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Les cartes, planches, tableaux, etc., peuvent etre film^s d des taux de reduction diffdrents. Lorsque le document est trop grand pour dtre reproduit en un seul clich6, il est film6 d partir de Tangle sup^rieur gauche, de gauche h droite, et de haut en bas, en prenant le nombre d'images ndcessaire. Les diagrammes suivants iltustrent la m6thode. D 32 X 1 2 3 4 5 6 t* Na»naajl,a,v B.Ohomeque na„o„ale au Ldiiada of Canada / lA] CANy^ nOYAL CANADIAN SIUUIIS. ARITHMETIC FOIt PUBLIC SCHOOLS. rAM, TORONTO: CANADA PUBLISH, KG COMPANY (i-tmited) I8S2. '■^hi:. -■I Entered acconUuf^ to Act of the PayVuimcnt of (amut,t yi the Year One Thonsiuui E^ht lluudrcd ond Eii;hty-iKH>, by t,ie Canada 1'uiii.isiiinc, Comi-any (Limited), tn the Ojjuc oj the Ministey oJ Agriculture. f PREFACE much value i he nracdc/r.r^'"^''*?" '" ""'^""btcdly of tbe habit of nestSh.^ hI""1 "^ "^"'>' ^''^>' ^^^^lin^t based is no of nfeor infoortaC'^^rf 7 ^"^"'^^^ *^"^ ^^'^ ^^ student a mastcrVS Z "' , , ^^\^/°"^^"^ffives tc, the commercial and ^ientifct.y^'^fl.'''^/ '^" serviceable in ccntrate his rtleXn h.^n 1 J ^^^ ^^"^"'" ^^^^^ to cn- tion and accuratftlouH ^ t^T' r'^'^' "/ I^^''^"^ ^l^-'^t'-^^" of reasoning's to lead I; ° ^^^^1^"-^ Inm with the laws of every inference he drawT "'""^"' ''^^^"">' '''' grounds metl'Llifarth? IheScarf "^S' " ^^^^'^^"^^ °" Arith- thepractica. But if the ?L '' — " "'' "^ subordinate to and%he principles are no ?'^ '' jmpcnectly understood, can only^be sXed n echaLari^' "'^'^'v/^"" ^^"^^^^""'^ first making the rules TntdH^^hlo* V.?""^ -^^'^ necessity of on the mind by copiouran Inrn r ^pressing them object in view^ 3°mpom ^.W^ ^^''^^^ '^'^' mind : (1) That The 'e^?rc£s ST?" ^'""''^ ^^^" ^^'^P^ in require thi pupil to Sanf/i t, ^° ';°"^tr,"cted as to largely of examples se ect^d wtl '^ '''■ /^'"^ "^^" ^«"«i«t pursuits of an agric^lttatlrl^^^^^^^^ ^° ^'^ speci^i;";^i:i^ronVctrt"on"^t -^^^^'-- which are the problems ttv . ''; '"1° / ? character and number of -.'=^iuuiems they contain will it is bohW./ V comprehends evervthi,,. that is'nsu'.^n'yty^.i.lt;/' tical th( importance in Arithmetic. '"tI work - • ly be enumerated as foil hat it prac- lie prominent features of OWS! (i) The invcstlf^ation of the principle on wlncli a rule in Aritliinctic depends always precedes tlie statement (^f the rule Itself. (2) Every process employed in the solution of a question IS reh.Tred to some general law or axiom in the theory of numbers. (3) These general truths, as they may be called, are dis- tinctly enunciated and are printed in italics. Jf self-evident, they are illustrated by simple numerical examples; if other- wise, more extended demonstrations are given : in every case, the truth itself is stated in a clear, concise form. (4) The solution of money problems, and the application of reduction to concrete quantities in their simplest form, leading the pupil gradually up from the abstract to the con- crete, are placed earlier in the course than is usual, and are thus made available in subsequent exercises. {5) The logical relations of the several parts of Arithmetic are lucidly marked by their arrangement. For example: Reduction is not treated as a separate rule, but so much of It as belongs to Multiplication falls under that head, while the rest takes its proper place as one of the practical appli- cations of Division, As will be apparent, considerable space has been taken up with exercises for rapid mental work, the importance of which, if the principles which underlie them are fully brought out by the teacher and grasped by the pupil, can hardly be over-estimated. It is confidently believed that the exercises will be found sufficiently numerous and varied, and that the examples solved in Ex. 26, 42, 43, 51, 57, 58 and 59, and in the Examination Papers i to 6, will aid in the illustration of the general principles which form the key to all problems in Arithmetic. The work, as a whole, it is hoped, will prove of the highest service to both teacher and student, and merit a permanent olace among our Canadian educational text- books. . ToKONTo, June, 1882. h a rulu in ent of the 1 a question tlieury of :cl, are dis- ilf-evidiiiit, ;; if otlier- : in every )rm. ipplication )lest form, :o the con- il, and are Arithmetic ■ example: JO much of cad, while ical appli- 3een taken ortance oi ly brought hardly be 3 exercises d that the 59, and in stration of roblems in will prove and merit ional text- CONTENTS. CHAPTER I.-SiMPLE Rules. Definitions ... Arabic Notation Roman Notation Numeration Addition Subtraction Multiplicati.jn Division ... Examples on all previous Principles CHAPTER II.-Factoring. Definitions r^ '" '" ••• Greatest Common Divisor... Lea:3t Common Dividend CHAPTER ni.-FRACTioNs. Definitions ... .._ / Reductim of Fractions... Addition «< Subtraction «« Multiplication " Division <* General Examples on Fraction^... ""' ... ■" CHAPTER IV.-CONCRETK OUANI Canadian Money Sterling Money ITllLO Paob V 2 9 i: i6 31 47 72 9« io6 107 III 116 120 126 128 132 134 143 148 ^53 VI CO\'TK.\TS. im Reduction of Sterling Money Addition " " Subtraction " " Multiplication " •' Division " " Avoirdupois Weight Apothecaries' " Troy •' J ••• ... ... Long Measure Square " Cubic " Time " Dry '< Liquid " General Table CHAPTER V.-Decim.\i.s. Definitions ... Addition of Decimals Subtraction " *** ••• •«• ••• Multiplication " Division " Circulating " CHAPTER VI.— Percentage. Definitions General Examples on Percentage CHA^'TER VII.— Measurements. Practical Problems in Measurements of Surfaces Solids CHAPTER VIII.— Bills or Accounts. General Examples of Accounts Examination Papers Answers... and Paok 154 156 157 159 i6r jG^ '65 165 1G9 171 172 176 179 180 182 iS: 186 187 3. 187 . ^87 ■ 192 ; L 'i 5. 194 ^ 195 ' 198 201 ... 203 i to XXV Paok 154 156 157 159 161 T^)^ • 65 165 169 171 172 176 179 180 182 185 186 187 187 187 192 ... X94 195 es and ... 198 201 203 ... i to XXV Arithmetic for Beginners. CHAPTER I. DEFINITIONS AND PRINCIPLES. 1. 2. i 4- 6. 7. NOTATION AND NUMERATION. A Unit is one, or a single thing. ^:f.— One : one boy ; one dollar. A Number is a unit, or a collection of units, and answers the question How many ? ^j:.— Three, Five, Twelve, Sixty. The unit of any number is one of the collection of units which form the number. ^ ^j:.— The unit of eight is one ; the unit of six horses is one horse ; the unit of twenty dollars is one dollar. A Unit may be cither abstract or concrete. An Abstract Unit is one that does not refer to any par- ticular object or thing. £x. — One. A Concrete Unit is one that is applied to some particular object or thing. ■fi"^.— One cent, one quart. A num?;cr is Abstract or Concrete, according as its unit IS abstract or concrete. ^jc.— Three,^ Sevf^n, Twelve. Forty, One Hundred. are Abstract Numbers. Three Dollars, Seven Gallons, Twelve IMcn, Forty liof.ks. One Hundred Cents, are Concrete Numbers. ^ AnilUMKTlC ran JiEGINNEIlS. 8. All numbers in ordinary use are formed from the charac ters I, 2, 3, 4, 5, 6, 7, 8, 9, o. £x — 384, 5072, 165. 9. The Figures i, 2, 3, 4, 5, 6, 7, 8, 9, are read one, two, three, four, five, six, seven, eight, nine, and are called Digits, or Significant figures. 10. The Figure 0, called nought, zero, or cipher, has no value, and is termed the Insignificant figure. 11. The Art of Expressing Numbers by means of these or other characters is called Negation. 