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Les diagrammes suivants illustrent la m^thode. 1 2 3 12 3 4 5 6 -fat^ Presented by G.W.Ballard, K.C., from the library of his father, the late Dr.W. H. Ballard, who served the Board of Education in Hamilton from 1874 to 1934 as a High School Teacher. Chief Inspector and also as an Educational Advisor ■ jIl, National Library Bibliotheque nationale ■ T^ of Canada du Canada I MATH AN KLEMENTARV TREATISE ON ARITHMETIC lOR USK IN THE Public and Model Schools of Ontario. BY WILSON TAYLOR, B.A., MATUKM.T,CA. ...STKH OK C„.VrH...M CO..KO,.TK ,S.rnrn, KOHMKRLV OK .^ AXl) STRATKOKD COLLKOIATK IXSTITCTISS. NOERSOLL << > Tis plain that number is the object of Arithmetick." -JOHN WILSON, 1741. TORONTO: ^^•ILLIAM BRIGGS, ^^'ESLEY BUILDINGS, 29 TO 33 Richmond Stkedt West. 1JJ98, ^N p I ^ 2^ Kuiered ftccordiii),' to Act of the Parliament of Canada, in the year one thousand eiifht hundred and niiiely-eiKht, li.v William Hhiuum, at the Deimrtnienl of Aifripulture. Prob. tioiial v( the stud quantity of two ( cjuuntitu the min(! ing corul facts iiit( This on scale whe The fa object is whieli m{ («) To( analysis a sutficientl; 5 stands. formed, th or symbol, more than ■5-^Vb-^V<^ M I 1 PREFACE, >ne thousnnd epartintiil of PHOi.A„LV no subject of study i„ ,.,„. schools is of ,„ore educa t.onal value tluu, Arithmetic. More than any other it re,,uires the student to form an accurate idea as to the «>a^autude of a quantity; to conceive the exact rdat^on between the magnitudes of two quantities ; by comparing two rehttions an.ong three quantities, to obtain the necessary third relation bv an act of the mind; and, finally, by repeating these processes un.ler vary- m« cond.tions, to bring a number of seennngly disconnected facts into relations, all tending towards a c.-tain desired Hid llus on a small scale is just what the pupil must do on a lar^e -scale when he enters in earnest upon the practical affairs of Hie The failure of Arithmetic, if there be any, to secure this object is due to no fault of the subject itself, but to causes winch may be easily removed. («) Too much time is spent in trying to teach " nun.ber analysis and synthesis" to young pupils, whose n.inds have not suthciently matured to form the "concept" for which the symbol u stands It may even be doubted that this concept ever is funned that the pure number 13 is anything but a mere word or symbol, and that operations upon pure numbers are anvthin. more than mere combinations of names, of symbols, or of sounds" IV PREFACE. It caniiut he claiiiKMl (hat the teaching of mere words contributes much to mental growth. A hirge part of the time so spent in the primary chisses would be better employed in teaching how to read and write the English language. (h) There is much misconceiition resulting tVom the want of a clear distinction between (piantity and number. When our present text-books call ";")" a number, and, at the same time, call "5 apples " a nunib(>i', they \ iolate the first principles of speech. And, fuither, when they attempt to explain the rules of Arith- metic, it is no wonder that they are lei(ls contributes ine so s])eiit in eat'hini,' liow to 111 the want of ■i". When our same time, call plos of speech, ■ules of Arith- is confusion of the teacher is t the pupil is ^rs are tiotnis, ibsurdity that t happens that imetic is nesr- urately is the vboring undci f, is made to ?se misconcep- mbers ; to lay 1 of all know- uental princi " PJtEFACE. .. U,c e.ulK.,. »u,.e« of the «,bj..o, whil. the p„,,i, i. e„„ ' „„ ; -'. " .» -itt..„ „,.. .„„ ,.„„„,. ,„ .,„ ;^,_. ^ = ..i.e.- ea»e „ .™t .h,U „. have p..«.,.ed the s„hje t i, ,„ - «„y that he who ,,,.,!» >W11 „„der,ta„,l Tl,e li„nt» t„ which the b„„l< i, ,,„t,.iete,l do „ot nennit the tull treatmcjit of m-.nv ,.f fi i- • • I'umu tne work will I T ''"^"-^----f the subject; but the .k -11 be found con.plete in itself, and suHicient to nK.t tl.e u..,„ro„.ents of the Public School Leavin, <.. LVn/l ^--„at.on It i« ,. „,„,,„ ^, ,.^ ^^^^,^; ^ - J- -;;--'• -<>n.p.te treatise of the .ubiect, for tli us^^ '-- ^^ ..> are preparing for Public School and Hi-d. School teachers' certificates. * ""' Wilson Taylou. Chatham, .hiiniiuy, 189S. :hnical words extended and CONTENTS. CHAPTER I. TiiK Orkjin and Use of Nl'mbkrs ''^"9 CHAPTER II. J READiN(i Axu \Vritix(; Ndmbkrs . . ,- 16 CHAPTER III. Addition - . . . • ■ • ■ ■ ■ - 17 CHAPTER IV. Subtraction - . - 20 CHAPTER V. Multiplication 25 CHAPTER VI. Division .... • • -33 CHAPTER VII. Reoulah Subdivision ok the Unit-Decimals - - - . 43 CHAPTER VIII. ''"-"T:'c,!:c:';;.:;;ut'cT'^" ■"" ^"■""— ^»-. 55 CHAPTER IX. Irregular Division of the Unit-Fractions - ■ - - (J4 via CONTEXTS. CHAPTER X. Ql-ANTITIKs IN rROI'OKTION- .... CHAPTKR XI. CUMPOIND QCA.NTlTIKS-KKI.rcTION ..... CHAPTKR XII. """LlrTla"' '^r""^-^''-"'"'-->' Notes-Con,p„un., In- tciesl-E.juat.„n of Payments -Averaging Accounts - . CHAPTER XIII. Bcvix. .N,, Sk,um,-(U,, and Loss: T.a.le Discount-Partnor- ^lnp-Con,n,.s«ion-Stock-I)utie,s-I„surance-Taxes . Sqi'akk AM) Ci-HK Roots CHAPTER XIV. CHAPTER XV. Bock T °rM T ^'•'T"""-'^'''' Cncle-The Rectangular Block-Il.e Cyhndcr-The Pyranu.l-Tl.e Cone-Tho Sphere CHAPTER XVI. The Metric Un.ts : Contracted Multiplication an.l Division . . rxoK 80 8fi 06 116 140 151 176 MiSCKLLANEOUS ExERCl.SE Answers ... 186 200 PAOR 80 86 ELEMENTARY ARITHMETIC. npoiiiid In- 06 CHAPTER I. t— Partner- ectangle — t-etaiigu'ar Sphere .- ion 116 140 151 176 186 200 THE ORIGIN AND USE CF NUMBERS. 1. Of the things jihout us iiiiuiy ;ir(> i-alled l)y the same name. The persons in tho school-room are called pupils ; there are trees in the orchard, leaves in this book, houses in a town, wheels on a waggon and legs to a table. Let us now group some of the things which are of the same name-for instance, tlie matclies in a box. This we will do as follows ; Tie the matclies into bundles, each containing ten matches ; there will be left a few inatc'->s less than ten, which w(^ shall lay to one side. Next, tie these bundles into larger bundl.>s, so that eacli contains ten of the small l)undles ; there will be left a few small bundles less than ten, which we shall also lay to one side. Again, tie these larger bundles into still larger bundles, so that each contains ten of the larger buiuiles ; there will be left a few larger bundles less than t<'n, which we shall also lay to one side. Continue this process till all the matches are laid aside. Each match is called a unit. Ei'di bundle which contains ten matches is called a iimlflple imit of the 1st order. Each bundle which contains ten of these bundles is called a multiple unit of the 2nd order. Each bundle which contains ten multiple units u£ the 2iid order is called a multiple unit of the 3rd order ; and so on. 10 KLEMENTAIIV AfUTHMETlC. "f unit, laid *, ,■.'-,! ."'■ "■"'■''■"' '"■PP""' "'o ""'"'"■.■ I I, I ,.,1,. ,» o ; the „„,„,,„ .,f „,„„;,,„ „„(„ „, .,' shall he written to tlie left of .,„ J no . . , -;;;u.i.n„i.of.he,.,uJe,.:i;r:i:r "■ '"^ '"""''-"' rhm.f>,„ we »h„„l,| „,-ite these ,u„„be,,, tl.u,- 4.!3- We simple number may he with one or more 0'^ '"m»)ei but a sists of more than one of these sym ho it i ' T , "'"^ '' "'"- number Thus 700 ; ^^^'^ ''>'•" ^"^N 't i« called a compound uinoei. ijuis, (00 ,s a simple number, but 4;]j7 i^ .. THt ORIGIN AND USE OF NUMBEUs. 11 y tho same name f the Ist Older Any one of the words one, two, Lfht, are 1, 2, ,3, 'I 0, is culled a of matches less number of bun- i to one side is ; ten small bun- number of the 'se the number Jnits of the 1st lits of the 2nd 'le units of the le units of tlie to, the number the 2nd order he number of : ^35 7. We find the box ne symbols — umber, but a when it con- I a compound >7 is a com- = -.vi ■ 5. How the operation of counting is represented —Let it be a^'reed in- all that tiie operation of grouping and counting, which we have described in the ])receding articles, shall be repre- sented by a horizontal line drawji in this manner : tlie matches in the box one match ' and that wlu'ii we have performed the operation we mav write I the matches in the box one match ~ ' ' ' ' This written statement may be read: "The numljcr of matches in the b()x when counted l)y one match is 4357." In the same way, the operation of counting other things which are of the same name may be represented. For instance, the pupils in this room one pupil that is, the number of pupils in this room when counted by one pupil is 32. «. Quantity and Number— When we have a collection of things of the same name, we call the whole collec-tion a (piantity, and each individual thing we call a unit. Thus we say, "the (luantity of matches " in the box, when one match is the unit by which the (juantity is counted. We may also say, " the (piantity of pupils" in this room, where one pupil is the unit by which this quantity is counted. When we say the quantiti/ of matches we do not mean the same as when we say the number of matches. When we say the (luantity of matches, we mean the matches which we can' feel and see ; but when we say the number of matches, we mean, not the matches, but the number 4.357, which at present is not much ■ nore than a name. Thus we say, the (juantity of matches =4.357 nuitches, but the number of matches = 4.357. 12 KLKMENT.AUV ARITHMETIC. :: ^"•;"'^"'tlK'n.,.,M =49 chair., tl'e VHluf of my j.orsr == Or,,!,.]!., ' and NO on. . This written statement we shall read, "the length of the liel.l is derived from a f.x.t by the numlier |;IV_'," and we shall call the sign " X " the sign of (h'rivatioii. Aceordingly, w,. shall ivpresent that a .jnantitv is d.-riNed fi'om itself, tims : H yard = a yard x I. It thus seems that when a numb(.r as l.'{r,2 is written ah.ne it lias no m.'am-ng, hut when it is written in eonn..etion with a unit, as a foot, it has a meaning whieh ean alwavs be explained. 'I bus, when we are told that ii tub of butter = a pound of butter x ;ir), we mean that 5 one-ix.und rolls and :{ ten-pound rolls together make up the whole (piantity of butter. t>. Number Defined. -When we think of a number as lieuig ohtame.! l,y counting the units which make up a -luantitv vve call the number tl,e meamre of th. <,uni,tittj ; but when we tinnk ot It as telling us how a .piantitv is derived from a unit we call the number a rate. We therefore d,4ine a number to be : (1) The nieasure (♦f a (luantity of units. {■!) The rate which tells how a ,,uantity is .lerived from a unit. I«. NOTK.-(.0 Tt will he seen later on that their is a pecu- i.tr advantage in writing the numb.-r after the unit, and not >"' ore It, as in ordinary language. Besides, it is th(> natural onler of thought, first to regard the quantity as a whole, then t. tinnk ot some km.wn unit with which to measure it, ami lastly to conceive tlu- exact relation lu^tween the ,,uantitv an.l the unit, which IS exp.vssed by the number. It is worthy of note also that the word rate is here used in the same sense a« it is used m commerce, and that it is synonymous with ratio o 14 ELEMENTAHY AIMTIIMKTIC. , ' ;"'l "•'Wo'V'ul "uyunlx :{:.!,,-■ w. ...ay suv in full " tl. •'•"<- '>-v.ty, ... ,nny say ''u yanl Uy ;L>.,." Sona- w ^c, ^-"l .vml it, ";{i>U ti.,M.s H yanl - JO It is Hssu,„,..|, at Mm. (irst of this ..ha,,t,.r, that tho you,.. '7"' '"'T "'" "-"'-'•"-'"-«>...•. two, thr...., four, five six" — ,n.ht, ,un,. ami tcn-i,. their proper or.le and t . at 'i ::::z:''t''t: n.!,..utno't:i;":L: ' ll„„Ks. ||„s ,s ,.|| ,|„.l i, ,„h;,.,.„u.1, ■|.|,, ,,,„,,„., ,.. : 't'lr. ';;: 1'"1- '""" ' '-'" ■"■"•■■ '■<• i- ''^ ■ nil A(l(lili,>,i ,uii] .Multl|ilifiitic,ii T.ll,l,.,s. „,','," ";'""•"";''• "'»"• """ ""• i-'M'M ,.1,1 1„. „i,i„ t„ ,,,„., „„,, ,» .i.o,„o,v,hm™iti„„«„,«,.„f „„„,,„„ ^ ,,^ .^ ; ^,„., ;,t::;!:':;^,;:r'""' "" ■ "-^ ■ -'"-■ '"- "^ ■-• EXERCISE I. 1 . Deseriho or perfonn the operation whieh is represented hv the hTie HI each of the fc'lowin^r ■ ' tsenttu in (a) :^j"""'^»' <»f l^eans (/') one bean. A basket of marbles one marble. Th(. l(.n«th of the desk (»ne inch. An ai'iiiful of wood (d) one stick. :^. Describe the operation which is represented by the si^n X in each of the foIlowin<,' : ^ (a) A loa,l of wheat = a bushel of wheat x 25 (6) The length of a rope = a yard x iT..! (f) The value of a farn, = a dollar x 5:520. V') A box of marbles = a nuirble x 141. 3- Head each statement in 2. 4. Complete the statements in 1, and read them. ><'iy ill full, "the l)y till' rate .'L'O.' >^<»iiii', lidwcvef, READINO AND WUITINCJ NUMHERS. 15 '', tlijit the youiiu ro, four, five, six. iltT, and tliat Ii.' no farther than, The analysis oi' VI he has learned ahle to read and curacy befoic he >r it is throuyli II, that he must represented by h of the desk y inch. 1 of wood tick. 'd by the sign < 25. i CHAPTER II. READING AND WRITING NUMBERS. II. In the preeedin;,' chapter we have supposed that nunilM'rs are lead by giving the digits in their order; thus, 4l)5U.J would hr read four, nine, five, nought, three ; and nearly every purpose ill Arithmetic would be served by reading them in this way. Hut ordinarily numbers are read as follows : 10 is ten 11 is eleven 12 is twelve 13 is thirteen 14 is foui'teen 15 is fifteen 16 is sixteen 1 7 is seventeen 18 is ei<'hteen 19 is nineteen 20 is twenty 30 is thirty 40 is forty .")0 is fifty GO is sixty 70 is seventy SO is eighty 100 is one hundred 300 is three hundred 28 is twenty-eight 64 is sixty-four 73 is seventy-three 96 is ninety-six 348 is three hundred and forty-eight and so on. 90 is ninety For numbers with more than three figures, the following scheme shows the manner of ri'ading and writing them : -J o 5 S- 3 = ^ P 0,0 0,0 0,0 0,0 0,0 16 Kr-EMENTAKY Arfnif.MKTrn. ■'■'■••- "^- ":r::r :::;::-;;r";r'^"''-'- ^^"t< iu sun!,ols Hiiy mm.brr proposed. EXERCISE II. 1. ii«"l tlH•f..ll..uiM-,MM,>l„.,•S: («) 532; .120; G.SO; W.i ; 207 (/') 182314; 510SI2; r,00;{04 ;' ,S->iG.ri {0) I234;)67890; 3204030001 ; aS2 1 4UOUOU4 ('/) 1000000020004; 5003000472031. 2. VVrito in sy,„l.ol,s the following, nu.nhors : («) Iwo huruJied .uid Hixty-ei^ht. ('>) Nine Iiundivd u.id thiny-.s'ix (^) Ni,„.|„„„,,,.,| I „,i„^.,i; ^,,„„^^,, ^^^,^_ and sixtv-('i<,dit. (.» ^»" "■■'"■'l-lMey-"i.u.,,,iiii,.,,,woi,„,,,,,.„u„,, thnty-fuu,. thuusami, five hundred and thirteen. f 14. Th and count basket =- i are 6 app 2 I ADDITION. liivc liiiiMlivd and luce. A^'iiin, \m (Is (IllCf liiMidivd iius, ;{0!»,(ioo,5cr). lit' <'Hii i-uirt'ctK )r(»po,s(*(l. jr •004. I, two luiiidivd ind two. u huiidiod and md thirteen. OfiAPTRR rri. ADDITION. I'v* ^-.I.poso there are 5 dmirs in the kitohon and 8 chairs in ''• uuun,M.oon. If „„w we put all the ehairs in these two It re a 1. ,,,,, j ,„.^ ^^.^. ,^,.,. ,^,^j^, ^^^ ^^^^ ^^^^^^ ^^^^ ^^^^ '»' .' .t".l he nun.her ,s t.^eth.-r make the ninnber l;{ We .nuy wnte th.s. -1 + 8= 1.3," which we read, ''5 and 8 are 13 '' - "5 P us 8 . 13." Now in order to avoid the tedious pro ess f eountn.,. the student shouM learn the f<.Ilowin. res.dts so tJius. 5 + 8=13, S + 5=13, 13 = 8 + 5 13 -'i^^ \vi s.iy is + J = 13 ^^,e j^j J g - IS The Addition Table. 2 + 2= 4 2 + 3= 5 2 + 4= 6 2 + 5= 7 2 + 6= 8 2 + 7= 9 2 + 8=10 no=ii 3 + 3== 6 3 + 4= 7 3+5= 8 3 + 6= 9 3 + 7 = 10 ^=11 o J=12 4 + 4= 8 4 + 5= 9 4 + 6=10 4 + 7 = 11 4 + 8 = 12 4 + 9=13 5 + 5=10 5 + 6 = 11 5 + 7 = 12 5 + 8=13 5 + 9=14 6 + 6 = 12 6 + 7=13 6 + 8^14 6 + 9=15 7 + 7=14 7 + 8 = 15 7 + 9=16 8 + 8=16 8 + 9=17 9 + 9=18 14. The next step in Addition. -Suppose now we -roup and count the apples in a .,asket, and find that the apples in the basket =- an annlo " ~fi ^"1 • i . , I n.. « ; . " ' ""'""'^ ^^'""^^ "» the basket, there i are 6 apples, and 7 piles eacfi containing ten apples. Let us '"' ii 18 ELEMENTAIIV ARITHMETIC. Il!!l: !ili, P h> I now put witl, these 8 apples uunv, tlu-n there will l,e in th. basket 14 apples an.l 7 piles. But the U apples ccnsist of 4 apples and 1 pile. Therefore all the apples will consist of i apples and 8 piles, or as we have agreed to write it, all the apples = an apple x 84 Now, we obtain 84 by adding 76 and 8 by me.nory of th. arld.tion table, in two steps, as follows : 8 + 6 = 14, and 1 + 7 --. 8 Here we call 84 the su>n of 76 and 8. The student „u.st now learn to give accurately all such results as the followino- • 43 + 4 = 47, 59 + 6 = 65, 27 + 3 = 30, 84 + 8 = 92. 15. Addition of Compound Numbers—Suppose that the matches in one box = a match x 824, ^ the matches in a seccmd box = a match x 596,' | the matches in a third box = a match x S5d ' It now we put into one lot all these matches, we shall find how ^ many there are, not by grouping and counting them, but bv nddu>, the numbers 824, 596 and 859, which we do by ou'r memory of the addition table. Beginning at the ri^ht tho manner of adding the numbers of units is : « - " 9 + 6=1.5, 15 + 4 = 19. llie manner of a.lding the numbers of multiple units of the 1st orderis: 1+5 = 6, 6 + 9=1.5, 15 + 2=17 Also the manner of adding the numbers of multiple units of th. 2nd order is: 1+8 = 9, 9 + 5=14, 14 + 8 = 2'> Ihiis we are able to say, all the matches = a match x 2279 In p,-actice the nund)ers are set down as below, and the addi- tion is performed mentally, thus : 824 i 596 ^ 859 '■ (2) tities ; by niei Wv result ( The until tl 1. S( 2. Fi tities ar X 963, 3. Se number! and 521 1. In and net row, anf numbers to the ri corner sj 2279 m7^^ ^^^ of Addition.--There are thus two operations (1) rutting quantities together, which operation we may per form witli our hands ; and ^ ^ 5. Mai numbers, ADDITION. 19 'fc will l)p in the apples cimsist of es will consist of rite it, [. y ineiiioi-y of the 14, anrl 1 + 7 --^ 8. ituflent must now bllowing : 84 + 8 = 92. .—Suppose that h X 824, h X 590, b X 859. e shall find how :| ig them, hut by h we do by our : the right, tho units of the Lst pic units of thr :ch X 2279. ', and the addi- (2) Adding the nmnl.ers wind, are the measures of the quan- tities put together, which operation we perform with our minds hy memory ot the adcHtion table. We perform the 2nd operatio.i so that we may know the result of the 1st, without actuady performing it The examples in the folh.wing exercise ma^ be easily increased, until the student can add with accuracy. EXERCISE III. 1. Set down in columns and add : (o) 27 ami 38 ; 9:3 and 27 ; 143 and .^71 ; 2965 and 2186 {'') 123, 421, 561, 329, 244, 052, 531 and 508 (c) 32475, 2190, 821, 599, 23 ami 7. (d) 44, 9999, 53216, 28, 214 and 9102043. 2. Find the total (juantity <,f apples when the following quan- tities are put t.>gether : an apple x 97, an apple x 532, an apple X 963, an apple x 301, an units of^^the 1st order. Therefore the wheat left in the pail con- sists of / units, and 4 multiple units of the 1st order; that is the wheat left in the pail = a grain x 47. ' , Again, suppose ■■ the wheat in the pail at first = a grain x 4072, Nc and the wheat taken out = a grain x 2785. i SUBTRACTION. 21 ns of nuniliors in cm in a different 3 give this in the s 7 is 8," or "7 1. 1 5 ; and we call .—Suppose that eat, and that a low to find out, That is, suppose X 72, X 25. lumber 72 tells ■ 2 units, and 7 ', we cliange 1 ve see that the >i the 1st order. and 2 multiple in the pail con- der ; that i.s, 7. I X 4072, < 2785. I 4 multiple units of the .'Jrd order, 7 multiple units of the 1st order,' iuid 2 units. These can be changed, so that the who],, wheat consists of ^ multiple units of the 3rd order, 9 multiple units of the 2nd order', IG multiple units of the 1st o.'der,' and 1 2 units. Now, the wheat taken away is to consist <.f 2 multiple units of the 3rd ui-der, 7 multiple units of the 2nd ordei', ;j 8 multiple units of the 1st order,' !ind 5 units. Therefore, the wheat left in the pail consists of 1 multiple unit of the 3rd order, 2 multiple units of the 2nd order, 8 multiple units of the 1st order,' and 7 units. Tl«t i», tlu. „-l„,at l,,ft in the ,«il = a ...tti,, x li»7 ... 'ir:.:"::- "- -"""■^ :;-■' "••»"• ' - '-'«■ >'.» 2785 l--j = 7, 1C-S = ,S, 9-7-2, aTi.l 3-2=1. "OH, tliat there are two operations : (1)^ Taking a part of a quantity away from the whole quan- tity, which operation we perform with our hands ; and (2) Subtracting the measure of the part from 'the measure of ' ill 5 22 ELEMENTARY A RITHMETIC. Hi tl.e wholo which operation we porforni with om- .ni.uls bv memory of the addition table. ' ' We perfor,„ the second operation that we may know the result of the first without actually performing it. m The Second Method of Subtraction. -Then, is no ...ethod of .d,tn.tion in eomn.on u.e, which lea! ^ tl! d g. ts of the larger number unchanged, but changes those in the smaller nun.ber. This we .shall now explain. The wheat in the pail at first = a grain x 407'> and the wheat taken out = a grain x 2785' ^^_L.t^us increase the wheat in the pail at first, by putting 10 units, 10 multiple units of the 1st order and 10 of the 2nd order; ' so that it will then consist of' 12 units, 17 nmltiple units of the 1st order 10 of the 2nd orfler, and 4 of the 3rfl order. 1 multiple unit of the 1st order, 1 of the 2nd ordei-, and I of the 3rd order ; so that the quantity of wheat taken away consists of 5 units, y multiple units of the 1st order, 8 of the 2nd order, and 3 of the 3rd order. a J that Asl and pe 1. Si 70, 20, 2. F] 3. Fi i ( 4. W to make 5. Su How nu 6. H( liumber SUBTRACTION. 23 tfi our miiuls, l)v 'e may know tlie it. Ction.— Thciv is . which leases the an.ifes those in the n X 4072, II X 2785. first, hv piittini-' way by the sano t.s of same as before. 1 7 units, 8 multiple units of the 1st ordfci', 2 of the 2nd ordei-, and 1 of the 3rd order ; that is, the wheat left = a grain x 1287. As before, we set the numbers down thus : 4072 2785 1 287 land perfoi'm mentally the following operations : 12-5 = 7, 17-9 = 8, 10-8 = 2 ami 4-3=1. EXERCISE IV. 1. Subtract from 100 each of the following numbers : 80, 60, !70, 20, 25, 50, 75, 64, 21,19, 36, 42, 85, 99, 79, 88 and 11. ' 2. P^ind the difference between the following pairs of numbers : , (a) 2315 and 6913. (e) 135791 113 ami 24681012. j (b) 2008 and 1963. (/) 1003005 and 300105. I (c) 83143 and 9406. {(/) 32145 and 9614835. ((/) 100000 and 12345. (h) 214 and lOOCOO. 3. Find what is left when (a) A foot X 532 is taken from a foot x 934. (b) A dollar x 1035 is taken from a dollar x 2110. (c) A book X 29 is taken from a book x 53. (d) A grain of sand x 1934876 is taken from a grain of sand X 2043798. 4. What quantity of apples nmst be put with an apple x 203 to make an apple x 501 ? 5. Subtract 43972 from 307804 as many times as you can. How many times ? 6. How often can 837496 be subtracted from 4096382 '/ What number will be left 1 24 ELEMENTARY ARITHMETIC. I 7. 0„ Monday I l„„| „ cent x 4323 ; „„ Tuesday I ,pe„t a „■„, X 1320, „n Wednesday I „,,e„t a cent x 931 ; on Thursda v :^ ;r:rt;':ofnT ' -n - -" ^ ^« ^ -■"■^" ^ I it a cent x 400. How much money had I then left / ?JI. The Roman System of Writing: Numbers uh; i oie V oi A, and X may be written before L or C and C n,... bewntten before D or M. Then the combination oh^ letter, stands for the difference between the nun.ber: inLt by the separate letters. Thus, I V =. 5 _ i _ 4 .,„. .. r '"^''^'^\''"' = 40,andsoon. ^ -+, and XL = 50 - lu sum\^"tr'''' T' '''' '''^'"'^'"^^*-" »f letters stands for the sum of the numbers indicated by the seoar-.f. l.ff lu XXV = 10 + 10 + 5 = 25, XLTX = lol 9T 9 No "'tt '• ' wntten in succession more than thre . times A ^^sh " letter increases the number a thousandfol ""'' ' : JJ«. Si Here t bles toge tlie pile. 0f niarblt Here t together, ttp the b these opei EXERCISE V. 1. Write all the numbers fron. 1 to 100 in Roman symbols. — Write the numbers: (a) 3-^0 n.\ iaq / \ /"no /, XPTV^ I'^^V^r '!'" ^"""'''"^ numbers: (a) MDCLXVI (M XCI\ , (c) CML, (,/) CXTX and (.) CXLTV ? ^ If now iddition, 1 ; We say the same ( are used ii 1^'hich is n in the san i ■which is re j The sing fity from t fuccession, fi'ocess of f c. 'wlay I spent u cent '31 ; on Thursday I X 9G; and on Satin had I then left / Numbers, whieii md the liours on u he letters 1, V, X, « 1, 5, 10,50, 100, 1 may be written L or C, and C nuiv nation of the two lumbers indicated and XL = 50- lo Multiplication. 25 CHAPTER V. MULTIPLICATION. 'vs stands for tlie ate Jettei-s, thus, 9- No letter is A dash over a Oman symbols. ) 693, (d) 1437, /) 777, (^•) 358, VIDCLXVI, (i) I T4 Suppose that a pile of marbles = a marl^le x 7 i Here tlu. sign " x " represents the operation of putting niar- Bles together, and 7 tells how many are put together to make up the pile Suppose, also, that in a box there are 6 of these piles pt marbles, so that ^ ■ the box of marbles = a pile of marljles x G ^ Here the sign " >. " represents the operation of putting piles ^gether and C tells how many piles are put together to nLe ep the box of n,arbles. We „.ay, therefore, represent both tnese operations in one statement, thus : the box of marljles = a marble x 7 x G ^ If now we perform both operations and count, or find out by fcldition, the number of marbles in the box, we shall be al>le to say the box of marbles = a marl)le x 42. , We say, then, that the single rate (or number) 42 will derive the same quantity from the unit as the two rates 7 and G, which are used in succession. This is what we mean when we say, ? 7x6 = 42, Vhich is read, " G times 7 is 42." In the same way we may show that I . Gx7 = 42, I'hich is read, "7 times G is 42." I The single rate (or number) which will derive the same quan- tity from the unit as the two rates 7 and G which are used in krocess of finding ,t is called multiplication. ^6 ELEMENTAin- AKfTUMETIC. f ll (1) VVhen it is wi-ittcii I)et\vt'en tlu> nnif ... i i miiiiber our a„... n i, ,.„„ „„., „„ :;r::r;,:L'z.'"'"'™ -'"'i- « .i- Now we „.,,,,,„ t",e ,,:";'"'*''""' """"l"-"''- only one '" ^ '"^ P^'*™' ^o opor..ti„„., „f .he ,i,.,t ki„.i, b... -«..,.,.= yone. I .Y'0"s.»te„t then t„ ,,,ul the .sutement, „....,ll,:,. thns ■ .. the 7 " T,"'*'"' = " '""'■'''« ^ " - «. ■ «"i'» "I 7 Ii.i;L,t a' "" ' '' '^'"" '"'" " '""'■'"'• ''^' '^^ - '^" "' ■' till' ordei -•<• The Multiplication Table. f<" -l'<'rt 2x 2= 4 3= 6 2x 4= 8 2x 5=10 2x G = 12 2x 7=U 2x 8=16 2x 9=18 2x 10 = 20 2x11 = 22 2 x 12 = 24 'Sx 3= 9 3x 4 = 12 3x 5=1') 3x = 18 .'5 X 7-21 3 X 8 = 24 3x 9 = 27 3x10 = 30 3x11=33 3x 12 = 36 4x 4 = 16 4x 5 = 20 4x 6 = 24 4 X 7 = 28 4 X 8 = 32 4x 9 = 36 4x 10 = 40 4 X 11 =44 4 X 1 2 = 48 5 X 5 = 25 5x 6 = 30 5 X 7 = 35 5x 8 = 40 •'>x 9 = 45 5x 10 = 50 5x11=55 5 X 12 = 60 Gx 6 = 36 Gx 7 = 42 Gx 8 = 48 Gx 9 = 54 Gx 10=60 G X 1 1 = 66 Gx 12 = 72 7x 7 = 49 7x 8 = 56 "x 9 = 63 7x 10 =.70 / X 11= <7 7x 12= 84 ^x 8= 64 ■^x 9= 72 «x 10= 80 8x 11= 88 8x12= 96 9x 9= 81 9x 10= 90 9x11= 99 9x 12=108 lOx 10=100 10x11 = 110 lOx 12=120 11 X 11 = 121 11 X 12=132 12 X 12 = 144 here \vt ^gure." |iiits figii Cautio |s sooo, t'liev sho illDuld 001 J Thustl Is + 2 ; w the orfh>r BOOOOOO. « ric. MtTLTI PLICATION. 27 'ippoars that the si-i, i .,,, : , , , • i , f Jlus tahlo mdu.les the r.suhs of .nultiplyi,.., two sinn.le it a,ul ..un.her, a.s .,?' '"■', '"^" ''" '" ''" ''""*" ^^■'^>' ^^^ "' ^^''^i^l*^ 22. Those •l--ntion of puttin, t' '\"''''"V""'' '" '''■' '"^ ^''' '^-'"'-'^teb- rapi.lly and i„ ■^' '"avporfonnwi.^ , '".''"'''•. '"'•■ "^ ^'"'*''^ « >'^ 42," "G ti.nos 7 is 42," "42 ivatiou. 'i|MMnes/," and " 42 is 7 times 6." ^rs, as 7 X 6, the si.,. | Xotk.-No sui^^estion is here offered as to the best ,„ea„s of by r.. w„ch opera leanun^ tins tahU-, hut eaeh teacher or pupil will a.lopt the y ot the table given met hod Ijest suited to himself. :n of multiplication, 'mtions, so that u. '^5- The Order of the Digits of a Compound Number the hrst k.nd, bu, ^Suppose f.e matches in a box = a match x 87r.3 T7tl'; tateu^nt, «umber 87G3 we s.y that, since G is the number of mJltip , »u,ble by the rate |tt of 3 ; the order of 7 is the 2nr shortness, let us say that tlie order of G is + 1, the order of 7 is + 2, and the order of 8 is + 3 .here we let ''V stand for the words, « tL the left of tlie units J,.ure Then '1 + 4" moans "the 4th place to the left of the »n.ts figure. So also the order of the units figure is 0. 9 X n " It ?'^ ?' 'T ''" '" "'"' '"""^^ '''' '■ ^''^'^ !-'« to error. J X 11= 99 llH.y should count the figures and O's before the last 0, and Should count from right to left, in the following manner : (7X6K5X*X3X2X1X0) 8 3 2 Thus the order of 8 in 80000 is + 4, and the order of 7 in 700 s + - ; while G written hi the order + 3 is GOOO, 4 written in |he ^,. + o is 400000, and 300 written in the order + 4 is 7x 12 = 8x 8 = 8x 9- 8x 10 = 8x11= 88 8x12= 9G 9x 9= 81 84 G4 72 SO 9x 12=108 lOx 10 = 100 10x11 = 110 lOx 12 = 120 11 X 11 = 121 11 X 12 = 132 12 X 12 = 144 J J ) ) y u ^8 ELEMENtAllV AlUTIIMMTin. 'iil The Rule of Order in Multiplication. Since, a match bein^ tlit- iiiiil, A multiple unit of the ~)t]\ onlt-r =-t\, multiple unit of the L'lid oi'der x 1000 = a match X lOOx 1000. liut also a multifile unit of the 0th order = a match x 1 00000. (•■;»( 1) (:iX--iXl» (^^)(iX:tX-!Xl) Hence, lOOx 1000-10000 where the orders are indicated ahove tl Now the order of 1 in the 1st numl le numbers. )er IS + 1' the oi'der of 1 in the liiid niimher is + .'{, and the order of 1 x 1, or 1, in (he luddiicl is + ,"», which is obtained h\' addin^f + '_' and + .'{. Therefore, fhe order of thf pnx/iirt of tiro simple nnmhern is found by adding the orders of the simple nni)ibevi>. We shall show how to use this rule in the next article. I of 6 in noO i I set down tin we }iei'form i Here we sha "in. To Multiply two Simple Numbers. - For instance, to multiply to en ^. th Irii ,'tli i^i Ml X .|:{; liiii i i ! 1' and tlif length of Mi = an iiuli x ,s^[\\ (.f Cl\ =-- (111 inch X 8!»;{ x 4.'{" I. Jlcir the nuinlKT 803 UOIs us how to dciivc the l<'n;,4li of A I, from ;in inch, and the; nunih.-r l.