2 r. 8 rd. 26 yd. 8 ft. 8. 4 da. 23 h. 28 min. 1. 3 pk. 1 qt. li pt. 3. 4 yd. 2 ft. 5| in. 5 17 cwt. 2 qr. 7. £1 128. lO^Jd, ; £5 2s. 8H<1- 9. 1 lb. 9 oz. Exercise XO.-Page 129. 1. 6. 9 2- Toitro' 10. fU. 3. 7. 11. ■ T • 4. 8. 12. Hij* 7 1 S ffOD»* H- 1. 3r. 81 rd. 4. 47 min. 6 sec. 7, 7 fur. 29 per. 16. 3s. 5id. Exercise xci.-Page 130. 2. 9 oz. 15 dwt. IS gr. 5. 11 h. 55 min. 40$ see. 8. 6. lOid. 8b. 9d. 8 15cwt.2qr. 61b. 4oz. ?;, l^s. 6|d 11. 2 da. 12 h do mia. 21 seo. a-, t^i- ANSWERS. Exercise xon.— Page iso. 1. £526. 5. -SlSSpk. K -625 fath. 1. »193. B. »2667.60. 9. »381.75. 13. $2388.50. a. •282t. 6. £9-26876. 10. -71. 8. '78126 oz. 4. 7. 17-895 owt. 8. 11. 129-78 hr. 12. 17« •775 mi. 7-875 bu. •001625 t. 2. 12 H lb. 6. l^f in. 9. Had. 12. »7162 31JH- 16. $2.10. 19. 41661 yd. 23. »1736.23|. 27. »8400. 31. 64 oenta. Exercise xcni.— Page 131. 2. »148.50. 3. $436.80. 4. $388. 6. $.615. 7. »496.12i. 8. »308. 10. »35.65. 11. $101.85. 12. $124.20. 14. $44.04. 16. $32753.12J. Exercise XCIV.— Page 132. 3. $65.10/,. 4. 7. £34 12s. 4d. 8. 10. 6Joz. 11. 13. $i73-74TJ^. 14. 17. 98 yd. 18. 21. $1108.80. 22. 24. $880. 26. 28. $3000. 30. 58. 2d. $567,526. 7 hr. 11 min. 8 seo. 30 f yd. 3b. 7WVd. ,», ct.; $5.76. 6 oeuts. 1. 84-25. 6. 44. Exercise XCV.— Page 136. 8. 493.33. 2. 6. 32.22. 43«.99. Exercise XGVI.— Page 136. 1. 72. 2. $16. 8. 23 sheep. 6. 14 men. 6. 45. 7. 40%. 9. 60%; SQjY/c. 10. 1100. 11. $1200. Exercise xcvn.— Page 137. 2. $11.20. 3» $15.20. 6. $110.40. 7. $166.25. 10. $7000. 11. $70000. 4. .64.40. 4. $10.50. 8. 75%. 4. 8. 12. 1. $18. 6. $100. 9. $2500. 13. $6276. Exercise XCVIII.— Page 138. 1. $14.40. 2. $15.e0. 8. $10. 6. $247. 6. $112. 7. $2500. 9. $788.76. 10. $1.75. 11. 3J per cent. Exercise XCIX.— Page 139. 8. »236-412. 7. $236.64. 10. $842.19|. 12. $31^.64. 16. $882. 18. $14-858... $11.25. $65.20. $9000. 4. $30. 8. 7500 bu. 1. $48. 4. $668.06. 8. $503.36. 11. $198.66. 14. $192-226. 17. $934.92. SO 2. $£8.50. 6. $451.60. 9. $8t). 13. $164.78. 16. $2076.36. 19. $287.67.. A AIQAA.Q^O » IT-*--" Ol fi n^w OO "Ti 17« ELEMENTARY ABITHMETTO. 23. 6 p«r cent. 24. $3500. 26. 19000. 27. 3 yr. 29. Oct. 4, 1877. 30. 14» yr. 86. 93250C. 28. 8 yr. 1. «1168.70. 5. $2100. 9. 11.25. 1. 17. 5. 36. 9. 64. 13. 625. 17. 6326. 21. -32. 26. 8-4261... 1. 96 sq. ft. 4. 18| sq. yd. 1. 84 yd. 5. 948. Bxercise 0.— Page 143. 2. $457.50. 8. »900. 6. Gain $50. 7. $15582. 10. $3.60. 11. $10.35... Exercise CI,— Page 146. 2. 19. 6. 75. 4. Tbe latter. 8. ^242.32. 10. 37. 14. 612. 18. 5008. 22. -8449... 26. 2 5298... Exercise cn.— Page 147. 2. 91 sq. ft. 6. 351^ sq. vd. Exercise oiii.— Page 148. 2. 26§ yd. 3. 64 yd. 6. «57.60. 7. f29.16|. Exercise CIV.— Page 148. 4. 25. 8. 49. 12. 56. 16. 2401. 20. -27. 3. 24. 7. 95. 11. 47. 15. 343. 19. -47. -. -. 23. -946.... 24. -9486... 27. 3-794 8. 626 sq. ft. 6. 470 gq. ft. 4. 138§yd. 8. 2i ft. 5. $6.80. 1. 126 yd. 2. 678 sq. ft. 3. 115J yd. 4. 57} yd. Exercise CV.— Page 149. 1 940 on ft 2. 95 cu. ft. 3. 187^ cu. ft. Vm^l 6.16000.. 6.$14.51H. Miscellaneous Problems.-Page 150 2. 5. 16 days. 5 days. 9. 3/t ^^y^' 12. 13 f days. 24 days. 13|i days. 16. 20. 23. 25. 28. 32. 36. Man 90 da; boy 180 da Woman 30 da. ; boy 40 da. . . nn no 1 3. 5 days. 7. C| hours. 10. 13i days. 14. 5tH days. 17. 8t«r hours. 22. 24. 27. 24 hours. 180 min. 35 ft. 40. «5000. 44. 36 da|B. 48. $192. K1 US.. ft1Q 29 83. 37. 42. 45. 49. 62 2d^\g min. 72 min. $1582. 8 days. $94.50. 300 men. $10.31i. 55. irper cent. 66. 20 per cent. 4. 9 months. 8. 8 days. 11. U} days. 16. 