12. Notation is of two kinds, Arabic Notation and Roman Notation. ARABIC NOTATION. 13. Arabic Notation is the art of expressing numbers by means of the characters i, 2, 3, 4, 5, 6, 7, 8, 9, o. (It is so called because it was introduced by the Arabs.) 14. The next number above nine is ten, which is expressed by the figure one placed to the left of the figure nought, meaning one ten and no units; in the same way eleven, or ten units and one unit, may be ex- pressed by the figure one (meaning one ten) placed to the left of the figure one (meaning one unit), and so on for the rest of this order. 15. The complete order will then be : Fifteen, represented by 15 Sixteen, " " 16 Seventeen '• "17 Ten, represented by Elevfn, " " Twelve, Thirteen, ' Fourteen, " ii 10 II 12 13 14 Eighteen, Nineteen, K (< << <( 18 19 The pupil will notice that the first part of each won] agrees with the right-hand figure in the number ex pressing its value. 16. In like manner, all numbers in the next group aro expressed by using the figure two, followed by the I. Wri 2. Wrii 3. Writ rom the cliarac- s read one, two, , and are called cipher, has no figure. means of these on and Roman ng numbers by . 7. S, 9» o- oduced by the ch is expressed ft of the figure ts; in the same it, may be ex- •ne ten) placed one unit), and ented by 15 " 16 " 17 " 18 " 19 t of each word the number ex ext group are )llowed by the ARABIC NOTATlOlf. g original digils in order, the number twenty (written I 20) meaning two tens and no units, the rest beinc I formed by proceeding as befc)re, thus : 7,^^'''"^>''^'"'^' represented by 21 Twenty-two, " « 22 Twenty-three, " «< 23, etc., etc. 17. The next group will express three tens, called thirtv anu a certani number of units, the successive numbeis 1 o n^x^ i^nnr^A in the same manner as before. la. I he remaining groups will commence with the following key wtrids: — '^ Forty, represented by 40 Fifty. Sixty, Seventy, represented by 70 50 Kighty, .. .. 80 60 Ninety, •« « qq The pupil will again notice the likeness the first part of each word bears to the digit in the number opposite 19. We now come to the greatest number that can be ex- ninetn^^ 'T ^?^"'"^^'.^''^" 99. meaning, of course, nine tens and nine units. EXERCISE 1. I. Write down the following numbers in figures :_Four Eleven. Twenty-six, Thirty-seven, Forty-five Six"v eight,Seventy-seven. Fifty^nine, Eighty,V]nely-two '■ Txpr^sed '' '"^""^^^ ""'^ ^^^" '^^ ^''^'''' "^^"^ber by one figure, by two figures, by the figures two and three, by the figures three and nine, by the figures nine and one. by the figures one and seven, by tlie figures seven and eight, by tho figures eight and two. 3- Write down in order all the numbers From twenty-five tn forty-seven, si.xty-three to seventy-(>ight. seven loen to eleven. • forty-seven to thirty-three. AfiiTmnETic ron nr:oi\XERs. 4. Write down the number of words, and also the nuinbe of letters, in tliis sentence. 5. Count the numbers between fifty and seventy-five. 6. Write down any numbers you can make by using both the figures 6, 8. by using both the figures 7, 6. by using both the figures 8, 7. "^ t 20. We left off at the number 99, and must now show ho\ to represent a number having one more unit than gc The required number is called one hundred, and i written 100, — the figure i meaning one bundled unit> the next figure meaning no ten units, and the lastc meaning no units. 21. We thus arrive at one hundred, two hundred, etc., am by combining the hundreds group, and the precedin; group of tens, and the simple group of units, w complete all numbers that can be formed tvith thrr figures. ^.r.— One hundred and sixty-four is written iG: meaning one hundred units, six tens of units (or si.\t units), and four unil'^. | Four hundred and four would be written 404, tl zero meaning that there are no units in the tens groiij Ei^ht hundred and sixty is written 860, the zero in plying that there are no units in the units group. 22. We now see the importance of the figure as a meai: of keeping other figures in their places, for without i 404 would be 44, four tens and four units ; 860 woui v be 86, or eight tens and six units. | 23. The hundreds group or order brings us up to the nunibf 999, the largest that can be written with three figure' J^x. — Write down the number Three Hundred an osventy-nine. The three, standing for hundreds, must be place on the left of the required number, in the hundrff^ ^ \1. J It A lilC SOT A rioN. d also the numhe H seventy-fivp. ike i, 8. % 6. i. 7. St now show ho\ norp miit than gc ' luindrecJ. and i ine hundied unit- lits, and the lastc inndred, etc., am and the piecediiii lup of units, \v ormed with thrr ir is written i(k I of units (or sixt : written 404, tli in the tens grou] 1 860, the zero in units Rrnup. ure as a meai; ;cs, for without \ units ; 860 woni up tothenunibf vith three fij;iiie> ree Hundred an , nuist be place in the hundiffl place ; the seventy or, seven tens, we must put in the nox't place, to the right of tlie three— tliat is, in the tens' place; and the nine, which means just nine units, must be in the next or units' place. The num- ber then stands thus : — Hundreds. Ten^v Units. 379 or, 379- If the number had been three hundred and nine, then, there being no tens, the tens' place would be filled by zero, as 309. If the number had been three hundred and seventy, then, there being no units, the units' place would be filled by the zero, as 370. The number 083, meaning no hundreds and eight tens and three units, should be written 83, for it can be of no use to express hundreds when there are none. EXERCISE 2. 1. Write down the following numbers :— One hundred and seventeen ; three hundred and eleven ; five hundred and eleven; five hundred and seventy-five; eight ^ hundred and ninety-nine; four hundred; sixty-nine; \ five hundred and seven; eight hundred and sixty; four hundred and ten ; nine hundred and nine; seven hundred and eighty-seven. 2. Write down, in order, the numbers between one hun- dred and five and one hundred and twenty. 3. Write down all the numbers of three figures in which if- ^^^ ^^^* ^^^"^ figures are seven and nine ; in which the two right hand figures are eight and nought; in which the outside figures are six and seven ; in which the outside figures are seven and nought. 4 Write down the greatest and least number composed of three figures, 5. Write down all the numbers formed from the fieuicbaix eight and nine. \ 1 ''M I ARIXnUETIC foil HEOTNNmS. ' 6. Write down, in order, all the numbers made up of tlir ^St i;^:ir"^'*' "^"^-^-"-"^^"^ -^' ';< ^' "anTlonr^""!i'^T ""? tl^^'-e between one hundrer seven^.^ o"^'^^ ^"^ ninety-nine ; between ninety seven and one hundred and nine ji) the three figures 7, o, o (2 ' 8, o, 6 P " " " 9.9.8 (4) 7. 7, 7 f 24. We next proceed to numbers of four fifjures the fir.t thousand '?, '""^ ""^^^" ^-°' - "-1 ed ^^^ thousand The remaining numbers of four firuu-s are formc.d by writing thousands in the fourth place, hundreds in the third place tenJ iTt\t second place and units in the ^first 'place from right. Thus 76og represents nine units of the first order.no units of the second order or tens ei^M unS : \t '?iurth"'".°^ hundred;;::;^' set! umib 01 tne lourth order or thousanH*! ti,„ wliole. n^umber is read seven thousanS"e!g\"t'hundTed 25. The pupil will by this time see that one unit of nv of these orders has the same value as ten units of^th the's'^l'^^'^^f I '''^' ? to say:-One thotisand i s the same as t?n'f '"" h^n/reds ; one hundred stenunTs! '" ''"' = "^^ °"^ ^^" ^he same 30. 