^Tl tells us how to dt'rivo tin length of CD from tlit« Icn^^li of All. Now, when we lind l>y multiplying (hat HU.'} x |.;71 .'JOO.'J.'IO.'J, the numhcr ;{903;{0:{ tells us how to derive the \v\vn\ of CD from an inch without usim,', or tIiinler 20000U what is the order of 2 ,' of 20 I o 200 ? of 20000 ? 3. Wi-ite down : {n) 6 in the order + 2. {h) 5 in the order + 4. (c) 7 in th(! order + 1. {d) 9 in the order + 0. (e) 1 in the order + 5. (y ) 4 in the ordi-r 0. (,'/) 13 in the order 0. {h) 28 in the order + 3. {i) 14'J in the ordei' + 5. (,/) 200 in the order + 1. [k) 1 20 in the order + 2. {I) 56 ill the oi'der + ]. B. 4. What is the order of : {n) 2x3 in 200 x 30. {f>) 4 X 3 in 4000 x 300. (c) 5x5 in 50 x 50. (o?) 8x9 in 8000 x 90. 5. Find the product in each of the following {a) 400x80. (,,) 10000x100. (6) 300x300. (/) 9000 X 8000. (o) 2000 X 70. {,j) 500 x 40000. {d) 60 X 50000. \h) 2000 x 80. (e) 6x5 in 60 x 500. (/) 7 X 7 in 7 X 7. {g) 8x1 in 800000 x lOOOO, (A) 3x8 in 300 x 800000. I I. Mu {a) ('•) (d) 15. Wh 'JOO ? Wli of 35296 a 16. Fin( (^) (b) ('•) 17. A n what is tht If^. Tf a stick X 14 MFLTir'rjr ATION. 31 6. I iy wliiit, rciiMiiiiii^' ditl yuii ulitaiii (lie ifsulls in 1 ;iri(l '> J (inlcr 0. II order + -3. 3 order + 5, L' order + 4. B order + 2. order + ] . 15. 296? f 1 of 35 X 500. 1 16. X 7. DOOOOx 10000. 00 X 800000. 00. 00. 00. 17. MJiat 18. 7. Multiply tlie fullctwiiifj; jwiirs of nuiuhers to;,'etlier : 12x3, 2131 x2, 10321 x3fmd 101 x I. 8. Kind .sin^'l«> nites (Mniivalent to 132 x 1, 51(ix 1, 321 x 1, in(U3x 4 and 5217() < t. 'J. Multii.ly 1234;i()7 by 5, l,y t and by 3. 10. .Multiply 23476 by (5 and tli(- product by 6. 11. .Multiply 3215!)87 by 7, by (i. ' 12. .Multiply 28312 by !) and the product by 9. 13. .Multiply 80357 by 8 and tlie product by 8. These examples should be continued until the .Multiplication Table is learned accurately. D. ]\rultiply the following pairs of numbers to^rether; (a) 512 and 23. (c) 517G anfl 214. {h) 1234 and 21. (/) 835 and 29G5. ('•) 476 and 34. {,,) 246 and 8409, {d) 2030 and 504. (A) 3215 and 809. What is the order of 2x3 in the ju'oduct of 35417 and Why? How many Hj^ures will there be in the product 296 and 2473 ? How do you tell without multiplying? Find the product of: {d) 793, 257 luA 578. (e) 314159 and 27828. (/) 95329 X 498(17. A mile = a yard x 17G0, and a yard = an inch x 36 ; is the number of inches in a mile ? a pile of w(K)d = ail armful x 35, and an armful = a {a) 9876 and 3987, (6) 99893 ami 976. ('•) 87969 and 9596. •4 .J .i ..J ...I .') 1.) D Tf stick X 14 ; h ow many sticks will make up the pile of wood ? ill:: = 82 ir>. Tf ELEMENTARY ARITHMETIC. •ne i-ow „f s,,n.ares = a square x 43, and row of squares x 26 ; h 20. Tf the apple-tree 0\V 111 a field = a of tret' « = a tree x 26 ; how 21. Tf a larjre box of matclies = any squares will make up the field ?| a row X 38, and a row many trees are there in the orchard' '« in an orchard = ai >d a small box of ties = a small box of matches x 27 matches does matches = a match x 144 ; how 99 a large box contain ? many If the distance to tlu distance to the the distance to tl 23. I^educe to sun the "i«>'>» = a mile x 237125, and the le sun. moon's distance x 391 ; find in miles! one number each of the foil (a) 792 X3.S + 421x69 -803x60 {h) 532x693 -216x257 + 125x160 (c) 2395 X 999 - 2396 x 998 owinir 24. Find the product of ; («) 5x5x5x5x5. {b) 6x6x6x6x6x6. ic) I X 7x7x7x7x7x7. (d) 8x8x8x8x8x8x8x8. (e) 9x9x9x9x9x9x9x9x9. 25. Fhid the sum of all the numbers from 1 to 199 inclusive. -6. iMnd the sum of all the numbers from 35 to 69 inclusi^•e -'. l^nd the sum of all the numbers between 321 and 563 L8. Find the sum of all the numbers bet^veen 1893 and 3753 ua y 1st 1 cent, on January 2nd 2 cents, on January 3rd 3 cents and so on for a year. How much in all did lie promise ^ 30. A small box of matches contains 63 matches, a lar^e l,ox con ains 27 small boxes, a case contains 36 large .oxe^a nd •n. If you are told that a bushel of wheat contains 19347-> ' gmins, what does the figure 9 in. tliis number tell you FxH u. detail the information given you by the number 193472 ' [barrel, an (n) To tlie long ( the table. foacli as 1( I suppose tl ? of the tab] I by a horizi This sta -■table, whei » Here th }.nnit, by n '.< pencil ; the I denotes the I that them( I (b) To n 3 use a watcl I fhe time in to do is to 1 3 DIVISION. 33 . and a- field = ;, lake up the field .' X 3H, and a row e in the orchard ' ^f matches x 27, 44 ; how many 2.37125, and tho 1 ; find in miles CHAPTER VI. DIVISION. 31 'x7 X 7 X 7. ^x8x8x8x8. 199 inclusive. 69 inclusi\e. i21 and 56.'3. .^93 and 3753. 'Hows : on Jan- iry 3rd 3 cents, omise ? es, a large ho.x :■ boxes, and ;i Hatches arc in ntains 193472 'ou? Explain A\ e have shown how to group and count things which are of t e same name. We shall now show how such tlings as the ongth of the table, the flat surface of the table, the wl. in ! barrel, and the tmie from now till noon are measured (.) To .neasure the length of the .«6/.. -Place a pencil along 'bf t " T' ''"" '" "•" «" "P *'- -hole length J? I the tabl. By so doxng, the length of the table is cut into%rts, |oach as ong as the pencil. Having counted the parts, l^t u iilahriS"' T '^'''- ''^' ''^'^' ""^ "'^^--' ^''J length ,of the table, and we represent the act or operation of measu.^. I by a horizontal line, thus : " I the^lengtli^ of the table J the length of^ the^^encil ' ; This staternent we read, '^ The measure of the length of the I table, when the length of the pencil is the unit, is 13 '' I Here the ^na.^i^y measured is the length of the table; the '-^ by means of which it is n.easured, is the length of the I pencil ; the measure of the quantity is the nun.ber 13 ; the line iltr^ the operati.m of measuring ; and the sign '< = "indicl:: j tn.it the measuring is completed. I {f>} To measure the time from now till noon. -To do this we I Z IZ , " '^'f ' "'"' " "^"^^^"^ "^^^^' -^ -'^ to divide I !'::,! "^:"!"!^?^ ^^^'^ '^"^ '^ -""^ the parts, ill we have The ;=i3. 193472. f ^" ^'"^ '« to read the number of parts indicated on its face. » 3 ::5 .J ..J ■ J ! I, u ELEMEiNTARY ARITHMETIC. time It takes the second-hand to «o once around is the unit of time, which is called a minute. Suppose the number of thes.. units required to make up the time from now till noon is 96 then, as before, we represent the operation of measuring by n horizontal line, thus : the time from now till noon r— — OR a minute • (r) In the same way any quantity may be measured. A part of the quantity is chosen as a unit, and it is found out how nianv times this unit occurs to make up the quantity. Then always ' the (juantity the unit = a number. ^l Division Defined.— By multiplying we find that a yard x 1.3 x 28 = a yard x 364, whicli we may write backwards, thus, a yard x 364 == a yard x 13 x 28. Tf now we regard a yard x 364 as the quantity to be meas ured and a yard x 13 as the unit by which it is measured, w. shall have, a yard x 364 — . ox • a yard x 13 "" ' where the horizontal line represents the operation of measurin.' described in the preceding article. Again, speaking <,nly of numbers, since 364 was obtained bv multiplying 28 by 13 ; let us agree that, 364 'Jnntip.li III' read, g/.intal li I ^'^ ^^ -■^lf diviaioi I ^'-■>', b "1(1, by di Tliei'efoi [isually dit ft by p(>rf() lie ineasu, licnt, whei |i't' the secoi XoTK. — t"iies used. m DIVISION. 35 und is the unit of number of thosf till noon is 96 ; f measuring by n 6. easured. A part nd out how nianv Then always nuntievt, whic-h in tliis rase is 28. The statement ^^^28 may !"■ read, "the ([uotient of ;56i by 13 is 28." ;{:{. The Use Oi Division.-It thus appears that the hori- /niital hne has two meaiiinrrs : (1) When it is drawn between two quantities of the same Kiiul. thus, the^surface^)f a lioard a S(juare foot ' it represents the operation of n,easuring the quantity al>ove it 1- means of the quantity below it as a unit; whieh" operation - perforn. wUh our hands, assisted by instrun.ents. The line, |tlicn, IS called the st^r,^ o/;,,,,,j,,.„,.^-,j^^ (2) When the line is drawn Ix-tween two numbers, thus ■ 364 IT' itity to be meas t ^'^f^''^^^^ /';^' "Poration of ,livi,lin. the nun.ber above it by is measured, we | ^ n"^" '"'"";*; ^^'"^'' "P-^^^-" -- P-'f-n, with our I uHlsl^. niemory of the tables. Here the line is calle.l the .i,jn mif division. '^ Xo»/, by measuring, we find that find that a yard x 361 a yard x 13 "" ■'^' jn of measurin'' I I vas obtained by fi"d, by dividing, that ^ == o^ i, ' n''1-r' f'"" "'' "'■'' '•'"•'•'^^■""' ^'''^^ i«' measuring is T :tr^ ^p..,.,,,,,, ,,,„ i.npossible, we may .:;.,; n which nndo..< f, ' l"^'-*"''"^"'^ the second ..peratior., that is, division. No th.t 3s the operation I rT'"" i'^ ''"' •lU'^"tity, by another as a unit, is the cmo -• ^'- ^--'-' If til; :';";" ""'^"'^ '^ ^^'^^ ''-^ ^^''•-<'^"' l>y tl. n^as^a. low the line is | ^■* i. 11 i .i ::j .J ..J ..J ::) D 1 1 86 ELEMENTAIIV A RITHMETIC. ^i' ( - ,a fo„t X -t," the sign " ~ " denotes tl.e operation of nieasurin.- It 2 feet by 4 feet as a unit; while in " 172 -h 4," the si-m " ^" denotes the operation of dividing 172 Ijy 4. ' * 34. Rule of Order in Division. -Since the order of tl,r product of two simple nunil>ers is found by addi.ig the orders of tl.o simple numbers, therefore the order of one of the simple numbers is found by subtracting the order of the other simpl,. number from the order of the product. In other words t/> order of a simple quotient hy a simple munber Is found hj sub- tractiwj the order of the Divisor from the order of the Dividend :«. To Divide, by a Simple Number, such numbers a. are found in the Multiplication Table. For instance, to divi.l. 3500000 by 700. ,tounm, From the table, the student knows that — = 5. Again, since the order of 35 is + 5, and the order of 7 i,s + v tlierefore the order of the ([uotient 5 is + 3. So that 3500000 -700- = '5000. Article 31 34». To Divide a Compound Number by a Simple Number— For instance, to divide 5S416 by G. The studeiii knows, from the Multiplication Table, the numbers which 6 will divide, namely, 6, 12, 18, 24, 30, 36, 42, 48 and 54; and th, number, 58116, is made up of these, as follows : 54 in the order + 3, that is, 54000, 42 in the order + 2, that is, 4200, 18 in the order + 1, that is, 180, and 36 in the order 0, that is, 36.' Now the divisor 6 is in the order 0. Therefore, by ropeatin.^ the process of Article 35, the (juotient will consist of " .*; that is, in pr; ■ pi'iformi :{8. Ti \purk. — Ti into piece the piecei when we 83 f It is nu in Article tion of nioasurinjr; i i," the sign "-;- "j the order of thoi ling the orders of| ne of the simple the other simple other words, the. 131 VISION'. in the ordei- + 3, 7 in the order + '2, 'i in the order + 1, iind G in the order 0, [that is, the ([uotient is 97;5G. In practice the student sets down the mmiher thus : C)j-J841G "973G 37 i^ found by s«6.|l"'''f*"™j"^' mentally the operations as follows : y' the Dividend such numbers asi istance, to divide -^- = 9, 58-54 = 4, 21-18 = 3, 3G 42 ^ l,s ^.^7,44-42 = 2, 12.3, = G. o. :n. Inexact Division—To divi(h. 473 hy 7. As before 4.3 IS made up of 42 in the order + 1, 49 in the order 0, and 4 hn the order 0. Then the divisor 7, being in the order 0, the .,uot,ent consists <,f G in the order + 1, 7 in the order 0, that is Article 34. l^' 5 ^^"t the remaining 4 is not divisible by 7 at present We n shall show later on how 4 can be .livided by 7, but now, we shall I only uuhcate tluit it is to be done, thus : J 473 4 4 by a SimpleB -y- = 67 + — or Q>1-. 1 The student ■ Z ^^ ^ H " i ?■ ? ""T '''' ^"^"'' '^^"•' '^ '^^ «-^ 5 pounds of Kl 54 ; and the tpork.So actually measure this we should have to cut the pork 1 H-to pieces ead. ccmtaining 5 pounds ; and then group and count I the pieces. Ihis operation is indicated by the horizontal line • m when we write 8325j)ounds of pork a po und o f pork x 8325 re, by i-epeatin J .- ^ ^^-''-'^ P-k^ ' "' a pound7.rp;;Hr^- t of ^ J :, , ^ not ,h>sirable to perform this operation, and it is shown HI Article 33 how we may a^oid doing it, by performing in our 38 ELEMENTA RY AIUTHMETIC. ' i V I minds anotlier oporation (|uit(> (liflForont, niunely, by dividing tin nuiiihcr 8.325 by tho nuniber 5. This operation is indicated by the line in ' J" , 8325 and liaving performed it, we may say, -— ^- = 1065. Therefore, tlie measui'e of S325 pounds of pork when the unit i 5 pounds of poi'k is 10(55 ; that is, a pound of i)ork x 8325 ^ — _ 1Q(;5 a pound or pork x 5 Further, we may say, 8325 pounds of pork = 5 pounds of poil | X 1GG5, that is, "8325 pounfls of pork is derived from 5 pound of pork by the rate 1665." 1. EXERCISE VII. Divide : (a) 500000 l)y 80. (c) 210000 by 7000. (h) 8100 by 90. (d) 30000 by 000. (e) 420000000 by 700000. (./') Tlu-ee liundred and fifty million by seventy thousand | (g) Three hundred trillion by sixty million. Give the reason for the order of the simple number in tliJ quotient in each case. 2. Show how the number 435 is made up of the products of found iji the Multiplication Talde, written in their proper order- Hence, write down the (juotient when 435 is divided by 5. 3. Show how the number 439821 is made up of the produt i J of 9, found in the table. Hence, find the quotient when tlii^i number is divided by 9. 4. Divide : (a) 2139216 by 2, by 3, by ■*, by 8. I (b) 31425 by 5 and the quotient by 5. (c) 593021 by 7. ((/) G1G8900 by 2, the quotient by ."., the second quoliei. by 4, and so on, by 5, 0, 7, 8 and 9. Division. 39 y, by dividing tli Heventy thousaiK e secuiui quotifi. the result of pefforniing the operation (e) 0108900 by 9, the quotient by 8, and so on, by 7 i), 4 and .). a. A mile = a hmt x 5280, and a yard = a foot x 3. Find a mile a yard' 0. A wheat field contains 420 shocks of nr,ain, each shock- contains 10 sheaves. How many sheavf^s are in the field, and how many loads, each consisting of 300 sheaves? How did you find out I 7. Tn a box are 2384 matches. Tf these are tied in bundles each containing 70 matches, how many matches are left which are not enough to make a bundle ? 8. What is the measure of a pound x 891, when the unit is a pound X 9? 9. Perform the following operations as far as possible • ia)'J^. (6)^il'«. (,)L0OLO 830O0_ 5 (e) 2143 (/) 8 4713 (9) 2103 • ^^/ 3 10, Find the result of performing the operations : a yard x 8321 , , 8816 ... 5191 cents (6) - i^) , -, 18321 men / cents ■ ^"'' a yard x 8 (/) 6 men a minute x 480000 («) a match x 8347 a match x 5 an hour 11. Distinguish between the operations indicated in No. 10 and those in No. 9. 12. Divide 47585 by 4, by 40, by 400 and by 4000. 13. Divide G03S071 by 60, by GOOO, by 800, by 90000 and bv 1000. ^ ::> .,i u ...1 .J ;> !') iv: 40 ELEMENTAUV ARITHMETIC. :W. To Divide by a Compound Number—For instanc. to divide 14324 by 593. It will l,e sulHcient to find the order of the first fi-ure of thr quotient, thus: Since the order of 14 in tlie dividend is + 3, and the order of 5 in the divisor is + 2- therefore the order of the first figure in the ,,uotient is + 1 • that IS, the quotient will consist of two figures. We next show hou to find these figures. By trial we find that 593 x 3 is more than 1432, and that 593 x 2 is less. Then the first partial dividend = 1432, and 593 x 2 =1186' therefore the remainder = 24 G,' which is less than 593. Again, by trial, we find that 593 x 5 is more than 2464, and that 593 x 4 is less. Then the second partial dividend = 2464, and 593 x 4 _ 2379' therefore the second remainder = 92 which is less than 593. Hence 14324 - 24 92 593 " ""593' In practice these operations are performed as follows : 593)14324(24 1186 2464 2372 92 It will be shown, later on, how 92 may be divided by 593. 40. Inexact Measurement of Quantities. -Lot us meas- ure the quantity 4325 inches by the unit, a yard, which is 30 inches. 4325 by Now, we have for inches b} I he yard -mall poi have beei show, in divisi(»n ii 1. Div (a 2. Divi («; (*) (0) 3. Stafc performed 4. Find tients : (a 5. If tl distance i iistance w 6. How DIVISION. 41 -For instance I'st figure of th» iiuheH. By Article 33, we find the retiuired me m 4325 l)y 3G. When this is <1 ijisure by diviciing •132.' ent is + 1 ; that i next allow how '2, lian 2464, Hows id by 593. -Let us nieas , which is 3G one, we find that -'-— == 120— 36 36- Now, we are not as jet able to divide .5 by 30. This we might have foreseen, had we actually measured the distance 4325 I inches by the yard length. For we should then have found that the yard would have been placed down 120 times, but that a small portion, 5 inches, would have been left, which it would lia\(' been impossible to measure with a yard as unit. We shall show, in the next chapter, how we may i)roceed both with the division and with the measuring in such cases. EXERCISE VIII. 1. Divide: (a) 144 by 24. (b) 1728 by 36. {(■) 2448 by 17. 2. Divide: (o) 139748 by 629. {b) 82143 by 5389. (c) 7218356 by 84162. (d) 83376 by 18. (e) 543125 by 125. (/) 31416 by 24. (d) 8000000 by 725. (e) 835129 by 6172.3. (/) 800405 by 90301. 3. State the result of the following operations, after you have performed them : 12345^ 203527 ^"^ 432T~- (^') -frr- 4. Find, without dividing, the number of figures in the quo- tients : 123476 596213 ^ ^ 100000000 241 4132 56042 -i 5. If the sun's distance is 92445600 miles, and the moon's distance is 2:^7040 miles; what is the measure of the sun's distance when tlic moon's distance is the unit i 6. How many 25 cent pieces will make up 29450 cents? LJ ..J J-;: I 42 KLKiMRMTAHV VIUTII.NUTlC. 7. A milo = a yani x I7(i0. 11, nv ,„u„v iniLs u,v tl.Mv in yanl X 1283040/ ' .^. A car is Ina, repeat tli.> unit until the other end is reached. It will h(. found that the last time the unit of len-rth is plaoeii dnwn its end reaches heyimd the (>nd of the line. There is, therefore, a ))ortion of the line which cannot be measured by the unit chosen. The greater part of it is measured, but a small I part less than the unit remains unmeasured. To measure this small part, we proceed to choose a more con- venient unit. Let the original unit be cat into 10 equal i)arts, each of which w(> shall call a mh-nnit of the 1st ordn: Pro- ceeding with this sub-unit to measure the part of the line still unmeasured, we find, as before, that the greater part of it is measured, but a small part less than this sub-unit is still unn.easured. Again, to measure this small part, let us cut this sub-unit of the 1st order into 10 equal parts, each of which we shall call a snh-nnit of the ,2nd order. Proceeding with this sub-unit to measure the part still unmeasured, we find, as before, that the greater part of it is measured, but a small part less than this sub-unit is still unmeasured. liy coiitinuing this process, we may measure the line until the part unmea-iiretl is so small that it may b(> neglected. It may I I ■J • I : ) \ ) : 5 O ;; 44 ELEMENTAIfV AnTTIIMRTiC Jj'JPl-". '"'wover, that a sul,-,nut oxaetly .noasu.vs tlu- part l-H .n wluch ca,s(. th.- lin.- is acc-umtHy n.msun.l 4'^. S„pposo, ,..nv, .just lH.fon. tho .,.,1 .,f th,. lin,. was r..ac-lu..| l-......nal unit was pla.....l ,l.>wn 7 ti.nes, U,,. sub-unit „f tl,.: Is .>r,l,.r 5 tnn,.s, tl... suh-unit ut tl... 2n,l on|„,. 8 tinu-s tl... sul. un.t..f tl... ,,.,.,,.,.., un.,.s,an.l that w.,.n,.,l..etth..', A,-cor,li,,,, to what was a^r.,.,! upon in Articl,. ;{, „,. wri„. h.s,. s.n.pl,. nun.lK...s in th .,|.,. 75H4 ; an,l wh.-n w,- hav. I..st.ngu.she, th,. fi.an,. which is th,^ nun.l,.... of on.Mnai units from the ..th,.r figufos, w.. shall hav. ,..,n.pl,.t,.|y ...v^vss,., n>easuro of th,. ..uantity. This is usually ,|o,u. 1,1 wntil po.nt after the 7 thus, 7-5S4. ,U.t this way of witin, th." :•.; IS son.owhat nusl,.a.lin. an.l we shall, in this ,.hapt,.r: n.ark th., units hgure by writin. the point above it, thus, 75.S4. In ,.ith..r case we rea,J the nun.be., ' jiouud X 2 ■•5. A me .1 $ X 893 8279; on What is th it^mvs the p.iit I. nc wiiH readied, sulMinit of til.. ' times, llie siili ct tli<' |)iirt stil cle .'{, we svritc when \\(. }i,i\,.j f <)n';,'iii;il units ■ ox pressed tl,,. * •)>' writing' a I'itin;^' the jHiint -pter, iiifirk tlic ^^^- In either t, four." le line by the RE(UJLAIl SCIJDIVISION Ol' Till.; I WIT— nKclMALS. 45 I Nvei-hts of whieh 10 make one of thr pr.ve.hn-, all p„t to^r,.th(.r jinake up the weight of the iron Imll. This operation of (h-rivin- |tlie weight of the iron hall fn.m the pound weij,dit is denot..!^ as before, by the si-n " x ," which is hero cuUed the sign of jderivation. Further, when we write a unit x S;i }02, we mean that : H nudtiple units of the 1st order, 3 units, 4 sub-units of the 1st order, and 2 sub-units of the 3r(l order, |.uv put together to make up a certain (juantity. 44. In woiking the following examples, it is sometimes neces- sary to remember that, wlien the units digit is marked, O's may be wiitten either befi.re or after a number witliout changing it. |T1ius, 123 = 00123000, but 72 is not the same as 7200. EXERCISE IX. 1. T)esci-ibe the process hy which the (juantity is derived from the unit in each of the following : (ii) A rod = a yard x 55. (if) The length of a field = a r. >, ^ 326. (r) A roll of butter = a i)ound of l»utter x 525. ((/) The surface of the table = a square foot x 5934. ((') The cost of a yard of doth = a dollar x 1875. (/) The cost of a bicycle = a dollar x 645. 2. Find the total of the followin« <|uantities : a pouiul x 361 I anc a p.mnd x 8975, a pound x 80213, a pound x 51: pound X 253. 3. A merchant receixed, on Monday, a | x 3456 ; on Tuesday, a $ X 8931 ; on Wednesday, a $ x 8235; on Thursday, a $ x 7; and on Saturday, a $ x li>389. 8279 79 ; on Friday, a f x 426 M u ..I • I ■ .) 1 ) What is the total sum he received during the week ? 46 ELEMEX'l'A R Y A HITHMETfC. UEG 4. Add tojrretlier tlic rates 42390, 531, 01197, 8001, 56021.; 000008, 500, 89601, 321 and 03. 5. Find the sum of 1321596, 30097, 000146, 3962, 8009321 5037968 and 987654. 6. Find the difference of the following (juantitie.s : (n) A pound x 87239 and a pound x 936. (b) A minute x 7238 and a minute x 81627. (c) A yard x 1823714 and a yard x 8971*384. (d) A dollar x 176312 and a dollar x 31862. (e) A cubic foot x 3904 and a cubic foot x 4372. 7. Subtract : {") 05 from 1. (b) 83 from 10. (c) i ^rom 32. (d) 032 from 041. (e) 0062 from 0532. (./■) 09999 from 1. {[/) 98765 from 12376. (A) 0005 from 00061. (i) 2003042 from 3901621 0") 123456 from 123456. 8. Tf out of a barrel of water whidi consists of a galh.n x 320G, there be taken a pailful which contains a gallon x 371, as many tunes as possible ; how much will there be left? 45. The Order of the Digrits of a Number—In anv number, such as 135798642, as in ArticU> 25, we say that (lir order of 9 is the 1st place to the right of the units digit, the oi-der of 8 is the 2nd i>lace to the right of the units digit, and s., on. In other words, for shortness, let us say that, the order of 9 is - 1, the order of 8 is - 2, the order of 6 is - 3, and so on, where we let the sign "-" stand for the words " t.. the right of the units digit ; " then - 5 denotes the 5th place to the right of thr ^ units digit. The orders of the digits are shown in the sclicnH- B +() "W from the g ■^;,'reater." 4r. To stance, to 'place, we c '^llieii from -las in addi he left. Again, t fifth place the right (j ;it the eigh Hence a another : " Changt Addition 1^ Thus, to IlEGULAU SUBDIVISION OF THE UNIT— DECIMALS. 47 , HOOl, 5602 l:i 31)62, H00932I. cs : 27. 3S4. .62. X 4372. • m 1. in 12376. I 00061. "roin 39016: !H om 123156. b gallon X 3l 06 < 371, as mam iber. — In any ' say that tlir iiiit.s (lii^'it, tlic •s digit, and sn to the right of hi' right of the n tho sduMiie ; 4«. To Find the Sum of Two Orders. -For instance, to Jaild -5 and -3. Beginning />o;h tho units place we count to ^thc riglit 5 places, arriving at the fifth place ; then,yro»i the 5th jplace we count to the right three places, arriving at the eighth place. Thus we are able to say ; - 5 - 3 = - 8. To add - 5 and + 3. As before, we arri\e at the 5tli place ; tlien, from the 5th place we count to the left 3 places, arriving |at the 2nd place to the right. Thus also we can say : - 5 + 3 = - 2. [Tn the same way we may show that +5+ +8, and that + 5 _ 3 = +2. We have, therefore, the following rules for finding the sum of two orders : (a) "When the signs are alike, add the numbers, and write before the sum the same si) "Wlien the signs are unlike, subtract the less number J from the greater, and write before the difference tlie sign of the gi-eater." 4T. To Subtract one Order from another.— For in- jstance, to subtract - 3 from - 5. Beginning J'ruin the units jplace, we count to the right 5 places, arriving at the fifth place ; nhcn from the fifth place we count to the left (not to the right, fas in adding orders) 3 places, arri\ing at the second place to -n the left. Again, to subtract + S from - 5. As before, we arrive at the Ififth place to the right ; then, from the fifth place we count to the right (not to the left, as in adding (jrders) 3 places, arriving [at the eighth place to the right. TTenee we have the rule for subtracting one order from another : "Change the sign of the order to be subtracted, and use the [Addition Rule." Thus, to subtract - 5 from + 2, we have + 2 + 5 = +7. n1 I ! I.. c;> L.J '■",> '.{ I 1 > 48 ELKMENTARY ARITHMETIC. li/ir' EXERCISE X. ^ 1. What is the order of each digit in the numbers 1000301, ^ 0300082, 1800001, 7, 1708 and 8000010000007 ? 2. Write 3 in the orders +3, -3, -5, +1,0, -land +2. 3. Write 72 in the orders +2, - 2, +1, - 1, - 5, +4 and 0. 4. Write 30 in the orders +2, +5, - 2, 0, -8, +1, -1 and - 3. 5. Write 293 in the orders +4, - 4, +2, - 2, +1 - 1 and 0. 6. Write 8200 in the orders -7, +1, -8, +4, -3, _ 1, o and +2. 7. Find the sum of (a) - 3 and + 2. (/) + 2 and + 3. (k) + 8 and - 8. (h) - 8 and + 6. (y) + 3 and - 7. (/) - 3 and - 4. {c) + 5 and - 3. (h) - 5 and + 7. (m) - 348 and + 962. (d) -Sand -1. (i) + 7 and +5. (?<) - 1203 and -481. (e) Oand -2. {j) +1 and -1. 8. Subtract the 2nd order from the 1st in each of the above pairs. 9. Subtract the 1st order from the 2nd in each pair of No. 7. 48. The Rule of Order in Multiplication.— Since a sul, unit of the 3rd order = a multiple unit of the 2nd order x 000001 = the imitx 100 X 000001, and a sub-unit of the 3rd order = the unit x 0001. (■^XD . . (iX-'XiiXlXr') . dXiiXii) Therefore 100 x 000001 = 0001. Now, the order of 1 in the l.st number is + 2, tlie order of 1 in the 2nd number is - 5, and tlie order of 1 x 1 oi- 1 in the product is ~ 3, which is the sum of + 2 and - 5. .■Li REGULAR SUBDIVISION OF THE UNIT— DECIMALS. 49 hem 1000301, - 1 and + 2. - 5, + 4 and 0. -8, +1, -1 -2, +1, -1 i, - 3, -1,0 id -8. id -4. and +962. 3 and -481. of the above oair of No. 7. — Hince a suli DOOOOl Therefore, in tliis case, as in Article 26, the order of the pro- Jduct of two simple numbers is found by adding the orders of the ^ksiinple numbers. I Simihirly every case, which we may examine as above, is found 4u) l)c compreliended in this rule. This rule governs all the opera- Itions of compound numbers, and the student should master it. 1^ 4». To Multiply 43052 by 80076. J Since the order of 2 in 43052 is - 3, Jiuul the order of 6 in 80076 is - 3, '^therefore the order of 12 in the product is - 6. ^ Again, the order of 4 in 43052 is+ 1, ^aiid the order of 6 in 80076 is - 3, therefore the order of 24 in the product is - 2, and so on. Now, each simple number in 43052 is multiplied by each simple number in 8007^ • orders of the products are found as [above, the products art . y^en in these orders and added. This is done conveniently, as follows : 43052 80076 0258312 = 301364 = 344416 The product by 6 Itlie product by 7 llio product by 8 therefore 43052 x 80076 = 3447431952 EXERCISE XI. 1. Write down the product of each of the following pairs of miiubers, and give the reasoning by which you find the order : (a) 300x004. (e) 7000x000008. {b) 07 X 008. (/) 4000 x 0004. (c) 0003x400. {g) OlxOl. {d) 8000 X 002. {h) 002 x 002. 4 M u.. f.:3 1.1 • J :;5 : 1 «:;•" ^..:, *:;!; :;;:» 50 \'y I ELEMENTAUY ARITHMETIC. 2. Multiply together (a) 1004 and 03. (k) 0703 and i02. (c 729x531. (d) 1382x2976. (e) 14896x00342. (/) 7698 X 372. (y) 2190 and 827. (h) 78912 and 00397. (0 9437 and 8647. U) 1097083 and 407301. Whiit 3. A rod = a foot x 165, and a foot = an inch x 12 rate will derive a rod from an incli ? 4. A barrel of water = a gallon x 315, and a gallon = a cubic :nch^x^2.7274. What is the capacity of the barrel in cubic 5. The oircumterence of a circle = its dian,et«r x 31416, and U,e diameter = a foot x 2575, How many feet does its circun.- lerence consist of. 6. Simplify 0125 X 612 X 2i3 - 0375 x 504 x 312. 50. The Rule of Order in Division. -Since Division i. he operation which reverses the operation of Multiplication LViL D- ^"'"f ""' ''-'' '^'^^ ^"^ ^- «^ ^'- rates found! "^ '' P'"'''' ^^ ^^^"^'^ ^^« «^h«r rate is Moreover, sin .e f he order of the product of two simple num- bers IS found by adding the orders of the simple numbers- therefore, the order of one of the simple numbers is found W ubti-acting the order of the other simple number from the orde ot the product. In other words : The order of the quotient by a simple number is found by sub- tra.t^ng the order of the Divisor from the order of L Divil^ 51. Examples.—!. To divide QQ^2 by 1Q. rrom the tables, -^ fi REGULAR SUBDIVISION OF THE UNIT— DEC.'IMALS. 51 Now, since the order of 42 in 0042 is - 3, xiid the order of 7 in 70 is + j therefore the order of G in the quotient is - 4, which is - 3 - 1. bo tliat the quotient = 00006. 