15 days. 19. m days. Man 27^", da. ; boy 120 da. Man 28 da. ; woman 40 da. 24i^ min. 30. 17^ min. 35. 923030. 88. $2843.76. 43. $1827. 47. $32. 60. 12 o«. 63. $350. 67. S3f per oent. r Mu ANSWSB8 \. 177 59. $1.60. 60. 17226. «1. »136.70. 63. 1^45.80. 04. »1.45. 66. 676,664. 67. 9125 ; 1226 ; »160. 68. 1160; $200; »260. 70. 112 a. 5 98 a. 71. 1620 bu. ; 2280 bu. 73. »G0; «I144; 896. 74. $39uO; 94800; $3300 76. 814.40 ; f.d.QO. 76. 96.30 ; 96.26 ; 94.20. 78. »21.60; 123.04; 926.92. 79. 9525; 9500; 9480. 80. S1440; »2385. 82. $124.16.. * • 83. 91216.60.... 84. 1^6.8992. 86. 466.56. 66. 9161.02.... 87. U^'A,75i. m I :i ^f^.% .c»^. \^ IMAGE EVALUATION TEST TARGET (MT-3} ^^ y 1.0 I.I l^|28 |25 HI z:zzz 1^ ftw 12.2 w nA liaiB m 1.25 III 1.4 ill 1-6 *l 6" - ^ Photographic Sdences Corporation 23 WEST MAIN STREET WEBSTER, N.Y. 14530 (716, 872-4503 ^\7 ^ %0 '<> vssssm EXAMINATION PAPBE8. JCM EXAMINATI0K8, 1877. ADMISSION TO HIGH SCHOOLS. TIMIS — TWW HOURS. Examiner^-h A. MoL^LLiN, LL.D. 1. What is the least numb- tba^^^^^^^^ millions to make the sum exactly diMsibie oy b«>b and nineteen ? 20 im + 7 J - i6i ^ y A . 2. Simplify 21— ^16i x U| x 121" ' Tj/ £14 12«. lid. £10J5i-^— 8. Simplify -lotTTsf"' 10«- ^i^- „ •"5 \\ 'Zds of cloth cost $12.50 ; what will 23^ y«a. costt certean quantity of silk; ^^*X^ "P^^^ re^-4lJWorytrp:°lw .«ae M silU could Bhe hav^ b°„ght at tot? ^^ ^^j_ 8. V°°^VV»lswhr"ald both! Expense ol ooyering *^trti.r"le^Sy oYmXg, a room a yard lor.ger and TT^'lvLago of fonr quantities U IBA', ,. the ^t 's Jm^^JZI^ 8.592. and the third U 88.06. Fu^d the fourth. ^ ^ g ^^612.80 ; if A 10. A bankrupt owes to A $108H.»4, ana w « receives $357.44. ^^^^ ^''^^ ^ '^'^'''^^ 10 marks to each question. EXAMINATION PAPERS. DECEMBER EXAMINATION. 1877. 179 ADMISSION TO HIGH SCHOOLS. Values. 12 12 12 12 12 12 16 12 TIME — TWO H0UB8. Examiner. — John J. Tillbt. 1. How often is 6 yds. 2 ft. contained in 26 ruHongs ? 2. If I buy 3 bushels, paying 6 cents for *»very B quarts, and sell at a profit of 10 cents per gallon find the selling price of the whole. * 3. Simplify ^i+t°tl 2-i„ll,. 18n+Sy.-»2» 4. Reduce 2 hrs. 20 min. to the decimal of 3 J weeks. 6. A sum of money was divided among A, B, «nd C. A received f of the sam ; B, $20 less than f of what wa.^ left ; and the remainder, which was f of A's share, was given to C. Find the sum divided, 6. Trees are planted 12 feet apart around the iWes ol a rectangular field 40 rods long, containing two acres. Find the number of trees. 7. I buy a farm containing 80 acres, and sell f of it for I of the cost of the farm ; then sell thjremaiuder at $60 per acre, and neither giai nor loose by the w*iole transaction. Find the cost of the farm. 8. Find the amount of the following bill of goods i— 18| cords of wood, at 83.60 per cord. 16 yards of cloth, at $1.12^ per yard. 12 bus. 25 lbs. of wheat, at |l 20 per bus. 1,400 feet of lumber, at $12.50 per thou8a«4. 65 tons 12 owt. of coal, at $0.30 per owt. 180 EXAMINATION PAPERS. JULY EXAMINATIONS, 1878. Values. 12 12 12 12 12 12 16 12 ADMISSION TO HIGH SCHOOLS. TIME — TWO HOURS. Exami7ier. — John J. Tillet. 1. Define prime number, multiple of a number, high- est common factor of two or more numbars. 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 three together, plus 45, amount to 561. Find the dividend. 3. I sell 12§ tons of coal for $80, which is one- seventh more than the cost. Find the gain per cwt. 4. .001 X. 001^.0001. 6. A cistern ya 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 bo half full ? 