26. We sliould therefore naturally expect to meet after the order of tliousands a fifth"^ order repr^sentinrten ni^itT;'' ""^ ''''' '' '''' ^'''' but^ns ead o'f gn of^ housandrr!' '' «r^" ^' '^'' ^'^'^ «f '^^^ of tlK?u"anSs.' six'^'thous'n'd's 'Z'TI ^7'* *^"^ tens pnH ni«« , "luiJsanas, no hundreds, seven seventy-nine '"'' °' "'^^^^^'^ "^°"^^"^ «'- 31. rs made up of the inencing with tlu en one hundred ; between ninety numbers formed figures, the first and called one s of four figures in the fourth :e, tens in the t place from the nits of the first or tens, eight eds, and seven 3usands. The i eight hundred 5 unit of ny of en units of the ne thousand is ; one hundred ■ ten the same meet after the ^resenting ten instead of giv- order of tens nts eight tens ndreds, seven thousand an(! I ^ ARABIC NOT A T JO \. 7 27. As the order hundreds followed in value the order tens (Art. 20), so now wc have hundreds of thou- sands fi)lli)witig in value tens of thousands, and forming the sixth order from the right. 28. PKcecding as before, we obtain (after one hundred thousand), two hundred thousand, three hundred thousand, and so on until we reach ten hundred i thousand. This is expressed by a new name, and ' called a million, 29. After counting to a million, we proceed to count one million, two millions, three millions, and so on as far ■^ as ten millions, which forms the seventh place from the right, or the seventh order. 30. After tliis hundreds of millions will follow tens of millions, just as hundreds of thousands followed tens of thousands (Art. 27) ; and hundreds fol- lowed tens (Art. 20). We then come to ten hun- dreds of millions, which is called a billion. 31. The figures 1,2,3, 4' etc., when they stand alone, or when they occupy the first place, denote simply so many units or ones, and are called units of the first order. When they occupy the second place, they represent tens, and are called units of the second order. WHien found in the third place, they stand for hundreds, and are called units of the third order, and so on. This may be illustrated by the following table: — The 1st order of units is called units. 2nd 3rd 4th 5th 6th 7th 8th 9th loth nth 12th tens. hundreds. thousands. ten-thousands. hundred-thousands. millions. ten millions. hundred-millions. billions. ten-billions. hundred-billions. And we may extend this table to trillions, quadrillions, etc 8 ARITHMETIC FOR RmiySERS. th^fc! ?^ ^"'?^^^* "'■'^^^ of ""its is hundreds of thousands, and there are nine of thpm m.! : order is tens of thousands^of which her" ^re seven he next order ,s thousands, of winch there are none- he next hundreds, of which we have six ; he next' tens, of which there are none • the npvt ,,«?♦»! th^oH''^^?'^""^'' h""dreds,tens!unitl, one after the other, a ways putting in a cipher or zero for thi ttrbe^^r'ilteV'-^ ^^ ^ ^^^ ' -^^^nZLiVu^t 970608. IS- Too much attention cannot be paid to the placing of the zero. ^ ^"^ kVrite Th; Ele Six 2. Wr 4- 5- Wri 1 I Wri I Wri: f EXEECISE 8. I. Write the following numbers in figures: Nine thousand and forty-eight. Five thousand and seven. Forty-three thousand six hundred and fifty-nine :n5"tf '"' *'^^^y-^'^ ^^— d th'ree hundred ^^"hir^een.'''^ '"^ ^'' '^°"'^"^ '^''' ^""^red and Five million forty-three thousand and thirty-seven iiree hundred and fifty-six thousand and ninetv-seven """tlemy^s":'"' '""''■^' andforty-fivethousLdrd Eighty thousand and fifty-six Nine million ninety thousand nine hundred. Eighty-three thousand and seven Nine thousand and ninety. 33. In Ri expn All oth letter the e; NOTB TO T requires a m pupil haa yel have been lea 1 on the slak nine hundred Jundreds of 11 ; the next e are seven; re are none; X ; the next xt units, of iber we first and this is IS of thou- :s, one after zero for the umber must >aid to the P.OHa\ VOTATIOX. 9 Write h J lollowing numbers in figures: Three hundred and seventy-five million eight hundred and sixty-seven thousand seven Imndred and ninetv- nine. ■' Eleven thousand and seventy-one. Six million eight thousand seven hundred and four. 2. Write down the greatest and least numbers that can be lormed by using 4 figures. ■ 5 " 6 " 4- ti Write down, in order, all the numbers of four figures having 3 f )r the left hand figure and 72 for the two right hand figures. Write down the greatest and least numbers that can be formed by using all the figures 7, 3, o, 5. Write down all the different -umbers that can be formed by using all the figures 7, o, o, 8, and name the greatest and the least. line. !e hundred tidred and seven, y-two. ety-seven. usand and 33. ROMAN NOTATION. In Roman Notation seven Capital letters were used tc express numbers. The letters were I standing for the number V X L C D M II II << i< I 5 10 50 100 500 1000 All other numbers being expressed by combining these letters in different ways. Hence Roman Notation is the expression of numbers by letters. r.^wv!a'° Teacheu -As the writing of numbers by Roman Notation requ res a more extended idea of Addition and Subtraction than the K^beJJLrntJ?" '•' "^'^ ""' °'°'"''' "'^'" "'*^^ these operation. ^ J 10 AJtJTUMKT/C FOU LEOllfNBHS. 34. Tliey are combined with the foll( 35. I II 11 :f ,11 i I ng results : 1. When any letter is repeated its value is repeated 1 hus, X stands for lo, and XXX stands for ao • C stands for loo, CC stands for 200. i > ^ 2. When a letter of less value Allows one of greater value, Its own value must be added to that of the greater, 3. When a letter of less value comes before one of greater value it takes away its value from that ol tne greater. T])us, X = 10, IX = 9. L = 50, XL = 40. 4. When a letter of less value stands between two of greater value the less must be taken from the on. that follows It, and the remainder must be added to the ore that precedes it. Thus XIX = 19, CXL = 140, CXC == 190. ^' fhnn" 7^' fu ^^"^^ ^^ ^^^^^'^ "^^kes it as many thousands as there arejmits in the letter or lettei- . Thus V = 5000, L = 50,000, C = 100,000, IX = 9,000. ti^ Vf "^' '' "°* "?'i4^>' repeated more than thre. ' "Hiese methods will be fbund quite sufficient to form ^ all ordinary numbers, and although this Notation couW " be used ,n busmess and other calculations, the piSe^ ^ would be very tedious, and fbr this reason snS^S ! The characters are chiefly employed to mark the hours on clocks and watches, to number voCS and banSs:^t^°'^' ^" ^"'^^^^^ ''^ -^"- ^^ -^ I ROMAN NOTATION ^r Ten. J{ • • • • . Eleven. Oi; • • • . Fourteen. $^, Fifteen. XvJtV • • • • Sixteen. XVIII .... Eighteen. ^^^ .... Nineteen. ir. TABLE OF • • . . One. • • • . Two. Ill Three. lY Four. y Five. VI Six. ^X Nine. iH. 19. Eight cig nOMAS yOTATIOff. 'esults : ilue is repeated, tands for 30 ; C 3 one of greater to that of the 00, cm = 103, i before one of le from that ul '» XL = 40. between two of 1 from the one ist be added to -XC == 190. :es it as many tter or letter.,. = 100,000, IX lore than three D rather than cient to form 'dotation could IIS, the process on is not used, to mark the r volumes and lues of coins, I XX . XXX XL . L LX . XC . c XXV cxx_ , cLxiv Twenty. Twenty-one. Thirty. Forty. Fifty. Sixty, Ninety. One hundred. 25000 I y . . 1 20CXX) M . . 1 64000 1 CD . D . DC . Dec DCCC CM . M . MM . 5000 I 000000 11 Four hundred. Five hundred. Six hundred. Seven hundred. Eight hundreil. Nine hundred. One thousand. Two thousand. DLCXL . 