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Les diarrammes suivants illustrent la m^thode. 1 2 3 3:x 1 2 3 4 5 6 *(i«ass8!fii^«^i(' :i(i;3ii!!a. >'« :'Vrr;-?.«f5g^g^??^»©,'SS*il«!!^i^P««P 1 '1" M E PUBLIC SCHOOL ARITHMETIC TORONTO: CANADA PUBLISHING COMPANY (LIMITED) h 6 Entered according to Act of Parliament, it. the ofTice of the Minister of Agriculture, in the year 1887, ^V the Canada I'iulimuni; Comi an v (Limited). PREFACE. culture, This book (litters from tlio or.linary school textbooks on cUanciit- ary imtlmiotio in tlio following rcspocts:— Ist. It omits the r,.k..s, solved examples and explanations usually given under the headings of dotation and Numeration, the Simple and the Compoun.l Rules. These rules an.l explanations are never stu.he.l ),y the ,,upil who is not sutfieiently a.lvaneed in the art of readmg and in las general studies to follow explanations and direc- tions gnen in print. They must be presente.l to him orally, with all tiie advantages of variation of expre-^aion and of emphasis pos- sessed by the living voice, and he im.st see every solution develope.l step by step. The space gained by the omission of these useless rules has been devoted t(, the practical problems given in Chapter IV _ 2nd. From the Tables of Measures all weights and measures not in general u.se in Canada have been omitte.l, and in those defined by Imved '''"''"* ^^''^ Domhiion Statutes have been carefully fol- 3rd. The extremely complicated expressions which it has become ot late the custom to introduce under the head of Complex Fractions are represented by the last lialf-dozen questions in Exercise LXIII and even these may well be passed over by the teacher. These fasli! loimble conundrums in symbols are out of place in an elementary antlimetic. ■^ 4th. No mention is anywhere made of the so-called True Discount ot the text-books. In business transactions the word dlsronnt bears one ineamng and one only, viz., that given on page 166 of this work. 1 he text-book problems in True Discount are nothing more than lestions in Interest, and to call them by any other name is merely introducing needless confusion. speeia cla.B of fractions, are presents as an easy and natural exten- sion of our ordinary system of numeration. Too often the result of deriving the rules for decimals from those for fractions is, that the IV 1MILI'A( E. pupil reduces all decimals occurring in liis work to the fractional form and oporatcH witli or upon these, thus losing tiio enormous advantages ot declnial caluulatioii. By the metliod h(^rtPiic'ti()n Multiidiiiition ."* Division CoMi'OT'ND Notation- Svstkms— ^^ Tables of Measures Reduction ' " Compound Addition | , | ' ' Compound Sul)traotion ,',".' Compound Multiplication j- Coni{)ound Division _ Applications of Pkkcedino RrLEs— '^' Values Bills and Accounts ', „ Aggregates and Averatjes. "'* Sharing ....'.[ Measurements... ^ J Linear Measurements I. Areas of Rectangles .' ^'* Volumes of Quads '^, Factors, Measures and Mi'ltiples- ^^ Integral Factors Measures ^'- Multiples ■"' Fractions— ^^ Notation and Numeration Reduction of Fractions ^ '•* Reduction of Improper Fi^Jti-ins to' Mi;;ed N«mb;;s '"120 Intercon version of Denominators . * " ,„, Reduction to Common Denominators j gg vi CONTKNTS. VnAimiosH—Contmueft. Achlition of FnictionH 1'>H Suhtnictlon of Kriutioim I ;{ 1 Multipliciitiou of Fractions |.'{;{ I )i vinioii of Fraotious I H7 Dfiitiiiiinato Fiiu-tions 110 Applications of I'rucciliiig Rules 144 I)i;c'i.MAi.s — Notutiou and Nninemtioii ir»2 Addition and Subtraction of Decimals . . IM Midtij)lication of J)ccinials lAa Divioion of Decimals •.,,.. \'u Interconversion of Decimals and Fiuctions ir)!( Denoniinato Decimals , UK) Al'l'l-ir.VTIUNS OK DkCIMALS — Percentages 161 Applications of Percentage Kj.'J Pi'otit and Loss > 1 (!,•{ Conuuissiou .... Km Trade Discount 166 Interest HIS Hank Discount 1 70 AxswKRs 1 7.S Ari'KNuix 183 Eiitorotl, aeeoKiing to the Act of tht; Parliamunt of Canada, in the year of our Lord one thousand eight hundred and eighty-seven, by the Canada PuBLisinsa Company (Limited), in the Office of the Minister of Agriculture, at Ottawa. I'2S I Ml 110 144 ir>'2 ]■)-) 157 159 100 ii;i Dili lOM !().-) 160 i(;s 170 17M 183 ;hc year of our ADA Publishing at Ottawa. ARITHMETIC. CHAPTER I. OP NUMBERS AND NOTATION, Tho ten iiiarkH or clianicters , ,. ^\, ^' 2, 3, 4, 6, 6, 7, 8, 0, denoting n,n,,,hf, one, t,ro, fhnr,four,Jir,; ../,,•, srrn,^ cu,hi, nine respectively, are culled Arabic Numerals or Figures The tnsfc IS culled nought, cipher or zero, Tho reuiuining nine uro fulled digits. A number expressed in Arubic uumcruls is said to bo writtGli 111 Arabic Notation. The letters I, V, X, L, C, D, M, .lonotmg aH^> r., #n^//^,V, 0,1. 7nr>,r/m?,./?,r 7, «m/m/, ,>>,c mi oTf°" '" '"^'"'' ''°*'"''" °' "'""^"^ l"-*=«"'*«l Objectively; id Reiiinr ''T"''*"" °' """''-'" "^»"'^««'^'' '" Arabic Notatio"^; .!rd. Read.ng numbers written in Arabic Notation, and writing them i . wo«i3 • m. P x-press.on ,„ Arabic Notation of numbers written in wo!;is ; ' oth. eadn,g numbers written in Roman Notation, and writing them in words • Cth. Lxpressior ,n Roman Notation of numbers written in worfls • l.„.,ir 15,'"^' "'" "'""'"' '^'"^^*^'0" t" Arabic Notation, and vice verm 8 .NUMBERS AND NOTATION. In coxmiing objects and in measuring magnitudes the standard hj ivhich xve count or xoe ineasure is called a Unit. Thus in counting the pupils in a class the unit is one pupil; in counting the pages in a book the unit is one page; in counting eggs by the dozen the unit is one dozen eggs; in selling bricks by the thousand the unit is a thousand, bricks; in measuring cloth by the yard the unit of length is one yard; in weighing sugar by the pound the unit of weight is one p:und; in measuring apples by the bushel the unit of volume is one bushel. Every number is either abstract or concrete. An Abstract number is one that docs not specify what the objects are that are counted, or of xchat kind the magnitude is that is mcasiwed. Thus 4, 7, 12, 25 pairs, 9 d'jzen, 37 thousand, are abstract numbers. An abstract number, therefore, signifies only the number of times some imit is repeated. A Concrete Number is one that specifies not only the numeri- cal value of the quantity, but also what the objects are (liat are counted, or of what kind the magnitude is that is measured,. Thus 4 boys, 7 books, 12 pencils, 25 pairs of skates, 9 dozen oranges, 37 thou- sand bricks, are concrete numbers. A concrete number is not, strictly speaking, a mere number, but is rather a concrete quan- tity ; and its complete representation must conse(iuently consist of two parts — the one representing the numerical value (the number proper), the other naming the things counted or the standard of measurement used. Like numbers are numbers that hav the same unit. Unlike numbers are numbers that have different units. [Pupils should be practised in the use of the above-defined terms till they understand them clearly and are thoroughlj' familiar with them. The test of sufficient kno\vledj.'e is not the ability to repeat, however glibly, the definition of any term, but the unhesitating employment of each tenn wherever it ought to bo used and nowhere else. Kxercises should be given — 1st. In namiiiu' the units in proposed numbers ; 2nd. In distinguishing the abstract from the concrete numbers in mixed groups of these ; • d. In distinguishing groups of like numbers from groups of unlike numbers ; 4th. In assorting into separate sets the several classes of like numbers contained in a miscellaneous group.] ! standard hj in counting ig the pages lio dozen tlie iintl the unit I the unit of . the unit of shol the unii CHAPTER II. THE POUR FUNDAMENTAL OPERATIONS. )ify what the litude is that housand, are LB, signifies ted. y the nuvuri- ',t are counted, rhus 4 boys, ges, 37 thou- uiber is not, ncrete quan- iently consist 1 value (the anted or the unit. nt units. terms till they 111. The test of the definition of :r it ought to be in mixed groups inlike numbers ; iiiibers containeil I. ADDITION. A nnmher ichich as a vhole i. v,ade np of two or more numbers a. part, ^s called the Sum of these numbers. u^^^:' ''' '''''''-'' 'y ^'''^ -^-^ ^^^^ ^- of two or The numbers to be added together are called Addends. The sign of addition is + ro-id «/7/« Ti,;„ • t , , ^ '» 1-, it-ad ^jf?(s. 1 his sign + written Hp- orc^.„u,nber denotes that the number is an addend Thu. .his'thtC'' .^f 1 /' T^ "--ty-three plus twenty-two 0>>hj like numbers can be added together. Unlike numbers can- fro!n lloh ^^uj;:":^ '°"°^^'"^ '"''''^ ^^ ^'-- °f -ereises have been o.itt«, -e;L!o;^ea,rtl:r'"^°"^'^*''""'"''^'^ '"*'<^-*-y to the several rules or the text-book and the es^IZ^^ ^^''^' ''!" ''^^-^ -'^ reference i,, made to valuo, must therefore be supplLd bv h.. , k- ^''''^ P''°''''""«' *° ''^ «' '^"X in variety and „„n,h.r .^^^1-1,^ . "" ''"'""' """^ ^''"""^ ^e adapted 9 10 AHITHMKTK". ilMMTION rdtl.K. <> } I I 1 2 l-^ •2 4 1 'A 4 ;< 4 I 'J i: 4 / 2 (} S ;{ fi 4 () 10 10 4 / 11 <) • ) s 10 ;{ s 11 4 8 12 I 9 !l II .*{ 12 i 4 !) 13 I 1 2 ,> 4 5 () 1 s .) f) « 1 8 9 . 11 12 i;{ u ; (i (i (> G () 7 8 9 12 U 14 15 / 1 / 1 8 9 13 14 If) 16 S S 8 8 () 7 8 9 14 9 15 15 9 / 10 1 10 10 10 10 10 10 10 10 10. V 1 2 4 5 7 10 11 12 13 14 15 16 17 16 17 ' <) 9 8 9 17 18 10 10 8 9 18 19 ADDITION. 11 1 8 1 9 •) 10 .) •) >S !» U) " ;{ •.i s <) II 12 4 4 8 <) 12 l.S 8 9 i;{ 14 (i 8 9 14 1.-) S t 9 15 10 8 8 8 9 I(> -"I7"i <) 9 ! 8 ^ 1 17 18 10 10 8 9 18 19 EXE3CISE I. ^- ^» t';« reopening of school after the mi.ls.mnner lK,Ii= h i y luwniany , 3 Tn tin. Third Keader there were 7248 words ; i„ the arithmetic. -1/.); in the grammar, 2;{!)(i ; and in the g<,.ography imj How many words were there altogether in the four hooL' ' J.^"^ '^'l?^ ^'"''"" *'"'" ""■« '''-'''^ •^•"'--; in the arith- mctK, 98.0; in tlie granm.ar, i;{,789; and in the geography :U 874 How many letters were there altogether in the four LL' o. In the school there were 19 pupiLs in the first class,"]5 in the second, 17 m the third, 9 in the fourth, and in the fi t iW many pupds were there in the school ? 0. JolnUiad 8 marbles, his uncle gave him 19, and on his way to school he hougiit 20. How many had he then » ^ 7 The other boys in John's class were William, James. Edward Thomas Henry and George. How many letters were th rel t le ;;:;;;:six 'r "' '"' "^" ^-'^ '-- -^ ^^"-« ^" ^^^^ S. During the first week of school John received 27 merit «,arks .U.rmg the second week 29, during the third week :U, and 2 i."; :;;: ;rt:s ''- ^^^ -^ -'^ -^^ ^^^ ^- --^ ^^ .^. He receive.1 109 merit marks in September, l.-i7 in October 28 in Noven.ber, and 97 in December. How maiy d d L rec h ' (luring the whole four months ? ^ "" 10. Having been first on the Iionor-roII for Octol,er ].;« f .v. 11. John attended school lOdavsin Tin..,,., mi ■ .. , .ju,.«. „»„ „.a„, „.,, ,ia he ,'..„a »,i, ;;:!;! ',:';',::;!;,^: ■■; 12 AJUTIIMKTIC IJ. The Hohool-rooni is 17 8tei)a long and 14 steps wide. How many stops would go completely round it? 1:1. 'J'hc school-yard fence has '.VM pickets on tlie front, 'i.')! on each end, and TiTS on the back, ilow many pickets arc there on tiie wliole fence ? 1.4. In a game of baseball one side made 17 runs tlie first innings, 14 runs tlie secon.l innings, H) runs tlie third innings, 8 runs tiie fourth innings, and "id runs the lifth innings. Tlio other side made ill their corresponding innings 4 runs, 1!) runs, 'IW runs, '2!l runs and 8 runs respectively. How many runs did each side make? l'>. In tlie first innings in a game of cricket the first boy out made 2.S runs; tlie second boy made lit runs; the third, 7 runs; the fourth, .'W runs; the fifth, 19 runs; the sixth, 3 runs; the seventh, no runs; the eiglith, 9 runs; the ninth, 7 rura; the tentli, 2 runs; and tlic last boy carried out his bat for r> luns. There were 4 wides and 9 byes. What was the total of the innings ? X6. In a game of cricket one bowler bowled 'M)'> balls, a second bowled '259 balls, a third bowled 179 balls, a fourth bowled 19S balls, a lifth bowled 97 balls, and a sixth bowled' 09 balls. How many balls were bowled in all? 17. Annie has 3 cents, a five-cent piece, a ten-cent piece, a twenty- five-cent piece, and a fifty-cent piece. How niueh has she in all ? IS. Annie's hen has 9 chickens, Jane's lias 8 chickens, Fannie 's two have 19 chickens together, and liertha's three have 27 chickens. How many hens and how many chickens are there altogether? 19. John costs his fiither .*;()7 a year for food, §2.") for lodging, >}'M\ for clothes, ^7 for pocket money, and 1?47 for other expenses. How much a year does he cost his father in all ? ■JO. Thomas cost his parents §r)7 the first year of his life, $49 the second year, SliS the third year, S7S the fourth year, .$(19 the fifth year, $74 the sixth year, and $85 the seventh year. How much did he cost them the wliole seven years ? m. Annie paid 18 cents for milk, 35 cents for cream, 48 cents for eggs, 37 cents for butter, and 26 cents for cheese. How much did fche pay in all ? S2. How many letters in the names of the days of the week ? JS. How many letters in the names of the months ? on Tuesd ^4- Martha gathered 37 eggs ;day, ■«y, Wednesday, 24 on Thursday, and 15 on Friday. How many did she gather during the five days ? ADDITION. 13 I wide. How front, 'i.lt on iiro thorc on J first innings, j;.s, 8 runs tlio hor sidi! niiidn s, 2!l riinH and akc ? boy out niiidc is; the fourtli, enth, no runs; runs ; and tlu; 4 widcs and I) l)alls, a sct'ond ;h bowled liKS [) balls. How ieco, a tvventy- s she in all ? kens, Fannie's ,'0 27 chickens, ogether ? )r lodging, !?.S!( :penses. How is life, $49 the , $m the fifth How nmcli did in, 48 cents for iow much did he week ? ^uesday, 18 on low many did ^o. Durmg tlie week of the fair Harry spent .'H cents for peaches 18 cents for pears. 17 cents for apples, 24 cents for orungo«, an.l cents for candy, and had 13 centr left. Ho,v r.mch did he spen,l and Jiow nnich had he at first? ^ -V;. A n,an paid $1!) for a suit of clothes, §1.-, for an overcoat, .$.3 .or a hat, .^4 for a pair of boots, .$24 for un.lerciothes, an.l ^7 for other articles. How much did he pay for the whole ? .J7 A man .spent .^174 a year on clothing for his family, S.369 for food, $U\ti for house-rent. $Gi) for fuel, ^'27 for light, .^77 fo"r furni- ture, .S84 for wages, and $G7 for incidentals; he also paid .^18 to a doctor and $24 for ta.xes. flow much a year did he spend on all these things together ? US. A merchant sold .$278 worth of goods on Monday, .«!;«),-, worth on Tues.lay, mi worth on Wednesday, $594 worth on Thursday *19;, worth on Friday, and .$947 worth on Saturday. What was the total value of his week's sales ? U'J. Four men built and equipped a mill. The first paid oa it */418, the second ,?9475, the third $8043, and the fourth $7464 How much did the null cosit them ? 30 A (hover bought 78 sheep on Monday for $:m, 49 sheep on Tuesday for $313, .36 sheep on Wednesday for $194. 57 sheep on Ihursday for $328, 65 sheep ou Friday for $347, and 193 sheep on ha unlay for $978. How n.any sheep did he buy during the week and how much did they cost him ? 31 In the first car of a railway train there were 27 passengers m the second car 36, in the third car 29. and in the drawing-room car 18. How many persons were on the train, counting in the con- uctor, the drawing-room car conductor, the driver, the fireman, the mail clerk, the express clerk, two brakesmen and the newsboy ' 3.. in a cattle train there were two cars with 17 head of cattle in each, three cars with 19 head in each, one car with 22 head, and two cars with 21 head in each. How ma. y head were there in Lll v 33. A farmer had 27 acres of land under wheat, 15 acres under oats, 14 acres in meadow, 19 acres in pasture, 9 acres under peas, 6 acres m potatoes, 7 acres in turnips, 9 acres under In.lian corn 5 acres in orchard. 3 acres for house, garden, stables, barns and barn- yards, and 29 acres of woods. How many acres in his farm ' oJ' iu^'T!'^'^ ''* '°'"'' ^ ^•'■*'^'*^«' " y«"»g ^'-^"le. 5 horses, a colt, a filly 3/ sheep, 14 lambs, and 19 swine. What was the total number of his live-stock ? 14 AUlTHME'nC. 35. A farmer bought three farms with tlic standing crops and tho live-stock on tlicm. For tlie first farm lie paid ^A1S)•^ for the land, $47!) for tho crop, and 8008 for tlui live-stock ; for the second he paid $r)98r> for tho land, !?!)7a for the crop, and .S")4(; for the live-stock; for the third he pai.l $8078 for the land, 810!»4 for tho crop, and §783 for the live-stock. What was the total amount he paid for the land, for the crops, and for the live-stock, respectively ? What did the whole cost him? SC. At an inspection of the Queen's Own Rifles in Toronto there were present 2 otlicers and 77 non-commissioned oHlcers and men in No. 1 Company; 2 O. and o9 N-C. O. and M. in No. 2 Company; 3 0. and GO N-C. (). and M. in No. 3 Company; 3 O. and 72 N-C. O. and M. in No. 4 Company; 3 O. and 64 N-C. O. and M. in No. 5 Company; 3 O. and 67 NC. O. and M. in No. 6 Company; 3 0. and 60 N-C. O. and M. in No. 7 Company; 3 0. and "i N-C. 0. and M. in No. 8 Company; 3 0. and 49 N-C. 0. and M. in No. 9 Company; 3 0. and 60 N-C!. O. and M. in No. 10 Company; 7 officers of tho Staff, and 37 musicians in the Band. What was the total strength of the battalion present at inspection ? 37. A man walked 29 miles on Monday, 37 on Tuesday, 28 on Wednesday, and 19 on Thursday. How many miles did he walk altogether ? 3S. A man travelled 79 miles by stage, 47 miles by water, 198 miles by rail, and then 67 miles on horseback. How far did ho travel ? 39. James hoed 29 rows of potatoes, William hoed 27 rows, Edward hoed 2.1 rows, and Henry hoed 47 rows. How many rows altogether did they hoe ? 40. Annie picked 7 quarts of berries, Jennie picked 9 (luarts. Bertha picked 5 (juarts, Mary picked 8 ipuuts, Harriet picked 13 (juarts, and Bella picked 14 quarts. How many (quarts did they pick altogether ? 41. A miller bought 897 bushels of wheat and sold 136 barrels of flour in September ; in October he bought 6r)5 bu.^hcls and sold 97 barrels ; in November he bought 768 bushels and sold 88 barrels ; in December he bought 596 bushels and sold 194 barrels. How many bushels of wheat did he buy in the four months, anil how many bar- rels of flour did he sell ? 42. James gave 8 apples to Henry, 7 to John, 9 to Thomas, 6 to Daniel, and had 17 left. How many had he at first? k V I -II ; crops and tho I.S for the land, ! second he paid tho live-8tock; ) crop, and §783 lid for the hin7 pupils present at a school examination 2[) Mere hoys. How many were girls ? 4. Out of a class of 43 hoys 26 were promoted. How many Mere left? r>. In a game of ball one side made 37 run.s and the otlier made 'JO. By how many runs did the first side win ? (J. In a game of ciicket both sides together made 235 runs; of these the winning side made 137. How many did the other lide make ? 7. Seventeen swallows were sitting on a telegraph wire. A num- ber flew away and there were *) left. How many flew away ? ,S'. James had 73 marbles. He lost 19 of them. How many had he then ? n. Maggie and Emma tried which could find a sunflower with the greatest number of seeds in it. Maggie found one with 279 seeds, hut Emma found one with 293 seeds. How many more seeds were tiiere in Emma's than in Maggie's? 10. There are 23,14,5 verses in the Old Testament and 7957 in the New Testament. How many verses in the whole Bible, and how many more in the Old Testament than in the New? n. The last chapter in Isaiah is numbered LXVI. ; the last Psalm CL. How many more Psalms are there than chapters in Isaiah ? IJ. John has read all the Psalms to the end of the XCVII. How many has he still to read to finish the whole hundred and fifty of them? ^ 13. Jane has read 157 pages of her book, which consists of 324 pages altogether. How many has she still to read to finish the l)ook? n. Harry bought a Third Reader for 36 cents and an arithmetic for 25 cents. He gave a dollar bill to pay for them. How much change should he get back ? In. Annie bought a Third Reader for 36 cents, a geography for 65 cents, and a slate for 27 cents. She gave a two-dollar bill to pay for them. How much change should she get back ? 2 18 ARITHMETIC. Itl, Out of a bag containing 225 marbles John took two handfuls, an school days in the six months, how many days was Willie absent from school ? ^'4. On Monday I started on a journey of 4000 miles, and made 457 miles that day, 468 miles on Tuesday, 528 miles on Wednesday, 509 miles on Thursday, 514 miles on Friday, and 579 on Satunlay. How many miles of my journey remained for me at the close of each day ? How many miles had I travelled at the close of each day ? i^5. A farmer had 127 acres of land, and he bought 87 acres. He afterwards sold 68 acres. How many had he left ? 26. A drover bought 123 sheep for $710 and sold 69 of them for $475. How many had he left, and for how much must he sell them in order to get back his $710 exactly ? 27. A cattle dealer bought 235 head of cattle 'or $5784. He sold 148 of them for $4375. How many had he left, and for how much must he sell them in order to gain $1250 on the whole transaction? SUPTUAf!TION. 10 ;wo haiulfuls, t? i foi- tin- loan. ,'i ut uiiuthcr. , ,S8 cents for !7 cents for .S ■ gave a tive- get hack? is fatlier gave spent S cents unt for candy. ) merit marks, 4, anil (luring ill during the lolm, who got > 8 more than Willie got 18 :. How many 1 February, 16 there were 12.") Willie absent I and made 457 I Wednesday, ) on Saturday, e close of each : each day ? 87 acres. He 39 of them for st he sell them j784. He sold for how much e transaction? 28. A butcher bought 47 oxen for !5!I630 and 107 sheep for 872r>. He solrl 29 of the oxen for .S124.-), and tlie remainder for .^SoO. Ho sol.l 48 of the sheep for mW, th.-n 17 more c.f th-m for Sir.O, and tlie remainder for .S12.-.. How much did he gain ultog..thei ? ..".>. Je.uiie had 23 chickens more than Kditli, but only 9 more than Mary. How nuiny had Mary more than Edith ? .!(!. Anni(! has one lien with 7 cliickcns, a second with 1 1 chickens and a third with chickens. Jane has two hens with 9 chickens each and a thir.l with only 1 chicken How many more chickens have Annie's liens than Jane's? ■ it. Bella had 173 nuts. She gave 19 to her sister, 17 to eaeli of iur two brothers, 23 to her father, 2.', to her mother, and 22 to her consul Ella. How many had she left for herself ? ■iJ. Annie ha.l f.l nuts. She gave 14 to Ikssie, 12 to Fannie, ate 1 1 herself, and gave the rest to her little brother Harry. How many did she give to Harry ? ■13. In tiie first car of a railway train there were, on starting 29 passengers; in the second, 27; an.l in tiie thir.l, 1.',. At tlie first stoppmg place 19 passengers got out and 7 others got in. How many passengers were then on the train ? -V^. In the first car of an excursion train from London through Hamilton to Toronto there were 27 passengers; in *lie second car, .J ; in he third, 31 ; in the fourth, 2.", ; in the fifth, 32 ; in the sixth J4 ; and m the scn er h, 26. At Hamilton 8 passengers got out of each of tiie seven cars, and 4 got into the fourth car and 3 into the seventh. How many passengers were there tlien on the several cars and how many on the whole seven ? ' .y-). A man had to put 73 head of cattle into four cars. He put 18 into the hrst, and 19 each into the second an.l thir.l cars. How many were left to go into the foui-tli ? 3n A man bought a horse for §97 and another for §85. He sold the two together for §163. How much did he lose on them ' 3. One farmer ha,l 157 bushels of wheat worth 172 .lollars, S,-,6 bushels of oat.s worth 102 dollars, and 163 bushels .,f barley worth \u r; 1 rf '' ^"""'^^ '^^"^ ^' ''"^'^^'^ «f -''-^t -orth 107 dollars, 311 bushels of oats worth 118 dollars, an.l 244 bushels of bar ey worth 146 dollars. Which farmer had the greater number of bushe s of gram, and how many bushels had he more than the other ■> h.ch farmer s grain was worth most, and how much was his worth more than the other's ? 20 .\KITHMK'n(\ .JS. 8«.h^< gofxls for $1225, gaining tlu uhy §248. How nnich .lid tlio goods cost ? ,iU. A man luul pioprity woitli $123,273. Of this imiount !:^15,274 was in mil estate. §27,310 \\ as in bank st, brings him in §225 a year. If his expenses arc §975 a v -rf baud for §825, and a mortgage for the balance. For ]"■«• IV. li M.is the mo'«-'..age? 51. Joaes-. v.d Smith §;;..i; in payment he gave a horse and §49 in cash. How mucli was the horse I'tickoned at? •i 1 .'^ srnTiurTiox. 21 ow much (li«l [iiomit .'"(15,274 w an ill railway i. H(»w much ho lost ^-tTM'i. t$;{l4.'l. How ;t week ho lost 10 gained $925. ith ? than William, imcs together 1 than Thomas, n Albert V •Ht I gave $725 liirtl, for which jt he paid §129, i third $27 less onrth ? mproved land. ), 11 acres weru len, house, and les were unini- i property tha^-, 75 a >t;ii.i' how u paid $1225 in v nmch was the He paid $19.50 for the balance. I horse and .$49 m I .5,'. Willie had 33 marbles j ho won 9 from .I«,hn ami lost 7 to Henry. How many had lie then ? .-./. .Jane had ;{5 ])iiimM, Annie hud 28, and Susan had 22. Jane gave Susan ono'igl, to n.aito her numher up to Annie's. How many had JuiK' left for hers. If ? .;.;. liiury hxl 25 i>igeons and Bessie luul 14 hens. Harry'gave BoHsie 9 of his pigeons in exchange for 5 of her hens. How many binls of eadi kind Jiad each of thom after the exchange ? o.i. Charlie had 29 nuirbles and Willie had 22. Charlie won (> from Willie. How many Iiad Cliarlie at first more than Willie, and how many more than Willie had lie after winning the six ? oa. Harry had 17 marldea and Joiui had l.'i. Hairy l)ought 14 and then j.layed witli .lohn, who won 9 from liim. Wiiieh of them hiwl m(c abovo examples, 1 pij^eon and 1 cherry respectively — while the latter quantities cannot be so expressed. ] In counting any collection of objects, the unit or standard by which wo count may be a single object of the kind counted, or may be a given number of such objects. Thus we may count a group of objects by one at a time, or by two at a time, or by three at a time, or by four at a time, or by any other number at a time, and say there are so many ones, or twos, or threes, or fours, or whatever the number may be that Ave used in counting. For example, eggs are generally counted by twelves, and the number of twelves, or dozens, as they are called, is stated, not the number of single eggs. Stockings are counted by twos^ and the number of pairs (twos) stated, not the number of single stock- ings. But if the number of twos, or threes, or fours, or what- ever the number in each count may be — that is, the number of imits-hG stated, it may be required to find how many single objects there are. The number of these is found by the opera- tion called multiplication. Multiplication is the operation hy which tve find a number which is equal to a given number ichose unit is itself a number. The number which is the tout of the other is called the Multipli- cand, and is said to be multiplied by that other. The number which has the multiplicand for its unit is called the Multiplier. The number resulting from the multiplication is called the Pro- duct. The multiplicand and multiplier, taken together, are called the Factors of the product. The sign of multiplication is x, read "multiplied hj." This sign X written before any number denotes that the number is a multiplier. Thus 3 x 2 is read "three multiplied by two," or MTTLTIPLIOATION. 23 rcises to apprehend ;uch as 2 i)uirs ot L'flnite jjjroup-uiiits, vo that the former roup— in the above • tiuatitities cannot or standard by lid ctninted, or ve may count a b a time, or by tlicr number at s, or threes, or ed in counting, elves, and the [, is stated, not id by twos^ and of single stock- fours, or what- the number of iw many single d by the opera- find a number a numher. I the Multipli- nit is called the called the Pro- r, a?-e called the lied by." This ;he number is a ed by two," or 'twice three," and denotes two threes, that is, the sum of two threes; 5x4 is read "five multiplied by four," or "four times five," and denotes four fives, that is, the sum of four fives- 7 pencils x 5 is read "seven pencils multiplied by five," or "five times seven pencils," and denotes 5 bundles of 7 pencils each- 12 bricks X 4 is read " twelve bricks multiplied by four, " or " four times twelve bricks," and denotes 4 lioaps of twelve bricks each. Since a unit may be either abstract or concrete, the multi- plicand, which is the unit of the multiplier, may be either abstract or concrete. Since the multiplier has the multiplicand for unit, the multi- plier taken without the multiplicand must be abstract Theiorodwt and the multiplicand must be like numbers. MIILTIPMCATIO!