6. Ten men can do a piece of work in twelve days. Aftei 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 $600, of which twelve boxes are unsaleable. I sell two-thirds of the remainder for 1400, 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, 6 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. — Plasteiing a room 24 ft. long, 16 ft. wide and 10 ft high, at 16c. per square yd. — Sawing 68C0 shingles at 40o per 1000. EXAMINATION PAPKBS. 181 PEOVINCIAL MODEL SCHOOL. SECOND DIVISION. 1. Among how many persona can you divide £962 6s. OAd.. givmg each of them £137 98. S^d. 2. Divide »6640 among A, B, and 0, so that A may have one-quarter of it, and B, four times as much as C. 3. If 6 men do as much work as 8 boys in a day, how many days will it take 32 boys to finish a piece of work of which 16 men did one-fourth in 16 days ? 4. If the dividend be 6060034, the quotient 660008, and the remainder 1, what is the divisor ? 6. Divide 478673 by 106 by using its component factors, and explain how the true remainder is obtained. 6. The price of 2 turkeys and 9 chickens is 914.40, and the price of 6 I .keys and 3 chickens is $20.40. Find the price of a turkey and of a chicken. 7. Eeduce to a deciudal, correct to 6 places, 1 -^ r + ^-^ + -^ +eto. 1x2x3 1x2x3x4 8. Find the value ot2^n of 13 acres. 9. A drover invested $30,450 in the purchase of horses at a certain average price. He sold a part of them for $12,000 at $400 each, and lost »35 per head. At what pi ice must he sell the remainder so as to gain $750 on the whole ? 10. If 14 men can do a piece of work in 14 days, working 10 hours per day, how long ought the work to occupy 12 men working 7 hours a day ? 11 Two watches are set together ; one loses 7 sec. and the other gains 8 sec. a day. When will one be a quarter of an hour before the other 7 m M I 182 EXAMINATION PAPERS. CITY OP TOBONTO. OOHBINBD BXAMINATI09. 1. Divide 640 tons, 10 owt , 3 qra., 15 lbs., by 376. 2. How many seconds are there from 9 o'clock a.m., on 13th January, 1876, to 16 minutes past 10 o'clock p.m.. on 27th June, 1877 7 f •. «" ^«iu 8. Beduce 3«. 6d. to the decimal of £7 10«, 3 4. Simplify 8+. 4+ 6 + J 6. Beduce 1 ton, to lbs., oz., Ac, Apotl eoarieg* weight. 6. Simplify i■*•|-^l 1 1 1 H H H 7. If 6 men will dig u trench 30 yds. long and 8 yds. broad in six days of 16 hours each, in how many days of 12 hours each will 8 men dig a trench 40 yds. long and 16 yds. broad f 8. Divide the difference of IOC and ^^ by their sum; and also their sam by their difference, and f. i the sum of the (juotients. 9. What is the interest on 5757.60 for four years and four months, at 6i per cent ? 10. Find the difference in the expense of carpeting a room 24 ft. 9 in. long and 15 ft. 6 in. broad, with carpet | of a yard wide, at 1^1.50 per yard, and with v ^rpet J of a yard wide, at 70 cents per yard. 11. A man sells a house for 5437.60, and loses 12J per cent, on the original cost. What was the original cost ? 12. How many flag stones, each 2-88 ft. wide and 8-30 ft. long, are required to pave a walk round a grass plot 137-31 ft. long and 126-79 ft. wide, the walk being 4 16 yds. wide ? EXAMINATION PAPEBS. 188 t;OUNTY OF WATERLOO. COMPETITIVK EXAMINATION. Promotion from 5th to 61 h Form. a^k ^^7^° ^^? ^°*° *^° P^'^8, one of which shall be three- filths of the other. 2. The figures in the units and millions places are 8 and 6 respectively ; what will they become when 999,999 is added to the number. ao?" J"^^ *^® P"^® ®^ ^ *°°^' ^^ <'^-' 3 qre.. 12 lbs., 8 oz., at »86.24 a top ? Eeduce the answer to £ s. d., taking the value of the £ to be 64.867. 