550140 MDXC . 1000590 EXERCISE 4. Write the following in Arabic and also in Roman Notation I. 2. 3- 4- 5- 6. Ten. Eleven. Fourteen. Fifteen. Sixteen. Eighteen. Nineteen. Thirteen. Seventeen. Nineteen. Twenty-six. Thirty-eight. Forty-four. 7. Ninety-seven. H. One hundred and fifty. 9. Two hundred and eighty. 10. Seven hundred and thir- ty-eight. 11. Eight hundred and forty- four. 12. Twelve hundred, 13. Eighty-seven. 14. Six thousand. 15. F'ifteen hundred. 16. Eleven thousand. 1 7- Eight hundred and eighty- eight. 18, Seven thousand five hun- dred and ninety-two. [19. Four thousand seven Imn dred and eleven. Fifty=two. Thirty-nine. Forty-three. 23. Sixty-seven. 24. Ninety-one. 25. One thousand eight hun- dred and eighty-one. 26. Twenty-seven. 27. Forty-nine. 28. Seventy-three. 29. Sixty-eight. 30. Eighty-four. 31. Ninety-seven. 32. One hundred and ten. 33- -Five hundred and fifty. 34. Seven hundred and forty. 35. Nine hundred and ninety. 36. Sixteen hundred. 37. Fifty thousand and five. 38. Three hundred and eigh- teen. 39- Seven hundred and nine- ty-six. 40. One thousand and ninety- six. 41. Twenty-five thousand. 42. Fifty-nine thousand three hundred. 43- Eighty-seven thousand and forty. 7 • n: ARlTHMETiC FOR DBQtNNERS, NUMERATION. 36. N Ulceration is the art of expressing in words tliose numbers tliat may be given in figures or letters. 37. If tlie pupil clearly understands the meth(;d of express- ing numbers in Notation, there will nut be nmch trouble in reading any number that may be set down in figures. To read large numbers more easily, the figures are separated by commas into periods or groups, commencing from the right hand; and the method that is nearly always adopted is that in which each period or group consists of three figures. 38. The first or right hand period contains units, tens, and hundreds, and is called the period of units ; the second period contains thousands, ten-thousands, ard hundred-thousands, and is called tiie period of thou- sands ; the third period contains millions, ten-millions, and hundred-millions, and is called the period of millions; and so on for the others. This can be more clearly seen from the following table : — 4 c 39. C o a o « w w r— 4 H C o •p-4 •f-4 'A a o 1 G o s o 13 G w B c '.ei: to ...ly nun^ber of orders. The periods after tnlh: ;s .o, in ilair order, quadrillions, quintill- ion;., sxtillions, ;.ptillions, octillions, and non- ihions. (a) Point 3- 4- 5- 6. (^) "III yrMFitATrn.v 18 1 words those ■ letters. jd (jf express- not be much y be set down re easily, the :o periods or iiid ; and the that in wiiich ures. lits, tens, and f units ; the lousands, and *iod of thou- , toii-inilliuns, e period of This can be le:— 39. Hence wo have the following rule for reading large num hers easily : RULE FOR NUMERATION. Paiftf cff the number into periods of three figures each, beginning at the right hand; then begin at the left hand, and read the figures ofi each period separately, adding the name of each period except the units' period. Ex. I.— Kead 261034. First point off by commas, thus: 261,034. The number will then read 261 thousand, and 34. Ex. 2. — Pead 4604792816. Point off thus ; 4,604,792,816, and read : 4 biUion, 604 million, 792 thousand, 816. EXERCISE 5. tn u C 3 B H I 4 5 iBtP ir'.' U: .,. sxpressed in trillion, five hundred and fty-six thou- table may be periods after ■js, quintill- s, and non- {a) Point off, read and write : I. 2. 3. 4- 5- 6. 7- 11G234. 65231, 20703. 7T005. 3104. 48000. 60029. 8. 141 120. 9. 101207. 10. 68978. 11. 72020. 12. 80001. 13. 857000. 14. 91029. 15- 16. 17- 18. 19. 20. 21. 7640. 800900. 2568242. 1008003. 212375647. 6090(^3588. 897856846. {b) Write in figures and read : 1. Nine in tlie ist period. 2. Two hundred in the 1st perifxi. 3. Sixty in the 2nd period, two in the ist. 4. Seven hundred in the 3rd period. 14 AHITHMETIG FOR BEOINNESS. (p) Write in figures and read : ^* lZ\^r^'"^ '""^ '^'''^ '" '^"^ ^'^ P^"«^' ^'^ty in 6. Eighty-one in the 4th period, five hundred and one in the 3rd, seven m the and, twelve in thq 1st ^" Ihellt!'' *^^ ^^^ ^^"°'^' ''^ ^""^'"^^ ^"^ three in 8. Seven hundred in the 5th period, eighty in the 4th. 9- Eight in the 4th period, seven in the 3rd, fourte 1 in the and, and ten in the ist. '°* ^/"f "V" ^}^ ^*^ P^"°^' eighteen in the 4th two hundred and seven in the 3rd. and eighty-one in Vhe {c) Point off, read and write: 1. 60701892. 2. 50607801. 3. 600000. 4. 49000000. 5- 593006070500. 6. 190190001900. 7. 1 63 1 94568. 8. 3050050183. 9. 5000204. 10. 594900. 11. 12000012. 12. 2007980134, {a) Write in figures I 2 Eighteen in the and period. In7f{fV'hri;'iSre';sr'^ '" ^'^ ^"'' ^"^ ^-^-^' Sixty in each of the 4th, 3rd, 2nd and ist periods. 60 million, 200 thousand, 500. 402 billion, 348 million, 213 thousand, ao. 78 trillion, 640 billion, o million, 6 thousand, 16. 6 billion, 542 million, 25. 8. Su billion, five hundred and forty-two million, twenty- 9. Four hundred and two billion, three hundred and sS;Td';w:;'!r' *"° '""'^^' ^-^ tha-tceiftho-u' 3 5- 6. 7- " (^ WrifrE 10. F ni 11. T hi 12. T m 13. F( hi 14. Tl fo 15. Fi an ni 16. Ei th {e) Expre I. 2. 3- 4- 5. 7. 8. 9' 10. II. 12. 13. 14. 15. 16. 17. la J9. ao. 21. If UMBRA TION. n iod, sixty in red and one 1st. nd three in n the 4th. fourte 1 in le 4th, two -one in the e hundred Jriods. I, 16. n, twenty- idred and cen thou- r^ Write in figures : 10. Five million, eight thousand, nine hundred and fortv- nine. ' 11. Two hundred million, three hundred thousand, eight hundred. ° 12. Twenty-nine billion, five hundred and ninety-nine million, SIX hundred and one. 13. Four trillion five hundred and fifty-eight million, two hundred and forty-thousand and sevenjty. 14. Thirty-two billion, one million, three hundred and torty-three thousand, four hundred and four. 15. Five hundred and fifty-five hiillion, seven hundred and seventy-seven thousand, six hundred and sixtv- nine. ^ 16. Eight hundred and six billion, seventy million, five thousand, two hundred and six. Express in Arabic Notation, and also in words : 1. XIX. 2. XXI. 3. X. 4. XLV. 5. LXV. r-. Lxiv. 7. LXXiX. 8. LXXXV. 9. ex. 10. CXIX. 11. C. 12. CXIV. 13. CLX. 14. CXC. 15. CCLX. 16. CCXC. 17. DCXXIX. 18. DCCCXl. 19. CML. 20. MCCLIX. 21. LXXVI. 22. MDCCCLXIX. 23. VCXCIII. 24. xvir, 25. My cxxii 26. MIviMD. 27. CDL. 28. XLVIII. 29. DXXXVI. 30. MDCCXCIV. 31. XCXVI. 32. CCCLXXXI. 33. LCMXCIX. 34- MMDCXII. 35- VCDLXX. 36. CCLXV. 37. MMMDCXXVU 38. xix. 39- iJv. 40. cdii. 41. (ixxxvi. 'I ' • 10 .j,i ARtTIl^fETlC FOR nr.OlSflKM. {<) Express in Arabic Notation, and also in words: Ixxxv. 42. 43- 44- 45. 46. 47. 4.S. 49- 50- 51- i3. xviii. Ixxvii. LXVII. CLXIV. CXXXV. CXMX. MX IX. DCLIII. CXCIX. 52. VDLIX. 55- I>LX. 54. XXXID. 55- LIXCCCXLIV. 5^J. XVDCCXLIX. 57- MMMMXC. 55. VMDCCXLIX. 59- MnxxvcDLXXIX 60. RrDCCCLXXXlI. i '''■M-:s.7=-^T:ssx';r! ADDITION. 40. The Addition of two or more n«n,l,ors is the raethod „f Thus the sum of 4 and s is o for J., . «k r units and ,n 5 thcrc'arc fi4 u.?i't ?. d ti we aZ h" more units after we get four we obta.n cj uni'^ ' 41. Tliis is often written 4 + c = o the .;icr„ ^ . the nund>ers on e\th^id'^' o "it^i^^ lo'KS to' gctlier. It is called Plus. '^ *''' 42. The sign = means that the expressions on each side nf U are equal, or of the same v'alue. It is 'fad Equal Ex. 3 + 6 = 9 would be read three plus six eonai to ntne. and U means that the sun. of ^d 6 is X r 3 4 5 6 7 8 9 10 5 an I an 2 6. 10, and 3. 12, 10, 2, o, and 9. 