^ TABLE. '1 ^ 3 4 5 6 1 7 8 9 18 27 36 45 10 20 11 22 12 24 2 3 4 6 6 8 10 15 12 14 16 24 32 40 9 12 16 20 24 28 18 21 28 35 30 40 33 1 36 4 8 12 20 24 44 48 6 10 15 18 25 30 30 50 60 55 60 6 7 12 36 42 49 56 48 54 66 72 14 21 35 42 48 56 64 63 72 70 77 84 96 108 8 16 24 32 40 80 88 9 18 27 36 1 45 54 63 72 81 90 99 10 11 20 22 24 30 33 36 40 44 48 50 55 60 60 66 72 70 77 84 80 88 96 90 99 100 110 120 132 144 110 121 12 108 il20 ] 132 [Pupils sliould so learn the table as to be able to repeat it both by columns and by rows ; e.g., twice 1 is 2, twice 2 is 4, twice 3 is 6 twi,. . ^ )cT I ^°,^ '^"^ m in 2, twice is 4. thrioe 2 L r, in^r t^./e! o •! '.1 '-, '''1* '' '' ^"^^ '^"^ °"^« ^ m perceive the truth of the in.portant pHlicip,; !_ ' ''" ^'"''' ""''' "'"^ '^" ''^^ "^ 24 ARITHMETIC. EXERCISE III. 1 Willie was at school 6 hours each day for 22 days in March. How many hours was he aL school tkvt month? _ 2. There are 60 minutes in an hour. How many mmutes a,e there in six hours ? ,•?. How many minutes are there in a day . it. How many minutes are there in a week ? There are 24 hours in a day. How many hours are there in July' Ho A' many steps will hi How many steps will lu G. How many hours are there in a year-36o days . 7 Harry attended school on 17 days in January, and had to ualk 3 miles each day to do so. How many miles did he walk to attend school that January ? , , i -i / Annie walked a mile to school every school-day and a mile back again. How many miles did she thus walk m a week of ' ttS:iy train ran for 4 hours at tl. rate of 27 miles an hour. What distance did it run ? ID. George takes 2350 steps to the mile, take in walking 3 miles ? IL Fred takes 24()0 steps to the nnle. take'in walking 3 miles a day for 5 days ? n A cat has 18 toes. How many toes wdl 18 cats have 13 There are eight boys in Willie's class, includmg Willie hm>- self. Each boy has twenty-eight teeth. How many teeth hav. thev altoKether ? , .„ ,. U. A spider has 8 legs and a tty has 0. How many legs will .. spiders and 8 flies have? ^,.7 How many feet altogether have 3 horses, 4 cows, and 5 sheep? V; A mail-carrier drove daily from A to B, 4 miles ; from Btoi, 3 miles ; from C to /), 5 miles ; and from D back to . , 5 miles. How many miles did he drive every week, omitting Sundays . 17 James walked 8 miles a day on 25 days in January, 23 in Ftb- ruary, and 20 in March. How many miles in all did he walk during ^''ilAn^lcre of land contains 4840 scjuare yards. How mauyj square yards are there in 37 acres? 10. Find the cost of 27 tons of iron at $39 tlie ton. MULTIPLICATrON. 25 hours are there in many steps will he ow many legs will •> yards. How maiiY R i'O. At 27 bushels of wheat to the acre, how many bushels would there be on .SH acres ? .7. At 23 bushels to the acre, how many ))ushels would there be to the s.jHare mile of 640 acres, deducting 4;i acres for roads, fences and waste land ? ' J:.'. How many bushels of wl.eat could bo raised in a township contammg 78 square miles of (i40 acres each, allowing 47 ac^res in each square mile for roads, fences, and waste land, the wheat aver- aging 27 busliels to the acre ? .^3. A drover bought 37 head of cattle at $48 each. How much did he pay for them ? :^. A woman bought 6 pounds of tea at 07 cents the pound 18 o. is of sugar at 12 cents the pound, 2 pounds of coffee at ;« cents he pound 8 pounds of cheese at 14 cents the pound, 1.3 pounds of butter at 2.^ cents the pound, and 9 dozen eggs at H. cents the dozen. t md the price of the whole. ^^Jo.^ How much money would l^e required to pay .§500 each to 798 Ji;. Find the strength of an army consisting of 97 regiments of 8/3 men each. •- ":7 /"^"'y l^ricks will there be in 97 feet of wall if each foot require / 8 bricks ? ^S. Wliat will be the total issue of a newspaper in 13 weeks of « • lajs each, if the daily issue be 23,78o copies' .,,f ^^ '""tr"' ^*°"^^* ^^^ P°""'^' °^ '^''^' ^' 7 cents the pound, 287 pounds o butter at If) cents tlie pound, an.l 178 dozen eggs a 13 cents the dozen. How mucli did the whole cost him v .in A merchant bought 7 cliests of tea, eacli weighing 08 pounds .t 13 cent, he poun-.l; and 44 canisters of spices, each weighing 24 'ounds at 1 7 cents tlie pound. Find the cost of the whole JL A man has a chest of tea which at Hrst containe.l 87 pounds but 29 pounds have been taken out of it. How much is the re I'uuning tea worth at 03 cents the pound ? \J:' H "';? '^""«'^*.'^" ^^^""■^' ''"'^ <^o"tai„ing 107 acres at $73 the [acre the other containing 79 acres at $87 the acre. How m,u.h did ^he two cost him ? t get back out of a twenty-five-cent piece he gave in payment? 26 ARITHMETIC. .34. A man bought 9 cords of wood at $4 the cord, and gave 4 ten- dollar bills in payment. How mueli change should iie receive ? -?5. What is tlie weight of a train consisting of 17 cars, each weighing 22,.37r) pounds, and an engine and tender weighincr together 147,800 pounds? SG. James knows of a butternut tree with 34 bunches on it with f) nuts to tlie bunch, 47 bunches with 4 nuts to the bunch, 18 bunches with 3 nuts to tlie bunch, and 7 bunches with 2 nuts to the bunch. How many bunches were there on the tree, and how many butter, nuts were there ? 37. If out of a salary of .$1300 a year a man pay §180 for board, SI 97 for clothing, $1(17 for books, and §238 for other expenses, how much can he save in seven years ? ,3S. Charles had saved the sum of 29 cents. His father tiien gave him five times as much. How much had he then ? 39. Fannie had saved 19 cents. She was first on the Honor-Roll for May, so her father gave Iier five times and her mother twice as much as she had saved. How much had she then ? 40. Bertha picked 47 plums and her brother Thomas picked 9 more than five times as many. How many did Thomas, and how many did botli pick ? 4L Annie bought a book for 17 cents, and a box of imints for four times as nuich. How much did both cost her ? 4J. Emma bought a doll for 25 cents and a doll's carnage for five times as much. How nuich did both cost her ? 43. The furniture in a house was worth $1837, the house itself was worth twice as mucli, and a library in the house was worth twice as nuich as the house. How much was tiie whole worth ? 44- Jane's hen has 13 chickens; Annie's 5 hens have four times as many all but 4. How many chickens have Annie's hens ? 4o. If a certain farm-liouse be worth .S720, and the farm and barns be worth .'?400 less than five times as much, and the stock and stand- ing crops be worth ,$125 more than thrice as much as the house, how much will the whole be worth ? 4G. A mail had four rolls of five-dollar bills. In the first roll were 17 bills; in the second, 25 bills; in the tliird, 24 bills; and in the fourth, 33 bills. What was the total value of the four rolls ? 47. A man liad three rolls of five-cent pieces. In the first were 60 pieces; in the second, 75 pieces; and in the third, 118 pieces. What was the value of the whole ? MULTIPLICATION. 27 ther tlien gave iainta for four -rriage for five 4<^. In a certain book there are 239 pages of 37 lines eiel, -.vp,. ^'V. In a certain sehool-i.ouse there are 29 win.loM-s; in eacii vd„ dow there are 4 rows of panes .vith 3 panes in each row W n " panes in each window, an.l how many in tlie whole ' ^ evlt Jr ' h' °' ^""^Zr" *'"' ^^'"" '' '"^^-^ ^^'^h '« '•"!« i" eveiy ow. How many hills were there? If these avera-^e.l 7 eirs to a hill, how many ears di,l the field yield ? .7.^ In a certain house of 4 stories there are in each story ] ", win- chnvs ni the front 8 windows at each end, and 14 win.U s „ the tear In each window there are 12 panes of glass. H<,w many anes cneir \ alue at 17 cents each ? Jl A clrover bought 73 sheep at .^6 each and sold the whole for ^/o. How mucli di.l he gain tliereby ? i'>4- A man boujiht 27 horses nt «iq7 ^., i i i i , o"" -' ""ihts at .-^i.i/ eacii and sold them at «I''fi each. H,nv „„..,, ,c «»!„ „„ eao„ ,„„. „ „„„„ „„ ,", ^f, .5... A.ln.vcr bought 89 l.ca.l of enttio at S3!) tlio hui.l ,„.l „ll them at m the head. What was hi, gai,. .„; th„ !,;',:. ■"'" '""' at'tl rr"' !'°"'^'" "' '"""' -' """" '" «»•! «"=1' "'■■i »0M tho,„ or tnc cattle. What was his net gain ' soM 'u '""fT ^""°''* '^ ^'""'^ "^ '^^•^^^^ ^* 6'^ -"ts the yard and sold 14 yards to one man, 9 yards to a second man, 15 yLls oa ail at 9.) cents the yard; the remainder he sold at 87 cents the v.r.l How much did he gain on the whole ' ^ ' ' Weeh, he .„,, th.,„ „. ,..„, „,, ,.. ,,„ „, J^^^ til r!,:! 28 AHITHMETlC. IV. DIVISION. imt they ilntro(inptor.v exercises Hhonld lie triveii in lioth kiiula of Pivisioii, siiould not bo mingled indiscriminately. In ruiwliiij; tlio results the divisor and the (luotient sliould always be read as co-factors. Thus the result of the division R)15 cents should he read 3 times 5 cents is 15 cents; the result of the division 3"l)iiiins )ir. plums should he read n times 3 plums is 15 plums ; and the result of 3)15, in \vhiciri)oth divisor and dividend are abstract, should be read both as 3 fives and as 5 threes. ] Division i.s' the operation hij which ire. find the number vhieh, taken as co-factor with one of t>ro given numbers, would yield the other [liven numhcr as j^roduct. TJie number found hy the division is called the Quotient. That one of the given numbers which is co-factor of the quotient is called the Divisor. That one of the given numbers which is equal to the product of the divisor and the quotient is called the Dividend. Since in multiplication wo generally liavo so many tiitno.s repeated so many times, there will be two kinds of division, according as the number of things or the number of times is given as divisor. In the first kind of division we find the number which, taken a given number of times, would make up a given numlier. lu such case the divisor tells how many times the quotient nmst bo taken to make up the dividend. The divisor must therefore be abstract, and the dividend and the quotient must be like num- bers. In the second kind of division we find how many numbers, each equal to a given number, would, either by themselves or else along with a number (called the remainder) to be found, and less than the given number, make another given number. In this case the quotient tells how many times the divisor must be taken in order that the product increased by the remainder, if there be any, may be equal t,040 inches) in 3520 steps. How many inches did he take each step ? IS. If 7 dozen eggs cost 168 cents, what was the price per dozen? How much is tliat ])er egg? 19. Q'iie wages of 13 men for one week were $97.50, How much did a man earn per day ? 20. The expense of building a bridge was $^8743, and of opening a road was §2103, The total expense was borne equally Ijy seven townships. What was the share of each ? ■21. A man receives a salary of $1200 a year. Out of this he saves $212 eacli year. How much does he spend per week, counting 52 weeks to tlie year ? 2..:. A contractor requires a million bricks. He has 559941 already. How many loads of 437 bricks each does he need to make up the full number ? 23. How many bags of flour, each containing 25 pounds, can be made out of 75 barrels of flour, each containing 196 jjounds? 24. A bushel of wlieat weighs 60 jwunds and a bushel of oats .34 pounds. How many bushels of oats will ^v■eigh as much as 187 bushels of wheat ? 25. How numy 11 -foot panels in a mile (5280 feet) of fencing? 26. How many boards, each 12 feet long, will be re(]uired to l)uild 1320 feet of fencing .5 Ijoards high ? 27. Seven men have an equal interest in a farm of 107 acres. Tliej- sell it at $56 the acre. How much should each receive ? 2S. In a school-room there were 72 seats arranged in 6 rows. How many seats were there in each row ? 29. In a school-room there were 6 rows of seats, and 57 boys tilled all the seats but 3. How many seats were there in each row ? 30. Fred had 75 cents. He bought 2 dozen oranges and had 3 cents left. How "uch were the oranges apiece ? DIVISION. 31 iliat will 1)0 tlie I eggs were there :en. How much eps of 20 inches 9:),040 Indies) in J? irice per dozen? ,50. How much md of opening a ijually l)y seven t of tliis he saves lek, counting 02 s r)59941 ah-eady. make up the full pounds, can be pounds ? ushel of oats 34 as much as 187 ) of fencing ? ■eciuired to l)uild 107 acres. They ive ? iged in 6 rows. ;s, and 57 boys i there in each ,nges and had 3 I .?/. Willie lias 60 nuts. He gives 12 to each of his brothers and keeps tlie smallest share for himself. How many brotlil hi he How many did lio keep for himself ? " oV. A la.ly sent a bag of apples to be divide -- -" ^^o the ^f^. James has 72 marbles, John has half as many, and Willie one-third as many as John. How many has Willie ? "^ ^" •is Thomrn '" ^^ ""*'' ''""^^ ^"" ^'"^ '""- t^'-' '-^If - many 3 mts'lrhl:.'; """'' ^ '"^" *^'^ *^ ^^^^'^ ^« -^^^ ^^ ^^^ -te of ,ni,e^, ' *= °"' iiuttdlo to (.oderich, a distance of 160 //'k How many days would rate of ,3 miles an hour for 8 hours a d man take to walk 156 miles at the ay CHAPTEK III. COMPOUND NOTATION SYSTEMS. I ■:■ ' ' 1. TABLES OF MEASURES. MONEY, OR MEA^^I REH OF VALI E. 100 cents (ct.) = 1 dollar ($) MEASIRES »V MEKJIIT. Avoirdupois Weight is used for all the ordinary purposes of .weighing. 10 ounces (oz.) = 1 pound . . . .^. . , (lb.) 2000 pounds = 1 ton (T.) 100 pounds is called a cental or hundredweight, denoted by cwt. In weighing very small () lb. Turnips, ^ Parsni])8, ul Onions. ('»() lb. IS C(xil . 70 lb. (uiin.) (hr.) («.-™tn.„K.r..„,.„.,„,M.::::,:;;;i:,t':;;;:t;:.,,.-"'' A.\«ri.iK MK.l.siKK. «0 seconds C") = 1 minute . ..v »»1« circuit. hi counting certain classes of articles 12 articles = 1 . 274.S 11). of wiieat ? 27. 1679 lb. of Indian corn ? 28. 7236 11). of peas? 29. 1763 lb. of beans ? SO. 3996 1b. of carrots? •W. 1843 lb. of oats? J.?. 4444 lb. of wheat ? Al A cubic foot of water weighs 1000 oz. How many pounds will a cubic yard -weigh ? 34. A cubic foot of granite weighs 168 lb. Hew many tons would a solid cord of granite weigh ? .•?.'7. How many tons of provisions would l)e required to feed 379 men for 3 years if each man be allowed 52 oz, a day ? _ 3>1. How many acres will be required to raise 5000 })u. of carrots if each square rod yield 4 bu. ? 37. A grocer paid $7.20 for a barrel of vinegar, and found that it cost him 3 cents the pint. How many gallons were there in tlie l)arrel ? 38. A man sold at 20 cents the quart a barrel of molasses wliich cost him $23.40, and gained tliereby .$5.40. How many gallons were there in the barrel ? 39. A brick weighs about 4 lb. What would be tlic total excess leau. 2 i)k. 1 gal. 2 qt. of wheat, 379 bu. 2 pk. of oats, 134 bu. 1 gal. 1 qt. of rye, 97 Ini, 3 pk. 3 qt. of barley, • and 196 bu. of Indian corn. Wliat was his total grain crop? /';. A man \Yalk3 7219 yd., 6!U7 yd., 6894 yd., 6748 yd., 6536 yd., and 5977 yd. in six successive hours. How many miles, etc., did he w alk in all ? 17. Find the total weight of the following nine loads of wheat, and also the total numljcr of l)V.sluls in them: — No. 1. 27 l>u. 18 lb. No. 4. 25 bu. 54 lb. No. 7. 24 bu. 47 lb. No. 2. 19 bu. 44 lb. No. 5. 26 bu. 17 lb. No. 8. 22 bu. 36 lb. No. 3. 25 bu. 31 lb. No. 6. 21 bu. 35 lb. No. 9. 29 bu. 48 lb. IS, A rectangular playground is 38 yd. 2 ft. 6 in. long ".ml 32 yd. I ft. 9 in. wide. What is the total length around it ? I'J. A school-rorun is 29 ft. 3 in. long by 24 ft. 7 in. wide. Find the total length around it in yards, etc. ;^IK In building a house the cost was as follows: — Bricks, $148.75; lime, $38.,")0; sand, $8.40; woodwork, $374.98; cartage, $94.65; wages, $^974. 57; and extras and miscellaneous, $173.48, The site cost $325, and fencing and draining it cost $49.64, What was the total cost ? 21. A man travelled 38 mi. 429 yd, cue day, 24 mi, 785 yd. the next day, and still had 46 mi. 376 yd. to go to finish his journey. What was tlie length of that joui-ney ? A farm consists of eight fields of the 'following areas: — No. 1. 7 A. 127 sq. rd. No. 2. 13 A. 45 s(i. rd. No. 3. 19 A. 55 sq. rd. No. 4, 19 A, 119 pq. rd. What is the totul area of the farm ? No. 5. 16 A, 95 sq. rd. No, 6. 13 A. 68 sq, rd. No. 7. 9A. 137 3q. rd. No. 8. 5 A. 88 sq. rd. I. yil. of Icind under 9 sq. rd. undei' liay, 97 sq. rd. woodland, I, tuid 1>uilding sites. omit. It was made e cori'eot amount by i,y, 45 mi. 197 rd. by iw far did he travel (£t. of wheat, .S79 bu. 3 pk. 3 qt. of barley, I grain crop ? ,67'18yd.,6536yd., ly miles, etc., did he line loads of wheat, Fo. 7. 24 bu. 47 lb. Fo. 8. 22 bu. 36 lb. U>. 9. 29 bu. 48 lb. i in. long -^nd 32 yd. id it? "t. 7 in, wide. Find s:— Bricks, $148.75; 8; cartage, $94.65; , $173.48. The site 64. What was the r, 24 mi. 785 yd. the > finish his journey. ving areas : — 6 A. 95 sq. rd. 3 A. 68 sq. rd. 9 A. 137 sq. rd. 5 A. 88 sq. rd. COMPOUND SUBTKACTION. 48 IV. COMPOUND SUBTRACTION. Compound Subtraction is the operation of finding the difrer ence hetxveen two similar compound numbers. '' EXERCISE X. 19 T.'IoS^lbT' "'"* '" "''"' *" ' ''■ '''' "'• *■' ""^^^^ «- -hole f Wl,'r "-f; 'V!" f '• ' '''■ ' P*- ^^'«« «-'^ 20 gal. 2 .,t.v 3. What weight added to 3 T ITfU H. .-. n ■ weight as 5 T 104^ IV fi 1 ,/ '''■ ^''^^ «'''« ^^e same wugnt as .) I. 1943 lb, 8 oz. a ^""^ e A crock of butter weighed 39 1]>. 7 oz., an.l tiie crock ^vei.lie.l 6 lb. 12 oz. How much did the butter weigli ? ^ 7. What is the final remainder on taking 3 do/ and ■-. .,« u possible from 1 1 doz.? ^ ' '''' "^**^" '^« S, A owes B $73.64; B owes A $29.33. B pavs A «!<{ .1- i * navq /? «J.- fi- -v-^T-u- !.••,, T pays yi !5>lb.4/, and A if. How long is it from 24 inin. 35 sec mst H i., tu. 12 min. 30 sec. past 4 in the afternoonT ""™"'^' *" ii. A wheat buyer bought 196 bu. 48 lb. of wheal- nn \i i 473 bu. .351b. on Tuesday, but sold 600 lu^Tl^f'7' '•ought 847 bu. 19 lb. on Thursday, and 1573 bu lo lb . ^^\ but sold 2000 bu. 39 lb. on the Ler ITueZZ^^^' U hand at the beginning of the week. How muX wf h^rhe 1^. Take a million inches from a hundred miles u AlUTHMETTC. :|l III 14. A farmer luid 724 liti. of oats. Hi! sold 429 1)U. 1 pk. and fed to liis horses 9.3 l)u. 2 pk. 1 gal. 1 qt. How much IknI lie left? 1'k Three i)iles f)f wood contained respectively 12 cords 72 cu. ft., 27 corda 4.3 eu. ft., and .31 cords 90 cu. ft. There ^^ils sold from them 57 cords 100 cu. ft. What quantity remained ? 10. A farm of 110 A. 75 sq. rd. 20 sq. yd. consists partly of wood- land and i^artly of cleared fields. The cleared fields cover an area of 63 A. 118 S(|. rd, 30 sq. yd. What is the area of the woodland ? 17. A man had a farm measuring 125 A. 80 Bi[. rd., of which 88 A, 110 sq. rd. was cleared, the rest being in woodland. He sold 31 A. 97 sq. rd. 12 sq. yd. of the cleared land, and 7 A. 43 sq. rd, 25 sq. yd. of the woodland. How many acres of cleared land and how many of woodland had he left ? 15. St. Paul's Cathedral, in London, England, is in latitude 51° 30' 48"; St, Peter's, in Rome, Italy, is in latitude 41' 5.3' 54". What is the difference in their latitudes ? 19. What is the differeiice in latitude and longitude between Maflrid in 40° 24' 35" N. Lat. and 3' 41' 51" W, Loii„'. and Montreal in 45" 31' 27" N, Lat. and 73° 32' 30" W, Long. ? m Berlin is in 52° 30' 16" N, Lat.; Toronto Is in 4.3° 31' 45" N. Lat. How much farther north is Berlin than Toronto ? 21. A farmer had 75 cords of wood for sale. He sold at different times 7 cords 48 cu, ft., 15 cords 36 cu. ft., 5 cords 60 cu. ft., 18 cords 96 cu, ft., and 2.) cords 64 eu. ft. How much had he still for sale ? B3. A coal dealer agreed to deliver 22 T, 1000 lb, between the 1st July and the 1st September, He delivered 13 T. 1749 lb. in July, How mucli had he to deliver in August ? :23. A sold to B on 3rd March goods amounting to .$15,48, on 19th March goods amounting to $37. 74, on 7th April goods amounting to $28.63, and on 28th April goods amounting to $45.63. B paid to A on 19th March $31.40, on 18th April $23.6,5, and on 1st May $50. How much did B still owe A after the last payment ? 24- A merchant's accounts showed for July: receipts, $1746; ex- penditure, $1423.47. How much more did he receive than expend ? ^5, I sold goods for $97.48, gaining thereon $19.50. How much i did the goods cost me ? j 36. Out of a cistern containing 1000 gal. of water 100 cu. ft. off water were drawn. Find the weight of the water remaining in the cistern. pro poii 11 17 c, did ] 1.3 How U 148 ] How ir>. in 99 Ifl. tions 17. of o r IS. same 5yeai 10. eertai Find 1 20. fields, tains much 21. each. 429 Im. 1 pk. and fed iich h.idhelcft? !ly 12 cords 7'2 cu. ft., Thcro was sold from aincd ? insists partly of ■wood- fields cover an area of of t]i(3 woodland ? 80 8<|. rd., of which 1 woodland. He sold , and 7 A. 43 &(£. rd. i of cleared land and nd, is in latitude 51" ide 4V 5.3' 54". What d longitude between . LoiiLC. and jSIontreal 9 to Is ill 43" 31' 45" N. loronto ? He sold at different ., 5 coi'ds 60 cu. ft., >w much had he still )0 11). between the lat 13 T. 1749 lb. in July. , ing to $15.48, on 19th 1 goods amounting Uy $45.03. ^paid to A\ and on 1st May $50. ment ? : receipts, $1746; ex- receive than expend ? 1 $19.50. How much k V. (;oMpor:;i) MiTi/npijcATrox. COMPOaND MULTIPLICATION. 45 Compound Multiplication is the operation of flndinn the EXERCISE XI. !■ 7 lb. 5 ()/. s ,3. J. 18 lb. 9 oz. X 4. 3. 3 gal. 2 <|t. X 5. 4' 5 ft. 7 in. X 6. '^. 9 da. 13 ]ir. x 7. f water 100 cu. ft. of*^ ater remaining in the 0. 38 bu. 3 j;k. 3 (jt. X 49. 7. 47 da. 18 lir. 36 min. x 81. •S'. 5 da. 13 mill. 7 sec. x 100. .9. 7 mi. 1140 yd. X 23. ,, „ ^^- 3""- 147 rd. 3 yd. Ift.x2. ntsrt/'""' ' '''-' '' ''-' p^^-' -^' -"«"« 1^^ A farmer plowed 1 A. 50 sq. nl a day for 6 days. How much did he plow during the whole six days V -now muUi Hnw ^ ';"^.f/''^™'^ 1 pk. 3 qt. of berries each day for 5 days How much did he gather altogether •> ^ ' -^ "-^ys. U. A grocer bought im lb. of butter at 18 ct. tlie lb. He sold 148 lb. of 1 at 23 ct. tlie lb., and tJie rest of it at 12 ct the 1 How much did he gain on the whole ? ^ ^^'' 1<>. What idstance will a whppl 19f+ lo:., • • <• in 999 revolutions? ' ^^ "'* '" ^rcumference roll ir,. A sulky wheel 14 ft. 8 in. in circumference made 3600 revolu- 17 ITar'Xel "^^^y^^^f^^ ^f ^^^y ." ^Uu-ing the ho^^" of -.V Tl' '"• '" ^»r«"mf''rence is rolling at the rate of .revolutions jk,. second. How far does it go per ho^r ? s J; t .^^ 'T ""• '^""^ ^''- ^'^ ^'^'^""^ «^«f» '"orning, and the same distance home each afternoon on 211 days in each year for ' C"^ o "'/''' '''" ""' *'"" ^^■^^^'^ ''--S *»-^t time ? "^ ^a If 9 mi 168 rd. 2 yd. 1 ft. 3 in. be taken forcr times from ■. oertam quantity, there will still be 3 mi. 137 rd. 1 yd 7 in Tft Find the quantity. ^ ^"- fieS \tl2 1 '^^: '"^ "'• "^- ''^ ^^i- y'l- - ^»-'de^l into four t^l\^ r """ ' ^- '^ "^- '•'^- '^ «^1- y^l-5 the second con- much as the first. How much does the fonrth field contain ? each ntr" r^M? ?^ ''^''P ''^^ ^'•'•' ^^^''' '-'1 -^fi "thcrs at .$4. 12 each. How much wdl he gain by selling them all at $4..37oach^ 46 AKITHMKTIC 22. A in«!ithaiit hoiiglit 24 pieces f)f clotli meuHuring 36 yd. each at )!S18.7'2 tlie piece, and sold tlie wliol*; at $1.07 tli« yard. How iiuich did he gain on the wholt! V iiS, How nuich coal oil is contained in 30 lianelH, each containing 30 gals. 1 qt. 1 pt.? ,,'4. A woman sells a grocer 23 lb. of butter at 19 ct. the lb., 63 lb. of cheese at 9 ct. the lb., and 13 doz. eggs at 14 ct. the doz. ; and buys from him 3 lb. of tea at 5') ct. tlie lb., 12 lb. of sugar at 9 ct. the lb., 2 gal. mohiMsea at 23 ct. the qt., 8 lb. of currants at 8 ct. the lb., 11 lb. of raisins at 13 ct. the lb., and 3 doz. oranges at 23 ct. the doz. The difference between what the w. of sugar at 9 ut. , of currants at 8 ct, do/, oranges at 23 ct. nan hought and what fe it? Wliich liad to I). 3 o/. each, and the the hmdi weigh '/ holds i.n pailfids of rk was 196 lb. 12 oz. ti 20<) lb. How much '• 25 days. How far e travelled 2r)00 miles I. Avi6 min. 24 sec. -f 11. 97" 37' 36" ^4. 9. 12. $73,264-9. 13. $183 -r 4. 14' 19 mi. 246 rd. 1 yd. -^6. lo. 129 mi. 187 rd. 2 yd. 4- 7. 1(J. 193 mi. 266 rd. 4 yd. -f 9. 17. 49 mi. 118 rd. 6in.-^5. i.V. 47 cu. yd. 11 cu. ft. -fS. 10. 104 cu. yd. ocu. ft, 4-9. M. 48 A, 7 8(1. ch, 2464 sq, 1. -18, ■Jl. 10 A. 44 sq. rd. 12 sq. yd. -j-S, .1^. 497 A. 89 sq. rd. 23 sq. yd.'-5.9. ^^,V Twelve boys gatliercd 11 bu. 2 qt. of nuts and divic'ed them er ually among themselves. How much did each receive v 'M. If 11 men can mow 24 A. 32 sq. r.l. of grass in a tiout will bo an Jihstnict number ux- i)rossiug how luuay tiiuus tho dividend cuntuins tin aiviaur. EXERCISE XIII. 1. 21b. 8oz.-f4oz. 2. 101b. Soz.-T-12oz. B. i;J7T. 11891b. 4 oz. -7-304 lb. 12 S. 151b. lOoz.-r-llb. 9oz. 4. 7251b. 5 oz. ^3 lb. 7 oz. oz. 7. 2 da. 2hr,-r-50min. o seo. 6. 15 da. 18hr.-f9hr. 8. 6 da. 6 hr. 20 sec. -f 1 min. 9. 13 wk. 1 da. ^3 hr. 50 min. 10. 12 gal. 1 qt. 1 pt.-=-l gal. 1 qt. 1 pt. 11. 4851gal.^31gal. 2qt. 12. 119 bu. 2 pk. 1 qt. -=- 1 pk. 1 qt. 13. 102,336 bu. 2 pk. 3 qt. 1 pt.^11 bu. 1 pk. 3 qt. 1 pt. 14. 8 yd. 2 in. -f 2 ft. Sin. 16. 1 mi. -r 2 ft. 6 in. 16. 3 mi. 100rd.-r2ft. 9 in. 17. 25 mi. 100 yd. -f 2 yd. 1 ft. in. 18. 999 mi. 99 rd. 9 in. -f 10 mi. 76 rd'. 1 in. 19. 13,900 8(1. yd. 2sq. ft. 127 sq. in. -f 116 sq. yd, 7 sq. ft. 41 sq. in. SO. 1254 A. 80 sq. rd. 15 sq. yd. 2 sq. ft. 36 sq. in. -f-11 A. 115 sq. rd. 27 s.j. yd. m. 64,447 A. 18 sq. rd. 29 sq. yd. 3 sq. ft. 34 sq. in, -f 12 A, 133 sq. rd. 20 sq. yd. 5 sq. ft. 110 aq. in. 22. 1,764,578 cu. yd. 18 cu. ft. 1129cu. in, -r 19 cu. yd. 1 1 cu. ft. 1 19 cu. in. 23. $14.50^$290, 24. $1110-r$3.70. 25. $1001 -f 13 cts. SG. How many yards of sateen at 15 cents the yard can be pur- chased for $4.95? 27. How often can 77 sq. yd. be subtracted from 1 A. 120 sq. rd.? 28. How many posts placed 7 ft. apart will be required to support a fence round a field, the length of the fence being 64 rd. 5 yd.? How many posts would have been required had the fence been straight ? 29. How many sleepers laid 2 ft. 6 in. from centre ta centre will be required for a railway 56 mi. iOO rd. long '{ 45. rf)MPoi'Nn DIVISION'. 40 rE NUMBER. I. -r 2 yd. 1ft. 6 in. yard can be pur- re ta centre will SO. Uow many Arenn piucoH each l.-,i yds. l„ng can l.n out from a pieue of goods 40.'{ yds. long? M nn^v lung ^,m l,j !.„. 2 pk. of o.ts last a horse, giving lum .i feeds a day of r, ,,t. 1 pt. eacli ? .W. Ilowlong will i;u T. 100 U,. of food last 9.i0 men. allowing tlicm J II). 4 oz. per day per mun ? •U How n.any l.ars of lead each weighing l.S lh.s. 7 o/. will be required to make up a weiglit of 20 T. 1428 11). 2 o/ •' S4. How many loads of coal weighin;,' 1 T. 80 lb;'caoi, are there in hi car loads weighing 1(5 T. KiOO 11). each ? I -W. How many barrels hulding 1 bu. ;{ pk. 6 ,jt. each will a farmer require to pack 310 bushels of apples fur nuirktt » S(J Has. nmnycans hol.ling 4 gal. 1 qt. 1 pt. each can be filled out ot 5 barroLs containing .'51 gal. 2 (jt. each ? 37. How long woidd a cannon ball travelling at the rate of 1 '{^O ft TZ ZTV''^^ *" P"'' ^''"'" *'"^ "'^'■^'^ ^•^ *'^« •"'^""' '^ distan"ce of -J. sec ' 4^'. How many turns will a wheel 14 ft. 3 in. in circumference make in rolling a distance of 1 1 n.i. 15o9 yd.? umiercnce 43. How many pieces of ribbon ea.h .-) yd! 9 in. long can be cut horn a ribbon 100 yd. long, and h.t length will remabi over ' 4^. How many bottles each holding 1 qt. 1 pt. can be filled from a barrel containing 31 gal.- 2 (|t.? 45. A regiment in close column occupied II sq. rd. 2(i sq. yd 8 sq. t How many men were there in the regiment if each nian occupied 3 sq. ft. 52 sq. in.? ^6. How long will it take to plough 50 A. 100 sq. rd. at the rate 01 4 A. ,ii) sq. r(7ct.) = 1372 ct. =813.72. Ex. '?.— Find the price of 72 marbles at eight for a cent. 72 marbles = (8 marbles.) Price of 8 marb]es=l ct. Price of {»(« marbles) =-0(ict.> = <>ct. EXERCISE XIV. Find the price of - /. 8 lb. of beef @ 12 ct. tlie lb. ■J. 17 yd. of calico @ 1,3 ct. the yd. 3. 6 pair of chickens @ 65 ct. the pair. 4. 27 doz. eggs (a 17 ct. the doz. o. 19 doz. clothespins @. 7 ct. the doz. 0. Two fish, the one weighing 9 lb., the otlier weighing 12 lb both ® 14 ct. the lb. ' e & •, 7. Three crocks of butter weighing 27 lb., 2.5 lb. and 24 lb. respec- tively, all @ 19 ct. the lb. 8. 4 pair of chickens @ 5,5 ct. the pair, 3 pair of ducks (a 75 ct. the pair, 8 geese (&.. ()5 ct. each, and 5 turkeys («; §1.05 each. 9. 7 lb. of black tea @ 65 ct. the lb. , 4 lb. of coffee (a '.^'y ct. the lb. , 7 lb. loaf sugar @ 12 ct. the lb , 8 lb. crushed sugar (a- 9 ct. the lb., 8 lb. of cheese @ 14 ct. the lb., and 13 lb. of Carolina rice (a: 9 ct' the lb. 10. 3 doz. handkerchiefs (« 45 ct. each. 2 doz. tins of tomatoes (a 9 ct. each. 5 doz. tins of sweet corn @ 11 ct. each. 3 gal. 2 (jt. of molasses @ 18 ct. the qt. 4 lb. 7 oz. of rhubarb @ 25 ct. the oz. 3 lb. 11 oz. of iodide of potassium @ 55 ct. the oz. 16. 5 lb. 5 oz. of quinine (« $2.25 the oz. 17. Find the cost of cravellinsr 3 mi. 11. 13. u. 15. grav< 50 18, Find the cost of 48 rods of fencing r< id. of road & $8. .50 the rd. 'ii', 75 ct. the yd. 52 ARITHMETIC. i'S. SO. Find the value of — . 19. A steam-hammer weighing 189 T. @ 28 11). to the dollar. 20. 60 packages of dried yeast On '20 ct. the doz. ;?