4. A car is exactly filled by barrows that hold each 9 cwt., and emptiei^ by sacks that hold each .5 cwt. ; given that it holds between 8 and 10 tons, find the exact amount. 6. Simplify •004-4- -0006 2.423 + 3.576 + 2.0001911 6. A man borrowed 5500 on the Slst Januarv, 1874, at 8 per cent, interest, payable half yearly; every half-year he payg |80. How much will he owe on the 21st July 1876, whatever he pays each half year over and above the interest due being deducted from the principal ? 7. A, who owes B »1000, sends him 725 M ft. of lumber, which B sells at 616 per M, charging 3 per cent, commission, and paying out of the proceeds expenses amounting to ^161. 50. How much is coming to A? 8. If B buys for A, with the sura you found due him from No. 7, flour at 14.50 a barrel, charging 3^ per cent, commis- sion, how many barrels will A. receive ? ^ 9. If I buy broadcloth at »3.60 a yard ; what must I ask for it a yard that I may be able to throw off 10 per cent, from my price and yet make 25 per cent, profit ? 10. How much Dominion 6 per cents at 106 can I buy foi $16,200, and what yearly income shall I have therefrom ? 11. The population of a town has increased 5 per cent, ginca last year. It. has now 3780 inhabitants; how man* had ii then? 12. I borrowed 6250 on the Ist October, 1875, and paid back $268 on the 24tn February, 1876; what rate of interest per annum had I been charged 7 184 EXAMINATION PAPBB8. ■r Values. COUNTY OF DURHAM. COMPETITIVE EXAMINATION . Senior, 16 16 16 18 16 18 18 18 1. A can do i of a piece of work in one honr, B can do f of the remainder in 1 hour, and C can finish it in 20 minutes. How long will it take A, B and C together to do the work ? * 2. If it costs $70.40 to carpet a room 24 feet long with carpet 2 J feet wide, at $1.10 per yard, find the width of the room. 3. By selling tea at 96 cts. per lb., a merchant gains f of the cost ; he then raises the price to $1.06 per lb. ; what does he clear on every $8.40 of his outlay by the latter price ? 4. A bankrupt who is paying 37^ cts. in the $ divides among his creditors $6300, and secretly retains $2100.- What do his debts amount to, and how* much in the $ would his creditors receive had they obtained all his assets f 5. Find the price of 423 lbs. of peas and oats mixed equally by measure, when peas are 70 ots. and oats 40 cts. per bush. 6. A merchant sells 60 lbs. of tea and coffee for $43.50 — the tea at 90 cts. and the coffee at 40 cts. per lb- How many lbs. oi each did he sell ? 7. John McCromb rents a house, the cash value of which is t?400, and which is kept in repair by the owner at an annual outlay of $20 paid at the end of the year. He pays in advance an annual rent of $200. How much a year would he save by buying the house and paying for it in cash, money being worth 7 per cent, per annum f 8. Simplify {a) 2-23 + 3-76 5.5 ->- -69^ + 8-24— 1-21 •^7* + 1-36 1-34 + 2-466 .8 X ^ -m. X 3Ji (6) (^ — i of 3) of £4 lis. 4d. is what fraction of (J of . i — i) of f 164.4f), £1 being worth $4^ 2.20 marks ti iaH paper. T ANSWERS TO Ej^INATION PAPERS. 186 ANSWERS TO ExMnATION PAPERS. 1. 4. 7. 10. 1. 4. 7. 1. 4. 7. 4 547. 222i cwt. 37i yds. »210.66. 825 times. •00416. »3000. 2, 2, 3, 3, 6, 7. •01. 120 boxes,- AD1I1S8I0N TO niQH SCHOOLS. July, 1877, 6. 11)3.76* • o. $36. December, 1877, 2. HM, 6. »200. 8. 609.625. July, 1878, 2. 31116. 6. 8 hrs. 8. 5101.85J. PKOyiNCIAL MODEL SOHOOL. 