27. 10, 3, 8, and 7. 36, 12, 7, 4, aid ro. II, 12, 10, 9, and 8. s/ I . AIHTirjfETW FOR BEGINNERS. Add alternately, 5.6,5, 6,5,6,5, 6, 5, 6, etc.. till the sum ^' 4. 6, 4, 6, 4, 6, 4, 6, 4, etc., till the sum 7.5,7, 5-7,5. 7.5. 7. 5- etc., till the sum «. 9. 8, 9, 8, 9, 8, 9, 8. 9, etc., till the sum 88. 100. 120, 119. ■•% i SI ,! the su. «f eac^h ol;tX,f,-',-^,™-^-. -'>«> ^^' ""sam/unT""^ '°^="'"' *ey must be of the way, 5 tens and , units „aKe n'eiti^S^ te„l"nor ^ Ss^ '• oT^th'etmeTind'f^rhVs'""'" ^^^= P'"^ «8«- under units, teSs under S"^ u"'"?"' "'=" i''' """s dreds, etc '^"=' hundreds under hun. 48. We thus have the following RULE FOR ADDITION. 3. /'/rrr*' «;/^/'"^' ^^'^^" ^»"r "lore joined tiiem: how many ducks were then in the flock? 12. How m.ny are eight cents, six cents, and five cents? 13. There are seven books on one desk and six on another- how many books are on the two desks ? 14. Joseph had three cents, his aunt gave him five, and his brother gave him eight : how many had he then ? 15. One hen had four chickens, and another had nine - liow many cJiickens were there altogether? 16. Herbert had four apples, his brother gave him three and his sister two : how many did he then have ? 17. One word contains ten letters, and another seven: how many letters are there in the two W(;rds? 18. Mary had nine books, and her mother gave her three more : how many had she then ? 19- A man gave nine dollars for a plough, eight dollars for a rake and six for a harrow : how much did he give for ail ? s I 20. Ho do] 11. Da pea did 22. Ho 23. A I five 24. Ho bin 25. Wi plu 26. In oth 27. Ho 28. Sus her 29. Hoi 30. Jan thn hov 31. A r the foul I ih) Add t (1 2 6! (7) 18: 71: (13) 784c 21OJ 3^ ADDITION. n md his brotlier get in all ? < days tlie next he two weeks ? fht cherries in mds? X and nine in ^en in another: liars, and two out five more: 311? nd some pens d? ne, and John en four more I in the flock? d five cents? x on another: II five, and his he then ? -r had nine : r? ^e him three, n have ? ■ seven : how ve her three It dollars for did lie give 20. How many are nine dollars, three dollars, and four dollars ? 21. David gave seven cents for apples, eleven cents for pears, and eight cents for peaches; how many cents did he spend ? 22. How many are six and three and five ? 23. A boy bought a pencil for ten cents and some pens for five; what did both cost? 24. How many are eight birds, seven birds, and two birds? 25. William spent nine cents for pears and eight for plums ; how many cents did he spend altogether ? a6. In one window there are nine panes of glass, in an- other six ; how many are there in the two ? 27. How many are eight and five and three ? 28. Susan had eleven pears ; her father gave her five, and her mother three ; how many had she then ? 29. How many are seven and five and six? 30. Jane paid six cents for silk, seven cents for a spool of thread, nine cents for pins, and four cents for tapej how much did she pay for all? 31. A man owns 4 farms; the first contains 1143 acres, the second 2320 acres, the third 3425 acres, and the fourth 2010. How many acres does he own ? ^h) Add together i^) (2) (3) (4) (5) (6) 23 43 27 36 72 123 02 36 31 22 17 241 (7) (8) (9) (10) (11) (12) 181 5431 7654 5346 6135 4523 712 23G4 1235 2453 3844 2236 (13) (14) (15) (iC) (17) (18) 7840 I 122 1216 3701 21020 2105 4314 2203 2701 1293 34917 33 2432 3322 1082 2005 22032 •/ i 1 1 ! r ^^^^1 !* ■{ . » ^^^1 14 ARITHMETIC FOR BEGINNERS. (^) Add together : J (19) 313291 201306 21 1002 1 23 100 (20) 133072 lOI 3303 12322 (21) 3093124 2101003 3003251 2020 '51. (22) 202020 333222 262626 102101 (23) 9331567 4oroi 623311 2020 24) 121 202 1 2301304 33330 201000 50. -ff^.-Add together 378, 691, and 421. in 1490 Writing units under unit.! f»«e j as before, we sav t ,mi? o ^ ^- """^^^ ^^ns, c^c, units are' 10 unfts wh ch 1' ""'' f' ^ ''''''''' ^-'^ « units. Set down ihe nm-fc T^\ *° ' ^^" ^"d and take the 1 ten toth^n" . ""'^^^ V^^ ""'^s' column • . 7 1 , " ^^^"snext, or tens rnliimn ti I ten (which we carry) and 2 tens ar^ oT. T' tens are 12 tens, and 7 tens are ?. / ^ ^1'' f""* 9 equalto I hundred and tens V?*!^""' ''''''^' ^''^ the tens' column and carrv th^ ^'^ ^^'^^ *^"^ ""d'^'' or the hundreds' cobmn Vl^'n H.undrolf? V","^^^ carry and 4 hundreds ar^ Vl J """died \vhich we are 11 hundred fnd, V i'"'^''^^''^"^ ^^''"dreds which are equal to 1 thousanH ''^.^ T '^ ^'''^dreds, down the 4 hundreds u"derfts'l'^ hundreds. Set there is no thousanri;' ,~ii °''^" column, and as the I thousand mt^ttX^'w^'r^n,""^^^^ P^^^ column would be Thus wp h? ., '^ *''^ thousands' " ue. I nus we have the answer, 1400 In short : ' carVJhV.^^Vhe^,?/; -3 f ^„^-n the and 7 are 19. Set dow>; the 9 and ca;rv thcfr'^^^''-^"'^ and 4 are 5, and 6 are ir Tr.A ^ ^ '• ^°^'"' ^ the 4%nd c'arry ?he 10 ti; ZJ T^ '^^ ,^"^ ^own in other words.^set down the 14 ^^"'' '° *^^^ ^^f*' °'' Now 1 52. 53. 54. f23) 'I567 OTOI 33II 2020 24) 12I2(J21 2301304 33330 201000 e sum in any 421. r tens, etc, units, and 8 I ten and inits' column imn. Then, I tens, and 9 I, wliich are • tens under I to the next d (which we 1 6 hundreds I- hundreds, Ireds. Set iin, and as mply place thousands' ^er, 1490. the and 're 12, and Again, i Set down he left, or, ADUITIOS. »51. The work might be written thus: 3 hundreds + 7 tens -f 8 units 6 " -)- g " _i_ I 11 4 " + 2 " -j- 1 »• '25 13 hundreds + 18 tens + 10 units [Now 10 units = T ♦«« I ^ -i P . I ten +0 units ,1 ^^ J ^ , I Inindred +8tens-founits 13 hundreds =:ijhou^nd + 3 hundreds + o tens + o units I thousand + 4hundreds + g tens+o units This is exactly the same result as we had before. 1 52. We thus obtain the following complete RULE FOR ADDITION. I . Write the numbers to be added, placing figures of the same kind in the same column. „ 2. Begin at the right hand and add each column separately If , the amount of any column be less than 10, place it under the I column added; but if the amount be 10 or more, place the I rtghl-hand figure of the amount under the column added and I carry the left-hand figure or figures to the next column ' I 3- Proceed in the same 7my through all th, columns, and set doicm the whole amount of the last column. I 53. As in the former case, the best way of proving the cor ^^' ^"f^^^'?^ nijmbers, the pupil should be always taught '^i:ri%ztf ''''-''''' ^'^ ^""^ -^^ ^'- ^ -w ^^.— Add together 869, 4931, 2687, 1072. 869 4931 2687 1072 Ans. 9559 ^t^ffinning at the right, n-osav^ - i ^o, and 9 an. 19. instead ^f^avin^/^^ 9. and , an- 9 and I are ,0, 10 and 9 are 19^ "^' ' ^"^ 7 are 9, '-d^^:rfotd"t,:!::^„:T^r« -n, the , ten f y ng. I and 7 are 8 and « a i T '" I''^ ^''^"^^ ^vay 6 are 25. and so' on for ail the c'l inn?:' ' "'^ ^^' ^"'^ !■ Add together: 756 425 ^95 143 7^4 231 7856 ili 8943 483 6789 874 4584 965 ^") (12) i^7 251 8^9 432 721 897 (17) 12734 63741 32347 87698 37 794 EXERCISE 8. (3) 127 341 210 (8) 748 249 838 749 (13) 4376 8231 ^3i3 65S (18) 3786 97643 278 89784 3264 1640 (4) 106 341 121 (9) 4681 7362 8428 1697 (H) 5438 7846 829 9731 96 6204 2413 123J (10) 36487 10462 38420 79549 (ao) 1379463 207839 999 7638 72109 367294 ]i! 14. are 9, and i arc ?. a and 7 are 9, i^^y the r ten, I" the samo way "'''3axe 19, and ADDITION. Fiiul tile sum of 27 Find the Find 68473 th )f9863 789632 + 4 + 67 + 879002 + 876 + 970. + 832 + 97+10029 + 7384, sum e sum of 1324+433 6204 2413 1231 (to) 36487 10462 38420 79549 (20) 379463 207839 999 7638 . 