/. One million bricks (a] ijiT. T") the M. L'.J. 087,000 ft. of lumber (w, $l.-..7,-» the M. 2,^1. 1.380 lb. of wheat @ 87 ct. tlu; bu. . i 470 lb. " @08ct. " 0480 lb. " @$1.17 in:?H lb. of oats @, 37 ct. " '2340 lb. " @ 45 ct. " 9480 lb. " (a 'JOct. 1872 lb. of barley (d, .17 ct. " '28.32 lb. " (a 0.3 ct. " .?/. 47,S.")0 1b. " ([, .-,9ct. 3 J. ]0'20,lb. of peas 6/ 77 ct. " 33. '2.'?401b. " ((/, ()9ct. 3/,. 40,()80 lb. " (u .-,7 ct. '« 35. 1") 12 lb. of rye (5 (kS ct. " 50. 24(>4 1b. " (i/ ().-{ ct. " 37. 2744 11). of Indian corn (w, .')7 ct. the bu. 3S. .")l,404 1b. " '« @49ct. " SO. 030 lb. of bituminous coal (i/ 32 ct. the bu. 40. 1740 lb. of carrots at 17 ct. the bu. 41. 8 burners consuming T) cubic fc('t of gas each per hour are used at the rate of r> hi-s. a day for 310 days. Find the cost of the ga burned @ $2,2.-) per 1000 cu. ft. 4'. What -will be the amount of a man's wages for days ot hr each(») KSct. tlielnmr? 43. How much will a nuin earn in tlirec weeks (q>, ,$2.2") a day, omitting Sundays ? ' ''' P''*'^ 44. A man's wages are $2.2.") a day of 10 hr. and .T) ct. an hour S^', for over-tin]^. How much ought he to receive for 10 full days and f 2.J hr. over-time ? **"" '-^ 4o. A mechanic receives .$2.70 a day of 10 hr. and 45 ct. an hour ""'^r-^ for over-time. What were his wages for a week on whicii he worked : /"'" Monday, 11 hr.; Tuesday, 13 hr.; Wednesday, 10 hr.; Thursday "'" 12 hr. ; Friday, 10 hr. ; Saturday, 14 hr.? C'P'' 4": A man workej *)! ' included, @ $1 . 90 a day, Sundays omitte' Find the cost of haulintr 17 t to *. t ® 3 ct. per cM-t. per mile ' '"*" " ''"'""*' "' "^ ""'- 49. Find the expenses of 7 persons for a journey of "(il ,„iu^ ,).„ Uuld amoun to per day. the postage averaging , !t. o, 'Hie tter .>./. A merchant sells 13 yd. of calico (T.l 12 ct the vd 1) v, f . Imushn @ 23 ct. the yd., and 17 yd. of flannel ® 4Srthe yd' . lltakes m e.xchange 38 bu. of potato, (^ 37 ct. the bu. and tl e baian e f m cash. How n.uch cash d. .«. (.e receive ' 47lb fit^l n'V'.r " ' ^f ^--f eggs® 18ct.the doz., 47 n. of lard @ 3 ct. the ib., and ] 17 lb. of beef @ 8 ct. the lb tc:!;:!;:h:::h ;:r^;:;^^^^-°^*'^^^-^«' -^ ^'- '-^-^ in cash. ■-?. A woman sells to a grocer 15 do? efftrc (77, i« ^f +u , +1.« 11. 1 It e . J ^*-' * ^"' °f raisins (^^^ 12 ct the lb., 4 lb. of currants @ 7 ct. the lb., 2 oz. of cinnamon @ 3 ct he oz. and 8 oz. of allspice (T. 3 ct. the oz. How much is stUl du; wages for days ot 9 hr. ,'"^^^ /^ '^' ''''''''' *'"« «"'» "' •'5-cent pieces, how many ought she J.^. A woman sells a merchant 7 pair of chickens f,^ 56 ct. the pair ^pair of ducks @ 73 ct. the pair, 4 geese ,. 93 ct. eaJh. an.l ,3 turke !^ hr. and .T, ct. an hourf,tVT r '°A .7' ^'■"'" ^"'" ''' ^'•- "^ ^''^ ^' l'' ^t- the J eive for l.i full days and Zit ^7^® ^ "'■ *'' ^'^^ '^ ^'^^ "^ «-"^l ^^ 45 ct. the yd" '^and 29 yd of chintz @ 26 ct. the yd. How much is due the m r' hr. and 45 ct. an hou/';'!* ?;'"^ ^^'^"fr*"" ' eek on whicii he worked : ^Cttt'lZ 'h f ' "' "^ "'''' ® '' ^*- ^^° ^"- ^'^ 2686 Ib. .day, 10 hr.; Thursday, l°f,^f.f ,?:*''" ^^"•' '^"'^ ^^^'^ **'« P'-«eeeds he buys 49 y,l. of ^carpet (^, $1 15 the yard and 24 rolls of wall paper @ 37 ct. the roU lOfi n f ' 1 ^, J ^"'^' "'""li ''as he left of thn o..^.h rcroivpd f • ' ■ i ^ HItli Oetoner, both da\.»i /-,. tt , — '•■" '^'^^f'J^ed lur ms sales? How much . f-v ards can he liought can be bought for yards ought to be 3W many bu. of oats tt, how many bu. of o\v many pounds of r$117? li $7C0 and have $28 for 45 lb. of cheese pound? he Ib."^, and bought fee @- 35 ct. the lb. she buy with what r? 1.18 the bu., for 10 n? expenses are $45 a a farm of 80 acres t. the doz. , and sells •s, how much will he 1 5 ct. the doz. , and imainder @ 2 ct. an )z. can she buy with BILLS AND ACCOUNTS. 55 II. BILLS AND ACCOUNTS. A Bill of Parcels (called also a Bill of Goods) is a written statenient of goods sold and of payments, if any, received there- for. The Bill should specify the quantities and prices of the goods, the place and time of each transaction, the names of the buyer and the seller, and any special terms agre«d on bv the parties. *^ A Bill of Services is a similar statement of services ren- dered or of labor performed. A Bill is also called an Account. _ A Statement of Account is a written statement of the total »ums due according to accounts already rendered. The seller of the goods is called the Creditor. The buyer of the goods is called the Debtor. The statements of the items due to the party rendering the iccounfc is called the Debit Side of the Account. The statement of the items due or the moneys received by ihe party rendering the account is called the Credit Side of he Account. The Balance of an account is the difference between its debit md credit sides. inen a bill is paid it should be receipted by writing at the bottom of the bill the date of payment and the words "Received )ayment," and under these words the creditor should sign his lame. ° If a clerk or other employe have authority to sign for his mployer, he should write his employer's name and directly eneath it his own name or initials, preceded by^er or bi/. (See Ixample 3.) He may, instead of signing this way, write his wn name and directly beneath it his employer's name, preceded yfor. 56 ARITHMETIC. Ex. /.—Specimen of a Bill of Parcels. GuELPii, IMh Oct., 1885. Mr. William Thompson Bought of Robert Brown. 1885 Sept. 22 24 28 12 lb. Biitter 15 lb. Sugar . 5 lb. Ttia . 3 lb. Coft'ee 1 F. Haddie % . . . @ 13 ct. 1 56 . • . @ 9ct. 1 35 . . . @, 55 ot. 2 75 . @ 35 ot. 1 05 • •#■»• 35 ' 0(J Ex. ^.—Specimen of a Receipted Bil' ,vith Credit Items. Hamilton, IGth Oct., 1885. Mr. James Robinson, fir. %a John R. Shav\ 1885 Sept. Oct. 28 30 1 Sept. 30 To 3 lb. Java Coffee "12 lb. B. L. Sugar ' ' 4 gal. Molasses . '« 7 lb. B. Tea . . " 9 lb. Butter . . " 3 oz. Nutmegs . " 15 lb. C. Rice . . Or. 5 Qr. Note Paper 3 Pck. Envelopes 1 Bot. Ink . . . 1 Box Pens . . @ 33 ct. @ 11 ct. @ 88 ct. @. 65 ct. @ 16 ct. @ Set. @ Oct. @ 18 ct. @ 15 ct. 1 3 4 1 Balance due 99 32 52 55 44 24 35 K J 188/ Oct Oct. 90 45 15 35 13 4: 1 8 Oct. 19th, 1885. Received payment. ^^J^K?-. ^~^. f^l^^ti*. Ma mg, V 1. ] 1885, @14 @$1. Nov. Skirt 2. I 1885, 4 lb. f 3 1b. S 51b. C of Chii 91b. S S. T side W 20 Slid 4. G !4 con Sawn ] LPii, Ifdh Oct., 1885. •f Robert Brown. @, 13 ct. @ 9ct. @, nf) ct. (n) 35 ct. % 1 56 1 35 2 75 1 05 35 7 06 ith Credit Items. LTON, IGth Oct., 18S5. John R. Shav\ 3ct. 1 ct. 8ct. 5ct. 6ct. 8ct. 9ct. 8ct. 5ct. 1 3 4 1 99 32 52 55 44 24 35 90 45 15 35 13 4 1 S BILLS AND ACCOUNTS. 57 Ex. -^ -Specimen of St. oment .,f Account receipted. Messrs. Jones c6 Co Bkantford. loth Oct., 1885. Tern^s: 30 day.,. ^° ^''''"■''' ^i'>f>iri80n & Co., ^X. 1885 Oct. 1 ' To Account rendered I I Oct. 16th, ISSo. Received payment. $47 50 -^. <^. EXERCISE XV, inf ttrf ^'"' ^"" *'' following.mentioned transactions, supply- mg, where necessary, names of places and dates of makmg out - i. Mr. James Thompson bought of William Smith: Nov. 2nd I880 14 yd^ Pnnt @ 13 ct. the yd.; Nov. 3rd. .33 yd. White Cotto^ ® SI V^7. ■ l^t "^"'^ ® ^'■'''' ^^°- 12th. 16 yd. Silk 51.8 9yd. Lming @ 13 ct., and 3 doz. Buttons @ 23 ct. the doz • Nov 14th 9 ycl Jersey Cloth @ 45 ct., 2 yd. Plush @ $1.95, 3 yd' 8kirt Lmmg @ 18 ct., 2 doz. Buttons @ 15 ct., 2 Spools @ 5 ct ,«;; ^^r Herbert Williamson bought of Thos. Acro'c Nov 19th 1885, 9 lb Roast Beef @ 12 ct.; Nov. 21st. 7 lb. Lamb @ is ct'' 4 lb. Suet @ 9 ct.; Nov. 23rd. 8 lb. Bl. Beef @ 8 ct.; nVv 25th 3 lb. Steak @ 14 ct. ; Nov. 26th, 13 lb. Lamb @ 13 ct. Nov 28th' 5 lb. Corned Beef @ 9 ct. and 2 Geese @ 65 ct. each ; Dec. 1st? 3 pa^; of Chickens @ 5o ct. the pair; Dec. 3rd, 6 lb. Venison @ 14 k Ld 9 lb. Sausages @ 12 ct. f • T^«- Sanson sold to Alfred Lawson on Oct. 28th, 1885 20 Out sKle Wmdow-Sash @ $3.50. 40 pieces of Window-Stops @ 3 ct and 20 Slide Ventilators @ .30 ct. ^_- -5 tt., ana i4 cords Maple (oj $3.oU. 4 cords Soft Wood @ $2.25, and 7 cords Sawn Hardwood @ $4.25. » "" / coras K8 ARITHMETIC. .5. Benj. Bradshaw bought of John Westover on 7th Jan., 1886, 700 lb. Flour @ $2.75 the cwt., 400 lb. Oatmeal @ $2.25, 300 lb. Cornmeal @ $2.25, and 200 lb. Buckwheat Flour @ .?2.o0. On Ist Feb., 1886, Beiij. Bradshaw paid $25 on the above account. 6. William Atkinson bought of Messrs. Moore & Co., Ap. 8th, 1886, 100 ft. g-in. .'i-ply Rubber Hose @, 20 ct. tlie ft., 2 pr. g in. Couplings and fitting ^ 50 ct. the pi, 1 Comp. Hose Pipe, $1.25; Ap. 16th, 3 Step-Ladders @ $1.50 each. The above account was paid in full on Apr. 16th. ' 7. Messrs. Hughes & Son sold to M. Stonehouse: Dec. 9th, 1885, 19 yd. Calico @ 17 ct., 17 yd. Linen @ 47 ct., 16 yd. Lining @ 9 ct.,*| Dec. 21st, 8 yd. Flannel @ 48 ct., 23 yd. Braid @ 3 ct.; Dec. 26th, 7 pr. Stockings @ 25 ct. and 3 pr. Gloves @ 65 ct. l*aid in full on 2nd Jan., 1886. 8. Peter Simpson bought of Jamieson Bros. : 14th Ap., 1885,31b.! Bl. Tea @ 75 ct. and 13 lb. B. L. Sugar @ 11 ct.; Ap. 16th, 5 lb.! Gran. Sugar @ 9 ct. ; Ap. 18th, 3 bars Soap @ 23 ct., 3 boxes Starcli @ 15 ct.; Ap. 21st, 1 Bath-brick @ 8 ct., 3 dpz. Eggs @ 17 ct., anJ 4 lb. Butter @ 19 ct.; Ap. 23rd, 12 lb. Flour @ 3 ct., 1 box Soda Biscuits @ 30 ct. ; Ap. 25th, 4 lb. Currants @ 8 ct. and 7 lb. Raisins @ 9 ct. On Ap. 2 Ist the sum of $5 was paid on the above account, " ' and the balance was paid on 1st May, \rrow .9, Edward Lawson bought of Bruce, Playfair & Co.: 5th Jan., ^ ^*' 1886, 9 Diaries @ 57 ct., 3 boxes Elastic Bands @ 25 ct., 5 Rms. ^^'^^''^^^ harge te-pencils @ 17 ct.; Jan. 22nd, 16 doz.L^'^" 6x9 Slates @ 95 ct.; Jan. 2."th, 5 qt. Ink @ 37 ct., 5 qr. Wrapping , ' Paper® 30 ct., 6 Col. Pencils @, 9 ct.; Feb. 3ru, 2 Rm. Acct. Cap '^^i^^' @ $6.00; 1 Rm. Letter Paper @ '''* ""^ " ^"- ^""'-~ ^ k ^^ . t..,. 16.50, F'scap @ $3.45; Jan. 13tl 3 qr. Blotting Paper @ 37 ct., 7 boxes Pens @ 36 ct., 5 boxes Si .50, 3 Pass-Books @ 5 ct.; Feb, 11th, 3 boxes Envelopes @ $1.25, Postage Stamps, On this / accc 188.' .Srd, Suel 10th @1] ur. 45 ci iOot @. i;: SOth, there "g i ava lb,] Bar; Tin iiiilet ,M; account the sum of $15 was paid on Jan, 16th, and a further sum ot ^^'^ ^ $25 was paid on Feb. 15th. 10. Simon Tomlinson bought of H. Ward & Co. , of Guelph, in 1885 July 4th, 5 doz. Hat and Coat Hooks @ 40 ct. , 3 Door Knobs @ 15 ct., and 3 Rack Pulleys @ 20 ct.; 7th, 25 lb. Cut Nails @ 4 ct., 3 pr Hinges (Sj 23 ct., and 2 Door Locks @ 30 ct.; 18th, 7 lb. Pressed Nails @ 8 ct., 9 doz. Screws @ 6 ct. ; Aug. 1st, 3 Padlocks @ 25 ct., 3 Hasps and Staples @ 15 ct. ; 2 doz. Bolts @ 20 ct. ; 20th, 5 lb. S, L, Cord @ 90 ct, ; 5 yd. Brass Chain @ 33 ct, Aug. 29fch, Accoun paid in full. 14./;,, loz, ( doz. J Feb, ilson U. ^' Tor ar. 16 lis, di •uart A over on 7th Jan., 1886, itmeal (a' $2.25, 300 lb. It Flour (o! .?2.o0. On the above account. Moore & Co., Ap. 8th, ct. the ft., 2 pr. ^ in. /oinp. Hose Pipe, $1.25; The above account was' BILLS AND ACCOUNTS. 59 jhouse: Dec. 9th, 1885, . , 16 yd. Lining @ 9 ct. ; aid @ 3 ct. ; Dec. 26th, 65 ct. l*aid in full on 3.: 14th Ap., 1885, 31b.J 11 ct.; Ap. 16th, 5 IbJ @23ct.,3boxesStarcli| Jqz. Eggs @ 17 ct., anc^I n. George Stevens owed .Samuel Crear on ^Of I. K account rendered that day, thesumof $14 c^ i^"''" ^^'^' ''' ^'' 1885, he bouglit of S Cre Jr nl f / ' '""' ''"""« ^^ecenibor. 8uetr«- ]2<.t . 8ti. 1-^ 1) w Z ' ^^""''^ ® ^4 ^t-. '-i"'! 1 11>. 10th, 2 lb. ^:^^ ^ ^ f-;^fy^ «;•. J 1^- Corned Beef 0, ct.; IQur, Lamb (?/^ 13 ^t 10 11. R^ t* « ^ ,!, ' ' ^^ ''*• Hind u- ^ K „ ' ^^ ^'>- ^o. Beef ® 12 ct • IKfl, o 'r Ao ct., 7 .11,. Shank (a). 4 ct • 2 lb T n.. 1 r. , r ' -lo>'g"es r?'. !n J. 1 rn , ^ ■* Ct. , .^ 11). Liilrd (a), 15 ct, • 99n<1 I (^ iOct., 1 Turkey, .S3. 50- 2-^r Stamps, $4. On this '"»'^t« @ 60 ct.. 2 doz. Draw-knives (a^ $8 50 3 1 U ' 'J '' ;h, and a further sum ot ^^^'^ ® ^5-75, 3 doz. Door Locks (S) $4 25 2 dl l"'"- T f^""" 14.75, 4 doz. Padlocks (w S'2 9^ r V ^ ' '^P""S ^°^"^« @ iut Nails ® 4 ct., 3 wt, '" ' ''"' '''^ I""'> »" March 9th an.l reoeiDtcl 1,1 TK - ct., 18th,71b. Pre,sc,lf ''*'"''"'>''''''» »fMe>,,r». Kent, Lewis it Co '^' ''^ """""" it, 3 Padlocks @ 25 ct.,r''*„«''- C'Jv.ry & Co., Berlin, purchased of Messrs Sl„.rf .. '■ ^--'"■>-— |us; ditifr-i^tr-tiiti^t ftTA^ iC-if r^ iuart&Co. * P* ^""^ "" I'^lialf of 60 ARITHMETIC. III. AGGREGATES AND AVERAGES. The Total or Aggregate of any iinmber of «iuaiit,ities of the saino kind is simply thoir sum. Hence — Tu find the Total or Agyreyate vf aitij number uf qiiuntitiea of the same kind, add the quantities together. Thus if a pupil receives 6 merit marks on Monday, 8 on Tuesday, 8 on Wednusduy, 7 on Thursday, and 6 on Friday, the Total or A^'fe'regate number of lii.s marks for the five days will be 35 ; for, atldin;,' together the numbers received on the several days, G+8+8+7+6-35. The Average or Mean of any number of quantities of the same kind is that quantitity which, if put in i)laco of each oi the given quantities, will yield a sum the same as that of these quantities. Hence — To find the Average or Mean of any 7i,nmJ>er ofquantitks of the same kind, divide the sum of the quantities hg the number of them. Ex. 1. — A pupil received 6 merit marks on Monday, 8 on Tuesday, 8 on Wednesday, 7 on Thursday, and C on Friday. What was the average number of marks he received per day ? Caluulation. 6 nurkK. 8 8 7 35 Calculation. Proof. 6 marks. 7 8 7 8 7 7 7 6 7 5)35 7 35 The total number of his marks for the five days was 35. Dividing tliis total by 5, tlie number of days he got marks on, gives 7 us the Average num- berhe received per day, tliat is, had he received 7 marks each day instead of the numbers he did receive, he would, at the end of the five d.ys, have received exactly the same number as he actually received. Ex. 2.—K farmer sold 3 cows for $46 each and 5 cows for $52 each. What v/as the total price and what the average price each of the 8 cows ? 3 cows @ $46 each 5 " " 52 ** 8 ) 8 cows, in all, _aTO_worth_|398 Total price. "l cow, o?t. an average, is worth $49.75 Average price. The total price of the eight cows is 8398; hence the average price jier cow, got by supposing the 8 cows to be all of equal value, is found by dividing the total price by 8, and is $49.75. are worth $138 " *< 260 are worth $398 AGORKOATES AND AA'KRAGES. 61 ERAGES. f quuutitiea of the er of (pMHtities of Caluulation. 8 8 7 35 P quantities of the 11 jjlace of each of ne as that of these of^qiumtitieti of the lie number of them. on Monday, 8 on and 6 on Friday, received per day? LCCLATION. Proof. 6 luurks. 8 8 7 6 5)35 au 7 and 5 cows for $52 ! average price each Total price. '5 Average price. erage price \^er cow, got i ly dividing the total price! EXERCISE XVI. Complete the following tabulated statements by filling in the totals and, where they occur, the columns of .lifFerences. J. Claasificati(,n of pupils, Cities of Ontario, 1883. Belleville . . . . Brantford (hielph Hamilton . . . . Kingston . . . . London Ottawa iSt. Catharines St. Thomas , . 'J'oronto Total i •/' o C "E JB ^ Nt-'MIIISR Of PlPILS I\ TUB -_l 2(33 374 415 91G 560 .'564 558 439 347 1940 J 14 104 i>40 51 402 87 I Total. 120 19 4 849 ; 158 ^Statement of receipts of grain in car-loads. 62 AKITHMKTIC. .9. Statement of Canadian live stock, 1881. II: . PROVINCKS. HorHCH. Cattle. Prince Edward Is. Nova Scotia New Brunswicli . . . Quebec 31,335 57,167 52,975 273,852 590 298 90,722 325,603 212,565 1,0.30,333 Onturifi 1,702,167 60,281 80,451 12,872 Manitoba British Colund)ia. . Territories 16,739 26,122 10,870 Total Sheep. Swine. Total. 166,496 40,181 377,801 47,256 221,163 63,087 889,833 329,19!,' 1,. 359, 178 700,92l» 6,073 17,358 27,788 16,841 346 2,775 4- Statement of school attendance during the year 1884, Number ok Pitils who Attbndbd TOWNSHIP. Cliarlottenburgh . Kenyon Lancaster Lochiel Total, 1884. Total, 1883. Increase . Decrease. 108 121 89 105 255 256 218 175 ^4 ii 336 315 317 233 ^■3 B ■-I |5 287 280 255 276 528 871 1207 963 1^ o;^ : Total. Li -' 171 195 179 136 j 42 I 25 I 32 20 765 162 /!. School Trustees' Financial Statement. TOWNSHIP. Cliarlottenburgh . Kenyon Lancaster Lochiel Total liiiluiice from 188;5. $593 96 194 .57 546 37 396 78 Receipts duriiit,' 1884. $6116 15 6294 91 6.')27 74 4482 99 Total Receipts. Expenditure Huluiice on during 1884. liuiid $6154 39 5917 53 6230 89 3961 62 p- Swine. Total. im 40,181 iO\ •t7,2r)6 103 53,087 iXi 329, 191? 178 700,92'.' 373 17,35ts 788 16,841 iiii 2,775 le year 1884. ACOUEGATLS AND AVEUA(;ES. 6S Attknded ^5 s4 111 Total. 171 195 179 136 42 25 32 20 765 162 8. What is the mean of 3 ami 7? Of 5 aii.l 1 1 » nr - i ..-. Of 10 and 20? Of o and 100? ' ''"'' '^''• ». What is the mean of 2, 5 an.l 11 ? Of :i «i ,,„,i i.,v nf r , andO? Of 0, 8and 10? Of 2, 2and20? ' ' /«. What ia the average of two wciirhts of -i 11. n.. ' u respectively? / v/i ; u>. an, lO lb. /i. Wlmt is the average of three lengths resp -.t, oK of i ft *> ft. and 7 ft.? Of 10 ft., 25 ft. and .W ft' ' "" /^. What is the average of four weights of 7 II... .. ih v, jk and 19 lb. respectively? » ^ ■, j id., d ib. ^5. What is the average of four lengths of r. ft Ift ff on u j *0 ft. respectively ? '^ ' " ^^^ ^^ ^*- '^nd /4. Four vessels holdi.ig respectively 2 gal. 2 qt., 1 gal 1 ot lr^r^ft?;il;::"""°^-- -wiar^Lviei^i 10. F.nd the average of 0, 1, 4, 9, 1.;, 25. .36. 49. 64, 81, 100. |. i''9^i;^ s'lTa"" °' *'" '"^""""' '""" '' ""'^'^ = **'' ^«' •»'^' 2.0. The aggregate weight of 8 oxeii was 12,376 lb. What was heir average -weight ? »>"a^i^^a8 Complete the following tabulated statements. ■oTnty Im"' °' ""'"■ '^ '^^'""'^ '^"'^ ^' ^'^^-^ P^I-'''^"'^" i" Expt'iidituro Hulaiioe on (luring 1884. ; hand. 16154 39 5917 53 6230 89 3961 62 TOWNSHIP. Number of Number of _ Schools Children m Township, in Township 13 ' 585 17 663 12 ! 588 14 ! 602 19 i 627 10 4 'JO Average I>fr School. Total , 64 ARITHMETIC. 21. Statement of monthly school attendance. SCHOOL. s A B C D E F Total 517 226 238 331 328 < I a s ■^ 1 Vi o 514 495 222 I 221 251 254 345 ; 341 324 ! 330 74 I 83 493 214 262 406 381 93 468 200 242 420 394 92 455 186: 235 ' 420 378 85: 533 217 215 431 376 77 541 212 222 432 395 74 Nov. Dec. H 536 518 206, 196 235 226 423 401 282 272 72 65 1 1 22. Statement of receipts of school moneys. YEAR. Receipts from Assessments. Receijits from County Fund. Receipts from Kent, Int., etc. T0T.\L Rkceipts. 1872 1873 S13,869 50 47,633 16 52,090 02 42,493 68 59,299 12 41,794 42 36,736 95 74,749 28 37,158 00 47,040 72 50,802 86 50,965 01 46,953 72 §5,019 57 9,035 50 8,977 14 9,108 03 5,295 2() 11,243 60 3,904 29 12,078 96 8,231 65 7,824 32 7,896 37 7,881 31 7,821 33 $126 00 202 00 216 32 560 80 1,925 77 1,300 00 50 00 50 00 37 50 619 83 520 00 513 72 576 44 1874 1875 1876 1877 1878 1879 1880 1881 1882 1883 1884 Aggregate. 1 i 1 Average . . 1 i 23. A grocer's daily receipts were: Monday, $219.57; Tuesday'| $247.38; Wednesday, $213.45; Thursday, $.368.72; Friday, $245.19] Saturday, 4/3.77. Find his average daily receipts for the week. 24. The daily receipts of a hardware merchant were : Monday! $47.3.67; T.sday, .$.594.68; Wednesday, .$.371.93; Thursday (a holi! day), nothing ; Friday, $687.-55 ; Saturday, $749.47. Find his averag5 Q <" 536 518 212 206, 196 222 235 226 432 423 401 395 282 j 272 74 72 66 i. tr'i^trx>-- R--"- H26 GO 202 00 216 32 560 80 ,925 77 ,300 00 50 00 50 00 37 50 619 83 520 00 513 72 570 44 lay, $219.57; Tuesday| 38.72; Friday, $245. 19 ceipts for the week, •chant were : Monday 1.93; Thursday (a holij 19.47. Find his averag< . includimj Thursday. AGGREGATES AND AVERAGES. (;5 S5. The monthly sales of a merchant were: January 84378 46- February, $375.3.69; March. $5685.75; April, $429738"' ltd the average sales per month for the four months. If the same averal tl?i:Tn ' *'r"^"""* *'^^ ^^^^' ^-'-^ --^^ have beenT total amount of his sales tliat year » :^G. If a man spend $142.31 in 19 weeks, how much does he spend rryerrXriTC) ; "^ *^" ^"^ '-- -' -^^^ ^« «-^ llflb^8ti?Tn7,?^ respectively 109 lb., 105 lb., 103 lb., 97 lb., 11 lb 88 lb., 106 lb., and 102 lb. What is their average weight 4iiof\n':::;' '''- ^^^^^ ' ''• ' °^' -^^^ -^^ ^^ «- -We J^J" TT ^'""^^^^ ^ ""'' °"" ^'^"^'' ^ ^'l«« tl^« «^«^ond day 8 miles dtanc 1 'T\ '',""" *'^ '°^^*^ '^y- ^^'^^^ -- th! tot distance he walked and the average distance per day •> .?ft John IS 12 years old, his sister is 10, his eldest brother is 15 and h.s youngest brother is 7. What is the aggregate and what th: average of their ages? 3! A grocer bought a tub of butter weighing .34 lb @ 18 ot the lb. a second tub weighing 42 lb. (7, 19 ct., a third tub weighing 48 lb @ 21 ct.. and a fourth tub weighing 31 lb. (a- 22 ct Cat was the total weight and price of the four tubs, and what the average price per pound ? cverage heiriitr '^^"^''' "' """ '^^^'*^ ^"'^ ^^h^* *h-^ — g« 'f ^ "^^^^^'•k^^^ 10 hours on Monday, 8 on Tuesday, 9 on Wed- nesday 7 on Tluirsday, 9 on Friday, and 8 on Saturday."^ Wha was t]>e total and what the daily average time he worked during The a ilk^tT "T ?'•''' " ^''" ^°" '""'^h is that on an average a ^veek, taking ,52 wk. = 1 yr.? fe • J^. An express company carries 30,553 T. 604 lb. of merchandise iJm ^Tl'-" '''"'' ^"'" ^^'^^ ^ ^''y-' P^'-"^»'« '^re $1.75; Jones' p. 10; Ivobmson's, .$2.40; and Thomson'.. .^2 25- W!-.f i^ th.' aggregate and what the average of tlieir daily wa-es v 37. in9hamsweigh276 1b.lloz.,whatistherraveragev,.eightv 66 ARITHMKTJC. 38. If a man's salary be S12o0, how much may he spend on an average per day, and how mucli per week, to the nearest cent, so as not to run into debt? (Reckon 52 weeks to the year, also .365 days to the year. ) 39. The total weight of 17 cheese wus J29 lb. 11 oz. What was their average weiglit ? 40> If a grocer use 95 reams of wrapping paper in a year, how much will he use daily on an average, counting 304 business days to the year ? 41. A man -walked .S73 yd. 1 ft. in 480 steps. What was the average length of his steps ? 42. A man dug G7 rd. 1 ft. 6 in. of drain in 27 days. What length did he dig on an average per day ? 43. A man walked 500 miles in 24 days. How far did he walk on an average per day ? 44' A traveller left New York by the Pacific Express at 10 o'clock on Tuesday morning, and arrived at San Francisco at 11 o'clock, New York time, on tlie following Monday mdrning, having travelled a distance of 33G4 miles. At what average rate per hour did he travel ? 45. If 1 1 men have to mow 24 A. 32 sq. rd. of grass in 1 1 hrs. , how much must each man mow on an i\,verage per hour ? 46. A farmer drew 17 cords 99 cu. ft. of cordwood in 13 loads. What was the average quantity per load ? 47. A wall containing 412 cu. yd. of stone was built in 6 weeks. What wa5 the average amount built per day (G working days to tiie week) ? 4<>. Five men took turns to keep watch over a house for 1 3 da. 19 hr. If each niau kept watch thirty times, what was the average length of each \\ .itch ? 49. The following summary is taken from a book of cash sales : — Amount. Aug. 7, ."0^310® $1.09 eacli 8, " 470 @ 1.25 9, " 640 @ .95 10, " 430(0', 1.07 11, " 580® .99 12, " .*;t)0(^A 1.10 What was the average auriilK;i' boI'I daily, the average daily casli business, ;uid the average selling price ? AGGREGATES AND AVERAGES. 67 :h may he spend on an ;o the nearest cent, bo as ) the year, also .365 days 29 lb. 11 oz. What was ig paper in a year, how ing 304 busiue«s days to steps. What was the \ 27 days, ^\'hat length How far did he walk on [fie Express at 10 o'clock Francisco at 11 o'clock, idrning, having travelled ;e rate per hour did he rd. of grass in 11 hrs., i^e per hour ? : cordwood in 13 loads. ae was built in 6 weeks. f (G working days to tiie over a house for 1 "i da. s, what was the average a book of cash sales :- A mount. , the average daily cash oO. A man has a salary of $8oO a year; for the first 7 months of a sertam year he spent an average of §8.5 a month. How much can he ipend a month for the remainder of the year and not live beyond us salary ? •' 51. A grocer mixes together 40 lb. of tea @, 45 ct. the lb 48 lb 47ct.,and641b. @53ct. What is the price per 11). of themixtvre? 52. A mixture was made of three grades of barley, viz., 8 bu @ >9 ct., 15 bu. @ 58 ct., and 28 bu. @ 65 ct. What is the value per lushel of th e mixture ? 53. A stationer bought 72 reams of paper® $.3. GO the ream and :8 reams @ $0.60 the ream. Find the cost of the whole, and the verage price per quire and per sheet. 54. A grocer mixed 106 lb. of tea costing 38 ct. the lb 75 lb costing 42 ct. the lb., and 94 lb. costing 45 ct., and sold the mixture It 60 ct. the pound. What was his gain on the whole ? 55. A grocer mixed 19 11>. of coflee costing 28 ct. the lb and 26 lb iosting 23 ct. the lb. with 10 lb. of chicory costing 8 ct. the lb At vhat price the lb. must he sell the mixture to gain $5.50 on the vhole ? 56. Find the total value and the value per gal. of a mixture of 7 gal. of vinegar @ 60 ct., 27 gal. of vinegar («l 40 ct., and 6 ual of vater. 57. How much water must be added to a mixture of 16 qt of rmegar @ 13 ct. and 10 qt. at 10 ct., that the whole mixture may )e worth 11 ct. the qt.? ' 58. A barrel of vinegar containing 25 gal. was bought for $9. low much M-ater had to be added to allow the mixture to be sold Ivithout loss @ 25 ct. the gal.? ' 59. A barrel of vinegar containing 3a gal. cost $10. How much vater must be added that $2.96 may be gained on the whole by sell- ag the mixture @ 36 ct. the gal. ? 60. The mean height of six mountains is 10,.357 feet. Find tin ggregates of their heights. What must be the height of a seventh lountain if the mean height of the seven is 10,643 ft.? 61. In 400 civil years there are 303 years of 305 days each, and 7 years of 366 days each. Find the average length, to the nearest ccoud, of the 400 civil years. 62. In a certain school tiiore is one teacher at a salarv of 9.9.m ^er nnum, two at salaries of 400 each, and two at salaiies of $.{00 ea!;li. md the average salary of the five teachers. 68 ARITHMETIC. 63. A man bought two oows for 335 each ; he sold one of them for $43 and the other for §32. How mudi did he gain on the first cow ' How much did he K..se on the second ? How muoli did he gain on the two togetl ->r ? What was his average gain per cow ? 64. A man sold two horses, gaining $32 on one of them and lo&ing $15 on the other. How much did he gain on the two together? What was his average gain ? 65. A butcher sold three sheep; on the first he gained §1.25 on the second he lost 53 ct., and on the third he gained GO ct. How much did he gain on the three, and what was his average gain per sheep? 66. A man paid 40 ct. a day for his board. On Monday he earned §2.00, on Tuesday he earned $1.50, on W^ednesday he earned $3.30, on Thu .day, which was a holiday, he earned nothing. How much did he earn during the four days over and above his board. How much did he thus clear per day, including Thursday? 67. A merchant gained §2.336 in his first year of business, §1875 in his second year, .§619 in his third year, lost §987 '••^. his fourtli year, lost §11 78 in his fifth year, gained §293 in his si:- h year, and gained §1.361 in his seventh year. Find his average ^.m for the seven years. 68. A merchant bought 5 barrels of pork. Three of them weiehed more than 200 lb. each by 1 lb. 8 oz., 3 lb. 4 oz., and 5 lb. 12 o/. re- spectively, and two weighed less than 200 lb. by 2 lb. 4 oz. and 3 lb 4 oz. respectively. What was the total weight of the 5 barrels,' and what their average weight ? 69. At six successive tides the highest point reached by the water was 1 ft. 2 m. below, 9 in. below, 1 ft. 1 in. above, 2 ft. 4 in. above, 1 ft. 3 in. below, and 1 ft. 3 in. above high-water mark respectively,' \\ hat was the average above high-water mark for these six tides? 70. In the Great Trigonometrical Survey of India a standard length was measured ten times; tivo of the measurements m-de 'le standard too long by 6 units each time, ttoo made it too sh • : 50 each time, three made it too short by 2 each time, and thi , e mao 't too long by 58 each time. By how much was the standa' d too long according to V\e average of the ten measurements ? i Jan Expi ;herefo ord( ill ct. 11 [has no' ;o!?ethc rom tl: illowin ^Villie r SHARING. ; he sold one of them for he gain on the first cow ? >w much (lid he gain on J jain per cow ; n one of them and losing j li on the two together?' first he gained $1.25, on ' I he gained (50 ct. How as his average gain per On Monday he earned nesday he earned §3.30, d nothing. How much ibove his board. How hiir.sday ' year of business, $1875- lost $987 •■". his fourth 3 in his si> 'i year, and^ is average ^^m for the| Three of them weighed oz., and 5 lb, 12 o,:. re- by 2 lb. 4 oz. and 3 lb. ;ht of the 5 barrels, and it reached by the M'ater j ibove, 2 ft. 4 in. above, ater mark respectively, k for these six tides? r of Iiidia a standard) leasurements m''^«*--" J'^-- -"' -'"hn. which will ,ive apples to each. But Jainos has two apple.s alrea.ly, so h. will l.ave 7 apples i,. all. Jamen. 2 apples. 5 <^ FoHM OF Calculation Joli n. 5 apples. 5 u 12 apples. 2 " to James. 