8. £41 Is. 6jd. 6. »2.77?t. 9. 4* 8. 312j|. 6. 132 trees. 8. 4 cents. 6. j4,. 1. 3. 6. 8. 11. 7 persons. ' 30 days. 4. 1 _ Turkey, ^3.60 ; Chicken 80 centg» SiVra. 9. $480. 60 days. 2. A, §1660; B, ?3984; C, S996. 5. 4558,V,. 7. 2-718281. 10. 23i days. 1. 2. 6. 7. 10. 1. 4. 7. 10. CITY OP TORONTO. 1 ton 8 cwt. 3 qr. 7 lbs. 14 oz. 8,", dr. 45926160 sec. 3. •023. 2430 lb. 6 OZ. 5 dr. 1 scr. 16 days. »61.15. 300, and 180. 9 tongit »1009'0.50. »160p0 ; »900. 48 min on 4 11. »500. COUNTY OF WATERLOO. 2. 7 and 6. 5. 1. 8. 2166gjj bU. 11. 3600. COUNTY OF DURHAM. 2. 18 ft, 3. §2.10 4. Ui. €• I.'bV,. 9. »213.39x',. 12. 300. 3. £177 Is. 7Jd. 6. -fl75.02. 9. $5. ' 12. 18 fo. ^Zi 6. 14.95. 6. 89 lb. tea ; 21 lb. coffee. \yv^ iri ^'^ VQ^ THE EPOG^ PRIMER ; i ; Of ENOii Twr HISTORY. Being an Introductory Volume to the leriei of Epochs of Englith HMorvt by the Key. MANDELLOHEIGHTON, M. A., late FeUow and Tutor of Mer' ton College, Oxford ; Editor of ' Epochs of Engliih Hiatory.' Fcp.8T0.pp- 148, price 80 cts. cloth. ^ f f^ * In making history attraotlTe to the young the Author has proved his apti- tude in a department of literature in which few distinguish themselves The narative is «o sustained that those who take it up will have a desire to fead it to the end.' DUNDKE AdybBTISHB. ' This Tolume is intended tobe in- troductory to the Epochs of English History, and nothing could be better adapted for that purpose. The little book is admirably done in all respects, and ought to have the effect of sending pupils to other and fuller sources of historical knowledge. » Scotsman, 'Mr Ckkiohton's introduction to the EjDochs of Ejnglish History covers in a nundred and forty pages more than 1800 years, but having regard to its extreme condensation is weU worthy of noticri. On the whole the work is admirably done, and it will no doubt obtain a very considerable sale.' ATHBN.aiUM. ' An admirable little book that can scarcely fail to obtain a considerable popularity,! notwithstanding the great number of previous attempts made to relate the history of England in a very small compass In this epitome the epochs become chapters, but an in- teresting account is given of such events as are likely to be attractive, or even moderately intelligible to young readers.* Welshman. • The excellent series of little books published under the title of Epochs of English History, edited by the Eev. MAKDBiiii Crbiohton, M. A., and Writ- ten by various able and eminent writers being now complete, the Editor has prepared an introductory volume, cal- led the Epoch Primer, comprising a concise summary of the whole series. The special value of this historical out- line is that it gives the reader a com- prehensive view of the course of mem- orable events and epochs aiid enables him to see how they have each con- tributed to make the British Nation what it is at the present day. LrniRABT 'WOKUD. 'As all the leading features— political, social and popular — are given with much impartiality, it can hardly fail to become a school class-book of great utility.' WOHCBBTEB JOUBNAL. ' Th»Eev. MANDBiiii Cheiohton has really succetsded in making an admir-' able resume of the whole of tlie prin- ciple eveDts in English history, from the time of the Boman Invasion down to the passing of the Irish Land Act in 1870. Interecting, intelligible and clear, it will prove of great value in the elementary schools of the kingdom: and those advanced in years might find it very handy and useful for casual reference.' Nobthampton HsBAiiD. ' This