72109 67294 ^473423 + 7 f- 99910. 4653 + 12 + 876 + 97843 + !^Z:74. '''" + ■""°^34 + 97 + 9647.. + . W a, ,. .ho „,„, of 3,6H,S + 637+4.3+9S976+3«i' ■ '^'''' ■'*'+24+»97l564-S8i +7,512. What is .he sun, „f8. + H,3+g3^,g+S.4683+, 000.0,? F,„d.h. su,„ "f3a47+86430. + 84+56703+, 000,003, wha. will l,e .he .o.al sum ? ^"' ""'' 53=°'6. Add ,..ge.her .he answers in ques.ions 3, 7, ,. .0, .,. Add together: i (0 348037 272465 530634 I 0987 I 693036 764543 233638 428432 389763 21C045 76oh'o6 636215 253734 251600 575453 807720 930045 ^74173 626245 342734 460375 84 I 68 I 239724 763256 437891 825432 285678 310720 403521 687489 324061 530724 623452 487638 290731 803256 731463 379574 8231^6 928348 963172 300725 463248 721003 387356 241653 603280 532176 278321 829248 171320 206782 461027 589203 248639 730461 672398 246175 928340 731629 849652 361728 4I238I 635403 ^72545 406223 294867 811236 576037 213744 764368 305216 436720 823284 217436 592301 243762 739445 429374 684569 28 I M. Tlie year the 2210 lattc pg. An a 270H how ''■ ^-^^59^/1" ;874,^° ,^^^' 74« ash 639 beech, J ^'-w many trees havej'in ajli'^''"' '"'^ ^+7 poar tree] what did lu, ;,,^i,l ,;;',,;'?> *97. and by Jus oats $. ''*£ThtTt'n?i;:4'L1^^^^^^^^ ^'' whatS^oi^^^Sf 478 ^^^ ^— 1-t $273,1 20. James was born in r^rA • years oJd? '" "^^S: in what year will he be "• g 1;^ ^r^^l'^d Sh-?i^^^^'^ --^ is 67 mo, find tJie sum of the tl^rj^! '^ "^'"''^ *^^^" ^^^e s4ond| ""PoSr,!::^!^/-;;^^^ ^^^^ $5635: how mucl ^^^ving lost that sum ? ^ ^^'"'^^ $957 instead c1 23- Commencinrrwith^?^ „.? .• , I ^^^'^ bers below ssT?^'"^' ^^'^^^ '^^^^^ sum of all the nun^is- Three '^" iTliLn'"^^^"^? "f - village was :,.. „ I S^ ^-n at the cS^ol^lhe fiftl^^;;^-^ -- th/^:pm4. A ma ''• ^::;;iuJ^ ^^ -t ^° ^^^^ ^^^^' ^- sons .nd th i ^ h'iii-ib. -rlis Wife receiver? «;/- °"-"'i>. ana t/iiff a as fnr and each daughter «o^« , ^9527, each son $c-26 * ^ whole estate?^ ^^^84: what was the value of 'tl, 26. A. owns a farm worth «r^-, x, f 9Jn a pair of ^xe \S ^^^^^^^^ $15. apiece and sheep worl tsf.' '''i '"'^' ^^^'''^'^ 3^ value of his property ? ^ ^^ ' '^'^^^ '^ the (u(.r 2. A. de has !; as in tlie s all th all? 27. WasJlinrrtnn W3<; I", • 'f-; Nap„,eo"iS\';',,7,\%->;l Napoleon 36 ye„„. fhd he die? "'*^ ^&^ "^ 53 '• m what yea- as for for bui how m b- I^'ive p^ f first d ^6597; the hn they al I3G. The fir than ti: 'JIS. ADDITIOy. ish, 639 beecJi, i^. and 247 poar tree] ^'7.«f^s -f 364, by ii, u oy Jus oats S M ^8. Tlie population of a certain city was 23000 in the year 1S45; u, the next five years it gained 5630: in the next five, 8763; in the next 16420; in the next, 22109: h.w many inhabitants had it at the end of the latter time ? g. An army consists of 6450 cavalry, 27846 artillery, and 270874 more nifantry than both cavalry and artillery • ?r lias 25 more thai ""' "^''"^ "^^'" ^^'^^^ ^''^J"e in the army? >. A. has $5786 ; B., $6724; C, $10536; D. as much as and C. ; h. as mucli as A., B., and D.; F. as much as all tne rest : how much have they in all? >wner lost $273^ year will he be second is 67 moii than the secondl 5^35 •■ how muc| $957 instead ck ^ of all tJie num 7' tlie next yea J third year 845! vas tlie p,;pulai sons, and tlirnr ^ach son $5726' he vahie of th- ses worth $15!'! ■'-'ws Worth !ji2.C t is tijc iotM :jleon 36 yenn in what yea | (I. A. has $84 ; B. has $23 more than A. ; C. has as much as A and B. ; and D. as much as A., B., and C together : how many dollars have they in all ? A. deposits his money in five banks ; in the first he has ^897 m the second $673, m the third as much asm the first and second, in the fourth as much as in the second and third, and in the fifth as much as in all_tlie others: how much money has he deposited in '^" VT^ T" ''''*^'' l"l" partnership; the first man puts JS3845. the second $2375, and the third puts in $s8q ^/r^H H the sums put in by the other two : what sum did they all put in ? ■ c/; "'''". ^^"'J'^^ ^^^^" ''°"^^s ' ^°^ the first he receives J)647 for the second §799, for the third $949, for the fourth §1467, for the fifth $1986, lor the sixth as much as for he first and fourth, and for the seventh as much as for he third and fifth : how much does he receive for building the sixth and seventh respectively, and how much for building them all ? I5. Five persons deposited money in the same bank ; the first deposited $59^7, the second $12980, the Ihird ^65973, the fourth .«;37345, and the fifth as much as tie first and second together: how many dollars did they all deposit ? j ^ ^^' Jun^ ^/u^^^ IT"" """^b^'-s 's 3125 the second is greater than the first by 5108, the third is c(jual to the sum of # 1 , , so AHITHHETIG FOIt BEaiNNEr^.l. the first and second, and the fourth is enual toi nZ hers" ''''' ^"' '''' '' -J-^ ^^ ^^-' sun/oflhll 37- The ship (9;7,;?/ sailed from Marseilles to Buei- .s AvI distant 6375 miles, thence to Valparaiso SgI n?, thence to San Francisco 6346 m le ' ence^o L'«« r-i^' ^fu''^' ^^52 mileJ. thenc; to Melbou 5588 miles, thence to Yokohama 5434 miles The to Calcutta 51 rs miles, thence to Boml "y 22' mi hence to Suez 2006 miles, and thence l/ckMV se:lks^x3X4 miles: what was the entire distal. 38. Find the sum of four hundred and three ; 502. • s,^ 39. Find the sum of 2050; three hundred and seventv thi^ sand and two hundred; fbur million and five ^o*^ honninety thousand, seven hundred and ei^tv • J hundred thousand and seventy ; 98002 seven millJ nme thousand and one ; 70070. H 40. Find the sum of two hundred thousand, two hundred hree hundred million, six thousand and thil seventy million, seventy thousand and seventy nil hundred and four million, nine thousand an J fonf h n dj;.^^''"^'^"'"'^^ ^^^'^""' "i"« thou^anjtr hund^H- ^"'^"t""^ ^'^^y: ^^« thousand, s^f Si^^'t^^:"'"'^"' ^"^"^^ *^--"^' -^l^t h.^ 41. Find the sum of all the dififerent numbers vou r J make by using all the figures : ^'^"^^ers you c | 3- 3. O, 2, 8, 9. o. 7. 9> o. o. 42. Add together all the different numbers of five fi^rur each number beginning with 375. and ending whi^ 43. Lily has 17 roses: T.nur=> hn- -- r,% *t Charjco has ,8 dahiras m"ore'Yh;r; ^^ ts ^l and Jennie has 14 dahlias more than Charles t many dahlias has Jennie ? V'lwnes , ii, M ^ I, I irtli is equal to I tJje sum of the f| estoBueu !s Ay; Jaraisc) 27O4 mi| lies, thence to ince to Melboul i434 niiJes, the! ombay 2257 niif ence back to M 5 entire distai^ ihree ; 5025 ; si a 1 thousand ; 20s and seventy tli^ and five ; two 1 and eighty ; 32 ; seven millil nd.two hundra ind and thirj ud seventy ; n| 'usand and fari^ le thousand ; thousand, seJ sand, eight hij umbers you fUnTRACTlOS. ai 44. Henry's purse contains 329 cents; Edward's contains 43 more than Henry's, and Henry's contains as many cents as Sarah's, less 94 cents : how many cents docs Sarali's purse contain? 45. Mary bought a pencil for which she gave 95 cents; Maude bought one for which she gave 13 cents more than Mary ; and Maude's pencil cost as much as Jane's, less 23 cents : what was the cost of Jane's pencil? 46. Henry lends $913 to Thomas, $473 to Samuel, $576 to Theodore, and has $576 left : how many dollars had he at first? 47. A man was 37 years old when his son was born: how old will he be, when his son has reached the ace of 59? 