2)10 5 '' to each. .E-x. ^^ -Divide 120 cent.s among Annie, Willie and Harry giving Annie o ch nmro than Willi., .-uul /iUie 11 ct. more |tl'.an Harry. FoKM OK CAL(n'LATION. Annie. Willii: Jlarni. 5 ct. fL" 11 ct. 120- J?"'* ^ J!'u ==2^" t- Annie and Willie. 31 " 31 " 31 ct. 3)93 " *^ " ^^2 " 31 - 31 . to each. i the standa'd too lontt) lents ? EmANATiON.-G.ve .5 ct. to Annie. Since she is to receive 5 ct. more than Willie n order that she n,ay .till have r. ct. n>ore than WiUie. But Willie is to receive 1 ct. more than Harry. Give H ct. to Willie and an equal sum to Annie Annie J>as now received r.ct.+nct. =16 ct., and Willie has received 11 ct;rncetoth ;T;t':'r"r'''"''*- + ''=*-=''^*-""'"^*'-^^°'^*' Deducting. 7°. SI; ihem 'a! oT ""TT ■'■ i° '"^ '"^'•'' '^""^"^ ^"^°"^ *h^ *»»'- 'Children. Ilowing them 31 ct. each. Annie therefore receives 5 ct.+U ct.+31 ct =47 ct rtilhe receives 11 ct.+31ct. =42 ct.; Harry recenes 31 ct. -"^l ^'t-*? ct.. 'l*^*i 70 E. first ix. t>. ARITHMETJC. man jumped 27 ft. in thre .l"n)p was 2 ft. shorter tlmn the th,r.l, but 1 ft, the seccnd. Find tlie lengtl ■ f each. « Huccessive junipri TheJ onger than' Int. 1ft 1ft. 7 "_8j:j. 8 ft. 8 in,' FoK.M Of CAL(^rLATIO,v. 2 ft. 1 _^ 7 " 8 in. J ft. d in. 10 ft. 8 in." -^A ^.-Divide 45 apples between Annie and Hu 3 to Annie for every 2 t.. Harry. Give to Annie and to Harry out of every Nt)w FoKM OK Calculation.. 3ap., 2 " 5 " fi ap. )45 ap, 9 times. Hence give to Annie 3 ap. x 9 = 07 .^p and t,. Harry g u ><9_]g\V' EXERCISE XVII. 1. Divide 24 marbles between Henry an.l Fr1«,o 1 ., may have 4 more than Edward. ^''^ '° *'^^* "^^''y '?. Annie and Jane together havp 17 ni,;„i than Jane. How many Ls eachT "' ' "^""'^ '^"^ '' ™°^^ .?. Robert has 6 pigeons more than Donald- tn„ofi, * 20 pigeons. How many has each ' ^^ Uonald, together t . ave 7mi.t:n,::^t^^s:: tr r- '^^^ ^^- ■ -ai.ed day? «Hlth.hrst. How many miles did h. .,, ', ,ach Uiot ' mor( 10 I men jhegi 11 I $150 ■>'' giving I gr^; inves 27 ft. 3)28 " 7 ft. 8 in. SHARING. 71 Huccessive junipri The 1, but 1 ft, longer tlian >,v. 27 ft. 7 ft. 8 in. ie and Hairy, giving dward so that Henry ns; Annie has r> more ; together t ave :, giva r-;-, ;. 18 ^j^ econd fk ^ ^ walked ilea did h. v ■ >, ^ach the'secild .' ^' '"''"'''" *''° ""' ^'"'^"^ ^'^^ ^^^* ^« ^'- "'"- *>-n 9. Divule $7770 between a college and a hospital, giving $2000 more to the college than to the hospital. i^. A merchant gained $79.55 in two years. He gained $114.3 mere during the second year than during the first. How much y express in accent ckboard. c. I measure. l.TxT. .9. 3"x3' 2. 3" I'xfi" 4. r)"x4". /;. I'xr ^/. The floor of 7. VxV. in. In determining the numbfr of yards of carpeting rwjuired for a room, Jird liceiile vhcther the drips nhall run l>ii strips will therefore require 20 ft. 5 in. X = 122 ft. in. = 41 yd. all but in. Tl}fre will therefore he Jfl yd. of carpetintj required. Had 7 in. instead of 5 in. per strip been required for niatfhin"!', the length per strip would have been 20 ft. 7 in., and the lenffth of the 6 strips would have been 20 ft. 7 in. X = 123 ft. 6 in. =41 yd. C in. But in. can be spared off the last strip, so that only 41 yd. would be required. EXERCISE XIX. /. How many strips of carpeting 30 in. wide will be required for a rectangular floor 22' x 15', if the strips run lengthwise of the room ? J. How many strips of carpeting 27 in. wide w' ' be required for a rectangular i]r:nx- 24' « ],3' ?•", 'f the strip?; nm !"?igthwisc of the ? room ? What width will have to be turned under ? I is Bold by tlu' ynitl of n. and 'M in. aritotiiiLj I'tMjuired for rioi It iKjthii'isr of the )f utrips ihrdvd. The of strips trill (jim the Tiining tho Icn^jfth of Kntr in mati'hinij the. ; 27 in. wido will bo I ft. , if the strij)8 run r matching '} strip of floor 21 in. e carpeting, making will bo too wido by 3 in. wide will there- be added to this for . 6 in. C in. req\iired. nintohinfr, the length per i strips would have l)een Cin. yd. would he required. e will be required for ngthwise of the room ? B w! ' be required for sin lengthwise of the ider ? MKASIHEMi;\T.S. 77 X How many strips of carpeting .•};{ in. wi.le will be required for a rectangular roo.n 2.T !)'x ]8' 9". if the .strips nu. across tl.e room ? How many strips will h. required if they run lengthwise of the room / How nu.ch will need to ho turned under in each case? V. How many yar.ls of carpeting 40 in. wi.le will be required for a rectangular room 22' 8" . bV ir. if tl.e strips run long hwise of tlR. room and 9 m. per strip bo wa.ste.l in matching' ... How many yards of carpeting 27 in. wide will bo re.aured for a rectangular roon. 17' 6" x ,4' o', if the strips run across the roon. and 11 m. per strip be wasted in matching' wi the number of strips 21 in. wide required to go round the room, leaviruj out the full width of the doors and the windows; a double-roll, or two single rolls, vjill be required for every h strips. Rmmple.— Row many rolls of wall-paper will be required to cover the walls of a rectangular room 22' x 14' which has two doors and three windows, the door-frames being 3' 10" wide each and the window-frames 4' 3" wide each ? The perimeter of the room is (22' + 14') x 2 = 72' The width of the 2 dtjors is 3' 10" x 2= 7' 8" The width of the 3 windows is 4' 3" x 3 = 12' 0" The total width of doors and windows is 20' ,5" Deducting the 20' 5" from the perimeter 51' 7" 51' 7"-r21" = G10"H-21" = 29 times and 10" remaining over. Hence 20 strips Avould not be enough by a strip of JO"; there' will therefore be 30 strips needed. 30 strips -=G (5 strips) = (} i/ouble-rons = 12 single-Tolh. EXERCISE XX. 1. How many rolls of wall-paper will be required for n. rooml IS' 6" X 15' 4", making deduction for 1 door and 2 windows each! 4' wide and 1 door 3' 8" wide ? .. How many rolls of wall-paper will be required for a room ofj ordinary height 23' 4" x 14' 5", Avitli 2 doors and 3 windows eacbl 4' wide? 5. Find the cost of the wall-paper at 75 ct. per roll for a rooirj 21' 8" X 13' 6", with 2 doors each 3' 9" and 3 windows each 4' 2" wiile,' 4. Find the cost of the wall-paper at 45 ct. the roll and borderin J at 10 ct. tlie yard for a room 27' 9" x 17' 3", allowing for 2 doors each] 4' 2" wide and 4 windows each 3' 10" wide, (ilic allowance for doorj and windows is made on the paper, but not on the bordering.) 5. If a roll give only ^t-o strips, and 9 strips be deducted for dooni and windows, find the cost of papering a room 23' 6" x U' with papoi! at 05 ct. per roll and bordering at 7 ct. per yd., hanging the papeij costing 15 ct. per roll. 1 but: 1 son in t I lis s S ;u imn tanj [4"> [deni 6 [3 Ideni r4 in 1 c. S R( 10. I Ej 13 14 15 16 17 IS 19. 20. 31. 22. 23. yd, lonj ami in double- 1 being 21 in. quired to paper a room '/j3,s 21 ill. ivide required tvidth of the doors and oils, loill he required for per will be required to 22' X 14' which has two s being 3' 10" wide each MEASUREMENTS. AREAS OF RECTANGLES. 79 + 14')x2= 72' .T10"x2= 7' 8" 4'3"x3 = 12'ir i« 20' 5" r ^^ 51' 7" 10" remaining over, by a strip of iO"; there s — 12 .s/>tf//e-rolls. l)e rerj aired for a, room 3r and 2 windows each •rs and 3 windows each ct. per roll for a rooix: ivindows each 4' 2" wide,' t. the roll and bordering dlowing for 2 doors each (i'lic allowance for doorsj on the bordering.) ps he deducted for door.' 'm23'6"xl4'withpapei yd., hanging the papei The Area ol any surface-figure i.s the measure of the surface enclosed by tlie lines which bound the figure. The numerical value of tlie area expresses how many times some chosen surface-figure, called the unit of area, is contained in the measured figure. The unit of area generally selected is a squaue who.se .side LS SOME .STATED TTNIT OF LENGTH. Square brackets [] enclosing the dimensions of a surface- igure denote that the figure is a rectangle. A number written [immediately outside the brackets denotes that number of rec- ■;angles of the dimensions noted within the brackets. Thus 4" X 3"] denotes a rectangle 4 in. long by 3 in. wide ; [1' k 1'] 'denotes a square 1 ft. long by 1 ft. wide— that is, a scjuare foot ; 6 [3' X 2'] denotes rectangles each 3 ft. by 2 ft. ; 4x5 [4" x 4"] denotes 4 times 5 squares 4 in. by 4 in.— that is, 20 squares each 4 inches square. EXERCISE XXI, Read ti:ie following and draw the figures denoted:— [3"x2'j. 4. 2[l"xl"]. 7. [I'.Tx?"]. [l"xl']. r.. 3[2'x2"]. ,v. [l'2"xl'l']. 1. o i>. 2[l'G"x4"]. 3. [3"x.3"]. (J. 4[4"x2"]. Read- required for a room of|m [2yd. x 1 yd.] 11. [13 mi. x 22yd.] l.j. [12 mi. 880 yd. x 99ft.l Express the following in bracket notation :— 13. A rectangle 8 in. long by 5 in. wide. 14. A rectangle 1 ft. long by 3 in. wide, 15. Three rectangles 7 in, by 4 in, 16. A rectangle 4 ft. 3 in. long by 2 ft. wide. 17. A rectangle 2 ft. in. long by 1 ft. 9 in. wide, IS. A rectangle 25 yd. long by .5 yd. wide. 19. A rectangle 20 mi. long by 100 ft. wide. !20. A square inch. ^^. Six square inches. 31. A square foot, oj^ Six inches srju.are. ,?J, A square yard. ^ff. Three square feet. 23. A square rod. i?7. Three feet square. 80 AUITH.METIC. Example.—Let the figure ABCD bo a rect- angle whose length AB is 4 units and breadth AD is 3 units. Mark off AB into 4 parts, Ac, <'/•' f, vfhh,f-- 5x 3[l"xl"] 3. [6" X 4"] - 6 [1" X 4"J = 6 X 4 [1" x 1"] 4. [2'x2']= 2[rx2']= 2x 2[J'xl']. J. [I'x l']-12[l"x l'] = 12xl2[I"xl"]. G. [1 yd. X 1 yd.] = 3 [1' x 1 yd.] = 3 x 2 [1' x I'J. 7. What are the dimensio.is in inches of a square foot' How many square niches are there in a square foot? <^. What are the dimen.sions in inches of a square yard ? How many square inches are there in a square yard ? ,9. What are tlie dimensions in yards of a square mile? How many square yards are there in a square mile? JO. What are the dimensions in inches of 10 ft. square? How many snuare inehes £ira tlioro >" 'ft ff c-«n^^9 ti .,■',, .' ' It. square? How many square inches are there m 10 sq. ft.? MEASUKIJ.MKXTS, 81 I ruct- 'eadtli s, Ae, h e, f I'iding ii h I Fro.u the preceding examples we n.uy obtain ti.e follcwin-. ■ule for deternnniiiy the .-.r,"!, ,,f i r,.f.f..,wri .1 r ■ .re given •— =- ' ' '■ "^ <>■ lectangh^ wliose dnuensiona ^^^ Express the length and the breadth of the nctanr,Ir in units of 1 unit wide by 3 unitsi 4,,^' f.!' T'^f, "^ '""'^ '" ''" ^"•'"'"^" '"''^ ^« ''- liMBLR ,/ s,inare unds of that denomination in the area. t X 3 units.] 1 Hence also, if the nnmh.r of square nnits in tU am. of a ^«.^..,?. he d.raM hy the nn,nher of lin.ar vnits of tl. sLe . the quotient vill be the n near nnds of the same denomination in the other sid. D each one unit long, m,,,^^^^^,^;^.^^ ,.^^ . , ;^ ^- - -'•; ■""- -j uw sa.,e 1, dividing each of the|,(e«,. uin't. nf h' !1Z ! 'i'y_«^'«;* ^'"'^^ ^« the number of [iiaros. Each of these '" ' "^ 4 rectangles together 1 EXERCISE XXIII. ts} : the figures and express ape/l7x7.r f ''"" ""'" "■'' ^''^^^ '" ^ rectangular sheot of ibie??;^?"'' ''"" '"' '" *'"' "' "" ""'^^^^ ^'f '^ rectangular .onus" "7 ''"" '"' ""^ ''"" "^ *''^ ^'-^'- <'^- ^^ '-^-S^'l-- 4. Find the area cf a blackboanl [24' x 4'] ot.'xfi6T' fow many lots of this size would be equal in area to an acre V ' ^a^ square foot? Howl Find^the a^ea in acres, etc., of rectangular fields of the following a square yard? Howl 9. 25 rd. x 16 rd. .. 40 rd suuire /-- A 00 ^ ard? ■//) OA^A OA 1 ^- -*" ra. square. /,. , / yd. x 33 yd. ■^^- ^* ™. X 20 rd. i/ 23 ch y in ,.1, /... io/> 1 ■ .0 ft. .,„„,, „„„.|; ; : ■ r^ ,„ ;"■■ """"r^'""- * ""'■ """-«• Htjvv many square »<>> 00-,^ v "q ,„i z-:? m. - IJ ch. .;> in. k*. 40 rd. 3 yd. x 10 rd. 2 ft. ,^6. 12 oh. 50 In. .quare. 6 I'J 82 ARITHMETIC. Find tho area in »({. yd., etc., of rectangles of the foUowinj dimensions:— 27. 12'3"x9'. ,-?,. The top of a table is a rectangle 30" wide and its area 10 sq. ft. What is its length ? 4< - = ''4' «" I J M- f J ^'^® height of the walls above the skirtin<' is Is and ceding of a rotri 'i"- .>miliii<, is I 9' 10" -10"= 9' „ 11 c v VI- 11 I The total wall area is r74.'S"vO'i .'»-.> ^^ roll of English wall-paixl mi .,, , L'* « X9j = »>72sf). ft. I The area of the ceiling is [24' x 1 ,'3' 4"] - ;3'>0 s. ft The gross area is 072 sq. ft. + 320 s,,. ft. = 992 scirft! 1 The height of the doors above the skirting-board is 7' 4" -10" =.()'(;", Inch is the same as the height of the windows. The area of 2 doors and [^ windows is 5 [C 6" X 4'] = 20 sq. ft. X :, ==: i;jO .s,i. ft. Half of this is 130 s.. ft • 9 c- ex. mi, X • i.>L» .S.|. rt. — .'. How much will it cost at 20 ct. per sq. yd. to paint the wej of a liouse 29' x 22', with side-walls 15' high and gable-peaks visi 9' above the side-walls, counting tlie two gable-peaks equal to oi *'ull wall of equal height'; 3ct, ^rd fJO. wn 29. ps( d th wid ,■?/. w pre ' 1.5 let. hdd and flooring is a Square IG S(^. in., and 9 shingloi I. = 1 sq. ft. At this ratt >ut to allow for waste am KJO shiwjles to the Squarr-i each, so that 4 bmicht a roof 4 bimclies will b or every 25 eq. ft. Henc uired for any roof tJie area of all openings i MEASlRKMtiXTS, 8') U. How many s.^uares of shingling are there in a roof [50' x 20'] ' -A .w T 1 mf: u Tn^"^ '^"^'■'' ^'^ ^^^'^ '"^ ^ »«°f ^^ the form of two rect- in. wide, so that a shmgli iglea each 30 x 16' 8"? ri«' ^"m,'"^°^ ^"°''''' °^ '^^"^'^^^ ^^" '^^ ^«l"ired for a roof IS. How many bricks 8"x4"x2" laid flatwise will be needed to ■ve a rectangular courtyard 48' x 30' ' i:hole numberneurcstto tlimof \ 4.- •., , ,„„ "j»«c(. fet rTs^ " w ? "".';'■ °' "" '° ""^ " """' "* *-' 1 »vei mg [1^ X 9 J. What was tlie area of the roof ' If i ^''' ^°:;', '^ '^ ^*" ^^ t^^' '^ ^«'J- ««^ '"any squares of flooring :IV. #e there in it ? ^ 1/'?^. How many slates at 3 to the qnimrp fr.r.+ ,.„-n u • 1 - lastering as follows : - ,ver 1 7 squares of roof ? ^ ^' "'"'^"''""^ *° X 10' ; two doors 7' X 4' an( £y/ Whnf wJll Kn +i,„ j. c ^4. What wil be the cost of ceding a school-room 37' 6"x 24' at 1) ct. the sq. yd.? '; two doors 7 X ,3' 10", tAv 0/7 xr/«.r rv,„„u „mi -^ i"i. til vfofTlir " ''^' ^'^^ ^^- ''- " p^- ^ «*-^ .::!-at trl^Sn^-^^- ^«^ ^ ^^ -^^ t^ere be in a box of glass con- 7' X 4'. 3 windows G'6" x 4' J %' How mf"^ r"' T-n ''" ^°"^^ *^^^^ '« ^'^ ^"^'^ ^ '^^ ' [<-"x4"x3"] denotes a quad G in. long, 4 in. wide and 3 in. thick ; 4 [1' x 1' x 1'] denotes 4 cubes 1 ft. long on each edge — that is, 4 cu. ffc. Let the figure ABCDEF represent a quad whose length AB is units, breadth BO is 4 units, and height CD is 6 units. Mark off AB into C parts, BG into 4 parU, and CD into 5 parts, each part equal to a unit of leiujth, and througu the points of division draw planes cutting the (juad into cubes. Along AB iliere are 6 units, hence there will be 6 slices like BCDfjhk. Along BG there aro 4 units; hence in the slice BGDghk there will be 4 columns like Blmnhk, and as this column is 6 units high there will be 5 cubes in it. Hence in the whole quad there will be 6 slices, each con- taining 4 columns of 5 cubes, or 6 x 4 x 5 cubes — 120 cubes in all. Hence 1 [6 units X 4 units x 5 units] = G [1 unit X 4 units x 5 units] = 6 X 4 [1 unit X 1 unit x 5 units] =6x5x4 [1 unit x 1 unit x 1 unit J = 120 cubic units. MEASUREMENTS. 87 s the measure the figure. w many times . contained in CUBE WHOSE , solid denote er written im- er of quads of s [G"x4"x3"] 4[i'xrxi'] cu. ft. se length AB 'D is 5 units. g / A 7 7 A / /) /\ ._. / ' / — / / / / 'Y Y k B rill be 5 cubes ces, each con- cubes in all. .5. 3[8"x4"x2"]. 6'. 6[2"x6"x2"J. EXERCISE XXV, Read the following: — ^. [l"xl"xl"]. 3. [4"x3"x21 ^. [3"x2"x2"J. ^. 5[3"x3"x.3']. Express the following in bracket notation :- 7. A quad 8 in. long, 3 in. wide and 2 in. thick 8. A quad 4 ft. long, 9 in. wide and 4 in. thick 9. A quad 16 ft. long by 10 in. wide by 3 in thick 10. Five quads 12 ft. long by 6 in. wide by 3 in. thick n. 4786 quads 8 in. long, 4 in. wide and 2 in. thick 13. A cubic inch, i^. A cubic foot. 13. 24 cubic inches. js. 20 cu yd ^^16. A four-inch cube-that is, a cube of which each edge is 4 in. 17. Seven 2 ft. cubes. Prove tiie following statements by cutting the solids and express the final results in cubic measure :— 18. [3" X 2" X 2"] =3 [1" X 2" x 2"] = 3 x 2 [1" x 1" x 2"] = 3x2x 2 n "x l"x I'T 19. [4"x3"x2'a=4[l"x3"x2"]=4x3[l"xl"x2"] ^' = 4x3x2 n"x l"x 1"! ^0. [6"x4"x3"] = 6[l"x4"x3"] = 6x4[rxl"x3"] ^' oi TT-u =6x4x3 [l"xl"xl"]. ^1. What are the dimensions in inches of a cubic f? t -,.k:- __!i_ .. »-. by the bracket notation a cubic mile in yard dimensions. Redi^e to cubic yards the cubic mile thus expressed, explaining each step HS ARITHMETIC. From examples such as 18, 19 ami 20 of the preceding exercise we may ohtaiti the following rule for determiniug the volume of a (juad whose dimensions are given: — Expreas tlie leiujth, the breadth and the thickness of the quad in units of the same denomination ; the nmtinued product of the NUMBER of units in the length, the numki il (f units in the breadth, and the number of units in ilie thickness loill give the number of cubic units of that denomination in the volume. Hence, also, if the number of cufiic units in the volume of a qu.ul be divided bij the prodnct of the numbers of linear units in any two dimensions, the quotient will be the number of linear units in the third dimension. The unit of measurement of excavations and embankments is the cubic yard. A cubic yard 'jf earth (.s caJVd a load. Hewn timber i-- i^nn; rally measured by tlie cubic foot. Lum- ber of an inch or iMom of thickness is measured by the board- foot, which in [1 x.i'xl"], 12 board-feet making a cubic foot. Lumber less than ^n iacli thick is reckoned as if it were an inch thick. Bricklaying is estimated by the thousand bricks, determined either by actual count or else by reckoning 22 bricks laid in mortar to the cubic foot. Masonry is generally measured by the cubic yard, but sometimes by the perch. A perch of masonry is not a fixed measure, but differs in different places. In measuring the materials in walls, deductions must be made for doors, windows and all other openings. A gallon of pure water weighs 10 lb. A cubic foot of water weighs 1000 oz. and contains 25 quarts. A ton of anthracite or hard coal measures SS cti. //. A ton of bituminous or soft coal measures J^ cu. ft. EXERCISE XXVI. 1. How many cubic inches are there in a brick 8" x 4" x 2"? 8. How many cu. ft. are there in a rectangular box 4' x 3' x 2' ? 5. How many cu. ft. are there in a rectangular bin 8' x 5' x 4' ? 4. Howmany cu. ft. are contained in a pile of cordwood 8' x 4' x 4'? 6. How many bricks 8" x 4" x 2" would measure a cubic foot ? MKASUKEMENTS. 89 )rececling exercise ny the volume of ona must be made Find the volumr,.s of quadrate solids „f the following dimensions :- 7. 2' X 6" X 4". 4'y o' ft " V I' «" // , .r , • 24t'l5 "l^' '"''" ""''"'' "^ ' ''''^ '^^ ^'l"'^^^ "'"be; celiar IZ^^^ZT ''' ^'^ *'"" '" ^ ^^^•*'^"«"''"- ^'^^'^-^-'^ ^^ ^ mt fr'.*°"« °; ««ft coal can be put into a rectangular bin 7 ft. long, 4 ft. wide and .3 ft. deep ? W. A rectangular cistern is 6' x 4' x 4'. What will be the weight .U A rectangular bin 8'x6'x4' is full of wheat. How many l.ushels of wheat by measure are there in the bin. and how much Mould the whole weigh at 61 lb. to the measurea bushel' JI: 5T ""''"^ ''"''^^ "^'^ ^^ required to build a wall 124 ft. lone %. "% '<«> N*', 4>\ ». Q c\ 4^ '%' w 90 ARITHMETIC. How many feet, board -measure, are there in — 39. 1 board IC long, 12" wide, I' thick? ::i! /JO. 3 :( 12' (( 10" (( 1' SI. 12 i( 10' (1 «" (( 1' s?.. 150 ( \ 16' (( 10" ii 1" .w. 720 1 ( 14' i< 8" (( 1" ,?4. 45 (t 16' i( 9" (( 2" S5. 24 it 16' If 10" (( 2" .%. 40 planks I( 10" (( 3" ay. 2.-) << 18' K 9" (( 3" ss. 24 scantlinga 18' ' 7a What must be the width of'. ..^.fc^n.r,,!^-, , • « - , f ft. in. deep to contain 6 T. of h.nloo^T'''''"' "^^ ' ''■ ^"°^' ''^ Wlv CHAPTEK V. FACTORS, MEASUBES AND MULTIPLES. I. INTEGRAL FACTORS. The numbers 1, 2, 3, 4, are called Integers, Integral Numbers or Whole Numbers. They are classified into Even Numbers and Odd Numbers. An Even Numbor is an integer which is exactly divisible by 2. RrampI('H.-—G, 10, 18. An Odd Number is an integer which is not -exactly divisible by 2. Examples.— 5, 9, 21. i . EXERCISE XXVII. 1. Name all the even nunibeis less than 20. ^. Add 100 to each of the even numbers you have just named and prove that the resulting sums are all even. 3. Name all the odd numbers between 20 and 40. 4. Add 100 to each of the odd numbers you have just named and prove that the resulting sums are all odd. 5. Write down all the even numbers between 125 an I 1.S5. 0. Subtract 99 from eacli of these and prove that the remainders are all odd. 7. Write down all the odd numbers between 144 and 154. S. Subtract 99 from each of these and prove that the remainders are all even. 9. Write down all the even numbers between 1001 and 1011, and divide each of them by 2. Which of the (quotients are even and which are odd ? 10. Divide 1056 by 2, divide the quotient by 2, divide the second quotient by 2, and thus continue dividing till you come to an odd number as qu^-tient. This dons, how many diviaions by 2 have you made ' 92 ULTIPLES. INTEGRAL FACTOKS. p^ Th„ Integral Factors of a ...nnhor an- anv intc-^ers othn- tliini, line ,ini f,,' k i/i/j/.,.- ,7 If i , " ""•-f,ers otiu > Tl,„» ;! ,u„I 6 „ro i„k.g,,.l f„ct,„-., „f 16, for 3 x 6 = 16 I,, lik„ Z7 ;'"- "" "'■" '"'°»"" '"^'™ <■' '». for 0x10= W. lave just named aud EXERCISE XXVIII. Resolve the following „u,nl,ers each into a pair of integral ^- «• .'/. 14. ,.; or, ^ .. Resolve each of the following nunahers into three integral ^'' 12- 1-<. 28. /.;. r,0. /7 ;V> "., Resolve the following nun.bers int<, fac-tor and oofactor as many Mays as possible :- <^oractor, '-'■ 12. .>:?. 18. .,, 30. >4. 60. Write ,lown all the number.s less than 5.3 of which- - y- o ,s a factor. ,,. 12 i« a factor. .'/. Both . and , are Find the least and the greatest integral factor of - A>. 24. ,u 847. .i^. 725. ,^, 1,13. .,,, factors : 51. i;?.3. factors : 170. iOOI. •Mch in 420. factors. :i.'?.3.'?. "itegial factor in co.umon-that is, if no „,t,ua-al factor of the "MO IS an integral factor of the other ' Th„3 21 is prime to 10, for the only integral factors of 21 are , 3^ 7, .na neither of these is found among the inU^J^Z I 10. But 21 IS not prime to 12, for 21 = 3 x 7 and'l2 = 3 x 4 ■ nonce d IS a common factor of 21 and 12. 94 ARITHMETIC. i I ^« I A Primo Number or Prime is an integer that has no iu- tegriil fiietors ; it is, tliorofore, prime, to iiU. integers less thun itself. A Composite Number is an integer that can be resolved into two or more integral factors. Thus, of the numbers less than ten, 1, 2, 3, 5 and 7 arc primes, but 4,0,8 and are composite ; for 4 - 2 x 2, = 2 x 3, 8 - 2 x 4, ana{) = .'{x3. EXERCISE XXIX. Which of the following numbers are prime and which are com- posite:— 1. f). S. 11. f>. 35. 7. m. 9. 91. 2. 5. 4. 27. 0. 39. 8. 63. li>. (>1. 11. Write down all the prime Tiumbers less than flO. JJ. Which of these primes ui c not found in the common nmltipli- cation table extending to 10 x 10 ? Why ? J3. Write in a column all the composite numbers between 31 and 01, and opposite each number write its smallest integral factor and the cofaator of such integral factor. 14. Which of these composite numbers are not found in the com- mon multiplication table extending to 10 x 10 ? Why ? 15. Which would not be found in tlie multiplication table extended to 20x20? Why? Id. To .30x30? Why? Which of the following pairs of numbers are prime to each other, and which have a common factor ? 17. 24 and .35. IS. 40 and 66. Jf). 91 and OS. JO. 231 and 260. A Prime Factor is a factor which is a prime number. To resolve a composite number into its prime factors — that is, to find the prime nund)ers, each repeated as often as necessary, whoso product is ec^ual to the given number — Divide the gioeti inmiher hij a nij jyrime factor. If tlie quotient he composite, divide it in lihe tiuDDiv.r, mid so omtinue until a prime qtudient is obtained. The several divijors and the lad quotient put into the form of al cmxtinued prod net trill expresA the re.sulntimt of tlw ijiiH'u- t)>imhi'i into its prime ftctors. at can be resolved *nd which are com- INTEGIUL FACTons. do In ^3 .n« for a pnme factor, it is le.t to tnj tkr ,.,,...■ n.unl.rs .nrticul.;i!; ^;?'f ''.';' ^'^'^^"""'^'^vitlx the H,uallest, l.oip.. st./"uo tf ; f '"'•^ '^ ""^^ '" '^^^^'^ "^ i^--V./;befor: p.issing on to tlie next larcer. U^amplc-Re^olvo 12G0 int.. its pri.no factors. 2)12(50 2)(;30 3)315 3)105 0)35 7. Tlierefore 12(50= 2 x 2 x 3 x 3 x 5 x 7 prime to each other, ;. .JO. 231 and 200. 'ih' uruimier, (md 8v\ EXERCISE XXX. Resolve into prime factors — 15. J. 12. .?. 36. 4. 108. ''. 112. G. 90. 7. 128. cV. 324. !>. 252. IC. .'55.-). //. 53!). I..', mo. lli. 1089, I'f. 289. io. 437. 16. 24. i7. 45. i^. 60. ,,, 144. ..,. ,40 i[; l'!^'] ^" *^^ ^*«gers less than 16 and prime to it ;;■ * ! , , ^^"^ iritegevs less than 36 and prime to it. i; f/"u *^^ '"^^^^"^ ^^'' *^^"^ 48 ami prime to it ~^. If the number of integers less than 16 and prime to it be mul tjphec by the number of integers less than 3 and'prin. o t prvo" h. t the product wdl be equal to the number of /„teger« less than 1 48 (= 16 X 3) and prime to it, L f ■ ^T *\^V^ *^'^ ""'"^''" °^ ^"^^^^'-^ '^^'^ th-^" ^' and prime to l^ the produc wdl be equal to the number of integers le'ss than 140 ( = 9 X 5) and prnne to it. I7 wm LT? 'T ^f ?' "^^'^ '° '•" P'"^"'^* «^ *•- «^«* five digits |/ will be a factor of the sum. ° kvil7i ^""/^^^^VV" ^"^ '^'^'^'^'^ **' ^''*' l*'^^^"''"* "f *he nine digits U p ill be a factor of the sum. 'b"^ « 1 96 AllITHMETIC. I ' ! ' Mi 1 il 1 1 1 i' % 1 1 'A II. MEASURES. Ono number is a Measure <>f another number if it is con- tained in tliat other an exact number of times. Thus 4 is contained 5 times in 20, irithout remainder; there- fore 4 is a measure of 20. But 4 is not a measure of 2;>, for on dividing 23 by 4 there is a remainder, 3. The reason for calling 4 a measure of 20 but not a measure of 23 is this: With a rod 4 ft. long with no divisicm marks upon it, you could measure off a length of 20 ft., but not a length of 23 ft. Similarly, using only a 4-lb. weight you could weigh out 20 lb, of a commodity, but not 23 lb. ; with a 4-pt. measure you could measure out 20 pt., but not 23 pt, ; with nothing but four-dollar bills you could count out a sum of $20, but not a sum of 823. A Common Measure of two or more numbers is a number which measures each of them. ^ Thu3 is a common measure of 24 and 30, $5 is a common measure of $25 and $-iO, and 1 ft. is the only integral common measure of the three lengths, C ft. , 10 ft. and 15 ft. The Greatest Common Measure of two or more numbers is the GREATEST number that measures each f the. circle, .although botl!' quantities are areas. MKASL'UES. .Lu'l*"-^''™'"-"-' "'" «■-' -"'-» i" a I.ori...„t.l line, 2 3 3 5 i^ 1155 1576 315 105^ 36 ~ 7' ■•?'•'/ .sf*-/).— Use tliu T.riiiie factors of a-m ^^otors of the other nu.nbor. 155 lat^ ""T"^ 'r'' I triul-factors that fail as -u-f...! f ' ^'"^^«"'»S t'^ose I either 1155 or 1575 thus '"' " ^""^"'' ""^^--" "^ '^ 1 .630_1L55__1576 3 1 315 1155^J576 3 1 105__38o__525 ^1 7 _77_ 35 1 1 11 " 6 Hero 2. tl>e smallest ,irin,e factor of 030 is not . actor of either 11.5 or ,575 ; therefore ea, e ": „' .nn, 1155 un.I 1575 ,lovv„ to the first hn f", Z tients. The next trial-faetor is ;{. It is a fll ; "oth 1,55 and 1575; therefore .11 ide ench o t :;un.bersh,aandwritetheir.,„otie!:t;;S:^.5r -"'» '.hutely heneath then, i„ the «ee.,nd line of -luofents The next trial-divisor is the see!:,'', f •'-'K down 385 to the next ^ :;*;^Jr' S' ' ^'"t" '^^"'•^' ''"' ' -" factor of 525; so divide 525 l.v •Jam wW «?>. "'"'•^■"'-d 3 is, however, a -« irn; therefore divide'hoth o tse n^S "r liV't ," ^- '''"''' °' ""*'' ^^«'^ [. and 35, immediately be,.eath thenr The 'If •- /" ""'''' '''''' ''"°*''="t«. Jf -oth 77 and 35 ; therefore divide both of l^:^ l:::^-:^ V' "'"' " " ''''''" ^m^tep.-Coneot the nncanceU.l factors and form their pro- 3x5x7 = 105. Jienalt. —105 is the G. C. M. of 030, 1155 and 1575 . II i i I Jl 98 AUITIIMETIC. From tho preceding example wo may see that this method of finding the G. C. M. of two or more like inti-gral numbers may be stated as follows :— Arramje the given numbers in a horizontal line. Resolve one of the given numhera into^ ittt jmme factors. Use these prime factors as succkssive trial-factors of the other given 7iumbers, cancelling those trial-factors that do not measure, in PROPER SUCCESSION, GVery one of the given numbers. The product of the uncancelled factors and the common unit of THE GIVEN NUMBERS will be the G. 0. M. required. If the given numbers are prime to each other, their G. C. M. is their common unit. If the given numbers are unlike, they must, if possible, be reduced to equivalent like numbers. If such reduction is not possible, the numbers are incommensurable. EXERCISE XXXI. Write down all the integral measures of— 1, 6. 2. 12. 3. 20. 4. 30. 5. 