2. To divide 4768 h;) 008. As in Article 36, the Dividend 4768 is made up of the pro- jciucts of 8 found in tlie Table, as follows : 40 in the order - 1 , that is, 40, 72 in the order - 2, that is, 072, liUKi 48 in the order - 3, that is, 0048. And since the order of 8 in the divisor is - 2 ; therefore the jiuotient consists of 5 in the order +1,9 in the order 0, and 6 Jn the order - 1. The quotient then is 5<36. In practice the operation may be sot down thus : 008 ) 4768 4768 • or -; — = 596. 596 008 3. To divide 1472036 by 0562. It will be sufficient to determine the order of the 1st fi^^ure of the quotient, thus : '^ Since the order of 14 in the dividend is f 2, luid the order of 5 in the divisor is - 1 itl.orefore the order of the 1st figure of the ciuotient is + 3 ; that lis, the units figure is the 4th figure of the (juotient. Next, we divide, as in Article 39, as follows : 0562 ) 1472036 ( 2G1928 1124 3480 3372 1083 1083 562 5216 5058 156 1580 1124 4560 4496 64 ONTARIO COLLEGE OF EDUCATION I) ^^ : :t ' I 1 ';i ^ t I '.'> 1 . i 1 ::■),: ■, 1 • .-i . ' ^ '.: ■ il -•ir, ■7 o,,i; ,;# Jiii: 52 ill ELEMENTARY ARITHMtTIC. Hen We ce - 11720;}6 0562 = 261928. f,V.l f/""^"^"*^^ ^''<^ division as far as we wish, hut in pra. to t Jk h"'""-! t ^^"'"""" "^ ^''^■'^^^ ''^' ^" ^^« ^>-'« to do . 593 ) 143240 ^ 2415 ri86 2464 2372 920 593 3270 2965 305 NoTK.-Iu the following chapters we shall mark the units figure by wr.tu.g the point after it, as is the custom, thus : 1357 = 135-7. 0032 = 0-032 =-032. EXERCISE XII. 1. Obtain the .juotient in each of the following, givin.^ the reasoning by whicli its order is known. " (a) ~~ 08 , ,, 4200 07 ,,, 00072 009 / , 240 004 (') A 63 {/) 0009 A 800' (h) 0009 036 9000' 280 ^'^ 7000- U) -. 005 REC 2. Red 3. Fine {a (^• 4. Obta (^• 5. Obta (a (&) («) G. A m Find to 6 7. A m yards (6 fi, 8. A ro< from a rod 9. inch Thel X 183 UEOlfLAR SUBDIVISION* OF THE UNIT— DECIMALS. 2. Reduce the following to single rates : .53 (a) 16- 1 1237 X Find the order of the first figures in the quotients when (a) 723 is divided by G24. (b) 1083 is divided by 7093. (c) yHia divided by ;iG02. (d) 001379 is divided by 00000' 35. 4. Obtain the quotients, each to six figures, in (^') G24 723 (c) 18964312 50372 , , 2345G (e) — ;-- . 234 (/.) . 953 ,„ 100000 (/) 2345 0037 ^ ' 15G25 ' 1234 5. Obtain the follov/ing quotients, eaoh to 5 figures : («) I (^) -3 i'i) 1 7 1 9 (//) 1 T3" (i) 1 TT7' (/O -jy ^^•) rm- (/)A- (^) 2^ l9" 1 (0 HiiTi 999" G. A metre = an inch x 3937, and a yard = an incli x 36. Find to 6 figures the number of metres in a yard. 7. A mile = a yard x 17G0. How many miles in 407821 yards (6 figures) ? 8. A rod c= a foot x 165. What rate will derive 4356 feet from a rod ? 9. The length of a desk = an inch x 457, and its width = an moh X 183. How will its length be obtained from its width? I ' . I' U i ! .1 ■ .1 - I : :) 1 > • -« i !■■- !;, ,•: 54 KLRMENTAItV ARITHMETIC 10. Jiy division and uddit ion siiujtlify •i IC + .-L' + Gl + nS- 11. Find a single rate of 5 figures equivalent to ^••n rm L'i7 I'^tL^ 79 X 007 :m • 1.1 Find five figures of ^l^^l^ -009143 x 21 -57 U. If the length J/i = the length /?(7x ;?■•)•) the length JiC= the length CD x .oQr, ' the length 6'Z)= the length i>A^xG.10,' and the length Z>^= the length J^ x ■OOi;;32 1 nnd the siiiir e rate whieh tollc, i. ^ i . ' ^ the length ?Q. "'^ '"" '" ''^'^^'^ *''^^ '^^"^'tl^ ^^^ fr-n J ir>. Find to the order - 5 each of the following • , . 800x111250 (b) 1:^858 -25704 -}627 X 58235 490x525 (c) _i01^li05^x 105_ io5xio5xio^rT" 10. What is the use of the point above a di prmlnct ..f iu factor. T. • factors of a Iarirenui,.h..r,.,. T " '■;'"'" "''''^''' ^^" -tural n«.nhers are Ji :!:,';::/ '^ '"'"'• '""' '^" ^^^- EXERCISE XIII. 4. factor into prime factors 50009 a„d -8105 («) 8, 12. (f^) .36, 24. (^•) 30, 40. (d) 24, 32. (e) G8, 51. (/) 21, 03. iff) 16, 40. (A) 2H, 91. (*■) 111, 75. (i) 24, 42. W ''">0, 03. (0 38, 70. (w) 42, 00. (o) 250, 150. (/>) 91, 52. (y) 135, 75. ('•) 32, 52. (») 1200, 1800. (0 135, 105. («) 10000, 15025. r ^^^ -. ,. ^"^ •'"'''^' ^7. (/i 0. \\ „te all the prime numbers helow 100 '• ;^ nte all the prime nu.nbers between 100 and ^00 8. ^nte down all the sub-„.ultiples of 210 feet 9^ Wnte down all the sub-multip.es of a , > ^d . 13^ 10. Ascertain whether or not a yard x 34-50 ^n" exactly a yard x 241 Qo Jt . ^^ "^'^ measure ^a^WUI a gallon x 2-93 exactly measure a gallon x 23-54'^ 12. What quantity must be taken from a yard x 7321 in THE GREATEST COMMON DIVISOR. 57 ■H. The prime tlu'v the prime iet!( Hfuiry. An and all other ^, a-' for 3 X .1 mnher.s 9, IG, '. 51, OS, 111, '>C0, (b) 2880, vo factors, r,f' 150. 52. 75. >2. ', 1800. 105. 0, 15G25. fOO. 132. 11 measure ? 1 X 23-54 ; ■Ji I (.nirr that the n'mainder may he measured exactly by a yard i:i. What is the smallest number wliith must be added to 7.U01 t(» make the sum a multiple of 834 '/ 14. Find sub-multiples of the following' (piantities, the length (a) and the surface {!>) : (a) (^) .'»4. The Greatest Common Divisor of Two Numbers. —The <,'reatest (piantity which will measure exactly two or more otlu'rs is called the dreatest Common Sub-multiple of them ; or for shortm'ss, the O. C. S. of them. It is evident, then, that we shall find this quantity by finding the greatest number which will exactly divide the measures of the (piantities. This number we shall call the Greatest Common Divimr of their measures ; or fi'i' shortness, the G. C. D. of their measures. 55. The G. C D. of Small Numbers is easily found if the student knows their factors. Thus, since 91 = 7x13, and 65 = 5 x 13, therefore the G. C. D. of 91 and 05 is 13. S( ), also, let the length AH = CDx\)\, and the length MI^^ CD x G5, where 91 and 65 are the measures of AJJ and MN with CD as unit ; then the greatest length which will exactly measure both A/i and JLV is CD X 13, which we call the G. C. S. of Ali and MX. 5C. Let AB and .4C be two quantities (lengths), each of which is measured exactly by the unit MX ; ' C B vS X 7321 in Af X 58 ELEMENTAKY AlUTHMETIC. then it is evident that the .hfterence liC is also Measured exactiv ^yM^. In a suu.lar n.anruT we n.ay see that J/.V will also measure the sum of A li and AC, the su.n of AB x 3 and AC x 5 n^d the (hrterenco of A/i x 1 1 and AC x 7. otl!!!"'"-?' '•'' 'f """' 'T"'''^ " " -iMnultiple of each of twu stn ; " :• '; ^"'"""'^'>'" "*• ^'- ->. tl- difference, th. ua of any n.ult.ples. or the .li/Ference <,f any nndtiple of these two «iuantitie8. ^ Therefore, also, if one nund)er is a divisor (factor) of each of two nun.bers, it is also a divisor of the sun, the difference the sun, ot any n.ultiples, or the diffen-nce of any n.ultiples of t hes. two nunibei's. J^T\T\ ^.'"''■'" '' •""'' '^'"' ''^ ^'"'^^^-« ' 'divides ^- X 9 - 03 X 4 ; that ,s, 1 20. So, also, sinc<, an inch x 7 meas- ures exactly an inch x 42 and also an inch x 03, therefore an rind, Vl2r'" "'^'"' "' '"' ^^''^'' '' - '^ ■' ^'-^ ^^' r,7. To find the G. C D. of Two Large Numbers- therefore C d.vules 1752x2-2701, which is 803, a nun.be less than either 1752 or 2701. nt'tof " ^/^'f '" '^'' '^"'^ '"^'^^ '"'' ^h'^-f-^ ^ divides 1 /.). - 803 X 2, which IS 140, a number less than 803 Agam since G divi.lc-s 803 and also 140, therefore G divides 803 - 140 X 5, which IS 73. Now, 73 divides 140. We shall now prove that 73 divides both 1752 and '^701 8mce 73 divides both 140 and 73, therefore 73 divides 140x5 + (0, which IS 803. Since 73 divides both 803 and 140, therefore 73 divides 803 x - + 140, which IS 1752, one of the numbers. Finallv-, since 73 divides both 803 and 1752, therefore 73 '^2 X 2 - 803, which is 2701, the other number i Hence In prac The nui Imts next [ii'iiducts i nllicr side 58. To We iind Hence t (I C. S. of divide THE fUlEATEST COMMON niVlSOR. 59 '/iV will also and AC X n, each of tw(. ilU'icncc, the ipli- of tlicst' ) of f>ach ot' fll'ienct', the jU'h of these B 7 divides » X 7 nieas- hert'fort' ,iii 4} ; tiiat i.s. Limbers- will dt'iioU- I also 2701, a number 5 G divides ! G divides 3701. les 14G X 5 HencH the G. C. D. of 1752 and 2701 is 73. in practice these operations are conveniently perfonned thus 1752 I 2701 j IGOG ' 3504 ! 73 2 Th(> numbers outside the lines are the mutipbc y of tlie num- ixTs next them on the inside, and they re so clu, .>n that the products are as nearly as possible e.pial ,n Is (^ num'-ers on the other side under which the products are pi act ^ 58. To find the G. C. S. o/ 159531 inches and 70479 inches. We find the G. C. 1). of 159531 and 70479 thus: 8 159531 140958 70479 74292 18573 1 90G5 3813 3930 492 492 123 4 Hence the G. C. 1). of the measures is 123. G. C. S. of the (luantities is 1 23 inches. Therefore the EXERCISE XIV. 1. Find the G. C. D. of the following pairs of numbers : (a) 3G and 48. (e) 1G539 and 27417. (b) 32 and 54. (/) 835125 and G9375. (o) 120 and G4. (r,) 3378 !s.3 and 3052575. )12345G and 9999999 90 Find the G. C. S. of r. GG429. (A) 79012345G i an inch x 1G9037 and an inch x x:;. 60 Elementary AiiiTHMETfc. 3. A field r,4G feet wi.le and 086 feet Ion. i.s to be fenced witi, boards as long a.s possible without cuttin, then. How on " the boards, and ho. n.an, .ill it take if the fence is t::Z 4. Find the G. C. D. of 650935, 530620 and 947095 o. A held 91 rods long by 65 r<.,ls wide is to be divided off nto the la.^.est possible squares. Wh.t is the sixe of eac square, and how many will there be ? 5». The Least Common Multiple of two or more quan t.t.es xs the least quantity which each of these quantities wHI .n^s^e exact^. It is evident, as before, that L find t, t! hall find the least number which the measures of these oulnt titj which 3 feet. 4 feet and 8 feet will measure exactly The measures are, of course, 8, 6 and 3, respectively. '^' «0. To find the L. C M. of Twd Numbers -For n..s ance to find the L. C. M. of 621 and 989. J.y "rmetl^ 1 Thus, 621 = 23x27, '""'"I. 989 = 23x43.' Wince 621 divides exactlv tho T P \r i • i +1 ♦ ... ' ^"^'^^ ""e tactors are necessary and they are sufficient; therefore, the L C M of f>,« , = 23 X 27 X 43 numbers ■^"^^^ 23x27 = 621, and 43 = — 989 THE LEAST COMMON MULTIPLE. 61 •e fenced witli How long ar(> ' is 5 boards )95. )e divided off size of eacli • more quan antities will fijifl it, we hese quanti ast nunibei' i least quan- actly. Tlie bers—For the method imbers, one re finding, two of its '. INI. must ssary, and numbers tiierefore the L. CM. = G21 x 989 "23"' Hence we havt; tlie rule : Tlie L. C. M. of two numlx'rs is J\mivl by vmltiplijuig one of lh>: numbers by the quotient tvhcn the other is divided by their G. CD. Example i.— Find the L. C. M. of 9G and 144. Since tlie G. C. D. of 96 and 144 is 48, jind the (juotient of 96 by 48 is 2, therefure, by the rule, the L. C. M. is 144 x 2, or 288. i*\. The L. C M. of more than Two Numbers is found by repeating the above jirocess, as follows : To find the L. C. M. of 96, 42, 70 and 60. Tlie G. C. D. of 96 and 42 is 6, the (juotient of 42 by 6 is 7 ; tiierefore the L. C. M. of 96 and 42 is 96 x 7. Again, the G. C. D. of 96 and 70 is 2, and the ([uotient of 70 by 2 is 35, that is, 7x5; but 7 is already written down in 96 x 7 ; therefore the L. C. M. of 96, 42 and 70 is 96 x 7 x 5. Further, it is seen that 60 lias its factors, 12 and 5 included in this, so that the L. C. M. of 96, 42, 70 and 60 is 96 x 7 x 5. For small numbers, this is set down as follows, where the reasoning is mentally performed. Thus, the L. C. M. of 24, 36, 40, 48, 60 = 60 x 2 x 3 x 2 = 720. iVt The L. C. M. of Large Numbers is found in the same way. Example l.—To find the L. C. M. of 621, 4209 and 2024. The G. C. D. of 621 and 4209 is found as usual, thus : Then the quotient of 621 by 69 is 9. 7 621 552 4209 4347 4 J -J 69 138 138 Therefore the L. C. M. of 621 and 4209 is 4209 x 9. 62 ELEMENTARV^ ARITHMETIC. Again, ^the G. C D. of 4209 and 2024 is found : Also we divide thus 23 ) 2024 ( 88 184 12 184 184 , 23 I 4 . Therefore the L. C. M. of the three nunibers = i209x 9x88 = 3333528. EXERCISE XV. 1. Find the L. C. M. of the nun,bers ii, each set : (a) iO, 50. (b) 72, 45 (c) 83, 47. (d) 28, 42. ^^) 16, 34. (/)81,54. (//) 121, 99. W 117, 65. (i) 250, 300. U) 1000, 325. (^•) 1287, 6281. (0 132288, 107328. (m) 94605, 96509. (n) 9534, 15663. ..rl38448323, o T- , , ^1-168695032. -■ Imd the L. C. M. of the numbers in each set • « 10,12, 16, 18, 24, 28. 30, 40, 42 and 48." if>) 24. 28, 36. 44, 55 and 60. (o) 32, 48, 64, 96, 80 and 108. i'i) 96, 120, 144, 84 and 90. (e) 120, 200, 240, 300 and 320. 3. Find the L. C. M of • ss8^';^^t::niS^^^^^'^^---- titL wni nitu^irr^'" '''''' '''-' "^ ''- ^^"^-^"^ ^"- A foot X 54, a foot x 72 and a foot x 84 5. The hind and fore wheels of a waggon are 16 feet .^ V^ feet m circumference. " " ■- -'^ -^^-r. nna i_ How far will the waci ggon go so that each wheel m will each 6. As tlie small How lor there be I 7. The their G. ( 8. The G. C. D. i 9. A c seconds, ( from the on the lin 10. Fir {(i) and (b (a) {'>) 11. Fin 12. Fin and 9933 13. The ijiches an< measure ej 14. The feet aroun makes an occur in gt 15. A n wide. It What is til 16. Fine 18 tt)s., 21 THE LEAST COMMON MULTU'LE. 63 thus : 88 G281. 8, 107328. , 96509. 15663. 8323, 3032. id 23360. ing quan- ; .and 12 Iiat each wheel may make an exact number of turns ? H will each make ? ()w many turns \ 6. A square field i.s of such a size that it can be fenced off into the smallest number of lots, each 66 feet wide and 72 feet inw^. How long is one side of the field, and how many lots will there be? 7. The L. C. M. of 391 and another number is 12121, and their G. C. D. is 23 ; find the other number. ^ 8. The L. C. M. of two numbers is 634938944491, and tlieir G. C. D. is 9187 ; one of the numbers is 85044059 ; find the other. 9. A can go around a race-course in 54 sec(mds, Ji in 03 seconds, C in 84 seconds, and I) in 91 seconds. If they all start fr.mi the same line, how long time will elapse before they are all on the line together ? How many rounds will each make ' 10. Find the G. C. S. of the followixig quantities, the lengths (a) and (b) : ^ («) (/^) ^ 11. Find the L. C. M. of (a) and {!>) in No. 10. 12. Find the smallest distance which will contain 15939 inches and 9933 inches exactly. 13. The length, width and height of a block are 28 inches 18 inches and 13 inches. AVhat is the greatest length that will measure each exactly ? 14. Th6 hind and fore wlieels of a waggon are 14 feet and 12 hrt around. How far will the waggon go until each wheel makes an exact number of turns? How many times will this occur in going a mile? 15. A rectangular field is 27264 inches long and 16512 inches wide. It is marked off into sfjuare plots as large as possible. Uhat IS the size of each plot, and how many are there? 16. Find the least quantity which will contain exa^^tK' 14 ft- 18 tt)s., 21 fts., 44 fts., 33 ll)s., 28 lbs. and 210 lbs. - I i: > ir-.:; ij:3 64 ELEMENTARY ARITHMETIC. a It ;i n CHAPTER IX. IRREGULAR DIVISION OF THE UNIT. FRACTIONS. 63. Let us now consider two (juantities of the same kind namely, the surfaces A and £; so that the surface A = the unit surface x 3. and the surface Ji = the unit surface x 5. A n Now, by Article 33, we know that the measure of the surface A by means of the surface li as a unit is the quotient when 3 is divided by 5, these numbers being the measures of A and B ; that is, thesurfacc^^^S the surface B 5 " 3 This ^ IS called a Fraction, which may be read, «'the quotient of 3 by 5," or "three fifths." We may, therefore, define a fraction to be an indicated quotient which represents the meas- ure of a quantity. 6-i. Fractions are Rates.— On refer ring to the quantl in the pi A from tl "Cut ]iarts tog( \\'e denot other, thi where " > Hence, derived f j);irts oth( Vn.) is a as a unit parti;. «5. Tl when we i the surfac into 5 equ made by c So also, therefore i In gene therefore 1 That is, if the unit is Thus, si therefore t FRACTIONS. 65 same kind, } quotient define a the meas- [uantl' he surface when 3 is A and Ji; I I in the preceding article, we see that we may derive the surface A from the surface Ji as follows : "Cut the surface B into 5 equal parts, and put 3 of these parts together." These will make up the surface A. As before, w(. denote this manner of deriving the one quantity from the other, thus : o The surface yl = the surface £x , where " x " is the sign of derivation. Hence, & fraction is a rate which tells how one quantity is derived from another as a unit, by cutting the unit into equal parts other than ten ; while a decimal (such as those in Chapter VII.) is a rate which tells how a quantity is derived from another as a unit, by dividing and subdividing the unit into ten equal parti:;. «5. The Quantity and the Unit Interchanged—Since, when we say the surface A = the surface B x I, y,o mean that the surface A consists of 3 parts, made by cutting the surface B mto 5 equal parts ; therefore the surface JB consists of 5 parts, made by cutting the surface A into 3 equal parts ; that is, the surface B = the surface A x ~ 3* So also, since the surface A = the unit surface x 3, therefore the unit surface = the surface A x ^. In general, then, since the quantity = the unit x the rate, therefore the unit = the quantity x ^ th( "ate' That is, if a quantity is derived from the unit by any rate, then the unit is derived from the (luantity by 1 -r that rate. Thus, since a yard = an inch x 36, 1 therefore an inch = a yard x '.. ■■> r;;- £";:» 36' 66 ELEMENTARY ARITHMETIC. inl^or^u" °n'^' ^^^^^ Numerator and Denom inator—lt would .seen,, tlien, that when speaking of i it : nuH ead.ng to call 4 the .umber or "nun.erator," and 7 the „ .n, or "denonunator." These words have arisen from the incon,- plete expression of an idea resulting in nn'sconception as to th. meaning an.l use of the i. When ^ is spoke. ,. alone, we mean no hn,g more than the quotient whoa t is dui...d by 7 (Artie), 63),_a,Kl when it is written or used as a rate, thu, "a. ,,,,. or T f an inch," 7 is the number of equal pait. int.. tiJ ; . " ': '"■^"'' '"'' ' " *^^ "'*"*'^- ^^^ these parts put together to m.k,- the quantity referred to by the above expres- sions. Tlie denonun.tor is not 7, but "an inch x i," or '-» of a.x mch We shall, the..fore, not use these wo;ds, but use instead the shorter aao mon expressive words DiviUeud and EXERCISE XVI. 1. Expla;M the meaning of the fractions in the following state- (a) A foot = a yard x - . o (b) A yard = a rod x -^ ir (c) A square yard ^ a square rod x ^. {d) The price of a pound of butter = a dollar x ~ 25* (e) A roll of butter = a pound of butter x -. 8 2. Find the decimal rates equivalent to the fractional rates •^ j^ ^ 17 49 V 16' 80' 125 •'"'' 3200- 3. Find the aggregate of the following quantities by redu. FRACTIONS. 67 id Denom- g of i, it is d 7 the !i i .36- U) (k) 96^ 112- 78 68" (i) 2. Reduce to their simplest forms (a) 1440 1728" (b) 221221 3T0370' (c) 32 48' 15(525 iooooo' 21 (^)35- (d) 2389 4576" 3. Change the following mixed numbers into fractions (a) 4 (b) 5 3 7" 1 9" (^) 82O3L. (e) 4196^. (9) 1 1496 8214" 216 U) 374 J|. ('•) U21_. if)S2ll (i) 8 ^^^ ^2l96- 48763 193214" (k) 6493 85 99" 4. Reduce to mixed numbers the following (l) 8437^^ . ^ ' 3571 («) -7- 23 /A 21740 (c) 4396 144' («) (/) 7926 loo"" 53217 ^"4T 6 (9) (A) (i) 2176184 12345 • 7964 7963" 99999 33332" 5. Prov 6. Find lent to «8. Ad Let the i^i and the w( Wlien the tVactional 1 Now, by the weight and the we In each ca siune numb size. The of 21 of th( the parts. Therefor( We have fractional r Change L and write a «» Exi The comi FRACTIONS. 69 .216 ^€ 99 3571 8" 4 32 5. Prove that , = — . 5 40 G. Find ii decimal rate to the 4th order to the right equiva- , , , 3 4 3 21 lent to - + - J 1 8^ 9^16^ 5" «8. Addition and Subtraction of Fractional Rates- Let the weight of one block = a pound x -, and the weight of another block = a pound x When these two blocks are put together, we wish to find a fractional rate which will derive the whole weight from a pound. Now, by Article 67, the weight of the 1st block = a pound x — . 56' and the weight of the 2nd block = a pound x ~. 56' In each case, therefore, the pound has been divided into tlie same number of equal parts, and hence the parts are of the same size. The first quantity consists of 40, and the second quantity of 21 of these parts; so tliat the total quantity consists of 61 of the parts. Therefore the total weight of the blocks = a pound x — 56' We have, then, the Rule for finding the sum of two or more fractional rates : Change the rates into equivalent rates having the same divisor, and write above this divisor the sum of the resulting dividends. <»9. Examples solved— (1) Add together—, — , — and — * 24' 3G' 48 54' The common divisor is the T., C. M. of 24, 36, 48 and 54, •■-.1 tS gj 70 ELEMENT.\RV A HITMMKTK;. Ml-r \ ,,... 91, m \ wind., by the method ,^ rli.-l.. r,o ,, f,.,„„, t„ ,,^. ^, j ^ 3 ^ ., ^ 3 ■ ''r \'^. '^''\"^"'^^^'i«'-'^ "*■ tJ»' 'livi.lcn.K thcrofure, arc 18,' i -, J and 8 ; so that ihe resulting dividcidH arc, 7x18, 5x12, 11x9 and i;{x8,' that is, li'O, GO, 99 and 101. The sum of these rates, then, is '^^- ' 1M2" In practice tl,e operations are conveniently arranged thus • ^ _5^ n 13 »= . • 24"*';5G"*'48'*"54' _^ 7xl8 + 5x l2+ll x94-13x8 24 x 1 8 ' __ 1 26 + €0 + 99 +104 389 24x18 "^432' 9 7 (2) To subtract — from Tt iu ovi.i., t n i • •. ;j2 24' " '^ ^'"'■^ "' Hunilar pro- cess must be employed Ix're, thus : — _ ^ _ liil - 9 X '"^ -^ - 27 1 24 32" 1. Add togethei («) 1 2 and 1 3" if') 1 3 and 3 4' (c) 2 |3 and 4 5' (d) 1 3 and 5 6' :i4 X 4 A '""id -. - 3 4 5 ^^'^ l2' 17; '^"'^ ^- (^) and 91* («) - and 12' (i) tit' 90 md -IL 144" 2. Sim (" ('') (e) if 3. Find X 5, a me 4. In a schools = t iisst'.' iUiem inent x jl Find what 5. Wha 6. A m received tl 7. A, }. work, B d 8. Find 9. AVhai 121-11^ 24 4t^ ■% ,., FHACTIONS. 71 I '2. Simplify if') 2 3 5 2 ■*" 3 ■*■ 4 "^ G" 132;|+29i;.312^. iff) 432^-231^ y (h) 82-J-3r,2. (0 1!^G^_43^ <^'4 12' 7 «<-"n- 13 126"*'lGO" w i«i^- .;c 21 32" 4 9 3 40" 153 96 Tl7~9T' 3. Find the aggregate of the following quantities : A metre X f, a metrr x {;, a metre x jj, a metre x ];,', and a metre x II. 1. In a certain town during one year the money required for schools = the a. (smeiit x . ^jj, that for roads and sidewalks = the as, I uuentXj^, that for the interest on the debt = the assess- ment X jIIj^, and ^i.tt for other purposes = the assessment x -\^jr. hmd what rate will deri^ the whole money needed. 0. What is left when .rd x ^\ is taken from a yard. 6. A man willed his property to his two sous. The eldest received the property x .V; ; what did the younger receive? 7. A, B and C perform a certain work. A did Ij' of the work, B did ^\ of the woi'k ; what did C do ? iS. Find the difference between , I 1 1 , 1 1 1 1 ^ ~ o + r - « and - + + - + . 3 5 7 2 4 6 8 7 n 9. What must be added to 24— -16 1: 17 16 to make the sum .1 .i -.1 I > K;, 24 48 72 Iff' 1 1 .I, ELEMENTAHY AlllTHMETrC. 10. Add m-^ 205 72U'^'^'^^'*'''"«' = .7^> ''y reduction. 1.4 . 15 Article G7. {■!) To multiply together 8^, 4'- and -^. b-I Theproduct = 8- x4- x^; 01' ^^ 24 15 '8-XyX52'by(5); 13x5x8x3x5x3 Article 67. 45 8x5x4 xT3"~ ^y *^^ ''"^^' by reduction, Article G7 (1). = 11-7, by division. 1*4. Division of Fractions.— Since division is the opera- tion which undoes the result of multiplying ; that is, when the product of two fractional rates and one of Uie /ates are given, division is the operation by which the other is found. T>et us suppose that the product is — , and that one of the rates is - • 11 9' ::> 74 ELEMENTARY ARITHMETIC. til: *i 1%. '1' then the other rate x - = — • 9 11 ' 5 7 that is, the quotient x - = — . Multiply these equal numbers by -^ ; 5 9 7 therefore the quotient x- x-= — x- that is, the quotient 11 9 .Since 5 1. Hence we have the rule : Division is turned into multiplicaf- n by inverting the divisor. liV Example in Division solved. 1,3 10 1. Simplify 7— -h 2- ^ -^ IG 28 13 19 The quotient = 7 j^-^ 2— ; 125 75 , = -Jg- ^ 2y> by Article 67 (5) ; 125 28 = Yq- X Yg> by the rule ; by multiplication ; _ 25x 5 X 4 X 7 ~4 X 4x25x3' 35 , = T--, by reduction ; = 2—, by division. 2. If Tom's money is I of Henry's money, and Fred's money is y'V of Henry's money ; compare Tom's money with Fred's. Since Tom's money = Henry's money x - , ^ FRACTIONS, and Fred s money = Henry's nioiiov x - . . , Tom's money 7 5 therefore =;; — - , ■^ =__:._ J^ red s money 8 " 1 2' Article 33. = y X y, by the rule, = — . 21 lO' Therefore, also, Tom's money = Fred's money x Thus Tom's money is compared with Fred's money. EXERCISE XIX. 1. Simplify , , 4 15 ^"^ 5 ^ 24- (/)3^x4[ (*>'-M^'f if»\4 (.) 5-^x8^1. o / ^'^ hx^j W2;ix4^^. Wl3^1l| ... 32 42 ^^^ 35 ^ 80- ... 2 5 12 25" , , 1225 1728 ^"^ 480 " 1001- ^. 21 84 (./) l^xSgX «'. 10 '> 2. Simplify into one fraction : 240 520 / ^ /J3 . 3\ /I on (/) 10^11 27 10 11 *3 W 12C;-^3^. 3. Prove that a pound x ^ x « = a pound x .', !/. 76 ELEMENTARY AlUTIIMETIC. C •I,., It: ue«f 4 A man gave r of his money to A, and | of the remainder to JJ. I£o\v much liad he left ? 5. A merchant sold | of his goods to A, =i of the remainder to Ji, and I of what then remained to C. How nmch did ho liave left ? C. ^, li and 6' reaped a field ; A reaped the field x : 11 reaped the field x jr. How much did C reap more than A?" 7. A boy spent for nuts, his money x !; for marbles, his money X jV ; for oranges, his money x A ; and for fireworks, his money X v/V- He had 139 cents left. What money did he have at first"; 8. A farmer had in pasture ^ of his farm ; in corn, ^ of hi. farm; in wheat, ;j of his farm; in oats, ..^ of his farm- in orchard, i of his farm. The rest, 19^ acres, was in wood How large was his farm ? 9. If my money = Henry's money x ^„ and Henry's monev = my brother's money x § ; find what rate will derive my mone^ from my brother's money. 10. If Tom's marbles = Dick's marbles x j\, and Harry'. marbles == Dick's marbles x J^-; what is the measure of Tom's marbles, when Harry's marbles is the unit ? 11. If A's farm = Ji's farm x f, and C's farm = B's farm x ? . show the relation between A's farm and C's farm. 12. One gallon = a cubic inch x 277], and a cubic foot = a cubic mch X 1 728. How many gallons are in a cubic foot ? 13. Divide $U between A and B, so that what £ gets = what A gets X i. 14. A man left his estate, valued at $10245, to be divided between his two sons, so that the younger would receive 5 of wjiat the elder received. Find the share of each. 15. Divide 250 lbs. of flour between two families, consisting of 9 persons and 7 persons, respectively, in such a way that esxch person will receive the same amount of flour. Ifi. Ap Ji's share 17. Ap . Ji's sha IS. Dis \u - 24) X (2 - A ) X (9^1,. _ 4 3. Keduce to one fraction rc. 4. Reducel^iziLlll)_l\.J 10^^5^131^x51- 5. Reduce '^aotH) + u 2if^Tf .1 hundredweiL'ht x ^— v ^"i%~ 'h i 5^x3i^5^-(3fx2U7^''TtT- 1 1 1 Gl : .' i 6. Simplify 1 - + . j 0^24 5040^72576" 7. Explain and roduce J a dollar x 0-25 a rod x ''■■' a pound x x ------ 'I a dollar x 7-5 a rod x --/!^* S. Find the aggregate of a £ X I X (3] + 11 ); a £ X 1 X 475 x Jlll^i-^LIIL • (^of3i) + ^^' 4-2 and a £ X 9. Simplify 012x240" 1234 X -4321 --01 •00481346" 10. Simplify ^i±^ix^^^ -y- 11. A man invests i of his fortune in land, I of it in bank stock, }: of it in railroad stock, and loses the remainder, 18000, in speculation. What w;is his fortune at first ? 12. Multiply -01019 by 23-04, and explain why the partial products are placed where you write them. 13. Divide -01342 l,y -0055, and explain how you find the order of the 1st figure of the quotient. 14. Reduce to its simplest form ^^'^'^^1 ^ 999999" FRACTIONS. 70 15. Find the G. C. D. of 94G0r, mid 9G509, and explain wliy your method gives the correct miniber. 16. Find the L. C. M. of 11, 7, L>1, 28, 22, 27, 81, 243 and 216, and explain fully your method. 17. Wliat purpose.s are .served by additicm, .subtraction, multi- plication, division, finding the G. C. D. and finding the L. C. M. ! 18. E.xplain fully how to subtract 16r)0]:|i, from 1761 •''-. 19. Give two definitions of a fraction, and from one of these definitions prove that I x •'' = ri. 20. Prove that 8 x V = ^. 21. Divide 2 (juadrillion, 18 million, 760 thousand, 681, by sixty-three million, two huiidred and forty-five thousand, fi\e hundred and fifty-three. 22. Find a decimal rate to the 7tli order to the right erjuiva- lent to 1 -t- ^ -{- 1 : + 1 1 -p + etc. 0x5 5x5x5 5x5x5x5 23. A man died, leaving 5 sons, A, B, C, D and E. J le willed his property, valued at $10000, to them in such a way that A would get $200 more than B, B $250 more than C, C $300 more tlian D, and D $350 more tlian £. Find tlie sums they get. 24. Gunpowder is composed of saltpetre, charcoal and sulphur in the proportion of 15, 3 and 2. A certain quantity of gun- powder is known to contain 325 lbs. of charcoal ; find its weight and also the weights of the saltpetre and sulphur. 25. In finding tlie value of an article from $1350 by the rate II, a boy used instead the rate t!^ and a girl the rate l^. Which made the greater error, and by how nmch ? 26. A man gave ;-. of his money to A, | of the remainder to /A /. of what t'len remained to C, and divided the rest eciually between P, Q and R Tf R received 143 cents, what did J, B find C each receive ? 80 ELEMENTARY ARITHMETIC. C p ■ '4 a.. c CHAPTER X. QUANTITIES IN PROPORTION. 7«. In tlie preceding Chapters we have shown that a number is the measure of one quantity when another of the same kind is the unit. We have also shown how a liumber is used to tell \mw one (luantity is de.