48. John throws a ball 30 yards up the road, and another 40 yards down the road : how far must he walk to bring them both back again ? SUBTRACTION. sof five figur u ending with :nure than Li -aura has r. s n Charles : L 67. By Addition we find that 7 units and 4 units make ii units. We will now find what 11 units become when 4 units are taken away. If i of the 4 units be taken from II, the result will be 10 ; if i of the remaining 3 units be taken from 10, there will be 9 units left. Again, take i of the remaining 2 units from the 9 and 8 will remain ; and, finally, take the i remaining unit from 8 and we have 7 units left. Thus we see, that if 4 be taken from 1 1 there will be 7 left. This process of finding the number of units left after taking a certain number from a greater number is called Subtraction. The greater number, as the 11 in the above example, is called the Minuend, and the lesser, as the 4 above, is called the Subtrahend. That which is left, as the 7 above, is called the Difference, Remainder, or Excess. I ) ri •.■i: 32 |f> AlilTUMETW FOR BEGlNSsWi. Tlius, lo - 3 = 7. ''^ ~ '^ "^^''oJ Minus. nieII;s%L'"''3~'^fs",,'f'"»f 'I;--- equal ,0 7, and will be left. Hew ,„ i. ,1 .." '""" '" ""i's? unit, ^'- "■':«:&, '^^'.l.f;;^^^''^ P-feCty understood and 160. The sul SUBTRACTION TABLE. 3 from 3 leaves o 4 from 4 leaves o 5 from 5 leaves o «4';n.SiS;r-^IL,^lX!;;" *oWn7^ tluH tiihio by lu n SUBTRACTION. 83 tign - between d Minus. qual to 7, and| f> "nits 7 unit^ 3 is the Sub nderstood and 5 from 5 leaves o 6 " 7 " 9 lo II 12 15 I 2 . I 3 I 4 I ■7 I 8 I 10 ' 10 fro m 10 leaves 11 i( I 12 t< 2 13 it 3 14 «( 4 '5 « 5 6 (t 6 7 (1 7 8 « 8 9 20 — 6? 21—4? 6. Five balls taken from 1 1 balls leave how many ? 7. Six cents from 20 cents leave how mafiy? 8. How many are 7 — 5? 17—5? 27 — 5? 9. How many are 9 - 6? 19-6? 29-6? 10 What numbejr added to 8 will make 12 ? u. What number and 9 make 13? 14? 15? 16? 12 Subtract by 2's from 24 to o. \ . i I 84 Ik 'II ' t i 13- 14. '5- 16. '7- 25 26, 27. 28. 29. 30- 3I' 32. ARrrriMETJc Fon deoisners. In the same manner, subtract By 2*s from 25 to r. ^ 1 m R„ .-o f ^» • ' -^ ' 19- tJy 4 s from 41 to i By 2's from 31 to 3. ^\v 3's from 30 to o. By 3's from 37 to i. By 3's from 40 to 4 By 4's from 44 to o, 20. By 4's from 51 to 3. 21. By 5's from 60 to o 22. By 5's from 63 to 3 23. By 6's from 66 to o.. 24. By 6's from 65 to 5,! 33- 34- 35- 36- 37. 38. 39- Count by 4-s from 2 to 58, and back from 58 to 2. Count by 5's from I to 61, and back to I. Count by 6's from 3 to 69, and back to 3. Count by 4's from 5 to 53, and back to 5. Count by 6's from 7 to 67, and back to 7. WhaHs the difference between nine dollars and fifte. ^^tr:h:;-r^^-^[s:^--f'-oth. ^em:in"" '^^'^ '^°" ^^'^^^^ >-^'^ ^ '-w many yards S■'!!".„^°"S^^,^''^^^^" '^•arbles, and brother seven of them: how he gave hisJ M .ary had fifteen examples to work nine, how many had she then to d Take seven b will remain? many did he keep? out: after finish; o? "g seven books from thirteen books: and how many! nierc^hant^had nineteen barrels of fl s, and kept the rest: ho twelve barrel keep? our; he soldi w many did he] 40. 41, 42. Ge( spe Tin and the] TaJ. niai 43. Mai oldt 44. A rr it a moi 45. Jam gav had 46. A b afte 47. Joh mot mai 48. A : doll he 49. A r sorr dol] hov 61. To SI the ■I nor cane 62. Henc ber; uni tho SUBTRACTION. 85 s from 41 to 1. s from 51 to 3. s from 60 to o. s from 63 to 3, s from 66 to o. 3 from 65 to 5, 1 )m 58 to 2. 5- 7- lars and fifteerf 'ek, and spenij old, the otherl leir ages ? w many yardsl ven dollars of'^ !r for paper: I market ; shp ■ ow many die 40. George had seventeen cents ; he lost nine cents, and spent the rest for ink: what did the ink cost? 41. There were fourteen books on one shelf of a book-case, and eight ou another: how many more books were there on one shelf than on the other? 42. Take eight cherries fn.-ni fifteen cherries: and how many will remain? 43. IMary is sixteen years old, and Jane seven : how much older is Mary than Jane? 44. A man sold a cart for eighteen dollars ; he received for it a barrel of flour worth nine dollars, and the rest in money: how much money did he receive? 45. James's father gave him ten cents, and his mother gave him nine: after spending eight cents, how much had he left? 46. A boy had thirteen marbles ; he bought five more, and afterwards lost ten: how many had he then? 47. John had eight books, hif father gave him five, his mother two; he then gave four to his brother; liow many had he left? 48. A man had sixteen dollars ; he gave away seven dollars and afterwards earned nine : how much had he then? 49. A merchant bought some cloth for nine dollars, and some silk for five dollars; he sold both for sixteen dollars: did he gain or lose by the bargain: if so, how much ? he gave hi^ ; keep? ifter finishing! id how many! ur; he sold nany did liel 61. To subtract one number from another they must be of the 3amc kind. .S;V'.-=~5 dollars from 8 apples leaves neither 3 dollars nor 3 api-ies. In h'c same way, 3 units from 7 thou- eando leaves neither 4 units tior 4 thousands. 62. Hence when we subtract, we should always write num- bers of the same kind in the same column, that is, units under units, tens ander tens, thousands under thousands, etc. 3A 63. We thus have the following RULE FOR SUBTRACTION. -ret [r„,i5j,e"i;Sef '"^ ^"'''^'-'^ ^-'^ -fi"^. I. —Subtract 238 from 749. Minuend 7 4 9 Subtrahend 238 Diiferenoe 5 ' ' ^nn^SlT *^^ "umbers, units under units, tens under tens, and hundreds under hundreds Then begin at the units' column and say: 8 xmits frJm n coCxh^ ""*; ^"'."^ ""^^ '"^'^ under the ur'ts^ column. Then 3 tens from 4 tens leave.; t f«^ 1 j we put the I under the ten's column Then 2 hundred. from 7 hundreds leaves 5 hundreds, and vve place the Sounder the hundreds' column. The answer bjng In practice we shorten the work thi,« • a t leaves i ; 3 from 4 leaves r ; 2 fTom 7 leaves /T ^ ing altogether 511 ' leaves 5 ; leav- To prove the correctness of the work, we add 2,8 749'andlt°a^*ny.^^ ' ^ ^' ^^^ -'^^-^ "L^ SUBTHA'niO.S. N. nder tfu greater, tc. , and draw a 't the one above, middle num- r, and if the same as the ct the lower ihouJd be tlie The reason for this is as follows : 4 from 9 leaves 5 ; and if we add the ^ back again to the 5, we must obtain the 9 we had at first. Again, if 4 taken from 9 leaves 5, the pupil will easily see that 5 taken from 9 must leave 4. Hence the proof of the work. Ex. 2. — Find the difference between 3065 and 78195, 78195 3065 75130 Here 5 from 5 leaves o, which is placed under the units' column ; 6 from 9 leaves 3 under the tens' col- umn ; o from 1 leaves i under the hundreds' column : 3 from 8 leaves 5 under the thousands' column ; o from 7 leaves 7 in the ten-thousands' column. EXERCISE 10. >nits, tens eds. Then nits from 9 2r the units' 1 ten, and 2 hundreds 2 place the swer being 8 from 9 5s 5 ; leav- « add 238 t 511 from (I) 469 327 (7) 6408 3207 (25) 546875 513213 (4) 8072, 3051 (5) 2741 1301 (6) 5462 1350 (8) (9) 8420 8742 3110 6331 (lO) 7839 5427 1243 123 (12; vi3) 4785 86493 1053 34272 (14) (IS) . (16) 972897 985094 987657899 120341 382040 123456789 (17) 99797' 36 89762^12 (18) 9892976 4730834 (20) 89487 32315 (26) 347985 323415 (21) (22) 75659 87392 32417 43181 (23) (24) 75285 88456 43151 32142 (27) 973856 951231 (28) 825944 8I25I2 1^9) (30) . 756345 914756 713125 902314 it .iiiiTHMETic Fun ni:oiNNj:jis. (31) (32) 41876 38789 31023 13321 (37) 418764 213321 (38) 13912 13311 (33) 64187 33123 (39) 67134 32132 (35) 187137 123013 (41) 91276 21230 (36) 69123 32122; 4i87(. I 31232 46- Thomas having 447 busliels nf „„.