84. G. Write all the integral measures of 24 in one line, all those ofj 36 in a second line, and in a third line all the measures common tcl the first two lines. Prove that the third line consists of all thel measures of the greatest number in it— that is, of the G. C. M. ofj 24 and 36? Form similar tables for- 7. 36 and 48. S. 45 and 60. .9. 108 and 144. 10. 84, 126 and 210. EXERCISI] XXXII. Find the G. C. M. of 1. 48 and 50. o 60 i( 75. .?. 45 {( 72. /. 1*20 ( ( 150. />. 210 <( .350. G. 440, 770, and 1210. 7. 560, 1008, " 1232. 8. 980, 1380, " 1960. .9. 1.386, 2:68, " 3150. 10. 1820, 6370, 8099, and lOlOlJ MEASURES. !)9 The i.recocling uiothcul „f (i„,ii„„ ^],^ <-, ., ,. . ^ numbers requires one of tl.,.,., T i ," ^ t^v„ „r ,n.,ro •litticult of resolution i„f,. fi tJ^* given numbers are « *-;»;;* »^'. »•« «-- ■.- .... ..... „.,...„, „ ,,, „.„ ,.,.. 51 Ond stej,.-Dniie 3r4, the divisor, by 51, the re.nainder " )37_4( 7 The remainder is 17, and therefore the (lev,,, . CI is also the G. C. M. of 61 ard ri , "' ^^ "'"^ G. C. M. of 374 and 2295 ' ' ''•°"««'i"«"t'.v the Sra .tep^^uio 51, the second divi.o. h, 17. the second ren.ainder 1. ).l (3 There i. no remainder, an.l therefore 17 i. the (; c M of 17 and 51, and conse,,uently of 374 u>id 2l'!).1. The steps of the calculation ^lay he collected thus— 61)374(7 357 3^.7 17 374)2205(0 2244 2{4 1(2 2nd arranijemcnt. I r24(>15 ■2.'«04 1311 18082() 1311 ^ro. 41)' 3033 ]03!M; < ^ 1311 1210 1 1 13 4 1210 i>2 23 02 02 209 =0. C. M. 270 01 1219 From tho above we may deduce the following rule: — To find the G. G. M. of tivn like nnmhers, divide the greater hy th'\ less; then the divisor hy the remainder, if there be any; thm thefir>^ remainder hy the second remainder, if therr be any; and so continnk to divide until there is no remainder. The last divisor will be th>\ G. G. M. required. Should it be required to tiud the G. C. M. of more than twc numbers, find the 0. C. M. of two of them ; then of this measure and a third of them ; then of this second measure and a fourtlj number ; and thus proceed throughout the given numbers. Tli( last measure found will bj the G. C. M. retniired. The two methods of findiii^r the G. C. M. may oftuii with ivdvaiitajfe he combine fj,^,„ i„ p., ._> above, the factor ". may l)e divided out o( 724(115, and then cancellt not being a factor of 180d2(). So 2 may be divided out of 180820 and cancelled. MKASirUKS. 7'-'4«lir» 101 //. ll'.'{41 ami 30401, IJ, 3SUS3 ami 80497. 13. 40603 ami 1)29!)». U. 4;)344»3aii(l 73>S1S30, U. 10330008 and 12220272 IG. 68, 102 and 153. i7. 1028, 28,S2 and 4.')43. /.v. 5040, 7770, 9012 and 1077.3. 19. e.'<.33, $.37andS8.r,l. SO. 1 mi., 1 rd. and 1 yd. 2:J-(}. 0. M. p. , „ „ EXERCISE XXXriI Find the G. C. M. of— 1- 111 and llim. ~. 279 ft, and 217 ft. .7. 830 lb, and 920 lb. 4. 0993 gr. and 8991 gr, -''>■ 1001 bu. and 2001 bn. «'>. 0307 gal. and 10812 gal. 7. 6;j4yd. and 034 rd. <9. 12341 ft. and 1394 rd. 0. 100 11 and 140909. 10. 14003 and 10013. ^/. 4^q.ft..3 8q.yd„2m,.rd.andlA.' ^T Divn' tl^"""'' !'" "'• '' "**'"• '^'"^ '^ -«k holding 25 gal -.7. Divulc the nu.nb.rH given in problem 9 aWo by their (f (' M .^-^. In problen. 9 above divide all the remainders by the G C AI I'lotienta ia prime to tho r C vr p*. , first and second or T« ,V :°,^*^- ^^•''^*''«««'''ond and third ti notion ts I ^6. In problem 18 divide the given numbers by their G C M M the greater hj tlM,,.,, fhrther^is^no '""'""^- ^"*''"*' "^*° ^'^''''^ P"-« ^-t-« . 5. «n,; , aii(i then caneellt )f 180820 and cancelled. . ii !! 102 ARITHMETIC 3. Three planks measuring respectively 12 ft., 16 ft. and 20 ft. in length were cut into the longest jiossible pieces of equal length. What was the length of each piece ? 4. A man had two loUs of bank bills, all of the same denomina- tion, one roll worth S140 and the other roll worth $27"). What was tlie denomination of thi bills if they were of the highest denomina- tion possible ? 5. A farmer had 18 turkeys, 36 goese, 54 ducks and 66 pullets, which he wished to send to market in coops, each coop to contain the same number of fowl, and all those together in any coop to be of the same kind. What w;v8 the greatest number he could put into a coop, fulfilling these conditions, and what was the fewest number of coops i-equired ? G. A farmer drew to market 1200 bu. of wheat, 864 bu. of barley and 1786 bu. of oats, each kind by itself, in bags of the greatest possible number of bushels. How many bushels did he put into each bag ? How many bags of each kind of grain did he draw to | market ? 7. Two vats contain respectively 7875 and 16,128 gallons. Find) the barrel of greatest capacity that will completely empty each vat. 8. A gardener bought three rectangular lots of ground— the first 72'x)44', the second 99' x 128', the third 126' x 96'— and divided them into rectangular beds all of the same length and all of the] same breadth. What was the greatest possible size per bed? 9. Two distances of 901 and 10;]7 miles respectively are portioned! off into daily journeys of equal lengths. Find the smallest number! of journeys into wliich these distances can be porti ned ofiF. 10. A farmer drew to market in loads all of equal tvdght, 385 bu. of barley, 270 bu. of rye and 196 bu. of wheat, drawing each kind of I grain by itself and making as few loads as possible. How many I loads of each kind of grain did he draw, and what was the Meightl of each load ? 11. What is the greatest length of the rails (all to be of the samel length) that can be used, without cutting, to enclose with a post and rail fence a farm 3588 ft. by 2880 ft.? How many rails will l)e| re(iuircd if the fence be 5 rails high ? 1.2. A man noticed that he had made an exact number of steps,! all of the same length, when he had walked 20 ft. 3 in., 27 ft.,| 33 ft. 9 in. and 49 ft. 6 in. What was tlie length of his steps if j they were more than 20 in. long each ? MEASURES. lO.S vhat was theweiAlitl »e*ri.e,. K„d the fcwe.t „„,„,,„; o"' ^tej he oo ,n"° '""■!,"' have used. ^ '^""''^ possibly i-^. A rectangular courtyard G yd. 2 ft. 7 in bv ", vd 9 f f - • • to be paved Avitli rmiar^ tilp« v- i xu , ^ ^ ^ "• " "^- "^ tue.aLu, „„ ,^<:;::l:;„f;r^l:r.,:,r"'^"°'"- 17. h ind the greatest number that will divide 2.5 and •?« 1. • the remainders 1 and 2 respectivelv Wh.f ., ' ^'^"""^ on dividing by i, leave thesfrellnderl^ ' "^'^ ""'"'^^ ^^^''' 18. Find the greatest number that will divide 590 and Qsr i • the remainders 4 and 6 respectively -> ^^'^' '"^'''"S J144, ^140 and 3148, leaving the remainders 19 o, ,,,,,, .,,, ^, ::t L/; '"-^ --^ '-'^^-^ ■"^- ''- "-- '-^' « -y -:: s "» o„.f. :.'™'^',""l""="™l)""i final remainders, ^^. A chest of tea weiLrhinir 70 IK ,..oc i . ' equal weight. Oa attempZg to l^e ,p T' - 7 '"'"'-'"'S"' »' ..e3 of the .™e ,„i,.,.t as t,.e^„rr;X« ;L ib tl"; ro:^- " liat wa, the welglit of one of tho padia-e/' '"•""'''«" »> «••• ...mber of paekage,. Fin.l the weight „( eaoh'^X' n'lt ."^ *'T T-" "P '"■" '"■"•" "' ""'"''''I """'aining 70 lb an,l .4 1 >. respectively into packages all of the same wei„l,t ^, , , ,'b I »P the packages he fou.ul he I,a,l II,. „v.r f„° "t ., ' , ., ' ? j 1«S, Imt nothing over from the two l,u»s toge hlr Fin V" 1 '■""■'"," ofapackagean,,the„„mherofpo„„,,soX:;-the,;:aWe:Cf' r I 104 ARITHMETIC. Kx ample. 1, 25 2, 50 3, 75 4, 100 5, 125 <^ 150 III. MULTIPLES. If a number be multiplied by an integer, the product is called a Multiple of the number. Thus $2x3 = 8G, therefore 80 i« a multiple of $2; 4 in. x5 = 20 in. , therefore 20 in. is a multiple of 4 in. If the multiplier bo 2, the product is called the second multiple; if 3, the third multiple; if 4, the fourth multiple, etc. ; the number itself is called the first or pime multiple. If a number and its multiples be arranged in a column, with the multiplying integers in a side column, the result is called a Multiple Table. The Multiplication Table ia simply a table of multiples. If a number be a multiple of each of two or more numbers, it is calld a Common Multiple of these numbers. Thus 12 is found among the multiples of 2, 3 and 4; it is therefore a common rmdtiple of 2, 3 and 4. The SMALLEST of all the common multiples of two or morel numbers is called the Least Common Multiple of these] numbers. Thus 12, 24, 36 and 48 are all found in the multiple tables of I both 4 and ; hence these are, all of them, common nmltiplesj of 4 and G. But no number less than 12 is found in boch tables ;| therefore 12 is the least common multiple of 4 and 6. The words "least common multiple" are usually abbreviated] into L. C. M. EXERCISE XXXV. Form a table of the first nine multiples of— J. 1.3. i>. 14. .1 1.'). 4- 48. S. 245. 6. Form tables of the first 12 multiples of .3, 4 and 6; select thf multiples common to the three tables, and form them into a table o| common multiples. Find a common multiple of — 7. 5 and 6. S. 6 and 8. 9. 9 and 12. MULTIPLES. lo; le product is called 3 of ^2; 4 in. x5 = Example. 1, 25 2, 50 3, 75 4, 100 5, 125 <}, 150 Jled the f 4, the lUod the nultiples integers le Table. lultiples. »r riore numbers., it ibers. f 2, 3 and 4 ; it ia | es of two or more multiple of these 3 multiple tables of I , common multiples j mud in boch tables;) and 6. usually abbreviated 48. r>. 245, }, 4 and 6 ; select th( I them into a table o 9. 9 and 12. 10. Of what two integers is 24 a common multiple ' n. Of what two, 30 ? n. 48 ? jj. (jo ? y ^ loo , of whi!h s^;f.''''"""''; '"f ^^'^' "^ ^ '^"•^ '''■ ^'^'"'^ ^» the integers of which 8 X 12 13 a multiple, and form u table of those wliich are common multiples of 8 and 12. 10. What is the L. C. M. of 8 and 12 ' of whicf l^r '""""u- ";"'''^'' "' ^'^ ""'^ ^- Find all the integers l: muit-lr;; rsr '' '^™ -^ "^^^ ^^ ^^^-^ -'^^ - - multipL • "' ' ""' '"' ^"'^ '"™ ^ *'^ble of its lirst live 1 rove b^ uc ual division tliat the common multiple thus found is a common multiple of 1 2 and 21 . On comparing the definitions of measure and multiple it is other, the second will be a multiple of the first, and he^fore that a multiple of any number is also a muluile of every measure of that number. From fcliis it follows that a factor contained m any one of the numbers, but a factor occur- nng m any one of the numbers need not be repeated on aecount of surable quantities can have a common multiple. Thus 12 hi. is a measure of 60 in., and CO in. is a m^at^pk of . in ^Tn '^'''T^'^y "^^ ^^«ry measure of 12 in., namely, of im., ^m., 3 m., 4 m. and C in. Also 60 in 18 a common multiple of 12 in. and 20 in. , hence all ^^t^^tV^^^II:'-^'''-'''''''-^ niust'bei:il' 't 00 in., but factors of 12 in. which also occur in 20 in need not be repeated, on account of occurring in both 12 in. and 20 in 12in. =1 in. x2x2v3. 20 in. =1 in. x2x2x5. 00in.=lin. x2x2x3x5. lOfi AUITHMETIf, TO FIND THE L. C. M. OF TWO OR MORE NUMBERS. First Method, — Example i,— Find fcho L. C. M. of 9, 24, 27 and 30. 1st step.— Strike out tlio 9, wliich is a measuro of 30, and arrange tho remaining numbers in a liorizontal line, thus — 24 27 30 3ii~5;_ I ; 5 Form the product of all the trial-divisors aiul tho n„n 1. rema.„,n, uncancelled in the last line; this ^rolet t n Tthe L. C. M. of the given numbers- ^ 2x2x3x3x2x2x5x7x5 = 2520. le uncancelled 9 in EXERCISE XXXVL Find the L. C. M, of— i. 4, 10, 12. -. 8, 12, 15. '"?• 4, 8, IG. -7. 12, 18, 30, 45. 4. 10,12,16. 0\ 8.28,21,35. »• 2.3,4,5,6,7,8,9,10,11,12. ^ft 36, 45, 88. 120, 54, 99, 60, 108 70 e^Q Ih 24, 42, .^'-., 52, .S6, 6.3, 273,' 112.' 126, 156 18 ^i?. 69. 147. 115, 1.54. 210. 207, 69.%. S85. " 3.3,9, 12,22. S. 13,7,11,9. llM lOS j^RITHMETIC. The above method of finding the L. C. M. of two or more numbers requires all but one of them to be resolved into their prime factors ; it is therefore applicable only when such resolu- tion can be easily eflfected. If two or more of the given numbers are difficult of resolution, the following method of linding the L. C. M. may be adopted : — Second Method.— To Jind the L. C. M. of tn-o i:ommensurablc nmnbertf, divide one of them by their G. C. M. and viultijdy the other by the quotient; the product icill be the L. C. M. required. Should there be more than two numbers, find tlie L. C. M. of two of them ; then of this connnon multii)le and another of the numbers; then of this second connnon multiple and a fourth number ; and so continne throughout the given numbers. The last common multiple found will be the L. C. M. recjuired. If the given numbers are unlike or compound, they must be reduced to equivalent like numbers. EXERCISE XXXVII. Find the L. C. M. of— 1. 217 and 279. -?. 1921 and 1469. .■?. 699.S anrl 10989. 4. 5724 and 77.33. a 7 ft. G in. and 4 ft. (5 in. 10. 7 lb. 7 o-A. and 17 lb. 11. 5 rd . and 2 yd. IS. 5 sq. rd. and 25 sq. yd. S. 6061 anil 73.37. C. .30401 and 12341. 7. 2318, 3111 and 3.55.3. .V. 5040, 7770, 9912, 10773. / 7. 3 rd. 2 yd. and 1 ft. L'/. 31 gal. 2qt. and 6 gal. /■'>. 1 .s(i. 1. and 1 sq. ft. IG. 910 gr. and 3 lb. 17. What is the least number which, as oofactor of 24, will yield a multiple of 30 as product ? IS. What is tlie least number by whicli 204 must be multiplied to yield a multiple of 650 ? 19. The multiplicand is 4095 ; the product is a multiple of .3906. Find the least multiplier. ^^0. Find the least multiplier which, with 8645 as multiplicand, will yield as product a number which is a multiple of both 1001 and 1045. MULTll'i.KS. 109 EXERCISE XXXVIII. tlurJ 7 ft 6 ta. l^gr * *" "'°°"'' " "■ '»"«■ «»'' the ■I- What is the least number ot men thnl- ,,„!,. remnants? '^ ' ^'' •^'^•' ^'^ ^0 yd., without or ho.es ® «,.S5 each, a'.,,, h3 tfhi„: ^e 'l^'e::;;!;? r'" many animals woiiHsueh a sum l,„j. "> every ease? How 4.rj,h':;:x\s:tit^-^^^^^^^ .y «nTr;:f :r:er^,::™:-;'rs"tr'r"Tr respectively? ^ ^ ' ^*- ^"'^ ^ S'^^- ^ 'it. 3^«». oroUOga,., or of i^ ;;tl ^t^eXL.y ^ ' ■^'"- ^ "' ".Five bells eommenoe lolling, the first tolls every second tb. _^.bet.een their tolling toUer:an„"Zi:T,lC4::h:: /.'. A can hoe a row of corn in 10 min. , fi in l^ min r ,•„ i - • 1.6 till thr^v ..ir« • I ' , '"^'^*^«'"' l'"w many hours will it :zt^:::^zvj: "' "" """° "■""'°"" "-^ --^ -- in 110 ARITHMETIC. 13. A can build 14 rods of fence per day, B 21 rods, C 18 rods, and D 20 rods. What is the least numl>er of rods of fence that would furnish an integral nund)er of days' work to any one of the four ? Prove that tliia lengtii of fence would also furnish an integral number of days' work to A and B working together, l)ut not to C and D working together. 14. Wliat is the smallest sum which I can completely expend eitiier on cherries @ 12 ct. the box or on raspberries Qn, 10 ct. the quart? 15. What is the smallest sum with which I can buy plums @ 40 ct. the gal., or peaches (o .§1.2.") tli( basket, or oranges (f/, .SO ct. tlie doz., and have no money left ? WHiat quantity of each kind of fruit could I buy for this si,jm ? 10. A dealer expended equal sums on eggs @ 20 ct. the doz., cheese @ 15 ct. the lb., and butter at 24 ct. tlie 11). V.'hat was the least sum he could have expended on each ? What quantity of each commodity would these sums purchase ? 17. Find the smallest integral number of pounds of sugar (a). 9 ct. the lb. that can be exchanged without either losb or gain for an integral number of pounds of cheese @ 12 ct. the lb. 18. Find the smallest integral number of pounds of coffee @ .35 ct the lb. that can be exchanged without either loss or gain for an integral number of pounds of tea @ 65 ct. the lb. ID. Find the smallest number of turkeys @, $1.50 each that can be exchanged without either loss or gain for an even number of chickens @ 55 ct. the pair. 20. A market-woman found that whether she counted her eggs by 4, by 6, or by 10 at a time, she had an exact number of counts. Find the least number of doz. she could have had. 21. A market-woman found that whether she counted her eggs by 6 or by 8 at a time she had an exact number of counts. Show that she must have had an even number of dozens. 23. What is the capacity of the smallest cistern that can be filled in an exact number of minutes by either of two pipes, the first of which runs 35 gal. pe.' minute and the second 42 gal. per minute ? ' How long w-ould each take to fill a cistern of that capacity ? 23. What is the capacity of the smallest cistern that can be filled! in an exact number of minutes by any one of three pipes, the first of I which runs 30 gai., the second 40 gal., and the thiid 45 gal. each perl minute ? '^ :l MULTIPLES. Ill an.l th. thi,,l 30gai±™if;f '„'■''■ T"" ' ""^ '""°""' '''"e"- take to mi the tank ! ^ ^°"' '""« ""'""'' "" *ree pipe, ^Jz s ;:r r ™ '»«="-■■■ *= -- ■'.■-«„„ „„„„ . P«tto,et,,e ,„„/„.,, --t'^ey^^-™^^^^^ aci:;,e?;reSxx::;^?:^.:\r^"-f™'^""'"' and the third every 20 mT T T ^2"""" *>'««*^^'"nd every ISn.iu., together again atl' 2 iin'^Po^^^^ T' """'^'^ "'" *^^^^ ^« '^» first boy have aained on fT" ^ , ,""'^ "'^"^ '"'^""^^ ^i" the -V T^,!. ^ . •*" '''^°"''' ^"^^ ^^^^^ "i^ny on the third v ^.y. Ihree men start toirethcr to w.n, ;« *i ,. * oval track 880 yd. around the fir Jf '"^ ''"■°"^"''° °'> »" the ,eoo..d ever^ 8 mZ, tf th « f e^Vio ''' ''°K^ ' I"' mmt they walk before thU will all bo tl! K '^ ."""■ '°"e post. How „..,.y „iie, .'iiC'th^sira r *"= *'"■■« at the rate of 332 yd., and the third at tU rate" 2«4 ^TT long will it bs between their once co,ni,g alt .„;!«!! '1 t^"-' conung all together again „t the same Bkce. \tT ' will each horse have trotted in that thne ' ' "'""^ ''°""<'' wh:irh:rw^"-t::i.t:f°''''-™-^-na«yd.. thf„th^en„';t:ir uii-^::'^'''- ;^ °"» ■■" '» ""»■ -^ .™ew.idtt;fiii 'j:;l'':n;^^^^^^^^^^^ - -- «-. A farmer could hoe a certain field of com in 21 w i- i,- j man could hoe it in 21 hr and th„ I. ° ' '" '"""' 28 hr. If each always ht'attte r^ Zb-r, "'"'" ""^ " *» what i. the least number of rows ther ."!::'«: ^^^^ "T t.me »„ld they hoe the corn were all three ^ ^ k ^her ' "' 112 AUITIIMFTK' Vim ill -■?. Tho first of three men could cut a certain pile of cordwood in 10 days, tho second coulil cut it in J ."» days, and tlie third in 12 days, uU working 9 hr. per day. If each can cut an exact nuinl»cr of cords per day, what is tho least number of cords there can be in tho pile? In wliat time could they cut it if all three were to work together? In wiiat time could tlie first and second cut it without tlie aid of tlie tiiird '! In wliat time could the third and second cut it without the aid of the first ? 34. Two cog-wheels containing 20 and 45 cogs respectively are working togetl jr. After how many revolutions of the smaller wheel will two cogs which once touch, touch a sccund time? 35. The circumference of the front wheel of a carriage is 10 ft. 6 in.; that of the hind wheel is H ft, A certain spoko in each wheel is pointing straight downwards at starting. ITow far will the carriage travel before the same two spokes will again point straight downwards at the same momei.t? 36. Find the least number which, divided by 3 or by 4, leaves in each case the remainder 2. 37. Find the least number which, divided by 6, by 8 or by 10, leaves in each case the remainder 6. 3S. Find tho three smallest numbers that, on division by 77, or by 99, or by 192, leaves in each case the remainder 46. 39. A market-woman who has an exact number of dozens of eggs finds that if she counts them by 8, or by iO, or by 20, there are always 4 eggs left. What is the least number of dozens she can have? 40. On counting out the marbles in a bag by 20 at a time, or by 24, or by 30, there are always 15 marbles left; but on counting them out by 25 at a time there are none left. Wliat is the least number of marbles there can bo in the bag ? 41. 480 grains is called a Troy ounce. Find the least number of ounces (Troy) that will weigh an exact number of pounds (Avoir- dupois). 43. A solar year is 365 da. 5 hr. 48 min. 46 sec. ; a lunar month is 29 da. 12 hr. 44 min. 3 sec. Show that 19 yr. is very nearly 235 lunar months, and that 1021 yr. is still nearer to an integral number of lunar months. Find the least number of solar years tl at are equal to an integral number of lunar months. i C'lTAPTKii \'I. FRACTIONS. 3 or by 4, leaves in 6, by 8 or by 10, I. NOTATION AND NUMERATION. ..ft f mto four, a quarter or fourth of it; if into tive, a fifth of fc; f xnto SIX, a s.xth of it ; and so on. These parts^a half ^::;::::r''^ - '^''^ ''— -"^^ ^-«-ai Part;!:; EXERCISE XXXIX. ^ is r ii::r r;i"r.° wi:'r ■, ir tT r- "'"-' of tiie.u ? "'vmea . \\ jiat jmrt of the whole line is each S. Draw a line 3 in. long and divide it into 3 equal parts UHv.t part of the whole line is each of these part. ' 4. Subdivide each of the 3 parts into 2 ec.ual narts Tnf« I 6'. Subdivide each of the o parts into 2 equal parts. Into how many equal parts is the line now divided ' VVhit n-u-. f /i 17 line is one of these parts t S of them ? 7 of tlLn ^ ' "' ''' ^'^"^^ 7. Draw a line 6 in. long and divide it into 3 equal parts AVhat part of the whole line is each of these parts' J. Subdivide the 3 parts, each into 4 equal parts. Into hn^ man,, equal parts is tJie line now df vjded y ^^•h,t part of the whole Ime is one of them? 5 of them? 11 of them' "«» the whole 8 113 114 AIMTfr>fKT[C. i .9. How iiijiiiy halvc'M of a Hlutepoiioil aro cinial to tlio whole of it ? 10. How iiiiiny thinly*? 1^, How iimiiy tontliM? 11. How iiiiiiiy (inarti'i.s? I;l. H)W nuiiiy twelfths? 14. Take a Btiiug tlio Iciigtli c)f your elato ami ilouhlo it at the middle; double it again, and yet a tliini time. What iKut of the lengtli of your slate ia the lengtli of the tiirice-foldcd string? What part would tlirce folds of the string ))e were they unfolded? !■'. If a Btring l)e cut into 9 eijual parts, wiiat part of the whole string is our of the 9 parts ? 2 of them ? 4 of them ? 7 of tiiem ? What iH the name of one of the parts of any (piantity divided into — i6'. 4 ecjual parts ? /.V. 10 e([Ual parts? i7. 6 ei^ual parts ? I'J. 19 ecjual parts? 20. How many halves of anything are ecjual to the whole of it? 21. How many quarters? 23. How many tenths? 22. How many eightlis? 24. How many hun.lredths? When anything is divided into 1*2 ec^ual parts, what is the name of - 2'). Three parts ? ..'7. Four parts ? 20. Five parts ? 2(i. One part? 2S. Ten parts? ,lu. Twelve parts? What is meant by — SI. Tiireri(|narters of an apple ? 33. Three-eighths of a yard ? 32. TwoOirds of a slate-pencil? 34. Seven-tenths of a dollar ? 35. In 2 apples how many halves of an apple are there? How many quarters? How many thirds? 30. Three oranges would yield how many quarters of an orange? How many eighths? 37. How many tenths of a dollar would be equal to $'> ? 3S. How many twelfths of a foot are there in 7 ft.? What is the common name 1 »• tlio twelftli of a foot? 3D. Which is -Ker, one-half of an apple or one-third of if Why? 40. Which is V,;^^ 5, ihir;i of a yard or a quarter of a yard' Why? 41. Which is hc-ViOi, u eighth of v pound or a sixteenth of a pound ? Why ? ,f ?. Which is the most, three-fifths of a bushel of wheat or three- quarters of a bushel of wheat ? Why ? 1 to tins wlioli! of it ? imiiy tentliM? iiiiny twflt'tlis? 1(1 (lotihlu it ut tlto Wliut [Kilt of tlie (led Btriiig? What ' mifoI(li!'« *° «•* '- -'^t'-ird of a certain bench. How many boy. of the san.e size would the whoLTe!:" ^ A^unit is any standard used in counting or in meas- In 3.ciuarters of a pound the unit is -. cjuarter of a noun. 5-eighths of a cupful ? ". 4-uinths of a load ? .V. 7-twelfths? !>. Half an hour? 10. 9 quarters of a year ? 11. How many fractional units are there in each of the numbers in the preceding ten questions ? 12. How many of the fractional units of the numbers in each of these ten questions would l)e required to make one of the corres- ponding prime units? 13. What is the prime unit and what the fractional unit in Ques- tion 1-1, Exercise LI. A Fractional Number or Fraction is a Number whose unit is fractional. A Fn«;tiunnl Number therefore t.rpressta one or more equal parts of some prime unit. To completely express a fraction both the number and tlie size of the fractional units must be stated. Hence to express a fraction in numerals reciuires two numbers —one called the Numerator, tlie other the Denominator. The Numerator (that in, the number-tellek) expresses the number of fractional units in the fraction. The Denominator (that As, the name-giver) denotes the size of the f]'actif)nal units by expressing how many of tliem are contained in the prime unit. The Numerator and Denominator together are called the Termg of the fraction. They are written, the Numerator a little above the Denominator, with a short lino between them, so that a fraction is written -^'""'^'•'^t"'--, Denoiniiiator. Thus the fraction fim-e.Ujhthn, which has live for its numerator and eight for its denominator, is written ^. A f; action expressed in figures i« read by first reailiug its numerator, and then its denominator with the ti'iniinatiou of tlie onal unit in Quos- k) exprosses the or its numerator DOTATION A.VD Ni;MERATIO>f OF FRACTIOXS. 117 copreapo„di„g ordm.l „„„,,„, except in the c«» of traction, with LeC^:"™'"""'' "■"°" "' '»"" - '»-» -' ■"."'«". -the ;B». i.-J is read three-tiuarters. The 4 exnrossiis tl,.,t n, pnme „n,t_hero staply the abstract nun.be.- l-Td vid d il M.e,ud parts or quarters; it thus denotes the sL „/ W C Jrtrt""""" "•^' *' '~'™ --'* °' "- "f tl,n^'";i~'~^'' "• " ""^ «™-t«IM,s Of a foot. The !•> expresses twelfths, It therefore denotes tlie length of the fracfionil n„it The 6 expresses that the fraction consists of >„ „f tiZZluZ. EXERCISE XLI. Read and analyse in the manner of Examples I and 2 above-- 10. {?j sq. ft. 1. h in. ^. f yd. 3. gib. 4- TTTgal. 5. IJ cwt. C. j^ cord. ■4 ■^■ Write in numerals — -?.?. Five-eighths of a lb. 14- Half an oz. i5. Three-quarters of a cord. iff. Eleven-twelfths of a yd. Express — .'>. I ou. yd. ^P". Thirteen thirty-seconds, i. HoM iiiany qiiaftcis of a dollar arc there in $2\- ? In $r)'j ? (J. How many eiglitlis of an inch are tliere in .Sg in.? In 7| in..' led into Proper REDTTCTION^ OF FRAOTIOXS 119 17S 8 Calculation. •41 eigliths = iii. Explanation. l=8-eighths, therefore]; = 17 (S-eighths) - I3(i-eightlis, therefore ITf^ iHfi-eighths an.l o-eighths = 141-eighths = -i|.i,. part of the mixed number """""atoi „f tlie fractional «*^r;:s;;;: i't:i:iT»*' ;■" '"" •"-»"■■ . 760ilAf^. EXERCISE XLIII. Reduce to improper fractions— ^'- 15H. 2i. 303t*jV ~ 36^. U. 343t#^. '^- 49ff. /^. ,03X. ^- 99A. /^. 333f^i i^;. 99A-. ij. 303glg. ^,, ,yy eai ' '^^'^ ' -''''-'■ ^« '-- --^ ^hil^I- can I give , '^ple ~ A Out of $;> I jjrive a ouarter of i -1«1U.. i. ^ .- , mauy ,,„arters hav^e I left ? "^ ^'^ ^ ' '^^^- ^^^^ ^'- ^''"'^'^ ^^ ^'"**«''' ^ «»• ^ ? «y how much i. it greater ? 120 AlUTUMETIC. Reduction of Improper Fractions to Mixed Numbers. Example. — Reduce ^^ in. to inches. 13 = 3 (4) and 1 over, therefore 13 (ju. in. =3 (4 qu. in.) and 1 qu. in. over =^ 3 in. and | in. = 3| in. EXERCISE XLIV. i. How many whole inches are there in 5 halves of an iiu li ? In 8 halves ? In 11 halves ? In 2 1 halves ? ii. How many quarts are there in \ qt.? In '^ qt.? In V' qt-? In V- qt.? 3. How many whole yards are there in 12 thirds of a yard ? In J^ yd.? In \'i yd.? What is the common name for ^ yd.? Answer the preceding questions, substituting this common name for "third of a yard." 4. How many feet are there in* ft.? In^ft.? In^fft.? Int^ft.? 5. How many pounds are there in J43- lb.? In V- lb.? In -'/ lb.? 6. How many dollars are there in $ V" ?. In Si J ? In $ V ? I" ^'i'i ? What coin is §1? tfV? $1? ^in'i How many wholes in — 7. 17-halves? IK K^thirds? 8. 29-quarters? 10. 23-eighths? To reduce an improper fraction to an etiuivalent mixed number, Divide the numerator by the denominator; the quotient tmll hr the integral part of the mixed number; the remainder will he the tbumerator, and the denominator of the given fraction will he the denominator of its fractional part. Shoidd there he no remai/tider, the quotient will be the irhole number equivalent to the given improper fraction. Example. — Reduce ^J- to a mixed number. Calculation. Explanation. 0)77 J7- = 77-ninth3 %aL—iJ.^ —S (O-iiiiitlis) and 5- ninths = 8 and 6-ninth8 = 8#. REDUCTION OF FRACTIONS. 121 d Numbers. 3 of ail iiuli ? In •' qt.? Ill V' r7"36' EXERCISE XLVII. Insert the numerators in— 1. 3 = nr- 4 _ V — Tlf Insert the denominat •^. ^s = ors m- 4 _ 7 2 . Twelfths? 7. Ninths? ,S. Sixths? t). Thirds? How many twelfths are ecjual to — 10. M? 11. IV. l-i- Vh'- IJ. i t? 14. 55? Reduce— 15. j"f to halves. 17. Uto sevenths. It',. iV to thirds. IS. Hto eighths. 1!). In -"- how many fractional parts are there ? " How many of 1 5 ' 5 8 these parts would make up one whole? In - - -— or - how many ^ ^ 204-5 4 fractional parts are there ? How many of these fractional parts would make up one whole? What is the effect on the vumhcr of 15 15 — 5 fractional parts of changing -'- into — '— 1 What is the effect on ^ ^20 204-5 their dze 'i 14 144-7 20. If — be changed into —^'- — , what has been done to the four- 35 35-4-7 14 tttn fractional parts making up the - ? How many of the new 35 fractional parts would make up one whole ? From the problems in the precedini^ exercise we see that Dividing the iiumeratcjr of a fracti(ju divides the jiuniber of parts making up the fraction ; while dividing the denominator groups the i)arts, for it divides the number of them required to make up one whole. |i| IIEDUCTIOX OF FRACTIONS. 