-ived from another of the same kiml. We shall now examine how numbers are used in connection with some quantities of different kinds. n. Suppose a farmer is taking a load of wheat to market, and that he knows that the load of wheat = a bushel x 45 The number 45 tells him how his load is made up of, or is derived from, a bushel of wheat. Now, the farmer is thinking about the price of a bushel and the price of his load of wheat-- two other quantities of a kind different from the wheat he has with him ; and the connection between these two kinds of quan- tities is such, that whatever rate is used to derive the load of wheat from a bushel of wheat, the same rate is used to find the price of the load from the price of a bushel. Thus, since the load of wheat = a bushel x 45 therefore the price of the load = the price of a bushel x 45. When (luantities of two different kinds are connected in this way, we say that (juantity of one kind is proportional to quaii tity of the other kind ; or that the quantity of the one kind varies as the quantity of the other kind. 78. Some Quantities which are in Proportion. Then aic many as in Arti (n) Tlu [l>) The (r) The ling. (d) The {e) The employed. (./■)The (.'/) The api'ed or vf till' use of Ex. 1. I l)iishels at Solution therefore ti P-iit the CO! therefore tl Ex. ,?. F weigh 13 oi Solution. therefore .■}( Therefore t liut the we Therefore t 6 QUANTITIES IN PROPORTION. 81 arc nm.iy (,iuuititit"s of dimnent kinds connected wit), each other us in Article 7G. TJius : ' (a) Tlie amount of a coinniodit}' varies as its price. {!>) The rent of a farm varies as the time it is rented. (V) Tlie distance a man travels varies as the time he is travel- liiit,'. (cJ) The work done l)y a man varies as the time he is working. {e) The work dom^ in a day varies as the (.(uantity of) men employed. (./•) The rent for the use of mon«^y varies as the money in use (//) The distance a train goes in a given time ^■aries as its !. Examples solved—The following examples illustrate tilt' use of numbers : Ex. 1. Find the cost rf a load of wheat consisting of 45 l)iishels at 85 cents a bushel. Solntion.—Hince the load of wheat = a bushel x 45 ; therefore the cost of the load = the cost of a bushel x 45. P.ut the cost of a bushel = a cent x 85 • therefore the cost of the load =3 a cent x 85 x 45 ; = a cent x 3725 = $.37-25. Er. ?. Find t'.e weight of 30 yards of wire, if 5 yards of it weigh 13 ounces. Solution.- ^ince 'IZl^^d^wire x^O^ ^^ a yard of wire x 5 Article 33. therefore 30 yanls of wire = 5 yards of wire x G. Therefore tli*- NNvight of the wire = the weight of 5 yards of it x 6. Hut the weijfht of 5 yards of wire = an ounce x 13. I'lierefon^ the weigfit of the wire ^ an ounce x 13 x 6, = jin ounce X 78. '.'J 82 ELEMENTARY ARITHMETIC. *•■ *> ■ «■; ■ I,.- I. C 1.' C c -ic ■mc. 'It Ex. J. Tf A can do a work in 11 days, and B in lM davs • 1 long will it take l)otli to do it \I\\V working toi;ether ? Solut ion. — Sinuo A's time to do the work = a day x 1 i Therefore a day = i4','* time to do the work IT" Article C. Therefore a day's work for A = the wlu)h> work x — . Art. 7 1 4 Similarly a day's work foi- /y = tlie whole work x — . Therefor •e a day's work for both = tlie whole work x ( — + J-^ • VU^21/ ' = the whole work x --, by addin^'. Therefore tlie whole woi-k 42 = a day's work for both x -f-. Article C" . Therefoi-e tlie time to do the woik 42 2 = a day x — = 8 - days. Article 77. Ex.^4. If 3 men can reap 25 acres in 7 days, how long will it take 7 men to reap G3 acres 1 Sulut ion. —Here the work of reaping an acre x 25 = a day's work for a inan x 3 x 7. Therefore the work of reaping an acre 01 = a day's work for a man x — . 2.) Therefore the work of reaping G3 acres 21 x63 25 ~' 1 2 men x _- x - = a day's work for a man x [Given, [xG:\ x&3 < 25 TlH'rcfort H0. N arc re(|ui \ciT nan liuil that to express tlie.se resii l)y curtail ill full an( 1. If 7 ■2. Tf9 i;in be boi :i. If 2; should 56 4. If -i should the '>. Tf 10 of water w (!. If a I he require 7. A cai hiiu to do '^. A far iK-eive for 9. A ma •^t'li alone a it working QUANTITIES IN PROPORTION. S3 =^ii (lily's work for 7 men x \K0 Therefore ti.e time for 7 men tu do it = a day x ---- 7^^ ,].,v. HO. Note on Solutions—Of course, if only practical results ui.' re.,u,re,l, the stu.Ient may contract the al,ov,> solutions to vvry narrow limits. But it is much, very n.uch, n.ore nnoor- ta.il that he shoul.i frain his miml, by strivinj,^ t<, conc-ive and to e.vi,ress in full lan^uao-e the elementary reasoning by which these results are obtaine.l. Therefore, instead of " savin- time " hy ourtaihng his solutions, he should rather seek to express them 111 full and accurate language. EXERCISE XXI. 1. Tf 7 yards of cloth cost 497 cents, find the cost of 10 yards 2. Tf 9 feet of hose cost 12U cents, how numv feet of hose laii he bought for 4242i cents ? :5. If 25 boxes of berries s.!l for one dollar, how many cents should 56 boxes .sell for ? 4. If ^ of my hay sells for $132 at $11 a ton, how much should the remainder sell for at $12 a ton ? ,'• ]^ ^^^!'-J ^^*^^' "^^^'"P>- 277] cubic inches, how many ll>s. oi water wdl fill a vessel whose capacity is 1728 cubic inches ? (i. If a man can walk 1800 yards in 25 minutes, how long will he recjuire to walk 1 20 yards ? 1. A cando a certain work in 7 days, how long will it take liiiu to do ^j of it? S. A farmer received .f 124 for ,; of his wheat, what should he "■'•eive for , ■, of it at the same price i>er bushel l 9. A man alone can do a certai.i work in 15 d.avs, ami his sun alone can do it in 25 days; how long will it take both to dg It working together ? * ^^ «v ::t 84 KLEMENTA UY A HI THM FITIC, « % c •c c 10. /I ciii do II work ill 24 duvs, uiid /I ill liO days. A work at it for .") days alonr, ln>w loii;i,' will it take /»' to (Inisli it 11. A can do as inucli work in 2 d, lys as /I can do II If .1 and /> woikin;' toifctlicr d .'{ dav' lont,' will it tako each to do it alone t o a certain joli in •) days, In )\V ll.'. A can reap a field in 12 days, // in IT) days, and (' in 2i days. In what time can all do it workiii;,' to;,'etlierf 1:5. A can run 2") yards in lU secon prr 2000 U.S., and takes in |.aynieiit (ilhs. of tea at 80 centM per lb.; ■22}. \hs. of c-oflee at 2(i cents per It).; X] ll.s. of siiirar at 12 Dis. for a dollar; :]2\ \hs. of raisins at IS',' c-ents per 11..; It];,; ths. of lia<<'n at l(i rents p.-r Ik, and the balance in cash Hovv niucli cash does the farmer recoiv*' / L'O. If 5 men or 7 women can do a jiiece of work in .'57 days. linw lon;< will a j)iece of work twice as ^reat occupy 7 men and women ? ■2\. A and 5 can do a work alone in 15 and 18 days, respec- tncly ; they work t(.j,'(>ther at it for ;i days, when Jl leaves, and after ;5 days A is joined by C ; these two then finish it in 4 days. In what time would C do the work by himself? •-'-'. A cistern can be filled in 18 hours by a pipe A, and can hv emptied in 12 hours by a pipe Ji. If the cistern be f full and both pipes are open, how long will it take to empty the ristern ? The following-' three examples are more difficult : 23. A can do a work in l:{i hours, and JJ in lG_:j hours. They commence the work together, but after i hours, ori account of an accident, A's efficiency is reduced by ^ of itself. How long is the work in doing? 24. Two ecpuil casks, A and B, are full of water. A can be emptied by a pipe in 4 hours, and />' by a pipe in 5 hours. If I'oth pipes be opened together, and closed when one cask con- tains twice as much water as the other, how long time will the liijies be rumiing ! 2o. T(mi and his father saw wood for a living. He finds out that he can split wood just as fast as his father saws it, but that liis fathei- can sulit wood four time.s as fast as he saws it. Now om wishes to spli liiey saw and split a cord of wood for .?l-20. T know how much of this money he should h -.1 I ) ..::» axe. Tell 1 um. IMAGE EVALUATION TEST TARGET (MT-S) k A '^Ij.^ 1.0 I.I |50 "'"^^ ^ 1^ illlL25 lllll 1.4 IM IM 1.6 I nOlUglcipiUL. Sdences Corporation 23 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) 872-4503 \ f^ ^<^~ o i/.A % % 86 ELEMENTARY ARITHMETIC. 1 i-' j: c i- :i Am. «•«. |;i c c 'C >■>' I.;' .!C ■'/. Thus, if the distance between tw(. stations be 13 miles, 5 furhjngs, '23 rous, 5 yards, we call tl distance a compound ipiantity. To show the manner by whi the distance is made up of these units, we write : The distance = a mile x 13 + a furlong x 5 + a rod x 23 + a yard x 5. el, COMPOUND QUANTITIES— REDUCTION. 87 8JJ. Table of Rates— Of the units which are used in I'jiglish-speaking countries to measure such quantities as lenj^th, surface, volume, weight, mass, vakie, time and angh*, the follow- ing tables give the names and manner of derivation. The student is expected to know these in the succeeding chai)ters I if this book. r. lot derived -sure a Ion;; lile is then lan a mile, rt, the mile chof whicli part is left ■e. Again, parts, eacli mtil a pait i the rod is make uji a )art is lefi L'asured bv ' has been ent names is said i« tween two ^e call the ' by which xl X 23 + (1) 2'o measure Length or Diatance. A league =:a mile x 3, therefore a mile =a league x J. A mile = a furlong x 8, therefore a furlong = a mile x |. A furlong = a rod x 40, therefore a rod = a furlong x ^^. A rod = a yard x 5^, therefore a yard = a rod x j\. A yard = a foot x 3, therefore a foot = a yard x \. A foot = an inch x 1 2, therefore an inch = a foot x yV. . A chain = a rod x 4. (2) To measure Surface or Area. A stjuare mile = an acre x GIO, therefore, etc. An acre =a scjuare rod x IGO. A scjuare rod =a square yard x 30 j. A square yard = a scjuare foot x 9. A stjuare foot =a square ii.^h x 144. An acre =a square chain x 10. (3) To measure Volume or Capacity. A cubic yard = a cubic foot x 27. A cubic foot =a cubic inch x 1728. A bushel =a cubic inch x 2218. A bushel = a peck x 4. A peck = a quart x 8. A gallon = a cubic inch x 277-274. A gallon = a cjuart x 4. A quart = a pint x 2. A barrel = a gallon x 3 1 ^. .) .1 I > > :.:3 88 ELEMENTA H Y A U [TH M ETIC. I.' <■. ■ I, C ;c ttm- A hogshead = n gallon x 63. A cord of wood or stone = a culiic foot x 128. In tlie United States, however, A gallon = a cubic inch x 231. (4) To measure the Wei,,ht or Mass of ordinary commodities the xVvou-dupois table of units is used. A ton = a liundredweight (cwt.) x 20. = a pound X 2000. A long ton = a pound x 22 10. A cwt. =a (juarter x 25. A pound =an ounce x 16. (.0) To measure the Weight of Gold, Silver, and Precious Stonrs tlie Iroy table of units is used. A Troy pound = a Ti'oy ounce x 1 2. A Troy ounce =a pennyweight (dwt.) x 20. A dwt. = a grain x 24. Therefore a Troy pound = a grain x 5760. But an Avoirdupois pound = a grain x 7000. (6) To measure Value. In Canada and the United States, A cent =a mill x 10. A dollar (|)=acentx 100. In England, A i)ound sterling (£) =a shilling (s.) x 20. A shilling = a penny (d.) x 12. A penny = a farthing x i. But a farthing is written, a penny x ], or {d. A guinea = a shilling x 21. Further, a pound sterling = a dollar x — , = $4-86.?. COMPOUND QUANTITIES— HEDUCTION. 89 28. ^nimodities ous Stonrs (7) To jtiensure Time. A year = a day x 335. A leap year = a day x ."JGG. A day =aii lunir x 24. All hour = a minute x GO. A minute = a second x GO. The length of the months is given in the foil (( "^ni Thirty days hath September, April, June, j lowing rli} me : and November ; ^vii me resi have thirty-one. Excepting February alone, Which has but twenty-eight days clear, And twenty-nine in each leap year." (8) To measure Aii;/le, Latitude, and Longitude. A quadrant = a right angle = a degree x 90. A sextant = a degree x 60. A circle = a degree x 360. A degree (l°)=:a minutt (1') x 60. A minute =a second (1") x 60. (9) Farmers' Pradtice.~The weight of a bushel of wheat, peas, beans, clover seed, potatoes, turnips, carrots, parsnips, beets, or (laions is 60 pounds; Indian corn, or rye, 56 pounds; Hax seed, ■JO pounds ; barley, buckwheat, or timothy seed, 48 pounds ; lienip seed, 44 pounds: oats, 34 pounds; blue grass seed, 14 pniiiids; dried ai)ples, 22 pounds; coal, 66 to 70 pounds. A barrel of pork or beef weighs 200 pounds, and a barrel of lli)ur, 196 pounds. (10) Miscellaneous Units. A gallon of water = a pound of water x 10. A cubic foot of water = a pound of water x 62| (nearly). A dozen = 1 2, a score = 20, a gross --- 144. A (juire of pap«>r = 24 sheets, a ream = 20 quires. i ■J .:;> 00 ELEMENTAUV AUITHMETIC. c: I.' c c i- 3k: H4. To Reduce a Compound Quantity to a Simple Quantity, that is, when a (luaiitity is fxiuvssed by namin-' more tlian one unit, to express it by only one unit. B.cample I. —To reduce £40 16s. id. to pence. Here the (juantity = a jfi X 40 + a shilling x 16 + a penny x 4, = a shilling X (20 X 40 + 1 6) + a penny x 4, = a shilling x 816 + a penny x 4, = a penny X (12x816 + 4), = a penny x 9796. ExampL ,.>.-To reduce 3 furlongs, 17 rods, 2^^ ^eet to miles. Here the (|uantity = a furlong x 3 + a rod x 1 7 + a foot x 2— . 14' = a furlong x 3 + a rod x 1 7 + a yard x - x -- •^ 3 14' = a furlong x W a rod x (l7 +— x - ), = a furlong x 3 + a rod x 17, = a furlong (3 + — x-^^. = a furlong x 3 ^ , ,3 7 ., 1 24 = a nnle x x -— , 8 7 = a mile x - . 85. To Reduce a Simple Quantity to a Compound Quantity. Example l.—To reduce 9if « acres to a compound (juantity as far as inches. COMPOUND QLTANTITIE8— REDUCTION. 91 120 Hrre 9-— acres an acic x (»^i). 1 on = an ac. x 9 + a s(i. rod x IGO x — - )!' 121 an at'. X 9 + a s(|. rod x ('-S). l-'l 8! = an ac. x 9 + a sq. rod x 158 + a s(|. vd x — = an ac. x 9 + a sc]. rod x 158 + a scj. yd. x 20- , 121' = an ac. x 9 + a sfj. rod x 1 58 + aaq. yd. x 20 + a .sq. ft. x 4 - , = 9 ac, 158 Ml rods, 20 .sq. yds., 4 sq. ft , 72 sij. in. Example I— To reduce 52132 ounces to tons, cwts., etc. 52132 ounces = an ounce x 52 1 32 ; , 52132 = a pound X -y^ , =- a pound x 3258 + an ounce x 4, 3258 = a quarter x - — - + an ounce x 4, = a quarter x 1 30 + a pound x 8 + an ounce x 4, 130 = a cwt. X — - + a pound x 8 + an ounce x 4, = a cwt. X 32 + a quarter x 2 + a pound x 8 + an oz. x 4, = 1 ton, 1 2 cwt., 2 qrs., 8 Itjs., 4 oz. 8«. To Derive a Compound Quantity from another by means of any Rate. Example l.—To simplify (£14 13^. 4(/.) x \'i. The derived quantity 13 . .,. 13 = a£xl4x — + a shilling x 13 x — + 15 Y^; + a penny x 4 x 13 15' :^ ill! 02 ELEMKNTAHY ARITHMETIC. i I c i t. c L. C L. .— To simplify (;5 acres, 14 yards) X 2-35. The derived (juantity S(iuare rods, 16 square = an ac. x 3 x 2-35 + a sq. rod x 14 x 2-35 + X 2-35. a scj. yd. x 16 = an ac. x 7-05 + a h^. rod x 329 + a s(i. yd. x 37-6, = an 11. ya. xsrb, ac. x 7 + a sq. rod x (160 x -05 + 32-9) + a sci. yd. x 37-6, ^ = an ac. x 7 + a sij. rod x 40-9 + a sq. yd. x 37-6, = an ac. x 7 + a scp rod x 40 + a aq yd. x (-^^ x '9 + 37-6 ) , = an ac. x 7 + a s.j. rod x 40 + a sq. yd. x 63-925, = an ac. x 7 + a stj. rod x 42 + a sq. yd. x 3-425, since 2 sq. rods = a sq. yd. x 60-5. 81. Examples solved. Ex. i.— To reduce #52-25 to £ .*. d. Here the money = a dollar x 52-25, COMPOUXI) QUANTITIES — REDUCTION. 98 15 L>OD ■2 1 5 = ai:x,.,x52-5, -^i^xf^x::^, =Hi:xlo-;;;, = a X' X 10 + a sliilli!!"' x 20x215 292 ~ =-a £ X 10 + a shilliiii,' x 11---, I -J = a .£ X 10 + a shilliii'' x 1 1 + !i ponnv x -"- — . = £10 \U S^^d. Tn the followiiijr example we give in full tlie mechanical labor iK'cessaiT to complete the reasoning. In the preceding examples this has heen suppressed for want of space; l)ut the student should do his multiplying, adding, and so on, always neatly, and pi'cservo it for inspection or correction. Ea: ,?.— To fhid the cost of l;{ ac, 2:} sq. rods, 21 sq. yds. of l;ind at 8120 per acre. Solution. — Here the land purchased = an ac. x l."5 + a sq. rod x 2.'? + a sq. yd. x 21, = an ac. x 13 +a S(|. rod x (2:? + x 24 I. ' V 121 /' ,., , 2879 ~ an ac. x l,i + a sq. rod x , = an ac. x 1 3 h x ■ V ^IGO 121 /' — an ac. x 25-^5 .59 160xl2"f Therefore the cost of the land = a dollar x -~ -"''':^:'l"!? 100x121 ' , ,, 254559x3 , „ fii = a dollar x — = a dollar x 1 577-81 , 484 I21' = 81 577 '85 nearly. -J ",( .1 .1 I > > S 94 ELEMENTARY AHFTHMETIC. f 24 1L>1 121 X 4 X -'."{ X 1 (50 242 121 278.3 19;{60 + 96 X 13 2879 -)S0m 19360 251 080 + 2879 254559 76;JG7; 121 484 ) 7()3077 ( 157781 484 2790 2420 3767 3388 3797 3388 4090 3872 2180 1936 ^4 EXERCISE XXII. to 1. Hoduco £174 10s. to pence. 2. Reduce £432 15s. 10^/. to shillinc/s. 3. Reduce £12 17s. Qd. to pounds sterling. 4. Reduce 5 ac, 137 sq. rods, 13 sq. yds., 6 scj. ft., 15 sq. inn. » sq. ins. 5. Reduce 7 ac, 15 sq. rods, 5 sq. yds., 3 sfj. ft. to sq. ins. 6. Reduce 15 sq. rods, 5 sq. yds., 3 s(i. ft. to acres. 7. Reduce 74237 sq. yds. to a compound (juantity. 8. Reduce 562934 s(|. ins. to sq. rcnLs, 9. Reduce 3 qrs., 14 lbs., S oz. to cwt. 10. Reduce 3 bush., 3 pecks, 3 qts., 1 pt. to bushels. U. Reduce 4930 cubic inches to gallons, COMPOUND QUANTITIES— REDUCTION. 95 iL'. How nmny s(«c-<. lids oliii)S(' fioin 2.:iO p.m. on Monday t(. S. 40 a.m. on Tuesday '. l:J. How many minutes from !) a.m. on May L'hli to 12 m. on June l.st. U. lieduoe '2 days, .5 hours, minutes to weeks. IT). What rate will derive 130 sq. rods, G.,V s.|. yds. from an acre ? 16. Reduce 11 cwt., 3 «|rs., 12i lt.,s. to tons. 17. Kxphiin tlie ineanin.' of, and simplify ^^.^J^'^lll'l- ' • ' £20 Us. 8-V." IH. Reduce 3/., acres to a comi^tund ([uantity as far as s(|uare inches. 19. Find the result of (3 days, U hours, 25 niin.) x ,^. 20. Express in acres tlie sum of }, of \ „f ] ^ of an acre :] of ' ',' "f I;:, man 1ms tlic use of another man's IxopiTty, that wliiih he pays for tlic use or the propj-rty is calh'd JifHf. 'I'hus, if r have tlic use of A's farm for a year, tlu' money T pay him for the use of his farm for this time is ealled a yo,nr's rent. .Just liow nuuh this rent is, and when 1 pay it, are previously a<,'reed upon between A and myself. 8!>. In the same way, if I have the use of Ifs money for a certain time, that which I pay him for the use of his monev may be ealled li,'nt. Thus, if T have the use of ^ViT) of /y's nionev for G months, the money T pay him for the use of this ,$435 may bo called U months' rent. .Just how much this rent is, and when I pay it, are previously a^'reed upon between li and myself. In such a case, we say that "I hired or borrowed the monev from />',' and that " // loaned or rented the money to me." Tlir money itself is called a Loan. Tt is usual for me to pay the rent when the money hired is paid back. Then the rent and the sum hired to,i,'ether is called the Amount at the end of six months. Tt is also the custom anumg business men that the rent of money shall be found by means of a numher used as a rat(\ J Jul they use the rate in two ways : 00. Interest and Discount Distinguished. (1) Interest,— \N\wn the rent of money is derived, by the rate UK NT INTEREST AND DISCOI'NT. 87 .iUm..l npon, hn„. th. s«.„ l.i.v.l ut the l„..i„Hin. of th. int.nul (L>) />tV.../.-nut wl.en the rent of .nonov is dorivo,!, l,y tho '"" '^«'; ' "1"'"' f'"'" tl'o a.nour.t pai.i l.a.-k ut the on.l of th,. -n.-.val ot time tho r.nt is calinl />;..,,,,,,, „„, „.,. ,,,,,. , ,., ni'- .s ..HI1..I the /^«,. ,/V>.Vo«„/. The an.ount t<, he ret:,-...! ^ the end (.f the time is ealled the /W>/ 'n... i>e,son who hired the ,no,.ey, tha"t is, the person who has -I'HyHdeht at aeertain time, is ealled the Debtor; and tho IJ-son who loaned hin. the n.oney, that is, the person to who„, ilK' debt IS to he paul, is called tne Creditor. JM. Rate of Rent (Interest and Discount). The rate " '•"Mt,s usually .iven whieh will derive .year', rent, and is '-1-1 the annual rate. The annual rate agreed upon l,v the ; ■•'t'"' -"1 --litur is spoken of, or ,dven, in different ways; 'ln.s, the annual rate n,ay he 5 p<.r eent., a per centun., 5 V, MM..;"- -Oo, wind, all mean the same, namely: that a year's rent ™-t. <,f o parts n.tde hy dividin,. the principal, or del.t, into I00e(|ual parts. In this oa*, then, a yeai-'s interest = tho princii)al x ~- ' 100' = tho principal x — , or = tho principal x -Oo. So, also, if the annual rate of discount is g| per cent., which 20 1 1 \M' may write — x ■ or — ••{ 100 15' 1 •t year's discount =-• the debt x — 15 .1 -I 1 > > 98 ELEAiENTAUV ARITHMETIC. Again, it" the annual rate of rent is 6 per cent. tlie rate for 73 d; ivs wil 1 •^ 4 lUU ^Gf)' or SO so that the intei-est for 73 (lavs = the t: : ijt. <; ; I,. =104 days. The rate of discount for 104 davs = -^ ^, 1^^ • 100 aO'V The di.scunt or rent = a $ x ^^^ "" ^ ^ i^i SGoOO • = a$xr)-128--=.r)-13, Tiie money hired ==$29t-S7. (3) If the amount of a sum hired for 4 months at 41 pvv cent interest was S.5;50, lind the i-ent. Solufinti.-The rate of interest for 4 mcmtlis^ — - y i-- A 200 1 'J ~ 200' Therefore the interest = the principal x - I'l't tlie pi'incipal = the princip/d x 1 . Therefore the aim.unt-the princiiial x ^ ' 200" Therefore th(> principal = the amount x -^"^ 20:5" Therefore the interest - the amount x ^ x -^ 203 200' 00' Article 8. Article 16. Article Go. = a|x.")30x.-^^-.f7-83. (0 .1 received from 7i mO for n.onths, agreeinc. to pay ■vnt for ,t at -T per cent, discount. Fhul the rent ;f .1 -I I ) > 'lif end of G month paid at 100 ELEMENTARY ARITHMETIf. I.: V it t" ■>« ■ "CI' Solution.- -The rate of discount = -' - x = — 100 2 40' Therefore the discount = the debt x But the del)t = the debt x 1 . 1 40' Article 8. 39 Therefore the money /I used = the debt x -" =-$520. Article I'J. 40 Therefore the debt = a 8 x 520 x 40 Tlierefore the rent (discount) = a S x 520 x ;, x — , • ) .) 40 = a 8x 1:^1 =.$13.1. (5) For how lonjL,' time will the interest of $500 at G per cent, be $23. \ ? SohUion. — The rate of interest for the required time is tin measure of $23' when $500 is the unit ; that is, a $ x 231 23 1^ 7 the rate for the required time = „ "" •' = --^ (Article 33) = — ^ a$x500 500^ ' 150 n But this rate = ~ -■- x the measure of the time. Therefore the measure of the time = G 150 ■ 100 !) Therefore the reijuired time = a year x ,;. EXERCISE XXIII. 1. Calculate the rent in the followinji; cases : (a) When $500 is hired for 2 years at 6 % interest. (6) When $325 is hired for G months at 4 % interest. (c) When $225-45 is hired for 9 months at 5,i % interest. {d) When $1234-56 is hired for U. years at 3^ % interest. {e) When $235-21 is hired for 315 days at 6 % interest. (/) When $1111-11 is hired tor HI days at 11 % interest, RENT— PROMISSORY NOTES. IQl ■2. Calculate the rent in the oases when l^') .*-lU pays a debt contracted 3 months ai^o at 71 7 Miscount. " ' a /o ('•) .1*1000 pays a del^t contracted ^7 days ago at 6\ v nisoount. ^ a -J /3 00 #13.rL>9 pays a dcl)t contracted «J months ago at 5 V nisoount. " /o («) I463-1;5 pays a debt contracted 14G days ago at 31 ^/ discount. J O "^ 'Jy /o .-i. Fiml the interest paid for .^500 hired on January loth and icturned on August 3rd, at r>i V 1. Tf the rent for a sum of money for a year be ,S4!)-i0 for wliut tnne will fche rent be $16-40 .' ' ^. If the interest of |290 be ^14-50 for a year, what is the iHte of interest 1 j , i is me 6. If the interest of $040 for a certain time is #10 of what sum ,s 142 the interest for the .same time ? ^^^^.Ktlie annual rate of rent be 7i %, for what time is the "'2 /o ' S. What rate will derive tlie principal from the amount, when the prmcipal is hired for 73 days at 13| % interest ? fiJm TT ''''!,"'" '^""■' '^' ""^'^""' ^^^' '^^ ^'"^ «^ ^ '--ths t.om the 8 months' interest, the rate of interest being 8 y ? 10^ What rate will derive the amount from the principal ^^ Inch is hired for 1 70 days at % interest ? IK What principal loaned for U years at 4 i y will amount 12 If the intei^st = the amount x ^, what rate will derive tlie intere.st from the principal ? 13. A man hired $.50 on the first day of each month of a I I I > > 102 ELEMENTARY ARITHMETIC. t: i certain year at 10 % interest, how much did he owe at the end of that year ? 14. Find the interest of |1l^5-G-J from April •2\}th to Sei)t(Mii her IGth at 5 %. 15. $000 was hired on ]May 9th at 6 '/^ interest, and thedelil was paid with $025. Find the day on whieh the debt was j'aid. 1(3. A debt was paid on May Sth with ,^020. The rent, whieh was at the rate of 5 % discount, was ."*:20. Find the day on wliich the debt was contracted. 17. ^1 received from />' ."?400, for which he is to pay rent at 6 y^ discount. At the end of G months A paid his debt ; find the rent. 18. A hired two ecjual sums of money, each for G months. For the one he paid 5 % interest, and for the other 6^, % interest. The total interest was S?4G. Find the sums hired. 19. A man hired two ecjual sums of money, each for G months. For the one he paid rent at G % interest ami for the other lie paid rent at 6 ^ discount. If the total rent paid was $40'GU, find the sums he hired. 20. A perscm borrow?! $500 on A\m\ 10th, and on June 2'2ni\ pays his debt with ^olO'^O. At what rate per cent, per annum was he charged interest ? •21. Find the rent of £24:5 G*'. Hd. for 97 days at G] %. 22. A sum amounts to .^.359'GO at the end of a year at 5 interest. Wliat was the amount at the end of G months 'I 2;3. If the amount of fB400 at the end of a year be f 430 at a certaic; rate of interest, what would be the amount at the end of 9 months at the same rate ? 24. A sum at S y interest amounts in 9 months to .^^o.'JO ; in how many niontlis will it amount to $540 1 25. A sum was borrowed for H months at 9^ per cent, interest : what is the equivalent late of discount ? . HENT— I'UOiM t.S:iOU V NOTtS. 108 J>3. Promissory Notes.— When one man, Jclin IJn.un, hires a sum of m<,ney (say .1i;r,00) from another man, Henry Smith, and agrees to pay him Kent for the money at G per cent. interest, it is the custom for John Brown to give to Heniy Smith a papei-, which reads as follows : ■^"^^-^^- ToKONTO, J% mh, IS'.)7. Six months after ,late, I j„-omLse to patj JIimr>/ Smith, or order, the sum of Five Hundred Dollars (-%500), with interest at per cent, pt r annum, m/ue receired. (Signed) John Bkowx, And when John Brown pays his deht, Henry Smith returns the paper to John Brown. This ])apor is called a pi'omissory note. When John Brown pays the debt, he is said to have "redeemed the note," or to have "paid the note." JM. Three Days of Grace.— Tt would seem tliat John lirown here promises to pay the de])t "G months after" May nth, that is, on November 9th ; Init, jy an Act of Parliament of the Dominion of Canada, this time of payment is extended ;] days, so that John Brown is not recpiired to pay the 5. Discounting Notes. -If, however, John Brown had asked for a loan of $r,00 from a Bank or Loan Company, with the understanding that he was to pay lent at G per cent, .liscount (or what is the same thing, to pay in advance, rent at fi per cent, interest), he would then give a paper, which reads as follows : 104 ELEMENTARY ARITHMETIC. J • r- ■II: I.. f ■Mm »c: ^^^^- Toronto, May 9th, 1807. Six months after date I promise to pay the Bank of Coimneree, or order, the mm of Five Hundred Dollars ($500), value received. (Signed) Jon\ Brown. Thus, as before, John Brown promises to pay |500 on Nuveni ber 12th. But of this a part is reiit, which is derived from the debt, -1500, by the rate, 6 per cent., thus : 500x6x 187 = $l5-;}7. the rent = a I x 36500 Therefore, Jolin Brown would not have the use of foOO, but of ($500 - $15-;37) .f484-G.'). Now, when the Bank pays |4S l-O;} \_u John Brown for his note, as above, the Bank is said to "discount John Brown's note." The |4H4-63 is called the Proceeds of the note, and the 815-37 is called the Discount of the note. In this case, then, John Brown uses $484-63 for 187 days, and pays $15-37 rent (discount) ; while in the case of Article 93, he uses 1500 for 187 days, and pays $15-37 rent (interest). 90. Selling Notes.— By virtue of the words "or order," a note, such as tliat in Article 93, may be sold by Henry Hmitli to a third party, George Taylor, for money. Henry Smith will then have the use of George Taylor's money from the day he sold the note til! the day John Brown pays it. For the use of this money Henry Smith usually agrees to pay rent at a certain rate of discount, so that the rent is derived from the amount of the note on November 12th by the rate agreed upon. On givino the note to George Taylor, Henry Smith signs his name ticros^ the hack of it ; and, by thus endorsing the note, will have to pa\ it in case John Brown fails to do so. Here, also, George Taylor is said to discount John Brown's note, and the money he pays for the note is called the proceeds of the note. RENT— PROMISSORY NOTES. 105 OT. Demand Notes. -Sometimes, however, a man may liiie money from another by giving a note, as follows : ^^"'>"- LoNDOx, June nth, 1807. On (h'lnand, I promise to imj Uriah llrrp, „r order, the .siua of One Thousand Dollarn (■'^lOOO), with interest at 7 per cent, per a)tnum, value received, WiLKlNS MlCAWHKH. Such a l.jan may be paid back part at a time. For instance, oil October IGth, .*450 may be returned to Uriah Ileep ; but it is agreed tliat the rent for tlie $1000 up to Octol)er 16th is first paid out of the ,$t50, that the balance of the 1*450 pays part of I ho loan, and that Wilkins Micawb?r pays rent for the use of the remaining principal. If, liowevor, the part payment is not • iiough to i)ay the lent due at the time of payment, no rent is to be paid for the use of the unpaid rent ; that is, the rent after the part payment is made, is derived from the same principal as it was before the part payment was made. 08. Examples solved. (1) A three months' note for .*s;5.")0 was given on INIay 29th, and was sold on June 26th at 7 per cent, discount. Find the proceeds. Solution. —T\\ii note matures on September 1st. The seller then has the use of the proceeds from June 26th to September 1st, that is, -for 77 days. 7 77 Tlie rate of discount for this time = x —— 100 365" Therefore the rent he is to pay September 1st, r)vl7. 7 77 = a % X .350 X — - X 100 365 The proceeds, then, ==a $ x (350 - 5T7) = |344-83. The student is to perform the operations as follows : 106 ELEMENTAUV AUI'l'tlMETrc. 1:: I.-. «c ■ .^ = #100. 