volume, taken with the eight small volumes containing the accounts of the different epochs, presents what maybe regarded as the most thorough course of elementary English History ever published Well suited for middle class schools, this series may also be studied with advantage by senior students, who will find, iistead of the mass of apparently unconnected facts which is too often presented in such works, a careful tracing-out of the real current of history, and an in- telligible account of the progress of the nation and its institutions.' Abkrdben JOTTONAIi. ' The whole series may be safely commended to the notice of parents and teachers anxious to find a suitable work on English history for their children, inasmuch as the several volumes are simply and intelligibly written, without being overloaded with details, and care has been taken to bring every subject treated on within the comprehension of the young. The namby-pamby element, which is so often cor ''lionous in histories for children is entirely absent, and the works in question are certainly amongst the best of the kind yet issued. The little volume now under notice, which brings the series to a close, is fully equal in every respect to the preceding ones, and it will be found exceedingly useful to every one who may have to teach English history.' LmAKiMOTOir Coxtbcbr. fh HMoryt itorof Mer" ^op. STO.pp- I — political, |{iven with hardly fail )ok of great JOTTBNAIj. iiOHTON haa 3t an admir-' )f the prin- Btory, from aBion down Land Act lligible and \,i value in le kingdom; 9 might find for casual Uebald. h the eight he accountM jsents what it thorough ish History suited for series may vantage by nd, iistead nconnected resented in cing-out of and an in- gress of th« roxniNAii. be safely of parents 1 a suitable for their he several intelligibly oaded with n taken to [ on within oung. The hich is so stories for t, and the ily amongst sued. The tice, which je, is fully i preceding ixceedingly iy have to COITSCBR. Imir/fa '# /^, ' / / / / .#' 3" V .^*^' * Q f^ HP n THIS SPOOH PRIMER Of KNOIJtSH HISTOKY. ' In mftking hiitory attraetiT* to the Tonng Ibe Author hM proved his apti- tude io a department of Uteratoxa in whioh few dCitiiiguiih themaelTei The acratiTe is so suiitained thai thoee vho take it up will hare a daiire to wad it to the end.* DtriTDiB Advebtiibb. jSslag an latroduetorr Yelmae to th« leHM ot EpoOut of ^JS*f%*2^ hrvSi Bev. MANDKjaiORKIGHTuN. M. A., Ute Fellow and Tutor ^Mer toncS)Sie,fl«Srdy»Utota«*B^^^ of BnglUh Hietorr.' Fcp.8T0.pp- 148, prioeSO ote. oloto. • A« all the le-diag featurea-^liM«il, ■ocinl and popular—are gi^a*^^™ much Impartiality, it can hAOT »«* to become a school dase-book of great utility.' WOBOBBTSB JOWUASs. ♦ The Bev. MAwnaiiiiCMBOHTON hai really succeeded in making an admir- able resume of the whole of the prin- ciple events in English hiatory, flrom the time of the Boman Invasion down to tho pas<)iug of the Irish Land Ad in 1870. Interesting, intelligible ana clear, it will prr-re or great value in the elementary t^.^ools of the kingdom; and those advanced in years might find it very handy and useful for casual reference. » NovrtLuanon HaBAiiD. • This volume, taken with the eight small volumes containing the accounts of the different epochs, presents whal maybe regarded as the most thorough course of elementary English History published WeU suited for ' This volume is intended tobe in- trodttotory to the Epochfi of EnglWi HiMtorift and nothing could be better Adaptod for that purpose. The little bo^ is admirably done in all respects, and ought to have the effect of sending pupils to other and fuller sources of historical knowledge.