125 Henco dmdiny hnth tcnns of a fradmi by 2, or 3, or /,, .,• nnu oth.r number ,1... not clu„.,e ike vaU.. of L fr^^tiJ:, lal merely ..,^<„„/,,,^ t, ^,,,,^,.,, ^J^^ ^ ' ^ • ^nto sets of 2, or .J, or J,, or other number, each as tL case ^nnX 8)48 i-^^423.0_7 6 48 48^(r8" ;/. Thirds? ? " How many of t is the effect on EXERCISE XLIX. Insert the numerators in Insert the denominators in 10. IH^ia. .V^-*. •'^- x*^=^^K C ."1291 1^- im-='y 1-^. un-'^-'-. The problems in t],e hist four exercises are examples ..f the Fundamental Principle ..k Fhaction.s, namely:- The value of a fka.tion is n.,t chanoei> if its terms be BOTH MULTIPLIED OR BOTH DIVIDED BY THE SAME M MBEiC. A fraction is reduced to lo^oer terms if a common factor be divided out of both nun.erator and denominator. A fraction is expressed in Lowest Terms if its terms are integral and prime to each other. Hence To reduce a fraction to its lowest terms, Divide both teiins bij their G. C. M. EXERCISE L. Reduce to equivalent fractions expressed in lowest terms ^3. mi 1. if J. .14 2. M. 6'. TTTI- 3. H. -Hi 3270 4' 41 TT5-5' 0. ID J''ij the deibomuuitor (rf thf, Jirxt fntc- tioit, ami iiuilfipJij both tefnia of tliis fraction bij thf (jnotlnit; do liktivim with all the other fractiom. EMmplc—Iieduce |, ^ and {.r to equivalent fracticms with least common denominator. L. C. M. of denominators 4, H and 12 is 24. 3_3x6 18 4.~4x'0 ~24' 5_r)x3_15_ 8~ 8x3 ~24' 7 _ 7x2 14 i2~ 12^x2" 24' 24-=- 4 = «) 24-=- 8 = 3 24-^12 = 2 To determine which of two fractions is the greater they must be reduced to the same fractional unit, and the fraction con- taining the greater number of such units will be the greater. Reduction to the same fractional unit is eflfected by reducing the given fractions to a common denominator. REDTJCTION OK FRACTIONS. 127 it fractions with EXERCISE LI. By what numbers must the terms be multiplied to reducc- '■ 4 tooths? ^. iamlAtoOOths? ■2. StoOths? 3. gto48ths? '}. 3, T^yimd ,"^10 I'iOths? <'• T. J. n. H, ii' and g^ t<> noOOths? What is tlio least common denominator to which can be reduced- 7. \ and \ ? S. k and i ? 9. -,', and ,\ ? -'■^- Ti. A and !,J? ^'^ 1?, U and ^4? What is the fractional unit of lowest denomination to which can be reduced — 13. *in., fjin., ?f in. a^U^in.? ■?-^- \% hr., fl hr. and /^ hr.? iJ. i oz., /y oz., fJ oz. and || oz.? Reduce to equivalent fractions with least common denominator- m. landg. 79. i, ^, ^. ^,^_ .i.\5. L i7. ^andA. i/). ^.J.A. ;^i. f i. ;. A, H. OO J7 II s 1 ■» '"'• IJl TliT> 7J> "JT- '■' ? 52 If. 2 « an '-^' f5» iT> -JJ. sf- Which is greater — ^6. AorH? ^■^- A, ij. m, im. £^5. 07 it or I?? i29. Find the greatest and the least of- 30 -^ ?I±- ?Z.Z^ ■ 25' 25 + 5' 25-5' 17 17 + 6, -- or ? 24 24 f 6 17 17-6, — or ? 24 24 - 6 3^. V, 3§, 3f. 5i. 17 17 + 5 17-5 33. K a as ^T7> -7 , 25 25 + 5' 25-5" 34. Of ^, II and f I, which is intermediate in vahie ? JJ. Find a fraction intermediate in value to § in. and I iu. m ith 24. as denominator. 3(]. Find a fraction intermediate in value to \} hr. and y^ hr with denominator 60. - iif - 37. Arrange in order of magnitude— h h h > ti h i> i> i) f. 6 » 6» 0) i 128 AUITHMKTIC. III. ADDITION OF FRACTIONS. Ki'fimplr. — Find tlio nuiii uf ^, I and fi. Tak(> thrcR slipH of paper, e(|ual to one anotliur in It-tiKth and in hn-adth, ami cut tlu'iii iwrosN, c'licli Into H f j.iocfs or viKlithM. Take ;< of tho cightlm of the; first Nlip, 7 of those of the necond slip, imd f) of ttiose of tlio tliird Hllp, and )mt thcni all to(fctlier. There will 1)0 3 t-T + S-lfi pivci-H or riiflitliH, •■i,ouk'Ii to make nji onu whole nlip and leave 7 pieccH over. Written in H.vnilioN, all tliis is 3 7 ^_^Jj-^_io 7 8 "^ « "*" 8" ^H~ ~ " "8 ~ H' If two or moro fractions to bo added together liave a coinnioii tlunoininator, oilil tlif niinirrofors iiuii'fliev j'vr flu- itmiwrntor of the 8um and take the cnmnum iliuomiiudor for its denominator. EXERCISE LII. Show by cutting slips of paper or pieces of twine tiiat — 6 4 I /. 3 3_3-*-:} 4'^4~~4" '^'^■ s. 3 7 1- 8 8 8 13 5 2 - + -H h-: (i (5 G 6 3 )-7_n 3 8 ~ 8 ~ 8' l+3+5+2_21_,^ ---- -. TT + /'r + /T+lV 0. l + l + l IV. II. li. ii+'TT' + V + V. ViT+li + ^l + iii. n 7 _L 3 « J. .'1 7 :i 1 fl 1 t 6 H Find the value of — 4. i + HS- 7'. A .^'. ti + i + S- ^' 11+2S + 3J. Find the sum of — 13. 6 pair, 3 pair and 2 pair, 14. 6 doz. , 3 doz. and 2 doz., 15. 6 score, 3 score and 2 score, W. 6 hundred, 3 hundred and 2 hundred, or 6( 100), 3( 100) and 2( 100). n. 6 halves, 3 halves and 2 halves, or S, 3 and %. IS. 6 quarters, 3 quarters and 2 quarters, or J, J and j. 19. 6-eighths, 3-eighths and 2-eighths, or J, g and |. :'J0. fi-tifths, 3fifths and 2-fifths, or i, f and ?. v*/. 6-thirty-fifths, S-thirty-fifths and 2-thirty-fifths, or ■^\, ^V and ^fj. or 6(2), 3(2) and 2(2). o;-ti(12), 3(12) and 2(12). or 6(20), 3(20) and 2(20). 'IONS. It and ill breadth, and I of thu fightliM of th«! II' third Hlip, and )iiit ithH, )'iioiit;h tn iimkc iIm, all thlti in r liavo a ciiunnoD till' iiitiiwrator of s denomhiatur. e that- ■[i+n + n- 4- '-' « j_ n 7 ;i a. R_i 1 T I fMT+ 1 (JT) +0T55' 3(2) and 2(2). 3(12) and 2(12). 3(20) and 2(20). ,3(100) and 2(100). or S, -3 and J. or n it J and 2 t or n g and t or n 2 and ? 8, or -A, inr and -^^j. AODITION or FIlAnTloVK. 129 i-V. i.— Add together .'^ and j{. pic«., or sixths. Take .oh." ?TT' """ ""' """'"' ^'"' '""^ « -""" Blip. You will nowh^ os,i r^^^^ Hi...lrHtHn,,„„d,..„ tho , th., ..,..,nd quarfrs and the nixth h. tl^i' w v t hof 'J' """""'*-' '*'" twelfths. an,I the 5 Hixth.^K r.^^ ml^f^'n :,*'''^'' '''^"■ length of a slip-that l/i, ' ' ""'"''•''' "''^-t^elfH' th.. "lip that i«, there are J.] of a nii,,. wntten in symholn. all this ia Eji. ^.— Find the vuliio <>f !';t +2^ + (JJl + i 7 , L. C. M. uf donominators 3, 8, 12 and 15 i.s 120 4 4xa~ia 5 5x2 10 «1 8x^~lij 120- 3 8 12 15 40 15 2 5 JO 11 2 5 11 8 7 fJiven Denoniinators. 3 8^12^15" tJuoticntM. <«ivuii NiuueratorM. 80 + 75+1 10 + ,')( J .'{21 81 •>- "~ = 2 - = 2 " ' 120 120 ^AO 120 9 + 2 + + l + 2.^,v = 20^,V- To find the sum of two or more fractions, add the resnlrnrj numerators together for the nnniemtor of the sum and take the com.^on d.un,.iaator for Its deno^niiZ ^uce thes^nnto its lorrest te,.,s, and If It f.e an n" fraction, reduce it to a mixed number. ^ If tliere be mixed numbers among the addends, add t],o frac- onal parts along with any fractional addends, and to tho 2n ^ckl the integral parts of tho mixed number. the!?"" ''V"^T" ^'"''^°"' "^"""= ^^- -^''-"^«. reduce tiiem to mixed numbers. 9 'H\ i 130 ARITHMETIC. EXERCISE LIII. in 'I It '^ • Show by dividing li lines drawn of paper or pieces of twine, that — 1 11 < 2 l_^+]_^ 1 1 1 1x3 1x2 1 3+2+1 ~. - + - + -=;r~7; + ;;— ^+T.=- on your slate, or by cutting slips 3 G '2x33x2 2 6 „ ., _ 3x3 5x2 2x4 !' + 10 + 8_'27 3 _.,1 ^- - + -^3^4^-*-G^^ + ^x4 = — 12-— ^-^■-— " l: '12' ^^ + H + \ 3x3 1x6 1x4 19 3 3 + 1 +1+— — = —— + " =5H =6 — -.i+i + i + ^^,^ 2x6 3xt 12 12 Add together-- ,1. 1 and 2. (;. § and §. Find the sum' of — /).]§, 2-1 and 3*. 10. 3^, 5h and 2-r"^. Find the value of — 13. h + h + i + i + i- 7, i, § and g. S. i. 11. {'i^lt and 3^. 1,?. 3./'i, 4and,V and i. ■i 'y Ti I ■" 1 ^ 1 ft i/. Tj5 + Tj!r+ 2S + TT' li). 4+A + tl+H- -?5. §+i^ + H+^. ;2i. $14 + §33 + S4i + $7fV + S17i^ £2. 1 i mi. + 3f r ""• + '-i mi- + 9r \ "li. ;?5. A A. + §■ A. + ^V A. + ih A. + i ^ A. ,'?^. f hu. + A bu. + ^■V bn. + ^ !>"• + 7u !)"• .^/J. A man bought at different times four lots containing respec- tively A A., .^- A., i A. and g A. How much land did he buy alto- gether ? 20. The difference in weight between two boxes of tea is 17U lb., and the lighter box weighs 49i'a U). What is the weight of the heavier ? 27. Tlie first of four measures holds 2-tqt., the second holds Ijl qt. moic tluui the first, the third holds I qt. more than the second, (lud the fourth holds I qt. more than t!-.^ first and the second to- gi ther. flow mucli do all four hold ''. yr by cutting slips ■_2'''.2i. 2 12 4 -a.'« r' 12 12 .v. J, 5. and i id 3V\,. ad s^. -h + l -if+ii n+n- ^N^+\n- 3 containing respec- ,nd did he buy alto- 468 of tea is 17H l^'-> 3 the \veight oC the j the second holds 1^ are than the second, b and the second to- 1 SIJBTHACTIOX OF FUACTlOxNS. IV. SUBTRACTION OF FRACTIONS. l.'U If the rnmuend and the subtmliend luive a co.nn.on donomi- ji; -Y^nd the nunu^rato. of the subtrahend be n..t greater tl I tl-t ..f the nunuend, t]xe question is one of simple subtraction EXERCISE LIV. From J. 7 pair ~. 7 dozen 3. 7 score tulvC 3 pair; 3 dozen ; 3 score ; M'ritfoi, la si/iiihols. 7(2) -3(2). 7(12) -3(12). 7(20) -3(20). 7(100)- .3(100). " _ ;! •I -1. i -i- iff ~ TTS- 4- 7 hundred 3 hundred ; 3. 7-(juartfrs 3-quarters; ^A 7-eighths 3-oigl)ths; ~. 7-tenths 3-tentli8; .V. 7-twelfths 3-t\velfths; r^f-r^. Find tlie vahie of What proper fractions a.lded to tlie following Mill i„ cu'li ca^,- 1 !• 1 D ' Jl 1 1 i 1 • 1,{. 7 .''v-' 3 1 •■' " give an integral .sum ': I--- \. 18 1'. I M. ■'■'■ nuA. trVVr. Pkin-ciple.- -Adding the .same nvmber to umu minuend and «UBTKAHEXI. DOES ^•0T CHANUE THE DI.EEKENCE OH KEMAIXDKK. To find the difference between two fractions, Add to both ndmend and subtrahend the pr.j.r frariL vhose mm with the subtrahend is integral; then subtract the now integral mbtrahend Jroni the integral part of the minuend. Ex. 7.— From 3^ take 1-,^ ^IJT- f"b = 2 3i^-l^ = ].1v-l Thin lint' to In- CDiiiiduttfil jU-mL I ((■ 132 AUITHMKTIC /&. ^.— Find the valuo of 7A - 2;^ IB Jifi- L. C. M. of 16 and 36 is 144. 7-+ — = 7 16 36 27^+;20 ~T44 ^31 2—4- 36 36 = 3 16 36 144 EXERCISE LV. lil' i Find the value of — 1. 3H-H. 0. l-h /7. 7f-6S. ,?. 20^V-3A. lu. l-l- 75. H-h\. •5. 17^-1 Hi. It. f-rV 19. 71/^-llU. 4. 15^s-14H- 12. U-H. 20. 93tf - oy^ij^. 5. 6T\"r-fM. IS. T 146' 21. •t7iVu -7TV G. 13Vf-3!^t u. Wl-AV 49A-39^. 7. h-h- 15. f-rVT.. 23. 235f-75TtT, . How much must be taken from I + f to leave ^ -i- J as re- mainder ? ,W. Find the difference between J I i and i t 7. .7/. Smith owns '^ of a section of land; Jones owns -j^j of it, and Ih'own owns the remainder. What fraction of the section does Brown own ? ,'i2. A teacher expended ^ of his salary for board, 1 of it for cloth- ing, It of it for books, and i\ of it for other purposes. How much I of his salary had he then left ? ,V.i. A piece of clotli measured 23| yd. before fulling, but only i 21 li yd. aftci fulling. Hew much did the cloth shrink? 34. Wilson agreed to sell 37^ cords of wood to .Jackson. He de livered 9.| cords one week, iS| tlie next, and 10|i the next. Howj many cords has he still to deliver ? 17. 7,?-6§. IS. Si-'l^'V. W. 71/.-11U. 20. 935i-395V <93 AQ 9 _ OQ 9 Jf as sum ? > leave \ -;■ J as re- i owns i^T of it, and af the section does | ml, 1 of it for cloth - iposes. How much i re fulling, but only shrink ? o Jackson. He de I0i5 the next. Howj Miri/np,,„.AT.«.N OF FRA.Tio.vs. ]:}:j V. MULTIPLICATION OF FRACTIONS. M,dtipHcation is the operation by Schick ve fin,l in / Exampk.-Uov,- nmcli is 7 times | ? 7 tune, H quarters^C li.nos 3) .,„artors = 21 .marters o.-, wnt.n, the deno,ni„atio„ .carters in sy.nbols. 7timesL^t''»««J_21 4 4 4" ''"■"" '""'"><'>./ /„, for ;«,«... this beeo.nos 3 quarters multiplied by 7-« luultinliprf )„r -^ o ^ "'"'"•''"''' V.)t.mes|Ib. .;. /, multiplied by 14 - ftm.esS,.,, 7. .\ multiplied by 28 -'• '*"»«« $fo- .S^ ii multiplied by 24. ^''•. i.— P'iud the value of J „f •! i„. Here we are require or th,-, „ h quarter of an inch. " *'"'° ^''"'-'»'''' ""e^^'- t'ci'i^r ^Ufh a i of 3:=1; nence, a.^ertin^ the unit nan,ely, a quarter of an inch ri of :l in. =4; ir. /i;/!„ -Fi^id 1 «,f I in. r of -• m. ^- of --— ill =_1_ ,--. ^ - B'iad the value at- ■^- i of I yd. ^^ ?tof*m. EXERCISE LVIL o'. I oi' U wk. 6- ioii bu.. if 1S4 ARITHMETIC. Examjik. —Find the vuluo of J of /j rd Analysis. -of -111.- of -ra. = -rd.; 9 U 9 11x9 11x0 therefore iofi-nl. 9 11 4 , , 4x7 , 28 , = rd. x7-- r7^~ 50" 8 7 lIcLe 2 has been cancelled out of the 10 and the IG, and 3 cancelled out of the 9 and the 21. Had these factors not been cancelled, the result would have been ^^^^, which can be reduced to 4-^ by dividing' both terms by C, which is the product of 2 and 3, the factors cancelled. EXERCISE LVIII. Find the value of — 1, § of i bu. W of \\. i3. Hof2§. o i of i ft. s. \\ of M. U. l-iofis. 3. |of }f lb. 9. \\ of t*. iJ. Uof2§. 4. T2 of i yd. JO. !-f of \\. IG. 3i of I J. J of * hr. 11. %\ of If. 17. UJofOiV 6. ^ofMT. l.i. I of li. IS. moiiu 10. 2 of ^ of if. 22. US of 4n o T of §S of 2.S. iJO. Aof H^of f|. 23. 4 of 1,T ^of -il. SI. f of jVof i*. 24. li of 23 of 32 of ^. VA , 4x7 , 28 , 11;-.!) IH) 28 1)9 rd. r. Wo iiiiKlit therefore inerator and de- oth terms with- , tlie ert'oct being 13. Thia is called and 3 cancelled out of , tlie result would have erins by C, which is the 13. H of n- I'h Uoit. 15. Hoi 21 10. H of f. 17. UJofO.V IS. ?Uof IH of U of 22'''j. ioiV '• of 3-' of H. MULTIPLICATION OF FRACTIONS. ]3i A fraction of a numbor-wliotlKT integral, niixcl ,or frac- tional - 13 called a Compound Fraction. Examplcs.~l uf 5, I <'f 1], '^ of -T. A Compound Fraction is therefore a Fraction vhosc Prime Unit IS itse// a nnmher. Now, the operation by ^vhich ;ve find the value of a number whose unit is itself a number is culled multiplication. Hence ::T;;f ;i '':^''f^ ^^ ^" - ^--^^^ --^^-^ way of sayln. a <'t ,. Ihe following are other examples of different ways of expressing one and the same statement. [In these examples l(-) IS to be read on.-^nir and 4(J2) ve^C. four-doze, , just as h is read one-half and j'. is read four-twelfths.] 1. Seven of four-dozen each = 7 times 4(12) = 4(12) multiplied by 7 = 4(12) x 7 = 28(12). 2. Three of five-sixths each = 3 times 5 = 5 multiplied by 3= •? x3 = J,''- 3. Three-dozen pairs = 3(12)(2) = 1(2) multiplied by 3(12) = l(2)x3(12) = 3(24). 4. Two-dozen of iive-i)airs each = 2(12) of 5(2) = 5(2) multiplied by 2(12) = 5(2) X 2(12) = 10(24) 5. Two-thirds of four-fifths =^ S of * = i multiplied by * = 4 x S = . ■^- EXERCISE LIX. Express the following products as compound fractions an.l find the value of each:— 1. § ft. X I 2. 3 in. X %. 3. 4 pt. X I. Ji. 3;i lb. X f . 5. fT ft. X If C'- ligal.xii. Express the following co.npoun.I fractions as products and find tne value of each : — 7. t- of h hr. S. 2 of 2,^ lb. 0. li of ^ (,z. 10. 3.^of2i|. Ji- v., of 3i. A^. Sj^of 3,V If 136 ARTTTTMETTC. -I.I n i 1^' To simplify a compound fraction, or to find the product of factors one or more of which are fractions or mixed numbers, If any of the fractions are mixed numbers, reduce these to equiva- lent improper fractions, and write integral factors in the form of fractions ivith 1 as denominator. The product of the numerators of the factors will be the numerator of their prodzcct. The product of their denominators will be the denominator of their prodiict. Factors common to both a numerator and a denominator should be cancelled. EXERCISE LX. Find the value of — 1. 3. 4. i n ST |-> 1 n "55 w 9 V R*. ifxl V V ■ 17 6. 3i of 3| of M X ^i- 7. 4^of2|of5ixi|. 9. 3J of 7 X 4i of -jVs. 10. I of 17 X ,\ of 63 X ,^*5. 7 1 • VTrTT' >|of 8^xf of IxV 5. Hxl7rVof31fxlH|-. Jl. Find the sum of f of § and .J of |. 13. Find the product of 1^ + § and 3 - §. 13. Find the nearest integer to the product of 3| and 12.\. 14. How far could a man walk in 2^ hr. at the rate of 3^ miles an hour? Find the price, to the nearest cent, of — 16. 3 J dozen eggs @ 17 ct. the doz. 16. 4^ lb. tea ® 65 ct. the lb. 17. 32 lb. sugar @ 8^ ct. the lb. 18. 17| yd. of calico at 11^ ct. the yard. 19. 4J doz. tins of tomatoes @ $1.00 the doz. L'O. 37-^ bu. oats @ 37i ct. the bu., and 4^ bu. wheat @ 85| ct. the bu. 21. Find the weight oi the water in a cistern containing 75i'V gal. 23. A man sold 17.? gross of boxes of matches, gaining 3§ ct. per doz. boxes. How much did he gain on the whole ? 23. Bronze consists of 1 part of tin to 4^ parts of copper. What weight of copper must be added to 1653^ lb. of tin to make bronze? r to find the 1 are fractions ce these to cquiva- rs in the form of I he the numerator lominator of their DIVISION OF FUACTKJxN'S. VI. DIVISION OF FRACTIONS. 1.S7 The Reciprocal of any given number is tlie number whose product wth the g:veu nun.ber is one. Thus 2 x i =. 1 ; there oe i IS the reciprocal of 2, and 2 is the reciprocal Jf |. |.' IT the..e^ore A .s the reci,,rocal of I and | is the rejprotal ^f / ^* X tl, = 1 ; theref.;ro ,',- is the reciprocal of 'i} or -'<>., and J «l or .'U IS the reciprocal of j'ij. j , ki ^ oi .Sy I a denominator 1* V J, TSa ^ 2 ^' : of ^V:i. ■ of 63 X rVs. i| and 12 i. o rate of 3Jt miles . wheat @ 855 ct. )ntaining 75i^Tf gal. gaining 3§ ct. per of copper. What n to make bronze ? EXERCISE LXI. Find the reciprocal of- 1. 3. J. 5. 5. 12. 4. h .7 1 D. 5i. 11. I - j^. Division is the operation hj vkich n^e find the n>onber which taken as cofactorM one of t,oo yicen ,nunJ>,^rs, n-ould ylM the other given number as product. (8ee imge 28.) Ex. i.— Divide 4 by h. SOLUTIOX, 4 =4x2x.', ; therefor This is merely another why of askiii;r 4 -r J = 4 X 2 X i ^ i ^'"^ "'="'3' halves are e-iual to 4, or what A \( Q " " Illiiilber TiiiiU!i>1i/irI 1... 1 i,i i Fx. ^^— Divide 5 by ^. Solution. nmnber multiplied by J would be ecjual ^*- Proof.— 8xi = 4. = 5x7 X -i therefore 5 -f- 5 = 5 x "r x i — ij = 5x^ = ¥ = lli Ex. 5.-A4-,V This is merely another way of asking how many f are ey V,. In hke manner (ho division of 2h by 3| Lay bo exprosaed either l.v ol ■ t,! i -i Sucl. » frHcti,.u ,. -J U called a c^,^,!., fraCion. £^c. i.-Reduce 1 to a simple fnvction-^A.^ i., fi„d its value. Ex. 5— Reduce ^| to a simple fraction. 2i 33 JA jXlo-f. lot are taken, and /eight of a single le of $1.60 for 56 7 for 10 ct. or at I doz. eggs bought ! lb. will pay for 3st ounce. ) parts of copper. ' copper to make be in 464| lb. of ;opper ? Eead— EXERCISE LXIII. 1. i. o 3| 4* Express as complex fractions— ^- ^^^■ 7. 24-f5i 6- f-A. S. ^^10. H 3 8 A ^- i-.\ divided by . I + i J'), hoi it divided by"^ -^ ?•. JM„ «. Fi„d «,e value of .ae„ „, the p.eoedi„, JZ^,^ Simplify — 3 21. ~ 12* 2A. -i A 2S. Li ^57 ^ ^ - nr ,■. / , , — . — . U + A 2G. ^0 2U-g 24-i r^-^- ^P. bu. 1 gal. 2 qt. 1 pt. i20. To the sum of ^| of 3 A. 2420 sq. yd. and ?;; of 1 A. 42S:J sq. yd. add che difference between 13 A. 3^ sq. rd. and TH ,' A. ^1. From 3^ lb. take ^3^ oz. 3:2. What length added to Y yl- will make /i rd.? ;!!?5. Find the difference between 3 sq. mi. and -^^g of 1000 A. ^4- IV ^low much is /- of 5 da. longer than /v "f 33^ lir.? i?5. Subtract 43.i times 45 cu. ft. from * of 43^ cords. Divide — 26. 3 T. 400 lb. by 7i. 27. 3 mi. 720 yd. l)y Ji. Find the quotient of — ;.-'cS". 4 A. 2360 sq. yd. ^ j\. 2i>. 17 bu. 3 pk. 1 gal. -=-3f. 30. 24 cu. ft. is 42V times a certain volume; find that volume. SI. Find the len^tli of time of which 3<3 da. 2 hr. is ^\. 32. Divide $45 by 3 + f. 13. II of 2 A. 620 sq. yd. 14. 4|of 3yr. 3 da. 2 hr. If). 4 cords 24 cu. ft. y. 4^%. KS. 54 gal. 3 qt. x 15}^. 33. Divide 7 lb. 1200 gr. by U f 2§ - 3?. ONS. :., etc. r. etc. 20 s(j. yd. t (la. 2 hr. u. ft. X ii%. V I'll'' > l)U. 1 gal. 2 qt. H of 1 A. 42S.i 111(1 7H4 A. )f 1000 A. XU, In-.? •(Is. 1. 1'y J?. 1 gal. -^3|. that volume. IS To* DEXOMIXATK FKACTIONS. 34. What fraction of a pound is 4 oz.? .•?.T. Express .S7 oz. in pounds. SfJ. How nuich of 1 yd. is (} in ' ^7. KcMhu:o 2 ft.;; in. to the fraction of a yard. ^d. \\ imt fraction of a mile is .'{ rd 1 yd ■> SU. Express 1127 rd. 2 ft. .'1 in. in miles " 40. \Vhat fr-.<.tion of an acre is 12.30 sq. rd. 13 aq. yd.? 41. How much of a dollar is 2^ jt.? 4..'' What fraction of a dollar is 1 4 ct.? 141 Express in bush ,1s— 4-i. 111,511). of wheat. 44. 1610 11). of barley. .^v. 1966 lb. of oats. 4'>- 1477 lb, of Indian corn. Divide — 51. ^ lb. by 41 oz. 6 J. 4 mi. 480 yd. by 1^ ,ni. Find the quotient of— 47. 1G40 lb. of buckwheat. 48. 1840 lb. of peas. 40. 14S0 11). of timothy seed. oO. 1.S70 11). of red clover seed. r>.i. ;'.00.sq. yd. by IJA. H- i yd. by /^. mi. JJ. ill). :- 1 oz. j^;. oz. ■Jib. 57. 22° 27 r-r 90°. 5.9. Divide -l of 4-i«r A. by ^ of 2.3 8(i. rd J/>. Divide I of .3 gal. l.a ,jt. by A of 2 bu. .3 pk. 6^7. What traction of 1 cwt. is .37 J lb.? 61. What fraction of 9.^ A. is 1628 sq. yd.? 6-2. What fraction of ({{ mi. is 375 yd.? 63. Reduce 17 (?a. 3 hr. to the fraction of 36r,i da ' 64. Express 27^3 lb. of wheat as a fraction of 63 bu ' 65. Express ^ of 13 mi. 3 rd. as a fraction of 20 mi ' Z' W.°?/' !"'• "K^"^' '" '''^'* ^'''"^"«" '^f ^ -'• y^'- "f water? 67. What fraction of 5 T. is 7^ bu. of soft coal ' e^. The profits of a certain business are divided into 104 e.iual dTireive ?"""^ ''- "' ''"^ ''-''■ ^'^^* ^^^^*'- '^^ ^'^^ P-fit« 69. How many twelfths of an inch are there in 2§ ft ' 70 How often is the third of an inch contained in ,V of A mi ' 71 How many lengths >f 3| yd. each are there in 44? yd and what fraction of a length would there be over > 7^^ How many kegs, each holding .3A gal., could be filled from two barrels, one containing 27i gal. and the other .30}f gal.? 142 AUnilMETKJ. 7-?. How many bottles, eacli holding » (jt., would W} h],]. of vinegar fill, reckoning 31 4 gal. to the full bl)l., and what fraotion of a lx>ttle- ful would tliore he overt I'^xpresH as the fraotion of a year (.Wr) da.)— 74. l''rom noon of 3rd April, ItSSd, to noon of 24th Aug., 1886. 7.7. From noon of 17th May, 188(i, to noon of 'Ah Doc, lN8(i. 7(1. From noon of Iflth Dec, 1884, to noon of 14th Dec, 188" 77. From noon of 23rd Oct., 1887, to noon of 12tli May, 1888. 75. A farmer sold 2.'U hu. of liis wlicat crop and kept for his own use the 78 busiielH remaining. Wliat fraction t)f lii.s wiicat crop did he sell ? 79. A man wlio liad .$42 .spent $2. 10 of tliat sum. Wliat fraction of his money did ho spend ? W'iiat fraction of it had he remaining? 6'0. A man bought a horse for .$80 and sold jiim for $96. What fraction of the cost of the horse did he gain ? SI. Smith bought a horse for ^120 and sold liim to Jones for .$150. Jones next sold the horse to Brown for $120. NVliat fraction of the cost of the horse to him did Smitli gain ? \Yha.t fraction of tiie cost of the horse to him did Jones lose ? S.J. A certain mine yields 113 11>. 5 oz. of metal from every 7,\ T. of ore. What fraction of tlie ore is the metal extracted V What weight of metal ought 274 T. 1 120 lb. to yield ? 8S. If 7i A. yield 101 ;| bu. of wheat, how many l)U8hels would 15 A. 1760 scj. yd. yield at the same rate ? SI4. Armstrong has $7.56 and Brown has $12. Armstrong gives J of hia money to Brown, and then $2.10 more, AVhat fraction of his (Armstrong's) numey did Armstrong give in all to Brown ? After Brown had received the money, what fraction of wliat lie then had had he received from Armstrong ? 85. Allan has $10.20; Barnes has .$24.50. Allan lends Barnes $1.10 more than a third of his (Allan's) money. Next day Barnes, wlio has meanwhile spent $1.50, repays Allan. What fraction of his (Harnes') money has he to give Allan to repay him? 86. A cistern can be filled by a pipe in 15 iir. H'»1. of vinegar jtioii of a bottle- I Aug., 1886. Dec, 1880. I Doc, 188- i Afay, 1888. kejit f(»r liiH own i wlicat crop ditl What fraction <1 lie remaining? for S9tj. Wiiat ) Jones for.'jil.'JO. t fraction of the ction of the cost om every 7 i% T, tracteil ? What / busliels would \rmstrong gives \'hut fraction of ) Brown ? After liat he then had .n lends Barnes ext day Barnes, I'hat fraction of ni? vv inucli of the •20mm.1 in 4i hr.? At rn? How long II cif the uiateru .9.V. From the end of a phu.k 14 ft. 7'. in. long 2^^ of ', of th - o 1.S cut away. W'luit length remain^ ? who! 8!K Tiireo persons received eighth of ;5!14.40. What sum respectively a fifth, a «ixth a il? What frfto. I owe $1.", 75 to Fraser, §2.99^ more than half a-s nuuh to May and to (.raham ^.m Icsh than half a.s n.uch again as J ouv to May. How much loss than .^.-iO do I owe to Fraser, May and r.raham together y VVhat fraction of SJ)!) would it re,,uire to pay the whole of tliese debts ? » t .) n/. A school-room is half as long again as it is wide. What frac tion of tlie perimeter is the width ? 9^. A man had to walk 10 miles. Me walked h of the way, rested one hour, ami then walked 2 mi. 720 yd. What fraction of his journey had he still to walk ? .'/./. A maii ,„ade a journey of 100 miles. He rode 7 nn', 340 yd travelled by rail J of the remainder of the way, and n.a.le all l.i; I-..0 y.l. of what still remained of his trip by steamboat. Wluvt traction of his trip was made by steamlxiat ? 94. In constructing a sewer 104,0r,0 bricks were supplie.l, and out of this number 90,(500 were used and the rest rejected. What frac tion of tlie whole did the rejected bricks form ? y.7. n is older tlian A l)y ^ of A 's age, which is r, i Find H's age and express the difference between the ages of .1 and B as a traction of B s aire. W; AN-hat is tlie difference between eleven times three-quarters of i of 3| nil. and three times four-elevenths of 7080 rd.? 97. How many steps, each having a Ci in, riser, woul.l be required tor a staircase reaching a perpendicular height of 12 ft.' What heigiit would liave to be disfn/nttcd to make the exact 12 ft "' 98 A geograplucal mile is the ^V, -f ..,',. part of the earth's cir- cumference. The e(]uatorial circumference is 1.31,48.3 200 ft How many common or statute miles are equal to 00 geographical miles on the eejuator ? 99. A knot or nautical mile contains 1000 fathoms of ft. each. How many statute miles are equal to 00 knots ? ^ 100. The area of (Greece is .'r of that of Britain. Spain has 2^ times the area Britain iuis. What fraction of the a.-»a -f «-,.in is the area of Greece ? 144 AllITHMETIC. « »l VIII. APPLICATIONS OF THE PRECEDING RULES. EXERCISE LXV. Ujxtmple.— Find the price of 7 lb. 5 oz. of cheese @ 13 ct the lb. The price-unit is 1 lb., hence the 5 oz. in the 7 lb. 5 oz. must be expressed as a fraction of a pound. 71b. 5oz.=7/„lb. 7tl3 lb. @ 13 ct. for 1 lb. =7v^, (13 ct.) = no,V ct. In commercial transactions reckon to the nearest cent; half a cent to he considered a ivhole cent. Find tlie value of — 1. 4 lb. f) oz. of butter @. 19 ct. the lb. ^. 8 lb. 7 oz. of mutton @. 1 1 ct. the lb. 3. 5i qt. of molasses @ $1.15 the gal. 4. Two hams, one weighing 14 lb. 6 oz. and the other weighing 17 lb. 12 oz., both @ 10| ct. the lb. 5. 1430 lb. of wheat (a 93 ct. the bu. 6. 1887 lb. of oats @ 43i ct. " 7. 17fl.5 lb. of wheat @ 89gct. S. 189(5 lb. of barley @ 63 § ct. " 0. 1678 lb. of hay @ $23.40 the T. Make out bills for the following-stated transactions, supplying dates and names of places where necessary : 10. Thos. Jones bought of E. B. Browne 3^ lb. of Butter @. 21 ct. the lb., 2i doz. Eggs @ 15 ct. the doz., ^ lb. Japan Tea @ 45 ct. the lb., 52 11). Sugar @ 9 ct., ^ lb. Peel @ .33 ct., 4| lb. Cheese @ 15i ct. 11. Messrs. Mason & Wright sold to James Cliamberlain, on May 1st, 1886, 240 lb. of Flour @ $3.10 the cwt.; .May 6th, 137 .V lb. of Oatmeal @, $2.35; May 11th, 366 lb. of Cornmeal @ $2.30"; May ISth, 245 lb. of Buckwheat Flour (a, $2.45; May 2Sth, 3.30 lb. of Flour @ $3.05. On 1st June, 1886, Mr. Chamberlain paid !S20 on tliis account to Timothy Webster, book-keeper for Messrs. Mason & Wright. (Makeout receipt for the payment.) ECEDING eese @ 18 ct. lb. 5 oz. must ct.) = 95^i, ct. t cent; ludf a ther weighing ms, supplying utter @ 21 ct. a (g( 45 ct. the sese @ 15J ct. rlain, on May h, \Tih lb. of ^ $2.30; May th, 330 II). of L paid .^'20 on ssrs. Mason & APPLICATIONS OF TJIK l-UI,Vl.;i)[.\( \Villiain Simpson bought of Alfn ; UIT.KS. 145 'Spencer (ii lb. Veal (a ]'>' :;• •;■■ * '"• ■^^™ *'/ ' > ct., li Ib. Bacon Ca V, ,1 --a ii. r "i i5. Henry Mitcliell sold to John Young, on A„il I2th lU v7 Print (S lU ct U\ v,l ^iii. ^,1 c..> ..- -. , ^ ' ^^i >"• 1-th Qi 1 %' V ^'^^®^^-^^^ 7.iya. Lining r«> 12.',c • Anril 1-th 9i yd. Tweed @ 97 ct.; 4^ yd. Cloaking ^ $o 87^ jf vd Plush (T, ..2.12^, 2i doz. Buttons (?, I8ct.; 9 Spools (« 50 "-^' was i5. A man is to receive wages at the rate of !^9 50 oer ^v.-pI- „f 7 days. What will be the amount of his wages W l^t M "to I December, both inclusive ? '^ *"" ^''^ of 'l5 s Ws?^ ' "'"'" " ' ""^""^ "^^ "'^^*'' «^«-25. fi"e barometer stood at 29 •- in on the fifth dav at 30. a. i„ +1... *,.ii„....v . , . .,_ '" "'•' o" day at 30 , the seventh day at 31 fr in., the following day at 30 in. Vv'luit was tl ^S. A man took 7501 steps in walking 3 average length of his step? }0 I ,i'o in., and on le Weekly average? IjV mi. What was the 146 ARITHMETIC. U. A man whose steps average 2^ ft. in length walked 9 mL in 2i hr. How many steps did he make on an average per minute ? 25. In emptying a cistern a tap discharged an average of 24-j- rral. per min. for tiie first 6,^ min,, an average of 19,3 gal.^per min.^for tlie next a^ min., an average of 1.3.