36y00 , , , , ^ 100 X 36500 Consequently, the debt =a ^ x — ^,-~^ n — "' = $408-46, which, of course, is the sum the note is made out for. (3) A demand note for $1200 was made on Jaiuiary 28t!i, drawing interest at 7 per cent. It was partly paid as follows ; RENT— COMPOUXD IXTEUEST. 107 X .'.■r = #lG-(><0. On April nth, 84125 ; on ()..t.,l.er 12tl,, S2:U>H. Wf.at ronmins due on December .'Ust ! Solution.— The interest up to April 1 |th = >;|->OU x - lUO";5(i5 When this interest is paid, the bahuuv of the payment re.hnr.s the principal to *791-f<0, for which rent is paid after April 11th. Ihe interest, then, up to October liJth Hut the payment, m-68, leaves $p-2G of this interest unpaid. N\ ith the same piincii)al as before, then, the interest up to December 3 1st = ^79 1-80 x -- - x '"^^ -^1 •>•! -. ir I. , , 100 305 ^'-^•^• Hence, on December 31st there is due f 4-20 + $l2-ir) + $791 -80 or #.sO,s-iM. EXERCISE XXIV. 1. Find the interest of the following note : *'^'^^- London, Septemlwr ICth, 1S07 Eiyht nwHths after date I promise to pay li. Johnson, or order, the snm of Eiyht Hundred and Fifty Dollars (^0), zoith interest at Gi per cent., value received. Petkr IUax. :.'. What rent is paid when the following note is sold on July "^th, at 7] per cent, discount? ■y^ooJ-oS. Sthatfoui), May '.nh 1S'.)7 Nme months after date we promise to pay to the order of Charles Smith .6 Co., Four Thousand Three hundred and Fifi,,^ ■•''■" i^iiDollars ('$4.j52-58), value received. Cash and Penny. 108 ELEMENTAUV AHITHMETIC. c it: tl or 3. I wish t(. liiro ji^oOO by selling,' my note <.ii .luiio Ist, f n'(« iiKiiitlis, ut 7 jxT cent, (liscoudt. Find wliiit diHcount I slifill pay, and wlial I shall niako out tlic note for, 1. A demand note for ,it!r)00 was <,dveii on .lainiary (ilh, 181)0, drawin<( interest at 7 per cent. On May IGth, iif2{)() was paid on the note. How mueli paid tho note on Octoher 12tli, iS'.Xi >. not te was discounted 7:5 days before it niatur(>d, at 7.', per cent, di.scount, and produced $:iOi. Find its face value (). A note for $H\0 is to run for n Find the debt at the end of that ti . year at .S pir cent, interest. me. Find also the rent the seller of the n(.te wouhl pay if lie sold it after G months, at 6 per cent, discount. 7. T hired $1l>0U at (5 per cent, interest, ayreeinnt. On what day was the note dated ? 1 1. What must be the face of a note made on January 19th, 1896, for 1 1 months, so that, when discounted at 7 per cent, tho was made i lay lay yield |48G-45? RENT— COMPOUND INTEREST. 109 I-'. Kin.l the intm'st ..f a „..(.■ fnv ^l^r^imir,, inutuiiii.Mi. I:.'6 (lays uf((,r clato, at 31 ,„.,■ crnt. Fin.l alsn .|„. ainuinU ..f tlic iu)t<' at iiiatufity. '■■'• ■^■'^''"''- To.iONTO. .A,m. /< AS'W. 0// f^!»ia«(; / promisr fn pa,f liirluinl lAlth; or onfn- Four Thonmmi DoUar. (-i^J/JOOj, u-ith inln'est at 7 per end., ruhii' ri'rfiivfil. ,„. . Stkpiikx TrioMPsoN. IhiH note was (Midoi-Hed as follows ; Septimher ITdh, IS'.tr, R,'cnred SJfOO-r,!!. A>. A. Decenihfr ir>th, IHUT, Rexeiml .S::fH)/K J{. L. March Isl, 1898, Received ■%',nif()l). R_ /,. What is the nott' worth .January 1st, 1S9<» ; !>!>. Compound Interest. A.i^'ain, A may lia\c V.w use of B's money for a longer time than a year. Tt is usual then foi- A t<. agi-e to pay the rent (interest) at the end of every year, oi- Ht the end of every six months. Tf he fails to do so, the rent, when due, is put with the principal, and then A pavs rent for tlie whole amount. Fn such a case, the rent A pays when he .liscl)arges his debt is called ('ompound Interest. If the rent is put with the i)rincipal at the end of every six months (say), the r.'iit (or interest) is said to be convertible into principal lialf- yeai'l}', or compound(>d half-yearly. 100. How Compound Interest is found. Thus, if a has the use of /y',v money at 6 per cent, interest, convertible yearly ; t he 1st year's rent = the principal x •06. Article !J I . I'.ut the principal = the principal x 1. This rent being unpaid, is put with the principal, so that the 2nd year's principal = the 1st year's principal x 1-00 ; that is, the 2nd year's principal is derived from tlu- 1st year's piincipal by the rate 1 -OG. ::» no FI-EMKNIAUV AKlTHMfiTIC. t::> f • •• »• ••. «:: (■• 5 i L 3j. Tliereforc, tli.' .'{nl v. n's jtriiK-ipal is dfiivt'd fioin (lie '2iu\ P"H'h imim[in\ Ity the saiii«' nitc, 1^ 'Mi ; that is, tilt' ''mi y^w's ]>i'iiici)ial = tlii' I'nd year's ittiiu'i|ial x 1.00. =^ the 1st yt'Hr's priiicijial x I UC x I •()(). Simil.iily, the Itli year's |iiincipul -the 1st yi'at's juiiuipal x I OC) x 1 (JO x 1 OO, ami sii (III. It lias Itt't'ii aifictMl to (icMotf tlu- product 1 00 x 1 -OH x lOO thus, (l-U(l)-'. I IciKc wf say that the amoiiiit of A's dcht at tlio end of 4 y«'ais = the principal x (1-Ufi)'. 'Further, this includes the jirincipal and the 1 years' rent ; therefore the A years' rent = the principal x [(i-OC.)' - 1 1. We shall call (1-OG)' " tho rate of th." aiiumnt ft.r 4 years," and ( I or))* 1 " the rate of interest for 4 years." Thus, if the rate of interest for 1 Vf ar is 1 j] per cent., the rate of amount for .") M-ius is ( 1 •Ol.'JT'))'', and the rate of interest for ') years is { l-04;{7rj)' - 1. 101. Since A'n debt at the end of 4 years = the principal x (|-U(i)^ ; therefoi-e the principal ^A'li debt at the end of 4 years x 1 (l-O(l)^" Article (i: 10?i. Since the 4 years' rent=the principal x [(l-06)^-i;; tiierefore the princiiial = the 4 years' rent x (l-0())^-r I0;{. Examples solved. (1) At per cent, what is the rate of amount for 100 days f So/vtinii. The interest for 100 days -= the principal x — — x 100 100 -365 UKNT — COMI'OI'NI) INTFREST. Ill ion K 1 -Wi of I ycjii- cars' rent ; '[• yvtivs. " and i; •■ the priiiciiiiil x TlitTcfuic tlit> jiiiiiiiitit at Ihc frill of lOO ilays ,, . . , 7 J " Ihc iiniicipal X ' ' 7.T -llic i>i'iiici|.al X lOl "(OChS.*), \,y i|i\ idinj,'. Ht'iicc flic iv.|uin>(| rutc is 1 OirjOdsn. (:.') To fiiiii the <'()iii]iiiiiiiil interest of !?."):L' for :! years, at I |Mi' cent. ion. ■ Tlie rate of amount for :{ years is (lOt)''; so tliat till' interest for M years == a ?« x 'r.\-2 x {{\-0[f - 1 ] -^itCri'A. The studenl will nuillii.ly as indicated, thus : Snhlf 4 1 • I 24H0i G32 24'j7l'S .*{74r)!L' (iL't:?-JO 6M27648 104 i-osh; i-ot lOSlO M24SG4 (3) What i)rin(;ipal will amount, at the end of 2 years 6 onths, at (•) per cent., to .*.")00 ? on. Sohditin. Thi^ i-ate of amount for I j-ear -r 1 the rate of nniount for fi niojiths= 1-03. Hence the debt at the end of 2 years months = the principal x (1 Ofi)-' x 1 -03 that is, the principal x (1 •00)'^ x 1-03 = $.-)00, •refoi-p the princip.-il =a # x IS seen from the followino' opeiat 500 (i!Hj)-x l-o: = S432.04, 2» ions 112 ELEMENTARY ARITHMETIC. «::> 5.; 106 1-06 0.30 100 M2.30 1 -O:} 1-157308 ) 5000000 ( 4:V2-0:\7 16292.32 .3707680 .3171924 .3.3708 11230 1 -157.308 2357500 2.314010 4294700 .3471924 822470 EXERCISE XXV. 1. At 7 per cont., wliat rate will dorivo tho debt at tho end of 2 yoarw from the sum hired ? 2. What rate will flerivo the compound interest from the principal l)orrowed for 2 years at 5 per cent. ? .3. What rate will derive. tho amount at the end of .3 yeai- from a principal hired at 8 per cent, per annum ? 4. What rate will derive the rent fi-om a principal which i. hired for 2 years at 8 per cent, per annum, convertible half yearly ? 5. Find the rate of amount for 3 years 4 mos. at per cent. 6. Find the rate of interest for 2 years 4 mos. at per cent. 7. A man hired .>?400 for 2 years at Oi per cent. Find h.i^ debt. 8. Calculate the compound interest in the followin;,' cases : (a) Wlien .f!300 is hired for .3 y(>ars at 10 per cent. (b) When .$4.35 is hired for 2i years at 8 per cent. (c) Wlien $180 is hired for 4 yeai's at 5 per cent. {d) When .f 1250 is hired for .3^; years at 4 per cent. (e) When .f 1234.50 is hired f(.r 2 years 140 days at 5 p.M cent. RENT— EQUATION OF PAYMENTS. 113 •2 -or, it tho end (jf i !). Calcuhito the rent when (^0 $:m is hired for I J, years at G percent., convertible lialr-yearly. (/>) IGOO is hired for 1 year at 8 per cent., convertible tjuarterl}'. (c) ^825 is hire VllOO / ^ Consequently, A could discharge his obligation by paying $110(J to li 32 days bc^fore August 15tli, that is, on July 14tli. I0«. Averaging Accounts. If, however, A pays /; in part payment, $150 on June 2Sth, and $300 on .luly 30tli ; then the statement of these debts and payments is called the Arcounf l)etween A and /J. As before, tliere is a day on whidi A may discharge the rest of liis obligation to />' by paying him $050, the balance of tlie debts. This day is called the "equated time of tlie account," and the process of finding it is called "averaging the account." Tojind this time.~Ii these part payments were withheld by A till August 15th then they would be worth $150 + the rent of (150 x 49) $7350 for 1 day, and $300 + tlie rent of (300 x 10) $4800 for 1 day, or in total, $450 + the rent of $12150 for 1 day. But on August 15th his debt was $1 100 + the rent of $35200 for 1 day. 8ee Art. 10-"-. Therefore he still owes on August 15tli, $ODU + the rent of $23050 for 1 day ; RENT— AVERAGING ACCOUNTS. 115 ust il^tt is, .SG.-iO + tlK. rent of .^GoO fur (''■^^■'^ \ .5- ,, , I.nc.' Ik. sl.ouM ,.uy Liu. 8(i50 to // ;35 d,u-s before Aug J-Jth, tluit i.s, on July Utli. " If A did not pay the Pm on July 1 Ith, a.s he evidently did •u uie I ate agreed ui)on. EXERCISE XXVI. 1. I owe Jan.es White m due in 4 numths, m due in ■> -"ths, .90 due in n.onths, and 165 due in 7 nK>nth ^nJ t'le ecjuated tune. 1 "\f''T? '^ ^''""'''"^' ''"''^'^^ ^^-'^ rh.e January 9th, . 14-^3 'iue March I8th, and .«2IS due May 2Sth. ^ --l ^ owes /^ the folh.wing debts ; $m due in 5 n,onths .*4-:> ue,nS.onths, .5G0duein9n.onths,..d^7.0du:'inV "".nths. Fnid the e(,uated time. ■^- Fiml the e,juated tinu- of the fallowing account: Peter r.-eehaw, Dr. to John Fitch, $13.^ due OctoW 1st, #590 dte ^oven.ber 29th, ^^ due Decen.l>er 16h ; Cr. bv /4OO pad l>eoe,nber ;3rd, and $:m paid Dece„d,er 27th. ^ "• ^^' KOBKKT Grxx it Co May 18th. To $]->:W-00 July 11th Aug. 20th. To 2:}00-00 Oct. iSth Hcpt. .30th. To 1250-00 ])ec. 5th Xov. 8th. To 2140-00 ^^'11 l>a> It off on December .•{Ist, at 7 per cent. : > th, and the other for #750, nurturing on Decen.ber 29th ' ^'''^■'^^' ^''" ""^^« I"' give:^ a note fur $13] (•lues this note niatui-e ? Cr. Hy 8 950-00 i>y lsOO-00 By 120000 U d ay TWW 116 ELEMENTARY ARITHMETIC. I:: 1.1*,. 8 It:- CHAPTER XIII. BUYING AND SELLING-GAIN AND LOSS. lOT. When A buys jui article from Ji, ho gives 5 money tui it; so that the aiticle Ji owned before A now owns, and the money A had befcjre /i now lias. A speaks of the money he paid forth(> article as its Cost Price ; while li speaks of the money lie got for it as its Selling Price. This exchange between A and A' is called a Jhisiness Tranmction. The person who makes (n produces the article at fii-st is called a MannJ'aclurer or Pro ducer : th" person who uses the article is called the Consnmer : while the person who buys the article from the produce)' and sells it to the consumer is called a Merchant. I©8. Gain and Loss- — Now the merchant usually gets more money for the article from the consumer than he pays fur it to the producer. The difference is what the merchant yaiti^ by buying and selling the article. The merchant, howe\er, may get less. The difference, then, is what the merchant lo^^es. 101>. The Rate of Gain or Loss. — Again, the merchant' gain or loss is thought of as being derived from the cost of the articl(> by means of a rate. 8o that the Hate of gain, or los-;, is that nundjer, or rate, by which his gain or loss is dei-i\cil tVom the cost price of the article. Thus, if a merchant says lu' lined i'.") per cent, by buying and selling an article, he meaii^ I Ui that his gain = the cost of the article x or if he loses 1-5 per cent., that his loss = tJie cost of the article x 4' 3 20" BUYING AND SELLING-GAIN AND LOSS. 117 no. Capital.--Si„co the purchase of an article nu.st take I'lace so„,e tnne before the .sale <.f it, a .Merchant nu.st have ...o-y toUuy .ith he^re he can ,ain n.nev hy huvin, L^ M Ihng. 1 h,,s money which he uses to connnence and carry ^n his H.s.ness IS called hi.s Ca,M, wind, is said to he invested in thc^ ■-ess. And. again, the whole.ain .>,• profit whid. a .ne..i.ant . k s dunn, a ^.-ar, or six ,..o..tl.s, is regard..! as derived from o^nt d unested hy ....ans of a rate. Thus, if, during the ,^<- 18J6 a .nerchant n.ade a profit of .S pe.- .e.it., we n.ean that . the year's gain or profit = his capital x -~ ' 100" III. Trade Discount. -Whe.i a merchant receives an cle winch he intends to .ell. he u.sually marks upon it the I nee for wh.ch he intends to sell it. Sometimes, however, he I oes not mark the price on the article, but sets it down in a h^^t, which he can conveniently consult. It may happen that, .0 ore he sells the article, he may see fit to reduce its marked cl^ hs price, in which case the amount he take.s oflF the price is called JxcdeBuco^nU. Moreover, as before, this trade discount is de- ;-o.l from the ma.-ked or list pi-ice by means of a rate. Thus 't a merchant adverti.ses a discount of 10 per cent., he means that the reduction in his price = the li,st price x ~ 10' so that the .selling price = the list price x — ^ 10" H?J. Examples solved. (1) A merchant niarked an article 16^ per cent, above its ' t, and .selling at this price, gained #1.60; find his co.st and ■H'lhng prices. Sohi(io7i.— Since 16" per cent =- - 3 '6' the advance above cost = the cost price x - 6" 118 ELEMKNTARY ARITHMETIC. c::> ••■•5 IT:* I. ..I — .^j •c::: This also is the gain, sd that the cost prico x , = a .? x I'GO, therefore the cost price = a $ x 1 -OO x (>, = .«!)-(iO. But the gain = !?l-()0, therefore the seHiiig price = $1 1-20. (2) A merchant marked his goods 20 per cent, above tlieir cost, hut sold them at a discount of 10 per cent. Find his rate of gain. Sulutivn. — The ad\ance above the co,st price - the costx - , o therefore the marked price = the cost x . Again, the discount = the marked price x — , therefore the selHng price = the marked price x — , .6 9 = the cost X y X --r, 27 = tlie cost X ~, 25 2 therefore the gain = the cost x ~. 20 2 8o that the rate of gain = ~ = 8 per cent. (3) During the 1st year of his business a merchant increased his capital by 12i per cent., and the 2nd year he increased it bv 15 per cent. ; he was then worth |4U0. What was his orijfinal capital ] Solution. T\w 1st year's gain = the original capital x - . Therefore the 2nd yoi.r s capital = the original capital x - . BUYING AND SELLING—GAIN AND LOSS. ]19 Again, tlK! 211(1 year's gain =. the 2ml years capital x ~ Tli(-rof(.re the final cai)ital = the I'nd ycvir's capital x ~ ■ 'J •^") -the original capital x x — S 20' that is, the (.riginal capital x , xf^ = $1140. ^^ 20 Therefore the original cai)ital = *U 40 x '""^ "^ y X 2:3' = 13200. (4) At wliat rate does a grocer reduce his price by giving \ an ounce with each pound, for good measure ? Solution. — The business transaction is : The grocer gives 1G| oz. of sugar (say), and gets the list price ot 16 oz. Therefore the selling price of 16i oz. of sugar = the list price of 16 oz.; that is, the selling price of an oz. of sugar x 10 » = the list price of an oz. x 16. Therefore the selling price of an oz. = the list price of an oz. x 16 x ;~. Therefore his reduction = the list price x — , and his rale of reduction is -- or 3^- per cent. EXERCISE XXVII. 1. A merchant paid #.3250 for a certain line of goods, which he sold at a gain of 7 per cent. Find his selling price. 2. A man gained $12 by selling an article at 12.V percent, above its cost. Find its cost. 120 ELEMEN'J'ARY ARITllMKTIC. . . ) W.I r -«. % .,« .'5. A merchant paid $4.50 for an article which lie sold for $G. Find his rate per cent, of gain. 4. The selling price of an article was $25 when the rate of gain was 25 per cent. Find its cost price. 5. A merchant paid $1432-25 for a certain line of goods, which he marked at 20 per cent, above cost. He disposed of them at a discount of 5 per cent. ; find his gain. G. A merchant marked an article .'50 per cent, above its cost, and sold it without reducticm for $22-10. Find its cost. 7. A merchant began business by investing $4525. He gained the first year at the rate of G per cent., and the second year at the rate of 8 per cent. What was he then worth ? H. A merchant's capital at first was $7500, at the end of the year it was $7000. Find his rate of loss. 9. The first year a merchant increased his capital by the rate of 12^ per cent., the second year by the rate of 10 per cent. ; his profit for the two years was $1520. Find his original capital. 10. A bought some oranges at the rate of 7 for 12 cents, and sold them at the rate of 2 for 5 cents. Find his rate of gain. 11. The manufacturer of a certain ai-ticle made a profit of 2U per cent., the merchant made a profit of 25 per cent, if thr merchant's selling price was 12 cents more than the manufac- turer's outlay, find what the consunier paid for the article. 12. I bought •■}25 barrels of apples at $140 per barrel ; I pre- paid the freight on them to Montreal at 6 per cent, of their cost. I sold them there at a profit of 12i per cent, of the whole outlay. Find my profit. 13. I sold two houses and lots for $1600 each, gaining on the one at the rate of 12i per cent., and losing on the other at the rate of l'2h per cent. Find my gain or loss on these transactions. 14. If $1-40 is gained by selling at 20 per cent, above cost. find what selling price would make the rate of gain 25 per cent, BUYING AND SKLLING— GAIN AM) LoSS. 121 1'). I marked my yoods ^^f, per a-nt. above cost, hut in si"lliii;,r ijavt" a discount of I),', per cent., ^'aininj,', however, !^V2'). Find what the goods cost. 10. If 4 artich's are sold for the cost of :], find the rate of ^'ain .' 17. A grocer sold Initter with a jiouiid weight .', an ounce Ught ; at what rate did he there])y increase his price I IS. A merchant measured off J?? inches of clotli for a yard ; by what rate did he thereby decrease liis list price? 1*J. A merchant decreases the list price by thive successive rates of discount, each e.pial to 10 per cent.; what single rate of (hscount is etjuivalent to tliese three rates? •20. Half of my goods T sold at a profit of 15 per cent., and the other half at a profit of 29 per cent. Find tiie rate of profit for the whole. 21. An egg buyer purchased 300 dozen I'ggs at 7 cents a dozen, 400 dozen at 7i- cents, 511. d„zen at 8 cents, and GOO dozen at 7-| cents. He sold them all at 7| cents a dozen. Find his gain and also his rate of gain. '22. I bought 4 barrels of molasses at :]2 cents a gallon. In NcUing 2 gallons were wasted. I made a profit, howe\er, of 1:5.^ per cent. Find my selling price per gallon. 2:i. A dry goods merchant marked his cloth 42L' per cent, in advance of its cost. He sold it at a disccamt of 10 per cent., and gained 15 cents a yard. Find the cost per yard. 24. During G months of business a merchant's gain wus at the late of 8 per cent. Of this gain the merchant withdrew .i?400 for private use. The next G months his gain was at the rate of 10 per cent. At the end oi the year lie was worth $11440. I'ind his original capital. 25. A merchant bought 225| yards of cloth at $1.17.1 a yard. His profit on selling was at the rate of 23^ per cent. Find his total selling price. 122 KLKMENTAKV AKI IIIMLTIC. of L>L'J, per criit,, 10 p,..- .vrit. ..f i) -;iiniii- at tl," rat., of is; ,„.,• (rut., ;{() per rvnl. of il ;;ai-.ii,;ir .,a the mt(- of |., piT cent., jiiid tli(- i(>st losing at the rate of :.() percent. Ilis ^aiii from these ti'unsaetion.s was $212. Kind the cost of the stock. 1'7. The 1st year ii merchant j^ained at the rate of IJ.', i,,!' cent., the L'nd year at tlie rate of 11,', j.er cent., the .hd'ye.u at tlie rate of 10 per cent., the 4th year at the rate of 'J ^\\>vr cent., and the oth year at the rate of f<\ percent. What raC will derive the 5 years' gain from the original capital 1 2f<. A disliouest merchant pretended to l)uy and .sell at tlic same price, hut cheated to tlie extent of A an ounce every time he hought or sold a pound. What rate will derive his dishonest gain from his outlay ? 29. A grain buyer bought 5000 bu.shels of whea, it Gtcenl- a bushel. When wheat had risen to 60 cents he sold it all. He then investeiit tnj,n't her tli .ondiKt the l)iisiiicss, tliiis funning' u J'orfinrs/nj>. It is fair that, it" tln'ir capitals arc in the hnsiticss (hiiini; thcsainc time, each should <,'aiii i>v lose at the same lat e : so that sv kliat uil deii\c the whole f the gain = $S00 x - = $400 114. Example solved. A, li and C form a partnership, .1 contributing $400, which was in business for S months; li .*r)00 for 6 months, and C $400 for 12 months. The whole gain was $900. How much of this belongs to each ? Solution. -A should get what $400 earns in S months ; that is, A .should get what $:5-200 earns in 1 month ; also, li should get what $;W00 earns in 1 month, and C should get what $4S00 earns in 1 month. Tlierefore, all should get what $11000 earns in 1 month. The whole gain is $900. Hence, the rate of gain for a month = '" = — - " $11000 110" Therefore J'.s share should = $3200 x - - = $2fil '* . •iiid so for the others. 124 KLKMICNTAltV .\I{ITIIMF;Tti\:\On. As s\uuv of the <,'aiii was .*! l;{ ; tin. I the shaivs ..f A' and C. •"). .1 and />' t'oiin a year's |taitnfr.sliii>, J .ontiibuting !ii<4'J0(). and n SIHOO. After (i months A eontrihutcd .foOUO luovv. Tlie whole -,^un was i<:VM){). How much of this helonys to .1 ? I. J and /i start a hiisiness, A putting in li^noOO and J! $100(1. After (I months they admit aiiother i)artnc>r, C, with a capital of S.'iOOU. At the end of the year the hooks show a loss of ,fO(i(i. The husiness is then wound uj). Mow much cai)ital does t-acli now possess '/ ■'). A \nu in a husiness .■^ISU and /i j^'jIIO. Their gains were •Sl'-iO anloyed i (). A and />' contrilmte, respectively, !i?L>;VlL>.;?r) and S-'^oTl.fiit, and gain .*(iJ.{r).L»|. How much was A'x profit? 7. J, /; and C rent a i)asture field for !t?100. A had 2") head "f . Find the share of each. 9. A, B and C contribute ecjual sums of money to carry on .1 business. At the end of •^ months A withdraws }, of his capital. at the end of C months H withdraws ' o^ his capital, and at the end of 9 month.s C withdraws \ of his capital. Divide $39r)0. the year's pi-ofit, equitably among them. mVINO AND SKLLINO— ()\|\ and j^ss. 12.- ) find $7<"t hole jirnlii, w ('a|tituls, *il;{; tiiKl Ling !i?HKK». iiioiv. Tlic A? (i/y!?K)oo. I capital (if ss of $W>(). (locH each gains wcir V long \va> $3571.0!*, I lT) head and C I'O :9.10. Oh capital (if i .f 185.1'.'). arry on ;i is capital, nd at tlic dt> ,^3950. II."*. Commission. .\ incivliiuii. .iipital wifji wliicli t( i<»\vc\c|-, iiiav li;t\c ti iMiiucl a liii.sincss tor liini.sclf. Im mkI ..ISC anclhcr pcis(.ii, ,.ullc(l a /'rinclj,,,/, finriislics ll uliicli t(» l)uv goods, and I a ic nioiK'v w itii receives the money for which the goods ch a merchant is called a Comniissioit M,rclt,i„l, uv \r»t, a.id the money he gets for his work in h.iving and s(.M'ini. are sold is called liis Commit moil. Of the iiHinev- I le receives from I Ms IVincipal to l.uy witli, it is the custom for him to take a p„r( t,.r his connni.ssion and to pay the rest for the g.mds ; and of ij,,. "or goods sold, to take a part for his coiiunis luonev lie receives f sion and to .send the rest to his J •ases, this commission is derived from tl 'lincipal. Kurtlier, as in (,ther le moiiev the goods Ixaight, or from the iiioiiev the A for goods .sold, I .\gent pays gent recfiv es toi , hy na-ans of ji rai,. previously agr I iii.oii. 'I'l if this rate l)e .'{ per cent., tl lUS. le comiiiission for l»ii\ ii g goods = the money the agent paid for the good .uid the connni.ssion for selling goods = tlu' money the agent receiwd f,,r the .n„.d ItJU" I UO" 110. Examples solved. (1) An agent received $m from his principal to invest, uu a cummis.si„n at th.> rate of 2 per cent. Find his connnission. Svhitiiin. — • The agent's commission - the cost of the goods x ^ , tlierefore the money he received, which includes tlie commission and the cost = the cost (jf the -oods x - • 50' that is, the cost of the goods x — = a $ x 4:{0. Therefore the cost of the goods = ft ij? x 43U x — or 126 ELEMENTARY ARTTHMETIC. Tlicrut'oic llic coini uissudi = ^ X l.")0 X — 1 51 ""So' a!ii;xi;30x^-$8-4;J ol (:2) An agent's rates are 2 \^e\• cent, for buying and 3 jier cent, for selling, l^pon advice from his principal he sold :]()() barrels of ajiples at .^1.7") per l)arrel, and with the pi'oceeds, less his charges, bought wheat at 0") cents jier bushel. Find tin- quantity of wheat purchased. s SolntioH.— The ai)ples sold for a $ x Iw") x .'500 ; that is, I ov a X 01';). The agent's charge for selling = a $ x 52") x -O."}, = a $x 15- /•). Therefore the net proceeds of the sale = a | x 509-25. Again, his charge for buying the wheat = its cost x This commission, together with the cost of the wl ')0' leat, = the cost of the wheat x 51 so that the cost of the wheat x —r = a | x 509 25 50 50 Therefore the cost of the wheat = a $ x 509-25 x 50 5l" But the cost of a bushel = a $ x — ^. 100 Therefore the cost of the wheat (Article 33) = the cost of a bushel x - 509-25 I' ;j X 50 G5 ] 51^l00'' Therefore the amount of wheat (Article .77) a bushel of wheat x | 509 25 50 G5 \ 51"^ Too/ s= a bushel of wheat V 509-25 X 50 X 100 51 X 65 ' nUYiNG AND SELLING-OAIX AND LOSS. 127 •)•) = H bushel of wlicat x 7G.S ~- ■2-2 V = 70S bushels ,i,„l j|,^_ iiearlv. V EXERCISE XXIX. 1. Cuh-uhito the e.mmiission in the followini^ cases- husheri'n '"'"' '''' '"'"'^ ''' ^'^'''^^' '''' -"ts per Du.snei, at j.\ per cent. " n- perL^;'" """' "'' ''™'^ '' '^""'' '^^ ^^-^^^ !-• -^t., at _ (<•) For l,uying 2S49 pounds of pork at 8o-S5 per cwt at 1,; per cent. * ' ^'- (^) l^>r selh-n, IS tons Cr20 pounds of hay at .fl2-G4 per tnii, at ,1] per cent. ^ 2. I received from n.y principal 840:30 to invest in t,>a at -^-H ;;::n::;;;;::;,r - - - ' j '- -'• ^''"<' "- ■.-■ b An a,<^ent sold i>S head of fit ....ffl,> ,.-i> „.,. i.>r., , . "^ t.ittle, whose averaw \vei<'ht Y liGi pounds, at $2-90 per cwt., on connnission >u the ^te "^ 'i I- cent. What sun, pani/. These men elect a few of them selves, whose work it is to direct the general affairs of tlir business, and wlu) are called the Directors, or the DirecUrrate. They also elect a man to manage the ])usiness in detail, who i- called th<> Jfnnnycr. The names of all the men who liave con tributed capital are entered in a book, with tlie sums they have contributed. These men are called Stockholders. At the time any one (say A. Ji.) of these stockholders paid in his money he was given a paper, which reads in effect as follows : Montreal, June Sth, 1SU7. lliis is to certify that Arthur Backus has standing in his 7iame ^otJiK) stock in the Dominion Express Company. (Signed) P. Q., Afanager. This is a Stock Certificate. 118. How Stock is Bought and Sold.— After awhile, A. B. may wish to sell his stock to C. D. for money. To do this A. B. and ('. D. employ a Broker, whose business is to buy and sell stocks, and who charges each man the same amount of money, which is called brokerage, for effecting tlie transactitm. Tin' manner of selling is as follows : A. B. gives the Broker his stock certificate, who, in turn, gives it to the Manager of the business. The Manager then removes A. B.'s name from the books, and, in its plac(% puts C. D.'s name ; he destroys A. B.'s certificate, ami issues a new one in C. D.'s name, which he gives to the Brokei'. The Broker then hands this certificate to C D., who pays hm lite of 5 per my charges, id".l .sell th.' I nnjuirod tn iber of 11 KM I, e\v of them Tails of tlir Directorate. letail, will) i- ho have con IS they ha\r At the time lis money lie 8th, 1SU7. ndiiig in liis Mimnger. tter a wliile, To (]o this to buy and nt of money, iction. The ier his stoek ihe business, )oks, and, in tificate, ai'jl the Broker, ho pay.s hini BUYING AND SELLING— GAIN AND LOSS. 129 the price agreed upon ; and finally, he takes out his own charges and gives the balance of the money to A. B. Il». Sharec. -The stock which a man has stamling in his nunie ,s usually .livided into Shares of $100 each, and this $100 IS called the par value of a share. (^0 Now, the pric(. agreed upon between A. B. and C D is ) Also the money (brokerage) the broker charges A. B. or CD. ,s derived from the par value of the stock by means of a rate. Ihus, ,f the broker's rate is \ per cent., the brokerage the buyer pays = the par value of the stock x ~ 400' 400' and also, the brokerage the seller pays = the par value of the stock x J) Further the profits of the business for a year or months .e distributed among the stockholders in such a way that each ■nans share of the profits is derived from the par value of his s ock by ineans of a rate. The whole distributed profits is called .1)0 Du.de,^d or a year or half-year ; but a stockholder would speak of his share of it as income, rent or interest. Thus if -i -nipany declares for distribution a dividend at the rate'of 6 }'«T cent., a stockholder understands that l:is income from his stock = the par value of hi.s stock x -1 yt) It a man purchased stock at the rate of 90 (per cent ) ^vl.lch declared an annual dividend at the rate of 7| per cent,' his interest = his investment x ~-- 90' which makes his rate of interest 8.1 per cent. 130 ELEMENTARY AlllTHMETIC. •' 'i ^ ■^0" 'it"- ; ■ '.I . i.^) «::f i Id ir*... , St""* I 124^. Therefore, if we say, "T bought ^oOOO six per cent, stuck at S."?, l)rokerage |^," we mean that r bought ;")() sliai'es of stock ; T paid fH.'i for eacli share ; I paid $^, in addition, to the broker ; my income from each share is •*() ; and mv rate of niterest = -— -r- r^l. Examples solved. (1) T bought .*r)000 stock in tiie G per cents, at 8.