* Sootskav. *Mr Obkiohtok'8 introduction to the Epochs of Singlish History covers In a nnndresl aLd forty pages more than 1800 yeait, but having regard to its eztnnte condensation is well worthy of notice. On the whole the work is admirably done, and it will no doubt obtain a very considerable sale.* ATHBiraBITH. •An admirable little book that can scarcely fail to obtain a considerable popularity,! notwithstanding the great number of previous attempts made to relate the hiatory of England in a very small compass . . ..In this epitome the epochs become chapters, but an in- teresting account is given of such eventa as are likelv to be attractive, or even iv . ever ^„ middle class schools, this series may also be studied with advantage by senior students, who v» "11 find, instead of the mass of apparently unconnected thcU which is too often ^ preseatod in such works, a eareftil tracmg-out of the real current of history, and an in- telligible account of the progress of ths nation audits institutions.* ABKRDaBN JOVKSAJj. • The whole series mi^ be safely commended to the notice of parents and teachers anxious to find a suitable work on English history for their children, inasmuch as the several volumes are simply and intelligiblv written, without being overloaded with details, and e«re has been taken to bring every subjltet treated on within the comprehension of the young. The namby-pamby element, which is so often conspicuous in histories for diildren is entirely absent, «nd the works in question are certainly amongst the best of the kind yet issued. The little volume now under notice, which brings the series to a close, is folly equal in every reqiect to the preceding ones, and it will be fonnd exceedingly useful to every one who may have to teach English history.' i IjBAKIHQTOIT Kutvumu. m I *»■ MILLER I m %m OF ELIIIIIIS, ^ Ufpd in Nearly all the principal High and" Public Schools of Canada ,^,^.... ti tf Hughes' Composition Blanks, No i—lOo. 2— 10a 3— 20a Canadian Spelling Blanks, No. 1— 7o. '; \ I 2-7o 3-7c. Dr Davies' Grammar Bla "is, No. l~7c. :' " " 2~7c. " " 3- 1 3c. " - - 4-1 3c. / PROM NOVA SCOTIA. ^ J. D. MoGILLIVBAY, Inflpeotor, Hunt Co., Nova SootUL // ^ "I have looked over oarefully your Spelling Blanks, Orammai BlankB and Composition Blanks, and consider them excellent. No school can be regarded as fully equipped \»hlch is without them, or books like them. I will, with pleasure, direct the attentionof our teachers to them." A. 0. A. DOANE, Inspector of Sobools, Shelbume Co., N. & " The Exercise Books are well arranged, of convenient sice, i seem suitably adapted for use in all our schools. As helps t» correct spelling, careful writing, and proper analysis and conston** tion sentences hey are really invaluable." W. 3. DABBAOH, M.A., Insp. P. 8., Cumberland Co., Hovft Boolfak SPBXJjINa BliAKXS AKD HUOHBB* COMPOSITIOK BOOKB.—" I Yltftm exaxBlned them with much pleasure. It is high, but certainly not exaggerated praise, to say that the books ar»in point of praotiiool utility unrivalled. I have perfect confidence in reoonunend^ig them to the teachen of our widely extended country. -"—•"^ D. H. SBOTHt ICJL* &u^. Colchester County, Nr ExBBOZBB Books.—" They are got up in excelleni ^ •ff eotinfl on the pupils and teachers a great economy &. tsndtonaoolOAtOAtMtoof neatness and oare in their w<