^ gal. per min. for tiie next .31- nnn., and a total of 29i gal. in the next 7i min., at tiie end of wliich time the cistern was empty. How many gal. did the cistern at first (■ontam, and what was tlie average rate of discharge per min. for the wliole time ? 2G. I bought 50 yd, of calico, part at 1.3 ct. the yd. and the remamder at 18 ct. the yd., and paid .$7.02 for the whole. How many yards did I buy at each price ? 27. How much tea costing 54 ct. the lb. must be mixed with 18 lb costing 45 ct. tiie lb. in order that if the whole be sold at 60 ct. the lb. there may be a gain of \ of tlie cost of the whole ? .?. shall have \ as much as A, B and C together (•alked 9 mi. in per niimite ? ■ago of 24| gal. d. per mill, for or tlie next 3} a end of Avliich cistern at first •er iniu. for the e yd. and the i whole. How- iced with 18 lb. d at 60 ct. the certain station 3 sec. , respect- I min. late on in? n. on Tuesday ing TJiursday, erage speed of lontreal time, irage speed of i. by 9.30 a.m. jy 12.05 p.m. ■h I walked at ^hich shall be svich that the I. longer than 3 first shall be length of the oportinn that liave li^ times [ C together. APPLICATIONS OF THE PIIECEDIN(} IIULES. 147 3o. At an election one candidate polled 39 votes more tlmn ^ of !o! """J'7f "'^ ^"' *''^ ''^^''' t^>« t«t'^l '■'«'»ber of votes ca.t being 1-4, . i ,nd the number of votes cast for each candidate. 30. Divide S79 among 8 men and 10 boys, giving each man $2.07i more than three-quarters of tiie amount given to each boy ^r Divide 74| bu. of wlieat between A and B so that if .( give iV of his share to B tl.ey shall have e.^ual quantities. .3S. A man who had three soils, aged respectively 18, 12 and 10 years, left hi. estate to be divide.l among them in proportion to their ages. ^A hat fraction of the estate is eacli to receive ' 39. Annie is 12 yr. 4 m. an.l James is 15 yr. 5 m. old. Divide .$9 between them so that Annie shall receive 50 ct. more than she would receive were the money divided in proportion to tlicir ages 40. A and B, wlio were 22^ mi. apart, commenced at tlie same moment to walk towards each other, A walking 1^- mi. per hour faster than B. They met in 3 hr. 18 min Wiiat were their respect- ive rates of walking ? ^/. Three townships have to r^ . .,aong them the sum of $7450, each township to raise a part c_ .a.s amount in proportion to its ITZT-J^' *''^ assessments are $1,745,080, $2,385,000 ami sS4,,G.3,o40 respectively, find to the nearest cent the amount to be raised by each township. /ot ^?'"^ ^^/ ''"'* "^ *^^ stair-carpet @ $1.35 the yd. for a flight of 23 s eps of Hi in. run and ^ in. riser, allowing l.^ yd. extra at top and 2i yd. extra for a turn in the stairs. (Reckon to nearest eightliof ayard.) 4'!. A map is drawn to the scale of 36 mi. to the inch. Find the total length of a railroad who«e several parts measure on the map St in., 4| in.,-2J in., l^r^ in., and l^^ in. respectively. The lentjth of the circumference of a circle is very nearhj 31- times the leiujth of its diameter. 44. Find the length of the circumfer- ence of a circle 3 ft. 4 in. in diameter. 43. Find the length of the diameter of a circle 7 ft. in circumference. 4. A locomotive wheel 4' U" in dianieter, making an average of 707 revoluti(ms per 5 min., travels for 4 hr. 10 min. How far does it go in that time ? 30. 'J'he front and hind wheels of a waggon being .S ft. 8 in. and 4 ft. 2 in. respectively in diameter, how many revolutions will a front wlieel make more than a hind wheel for every mile travelled? SJ. 'J'lie lengths of the diameters of the front and tlio hind wheels of a carriage beir.g 3 ft. 4 in. and 4 ft. respectively, how far will tlie carriage ha\e to travel hefor- the front wheel will have ma.le 100 revolutions more than the hind wheel ? S^. Of two rectangles of tlie san.e area, one; is 7' 6i" long by (j' ^" wide, and the other is 1 or wide. Find its length. .«. Find the area of a rectangle whose perimeter is 2r)0 yd., and whose length exceeds its breadth by 25;^ yd. S.j. Find the distance travelled in ploughing (i.j A. of land, the furrow averaging 9 in. wide. J.7. How long would it take to plough 7 A, 90 sq. rd., the horses travellmg 2J mi. per In-, and the furrow averaging 9.^ in. Vide? J.V. A field [02 r.l. 2 yd. x 41 rd. 4 yd.] yielded .SIO bu. of wheat. How many bushels was that per acre ? .17. The scale of a certain map is 40 mi. to the inch. Find the area represented by a rectangle on the map 2g in. long by 1}.', in. wide. ' " ' oS. Find the number of sq. yd. in the total surface of a recihangu- lar block of stt ne 7' ^" x 2' 8" x Gi". The area of a circle /,s ivr;/ rwro-hj 31 times the area of the square de- scribed on the radius of the circle. 50. Find the area of a circle of 2^ in. radius. 6'f7. Find the area of a circle of ^ in. in diameter. /^ SQihnE ON \ RADIUS. ^ y I'i 9 iriin. 9 sec. 38 radius, find 3un(l the sun. >y' in diameter ? an average of How far does .3 ft. 8 in. and lutioiLs will a lile travelled? o Jiind wheels iw far Mill the ave made 100 long by «' .S|" i 2r)0 yd., and . of land, the 1., the horses n. wide ? bu. of wheat. h. Find tiie ife' I'y 1 1 V in. f a reu.'-uiit'u- RAOIUS. AI'PLICATIOXS OF THK PKECEDLVO RfLKs. 14!) liJ whn1,''"* 'r '" 'T^ ""^ ^ P*-'""^' '^ ''^ ^'- i» *'i'^"'-ter. If a cent e .holly on top of a penny. Hn.l the area of the upper surface " the penny remaining uncovered * ^ ^ 0,A cube of chee.se 1^ in. on the edge was cut into cube.s g in on the edge. How many of tliese M-ere there » 0'.7 How many cubic feet of plaster wouhl be required to plaster a rectangular ceiling 18' 8" by 14' 4", the plaster to 1 e g n. thS 64. If from one end of a stick of s,uare tin.bor 21' x 7"^*" there be cut off 7;. cu. ft., find the length of the stick remaining. " Find the value of- o planks 12' x 9"x 2L" r«l $17.7,'5 " '« 6'cV. 1 , 7 r)0 scantlings 1 (i' x 7" x 31" H $2,S. 75 " 6Yy. 12,750 boards 16' x 5"x fVo), $;jl.50 " 70. A pile of cord wood 5^ x 26 V r«; $.3.75 tlie cord. ~/. A pile of bricks 12 bricks long bv 20 wido bv 9-, hi 1 the dimensions of an average-size.l brick. 7;^. How many tons of earth must be* removed to add l.V to the .lepth of a canal 7 mi. 425 yd. long an.l averaging 25 ft. wi le if a cubicyardofeartii weigh 2956 1!,.? • ^^Kie, it a 7.;. Find the weight of the sleepers for 37i mi. of railway if tlio sleepei. average 7 ft. 9 in. long. 10" broad and 8' thick, and are laM ?: .? ^ ""J^^S^ thickness of a slate 8"x 10" which weighs OK oz., if a cubic foot of slate weigh 180 lb. 7.7. How many gallons of water will pa.Js under a bri.Igo everv 10 tne lato or ,i| mi. per hour? 7ti. Into a rectangular cistern whose floor measures fi' 4 V bv 4' QX" tirr; "fiH°rj"'-^* '•" "^^^^ °' '"' ^'^^^^ p^^- ^--- «<^- '-'/-in t take to fill the cistern to a depth of .3' 10^' ' b i it 77. Find the cost @ $18.65 the M. of the lumber for a board fence five boards high to enclose a rectangular field 65 rd. x 36 rd tie lumber to be inch stuff 7 in. wide. 7S A lidless rectangular box, whose outside measurements are 4 ^ long X 2' 7^ wide x 2' deep, is ma.le of wood 1 J" thick. Find Its content in cubic feet. i.m 150 auithmktk;. The number of cubic unittt in, tho volume of a right cylin- der is equal to the product of the numbrr „/ ,s,ii«t,-e units in its (circular) base and the number of corrcxpundiufj linear units in the length of the cylinder. '^'9. Find the content of a cylindrical measure 8 in. deep by lOJ in. diameter. .9^ Find the content in gallons of a cylindrical measure 10^ in. deep by 8 in. diameter, taking 2.') pt. to the cu. ft. 81. A boy spent 2- of hia money and tlien had .«;i.20 left. How much had he at first? 82. A, working on piece-work, can do only f> as much work .la B, and so earns .55 ct. per day less than B. How much does eacli earn per week ? 8,1. A man sold ^ of his farm and then J of the remaindtjr. How mucli of Ids farm did he sell ? If he received $1210 from both sales, at tliat rate what was the value of his farm ? S',. A man paid .V of his money to B, i\ of it to O, and % of the remainder to J), and lia.l 15 ct. left. How much had he at P'-st? 85. A, 5 and C liave to do a certain piece of work. A v!oes A and then goes away; B does f of the remainder, and then G finishes it. What fraction of the work is done by C? If $I(J.50 be paid for the wliole work, how nuich should each receive ? 8n. By selling a house for $:.S!)90 I lost | of its cost. For what amount should I liave sold it to gain J of its cost? 87. If f; of 2 lb. of sugar cost as much as 2^ 11). of rice, and if 3.J lb. of rice cost 15 ct., what is the price of sugar per pound ? 88. A can walk 4 mi. while B walks 5 mi,, and /; can walk 6 mi. while C walks 5 mi. Compare yl's rate of walking with C"s rate. 89. How far will a train travel in 1 lir. 35 min. at the rate of Ox'V mi. in 14 j min.? 90. A watch is set right at 10.25 p.m., and it gains .3;^ sec. every hour. At what o'clock will it have gained exactly 3 of an hour, and what time will the watch then indicate ? 91. A boat's crew can row at the rate of 9J mi. an hour in still water. At what rate could they row, 1st up, 2nd down, a stream running at the rate of 2^ mi. an hour ? 92. An oarsman rowed 3 J- mi. down stream in 20 min., and bank again up stream in 36 min. Find his rate per hour each way, his rate in still water, and the rate of the stream. it. For what APPLICATIONS OF TMK PKECKDIXG RULES. 151 hour' ^r*''^"; ^^ ^'''n"' ^"""*^' '' '■"""'"^' •■^* '^ '•^t« '^f 24 „u. per while the train parses; 2..!, if he walk at the rate of S un. por ho r 3" V 7 • *'r *"'" ''' ""^'"^' •'^^*^' ^f he walk at thf a e o 3 nil. per hour ui tlio opposite .lire- .on v .0_J. A cistern which hol.Is 200 gal. can l>e filled l.y two tans of which one supplies rA gal. per sec, the other H J. per si I he first tap be turned on for 10 .„■„. and aftenvan L Wi" run ogethcr, in what length of time from the moment of op ^/the second tap will the cistern he fiHo.l ? optmng the ft-7. If 5 men or 16 hoys can do a certain piece of v ork in ] I hr m what time could .3 men an.l 48 boys do the'same work ' ' Jij. isvo men who are 12i mi anarf- «f.|.•^ .,<■ +k + J. 1 ^ , ^^:f I'll, apart stai t at the same mompnt +n to travel towards each other, one walking at the rate of ^ ' per hi., the other driving at the rate of 10\ mi ner hr Tf..^ 1 u 9S A can do a piece of work in 10 da. ; B can do it in 12 da A days'::!;: r v^t' '- *'^" ^-^-^^ ^^ ^- ^^ •>- -4 Clays ^^ 11 the two, working together, finish the job ' .A9. ^ and Z? start at the same moment to run in the same direc ^oii round a circular track. A making 8 rounds to /i's 5 W eTe will ^ overtake B the first time? the second time? the third il" Ho. many rounds will each have made on each occasion v nvll ; r. V '"""^ *^' '"'""*' '''"^"'^^ ^^ ^ ^l««^k are together at 12 o clock. At what times will they be together again" At whit "i^iiiJm;:^:;'---^^-^''--"^ JOl. At what times will the minute-hand be half as many minute- spaces ahead of the hour-hand as the hour-hand marks hoursT"*' gi tnacin! ""4 ""''^-^^ '' ' ''''■' ^ "^ 2 gal. and Cof 5 gal. capacitj-. A is eini,ty, B is full of water, ami C is full of z;s iii^v^"^;;^ ;: f • f v^^'r^^^^^^ ^•-" ^ -^ ^ ^'^-^ !.»• *i ; "-■ 'h™*" ^'""'«' the water and vinegar CHAPTER VII DECIMALS. I. NOTATION AND NUMERATION. In tho ordinary or Arabic notation a figure standing imme- diately to the right of another denotes so many units each ten times less than the unit of that other. Thus in 325 tlie unit of the 3 is a hundred ; that of the 2 is ten, wliich is a tenth of a hundred; and tliat of the 5 is one, which is a tenth of ten. By continuing this system beyond the ones, a figure immediately to the right of the ones would denote tenths; the next figu o to the right would denote tenths of tenths, or hundredths; the next figure to the right would denote tenths of hundredths, or thou- sandths, and so on. In the case of numbers thus containing figures denoting units less than ones, the figure which denotes ones is indicated by a dot (•) called the Decimal Point placed between it and the figure denoting tenths. Thus 4 hundreds, 2 tens, 7 ones, 8 tenths, 5 hundredths and 6 thousandths would be written 427*856. The units denoted by figures to the right of the decimal point •are called Decimal Units. A number containing decimal units is called a decimal number, or, briefly, a Decii..al; and the part to the right of the decimal point is called the Decimal Part,— the part to the left is integral. The Order of a Unit is its rank as determined by the number of times the prime unit must be multiplied by 10 or divided by 10, as the case may be, to produce one of that unit. Thus, tens are of the first integral order, for 10 = 1 x 10 ; hundreds are of the second integral order, for 100 = 1 x 10 x 10; thousands are of the third integral order, for 1000 = 1 x 10 x 10 x 10, and so on ; tenths are of the first decimal order, for 01 = 1 -MO; hundredths are 152 is; the next ths, or thun- NOTATION AND NUMERATION OF DECIMALS. 153 aLtf "rthi'lT'' "f ^'"" ^•^1 = 1^^0-^10; thousandths are of the th.rd doonnal order, for 0-001 =1-, 10 4- 10 -MO, and so on. 1 he prune units or oneh are of the zerotJ. order The greater the munber of nmlfciplioations, or the h.s the nund. o dmsu.ns by 10, the higher the order; the fewer the n d t i p oat. or he n.ore numerous the divisions, the lower t e lower order luu. thousands; while hundre.lths are o higher order than thousandths, but of lower order than tenths o ones, or tens. The number 324 -(507 represents 324 and 6 tenths 5 hundredths and 7 thousandths, and might be so read ; but since 1 of any order xs equal to 10 of the next lower order, tenths and 5 hundredths IS 05 hundredths, and 05 hundredths anct 7 Ton sand hs :s r.o7 thousandtlis. 324-057 is therefore reaa 324 Z 65. thousandths. Sinularly 4,023,148-478,602,0 .8 read 4 mil- hon 23 thousand 148 and 478 thousandths 6o2 naihot,^ tenths of milliontlis. nimian.ne . Another way of reading decimal numbers, ...u oi.e .Hat « very convenxent in practice, i.s to read the ..egrai pa^ rth! sual way, then to say "point" (<.r "decimal;, ana Jen 1 Ihus 127 -00435 IS read "127, point 0, 0, 4, 3, 5. " Read — J. 7 -06. i'. -756. 3. 75-6. EXERCISE LXVI. 4. -2304. o. 2-304. 6. -002304. 7. 1-0001. S. 1000-1. 010001. 'J. 10. 132.5000-625. 11. 13-25000625. 1^. 132500 0625. Write in Arabic notation— 13 Seven thousand three hundred and forty-nine and four hun dred and six thousandths. '^""" U. One million and seventy thousandths six milHonths "; 2r *\7^^"^'^h and one hundredth of a thousandth. i6 Ihree thousand and nine and two hundred and seventy thou sandths 8 millionths and one tenth of a millionth ^ l/aw"' "" °"^'" "^ *'" ""'''^^ '^^"'•^^ '"^ ^"««*-- «' '0 and 154 ARITHMETIC. II. ADDITION AND SUBTRACTION DECIMALS. OF Decimals ure added and subtracted exactly as in- tegers are. In urran^'ing numbers for addition or for subtraction, all figures denoting units of the same order, and only these, must stand in tlie same vertical column. To secure tluH, vrlte tho (jlcen immhvrs so that their decimal points shall be in a vertical column. The decimal point of tJie sum or the difference will be imder the other decimal points. EXERCISE LXVII. Add together — 1. 37 645, 283-039, 5847-036, 86-453 and 3768. 2. 459837, 4-59837, 45-9837, 459837 and -459837. S. -00876, 1-08972, 1000, -OOOD and 900 009. 4. 36400, -00364, 287 082, 578936 and 307-125. Subtract — r>. 97-46 from 368-24. 6. 109-87 from 193-857. Find the value of — 9. 37-5 + 48-26 + -00831 - 85-759. 10. 2-02 - -0909 - 1 -9009 + 19-009 - 7. -777 from 7. S. -9999 from 10. 9-029209. IJ. John had $7-38 more than James. John spent $29-13; JameS spent $19-45. How much had James more than .John then ? Ii2. A man sold -375 of his farm. How much of it had lie left ? /.■?, Three men did a, certain piece of work, The first did -.ST of it and the second did '33 of it. How much of it did the third man do? ' U. A farmer had 23-478 A. in one field, 29-38 A. in a second field, 18-076 A. in a third field, -875 A. occupied by barns and as barn- yard, and 1-305 A. taken up with house, garden and orchard. The rest of his farm, which consisted of 100 A. in all, was in woodland. How many acres of woodland had he ? MULTIPLICATION OF DECIMALS. 155 )N OF ;tly Eus in- traction, all these, must is, vrite thr in a vertical rence will be 29-13; James ;hen ? bcl he left ? •st did -37 of lie third man second field, md as barn- ■chard. The iu woodland. III. MULTIPLICATION OF DECIMALS. Decimal numbers are m,.ltiplied together exactly as integral numbers are The reasoning wluch proves tl>.t in n.ultipb'ng by any number of intejnd units the order of the units of the product rshujh.r than the order of tl^e units of tlie nmltiplioand by the order of the units of the nmltiplier, also proves that in mu It.plymg by any number of ckcimrd units tlie order of the umts of the product is hunr than the order of the units of the multiplicand hy the decimal order of the units of tlxe multiplier Thus nxult.plymg by hundreds raises hundredths to ones, tenths to tens ones to hundreds, tens to thousands, and so on; multi- P yu.g by unulredths lowers tens to tenths, ones to hundredths, tenths to thousandths, and so on, ^.«..,.,,;e. -Multiply 237 -G by two hundred and one, and also by one and two hundredths. (1) (2) 237 20 1 237 G 4 752 47757 -G 237 -G 1-02 237 G 4-7 52 242-3 52 Hence to multiply two dec.mal numbers together. Write the multiplier under the mnltipUouul so that the oms' ^Z^ilr^'^'-' ""'"" '' '''''-' ''' riyht-handjiyure of the MuWphj by eachfi,jure of the multiplier in regular succession begrnmug u.th the figure of lou.st order, and Zite each ^Hd product so that the right-hand figure shall he in the sarue ^ ^ column as the figure of the multiplier which produced it 4"^r''' ^'"^^ ''''''-'^ '-'' '^- -"'' '^ ''^ The decimal points of the multiplicand, the partial products aiul the total product wdl all be in tf^ same vertical cohJn. n i.-.d AltllllMKTir H.fiiin/ilvH — (I) (ii) I- '»n (») Ilia 14 T75 47jr» r)HM75 •2:1 ■I- •(M)ii>;{ •()(»() 1 JITi-) •000 \)i:,{) •004 725 ■oor)Hii7r. •J4I 75 •!»4r. 4^7i.'r. n-Hii 7o It is iisuiil to ..luit from tlm jHirtiai prodiictH llu-ir .l.-ciinal points aii.l tUv 1.011.1,'litH on tlioir loft,. Wluui tjiia \h dnn,;, \\h- nilo for tli,^ imiltii)licati(.ii of fart<.rH oontainiiij,' (IfiMiiiaJM may bo Hlati'd M>iUi,,l,, thr/,uf„rs fo,i,fl„r „s if thr,, „rrr Inlnjral, ,niil flow th,' rUjht Iniinl ,>/ (li, jnuuhiri in,trk ,>()\f„r ,l,rim,ilH an wnnij fhjmrx ii.s thny ,,l,l the niimh,r nt\f!,i,nrs in fhr pnuhivf l,r hsH than, the nnmhrr of p'llinrn to /.,. m,niy,l o{f\ siipplii thr ilrjirimrj, loj n'ritimj Hno,,hh on thr hft 0/ thr pnxhirt. Thr r,ihir of a drrimol is (m)^ rhomjril h>, irritiiuj noiojhts to thr ii ahsi'iico of tlu'so iioiij^lits lias no vW'wl on fhc ordi-r of the units of tlio oflu-r fis,niirs, and f Iicioforl. ],as no effect on tlioir vaUu-, and \W iionj,ditH thenisflvos liavo no value Tiuis7:{0 7:? 7^;?no. ., ,,. , EXERCISE LXVIII, Multiply — /. ;r 4r)i.y 10. 100, 1000, looooo, 01, ooi, 0001, o-ooooi. ..'. OOOT^J l)y 101), ()(»1, lOOOO, 00001, 1000000, O'l. ..'. 10 by 0-04, 00(j, 7000, 0-00002, 20 00. [Whi'ii a ihriiiial tonn'n- coiilahi^- no hifiynil inot thi^ witi/ be indicated by irritin;, a iuu<;,ht in fhr on,:,' i,hu-r, ,j,v /.v ,tonr in Ihr (lore prccedinf, 2>robleni^; but. an thin iwuiiht I's rralli/ of no iwc, it in cmtoniari/ to omit it.] Find the value of — 4. 7S:V4(5x-7. ■7. 7S:^4(5x70. (.'. 7.*<-'V46x •". 7. 78346 X 700^07. A'. ■047t)x4'2. .''. •047(ix4-2. !0. •047()x -42. IJ. •047*Jx4 02. /..'. 1 -476 X coo;?. /-?. 00031) X -.id. U. -079 -•; 300 X 03. J5. -004 X •COS X •5. DIVISION ,,!■ I»M(IMALH. 157 IV. ■DIVISION OF DECIMALS. ('ami.; I. - \VIh,„ Hu, .liviMcr in inttigrul, ah ,/,,,,,, /,, /, , ^,^^,.,. ,,^ ^^^ ^,^^^ ^^^^^^ ^ ^^^^ , ^^^ ^ ^^^ ';::'• ?;/-•« « d.r.,ua ruint ufUr Ik. JU,,.. tin. J..- LnL I, Hi' vi«ori.adedn.al, /'u '.V «/ /,v,.7 «., v,nny decimal j.laa-s as the divisor dors • an «./KaZ » »,«/..v of places to the ri.jht in (hn dlvvlnul ■ J hen divide as in Case I. K.moving tl.o duoi.ual points to tl.o rigl.t nu.lfiplic.H l.ofh '-•'-;• -f -I'vu on.l hy 10 a.s ...any ti.nes as the „ lint is o -H'od places. Diviso. and dividond a.o thu.s n.uItipHod H by tho Han.onund,or; tho ciuotiout will ti.eroforo not be afFoctod. Kxdviples — (2) 4r,-2-~-0H. M8)4(;20-0 CO 72-4.-, ^•!t. 0)724-5 80-5 (.'{) -001 . ooo;]. MW<.'i)fW10 In exam iscoiitiiiUfd TJ,l)t '*'^^"""' "°"»''ts are not a.:tually written dow, I if the .\' were there 3 •333 + 'I, I'lit the work 158 ARITHMETIC Removing the decimal point of aiuj number 1,2,8, .... jdace^i to the riijlht imiltiplius the number by 10, 100, lOOO, ; remov- ing the decimal point 1, 2, S, .... jyhices to the left divides the number by 10, 100, 1000, For by removing tlie decimal point o)ie place to tlie right the value of the unit of each figure composing the number is increased ten-fold, and therefore the whole— that is, the number— is increased ten-fold. Removing the decimal point tin, places to the right increases the value of the number ten times ten-fold, or an hundred-fold. In like manner the other cases may be proved. i ■ t ^. , EXERCISE LXIX. Divide — 1. 4.38-976 by 7, 8, 9, 11, 79,474. 5. 250-4.3 l)y 4, G, 7, 17, 2,1, to 4 decimal places each. .?. 40-04 by 10, 100, 1000, 7, 70, 700, 110, 1.3000, 1.300. 4. 72-09 by 10, Q-l, 100, 001, 1000, 0001, 0-00009, 0-0089 Find, correct to 4 decimal places, the value of — .1. l-07')-f-12.-). 9. 7 •29-^-030. 13 6. -004^-5. 10. 547^-007. I4 7. -04 -=--005. 11. -8^-0004. 15. S. 40^-0005. 7;A 6^-000725. 11 -02 -f 003-2. 8-0018^900. •006 ~ 70. •008 -kS -8. EXERCISE LXX. J. One hundred and twenty steps, each 5-875 in. high, lead from the foot to the top of a tower. What is tlie height of the tower ? 2. The side of a square plot of ground measures 13-3375 yd. What is its area ? 3. How many cubic feet of water M'ill fill to the depth of 6-75 ft. a rectangular tank 25*475 ft. long by 15-64 ft. wide? 4. The average annual death-rate in a city of 64 000 inhabitants is 23-56"25 per 1000. Find the total number of dcatiis in 7 years. 5. In every 1000 parts by weight turnips cont- in 905 parts water. How many gallons of water are there in 1000 bushels of turnips ? G. In every 1000 parts by weight rice contains 741 parts of starch, and potatoes contain 155 parts. How much starch would be con- tained in 1 lb. of eacli ? How m- y riounds of rice would (•nut.ain as much starch as 100 bu. of potatoes ? 3, .... places , . . . . ; rcmov- <'t divides the ; the decimal )f each figure therefore the Removing . the value of )\d. In like 0. 1-0089. •02-=- 0032. [)018-r900. 1)6^70. 08-K8-8. jh, lead from tlie tower ? 75 yd. Wliat thof 6-75 ft. inhabitants 11 7 years, parts water. f turnips? rts of starch, ould be con- nuld cniitain INTERCOXVERSION OF DECIMALS AND F|{ACTlONS. 150 V. INTERCONVERSION OF DECIMALS AND FRACTIONS. To express a decimal as a mixed number or a fraction, JVnte the decimal part for numerator, omittincf the decirnal point and for denominator write 1 followed by as many nonjhts as there are decimal places in the given number. Reduce the resulting frac- tion to lowest terms. Ex. l.~2-b = 2f^ = 2h. ^.•. 5.-00004 = „{J5oD = .m. A fraction whose denominator is 1 followed by one or more noughts IS called a Decimal Fraction. EXERCISE LXXI. Express as fractions in their lowest terms— 1. -2,-). s. 1-476. .;. -024. 7. 70-64. 1 -75. •1476. G. -0024. S. 7 064. .9. .S-62.-)00. 10. 3-0062,), To express a fraction as a decimal number correct to a given number of decimal places, Annex to the numerator a decimal nought for each decimal place required and divide by the denominator. Increase the last figure of the quotient by 1, if the next fi.M-.e would have been 5 or upwards had the division been continued. EXERCISE LXXII. Express as decimals correct to "> decimal places— 1' A 3. Ig. 4s; "61! -• Tlh! 4. ^rV'TT. G. Hf. S. fj«. 9. 10. i;; Solve the following problems by decimals, working to 4 decimal places and verifying your answers by reducing to decimals the answers given for the fractional solutions:— Exercise LIIL, Probs. 13 to 26; Exercise LV., Probs. 9 to 24- Exercise LX., Probs. 1 to 5; Exercise LXIL, Probs, 13 to 24. 160 ARITHMETIC. W «1 VI. DENOMINATE DECIMALS. nuinhin Express 7-2578125 mi. as a compound dtiiominate 7! -2578125 -mi-. 1700 yd. 15 4(i87o00 180 4(5875 2r>7_8125 4531 -75^. _3 ft. 2. -25 ft; 7'2r)78125 mi. -7 mi. + -2578125 mi. •2578125 lui. --. -2578125 of 17C0 yd. iyd. 25 ft. 12 = 453yd. + -75j-d. = -75 of 3 ft. = 2 ft. + -25 ft. = -25 of 12 in. =3 in. in. 7 mi. 45:; yd. 2 ft. 3 in. JE:r -Express 2 pk. 1 gal. 3 .jt. 1 j.t. as a decimal ..f a bushel. 2 ) 1 ].t. 4)3;5^(t. ■ 2 ) 1 -STSjal. 4)2;o;{75pk. •734375 bu. -■2pk. Igal. 3qt. 1 J, Sqt. 1 ].t. ^Sqt. + Jqt. -S-5qt. 1 gal. S.:. (jt. = 1 gal. +3-5 <,f J -;il. = 1 •S75 gal. 2 i>k. 1-S7.". gal. --2 pk. + l-S75 of ] jik. = 2-9375 pk. = 2-0375 of j"l)u. = -734375 bu. EXERCISE LXXIII. Express as a compound denominate number— /. .•J-47->8 T. ,;. 4-2G25 yd. ,7. 29-.-,30875 da. 4. How many seconds are there in -001108 da.? ■■>. llcdiice -OOOIiTo A. to sq. in. 6. Express 8.33 yd. 2 ft. 9 in. as a decimal of a mile, correct to 4 decimal places. 7. Express 1 da. 18 Iir. 28 min. .3.5-94.. sec. as a decimal of a day, correct to (5 decimal places. 5. Express 4 ch. 45 1. as a decimal of a chain n Express 12 A. 3 sr,. ch. 7o;)0 sq. 1. as a decimal of an acre. -n , V ,n ^ «'*'''^ ^''Tressed in .-u-rcs of a rectangular field 17 ch. i»0 1. by lOcii. 20 1. CHAPTEK VIII. APPLICATIONS OP DECIMALS. of a Lushel Jqt. fST) of J a^l 'al. 1-875 of . I'k. pk. Of ihu. >bn. I. PERCENTAGES. The plmu^e per cent., a sliortenod form „l the Latin »,r »*.», ,3 equivalent to tho E,„li.h word hundredths. ThC J p-r cent, of any quantity i, 3 hundredths of it; 124 L cen ' .3 12J n.,Ktadtl,3, and 135 per cent, is 13^ huX, h ' T- -0 2oT '."J"';,"' "°^,r "'"■ ■" l-dredths. Hence EXERCISE LXXIV. Read the following rates and write them deci.nally — ^- 5%. i^ 7i%. ,. 33d%. ,;. 1,0%. ■ 5. i;/. \Vrite the following decimals as percentages •- ^••07. 7.-70. ...375. ,.2-25. jo. -0075. How much is — //. 37otS700? L'l 125% of 120 yd' i^^ 10, ,' of $22.> ? 2,;. I2^% of 44 lb. ? What rate per cent, is i7. .S3 per §50 ? /cr 250 lb.? What percentage of — ■^0. $150 is $6? f/ 4Bft„^i -an i« ~A 480 gal. IS 60 gal? ^^. 750 A. is 18| A.' Jxpress the foUowin, percentage, as fractions in their lowest ^-^•25^ .,.20%. ...12^^. ,,,33^^_ ^^^^^^^^ 161 ^''- 102i% of §12.50? ^6". 1% of $75. 80? i9. 9 in. per 100 yd.? 102 ARITHMETIC. SS, Increase §225 by 8% of itself, 29. Decrease %?,m by G% of itself. 30. Decrease 1250 gal. by 8% of itself, and then increase the re- mainder by 8% of iVvr//. St. A farmer who had 8) sheep sold 20% of them. How many did he sell? 3.^. A man bought a house for S1760, For how much must ho rent it to obtain 12A% per annum on this price ? 33. A teacher spent on books !^i7'25, which pum was 7% of lua salary. Find the amount of his salary. 3.'f. The average attendance of -^upila at a certain school was .55, which was 62J% of the number of pupils enrolled. Find tlie number of pupils enrolled. 5.7. Willie Smith gained 8.^ lb. in weight in 12 months; this was an increase of 7.^% of hia weight at tlie beginning of the 12 months. What was his weight at the beginning of the 12 months ? SO. A liouse worth ^2750 rents for $320 a year. For what per- centage of 'ts value does it rent ? 37. Tl'.e total popillation of Canada in 1881 was 4,.324,810. Of this munber 609,318 were not born in Canada. What perceutage of tlie population was born outside of Canada ? .7,?. In 1884 the values of the several classes of exports from Canada of Canadian production were:— Produce of the mine, §3^,247,092; of the fisheries, $8,591 ,654 ; of the forest, $25,81 1 , 157 ; animals an.l their produce, $22,946,108; agricultural products, $12,.397,843; manufac- tures, $3,577,535; miscellaneous articles, $560,690. Find the per- centage which these separate values form of their total value. 3D. A man spent 85% of his income of $850. How much had he left? 40. A man who was receiving $8 -40 a week had his wages increased by 8%. Fi -i the amount of his wages per week after the increase. 41. A man wliose wages had been increased 10% was then in re- ceipt of $8-14 per week. How much did he receive per week before the increase ? 42. A house was sold for $3451, which was 15% less than it had cost to build. Find how much it had cost. 43. A man's wages were decreased from $7-80 a week to $7-20 a week. Find the rate % of decrease. 44. From a barrel of 30 gal. of oil 8 gal. woits drawn oif. What percentage of the original quantity remained ? AI'I'IJC.ATIONS OK PF:uckNTAGK. 163 increase the re- im. How many mui^h must Iks 1 was 7% of his 1 school was .W, ^intl tlie number lonths ; this was ' the 12 months, ths? For what per- I 4,324,810. Of it pereeatage of rts from Canada , $.