*?, the brokci' age being I per cent. Find my money invested, my annuitl income, the brokerage and tlie rate of interest, 1st Solution. — My stock = one share of stock x 50. Tiierefore my investment = the price of a share x oO, Article 77 = a$xS3i x50 = $415G-25. Therefore my annual income = the income from a share x 50, Article 77. = a.fx6x50 = .^300. Therefore the brokerage the brokerage for a share x 50, Article 7^ = a $ X 1 X 50 : 6-25. 6 132 Therefore the rate of interest = -— r = 6-—- per cent. 83^ 133 ' Qnd Solution. — My investment = $5000 x 831 100' = $415r)'25, on reducing. M y income = |5000 x -— = $300. Article 119 (r Article 119 (-•). SIX per cent. BUYING AND SELLING— GAIN AND LOSS. 131 1 The brokerage = iJioOOO x ~~ = U-'>5 broker \ g il per cent. k-estinent in >^ ipital and tlir r)8-2r), wliat is per cent, stock and puroliascil ' income, il l)y transfer per cent, stock :e ") per cent. r cent. Const il- Lgp l)eing ] p'l ?, and by h(iN\ 'r cents, at !'ii. ' cent, stock at ind the clianp' 13. Tlie whole capital stock of a comi)any is |50000 The profits for a year were |752S-;30, How much of this caii they leserve jifter dechiring a dividend of 12^, per cent. ? U. The capital stock of a compunv is ifefiOOOO. The whoh^ 'i.vidend for a year was .$4500. How much of this shoukl a stockhokler receive who hokls ^'y.iOO stock ! lo. A man increased his income $10 hy transferring his 5 per '•eiit. stock at 90 to 6 per cent, stock at 90. Find the capital he lias invested. IG. A person .sells $1200 stock in the 5 per cents, at 9G, and invests the money he gets in S per cent, stock, without changin-' lus income. Find the price of the S per cent, stock. Viz Duties. -When a merchant buys goods in a forei-ni country, he receives from the producer a paper, on which ""is ^v^tten an accurate description of these goods and their prices his paper is called an Invoice of the C!o<.ds. On the arrival of the goods in the place where the merchant c" by the owner of property, j,Mves the owner a paj.er on Nviiich IS written their pn.mise to pay the owner a sun. of moni-y H. case th<. property is destroyed by fire within a certain time. Ihis i,aper is called a J'o/ictj of Insurance. The sum the com- pany promises to pay is called the Face of the Policy, the Amount of Inmranc, or the Risk: So lon^ as the policy remains in force the company is said to carry the risk. The sum ,)f money the owners pay the company is called the Pre- "11 am of Insurant, and it is derived from the face of the policy by means of a rate. Thus, if a c.n.pany's rate for a certain class of buildings is i per cent., we mean that the premium a man pays for insurin«' = the risk x ~ 500' 1*45. Examples solved. (1) A man had his house insured for l of it,s value, ].avin- a premium of $12-60, which was at the rate <.f \ per cent. ' Fhid tiie value of his house. 'SW«-60 x ^^2 = jiioioo. (-') For what must I have my house, which T value at $1500, insured at 1] per cent., so that, in case it is burned, T may iccover my premium and I of its value? Solulion.^-Thii premium I pay = the risk x 4, and the con- (lition of the question is that 136 ELEMENTARY AUIThMEriC, u •^ .1 I:-- the risk = the value x ' + the preiniuiu ; 4 3 1 that is, the risk = if ir)00 x , + the; risk \' — 1 80 7<) •( Therefore the risk x -':-=- $1500 x - . 80 -1 Ti * u • I d. l-'JOOxaxHO ,,, llieretore the risk = a $ x = iU .'59 -U ■ which is what I must have my house insured for, EXERCISE XXXII. 1. Calculate the premium to be paid in the following cases : (a) At ^ per cent., for a risk of $2600. (h) At ^ per cent., for a risk of f 1 1 100. (c) At 1| per cent., for a risk of $6250. 2. My property was insured for JjilSOO at a premium of 4 per cent. It was destroyed. Find what I saved by insui-ing. 3. A company took a risk of $16000 on a building at g per cent. The building was destroyed. Find the loss to the com pany in taking the risk. 4. I insured my house for | of its value, paying a i)remium of $16, whicl. was at the rate of 1| per cent. Find the value of my house. 5. Find what I must insure my house for, which is wortl $5070, at I per cent., so that I may recover, in case of loss, both the value oi the house and the premium. 6. My store is worth '^ as much as my stock. The store is insured for q of its value, at i per cent. ; and the stock for ''- of its value, at 'j per cent. Tiie total premium I paid was $37.10. Find the value of my store and stock. 7. I insured my store so as to cover » of its value, and the HrvlN(i AM) SKLLlXfi— GAIN AM) LOSS. 137 im of I per ring. 1^^ at ^ per to tlie coiii- preinium of he value of h i.s wortl if I0.S.S, both he store is k for -,''j of vas #37.10. le, and the premium at '. W|.,a rate will .l.-rive the face of the poliey trom the value? . '' «. A company t.x.k a risk of <^:mOO upon a vessel at I 1 per '•'■Mt., hut afterwards place,! : „f this risk with another con.panv l-aynig then, a pre.niun. of 1 ] per c(>,.t. Kind the net sum eaJli comj)any would h.se in case the; \-essel we.v l„st. !)._ A company took a risk up.m some property worth .*IAOO »"!• A ot Its value, charging a premium of .*ll.JU. Find their rate of premium. 10. A cattle dealer purchased 1l>0 head of fat cattle at an -.'.•.ge cost of #.Sl.'o. J-Ie shipped then, to Liverpool at an H.l.ht.onal cost ot flO.-SO p.-r head. He had them insured f,.r t ."U- ong„.al cost, at U, percent. He sold then, in Liverpool at £11 b.. .W. ,,er head, and hauent, or the assessed value of his [..operty or income. The Council also fixes a Hate of Taxation l.y which the taxes a man must pay is derived from his a.ssess- •n^nt. Ihus, if the rate of taxation is fixed at 21 mills to the 'loJlar, the tax a property owner would pay = the asse.ssed value of his property x -ill ^ -^ lOOO" 138 ELK.MKNTAItY AIM IllMKTIC. >^ .1 l*i|. Example solved- A hkim's m't infomo \v;is $1 ISINO afk'i- lie |ifii(l an incctmc tax of lit mills to the (ittllar. Jt' $Hni of liis iiK'oiiu' waw not sul)jt'ft t«.> aMHUHsini'iit, find liiw total inconu'. t^oliiliuu. — TliL' tax liu itaid = liis assi-ssi'd income x !»S| lUUU" Thoirt'ori! his net income = $700 + his assessed income x Its I Therefore his assessed income x i()rv;^ = "' ^ ^ Ti'^l'SO. Therefore his assessed income = a $ x 784: -80 x -— - —$f< I/O I Therefore his total income = -^ 1500. EXERCISE XXXIII. 1. The whole assessed value of a town is .*'J(iOOOO, and tin money needed during a certain year is .*'i;")GOO. Find lo the nearest mill on the dollar the rate of taxation. How much more money will be collected than is needefl at this rate .' 2. A man's taxes were $4S'10 when the rate of taxation was IH.I mills on the dollar. Find his assessment. 3. In a certain town S.'JOO.'Jw") was raised from a tax at l'> mills to the dollar. What was the assessed value c»f the property in the town ? 4. The assessed value of the property of a town is $1493250. What sum of money can be raised by a rate of 20/, mills to tin dollar, after paying the collector of the t.ixes 2 per cent, of thein 5. The assessed value of a town is $2r)0'U0O. During the year the Council estimates the expenditure as follows: interest on the debt, 19575 ; the lioard of Works, $109.50 ; the Public ►Schools, $8570; the High School, $6825; miscellaneous expeii diture, $3295. If iL costs 2 per cent, of the taxes for collecting. mm HU VINU AND ,SKLMN(i— (JAIX AND LosS. 139 axation wus liM.I the Irust rut.', in ...xuct mills u„ the dollar, which will l.o nt'ct's.sary to raise the money rciiiiired. <■>. If a man's income up fo #700 is exempt from taxatii.n, and Ins net income, after paying a tax at the rate of I ! per cent is 8I1.MOIU, tind iiis tax. /. A school section, whose property is assessed for i^lNOOOO, iuiilt a school-house costing s.{ I. SQUARE ROOT. ViH. Wlicii !i single rate is the product of two ei[uti\ rates, the sini,'l(' rate is called the square of either of tlie e(jual rates, and each of the eiiual rates is called the stpiare root of their product. Thus, since G t = (S x S, Gt'is the sii .f 8 I [uare ot o, and may hi written 8- ; while S is the stiuare root of G4, and may be writtei \/()4. am, since -- = '.\ W <) tl . IS tlie scjuan of and nij ly written written .hile . is the scjuare root of — -, and may 1 2-) So also, -0081 =(-0!))-', and -Oi) = V-0081. V.t\^. Rule of Order in Square Root. Since the order of the {)roduct of two simple numbers is the sum of the orders of the simple numbers, therefore tlie order of the square of a simple number is twice the order of the simple number; that is, the order of the square of a simple number is even. 8ee Article 48. Thus, since the older of G in -OOG is -3, therefore the order of :i() in (-OOG)-' is - G ; so that (•00G)-'= •00003G. Ayain, since the order of 4 in 400 is + 2, therefore the order of IG in (400)- is + 4 ; so that (400)'= 160000. SQUARE ROOT. 141 re the order Now, from the Multiplication Table, the stiulent knows that 1, 4, 9, 16, 25, 36, 49, 64 and 81 are the s.juares of 1, -2, A, -[, r>, 6, 7, 8 and 9, respectively. Hence, if anv one of llu-s',. s,|u'are numbers be written in an even ordt-r, the onler of its snuure root is half this order. Thus, since the order of 49 in -0049 is - 1, tlwuefore tl,.. order of 7 in \/-0049 is - 2 ; so that V-OuTi) = -07. Ajrain, since the order of 81 in 81000000 is +(], therefur.> tlie ord.T of 9 in \/81 000000 is + :^ ■ s. hat VcSlWOOOO -^ UWH). }'^^^ P. ^.*"^ ^^^ Square Root of any Number. - Tn a compound lunnbei' othci' than owiri"' manner of oi'der to find the scjuare root of those in Article 129, we shall consider the foil ([uarmg a compound number, Since 47 = 4 we may multiply thus in the order + 1 and 7 in the ord( r 0- lU + 7 40 + 7 40 + 7 40 X 7 + 7 X ( 40 X 40 + 40 X X / X 2+ / X (, on addiuir Then 47' ^40x40 + 40 = 1600 + (40x2 + 7)x7 = 1600 + r)G0 + 49, from(«) = 2209. Tf now, in the number 2209 we mark the di (iiilcrs •0 even, thus, 2209, we find that 22 is the f number whose ord irst pai-t ot til also see that 22 is more than the ler is even, and that this order is +2. \\'( than the s(juare number 2"). Theref scjuare number 10 and 1. (|uare root of 2209 is 4 in ordei- + 1, that is, 40 Further, if (40)', that is, 1600, be sul)tract<-d 'ore the first part of tl le venini nder is 609; but, from line („) ;d remainder is also 10 from 2209, the )ove, w(> see that this X / X 2 + 7 X 7, th , tlie majoi' pai't of which i.s 142 ELEMENTARY ARITHMETIC. •■ > 40 X 7 X '2, as we can see in line (c). Therefore, when we divide 609 by 40x1', that is, by 4x2 in order +1, we obtain 7 in order 0, winch is probably the next figure of the Sijuai-e root. This 4x2 is called the Trial Divisor, which, we see, is doubi.- the pai't of the scfuare root already found. Finally, from line {l>) we see that the complete divisor i^ 40 X 2 + 7, that is, 87. When this is multiplied by 7, the pro duct is fiOO, thus verifying that 7 is the next figure of the root. 1 I _ Tn practice, the operations 2209 ( 47 described aboAe are conve- ^16 niently arranged thus : 87 609 609 KSl- Tf, however, the compound number be not a complete square, we shall proceed in the same way to find its scjuare root. Thus, to find the square root of -05742. Ill We mark the digits whose orders are even, thus, •05742U. Tiie first part of the number whose order is even is 5, in the order - 2. Therefore, the 1st figure of the sijuare root is 2, in the order - 1, that is -2. The first trial divisor by which the 2nd figure of the root is obtained is 2 x 2, in the order - 1, ami so on. The operations are as follows : •Oo7420 ( •2396 4 43 469 4786 174 129 4520 4221 29900^ 28716 1184 SQUARE ROOT. 143 en we divide obtain 7 in scjuare root, ee, is duubl./ e divisor i-^ ' 7, the pill Qf the root. Here 46 is the 2nd trial d ) a complete s(juare root. I I I us, •057-1-JU. is 5, in tilt' root is '2, ill y which the Jer - 1, and ivisor, by which 9 is obtained ; and )y which 6 is obtained. f 478 is the ."{rd trial divisor, 1 It would seem that, when the 1st trial ilivisor, 4, is divic; into 17, the ([uotient would be 4. But, on trial, the prwluct the coinplete divi.sor 44 and 4, namely, 176, is seen to be greater than 174. It is rarely neces.sary to find a square root to more than 6 figures, except when a very lai-ge (juantity is to l)e derived tVofu a very small one by using this scjuare rout as a rate. EXERCISE XXXIV. 1. Write down the sijuares of the following numbers, giving the reasons for the orders of the digits in the same : (") ^0. (,/■) -4. (k) -200. (^) 700. (y) -007. (/) -002, (c) 8000. (/>) -08. (,H) •!. ('-/) :3000r • (i) -OOOCI (n) -9. (e) I2OO0U. (j) -001. (o) -0000012. 2. Write down tlie square roots of the following numbers, giving the reasons for the orders of the digits in the roots : (a) 64. (f>) 6400. (c-) 810000. (d) 900. (e) 121000000. (./■) -09. if/) -0064. (A) 000004. (i) -0001. 0') -000121. (/•) 3600. (/) -36. (m) -000025. (n) -04. (o) -009. Extract the square roots of the following numbers : («) r,76 {/>) .•'.12o. (r) 15625. (d) 815409. (e) 687241. (/) 1522756. (//) 72900. (//) -09732. (i) -00004. 4. Find the square roots of : (/•) 231-29. (/) -2. (m) 2. (n) -00001. (n) -025. (-) (f') 25 64" ('■) 81 I2I* ^ '' 49 («) 5 16" 144 ELEMENTARY ARITHMETIC. I 5. Reduce the following fractional rates to decimal rates, and thenct^ find their square roots to 5 figures : 2496 ^'^' 38247' (/>) T3" 21 (9) 100 ImT' (^') :. ,.. -173 (A) '^. ^ ' •032 fl. An article was marked to sell for $1G0 ; but, on the price being reduced bj' two equal successive rates, it sold for #122.50. Find those rates. ' • 7. Tn marking goods for sale, a mercliant increased the cost price by two ecpial rates, and so gained at the rate of 44 per cent. Find these rates. S. A book contains 123201 words. Tt has as many pages as there are words on each page. Find how many pages in tlic book. II. CUBE ROOT. I.'W. Wlien a single rate is the product of three equal rates, the single rate is called the cnho of any one of the equal rates : and any one of the eijual rates is called the cnltp. root of theii' product. T, • 04 4 4 4 04 . , , 4 riius, since —= ^^ x ,. x y^, — is the cube of -, and may be wi'itten (^ - j : while , is the cube root of -^, ami may be written ^lyiyp. So, also, for decimal rates. CUBE ROOT. 145 I.U. Rule of Order in Cube Root. -Since the order of 2 P-'--t ot three si.nple nu.nhers is the sun. of the orders o 'iH' .unph. nun.hers, therefore, the onler of the euhe of a sin.ple Hun,ber ,s hree times the order of the simple number ; thati the^order of the cube of a simple number is a multiple of 3 a nnnT ''" '"'"' "^ ' ^" '^^ '' ■^- ^'--^-- ^he order of ••4 m (-100)' IS +G ; so that (400y' = 6+000000 8 inT'v' •''"?'" ":^"" f ' ^" •' " - ^' ^''^-^--^ the order of h m ( 2)' IS _ .J ; .so that (•2)-'= -OOcS Now since 1, H, 27, 64, 125, 216, :U3, 012 and 729 are the ;;^-oM, 2, 3, 4, 5, 6. 7, N and 9, respectively, if any one of esec.d>enum),ers be written in an order which 'is a Ll p of ^ .s cube root wdl be written in an order which is one-thL Thus since the order of 125 in -000125 is -6, therefore the order of 5 in v -000125 is - 2 ; so that ^ :000I25= -05 of -""^^''" ""'^^ "^ ' '" ^000 ^« +^. ^J-refore the order of 2 m V 8000 IS + 1 ; so tluit v 800"b= 20. ordeft'^n,^;"'^ ^^ ^"'^^ ^°°^ ^^ ^"^ Number.-In ule. to find the cube root of a compound number other than tl^ose m Article 1.3.3, we shall consider the following manner of 'ubing a compound number. Since (57)-'= (.50 + 7) x (50 + 7) Tl f /r.v,"^'^^^' +50x7x2 +7^. Article 130(a) Therefore (57)'= {(.50)-' +.50x7x2 +7^'}x(50 + 7) = (50)^ +(50)^x7x2 + 50 x7^ on multiplying by 50; ,,^,, +(-^0)-x. +50x7-'x2 + 7*, on mult, by 7 • 1^000^^'"^-^'^' '■'''''^' +7^ on adding (I; = 120000 +{(.50)^"x3 +50x7x3 + 7-"} x 7 M = 125000 ^(.,.,,00 +5x7x30 +7^x7."--- But (oO'= 185193 by ordinarv multiplication ^ It, now, in 185193 we mark the digits whose orders are multi- ples of 3, thus, 185193, we find that 185 i, the first part of the 14G ELEMENTARY ARITHMETIC. nunil^er whose order is a multiple of ."3, and that this order is + "i We also see that 185 is more tliaii the cube number 12:") ainl less than the cube number 21(5. Therefore, the first part of tin cube root of the number is f) in the order + 1 ; that is, 50. Further, if (oOf, that is, 125000, be subtracted from 1S519;!, I the remainder is 00193. But, from line (a) above, we see that this rem rem imc ler is also (50)- x 7 x 3 + 50 x 7- x 3 + 7'', the ma jnr part of which is (50)-' x 7 x ."^ or 5- x 7 x 300. Therefore, when "... 60193 is divided by 5'- x 300 the quotient is 7, which is prolMihh/ the next figure of the cube root. This 5- x 300 is called the (r'uil divisor, which, we see, is always formed by multiplying the square or the part of the root ah-eady found ]>y 300. From line (c) we see that the complete divisor is 5'- x 300 + Tj x 7 X 30 + 7'-, that is, 8599. When this is multiplied by 7 the pin duct is G0193, thus verifying that 7 is the next figure of the root. In practice, the operations described above are conveniently performed as follows : now a ; 185193 ^S'=. 125 5-'x300 =7500 G0193 5x7x30 = 1050 7- = 49 G0193 8599 (57 1!{5- — If, however, the number be not a complete cube, \\e proceed in the same way to find its cube root. Thus, to find the cube root of -0756. We first mark those digits in the number whose orders are multiples of 3, thus, -075000. Therefore, the first part of the number whose order is a mul- tiple of 3 is 75 in the order - 3. Therefore, the first digit in the cube root is 4 in the order - 1, that is, -4, CUBE ROOT. lete cube, \\t 147 Tin. Hrst trial clivis.r l,y which th. scco.ul .ii^it of tho root is tound ,s 4- X ;iOO, o.. i.sOO, and so on. The ,ru..Z is shown hcio. I I 4- X 300 4 X 2 X .'JO 4-' = 4,S00 240 4 l.t complete divisor is 5044 (42)-' X aoo 42 x2x30 2nd complete divisor is 5.31724 (422)-' X 300 422 xSx30 8- = 53425200 = 101280 64 3rd complete divisor is 5352G544 •075GOO ( -4228 04 11600 10088 1512000 1063448 448552000 428212352 20339648 1st ;fd T ,^^^? ""T ^'^"'^^ ^" *^« Root-Since the 1st an, ^,„, ^^,^^j^,^^^^, ^^^^.^^^^ ^^^^^ ^^^^.^ first digits the same and 2nd and 3rd complete divisors have their first Two di^it .J -me, ..may expect, then, that the 3rd and 4th compU:! , ^ - wdl ha., their first f.r.e digits the sa„,e. Therefore, Z^. s U^ree d.g.ts, 535, of the 3rd trial divisor as a divi.., anS last "'; ■ " ' '^"'^"'' ''-' '"^>' -P-^ to obta „ at least two more digits in the cube root, thus: 535 ) 20339 ( 38 1605 4289 4280 9 Therefore the cube root of -OToe is -422838 nearly, 148 ELEMENTARY ARITHMETIC. •:r; <.ri .: ) c:i, ir'-i EXERCISE XXXV. 1. Write down the cubes of the following numbers, giviii) 500. (c) 7000. (rf) 90000. (.) ■:]. (./■)-oi. 2. Write down the cube roots of the following numbers, givini; the reasons for the orders in which you write them ; (^0 8000000. (e) -064. (i) 729000. (/;) 125000. (./■)-21G. (,/) -000729. (c) 27000. (ff) -OOOOOS. (k) -125. (d) 512000000. (h) -000027. (0 125. 3. Write down the cube roots of the following fractional rates : (9) -004. (/) 70. {h) -0005. (m) 10000 (0 -1. (n) -000 1. U) 008. (o) -5. (k) 800. (/>) -2. (a) G4 125- (b) 729 512' ('•) 125 0*7 • (<0 8000 T29' 512 if/) 15 8' 343000' ^-^M 25000 26 (^0 I'V- (0 27" 000000729. 4. Find the cube roots of the following numbers : («) 91125. (d) 32282885G. (y) 74-088. (b) 32708. (e) -039304. (c) 1470G125. (/) -001092727. 5. Find to f(^ Tho, an,ount at the end of 2 years (Article 100) = -f;)00 X (the rate of amount f(.r a year)-' So that a .$ X 500 x (the rate of an.ount for"a year)-' = a S x GOO. 500 ~ "■ 0. 00. I. Therefore (the rate of amount for a year)' Therefore the rate of amount for a year = V^P^ = 1 -0954 15 Therefore the rate of interest = -095445 = 95445 per cent. ' in ISQ^-^^ the population of a town was 125000 individuals, HI 189 It was 21(5000 individuals. If the rate of increase of the 1-pula .on for each decade was the same, find what the popula- tion will be in 1901 at the same rate. Solu(ion.~The increase during the 1st decade (18G1-1871) = a person x 1 25000 x that rate. Therefore the population in 1871 = a person x 1 25000 x ( 1 + the rate). Snnilarly, the population in 1881 = a person x 125000 x (1 + the rate)-, and the population in 1891 = a person x 1 25000 x ( 1 + the rate)-^ • tlutt is, a person x 125000 x (1 +the rate)'' = a person x 216000. Therefore (1 -f-the rate of increase)-' = -~ ^ 125- Therefore (1 + the rate of increase) = ' /i^ _ ^ \ 1 25 ~ 5 ■ Therefore the population in 1901 will = 216000 X - individuals =: 259200 ind ividuals m 150 ELEMKNTANV AIUTHMETIC. ir'-J J::! *-.:■ EXERCISE XXXVI. I. At what latc foiiipouiul interest will tho interest for ihr use of .-^SOO f(,r i> years l)e .•?90-42 ? L'. When interest is convertible yearly, at what annual rale m interest will SIOOO amount t(» $12096 at the .-nd of .•{ years ? ;{. The iM)i)ulation of A in ISTI was 1;]6900, and its poi.ula- tion in l^ at the lower right-hand corner is a sub-unlf of the 2nd order. On counting these, we find that the surface consists of 6 units, 22 sub-units of th. 1st order, and 20 subunits of the 2nd order. 162 ELEMENTAUV AUITHMKTIC. P t::» if... '*«^ .'I Wlicn this is expressed in unntlu'r way, we may say, the surface of the plot = a square yard x ((» + 22 + 020), -a square yard x S' I. I'his is a tedious process; and we shall show in the next Article that, if tlie width and the Ieii<,'th he nieasuivd, as in Article l;)8, we may find what the measure of the surface is without actually measuring,' it, as we did ahovc. 140. The Rule for the Surface of a Rectangle. On observing,' the diagram of the preceding article, we see that the length i'l/y= a yard x ,\') ; therefore the surface yl£'=a sijuaie yard x .'VS. Again, since tlie width 7iC = ii yard x 21, therefore the surface AC = the surface /l£'x24. Therefore the surface of the rectan«de = a s the unit of surface .. the ,.n„h..t of tin nu-asure of the l)ase and the measure of half the hei«'h I4'^. The Rule for the Surface c^ a Trapezium. Ihe h-aue .1 AC/; :s a trape.iun, when the .side AB is parallel to the side DC; that i.s, when the distance hetween A /J and ^ CD is the same, wherever it is measured. Now, when JW is drawn, it is seen that the trapezium con- ^ sists of two triangles, AliD and S /WC, whose heights arc each e(|ual to /SE Now, by Article 141, when a foot is the unit of length, the surface of the triangle A liD = a square foot x ^}}1}^^^^'^ '>f HE * 2 ^"^^ measure of A B. Als(., the surface of the triangle hDC -•1 v,„i.,.. . v i. tlie measure of ^A' - a square toot x ^ 1 ^ ^j^^ ^^^^^^,^vii of DC. Therefore, when these triangles arc put t<.gether, the surface surface to be a s(|uare foot x;''jG. = a s( paiiitef luare foot x (SQl + 528 + ;39G), = a square foot X 17, Tlierefore the cost of the work (0) The length and width of a field are as 5 is to ;3, an' its surface is o acres. Find its length within an inch. .%/ution.—Ueve the measure of the widMi = the measure of the length x 1^ . tnt held ==a s.iuare rod x the measure of the len-^th X the measure of the lengtli x '] ^ = a s(iuare rod x (the measure of the length)- x '] . But 5 acres = a scjuare rod x 800. ^ Therefore (the measure of the length)-' x 'J = 800. o Therefore (the measure of the length)' = 800 x ^ Therefore the measure .So that the length. a rodx.36-5148 where the sub-units of ihe 11 and higher orders are ne''^''''^' '^^ ^'^^•^<' 17. Tf a ,s.,uare n.ile of territory is n.pre.sented <,n a n.ap l,v '■ ot a .s,uare nxcn, what will be tJie length and wi.lth of a n.ap U> '•^rlongs, . rods. 5 yards, and width S n.ile, G furlong 144. The 47th Proposition of Euclid. Upon a stiff piece of white paper draw a triangle A /JC so tliat It has a ,sc,uare corner, JJ .■ that is, so that the an-de at JJ is ■t right angle. On AC, ' ^ '' the hmgest .side of the t riangle, draw the s(|nare ACDF. Find the mid- dle points of the sides «f this .square, and from these points draw, as in the diagram, lines par- 'illel to vl/y and CJi, the "ther side.s (jf the tri- 'ingle ; thus marking off 158 ELEMENTARY ARITHMETIC. I S"5 this s(|U.ir(' iiiti) five parts, which are numbt'ird 1, 2, •"!, I aiid ."i. Xi'xt, with a sharj) knife, cut out these five pieces. Now, it will be found that, without turning,' these pieee^■, thev can be siid into the positions marked 1', 2', .'V, 4' and 5', respec tively ; tluis making squares drawn upon AJi and JJC, the othci sides of the triangle. Kence, if one corner of a triangle be a square corner, tlir surface of the s(iuare drawn upon the longest side is eijual to the suiface of the two si|uares drawn upon the two other sides of the triangle. This is Euclid's Proposition. 145. To Find the Diagonal of a Rectangle, when tlie sides base been measured. ose that the sides of the rectangle A BCD have been bupp measured, ami it is found that AB=: an inch x 91, and AD = an inch x GO. Therefore, the square drawn upon A J> = a square inch x 9 1 91, = a square inch x 82i ' . ai id tl le sq uare dr awn upon D a square inch x 3600. Now, since the triangle AJW has a square corner at A, thche two squares can be made into the square drawn upon £D. Article 1-U. 8o that the square drawn upon JiD = a scpiare inch x 11881. Therefore BD, the diagonal of the rectangle, = an inchxv/ 11881. = an inch x 109. Thus, we have found the length of £D without actually meas- uring it, ler sides ui n upon An 91 91, 82< ■. i..D 3600. at A, t R'ht' upon BD. MENSURATIOX— SUIIFACE. 159 Draw Ji) tho ht'i-ht of the triangk. Then ,IA', JC ^r /JC =im inch x 24, and nD = im inch x 12. Furtlier, since the tnaii = a Mjuare inch x (12)' tlierefore the .square upon AD = square inch x {(24)-' - (12) n Therefore AD = iu\ inchxV432. Hence the surface of the triangle = a square inch x 432. = a square inch x 24 X \/432 Article 139. = a .S()uare inch x 249-415 when the operations are performed as follows 40: 4148 41564 432-00 ( 20-7846 4 20-7846 12 3200 249-4152 2849 35100 33184 191600 166256 25344 160 ELEMENTARY ARITHMETIC. i L.J t.:> tT.i *^^, 1 ti-: ^-r, »-.. ^': > EXERCISE XXXVIII. I. A licld is lU !•(»(!>; long ;ui(l -"iO rods wide. J low iiu- is it lit'twccn its opposite coiners? H ow uiilcli fiirtl ler \vi 11 ii man \v idk in ^roiT!)' hali- ound u s({uare mile than in ,i,'oin<,r from fDiiior to coriicr acr HV OSS ar the field ? ■\. The longest side of a rigiit-angled triangh" is F-'i inch's, and one of the other sides is ";') inches. Find the lengih ot the .'ha side. 4. The side 'f a sit uare- an inch x 003.51. Find to (5 fiifuiis its diagona,!. T). Wliat is tliv 'oiigtii of the longest string which can l)c stretclied in a nx.ai 24 feet long, IS feet wide, and 10 fed high ? G. A boy walked 1.'^ rods north, 16 rods east, -M rods nortli, 43 rods east, .59 rods nortli, and finally 1 1 rods east. How fai' in a straight line is he from his starting point? 7. If a boy in taking a walk were to go 144 rods north, 210 rods west, 11.5 rods south, and then 323 rods east, what is the shortest distance he can now walk to reach home 5. Find the surface of a triangle, each side of which is 1 ") inches. 9. A boy, flying a kite with a string fiO yards long, found out that he was 48 yards from a pine tree when the kite caught fast in the tree top. How high was the tree 1 10. A rectangular field contains 2| acres, and its sides are iis 4 is to 9. How far is it from corner to corner across the field ? 141. The Circumference of a Circle- — One of the hard problems which puzzled the ancient people of the world, was how to find, without measuring, the Rate by which the lengl; '<' li]>' circumference of a circle is derived from its diameter; >■ nlv How fjii- is ji ir.g hali'-v.ay corruT across ') inch'is, and ih oi the .h a to (i figuri's rhich can he and 10 ft'ct 1 rod- noi'tli. St. How fill' ds north, 2 1 ( i , what is the which is 1 ■") ng, found out kite caught ;s sides are as sa the fiehl ! e of the hanl orhi, was how leiigl: !>;■ till- er; h - niv MEXSUHATION— SURFACE. ]61 vithin the last :^00 years has it been solved. We shall there ■Hie length of the circumference of a circle = it« diameter x;M 410 he f ,^rt '"«'"" "''""'""'■'■ "'""*' ''"' ■' '» ■"■"'■ """%* to give he fl,,t five «,.„,.» „,,,,„, i„ „„,,,,. „,.^,,^ _.^^__,^ „,L„ed „; Again, since --■-'= ;M4o,s.-,7 -- • ,. 1 . ■. • . ' ' i+-'^'J/, ; •« «erefore the surface of both triangles = a square inch x the measure of half the height A 1 , ^^^^ •^"'" '^^ <^he measures of "the basp^ A Md so when all the triangles are put together tlio surface of the many-sided figure = a square inch x the measure of half the height J J X the sum of the measures of 111 the bases. 162 ELEMENTARY AUITHMETTC. ->^ ti:: It, now, the triangles had been inade very thin and theic number very many, so that they still fill up tlie space about 0, then, in the most extreme case, the many-sided fi-^ure is a circle, the bases of the triangle form its circumference, the height of the triangle is the radius of the circle, and the statement above is The sui'face of the circle = a square inch x the measure of half its radius X the measure of its circumference. But the measure of the circumference = the measure of the radius x 2 x ;r. Therefore the surface of a circle the measure of the radius = a scjuare mch x — ; X the measure of the radius x 2 x tt, = a square inch x (the measure of the radius)'-' x tt. I4!>. Examples solved. (1) Find the surface of a circle whose radius is 24 inches. Solution.- The radius = an inch x 24. Therefore the surface = a sf|uaro inch x (24)- x 3'141 G, = a square inch x 1 809 -50, when the multiplication is performed. (2) The circumferences of two circus which have the same centi-e are 120 fctt and SO feet, respectively. Find the sur- face of the ring outside the smaller circle. Solution. — Since the circumference of the large circle = the radius x 2 x ;r, 1 therefore the radius = the circumference x = a foot X 120 X 2 X rr 1 . 60 = a foot X — . 2 X TT 7T inference (jf MENSUHATION—SUllFACE. 103 Therefore the .surface of the hirge circle = a s.,uare foot x ^^^) \^^ ,, „,„,,,, f,^,^^ ^ 3000 -Siinikrly, the radius of tlio small circle --. a foot x -. , Tl.eref..ro the surface of the small circle -- a square foot x (^^^^ " x ;r = a s.,uare foot : 1600 OQOO =^ a square foot x ~ — _, TT = a square foot x G.3G-G 1 8 when the student has performed the operatioiis, as follows : 3-U16) 200000 (6;iG6 18 18849G 115040 94248 207920 18849G 194240 2^49G 57440" 3141G 2G024 (3) The surface of a circular field is 10 acres. Find the cost of fencing it at GO cents a rod. Solufion.—The surface of the field = a square rod x 1 GOO, and also = a scpare rod x (the measure of the radius)^ x ;r. 1 herefore (the measure of tho radius)^ x ;r = 1 GOO. 104 ELEME\TA'»y vniTH.METIC. (100 »J4 f Theit't'orc (tlie iiieasmc uf tht- ih(1uis)-= ;r ThcMcforc the ineasuie of the rad Ills -4 1(500 40 V n Tliercfure tlie civcniinference = a rod x 40 - X ;^ X ;r. Tl = a rod X 80 X V TT, ic cost ot' fencing, then, since = a -^x-GOxHOx v/nr. = a .1itx48x\/;r. = |H5-0(S, as seen below I I I ••)-U16 ( 1-7724 1 27 214 189 347 2.') 10 2429 1-7724 48 141792 7089G 85-0752 3542 8700 7084 IfilO EXERCISE XXXIX. 1. Fitid the surfaces of .t •. ircles whose diameters we : («) 100 feet. (r) 12-34 inches, (e) ;• of an inch. {h) -032 yards. {d) -OIG miles. if, \% feet. 2. The circumference of a circle ■ ■ 1000 yards. Find its diameter. 3. Find the diameter of a circl ^h-' 'surface is 4 acres. (Gi^'^ result in rods.) MENSURATION— VOLUME. 265 J. Find the suj-fH.-,. of the l..u,..st cirH. which c-an be made out of a .s.,uar(- whos,> side is 10 i.iches h^e ..,,,1 ,.| i,,,_ ,,,»,,„„t„„|j. I,,.,,,, j|^_^ ^ liel>veei] tliuuiiiiumfcrii ». * cents [H,,- a,,ui,re r,«l ; ' ■" '•' la.rt ..u , .,„ f„||,„v., : A |mM, 4 feet wi.le i„ u, ,m a,T„„ it b<,tl, n..M. IH.1, f. t„„t ,„ ,l„.,„eter: a,„u„,| tl,e „ut»i,le, except at the P»t)i, ., .„t, ,„., „,„,„.„ j.„j^ „„j j,^^ J- . s.|iiaie y,i„i, ,„„ke out the l.ill. P"' 9. Kncl. 1. .,«!,, the difference between the . i,eun,fe,-ence ..f -t hat they teueh without overlappi,,,. Tell h1,„ how t' ^ aw h.» en-cle,, how many the-e wi„ U, and how tnucl ^^ ^ Will not be covered by circles. U. A cow i„ tethemi by .. rope 40 feet long to one corner of a .s,,„are enclosure whose side is 20 fe..^ „,;, ,„^ „,,.™;;J ow cannot «„. Find i„ „,„„,,. ^.,,,^ ^,,^ J -J *^ the COW can graze. "vti ^vnlcn wKlth of the track is 4 rods. Find the inner circuntference. 166 ELEMENTARY ARITHMETIC. 150. The Rule for the Volume of a Block. V-. •'•r, <.:> .; > In the ti«,'iir(', siippoHP that the I('ii;,'tli, AC, oi' the block, foot X ;>• the width, A /I, of the hlock, = a foot X \'3 ; I thi' hfi-'ht, AD, of tlir atu :j-4. I»l<)ck, =a toot X .) Now, whaU'vor nitr will derive the hottum .surface of the block from a .s(iuai(' foot, the same rate will derivi' the volume of the layer AE from the cubic foot at li Article 77. But the bottom surface of the block = a .sijuare foot x o-G x l',\. Article l.'5'J. Therefore the volume of the layer AE = a cubic footx 5'6 x 4"3. Ayain, in tlie same way, since AD = a foot x 3 4, therefore the volume of the block = the layer AE y, 3-4, Art. 77. = a cubic foot X .■)•(') x 4'.S X .'M- = a cul)ic foot x 81*872. Hence, the rate by which the Nolume of a block is derived from the unit of volume is the product of the measures of the length, width and height. 151- The Rule for the Volume of a Cylinder. Ijet the radius of the bottom surface of the cylinder = a foot X :2t>. Then the surface of the bottom = a s(]uare foot x ("i'O)- x .'{•UK). Therefore the volume of the \a.yev A liC D = a cubic foot x (2-9)- x 3-1416. And since th( igh i;ylinder = a foot X 3, MENSUKATIUN — VOLUME. 167 iippitHP that i tlic l»li)ck, *)•(; ; * the Ijlock, ■t'3; AD, of thr .tx;{-4. [• rate will jtu surface 11 a stjuarc iv AE from U'tide I'M). 5 '6 X 4 '3. , Art. 77. 1-87l>. is derivnl ires of the er. joni surface toni x.'iUlfi. nyevAfWD ;?U16. yliader J/' thort'for*' tlic volume of th« cylinder = a euhir foot X (2-!))- x :{• i 4 1 C x ;? = a t-uhir foot x 79-U()26, since only the first fi-,'ui(^s are eorrect. Hence tiie rate liy which tii,. volume of the cylinder is derived fr..Mi the unit of volume, is the i)roduct of the measures of the liei«,dit and the surface of its end. I.V4. The Rule for the Surface of a Cylinder. -Tt i.s left here for the student to prove that the Hate hy which the surface of a cylinder is derived from tlie ,s(,uare unit, ik the product of the measures of the heij,'ht and the di.stance around it. I.VJ. Examples solved. (1) How many gaUons will a cistern hold which is 7 feet deep and feet across ! The surface (»f the bottom Solution. a sijuare foot " Therefore the volume of the cistern = a cubic foot X 9 X ;r X 7, = a cubic inch x 6.5 x ;r x 1 7l>,^. ,, (J.'i X 7T X I72S = a wilUm X 277274 ' = a gallon x l2;J3-4, as .seen l)elow =-- ."3 X ;} X n. Article 33. 3-141G G3 94248 1H8496 197-9208 1728 158.3300)4 3958411) 13854456 1979208 34200H424 277-274 ) 342007 ( 1233-4 277274 64723 55455 9268 8318 950 832 118 111 168 ELEMENTARY AlllTHMETIC. •J I, Only the first 6 figures of the product are used for the di\ i- dend, and in dividing we check off the figures of the divisor, instetul of bringing down the figures of the dividend. iJut in nudtiplying the divisor by the figure in the (juotient, w^e "carry" from the figure checked off. (2) How many feet of lumber will it take to make a closed box 4 feet long, 3 feet 6 inches wide, and 2 feet 8 inclies high, out of boards 1 inch thick ? Solution. — If the box were a solitl block of wood, its volume would = a cubic inch x 48 x 42 x 32, Article LJU. and the volume of the inside of the box = a cubic inch- 6x40x30. Therefore the volume of the boards = a cubic inch x (48 x 42 x 32 - 4() x 40 x 30). But a foot of lumber = a cubic inch x 144. Therefore the lumber in the box 48x42x32-46x40x30 = a foot of lumber x (> = a foot of lumber x (1 4 x 32 2 = a foot of lumber x 64 " . o 144 46 X 5 X 5 3 ■). EXERCISE XL. 1. How many bushels will a bin hold which is 8 feet long, 7 feet wide, and 6 feet deep ? 2. How many bushels of corn will a waggon box hold whose length is 14 feet, width 3 feet 4 inches, and depth 14 inclies? 3. How many gallons of water will a cistern 6 feet deep and 5 feet in diameter hold ? 4. How many feet of lumber can be cut from a log 16 feet long and 26 inches in diameter, if in sawing I of it is used up in sawdust, slabs and edgings ? ake a ckispi Article 15U. loji; 16 feet MExNSURATION— VOLUME. 169 5. H 6 feet ? ow m any cords of stone in a pile 50 feet hy 30 feet by 0. H.vi"';--J the cost, at 75 cents per square foot, of polishi„. a marble column U feet high and 10 inches in diameter ^ 7. A fanner's roller is 9 feet long nnd 3 feet in dian.eter How .■ wd the farn.er drive to roll 10 acres, not reckoning tne turninirs ? " 8. m, water which fell during a shower upon the n>of of a house 6 feet long any a wmd-pun,p, which makes a stroke every 5 seconds, and which discharges a volume of water equal to that of a cylinder 4 inches in diameter and 5 inches long at every stroke How Ion IS it in filliRij 170 ELEMENTARY ARITHMETIC. 154. The Rule for the Volume of a Pyramid. if- c;:j li If (ino corner yl, of a cube he joiiied Kio. Flu. 3. to all the other corners, as in the fli;i j;rani, it will he seen, on ob.servation, that the cube consists of three eciuui pyramids, one of which is shown iit Fig. -2. Now, the volume of the cube = a cubic inch X the measure of the base x the measure of its heiylit. Therefore the volume of the pyraniiil = a cubic inch x the measure of the base the measure of its height Next, suppose the pyi'amid to !>. made up of very thin layers piled upon the base BCDE, and then carefully tilted till the vertex A is over the centre of the base, as in Fig. 3. Then the volume is the same as before. Finally, suppose each layer to swell or shrink in thickness, each by tlie same I'ate, so that the pyramid lie comes as it is in Fig. 4 ; then, what ever rate will derive the height in Fig. 1, from the height in Fig. 3, the same rate will derive the volunu; in Fig. 4 from the volume in Fig. .3. Now, the volume in Fig. 3 = a cubic inch x the measure of the base the measure of the height in Fig. 3 Therefore the volume in Fig. 4 = a cubic inch x the measure of the base the measure of the height in Fig. 4 X — Fio. 4. amid- IS in the dift observatidii, fc' three equal is shown iii e cube the measuic of its height. the pyramid ire of the base height 'ramid to li. rs piled upnu len carefully is over the ?\g. 3. T1h>ii ■i before, ayer to swell eacli by the pyramid lie ; then, whal le height in in Fig. 3, the le volume \u 1 Fig. 3. re of the base j;ht in Fig. .'! n Fig. 4 re of the base ght in Fig. 4 MENSURATION— VOLUME. 17] Hence the rate by which the volun.e of a pvra.nid is deri^ed from the cubjc unit is the product of the ni.sure of t 1 b s surface and the measure of one-third the height. 155. The Rule for the Volume of a Cone. If now up..n he base of the py.unid, in F,gu.. ,, ^^,1 j!*' ^ descr.be the largest circle possible, an.l join Us circu„.^.rel a all ^c^ts to the vertex ., we shall thus fonn a cone, as :n \^ It will be seen, also, that whatever rate will derive the surface of the circle from the surface of the square BCDE, the same rate will derive the volume of the cone from the volume of the pyramid. Now, the surface of the circle = the surface of the si.uare x -- Therefore the volume of the cone = the volume of the i)yramid x 7t -» cubic inch X ^'!^'i}!!a^»:^^f ^I^lieight 3 X the measure of the square base x '^, Art. 1.^. = a cubic inch x ^jjgjgg,^'^"'-^ of^hejieight 3 x the measure of the circulai- base. So then the rule for the vohnne of a cone is the same as th.t for the volume of a pyramid. 172 ELEMENTARY ARITHMETIC. 150. The Rule for the Curved Surface of a Cone -.J If a paper l)e carefully fitted abt)ut a cone so tliat the erlges are together along the Hue A/i, and the i)aper then unrolled, wo sliall find it to be of the shape A/iD, which is a part (sector) of a circle. It is left here for the .student to show, hy putting together thin tri- angles to fill up the space between A li and AD, as in Article 148, that The surface of the sector of a circle a s(juare inch x the X the measure of half the radius^ measure of the arc BD. But the arc BD = the circumference of the base of the cone, and the radius J/j -= the slant height of the cone. Therefore the curved surface of a cone = a scjuare inch x the measure of lialf the slant hei<'iit X the measure around the base. 151. Example solved.— A pile of wheat on the barn fhmr is the shape of a cone; it is 12 feet in diameter, and .3 feet ;; inches high, liecjuired the number of bushels in it. Solution. — The pile of the wheat •J (J = a cubic inch x 72 x 72 x 3-Ul(j x --, Article 15:>. and the pile of a bushel = a cubic inch x 2218. Therefore the pile of wheat = a bushel x I^iilil^^-ll^^-^ 2218x3 72x72x;M416x13 = a bushel x = a bushel X 95 151. 2218 MENSURATION — VOLUME. EXERCISE XLI. 173 1. A pyramid Ims a .s.,uaiv base whose side is lT, im-h^s • if the volun.^ is 1000 cubic i.u-hes, Hnd its heifrht. 2. How no^ny quarts will a vessel in the form ..f a.n in^erted ^•one hold, if it is 2 feet deep and 1 foot across the top ' ^ •■5. How ,nuch tin will it take t<. n.ake 'he vessel in ,,uestion -, with a circulai- cover, not allowing for seams ? 4. How n,any cubic feet in a pile of sand shaped like a .one ■>U feet across the base and 20 feet lii^di ? 5. Find the volume of the H.^^ure forn,e,l bv setting- sm.ure pyramuls, r> inches hi^h, on the si.les of a cubV whose ed'e is ( inches. 0. How many s.piare yards of canvas will be re(,uired t.> .nake a conical tent, '20 feet in dian.eter and Hi feet in slant iieight? 7. Omit the word slant, and read question fi. <^. A piece of lead, ]n inches by 1- inches by 2 inches, is moulded into a cone, whose l,ase is G inches in dian,eter. Find the height of the cone. 0. How many bu.shels of peas in a conical pile 10 feet in diameter and 2 feet high ? 10. How many cords of stone in a pyramidal pile which is - teet high, and covers a square piece of ground whose side is 1 > reet ? I5H. The Rules for the Sphere and Triangle— The following rules we give here, without proof. (a) The surface of a sphere is /our times the surface of a I ircle ot the same diameter. (A) The volume of a sphere - the volume of a cube into which the sphere fits x 6" 174 ELEMENTARY ARITHMETIC. —-J ((•) f.ct til.' measures of the three sides of a triangle be called a, h and r ; juid let s = .', x {a + b-\. c). Then the surface of the triangle = the unit sui-face x V'.s- x (.v -^)~><~(7'- b) x (7^7). Examplo. — Vmd the surface of a triangle whose sides are ;5 :, inches, 4-;{ inches and Tj-O indies. Solutinn.-Win'e .v = ;, x (;{-5 + [•?, + n-G) = Gw. Tl leref ore .s - » = ;j -2, s - i = -1 •[, s - c =- 1 • 1 . Thcrefoiv the surface of the trianirle = a square inch x \/G-7 x .'}'2 x2-4x 1 -l, = a S(|uare inch x \/oG-G01G, — a sijuare inch x 7'52."54. EXERCISE XLII. 1. A triangular field has its sides G5 rods, 70 rods and 7:. rods. Find its surface in acres. 2. How far is the longest side of the field in No. 1 from th.- opposite corner .' ■•3. Find the weight of a cannon hall G inches in diameter, if iron weighs 7^ times as heavy as water. 4. Find the surface of a triangle wliose sides are -05 inches, •12 inches and -l.'l inches. -■). I'^ind the surface of an equilateral triangle whose side is 30 inches. T). A sphei-e just fits into a hollow closed cylinder whose diain eler and height are each IG inches. How much space is occuj)it«d by air in the cylinder ? 7. Find the diameter of a sphere whose volume is 1000 cubic inches. .^. rf the diameter of one sphere A = the diameter of another sphere B x ."., find what rate will derive (rt) The surface of A from th(> surface of Ji : (//) The \olume of A from the volume of />'. £ 4. MENSURATION— VOLUME. y. rf tlio volume <.f the sphere .1 =the volume cf 175 th e sjiliere /ix(H, find what rate will (U'rive ('0 The diameter of A from the diameter of />' .• {f>) The siii'face of A from tlie surface of /i. 10. If the surface <,f the sphere ^ == the surface of the sphere /' X -', hiid what rate will (l(>ri\-,. (n) The diameter of A fi-om the diameter of // .• {/>) The voluuie of A from the volume of /I. 11. The sides of a ri.yht-anol,.,! trian-le are ,S0 inches and l.->0 •nehes. Innd the distance of the ri^ht an^de fron, the hypothe- 12. A ship 8ails east nO miles, and then north lU miles Tn the meantnne another ship, startin.i^ from the same port, sails ^v^st 2n nnles, then south 12 miles, and then east If. mih-s lUnv far are tlie si ips now apart '. l-l Find the surface of a re-ular he.xa-.m whose shle is inches. 11. By heating a block of metal its length, width and thick- ness are each increased by the rate -00028. iJv what rat,> is the volume increased i lo Find the surface of a triangle whos<. sides are 245 feet -'4() feet and ."? feet. 10. Three e.,ual circles, whose diameters are .30 inches, touch one another. Find the triangular space between them. 17 A cylinder is 21 inches in diameter and 24 inches Ion... T^.nd the length of a threa,! which passe, spirally once around it one end ,,,,. ,, ,,,. ,i,,,,,f,,.,^,,. ..^ ^,,^. ,^^^^^ ;^^^j ^,^^ ^^^^^^^ ' that i4 the top. 176 ELEMENTARY ARITHMETIC. CHAPTER XVI, It <:r> -.J THE METRIC UNITS. I5». The irre^^iilar system of ineasuriii- Len-tli, Surface Volume, and Wei,i,H,t ..r Mass, which is in use in En^rlisi,.speak- ing countries, and wliich we described in Chapter XT causes much unnecessary labor in addition, subtraction, multiplication .and diMsion. To aAoid this unnecessary labor, the French and other peoples of Eun.pe ,iow use the regular system of measuring these quantities, which we described in Chapters 1. and VI f The only difference is in the language used to describe the system. I«0. The Metre. -In English-speaking countries the stand- ard unit of length is the yard, and the student is supposed to have a nearly correct idea of its length. The French unit of length ,s the Mefr^-, which is derived from our vard by the rate I-09.3G;1:}; thatis, ' ^ a metre = a yard x 1 -Oe.'iri.-i'}, or a metre = an inch x .S9-;?7079. Now, the unit is their metre, a multiple unit of the 1st order is their DekamHre, a multiple unit of the 2nd or, we had ch.«sen the metre a< the unit of length, then the whole line or distance wouW consist THE METRIC t S {TS. 177 the length of the hue = a metre x 7m it IS also evident that tl'Hength of the lme:= a decimetre X 7584, «>»■ = a Kilometre x 0007584. Tl r., "''"'"''■ "■''""" "'''*• '« " Dek""ietre. Tin, unit i, "t the l.,t, J„,l and .-Ird „„Jera are culled the Dek-i,„ ,1, h . a.e.nd the Kil„,.e; while the ,ul,.„nit, If I ,,t ' i, ij'r'; We may say, then, tint the surface of a field = an are x 3476, or = a centiarex3476, or = a square Dekametre x 34 7G, or = a square metre x 3476, and so on. • l^^ 7^^ Litre. The standard unit of volume in France IS he volume of a cube whose edge is a decimetre. This unit il called the /..... As in the others, the n.ultiple units of the Is ..d am 3rd orders are called the Dekalitre, the Hectolitre anci the KUohtre ; whde the sul>-units of the 1st, 2nd and 3rd onier are calle.l the decilitre, the centilitre and the millilitre iJut It ,s as convenient and more expressive to say a cubic n.etre a culiic decimetre, a cubic millimetre, and so on. ;, '^***-.yj^e Gram. The standard unit of weight (or mass) IS a wtMght uhivh :h as lu-avy as a cubic centimetre of water measured just as it begins to expand in freezing. This unit is 178 ELEMENTARY AlUTHMETIC. •:r" .l , .- "'"■ '''"'^'"•^'""s are U.)0 Un,., .5 _> ,!,„., ,u,.i _'■;, ni. ineuHuml intn mhIIv. 4. Ki.uJ, in aies, the aivu of a circ-le wlu.so diam/is 4000 nu-tros •^. How ,„any cubic centiMietros of .opj.er a.v the.v . i,,. a Ivilcnetir lon.i, and ■:! cntirnetivs in .lianieter > ^>- TlH- ciivulai- shaft of a n.ino is 5 niotn-s in .lianu-tor ami I 1 ^aoincfes ,h.p. How ...any cnl.ic .ncties of ea..!. an.l roll li.i\e l)eci. oxiavatcd to ...akc it .' 7. If the .lian.eter of the earth at the e<,uat<,r is SOOO miles '-V n.a„y K,lo„.et.-es is it in circumference at the e.^uator^ ' ^es el 10 ,1.... .„ duuneter and S dm. dee,., if the vessel he full ? y. iMnd tJ.e diagonal of a .-ectangle whose sides are ■>-V, metres and .'M metres. "' ,,.'"■ '';";,',,;'"■ ; "■'■ "' " «"■«'•• ^'■'■- -•,•«,„„■..,.„,.,. ,, •j'JJi /b4 millimetres. eara tuM that.the leng.l, „f a line .l/;=a ,„rt,ex 370M-4 • hen we re««rd tl.e „,„„l,e,. 3700M4 as ahbreviate,! i„,tructio'„; I'l Z" 7 ';""" " ""■"■'^ '" ""*' "'> "- "-'S"' "f ^"' HI tull, these instructiuns are : JO Place a metre down ;3 times i„ the same straight line without missing or overlapping. JIP"' '^.■'"'"■' '"'' '^^^'1""' P"'-*^'-"^ place one of the pa t down . times, as before, in this same straight line e) Cut one of the parts placed down in (i) into "lO equal parts, aiKl place one of them down G tin.es, as before. (d) Cut one of the parts i.laoed d iown i parts, and place one of them do\vn 9 times, as befor 11 (c) into 10 equal IMAGE EVALUATION TEST TARGET (MT-3) 1.0 I.I |50 '™^^ ^ 1^ M 2.2 2.0 11:25 i 1.4 m 1.6 rlluiOgiapniL; Sciences Corporation 23 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) 872-4503 r '^y^^^ ^ ,^ ^'^^ 4^^y^ Q. C/a 180 ELEMENTARY AUITHMETIC. I ,31 J.; Sic: («) Cut one of the parts placed down iti (d) into 10 v([un\ parts, and place one of them down 5 times, as before; and soon. If these instructions he carried out, it will he found that one of tlie parts placed down in (e) is about as long as the thickness of ordinary i)ap(>r. Hence, the instructions given by the last two figures, L> and 4, may, or must, l)e disregarded. In ordinary cases, then, a rate with ;j figures v, ill give instructions fur deriving a ({uantity fron.' the unit sufficiently accurate. I««. Unnecessary Labor in Multiplication— If, then, we have to find a single rate eciuivalent to ;i7G9r)24 x 4021830, it is clear that the G figures at the left in the product are all that are needed, and that the labor in multiplying to get the figures after the Gth is unnecessary. Thus (see Article 48) : Since the order of 3 in the 1st number is 0, and the order of 4 in the 2nd number is + 1, tlierefore the order of 3x4 is + 1, so that 3 X 4 will be written 120. Again, the order of 4 in the first number is - G, and the ordei- of G in the 2nd number is - 5, so that the order of 4 x G, or 24, is in the order - 11. Therefore the whole product of the tw.i numbers extends from the order +2 to the order -11, and consists, conseciuently of 14 figures. Of these, the G at the left are needed and the other S are not. HJl. To Find the First Six Figures <.f 37G9524 x 402183G. The complete product is found by multiplying every figure in the 1st number by every figure in the 2nd, setting the products down in theii- proper orders, and adding these pnjducts. Now, since the product of the first figures of these nuinbers is 12 in the order + 1, that is, 120, therefore the first six figures of the whole product extends as far as 120000, that is, to the ordei- - 3. Hence, we may omit all those products which are in the orders -5, - G, -7, etc.; but we must multiply those figures together which give a product in the order - 4, for this product may have a figure in the order - 3. ntu 10 t'(|Uiil (J ; ciiid Hu oil. uikI tliat uTif the thickness 1 by the last Tn ordinjuv tructiuns fur ite. n.— If, then, X 4021830, it are all that 't the figures lie order of 4 X 4 is + 1 , so TUi the order 4 X G, or '21, • of the t\V(; T - 11, and G at tlie left 3769524 X plying e%ery , setting the ^se products, iinhers is \'2 gure.s of the o the order are in the hose figures this product CONTUACTED MULTIPLICATION 181 Again, since the order of 4 in the 2nd nund,er is + 1 and the order of 2 .n the l.st nun.ber is - 5, therefore 2 x 4 !,' 8 in the product, is in the order - 4 Tn fl. that .5x0. 9X2, 6xr..s ;.d 5x3 '"'^"T'' T ""^ '^'"^^ us then write : ^ "" '" '^'' ""'^'- " '■ Let i of tiie 2nd number under 2 of the 1st, of the 2nd number under 5 «.f the 1st' -' of the 2nd numi)er under 9 of the 1st,' and .so on. 3769(024 63812 04 Let us also draw a vertical line between the orders - 3 and - 4 of the upper nund.er. This is Hhown at the left. Then the product of each figure and the one abo^•e it is of the order - 4 and -dl tU } [ wluch fall to the right of the vertical 1^:^,^^^^"^ the manner of multiplying J ot, omitted, ^ow by 4 is : 2 X 4 = 8 in the order - 4 = 1 in the order - 3 (nearl v) 0x4+ 1=21 in the order -3 ^'''^''}h |;x-^ + -' = -5^mtheorder-2',andsoon- by 2 IS : 9 X 2 = 18 in the order - 4 = 2 in the order - 3 0x2 + 2= 14 in the order -3 .n.l • ,'f\'"^^^'^"'*-^'''''^""--^"idsoon; dnd smularly by I, 8 and 3. This work is then seen as below : If the last figure of a product which is in the order 4 is 5, 6, 7, 8 or 9, 1 is added to the hgure of this product which is in the order - 3 • l^ut if the last figure be 0, 1, 2, 3 or 4 1 \ not added. oi f, i ..^ It will be observed, also, that the multiplier is written backwards, and that the units figm-e in 151604 :;: :|r -- ^^--^-' Hne and to the 3739 524 63812 04 182 ELEMENTARY ARtTHMETIr. IC8. Examples solved. (1) To obtain the first five figums in 6o;i20l79 x 6;]|S296. ■Solutvm. -Hinca the order of .3 in the 1st ninn))cr is - 2, an) VO-'57694x 005218.349. («) 4-2135x62-3147 ('•) 0185132x000074.586.3. 4. Multiply 64.3.582x2576.39 to the order -1. 5. Multiply 12345678 by 12345678 to the order - 4. 6. Multiply 00006543219 by 5437692 to the order -3. 7. FimI the product of 783285961 and 0000008356923 to the order - 6. 184 ELEMENTARY ARITHMETIC. m : 8. Find the prcxluct of 00009037402 and 123452U7 to the order - I. 9. Obtain the first o figures of 30079421 x 0084321. 10. Obtain the first 6 figures of 0015700834 x 0217006894. 11. Explain fully how to obtain the first 6 figures of 03125679 ■< 123460072. 12. Find to 5 figures the surface of a rectangle whose len-th = a metre X 300-4279 and whose width = a metre x -008214687! 13. Obtain 5 figures of 603-5721 x 2-34786. 14. Obtain (1-045555. . .)« to the order -6. 15. The base of a cone is 3-4721 feet in diameter and 5-8234 feet in height. Find in cubic feet its volume to the order - 4. I«». To Obtain Five Figures in the Quotient when •0.3/34216 is divided by 59-216438. Since the order of 5 in the divisor is +1, and the order of 37 .u the dividend is - 3 ; therefore the order of the first figure of the' (juotient is - 4, and the figure itself is 6. Hence, the quotient to ha^•e five figures in it extends as far as 000060000, that is, to the order - 8. Again, the order of the last figure of the quotient is - 8, and the order of the first figure of the divisor is + 1, therefore the order of the last figure of the dividend which we need is -8 + 1, that is, -7. Hence, if we draw a vertical line between the orders - 7 and - 8 in the dividend, we may omit all the work which is set down at the right of the line. Further, it will be seen, then, that we need not use the figures of the divisor which are beyond the order - 3, except the one in order - 4, which we carry from in multiplying. The arrange- ment then is ; " CONTRACTKD DIVISION, 185 2147 to the tient when 59-2 164 ) O-Mimo ( -OOOG Tn dividing, each stop shcrton.s the divisor by one figure at the right, which IS checked off. The work then stands ; 5'j-2104 ) ■0-M:n2-2 ( 00063060 .'{5529,s lsT24 17765 359 355 4 In short then, to obtain 5 figures of the quotient onlv 5 figures of the .lisis^r are needed, and a corresponding nunibe; of figures of the dividend; but, in multiplying by the figure in the quotient, we carry from the 6th figure of the divisor. 110. Example solved— Divide 30o72438 by 47321 935 to 5 figures. Solntion.-Hince the order of 30 in the dividend is +1, and ot 4 in the divisor is +3, therefore the oi.ler of the first figure of tlie quotient is - 2. We need to use 5 figures of the diWsor and 6 of the dividend, the others are checked off, and the work tlien IS as follows : 4732-1 93r, ) .305-7243cS ( -064601 283931 21793 18928 2865 • 2839 26 47 Here 26 is more than half of 47; then we put 1 for the last figure of the quotient. 186 11 if** lw.4 P tj-; J ELKMEXTAHV AIUTHMETIC EXERCISE XLV. 1. Divide to 5 figures : (a) ll.';U-r)67Hby ;]4-2ir)96. (,l) 2o by .'^•141502(1. (/>) -00320718 by 8-r)7()92;n. ('') 1 by 3-1416. ii)i l.N by ^•;)7 {'■} -000415238 by -,-3164197. 2. Divide 300-215 by 12345-6789 to 6 figures. „ ^, , . 23-56421 X 51-315214 9-35284 X 2-9653721 "gu^es. 4. Divide 304-56 by 1-0422222. ... to 5 figures. 5. Obtain 6 figures of ^~~ ^ (1-0333...)-"- r 7. Obtain 8 figures of 3 -j- 2-78287828. 171. Miscellaneous Exercise.— We give here, in con elusion, an exercise which, we believe, will be a guide to the teacher in teaching, and a help to the student in reviewing the subject ; but it must not be regarded as furnishing a complete list of questions. EXERCISE XLVI. 1. Describe how to count toothpicks to obtain the number 2357. 2. If a carpet tack be the unit, how do you make a multiple unit of the 3rd order? 3. How is a multiple unit of the 5th order made from multiple units of the 3rd order 1 Express the manner in one sentence. 4. Tell in detail the information given by the number 1435 when we are told that the matches in a box = a match x 1 435. AVhat does the " x " denote here ? 5. Describe the operation denoted l,y the line in a pile of pebbles one pebble •MtSCELLAXEOL'S EXEllClSK. 6. Read in full Enj-Iish the sentences : 187 (") A pile of j)ol)l)l(.s one i)<'l)l)l(> = :\2: (/>) A pile of {.obblcs^,,,,,. pol)l)lex325. r. Why is a ninnber cullcl a rate, an.l whv is it rallori the measure of a (juantity ? N. Kxj.lain how you j.ut together a nmtoh x ;{-'0 an.l a match X 4<0 so as to obtain a match x HOo, Kxplain how you take a grain of wheat x 49.3 from a grain of wheat xO.-U, to find that the quantity left is a grain of wheat 10. How do you subtract U93 from olH2 ? 11. Of what use is .Subtraction and Addition? }Iow ,loes Addition save labor ? ^ 1:^. Make a diagram having 49 spaces arranged in 7 rows and < columns. In these spaces write numbers of 5 figures each, so that the numbers are all different, the figures in each number are different, and the figures 1, 2, 3, and do not occur. Find the sums of the rows and columns; then add the sums of th. rows and the sums of the columns. The two totals should agree 13. Repeat No. 12, with mor of a digit in a number? Write a scheme which shows the orders of all the digits of a number. 15. In the number 1976043U what are the orders of 43, of 7 of4314, of 19, of760, andof604? 16. What is the use of the point above the (J in '<4;{->7(iU dollars ( " 17. What is the "Rule of Order" in Multiplieati IS It proved 1 ion, and how 188 ELKM ENTA II V A RI'm M KTIC. 1.^. Multiply .-U-'l l)y Sd.U, usin- S first, (I m-xt, ;{ „ext, hi..! I next I!>. Multiply ;{|21- hy S(;;{4, '^oiw^ ii iiuihImts. I re verso iirder in l)Otll 20. Multiply lL>;Mr,r,7,S«J l,y USror, |:5i>l, ami then DS7G.M;{l'| »y |-J;{|r)(;7S!>, and see if your i»ro(lucts a<,'reo. M. Ht'peat No. 20, with modifications, till you make the pro- ducts a-,'ree the first time multiplyinj,' fixe time.s out of six •2-2. Why does 7 x <) = {\:] f le si;,'n of Derivation, and when is it th -*.{. When is " x " tl si-fn of Multiplication •2[. Of what use is Multiplication > How does it save lal)or ? '2'). Tell without multiplying how manv R j;ures arc in tln^ you set the single product 9xD m product 4.'?y76x 85714871 -'0. Tn what oi'der will <|uestion No. 2.*) ? 