%247,092; of tiimala and their ',843; manufac- Find the per- al value. w much had he wages increased i: the increase. waa tlicn in re- 3er week before 2SS tlian it haul 'eek to $7-20 a wii oif. What 11. APPLICATIONS OF PERCENTAGE. PROFIT AND LOSS. Tho Prime Cost of merc)iandi«o or otlu-r pro„erf v W H . suu. paid by the purchaser tiuu-cf to the BellJ^I^f" ''^' "^' The Gross Cost of merchandise or other property is the sun of theprnue c.t, all charges for purchasing, and .11 e x ■ Z fur freight, storage, handling, and such like. ^ Profit is the a,nount by whicli tlu, selling price exceeds t?,. est pnce. KH l>,^t or a.l. is the amount by d. d th « ii>g pnce exceeds the gross cost. ^^^' Tlie Rato of Profit i« ii«ii..n„ i the prime cost. ' "'^""'""^ '"' ^ Percentage of Thus, if j.co(i3 oostirifT ijo are «old /,„• ;<«■-„ ''"^^'■""•''^'^ SC-20-$5-C0 = il-.2„. uiifl tlie KATK of pro/it ia '^^ '-^ *i') IJO = ■24 = 2^:^. Loss is tlie amount bv whi(>h fl,« c it ■ . the cost T.ri,-P A- / 7 T ''"'"- I"'^"« ^'^"« slK-rt <:f uic cost puce. J\ .i la,s IS the ani.,unt by wliich th.. «..n;, tails short of the gross est. "° I^"^*^ The Rate of Loss is usually expressed as a percer .« of .1 prime cost. pLrccr. ge of the Thus, if floods costinfe' iu are soi.l for *;)(jii *'*=^'"^^'"'' «12-00-§O-C(. = S2.40. and tlu; KATK ,/ ;„,■, is j2-4f)_^ 4il2-00~ -"" 20/„. EXERCISE LXXV. Find the profit or the loss and the rate of - ?12. ''. 150. J. «225. Prolitoroflofss.givcn:— Sellin„, ./. 94o0. fr.rj-.TO. 1180. $198. • > (.). 6'. $500. S2-.00. $500 -.50. 104 AKITHMKTIC. Find the profit or the loss and the selling price, given :— Co.1t. Rate of Profit. Cost. RatcofLoM. ^' 9150. 6%. 70. $42-oU. lOr cS'. $225. , 5%. //. $2.-.0. ir/ 0. $137-50. .30%. J;:. $100(). 31%. 13. What will ho the rate of selling price if the nite of profit be 6%? 11%? 20%? 7A%? 33^%? 110%? 14. What will be the rate of selling price if the ra;< of loss I.- •{ ' ' 7%? 10%? 7i%? 33i%? 31%? Find the cost. Sellinrf Price. I''. $17-60. ir>. $38-00. 17. $3744. ~/. If a grocer wore to sell at.a l>'(t.:o/ Profit. 3:5i%. Selli'iij Price. IS. §n-40. n*. $;;,s-oo. ^''X $1094 -50. Hate of Loss. 10%. 2/0' ' ftti (>C 15% tea which cost him •-i.'e for H5 lb., and how much 48c. the 11)., how much would ho 1 of this woidd { 'C profit ? ^'2. Silk which cost 62-40 the y.l. is marked at 20% loss. Find tho, selling price and the rate of tliis selling price on the dollar of cost ;?.?. A merchant paid for freight and other expenses on certain stoves !;v"^ each over the cr„st price. He sol.l thcin for$;?5 eacli, which was 40% advance on the cost price. Find his net gain and his rate of pt ofit on the gross cost. .?4. A man hvys a 1)aidc. on tlie $1 of original cost. How much does he pay for it ': :Jo. a man buys at 55c. on the $1 a bankrupt stock which cost per invoices $'J400, aiul sells it at an average of 95c. on the $1. How much does he pay for it? Hom' much does he sell it for? What is his rate of profit ? Jd. A man buys a bankrupt stock at 60c. on the $1 per invoices, and sells it at an average of 5% advance on the invoice cost. Find his rate of gain. 27. A man buys at 68c. on the $1 a bankrupt stock which cost per invoices $5376, Half of it he sold at 5% above the invoice prices, a third of it he sold at 12% below the invoice prices, and the remaind'er he Bold at half the invoice prices. Find his total gain and his rate of gain. I en:— lintc of Low. 10%. 14%. 3i%. ite of prfjfil be of loss hti 4 '? liicle of Loss. m%. ^liich cost liim uul how much •% loss. Find . the «^^ct business for The Gross Proceeds ..f a sale or of a collection is the total aniount recen ed by an agent f..r his principal rho Eet Proceeds of a sale or of a collection i« the sun. due -mu ..11 other charges. These charges include freight, handling storage, advertising, and such like. ".maiing, C.m,,u-.../oH .-s «,sna/ij/ n-cfenerf a< a rate per cent, on the nro.. pocer,sj>f .a., and cottecfion., o. tl,e prinre rost ,>fp.J and on the net amount of inve.'itmcnfs. EXERCISE LXXVI. 1. An age..t bought §750 wortix of tea. Find the amount of his ■nnmnssumat.S-. At 1%. At 4^%. At ^%. At iV .1^:?^ 't ur^t^f ^, .S;'^^ *•'« ^-.mt of hi. com. J What sum v^ill a princii'al need to l-emit to his a-cnt to buv M..0 wo,,h of flonr if the agenfs rate of connnission be 1 ^ ^ 2 l 4. If an agent collect 8468 on a connnission of 2i- what s„m will be due fi'om him to his principal ? " 5 An agent charged §29.25 for collecting $130(1 Whnt was iu's rate of connnission ? " ",ir \\,is ins to h.s prmcipal. What rate of comn.ission did he change •> the vt '""f""" "--'-nt -Id 4000 yd. of white cotton at 7k. cue yd. \\ h, sum should he remit hi,-r,oi i • • . ..erateofl%? At the rate of 2% ? At the rate of 2^;. V IfHi AI{ITII.>fKTr(' TRADE DISCOUNT. Discount ;,s ,ut, ahahment or rvdmtmi from the notninal price orraiuiiofaniifhlaij; as, for example, from the cataloyue or list price of an article, from the amount of a bill or invoice of goods ur of a debt, or from tlie face value of a promissory note. The Rate of Discount is usually stated as a rate p.-r cunt, of the amount from which tlt^ discount is nuid,'. Thus a discotii.t of 20"/^ off $146 means that "M of the JU.i is to he dethioted from it. •2nof S140 = lJi29-20; Sil46-829-20 = $llU-sO. Trade Discounts are reductions made from tlie catalogue or list prices of g(j(jds. Ill some branches of husincss the niainifactnrcrs and the wholeaalo dealers cata- logue their ifoods at fixed prices, usually tlie retail stllin- luj,.,,, uiirt tlun allow n'tail dealers reductions or discounts from these catalo-ue prices. Those dis- counts {,'eiiovallydeiiond on the amount of the purchase and the terms of payment, whether cash or credit. By varying' the rate of discount, the manufacturer can raise or lower the price of his goods without issuinnf a now catalogue. Vi'ry often two or even more .successive trade discounts are to be deducted. In such cases the Jlrd rate denotes a percentage of the catalogue price; the second rate denotes a percentage of the remainder after the first discount has been viade; the'thlr.l rate, a percentage of the remaiwh-r after the second diseoimt has been made; and so on. Thus, discounts of 20% and 5% in succession off any amount, or, as it is generally exiiressed in busniess, ^'0 and r> off, means lliat -20 of the amount is to be deducted from it, and then from the remainder -05 of that remainder is to be taken. ^.ca)«;;/e.--Find the net cost of tfoods amountinii' per catalotruo price to S840 subject to 20 and :. off. ' ^840 -Cdtaloijue price, •20nf§s40-- ](i,S SG72 =Pro('eeds of 1st discount, •05of 5072=_;i;j(io $638-40^ Proceeds of 2nd discount = AV< cost. APPLICATIONS OF PEKCENTAGK. 167 3 pt-r cent, of to lie (lefluoterl catalocrne oi* price to $840, 1. 5. G. EXERCISE LXXVII. Find the net cost of goods invoiced at- $440, subject to 15 off ■? CQon i- ^. «- «— .n 1 • . . ^ • '^'^-"' subject to 35 and 15 off ?/.jO, suljject to 20 and 1 ". nff / ®o w» i • «d60, subject to 25, 10 and 5 off. ^ $144 -GO, subject to 25, 15 and 12^ off 7. §435-25, subject to 30, 22;^ and I'ii off 8. The gross amount of a bill of goods was $445-50 an.l ti of successive discounts were 25/ ami IT, vi^^.l ' *''" ''"*"' J. inid the difference between a single discount of 45°/ off and su cessu^ discounts of 25% and 20% off a bill of ,«500 ' 25^-in!rlmnl;^f°""^ '' ^^"^^^^-* - — ^ve discounts of pi ices ± nid the amount and the rate of his profit and in ''7"«;"^;;^^'''"^ %'-'* ^^^y^ machines at a discount of 25 10 itr^e o7;:t ''^'" ^* ^^^^ ''-- - --'^^^- P^^-- ^i"' 13. Purchased goods amountint,' to $12 4fi4 40 y 1 1 * in92dava«1l r.Qi o > -d i . fl'i^404.4U. .Sold from them m J^ days jl 1,631 . 20. Balance of goods remaining unsold So 760 1 5 Required the total gain, the average daily sales (lunday f^ pted) the average daily profits, and the average gain per cent" ^^' 14. bold merchandise at an advance of 307 on cost M„ * aii^d in business, and I lost 25% from the sfl W " te Whl?"'' the net gain or loss per cent. ? ^ ^ ^'** '^'•'^ cl^Ltr'-^'"''^ T^"^ ^'' ^°"^^^ ^* 25% advance on cost, but con- chiding to give up business he sold his stock at 20% discount fZ the marked price of the goods. What was his gain or loss p! c fnt" 16. An agent receives $14,000 to invest in wl.P.f ^^'"'^ '=^"*- • bushels at 85c. ought he to buy for his pr n ciml l\ . '"'''^ ^n be at the rat/of ,% ; 2nd.^if il tlTrl" ^ ^ prbTsht^ {In each case icork to the nearest bmhl. ) 17. An agent sold a consignment of sugar charrrinor oi^, sion^ Heinvcted part of L proceed, SbToTfl™'; a^S per bbl., charging 2% commi,»i„„ , ,„d after deductin, sni f pen«, other than hi, co„,„.i,U.n. he rendtllt tu" LlaTae balance, wh,ch „„ ,900. Por how much did .„. ,ell tu"^^^] l(j« AKITHMKTIC INTEREST. Interest is tlio sum wliich tho . ..lo- , : ,u, .ley chiii-fiea tlu- borrower for tlie us^o of thu sum borrowed, or wliich a «'e(lit..r cliar^^es a dobtor for all50 was on 60 days' credit, and the balance ^^.lo on 90 kays' credit. P'ind the equated time. Interest on §50(1 for .-JO days = Interest on §.500 x 30 = §15,0O0 for 1 day " 2.'-.0 " 00 " = •< .. 250x00= Ift.COO " 1 "■■ " "^ '^^J^L'L- " " 210x90- 18, 900 '■ ■■ -^ *^*50 960 )848,000"( .. Interest on .9960 for 50^-^ daya = Interest on «!9C0 x 50f| =$4^ for , y. Equated time = Sept. 10+51 da'..-. = Oct 31. In working, omit wnts and take the nearest nimher of iJoUars. APPLICATIONS OF PEU( KMAJJK. 169 y cliarjres tlic iih u creditor npaid affor it till) ratu per >r its USD fur at 7^'. :lue at differ- >e dis' harged :o ei' ser thr or Equated 00; of this sum ilanco \\,i on 9(i )r 1 day. for V. of iloUars. EXERCISE LXXVIII. Find tlio iiitere.sf uii — 1. 8150 ff.r 2 yr. at 6%. i.'. $21.) for H yr. at 5%. 3. .S.S47 •.-)() for 4 yr. at 4%. 4- SlG7>S0f„r liyr. at (r,;. o. .$84-7") for ^ yr. at 4.1,%. it. $1H8 •(!,") for 146 da. at 7%. 7. $37') for \r,\ dii. at 6%. 'V. §176-40 for 12(j da. at 5^%. I>. lMn.l the amounl, of .§44^44 at interest for ISS da. at 0.^ ^ - At what rate wouhl $12.-) yiehl $1,5 inter, st in 2 yr •> 1 . At M-hat rate of interest M-ouhl $225 amount to $2;{1 'in 1 .... da ' lo In what tune wouhl $401 -50 a nount to $410;{0 at 04 - ' ' i.iti. August. Fin.l the interest on it for that perio.l at G ' V.v. A merchant purchased on the 17th Senteml.er 188" ^'^ i -u,unth.g to $700.40. He was allowed 3 n!:;;:!^:^^!^';,^^ se aft^ wh:eh he was eha.^ed interest at tl,e rate o p^. ~:un^t:ir ''' ^^^°"" '' '- ^'-' ''-^'' ^-«- ^^-' JO. A merchant purch. 4 on the ]3th Fel)ruarv Iss", , 19th July. 1885 "^ *'" """""* "^ *'"' ^^««""t on Find the equated time of payment oi ^'- ^r.-)0. -'. 8470. ■''. $lM7-o(). iJaIr ,1/ Xntfl. Time. 3. Juno, 1HH(\. !K) (la. 2.". Ap., 188.-,. GO.la. U Sept., 1883. .Snio. 27 Ft),., 1887. 9().la. 'is .(an., 1888. 2 1110. Dale 0/ hlxcnuiit. June. 1 June. 2.3 Sej)t. •4 March. 2 Feh. Rate of Jh'ncuunt, 6%. NO/ /%. KV 6%. ."^''97 " ■'■ Isincty days after date I promise to pay Jan.es Thonison or order wo Hundred and .Mnety-seven ,% Dollar.s at the ^Futual Savings i>ankhere. Value recoive.l. h.uam Joxes. //.Find the proceed, of t!,e ahove not. dis..onnt..d in Toronto on 9tli ,Jan y, 1 887, at 7 ; exchange } '. $714/^^. f^XHSix, ;J7 Xor., 1S8G. lour mouths after date wc jointly and .severally proudse to pay to the order of John O. Willian. .. Co. Seven Hunched and Folnleen 1 1'ollar.s for value received. TTi.'vw^- T . .„, TiioMAr Doi;.. AN. 7. Find the proceed.s of the above note discounted 12 Dec, 188G, at at Hamilton on exchange l.jc. per .$1()0 or fraction thereof 172 AUlTHMKTia .$339 iVi). Pembkokk, 3 March, I8m. ' At thirty days' sight pay to the order of llrown, Jf)nes & Co., of Kingston, Three Hundred and lliiity-nine jVj Dollars for value re- ceived, and charge to the account of To Greer & HKynERsoN, LeMoink k Peterson. Kinyston. 8. Find the proceeds of the above draft discounted at 8% ; ex- change 4 %. 0. Complete the following discount sheet l.y filling in the blanks; rate of discount 7%, of exchange 4% : — BANK OF THE VORKTOWX DISTRICT. Toronto, 4 May, 18R7. Bills Discounted for .S.vndeks, Redford & Co. Drawi' Whore J'uyntjlo. I Uiiyii I Whnn due. ' h> \ run. 1. Alex. Blatchford ..., Stratford . . i July 2. i Fred. Meade & Co. . ' Parkhill . . . | Julv 3. , Oeo. Hart it Co I Berlin '■ Ausf. 4. : Geo. R. Tiyhe ' Guelph. . . . Au},'. 5. I Ab. S. Lewis it Co. . Chatham . . ' Sept. Gross Ain't. Interest 1 $44,'))on 149 80 '->().'■) 30 514 ()!» 390.34 Exe'ge. I'roe'Us. ExamiiKd . ID. Draw up and till iu a discount sheet for the following, arrang- ing the drafts in order of niaturing:— Messrs. Jones & Brown, whole- sale merchants, Montreal, take the following drafts to their liank on the 17th August, 1S87, to be discounted and tlio net proceeds placed to their credit: One at "^0 day.s from date on Wm. Brown, Brock- ville, for S2()0'r)0; one t 90 days from date on A. B. West, Pertli, for ,S114-40; one at 10 days from date on S. B. Wood & Co., Brant- ford, $440-2.'); one at (50 days from date on R. J. Stanford, Ottawa, $r)4-]2; one at 15 days from date on H. C. Bleasdell & Co., London, 89.1 -30; one at 6 days on J. K. Smith k Co., Hamilton, $314-65. How much sliould the Imnk place to Jones & Brown's credit, allow- ing the rate of dis.'onnt to be 7% ? Exchange i;/ on drafts for $200 or less, 1% on drafts for more than .$200. o'.K ANSAVEKS. ri the blanks • ;| Exo'ffe. Proc'ds. ing, arrang- ;; f;:'*'^'^- ■'■ ^^- ^'- ">-^- - ;^9. .V. ion. n o. "• /~' ''• '"'■ ''''■ ^'^- >^4; «;^. ''• ^^ °«^»t«- ^<^'. 7; (58. /.9. .S18.-,. ^r; -■ qL.'"' '*• "-^- '^^- -^- 11^ ^«»ts: 125 cents -'•310.7. ...§3156. ...833,000. ... 47;; ^SS. ■'■ 7.S1. 3. 18,770. ■''. 471. /.. lUcenta. !■'>. 140. /.;. 1197 $47"). ,.^i. 1(54 cents. -?e. $72. .?G. 697. 37 llf t',H •, !''''"'■ ^"^^ ^^^' ^^2-M«0. Exercise II.— (pa(/e 17^ / i. - L, . . I'^'ifet- I/). — /. 14 cents. ,.'. ill •? 90 / ,- -* Saturday, iW». 305.5.' i 'h^.' :;■ :^'.^'- f ,. -^; «■ «. :«,iino. /,.. Hi, I20.„t ••. "» ""l^- ."'.70.50. «. 13.080. ., »„,4 '*; .'i,; '*■ ./'•,f-.,/"- "«• "■^■'»- -.i. S1770. .',(. .S12.t«l. .'.;.I831|9("|00 ,. i';'!; •"•'■ '•™-'*'"«- .'fl I X-.T '>*M. ■ «^■».^^^«^0. ,.'6. SJ.I.SI men. ;.'7, T.ViO • '■"■'"•^•"' '"""»• "'• »>■"":<■ .."/. *500.04. .57. .S.lli.,-': 17;) 174 ANSWERS. .i7. /,S. 52. r,7. .Slfl,0o4. ,i3. 9 cents. ^. $4. .ir>. 028, 1 To pounds, m. KMi; 426 $3()2r). 3H. $1.74. 39. $1.,j2. Jfi. 244; 2f»l ///. 8, ■> cents. .$I.r.(). ^,7. .fl2,8o9. U. 48. ^J. §610.). J,6. §4!).-,. //;. §12.(>.-.. 884.<; 400,778. , 4'J. 42; 84. .->0. 12; H48. .7/. 5220; ,30,.-)82 r.40;2100;.S367.20. ,7,?. $137. J^. $10r,3. .^.J. $801. .W. $12,648. 5!;24.24. 5S. (Gained .$24. 5<). 37 years. 6W. 38 years. 18. ,.'4. .il. ,7,7. 3(1. 4. '.). 13. 17. :l. :ir,. 30. 34. 38. 4s. Exercise IV,— (Page 29). - 7. U. 3 pounds. 7. 36. 1. 14. 8. .$3. .?70l. 13. $2003. 14. 94. /J. 24cfcr.t3. i,'*. $1.25. ,>y. $1558. 330. ,.'.;. 480. ,.^tf. 550. 27. $856. 5 hrotliers; 9 nuts. 32. 16 to eacli boy; 20 to tlie girl 11 trips; 107 passengers. 34. 19 trips; 123 persons. 3.',. 8 cents 20 cents. ,77. 30 cents. ,7<<^. 14. ,7,'^. $64,04.3. M'. V2. 4/. 4 cents 6 hours. ^.7. 8 seconds. 44. 5 liours. .f-7. 6 days 4 liours. .'. 3 cents. ,7. 13 cents. 4. 6. />. 29. 10. $63. ii. 701. .$21.24. IG. 3168. 77. 27. ,.^/. $19. 22. 1007. ,.'.?. 588. 28. 12. ,7,''. 10. ./.v. Scents. Exercise VI.— (Page ,37). 70(^■loo ct. 1120 sq. rd 608,000 oz. 2880 sheets. 2,352 oz. 2,471,040 in, 1192 oz. -/. 800 ct. 97(W) ct. 10,000 ct. .;. 84 in. G. 12 ft. 7. 154,000 lb. 8. 672 hr. -?a .32 pk. 11. 192 oz. 7,7. 108,864 cu. in. 14. 15,840 ft. /,7. 192 pt. 11;. 40,320 niin. 7u. 36 lb. ,.'/. 54 bu. 7 lb. 32. 74 bu. 4 lb. 33. 1687 lb. 8 oz. 34. 10 T, 1504 lb. 35. 10.290 ^!\ 260 lb. 36. 7 A. 1,30 sq. rd. 37. 30 gal. .7.?. 36 .7,9. 31 T. .500 lb. ANSWKHS. SO. 106; 426. 41. S."i centH. ."). 47. §12.tM. r)2'26 ; 3(i,.")8-J. . fjG. $12,648. ^ears. 1 3 cents. //. 0. ^63. 11. 701. !168. 17. 27. 007. ,^3. 588. ). ■I'-i. 3 cents, to tlie gii'I. !. 5.7. Scents. 2. 41. 4 cents. s 4 lioiirs. 3. 10,000 ct. . S. 672 hr. )8,8(i4 cii. ill. 40,320 niin. 20. 1.104 (jt. ^. 46,080 oz. -.'r>r 40,010 ct. : 86,164 sec. ?6,240 sq. in. .'. §1,182,370. 14. 47. 317. (). .'J.?. 46 ct. ijal. 3 (jt. 1 pt. . 9. §41.10. 915 lb. 5 oz. . 171)11. 481b. . 4") bu. 47 lb. ")1 bu. 10 lb. 20 1)11. 36 11). '. 74 bu. 4 lb. t T. 260 lb. ^ rm lb. Exercise VIII. 4. 48,400. 9909. (Pago 40). — /. 768. 0. 7926 mi. 241 rd. 1 ft. ( 12,410. :?207.60. 3 mi. 720 v 1)11. .08 lb. 12. 84 mi. 69 rd. 2 yd. 2 ft 2 in 8 gal. 2 qt. 1 pt. 14. 201 bu. 3 qt in. 13 Jords m ou ft 4b A. 1 n sq. rd. 20 sq. yd. 2 sq. ft. .36 sq. in. Exercise XI.-(Pagc45).-i. 211b. 1.1, ,.. . 74 ih . ,,,, 17 gal. 2 qt 4. 33 ft 6 in. .7. 66 da 19 hr ^"^rvi-n , t .. 3869 da. 18 hr. 36 min. .. ^C^ ^21 ^1 "^ ';:;,^' "*• 175mU580yd. ... 6 mi. 295rd. 1 yd. 6in. ./: ^.::^40 LT io^ ?t; 'l-^:'^-^^' '^*«-«2. .7. 2 mi. 1371^: 10 mi ij 31 mi. 80 rd. per hr. m. 230.1 mi. 730 yd 41 mi. 170 rd. 5 yd. I ft. 1 i„. ,0. 13 a 1 s. ril 6 ., r $28.-44. Zi. §47r).20 >A o ^-, , V"'^" "'• ^' «'!• y'- ?5. 25 lb. 26. 285 gal. 3 1'9. 64 ft. 2 gal in. JC^ ,15 ft. 10 191 lb. 12 07 .■?5. $7.1.60. ,7,;. 1, qt. in. 31. 48 bu. 181b :;4.43: t ; tiic grocer. '^'- 28. 67 mi. 60 rd. ) nnn. ,38 s«c, 3.',. 144 rd. Si:. 1826 338 cords 7 cu. ft 'i274rd. 3 yd. 17(5 ANSWEKS. Exercise XII.— (P iige 47). — /. li lb. 6 oz. 2U ). / ()■/.. 10. U- 16. IS. M. •25. 27. 28. 29. SI. J. 13. 18. 22. 28. 32. 37. 4t. 47. 4. 10. 16. 27. 32. 37. 42. 47. 52. 57. 62. 68. 3 T. lo3!» Ih. 4. 3 T. 167r> lb. 10 oz. 5. 4 gal. 3 .jt. 6. 4 gal. 2 qt. 93 bii. 1 gal. 3 ([t. S. 47 bu. 1 pk. 1 gal. 2 (^t. 0. 4 da. 4 hr. 31 4 hr. 4(3 mm. 11. 24° 24' 24" $8. :0. / . mm. lb sec 3 mi. >/. $^ /. $31.10. .20. ic. $14.19. //. $18.73. /.'. $2.3.75. /,/. 1331.70. 14. $117.40. I. 'A yd. 1 ft. ft. 1)60 cu. in. 10. 4. 211. 54, LJ. 425. ANSWEKS. 177 67. fJ9. Exercise XVI. (Page 67). -J.. . $576; 24 ct.; 1 ct. .7.;. ,?50.92. 2qt. 5,S'. 11 gal. 5.9. 6 gal. 11. 49 ct. 62 ct. ;^2 ct. 56. §11; 22 ct. CO. 62,142 ft.; 12,359 ft. 365 da. 5 lir. 49' 12". 6?. $450. (J3. Av., $2 •SI. 32; 44 ct. 67. 3 in. 70. 8 units. XVII, $5.20; S1.30. 67. .$617. 6.9. 1005 r^O. 64. Av.,.«iS.50. 11«.; 201 D) 3. 6. 9. 12. 15. 17. 19. 31. 23. 25. 27. 28. 32. 37. 30. 4->. Exercise R. 13, D. 7. 4. W. 29. F. 4 (Pago 70). -i. H. 14, E. 10. 7. 5. 1st 15, 2nd 22. A. 11, J. 6. 1st $2. 75, 2nd .§2.25. 7. 1 408 lb. , 1232 Ih ' S C. $4885, H. .S2885. 10. 1st .$3406, 2nd' $4549 $3572. .50, $4372. 50. 1.3. $11.50, $7.50. I4. 41b 1 7 gal. 3 . 1 1. , 1 3 gal. 3 qt. 16. 17. 52 yd. 1 ft. 6 in., 47 yd. 1 ft. 6 in. .S. $5500, R. $4500. n. $213, $63. 10oz.,31b. 60Z. H. 14, A. 17, J. 1 921b., 1061))., 12211 7. JO. 45 ct., 35 ct., 20- cords 24 cu. ft., 9 cords 104 cu. ft. IS. E. llct.T. 7ct.,A. 7ct. $138, .$159, $123. 12 yd., 22 yd., 10 yd. 24. 13 qt., 10 qt., ifqt. 6 1b. 10oz.,91)>.,8 11). 6 ' 32 11>. 12 oz., 36 lb. 12 oz., 23 lb. 12 oz. J6. 90A.,70A.,40A. 0. 15, B. 10. $5.50, $16.50, $22. .'0. 36ct.,27ct. oz., 28 11). 12 oz. 10. 16ct.,8ct. 12 04- 35. 7. oz. of green, 1 1),. 4 oz. of black. ;W 36. 31. 75 ct. , 25 ct. 1225. $950. 40. 28, 32, 40. 4/. 9, 48 lb., 64 lb., 96 11) 8. 46. 4($5),S(.$2) 42. 63. 43. a. 9, B. 8. -//■ 10. Exercise XVIII.— (Page 7 ' '"• "• •'> JJ. ". 4.). fS lH(5<'t.), IldOct.), 14(25ct.). 48. $U 51' 7'. /5.70'10" 'age 75). —y/. 54', /,.>. 72'. j 16. $280. /7. .$26.40. 3. 58'. 1404. J^. $77.76. 21. 24", 20', 12" /'V. $128.1,- 56" Exercise XIX.— (P ■h s. 13. 39 yd. :igc76).— /. 6. J, 6; 4 in. r>. 9. $58.50; $60.75. io, $82.2 96 yd. Exercise XX.— (Pi . $15.35. 41 y " ^ J /- ,^■, -^ff. 40 yd. 47,48 yd. 4^. 18' 6". .^,9 1)0' 52. 192 sq. in. 53. $15,400. 12 122 s sq. ni. •^/'. 13'. 4,5. 4'. "^'^ 24. 5/. 81. 17cS ANSWERS. //. 18. 2Jf. 31. 19. 23. 28. 34. 41. 4n. .50. 3J. 61. 68. Exercise XXIV.-(Page 84).-/. S32. 12. ,.^ $12.10. ,?. .«;20 90 .$12.10. J. $7.26. 6. 27. 7. 10. S. 18. !). 10. ^' 10 7 S82.I0. X?. S20.40. X?.§;w.40. 7,^.10. ij. 10. 10.8 1?' :U 6480. 19. mm. -0. 324. ;2i. 900 sq. ft. ^i?. 30. .^,?. .'i'loo' $S0. ^.T. $8772.50. ^y;. ,'-,0. 27. 16. /.v. 8. ^-j.?. $7 74 jr; )q(). §18.70. 32. $12. -10. 5J. §18. ' ' "' Exercise XXVI.— (Page 88).— X5. 12. 16. 13i. 17. 6 7,9 2 60001b.; 600 gal. 20. 150 bu. ; 9150 lb. 21. 6o"',016. ';?..'. 26 254 44 en. yd. 2.}. 40. 2o. 15,000. »e. 2520. 27. 21,912. 252,450. ?9. 16. 30. 30. 57. 144. 5,?. 2000.' 33 6720 1080. JJ. 640. 36. 1600. 57. 900. 38. 432. 5.9. 900. ^r; 3«()0 3080 cu. y,l. 4J, 125. /^y. $56. .^.^. $33.25. /^5. ,«;446 25* 1500. .;?. 91 bbl. 21 gal. 48. 1392 lb. 49. 13,500 23 o'-rds 80 ou. ft. 51. $348.48. 19,360. 56. 3520. 57. 4 lai. ^2. $12. J5. $198. 54. 1815. .55. 25'. 59. 6' .3". 6VA 96'. 67. 15". tf^. 2'. CJ. 16'. 66. 16'. -.m 1 ft, 6 in. 21. 72 sq. in. 5. it). 75. 19. 24. 11. 16. 4- 10. 13. 19. 24. 27. 29. 82. ■ 48'. 62. ir2". 63. 5' 7", 24'. 69. 9'. 70. 5' 6". Exercise XXXIII.— (Page 101) 5 gal. Exercise XXXIV. -(Page 101).-7. 8 rd. 2. 5 yd.; 43 suits 4it. /,. .$5. .^.6; 29. 6. 2bu. 7. 63 gal. 5. ,S'xl6'. £). 53and61. 11 f barley, 9 m rye, 7 of Avheat. 77. 12 ft.; 5390. 12. 27 in !or-„i^-J"''''^^^- '"''• ''■^''^- ^'-12;6,4;3. 18.5. 125, 25; No. ^/. 7. ;?,^. 5 lb. .^^5. 11 lb. 16 lb. and 10 lb. orer, or 8 lb. and 2 lb. over. Exercise XXXVII.-(Page 108). -9. 22 ft. 6 in. 10 119 lb 20 nl. 7J. lOOsq.rd. 7.?. 12 rd. 7.^. 126 gal. 75. 4 sq. rd! 39 1b. 77. 5. 75. 325. 19. 62. 20. 11. Exercise XXXVIII.-(Pagel09).-7. 210 in. ?. 60ft. 5 3000 60 yd. 5. 50 ct. 6. |20. 7. $135. 8. 1680 lb. 9. 45 at 12,600 gal. 77. 1 hr. I?. 3 hr. ; A 18, 7^ 15, C 12 Z> 10 1260. 14. 60 ot. 15. $30. Z6'. $1.20. 77. 41b. 75. 13 1b 11. 20. 5 doz. 22. 210 gal. ; l.st 6 min. , 2nd 5 min. 23. 360 yal 1020gal. -•5.J)0,090gal.;16nnn.41sec. 26. 12 min.; 1st 3, 2nd 2 1 hr. ; 1 on 2nd, 2 on 3rd. 28. 2 la: ; 10 mi., 7 mi. 880 yd., 6 mi 1 hr. ; 5, 4, 3. 30. 20 min. ; 5, 4, 3 57. 30 in. ; 6 min. 168 rows; 8 hr. 33. 60 cords; 4 da., 10 da., 60 hr. 34. 9. 42 ft. 56'. 14. 57.125. 5,9, 44398, 88750, i:«! 02, ^.o. Udoz 40.375. -^7. 175oz. Troy:=121b. Avoii-. ^ J. 2,55*1,443 yr. ANSWKIIS. Exercise LIII.-(Page 130)._x OTj M). o^^ ^'fercise LX.-(Page 136). -~u 47. //,. sJ 179 17./f qt. I'l^'im. i?-. Slot. i5. S2.04. i,9S4l7 :^3. 70273 lb. " 1 ini. i,7. -)5 ct. -'^. Tol^-lb. ^^;?. §7.19. Exercise LXII._(Pagel.3.S).-,-5. 36; ]5?lb. o,, ,_^_ib .9705 31. 393.1 lb. 76J lb. 3. G. 9. 12. 15. IS. 21. 24. 27. 30. 33. 3D. 44. 40. 55. G2. GO. to. 82. SG. 30. 05. 00. Exercise LXIV.-(Pago 140).-/. no 0.. o. j 7.5,) lb I qt. 1 pt. 7. 1 pk. 1^ f,t. S. 1 hr. 58 nuu. 48 Hec. 19 cords 4.|t ,„ ft .6-. 872 gal. 2,^, ,t. I7. 4 lb. 6 o. "' II hr. 22 J mm. 25. 11 cords 04,«, cu. ft. 2G. 900 lb. ' ' 3 mi. 144, ,v yd. 28. 14 A. 404 sq. yd. ,0. o bu. 1,^, pj, 31. U-j da. 11 hr. 12 min. S2. $10 35. 2i\r lb. 3G. L 37 U ?A> • ' ^/. A. 4.?. tI}^. 43. 18^. bu. ^'^. 2fi,iV bu. 47. 51. 13,^. r> cu. ft. 1000 cu. in. 17 lb. 1480 gr. 34. \. 3?-§|Jmi. 40. 1\%%^. 33§ bu. 30^ bu. 15. n 320. 202 5G. G3. 70. 45. 571^* l)u. 50. 22-2 '>»• > t . G4. 2592. 7/ n 7 ,1 r .107 58. 04, 00 a 2 •'ft 50 65 4U * 1 ^'^- TTTf iilJ- 34i bu. .'!4 2 7 17P 48. 30gbu i- ^^. A. 73. 15.S5g Ijottles. 74, £?6\ TTo— 67 -.21 ■1-15 ^^r? II 80. J;. ,s/. 1. I 1 . 1 1 r. 9 -> ■117-Tf /-i. i/fij ?'^- B. 77. |ii|. 78. 3. 7^, mf.;2T. 328^-nb. A?. 207Abu. ^^U- .- ,,- ; f.T- i/. hr.; 4 hr. 48 min.; M S8 13 tt 11 •" $7.32; .V r).. 810,07; -x 01. i. 0.S1 ^^'4 o*? 60^y.-.i. P.. l8mi.0Qrd.3fyd^ .. ^^^1 ^^ ^^,^ .^1; 68t-x mi. 100. 1 s Exerase LXV.-(Page 144). -7. 87 ct. .^. 93 ct '^ ^1 ,« -^. «.>.22. ....$22.17. ^. mi4. 7. .$26.74. ....o.r" i ' ^^i /7 W . OS ,f 1^ OA„n. 'M+.H. 14, $o4.01. i-. n.,2S,3t. i^. .§46.64. i 7. 8258.82. W. $291 79 ,n^i- 20.m^.U. 21. 9U^ot. 22. 2i)U in. -?. 2^S" " " 333j^-gal.; U%Ui 27. 22' 11). •TiT'S" gal. '■a 28. 9 mi 11. 2i 2G. 274^1 13 ct., 22^ ni. 156,1 >S(ic. past 10 a.m. (" 18 ct. 180 ANSWERS. I 89. 32. 34. 3G. 39. v^. 43. 53. 57. 61. 67. 71. 73. 76. SO. 35. 88. 91. 93. 96. 99. 2r»U mi.; SOU ""■ 30. 3^ mi.; 2^ mi. .U. 2g lb., 46 lb. Hi y) + 10($3.90). 37. A, 39^ hu.; Ji, M\^ bu. $4.r)0 each. 40. A, 4 mi.; B, 2j\ mi. per hour. 35. 492; 755. ■5*- ^%, I'V, i 4i. $1462.22; $1997.73; $3990.05. .^ J. .$19.58. ^.?. 46.V, mi. ^.;. 10 ft 5? in' 2 ft. 2A in. 46. 50? yd. 47. 6G,606^AW.V "»■ per hour, 138}f. 49. 99lfiJ mi. 50. 54fi|. .w. 2095/j ft. 5;?. rA' 3740^1 sq. yd. 54. G32 mi 7837TrV sq. mi. 58. 5J1J sq. yd. 59. UUq-'ni. 6 J. 1000. 63. SJ^. C^. 102'. 6V7 $4942.27. 68. $1,357.71. 69. $2G77!50. 77/ 35hr. 12min. J6'. ig^VVAVbu. 19/^- sq. in. Cft $496.64 66. '0. $17.08. 5-ff sq. in- 177.24. 05,40(i/V4 T. 891i}Jcu. in.;[8i"x44"x21*"]. 6I86T. 932Hlb. 74. N'. 7-.;. l,79.S,4.KSU'al. lhr.28?.S-i'imin. 77. $181. .30. 7i'. 19/^^7^01.. ft. 7.9. 693 cu. in. 5f|. 6'i. $3. &?. ^,$8.80; /A $12. 10. &'?. ] • ; $1050. ,^.^.$2.25 U; A, $4.50; a, $4.80; 6^ .$7.20. 5C. .?ol,30. cS7. 9 ct 25 mi. to 24 mi. 89. 63 J, mi. .9^ 7.25p.m. of 17th day; 7.47J.p m Gx'^j mi. ; 12-,L „,-. cy,... 9. „,i, . r;^„^ „,; . 7 -^ ^^j . gj mi. 3f sec; 4? see.; 3^^ .sec. 94. 2 min. 23, J^ sec. 95. S^'- hr 53/v min. ; 2 mi. 1 -,7(i yd. 97. ,V, J,, '^0 ; Cg da. ftV. 3 da,. 1st time, § way round ; A , 2g rounds ; /?, 1§ rounds 2ncl " i " " ^,.r5j| .' ji^ ^ u 3nd " at starting point; yl, 8 " B, Ti " ^ Together. 100. 1 hr. 5/v mill. " 101? '■■ :6A " 21 r\ " 27fV " 32tSj. " 38x?j. " 43/r " 49rV " 54x'V Opp()f!itl>. 12hr. 32i\min. 1 " .38 ,\ «' 43A " 49A " 54tS- " At Right Angles. 12 hr. 16-/r min. 12 hr. 49:i'i min. 101 lOii 2 3 •• ;tJA- " « " 43vr " 2 " 27?, " 4 5 6 7 8 9 10 12 " . 1 hr. 6 mill., 2 hr, 12 min., 3 hr. 18 min., 4 hr. 24 min., 5 hr. .30 min., 6 hr. .36 min., 7 hr. 42 min., 8 hr. 48 min., 9 hr. 54 min. ' IrxnTgal. 3 4 6 7 8 9 10 11 5A lOf? 16r*T 21A 27 .'V 1 2 3 4 ;■) 6 7 9 10 11 21-A- 27/, 32/t 38rr 43,', 49tV 5A lOH 1 3 4 5 6 7 8 9 10 11 04xT 5A lOlf 21t''i 32A 38rt 43fT n AXSWKHS. Exercise LXIX. d'ago ir,H ).-/./. 7,s|4'2m.-.7I 8270-8621. n. .S41.r7.-,. IJ,. WHO. /.;. -mn. 181 12.-). . -0009. :J4. SI. 36. 39. u. 4. 10. 13. u. 18. 23. 1. .) 3. G. 4. 10. 13. 15. Exercise LXX.-(Page 158). -7. .w ft 9 in ./. 10,.Mr,. ... r,430gal. 6\ 12.-)-, 11,. 1 oz. nearly. . !T?!-^^"^-^^'"^"-' '"^"-^- -^ T. 951 lb. 9-0 o^ •^. .)l/3-/28s(j. in. 6. -4738 mi. 7. 1 76986 da oh. Exercise LXXIV.-(PagolGl).-2. -05. ,.>. -075 ,/ .33a / i-. -;005. ..7%. 7.70%. .V. 37i%. ^.2257 Va^''";^,o?- i.^. $22-50. iJ. 150 yd. 7.;!. 49' 11, /T tl9 -«1 '%. i'/ '' \^'%rj^'- -^^ V. ^. .243. ... ^8.4o:- 1:^1242; : i^i n:^'^ "-;, t,: '■■ '■ "-"■ '■ *"'^- ■■'■ m«. 1-06; 1-1 1; 1-20; 1-075; l.SS^i; 2-10 ' m. 19. m. ;JO.m()0. £>A.S4(}-92;$6-12 - Sl-q.2 .. f ; m%. 24. m09. -J. $1,320; $2280 72«'; 'r' "?• . $1191-68; 32, V,%. ' *^^«", /^i i./. 26.jo%. Exercise LXXVl._(Page 165).- $22-50; $7-50; .$33-75; .$3-75; $6-56 $135; $84 -.38; $.50-63; $47-25 • ^^"f •^^5/-i««S-7-": -$4008-.33; $4827-19. 4. ,<4o7-47 2J%. 7..$249.15. .V. .S5100; $5050; .f 5025 37 Exercise LXXVlI._(Page 167) -/ co-4 ^ c-,a *,«o.,o ,. j..,o«. ..."w.^: :.; t:,^: •^:'«^'- ^z^^- 40%. n. $58-70; 48%. 7;.'. 81"^. * $4(>26-95; $147 -; .ri237; 73i%''nearlv. /^. 2A loss Nothing. 7,;. 16,.348 1,u.; 16,.327 hu. /;. mm 010. LS2 ANHWEUS. Exercise LXXVI 1 1. -(Page 1(5<»).— /. Sl«. Jf. .?lo-IO * 1 2 •;. !riS42-rj2. /j- ,.^ -?- 7%. 7.?. 12S(lii. /,;.$. oth January. Exercise LXXIX.— (Page 171).— DateofMatiiritti. 1. 4 Sept., 1S86 «'. 27 June, 1885 .?. 17 Dec, 1883 4. 31 May, 1887 J. 31 Mar., 1888 Torm (if DUcnimt. Ducount. 91 da. 26 " 85 " 88 «« ,^8 " .S3 -74 2.34 2-18 •99 12 -50 G. $291-93. 7 Daiju. I'rnceedn. .'$24()-2(5 4()7 •()(■) 185 •,32 67-76 970-88 ■!fo, to run. Intfrost. Excl .$098-84. 6'. .$336-19. Prneccdg. 9. 1 62 S5-30 §1-00 e4.39-,30 74 100 114 11-24 213 -09 i) 122 9-13 •40 •60 1-20 •80 147-27 2,")9-61 501 -65 ■^rM 10. Proceeds: §258-25; $111 •86: .*i Total, §1269-51. •i»iS' hi -$1728-24. $53-22; $94-72;v.S313-31. la. = June 12. APPENDIX. Tl,o following tul.los arc given for ti.o.se teuchors who n.ay winh to «et ox.unplu« in tin^n as exercises in calculation .- ' ■t farthings =l,,o„„y 12 pence =lslulli„. . . " "; 20 shillings =lpoun,l . " ' ' [^\ TKOV WEIGHT. 4 grains =1 carat. 24 grains =1 pennyweight Mwf ^ 20 pennyweights = 1 ounce . - f''*-\ 12 ounces =lnoun,l ^ ""^ ^P°""'^ (tr. 11,.) tPOTHE«AKiE.S' nEliillT. 20 grains =1 scruple . , -x , 3 scruples -1 drachm . " " " L^ ' 8 drachms =1 ounce . . " " * z 12oimce3 =lt)ouT..] ^ I'""'"^ (tr. lb.) AIOTniX IKIES FLIID MEASIKE. 60 Huidmiuim,s(l1L)-lflui,i drachm . . (,1 -\ 8 Ihnd drachms = 1 fluid ounce " ' 1 z' 20 fluid ounces =ipi„t. . «• o) than the ounce avoirdupois Thctr^tT , t ^ "' '""'«' ^-^ ^"•'*"'« h^'-^vifr the apothecaries- pouncf^" ici of ^u'r't " "°* '^^'^" i" "^ ^^ -">■ 3--; writh.g out their prescriptions some nh v f ''."1^ '''-"^''^'' ""* '*'"^'-' !««'• I" woi..:tJH.tneithcr^heB;it^norL^^^^ "^'^ °' apothecaries- Weights to weigh pennyweights, dr c, ,ns S p u ",f r™"?'"'-^ ^''^°'''"'-« "• not adn.itted to verificatio^ b^ th. iZw' ' °^ '''"' "'^^^^^^^^ Bon^njion Weights .ndMea^lr^i^a^E;;;;' '''''''' ""' '"~- ^^^ All articles sold by weiirht shall t-o -old h- • ^ gold and siUer, platinum and precious stones anH'°'f?"" ''"''''^*' '""'"^^ ^^at sold by the ounce troy or b, aJj cf^^Z^] T 7 "''' ""^'' '^''''''' *"«i/ 1^^ person who acts in contravention of th-/«^1^^^t^^^ and every exceeding twenty-flve dollars f reach otnl'"" ' ^'^ "'*'^'' *° ' ^"•^^*^ "°* 183