27. The length of a furrow in a field is 438 yards, and the width of a Held = the width of a furrow x 416. Find the whole length of the furn)ws. 28. Describe how to perform the operation denoted by the line in '' i'i^-i^'J'I^'L:£^''*^\^^*''^ " the length of a pencil ' 29. Head the sentence " :^"^^ >''"'*^«- 56 " yards 30. When does the line denote the opei-ation of measuring and when does it denote the ojieration of dividing ? 31. Read"-^*^^=^^' 14 feet 14" 32. What is meant by saying " divide 1 496 hy 8 " ? MISCELLANEOUS EXEHCFSE. 189 •'•■5- What is M,,. ,,„ri....s,. of Divisi,,,, ,„ AnU„„,.tir ' •^j. SI.OW that iU. .nnnl,..,. ,S:-,1:|, js „„,,,. „^. „.,. .5Ji. Alulti,,ly 730s l,y G, 7, Sh.wI !», unci .livi,le the whole ;-'-t I'V h, 7, S .n.l U. usi„, the fi,u,.es i,. succession. Th h/»al (juotient shouhi be 7;{i'S. :jr. Repeat No .5, with .noditieutions, until you nu. ...ul.ipiv with ficcuniey and ruj.idity. ' ' feelisri;:^''''''''^^"'^'''''''^''^"'''^'''^^^ .3,S If a h-ne ^/y . . Ij.,, CZ> x r.^, and a line /^ = the line C Z> X I < ;,, express A II i„ terms of /V/ ^^39.^ Multiply 0.M7!) by S.7G.^ and divide the product by Jan)^r''''"'''*''"''"'^^''-''^^'--^'-^^'^^'+^. ted by the | ^ ^^^ J^^^ -thout nudtiplying the number of figures in 380472 4.3. Multiply 37-042 by 024. 44. Simplif^^ a dollar x-^'"^'^^:i! , 3 next, and nler in both 11 !i.s705i;L'i lake the pro of six. len is it the save labor ? are in the uct 9 X u in "ds, and the 1 the whole measuring •03G:i 45. Simplify .S275xl£'x^ ."3(35 ,S ' 47, Divide 1 by •0375 for 5 iigures, 100 KF.KMKNTARY AHITHMFTIC. <:^ tit l>*. Sim|i|ity (,/) •I7;j: ('>) MS ('•) •00.") :i-2 Uiviii^' III.- iciisoiiiii^r tor tlu' position ,.f the units point in t-iuli I'USC, l!t. Simplify ■20 + r,; + 00 • 603 ().') 50 01) •'»'). Define a tVuclion and a deeinial, and ion ions, ('0 IJt'fliicin^' a t'ract (/') Addin;,' two tVact ('•) Su!)tractin<( one fiaetion f ('0 Multiplying' two t'racti l»rove the rules for nwn another, oils, und {,-) Dividing,' one fraction hy anotl ler, lUid uce I KloJ I 'Ols ^'* '^'^ >ars ht Lountiny years '/ MISCELLANEOUS EXERCISE. 19, *5<^- •Mak,.ou. tlH.f.,l|..vvi,.. I.ill ,.t •.,„„,,. .,.{ , ,, '"•"'"'^•i<'tl' at .*f-.^(). ' - ' '■'• '"'«" '^^ ^"•■•^ . . 1 cents, 1 10 y,,,. ,i„t Ht !». ,.,,,t, v-i V ; " i'- ' '"' '^' ■^'^- «-'. A boy can ,|o ,1 j,itv,. „f w„rk in I-' I '^ ^vorkin. together ' ' ' '"'^' "*" '"^'' ""l"'''^' to -lo G4. Divide iDGOi) anion-^ /I /l .,,,,1 /- .. ^r. share, IS., n,.,. ,,,t of /; i ' "; '"' "- '' ^'' ^'^'"t. of share n..; be e,ual '^ '"■*' '"'•' ''^ I^' -t- of C". 06. How nmci. water must be aclde.i t<, <)■> ...ii gallon? ^ ''''"^' '" ""^^^ " '"-ture worth .S;5-G0 per «^.Findtheeostof nr^n^ :77^^^^'^'--^^-^- thousand. ^'^'^^ "^ '"'"'^^••. at .$17-2o per 69. A man's salary is i^Udo uer v<..... * - years he saved !. > , 1 ,',.^7 ,'''", ^'^'" '' >^^^'-«- ^n these -unting iuteresC 't^^ ^ Uif T :^'''' '-P-tively. Not years'/ ^'" '"'^ ^•^^'^^ ^^^'"fe's durin-. these r. 192 ELEMENTARY ARITHMETIC. I CD fa 70. Cliiin,ij[e £194 ISs. ;}(/. to Ciumdian nioncy, w-lioii £1 --■ 71. How often is 6 yards 2 feet, contained in 25 furlongs? 72. A man has 5 tons G cwt. of flour ; after selling 25 barrels of it, how many sacks, eacli holding 150 ll)s., can be filled with the remainder? 7.'{. A man lias 703 acres 142 S(juare rods 14| scjuare yards (jf land. He sold 19 acres 70 stjuare rods 2| s(juare yards. Hf then divided the remainder among his sons, giving each 45 acres 100 square rods 25 square yards. How many sons had he? 74. A person hired !^500 on April 10th, and on June 22nd he paid his debt with !S510-20. At what rate was he charged interest ? 75. For what time will the interest of i|30441 be «!221010, if, at the same rate, the interest of $24944*10 for 1 year 15 days is 8-i596-92 ? Also, what is the rate of interest ? 70. Calculate the interest of ii?9348-5r) from January 9tli, 189G, to September 18th, 1896, at 7f per cent. 77. On March 23rd a bank gave me |845 for a note of $860, charging discount at 8 per cent. When was the note due ? 78. On January 1st, 1897, a person borrowed #2417'50 at 6^' per cent, simple interest, promising to pay his debt as soon as it am. in ted to $2582*50. On what day did the loan expire? 79. Find the proceeds of a note for $1 389*25, drawn on Mii\- 8th, 1897, for 4 months, and discounted on July 21st, at S per cent. 80. $3420^"fjV Ottawa, September 9th, 1897. Nine months after date I promise to pay A. Ji., or order, the sum of Three Thousand Four Hundred and Twenty Dollars (.*3420), with interest at 6 per cent, per aqnuui, value received. W ng 25 barrels be filled with MISCELLANEOTTS EXERCISE. 193 The above note was sold on December 18th 1«97 at 7 per cent, discount. Find what was paid for it. ' Ma'v'lO^"" h'T T ■""' '" '^''-'^^ "'^^^ ^' '"-^^ «"^ f^'r on ^la> 19 so that when ,t is immediately sold it may yield $160 discount being at the rate of 8 per cent. ? ^ > ^ ^^^^, PavVn;'.o^'"'f "■ '''' ^''''-^" ^^-^"^-^^ I P-"»- to ^ceived ' "'""^' '^^ ''^ ''''' "' ^ P^'- -"^•' -'- This note was endorsed as follows : January 2.3rd, 1897. Received .$400. P Q August 20th, 1897. Receive^- 500. P Q ^^ hat was due on the note December 1st, i897 ? H.^ Find the accrued interest on a loan of .$600 at the end of 4 years at 6 per cent., convertible yearly. • ^^' r"*i f'' ^^^^^'"^"^^ between the simple and the compound interests of |9902» for -n vears -.f 'M ,... ^ "puuna -J lui _^, yeais, cit .i| per cent, per annum. 80. Explain clearly the distinction between discount, interest and compound interest. 86. A man has the choice of loaning his capital, $10000, for .5 >ears at 7i per cent, per annum compound interest, or at 8 per ^cent. simple interest. Which is the better investment ? S7. Find the accrued compound interest upon .$4530 borrowed Janmu^,6th, 189:5. at 6 per cent., when the debt is paid July «lonn^"-^'' ''""^'^ ''''•" ^* '''^''''' ^^SOO ^^ » P^"- cent., .$1-00 at ,\ per cent., and $1000 at 6 pe- cent. Find hi average rate of interest. 89. A man hired $1200 on May 1st, and paid it back July -oth, w,th rent at 7 per cent, (a) Calculate the rent at 7 per- cent, interest; (h) calculate the rent at 7 per cent, discount. i t J 104 ELEMENTARY ARITHMETIC. ri- : r-, f 3 J. js» ■ • ■ i)0. If tlu' (lifFcrence botwecn tlie simple and tlie coinpuuiid interests of a sum of money hii'ed for -"i years, at (1 per cent., is .|5.'Jsr).")f), what is the sum of money ? !)1. A man put in the bank SIO on the 1st day of each numtli for '^ years. A\'hat should be to his credit at the end of the "> years, if the bank pays ."i per cent, interest, convertible every C) mimths >. 92. Find the ecjuated time of the following del)ts : 8-')00 due January ir)th, .^600 due February 'Jlkh, S'^OO due iMarch l:Uh, and 1^900 due Julv 10th. 93. By selling an article for $10'S0 I gained 20 per cent. How much should I sell it for to gain 16§ per cent. ? 94. T marked my goods to sell at an advance of .30 per cent. of their cost. I sold them, however, at a discount of 10 per cent., and gained |.3-74. Find the cost of the goods. 95. A merchant buys his goods at two successive rates of disccnnit of 20 per cent, and 10 per cent, off the retail price. He gains by so doing $r96. Find the cost price. 96. A barrel of coal oil, containing 30 gallons, was bought at 12^ cents a gallon. In selling, 2 gallons were spilled. The retail price was IGi cents per gallon. Find the rate of gain. 97. 277-274 cubic inches of water weighs 10 lbs. How much will a cubic foot of ice weigh, if, in freezing, water increases in bulk by 10 per cent. ? 98. Ha'" my goods I sold at a gain of 2.5 per cent., a third at again of 20 per cent., and the rest at a gain of 1.5 per cent. Find my average rate of gain. 99. I bought a certain lot oi goods, half of them I marked 30 per cent, above cost, and the other half 20 per cent, above cost. MISCELLANEOUS EXERCISE. 595 In selling,', I ,^,i\v a discount of 10 per cent .II ,. 1 gained $1.3:3.20. Find wl..t thclot co.!t in^ ""'' "'' f e ,..te of ].>i pe, cent., an.l losin,. on the other at the rate of " ' P^-'' '•<'nt. Find my total c^ain or loss. '- '.ite of 101. If inter,>st is at the -.ate of ,S ,„.,• cent vvl.-.f ...l r niiivjf fht. •? ... 4.1 ) ,. ' ^^in., wnat relation nu,st he .? n.onths' cred.t price of an article bear to its cash price that the prices n,ay he e.,uivalent ? 102. Head "discount" instead of "interest" in No. 101 10:5. Brown purchased A of a tin.her linn't for .$40r,4-.^5 and Smith purchased ,'•• of the same property at -i rate -> , hi 1"' ^3 'it -i i.ite ;> per cent. ym u |,,,t du] Smith's part cost him, and how much of the property remains unsold? «37od'i.f' ^'' T^ f/"^'^'^*^^^' ••e«Pectively, .f2.300, .^2900 and $3.00 m a partnership. At the end of the year their combined capital was eossOwS Fi„d the cr-Jn f ,•>'''''"*" '^^""'^•'"''J • ^ "'" tlie gain of eacli jtartner. and^tltaff ^ 7^^'bute, respectively, #4295-25, ^56I2-:U and . IS ., 6-41 to conduct a business. At the end of a veu- A received ^593-21 as his share of the gain. Find h l^ ^ ,1 others received and the total capital. S4000"an' /Tir"" ' P-t--hip for half a year, contributing § 000 an,l .f 0000, respectively. ., withdraws .*90 and B ISO at ZmmtT: ^^ ^'^ -' "^ the time their capital 107. Tl„. .stook of an insurance company ,ell» at 1.17] an,l pay, ycari,v ,„vi,l,.n,l, at 1 per cent. U ,L b,,.l- --«- », the ^id::;^. z the building How much lumber, not counting the fence posts, lias it taki'n to fence the school-yaru ; 1:^2^ How much wood is there in the pile at the school-liouse. 'i"d Nv hat IS its cost at $:]-62h per cord ' ro.!m Hol'r ''' '' '' '^'^^"" '^ "PP"«^^^ ^---^ "^ ^^'e -'-ol- 1-M. How many gallons does a milk can hold which is •>4 inches m diameter and :]G inches deep? 125. A/iCD is an irregular field of 4 sides so tint J/' ; -i ZhZ °""'"'" "' '' """ '■ '"■" »'"■"- -•"- Find it! I -'7. An even bushel of cjal wei.'hs fifi tt,« tj i- . ■' •■ ^> f-t % 1.. feet „,„ ::;tZ^i, '"'"' "^ ■" avu.,e >,.,.,„ete.. ., ,, fcet and he«l.t 3 feet, i,.ide .ne^^Z 198 ELEMENTARY ARITHMETIC. o p 129. Calculate the cjuantity of wheat (in bushels) in a conical pile Hi feet in diameter and 4 feet G inches hiirh l-"50. A circle is 4321 inches in circumference. Find its diameter t inches wide and 2i inches thick, will be required to build a wall 45 feet long, 17 feet high and 4 feet thick, supposing the mortar to increase the volume of each brick 6] per cent. 139. Find the side of the largest .square stick of timber, that can be sawed from a 30-inch log. 140. A rectangular piece of ground, whose sides are as 2 is to 3, containing 15 acres, is fenced at a cost of 45 cents per rod. Find whole cost. in a conical Find its iclios. Find nic and also )6-624 cubic MISCELLANEOUS EXERCISE. 19f) in. A lu.llosv cylinder 4 feet Icn- has its outside and insi.le diameters .3 feet and 2 feet G inches. Find its wliole surface and its volume. 1 i-2. If iron is 7.' II iron cannon ball I foot in diametc times as heavy as water, find the weidit of 14-'}. A circul sijuare Held containing L'2.'i acr ar race-course, I rods wide, is to be laid out in a ■es, so as to be as long as possible. Find its length measured along its middl 144. Use the method of the last articles to Hud 5 fi (•3141.yj2())-xl-92043. 145. Simplify to 5 figures V 2 x \ 3. 14G. Simplify to figures (1-0345)'*. 'ures of 147. What is a number? 14.S. What is number? 149. AV'hat is a (piantity ? 150. What is (piantity .' 151. Has a number magnitude, so tiiat we may truthfully say that one number is greater than another ? 152. Has a (juantity magnitude? 153. AVhat is Arithmetic? timber, that t ANSWERS. I o i O ! p EXERCISE I. (Pa.;k 14.) 1. Let the student actually perform these operations. 3. (n) "A load of wheat is got (derived) from a bushel of wheat in the way tiiat is told by the number 25," •4. (a) The number of beans in the handful when counted by one bean is (say) 534. EXERCISE II. (Paoe 16.) 1. (a) Five hundred and thirty-two, etc. (h) One hundred ;uid eighty-two thousand, three hundred and fourteen, etc. (d) One billion, two hundred and thirty-four million, five hun- dred and sixty-seven thousand, eight hundred and ninety, etc. 2, (a) 268. (c) 936268. (e) 259234513. (b) 936. (d) 300004002. EXERCISE III. (Page 19.) 1. (a) 65, 120, 714, 5151. (c) 36121. (*) 3369. ((/) 9225544. 2. An apple X 2022. 3. 911456. 4. 80661. EXERCISE TV. (Paoe 23.) 1. 20, 40, 30, 80, 75, 50, 25, 36, 79, 81, 64, 58. 15. 1, 11, 12 and 89. ANSWERS. 201 counted by unrlred and (y) no8L>r)90. 2. {a) 4598. (r/) 870-)r). {^>) ^-^^ (e) 111110101. (c) 73737. {/) 70-'900. 3. (rt) A foot X 102. (,.) A l)ook X 24. (/>) A dollar X 1075. (,^) a grain of .sand x 108922. 4. An apple x 298, o, G times, 74G398. ^- 7 times. 7. a cent x 1273. EXERCISE V. (PA) CXLVIII. (c) DCXCIII. (d) MCDXXXVII. (c) MMCIX. (/) MMMXVI. (a) 1666. (fj) 94. 41-XLT. 45 = XLV. 49 = XLIX. 79 = LXXIX. 83 = LXXXIII. 90 = XC. 95 = XCV. 99 = XCIX. (^•) (9) MIX. (h) CMXCIX. (0 DCCCLXXXVni. (i) DCCLXXVII. (k) CCCLVIII. (0 CDXXI. 980. (d) 119. (e) 144. EXERCISE VI. (Pa(;e 30.) 1. (a) In 4200 the order of 4 is + 3, and of 2 is +2. (b) In 5321761 the order of 5 is + 6, of 7 is +2, etc. (c) 2 is in the order + 8, etc. 2. +5, +4, +.3, +1. 3. (a) 600. (d) 9000000. (g) 13. (j) 200OOOO (b) 50000. (e) 100000. (h) 28000. (k) 12000. (c) 70. (/)4. (/) 14900000. (/) 560. 202 ANSWERS. t:,i 1 i-: 1 ' *•■' 1 ti f ::3 \ R Uii k Ci i tj^ { Sa l4 " -.J • «j i f:) 1 1 t.5 5-3 h «it {iS ]' *jr *t::i •1. {n) +3. (..) +L>. (b) +.■>. (r/) + I. 5. (fi) ;i2000. (f) 1 touoo. {0) 90000. (7) --JOOOOOO. (c) 4- .I (y) +0. (/)0. (A) +7. (e) 1000000. (//) -JOOOOOOO. (/) 7J0000OO. (h) KiOOOO. 0. Ill 4 (a). Since the order of 2 in 1*00 is + 2 and of ."] in 30 is +1, therefore the order of G in the pro(Uict is ( + 1 + 2 or) + .3, so that :200 x 30 - GOOO, and so on. 7. 30, 4862, 30903, 1004. S. 1728, 2004, 1284, 78.-j72, 208704. 9. 6172835, 4938208, 3703701. 11. 22511909, 19295922. 12. 2295702. 14. (a) 11770. (f) 10184. («) 1100005. (/;) 25914. (d) 109020. (/) 2475775. 15. +0, see Artick 28. 8 figures. One more than the order of the left-hand ligure of the product. 16. (a) 39375012. (c) 814150521. ((0 117796978. 18. 490. 21. 3888. (6) 333164. (c) 823543. ((/) 16777210. 26. 2345. 29. 00795 cents. (b) 97495568. 17. 63360. 20. 988. 23. (a) 18995. 24. (a) 3125. {!)) 40656. 25. 19900. 28. 5247957. 10. 8151.30. 13. 5142848. {(/) 2068011. (A) 2625205. (c) 8742416052. 19. 1118. 22. 92715875. (<•) 1397. (l) 387420489. (/) 4753771243. 27. 100322. 30. 220244900 matches. EXERCISE VII. (Paoe 38.) 1. (a) 7000. (b) 90. (f) 30. ((0 60. (e) 600. (/) 5000. (v) 5000000. See Art 35. 2. See Article 36. 87. 3. 48869. ANSWERS. 203 ) +7. I ■JOOOUOOO. I KiOOOO. I of ;j in ;}o . + 2or)-f;]. . 81513(5. 51lL'8lS. 20680 U. ■2&2-)-20o. n the urder 211G():.2. B. 15,^75. i. 120189. 577 124;}. 522. .5000000. Art 35. 4. (a) lOfiOfiOS, 7i;i072, 5.11801, 267102. (b) 12;j7 5. 1760. 7. I matches. 0. (a) C)iU[. {b) 1)512^. (r) 11281. 10. {a} 201. ' {h) 71.3. (c) 8t8o;;. (,/) ami (r) ,in« llio 6. 1200 slimvcs, 11 ludds 8. Oil. same ('/) 9222^. ('•) 107U. {<•) 1010^. {d) 3053;;. 11. .See Article 33. 1^). 11896], 11 13. 100634ilA. 1006^I!IL 75i; (./■) 785;J {'J) 701. («) 1669: if) 8000.' flo» 1. {a) 6. flOOO> EXERCISP] 4> 17 1 89 JO' r M o 7 1 HOOO' 118 11 4 000' 6038 7 I lOOfl- VIII. (I»A(iE 41.) ('•) 1 (e) 4345. (/)1309. / /'\ S7 7ti;t7 (*) -18. (,;) J 032. 2. {«) 222\l-. (c) 85«|^-. e*) ^^Wll- {d) 11031vi;> ^- ('^) -^^'.W^-iSoJIJ^j. (6) .'"f ■^- («) 3. (/.) 3. (,) t. 5. 390. 6. 1178. 7. 729. 8. 102. 0. {a) 582978. (6) 400205. (c) 700403. (^) 7,ViAV (0) 8487-H-. 10. («) \2(S\U\. (^) 32';[}|^;jy. 1 1 8ee Article 33. 13 52763 _^^^ ■ 732 ~'"732- 12. 54 times, 2311 inches. EXERCISE IX. (Paoe 45.) 1. See Article 43. 2. A pound X 6881 13. 3. A $ x 8975. < 04346978. 5. 81397265256. 204 ANKWfeHs. c.i 1)4 Si !■; P. «) \ iiiiiiutf X !»L'i7. \v) A yurd x KiiOr)'?-,);. («) V''»- (r/) UOl). ('') !"• («) 047. ('•) --• . (./■) 00001 A g(illi)ii X L';)8. ('/) A $ X U I is. («) A I'ul.ic foot X 3 trios. ( 8-'6 S'^OO 820000. , - , - (") -I. (/>) -2. (f) +2. (d) -D. (") -•'. (f>) -11. ('•) +8. ('0 -7. (n) +'). {'>) +11. ('0 +7. (-) -2. (y) -t. (/') +2. ('■) +2. (./■) -I. (.'/) +10. (/') -12. ('■) --'. (./■) + 1. (y) -10. (A) +12. (0 +12. U) 0. (k) 0. {/) -7. (t) +2. (./) +2. (A) +16. (0 +1. (0 -± U) -2. (A-) -IG. (0 -1. (/'() +()11. (h) - IG84. (»0 - 1 3 1 0. (M) -722. (w) f 1310. (n) ;. 722. 1. ^ !l> EXERCISE XI. (PA(iE 49.) (c) i2. (e) 056. t'^<>. ((/) 160. (/) 16. ([/) 001. (/i) 00004. :x.'U()r)8. 311. iiioi. he order ~ I , • hikI 003. ■2'j:i. Si'O, 8200 + ()ll. - 1G81. -1310. - 7i>i'. + 1310. •f- 72:1. -■ («) 3Ul:>. (b) 67170<). ('•) 3S70!)!>. (il) 1 1 i 2H:\2. 3. 1(>8. T). H0f<\)ry2 fc't. ANSU'KHS. (/) L'8(;;U).-)(). (9) 181 li. 3. 205 {/i) U313:.',SU(;4. (/) 8H)0i73!t. U) n<)843UU:.".i83. f. A en). if in.li X 873] 131. *i. l63!)77. 1. (a) 04. (/>) 008. ('•) fiOOO. () b2is7r>. (<•) OlIlT). 3. (.0 4-1. I- ('/) 6oso;}070. (/>) 2575(17. T). (a) U"). (/') 033333, (f) 02. (d) 014285. G. 0914401. 1). 249720. 12. 26G070. 15. {'i) 6544297. KXKIiCISK Xli. (I'.AUK 52.) if') («) 7000. (./■) 0008. (!/) 40000. (//) 000004. ('/) 072. ('') 6(i40G25. (./•) 032. (r) 37(;48,5. ('/) 040. (^) 01 II I I. (./') 0090909. () G70S4S. (i) 004. (./) 4000. (k) 20. (/) 0005. (.'/) 0015025. (//) 24I0()|5(;l'5. ('0 +2. ('•) 100239. (./')00l9003u'. (i) 0052031. 0') 00085470. (^•) 000090009. (/) 000 loo 10. 8. 204. 11. 10497. 14. .39434784. (c) 734417. )01. )0004. EXERCISE XITI. (Pa.^k 50.) 1 . 9 = 3x3, 1 = 2 X •' X -^ V •' •)( .) ^ .) .1 ., , rp ^ ', , - ^ - X - ^ -, -4 = 2 X 2 X 2 X 3, and so on. lest your lesults hy imiltiplying a.s,'Hin. 2. 40 = 2 X 23, 7t; = i> X 2 X 1 9, and so on. 206 ANSWERS. «{S X ■). X •) X ;). ((/) -2' X :} ' X o- (e) L'-x.-i''. (y) 7x11x1; I. Let tho student prove liis factors •"). (a) 4 X -2, 4 X 3. (c) 10 x;^, 10x4. Tlie ffreatest divisors are {f>) 1 2 X :?, 12x2. (.) 17. (/)21. ({/) ^. if') 7. (i) I). (I 2. ;5. (/!■) 7. (/) :38. (m) G. (d) .*= (n) 11. (o) 50. (;>) i;i. X.., 8x4. and so on. (.s^) GOO. ■-), 7, 11, l:}, 17, 19, 2:^, 29, :?1, 37, 41, 4.3, 4 ■)n /. ;).i, o! Gl, G7, 71, 73, 79, 8.3, 89, 91 7. 101, 10.3, 107, 109, 11.3, 127, 131, 137, 139, 141, 149, 151 1"'>7, 1G3, 167, 181, 191, 193, 197, 199. •^. I foot, 2 feet, 3 feet, .") feet, 6 feet, 7 feet, 10 feet, 14 feet, 1.") feet, 21 feet, :V) feet, 30 feet, 42 feet, 70 feet, 10.') feet. 9. A If). X 1, a lb. X 2, a ft), x 3, a ft), x 4, a ft., x G. a ft), x 1 1, a ft). X 1 2, a ft). X 33, a ft), x 44, a ft), x G6. "^'- "• 11- No. 12. 5 yards. 1.3. 82.0. 1 4. Cut them into equal parts. EXERCISE XIV. (Paok 59.) ^- (") !-• {<^) »• (e) 111. (y) 40701. (^') 2. {<0 27. (/)37.5. (h) 12345679. 2. An inch x 121. 3. 14 feet, 880. 4. 3085. 5, 13 rods square, 35. EXERCISE XV. (Pa(ik (;2.) 1. («) 200. (6) 360. (c) 3901. (d) 84. X I 1 X 1 X (2. , anfl so on. ■) +. ) 000. ) 15. ') 025. ?, 47, r,:3, 59, 41, 149, 151, feet, 14 feet, '5 feet. , !i 11». X 1 1 , a 13. 825. 40701. 12345679. 3085. ANSWERS. (") 272. (/O .-,S5. (/.) j:,i^-- (./') 102. (/) I. -,00. (/) r,(j,ss:5S|. {) 27720. (,/) 10080. 2or ('/) 219282. (") 198257998530 (e) 4800. 4, 0. 8. 10. 12. 14. 15. (a) 137343405. (A) „5s^G40. (.) 105815505 A foot X 1512. 5. 48 feet, 3 ami 4 turns. 792 feet, 132 lots. 7. 7 ] ;}_ 08590142. a ou.,o , , ,,. , •'• -^'^-'^ ^^wjnds ; .4 1 82 rounds, /> loO rounds, r' 11 7 rounds, J) 108 rounds. Use ruler and compasses. H . s times the lon-er line. 085377incl.es. , ;}_ An inch. ^^4 feet, 62 tin.es. „. 384 inches, 3053 plots 27720 U.S. EXERCISE XVI. (P„;k (iC.) 1.8ee Article 05. ,. 75, .:5,25, .,,,5, .1:;,, .o,531->5 3. A It.. X 1 -4 1 25. 4 : - u . • , ^- ■■>■ •'• 'Set Article 05. 0. -390025. 8. ^ Warm X 1-0. 84. 1. (a) I if') I •^- («) ^. 3. (a) v.. if') MK id) -vy^i. EXERCISE XVII. (V.Kuv. (iS.) (./■) I in) \. {/>) 4. if') in- (e) ll^^ll if) ^^M. iy) ll\^. ih) mi H) I iJ) 'i. i^) ;!^ (0 i;. i<-) ^• (0 1 ■• ii 4 4 r .- 1 It .) 'J j 4 -• iJ) ^^J^M. (/) •lOlliS.HlJ 208 ANSWERS. 4. (a) 60i!. 5. 8ee Article G7. 00 16,\r^. G. 5.2009. 1 1 1. («) 2. (a) 3 EXERCISE XVIir. (Pake 70.) (d) (.') 7.'}6. A metre x .3iJ. 4 X () • .11 .-1 (T* (T(7- !• !■ (9) {h) \l 1 fi:i do- 14 4' (0 0") u :i 5 • .•1 ii .-. HO {ci) :10]ll ii) iriO,:^. (^.) 1 6 00* Tlie -tV A yard i f* s • I 1 ■ property x Jj. 7. /. of the work. 8. „"'•'- 10. 1 ti .1 8 7 1 lOtV-JCT' I'!' yc EXERCISE XrX. (PA.iE 75.) 1. («) 2. (o) 1 ;; 1 o 1 .-l- (/) 15; (^) -^' (,/) -> •)■) (w) 1G2 O.'i (A) 12,V (X-) Ui. (n) U,V^ 4. G. 8. 10. 12. 14. 15. IG. 17. 18. (b) 40. H Til 100 (r) 3) (d) I, (0 ^mi {") ^tA. if) 0. IS inonej' x ./'y e whole work 5. The good s X M j' acres. 7. 31 9. ij 5 cents. 1 04- 6.1 ■r- .18 1 To !'(• 11. J'.s-farm = C"sfi 13. ^, .^28 ; B, MG. arm x Tlie younger $42G8-75; the elder $5976-2-" 140| Ihs., 109^ lbs. ^,$380; ;?, .«G84; C, $399. A, .$700; /.', SI 120; C, $1G00. Father, $35 ; mother, $25 19. A, $240; B, $187-50; C, $150; D, $200 a Sim, $15; a daughter, $10. 6 •' ■*•**- " 1 li .-1 4 3 ■ 1 (0 :;V- ^ yard x j\. 8 1t_ J MO" (m) IG21 (^') ^toVt. [ w J 0" arm x :hter, SIO. 1. 4. 7. 10. 1.3. IG. 22. 2-0. _£flH8 ANSWERS, EXERCISE XX. (Pa.jk 77.) 209 2. 1 1 1^ •-'4 ()• ■i. 4^ pound 1 1 I ti . il I •-• L- 4 I O u • 0. A owt. X .'..^ -'^. A.£xi'i 1 1 . #00000. 14. .14' fi. ■"■»■'■> III .■iti : .y M o • I .! • 18. no. J iT- 2-44. 149008. 1-2490999. A,$-2rm; /y, .?2300; C, .92050 • D ■i^^ ll.s, Ofj Ihs., 4.3,1, ]hs ' ' 9. 9. 12. -2.347770. 15. 119. 21. 31022 III. i^l750; TS-, .fl400. .«< !) 1 ,S; .'0. A, 144 cents; J], 3.51 cents; C, 1 50 centh 1. 4. 7. 10. 12. 15. IS. 20. 23. .f!ll-.3G. .9210. ' day; ; day,' ays. EXERCISE XXI. (P.xcK 8:{.) 2. 3 1 4, V feet. 5. G2;''Li •3. .f2-24. 1 lOH- 8. .$03. 11. A, 15 d 0. 1; 9. 4 minutes y\ (iavs. ') ( 1-3. B, 100 IVS. 1 !■ The 17id 7«'.) 10. !23-4; merchant owes 80 cts. 21. 24 d iiys. lours. lys ; /i, 22^, d seconds. ] 4. •"). ■< ,] 17. #09-08. 19. $2-20. ivs. 1 0' 24. 3.1 I ays. 'lours. 00 rs. or. J7 hou 5. 40 cents. 1. 4. G. 8. 10. 12. 15. 18. 41880(7. EXERCISE XXII. (P voF, 94.) 80 )o: 30751875 .square inche," 3. £1211 ; !)_ i MONO 5. 44425044 M ■ acres. 14 U54 square inches. 7 ;! (I v'iKfo'.j square rods. s',{■-:>,. !•'{. $;{.•! I -G9. (e) $15;?-7G. (c) $.38-79. (d) $184-20. (/>) $49-4G. (.) S,S5-G5. (,/) .^^s-..;, 11. $200. 10. S1014-G1. In I EXERCrSK XXVI. (V^uv. l IllOIltllS. " ''/Vi IllOIltllS. ■"'• -'illy 21st, $.'30G2-84. 15.) 2. .Alai-di 27tli. 4. X()veii]l)ei' 1st. G. X(,vciiil)er I2tli. $.*{ 177-50. KXERCrSE XXVTT. (Puik 119.) 4. $20. $'")l80-22. ■^("Ki per cent. Loss $50-79. •'•'/i per (•<'iit. 27-1 per t-ent. •'^G-85 cents. $.'J2G-;?2. ^'.11 I""'' cent. ill."; iwr cent. loss. 2. $9G. •5. $200-51^. 8. Gj per cent. 11. 36 cents. 14. .>$8-7.5. 17. .'^.j'j- per cent. 20. 22 per cent. 2.3. 52i cents. 26. $1G00. 29. .$.356-15. •i. 2;) per cent. G. $17. 9. $6400. 12. .$64-26]. 15. $.375. 18. 2:-;L' per cent. 21. $7 2.3. 24. .$925f 26. 27. 62^, i)or cent. •30. $29-.37. 7. 8. 9. KXERCISE XXVIII. (P^.^k 124.) ^'. P>^ ; /y, $80 ; C, $9.3.1 /A $200-20 ; C, $286. ^. '^■tG85;; //, $3748^; C, 2905J. '•3^) months. /I, ■$24-.39; ^,.$51-22; C, $24-.39 ^, $|.31'GG ; n, $55-59. ^MlO.30-44; 7A $1.37.3-91 ; C, $1545^65 •5. $1480. G. $2469-57, 212 ANSWERS. ?]XERCISE XXTX. (Pm;k 127.) 1. (a) .^r»9-:?f]. (h) .SG9-89. (c) .$;M 2. (y ! pei' cent. 0. No clianye. ; 11. Tlie latter by $57-72. I-"'. $1278-30. I'y. $1440. 2. $5000 stock or 50 shares. 4. $210-25. T). 101'. 8. $92/,. 10." 120.' 1 2. $25 decrease. 14. $:}97-50. IG. 15;5v. EXERCISE XXXT. (Paok ];U.) 1- -Sieo-ys. 2. $iio-;is. ^ $778i-4s 4. $156-G;5. 5. $8-1. EXERCISE XXXII. (Paof. 1.%.) 1. (a) .*10-25. 2. $178-1-25. 5. $5101-89. {/>) $97-12^. 3. $1589.3'3.3. G. $3000. ('■) $70-31]. 4. $1920. 7 .1 -2 .-■ 1 M li ■ 8. $29500, $19750. 9. {J. i^n- cont. 10. $lil9-82,i EXERCISE XXXIII. (PA.iK i:?8.) 1. 27 mills, $320. 2. $2000. 3. $200-^50 4. $29999-39. 5. 10 mills. 0. $9-90. "• 5 mills. s. $825 •8:?. 9. $2471-22. 10. 19750. ANSWERS. 213 wt of wood. V of tea. •00 shares. ■4S. to. 9-82,^. 10. 00 EXPJRCISE XXX I Y. (n) 1600. (/>) 490000. (r) 01000000. (d) 90000000000 (e) 14100000000, (a) 8. (Paoe 14;J.) (/)"'<^- (/?■) 40000. (.'/) -000049. (/) -000004 (/O •00(34. (;;j) -oi. (0 -0000000009. (h) -81. (/>) 80. (e) 900. («J) 30. (e) 11000. («) 24. (6) 55-901 (c) 125. ((/) 903. (e) 829. (,/) -000001. (/) ■■■I {y) -08. (A) -002 •01. Oil U) if) 1234. iy) {h) (i) -316228. 270. 311961. 00632455. 4. {a) 5. (a) (^) id) 0. 12^ per ueut. •4. •78446. •70710. •089442. (^^) {o) -00000000000144. {k) 60. (0 -G. (w) -005. (n) -2. (o) -0948683. (/t) 15-2082. (0 -447213. ("01-414213. («) -00310228. •158114. i 1 (e) 4-4441. (./■) 1-051.-?. iff) 1-00503. {h) -81649. 7. 20 per cent. («) (0 •25546. (,/) -26306. (^•) ^47434. (0 -40000. 8. 351 pages -4- EXERCISE XXXV. (Paok l48.) («) 8000. (y) -000000064. {k) 512000000 (/O -000000000125. (/) 343000. (/m; 1000000000000. ,. .^.,_ ^. («) -000000000001 (/) -000001. (j) -000000512. 0>)-008. {h) 125000000. (c) 343000000000. ((/) 729000000000000. 214 It. lid -J ;5 1* ANSWERS. 2. (a) 200. (d) 800. (9) -02 (,/■) -09 (6) 50. («) '4. (A) •03 (k) -5. (c) 30. (/) •) 1-46459. («) 1-44225 (A) 4-64158. (c) 4-97932. (/) 1-58740 (i) -404158. EXERCISE XXXVI. (Paoe 150.) 1. 5^ per cent. 2. 20 per cent. 3. 142450. 1 9 per cent. G. 20 per cent. 4. 1174-80. 7. 8-24321G per cent. 8. 3-923 per cent. 9. (1-20061)'. 11. 11-2 per cent. 10. 18225. 12. GO rods. EXERCISE XXX VII. (Paue 156.) 1. UV«4-yd«- 2. 241 s(j. yds. 5. 292 s(i. inches. 6. 36' sq. feet. 8. 40 feet. 9. 64 sq. feet. 11. GO yards lon<,' Ijy 40 yards wide. 13. 16| miles. 14. !i?31-50. 16. 1787-88. 17. 10|- inches by 7^ inches. 19. A sq. mile x 102 + an acre x 633 + a sq. rod x 16 + a s(j. yard x 7|. 3. 55 acres. 7. $7-43^. 10. 66 feet. 12. $15-53^. 15. 16 sq. miles. 18. 13.1 i.^,^^_ AN'SWKHS. 215 EXERCISE XXXVIII. (Pa,;k KIO.) ^- '^0 ''^'*- -'• ^rro-> rods. ;i. lu iiK-l.es. 4. All incli X -00490389. •'• ■^^*"^*^*- •>• 124-530 mk 7. 1 HrfUJ ml,. «. A sq. inch X 97-42.S. ,j jq^ ^.^.^.^ 10. 32-829 rods. EXERCISE XXXIX. (Pa.ik l(i4.) 1. («) 78.54 sq. feet. (,i) -00020106 s,,. niiles (h) -0008042.5 sq. yards. (.) -110447 sq. inches, (c) 11 9-.:)97sq. inches. (/) 10-619 sq. feet. 2. 318-309 3-ards. 3. 28-540 rods. 4. 7854 s,,. ins. 0. 17-7245 inches. 0. 1979-2 sq. feet. 7 .«517-70 8. 130-69. 9. 9-102 rods. 10. 2^ feet diameter, 35 circles, 38-02 sq. feet. 11. 488-69 sq. yards. 12. 144-867 rods. 1. 261-77. 4. 472. 7. 9,^ miles. 9. 45U. 12. 22 inches 1. 4i inches. 4. 13090. 7. 65-86. 10. 101 EXERCISE XL. (Pa.^k 1(;8.) -'• 42-416. :i 734.2 •^- 70-3125. G. .^23-56. 8. 1-130 inches, 128-499 tons. 10. $19300. 11. 3 ft. 0-785 in.s. 13. 5 ft. Of ins. 14. 17-28 h..iiis. EXERCISE XLI. (Pa.;k 173.) 2. 13-0525. :}. .579-4. 5. 931 cubic ins. 8. 38-19 inches. 6. 55-85. 9. 40 79. EXERCISE XLII. (Pa,;e 174.) 1. 13^ acres. 2. 56 rods. a. 3309 lt>s. 4. -003 sq. inches. 5. 389-71 s,,. ins. 0. 1072-3 cub. in.s. :il 2I() ANSWEIIS. !'. {a) 4, (b) IG. ■. lL'-t07 inchi's. s. {o) !», (/,) 27. la {„) V-2,(i,) -ixV-i. "• '^'7 •''^••"•"^- 12. r)8 miles. l-i. •■574-12 stjuiirc iiichoH. 1 1. O'^OSt. I.'). .•U7-1H ,s(|uart' tVct. 1(1. :](;.l [ s.,uaro indies. 17. 70'2l' indies. J- 'jj -J 111' tJ KXPJHCLSK XLIII. (Pauk 178.) 1. 2o-y;}25 s(i. metres, 2. 201 -'Jir^ miles. .3. 4-48 litres. T). 70G8-() cm. (J. L'l.j98-rj cub. m. 4. 120604 ares. 7. 40447 Km. 10. 1121 mm. «. G28-32 Kt *J. ••5-9;J8 metres. EXERCLSE XLIV. (pA(;h; 183.) !• («) 12. (A) y. (,.) 8_ 2. («) -2. (/.) -4. (c) -9. (cO +£ (e) -;{. :i. («) 80r)0-82. (,.) -000361840. (e) 262-o6;3. {d) 242G9400000000. 5. 152-4156. G. 6:355. «• 11-1568. «j. 25-362. 11. 385-895. 12. A s(|uaie mile X 2-4679. l.-j. 14170. II. 1-306447. 15. 18-3793 cubic feet. {h) 16-2883. I. 1658-0. 7. -065457. 10. -003407 l.s. EXERCISE XLV. (Pace 186.) {a) 36-082. (,,) -000078105. M -31831 {!>) -00037976. {d) 7-9578. -0243178. 3. 43-600. 14753-9. 6. 1-0780206. 4. 292-23. ANSWEHS. 217 4, (/;) IG. Holies. litroH. i98-0 cub. in. 'is nietrt'w. (e) -3. ■56;3. 5. 5G2. 7 0. 31. 23. KXKRCISK XLVI. {I'u.y. 1H<;.) 3. S»'f' Artii-lf L'G. i; or. 12. 3. +1, •>, + .'{7. L>r,31-29 + 42. 13. 4;'). .$31108. 4«. (a) ) -IHU, (r) •00ir)G2r 51. 2G. +7. 38. /V^x2r). 43. l-'iSODGl. 40. -01222. .■J 8 ■ 53. 24 9G. 5G 779'01. 59. 221. 1 .-. I :; s .'I ; 02 Gl, 06. 69. 72, 75. 77. 80. 83. 86. 88. 90. 9.3. 9G. 99. 101. 102. 103. 104. 105. 100. 108 •)4. 49896. ' • .-I It tl • 00. iii!93-39.i. 27. 1M2208 yar.ls 39. 257307. M. $n9i). 17. -3 1 799. 49. 9O07U1. 5s. M III 65. !B39-35. OS. .>^202 09. 71. 825 times. 74. 10! per i-nit. 76. .l!502-2(). ' "f, 'lii.vs. 63, .SOOO, .S?700. ^i, $1580; /,', .$1896; f, .$213,!. 25;; friilUms. G7. $81 -15. $85748. 70. .«!)i8-57 ■• ■>»■ 73. 1.^. '■" 265 (lay.s,10 p^r cent. Juno 10th. 78. Jan, Otii, 1898, 79. $ 137.3~4-' $3.154-19. SI. .$102-00. s-> .«081--,6" $157-49. S4 $24-47. Th.. fornior l,y .$2-.30. .s;. ei373-os 7;U por cent. S9. (a) .$19-50, (M ,«10-89 94. $22. 97. 52-4S2 U.S. 100. Lo,ss$22- t-redit pi'ice = cash price x "i n I • • '' " ' i-^ivdit price = cash price x 'i^ §4070-16, I'.V.l of the property. vl, $253-46; /A.$3]9-.38; (7, .$407-74 /A $755-11 ; C, $1120-47, $20558-79 A. $4.')07-93; //. .$r,642-07. $10-50. 24)^ per cent. $1065-00. 92. April 7(h. 95. $5-00. 98. 21 J; j)er cent. $050, $3.50. 109. $32828. 10^ 110. $0. i , V PPI" <-enl. IR c H ^'' .-I u. O i.4 C3 i ■f tj ■^ h ' -J 8 8' ',! <::) t. «■. Ui ■ r,: r *:c i' IC ;) 5> <^ 1 11 AN.SWEU.S. 111. •*770()() si.H'k. 111'. m^. 11 1. 1 I5|.i tl.s. 115. (i |)«'r (■••lit. 118. )?-'02rr;{2. \'.\. r>H-74 ^s^als. 1 20. l-t!)-r.H I,„h1,. 127. 1 ft. .S.Hd ins. 1 2!». 2'i') l)iis|i. IMU. I-'mT)! i„s. I ••!-'. I-|NSS,.„1.. f.wt. IliT.OfUs,,. tVrt. I •■'I. !■'{ (J I ins. I '17. I 100 si|. y,|,s, 110. :iii!»0. I I-'. L'll-.'l U.S. Ml. IS-!)r)|. 1 17. See Articles, 11.1. l|M4it;{-66. NO. $10-77. I -25. .{^.'U s.|. yds. 12H. G7i;{ l.l)is. I-U. 02-S;{2 ins. I.i:{. \2-\ ins. l;?n. ,Jt!ll7-.s|. 1 ••{!>. 21-21 ins. I •'<•*». 7 cents. 1-iH. I!) I 52 l.i-ie|