^, bushels of them to IW L 1"°'='"^ ■ =<>" 234 Thomas remaMng? ^ ' '°" """>' '"'=''^'= ''"d *'• of oxeTLrZt' VP"' "f ""''-^fo' S346,and a yoke hors"th";„l;,".L'rc"r''' '""" ■'"' "^ «'- f- "- *'• ma^yT/d ttLit^l^ngl^^"' ^"'^ "33 of .hem. how 49- A gentleman owns a store worth ^^finc o«^ .heo.^'?h?„lf'rotS """" ■""" "'" '-^ S'- f- *'■ nervfor S.L'.'?'.^!:' '"""^ '^"<' ^^ ^''«97. and a tan- S the tt'Jfcfr;'?'"'" ™''^'' -"-^ '3'd the land cost 52. A nierchantthavingQ847vardsofrlr,f>i o^m o of it : how many /a?dfh^^ he remaining ? ^*** '""'' 53- A drover bought cattto to tbcj ario-nit of n/;.^ j 11 and sheep to the amount of j« .Mars ? htL ?' more dtd he give for the cattfe^'ilanloMhe sl*ep ? 54- 55- 57- 59- 62. 63. 65. In acDTRAcnos. 8'J (.36) 32122 (4= 4i87t 31232 ixn to his play- ^ ght 184 ; how w much more es, sold 234 bushels had 3, and a yoke give for the them ; how J and a grist- store worth )9. and for a I he fjive foi , and a tan- e land cost 5844 yards 547 dollars, how much sheep ? 54. Two men jui itlv built a mill fi r 7856 dollars; onefui nishcd 4520 dollars: w hat did the ether furnish ? 55. The earnings of a factory f^r a year were 456«9 dollars " and the expenses were 21352 d'^Hars: what were the profits ? ^6 The gross receipts of a railroad were 357845 dollars, ^ and the running expenses for the same time were 213423 dollars : what were the net earnings? S7 A. has a grist-mill worth 1875 dollars, and a saw-mill ^^ worth 1032 dollars: huw much more is the one worth than the other ? 5S. A farmer had 3672 sheep and 2312 lambs : how many more sheep had he than lambs ? so A man was driving 534 g ese to market, and on the way had 2 1 stolen from him : how many had he remam- ing ? 60 A farmer had 327 bushels, of oats, and sold 125 bushels of them : how many bushels had he remaining i 61. A merchant in one year sold 18972 barrels of »lour and 7370 barrels of sugar: how many more barrels ol tlour did he sell than sugar? 62. A ship is valued at 547B9 dollars, and its cargo at 40357 dollars: how much more is the ship valued at than the cargo ? 63. A gentleman having 57789? dollars gave to his eldest son 16805 dollars: how much had he remaining!' 65 In all the former examples, the figures of the larger number were either greater or c cpm to t je correspond- ng figure in the lesser number. We will now consider Ihose cases in which the figures of the larger number may be less than the corresponding figures ot the other. ^^.— Subtract 695 from 932 932 - 900 + huiidreils. 9 + 30 tens. 3 + 2 unity. + 2 M I m 40 ^niTHMETlC FOR DEOimERa. In the same way, huu.iro.i8. toDi 695 = 6 + The numbers then stand unit* 9 + 5 58. B. 9 6 T. n. + 3 + 2 + 9 + q ^ + 3 + 7 Now, we cannot take c from 2 ..n;' « ^'""• the 8 hundreds we do so and nl.i'?.^ hundreds from usual. ' ^"^ I"'^^^ the 2 hundreds as The work may be shown thus • °- "^^ ^- H. T. ' D. 932 = 9 + 3 + 2 = 9 + 2^,2 = ''37 = . . . _ , ^ b6. By another method, it is u-^.ml ir. 1^ necessary,tothefifure the figures .f the Subl^a^^ndf" ^^^at we only change 67. This system, known as carrvinff onf. Jc 1 wh,.e workin, a „.es.,on i^^^^^^Z,^^^^ '0 SdBTtUCTlOIf. 41 fienee we inJ lis maJvin;3f iti ■aves 7 units, we took I oi have only 21 id here, sincc-l e I of the 9I tens, makingP :aves 3 tens, taken i hun only 8 liun- ndreds from hundreds as T u. ■ 12 + 12 9+5 one, when i instead of inuend. 'ig 9 tens IS from i^ esuJt ; and ds, we say iy change ■ mentally d depends on Ihc fact thai i unit of any order is equal to 10 units of the next order to the right C)f it. ^8. We thus have the following RULE FOR SUBTRACTION. 1 . IVri/e the less number under the greater^ placing units under units, tens under tens, etc., and begin at the right to subtract. 2. Subtract, if possible, each figure in the lower line from the one above it, and set the remainder below. 3. If any figure }.n the lower line is greater than the one above it add 10 to the upper figure before subtracting, and diminish by I the rext left-hand figure in the upper line, and proceed as before. PRf'OF. — Add the remainder to the subtrahend; the sum should be equal to the minuend. '0 EXERCISE 11. 663 580 (2) 976 531 (3) 704 483 14> iSofi (5) 572 259 7238 4854 7580 4245 (la) 4300 3451 (8) 52836 28371 (13) 5491 4542 (9) 400500 215327 (10) 4236 3089 (11) 6170 5552 (14) 6180 2435 (15) 4192 1435 (16) 80502 38672 (17) 927381 345432 (18) 917183 421354 (19) 618190 234221 (20) 519080 324121 (21) 924390 432412 (22) 705180 443544 (23) 527082 232154 (28) 826041 434425 (24) 816141 135212 (25) 423453 141514 (26) 732250 241 341 (27) 734271 241342 (29) 46095 2873C (30) 555555 12345C i!i ib) I 2 3- 4- 5- 6. 7- 8. 9- lO. II. 12. 16. ^rt//7/.l/&Tyc/'0/Ji,AV/A.VA7,s. I. 2. 3- 4- 5- 6. 7- 8. 9- 10. II. 12. ^3- ^5 16. 17. 18. '9 20. 21. 22. • Fiom 854 take ^y^. ■ i'rom i7(jy take 1732. From 54y6 subtract 1492. I^rom 1584 subtract 920. ^rom 5672 subtract 2356. From 74760 subtract 39817 • From 8416 subtract 2918 From 30S11 subtract 13240. From 27880 subtract 9226 From 35846 subtract 12829 From 75901 subtract 17980. From 37229 subtract 17991 From 100304 subtract 62818. From 1000302 subtract 888772 From 892201 subtract 300998. i^rom loooooo subtract 322333. Find the value of 75S901 ~ 349806. 329500 - 54650. • 720065991 _ 12095890 . 1 0000 - 390. ^ ^^ 189501 _ 188605. 756-25 - 24319. 786 !9ooo - 17664508 i370426oi9-8205i2oc<; 97001 - 50077. ^^' 76734 - 977. 56400 - 1 00. 700000 -- 99. 5700 - 500. • 9777 - 89. . 76000 — I. 9(n for $29468^ I he gain ? I 23927 acres: lilTBin ACTION. 4o |i. If I borrow of my neighbour $9673, and pay him $ggg of it: how much remains unpaid? \2. A man has a farm of 400 acres; part is woodland, and part is cultivated ; the former part is 125 acres : how much is the latter ? The distance from the earth to the sun is about 95000000 miles ; the distance to the moon is about 240000 : how much farther is it to the sun than to the moon? [24. If I bought a ship for $42650, and sold it for $49000 : what did I gain ? I25. A gentleman gave $12462 for a house and some land ; the house alone was worth $9375 : what was the value of the land ? '26. A lumberman having 65* > ,et of boards, sold 162372 feet of them : how many ieet then remained? 27. The battle of Inkermann was fought in the year 1854 ; the Peninsular War was begun 46 years before this : in what year did the latter war begin? 28. A man having $100000, gave away $365 ; how much had he left? 29. A merchant owns property to the amount of $45563, and owes $21209: how much is he really worth ? 30. If two candidates for office received in the aggregate 73462 votes, and the successful one had 45309 votes; how many did the other have ? >ur, sold 492el| 31- 'hich was 50^ »ow many jjaJ 5^38967, wliicS ow much (jii J 7962 feet One province contains 55405 square miles, and another 50914 square miles : how many more square miles does the one contain than the other? 32. Mount Sorata, in South America, is 24812 feet high- and 21 241 feet higher than Mount Snowdon, in \Vales ' how high is Mount Snowdon ? 33. Bonaparte was declared emperor in the year 1804, when he was 35 years < " born? ige year 34 Sir Isaac Newton was born in the year 1642, and died 1111727: how old was he when he died? 44 35 ^r^rrnMETic foh nEGit^xE,,, 36 i7- 39- 40 41 43 44- 45- |!^^^it^ti:;^;i:-^^-ec,i„p,.,e houses the discovery of Hnc' °' '"^'"^^^ ^^s 516 years afl ;^at year wa^s ti'S^^^ T^Ldlf f^A Gunpowder was invented bv 9 ^^''' "^^^^"^ how Jong was this befb e fh??''" "-^ *^^ ^^^^ ^sJ which was in 1440 ? "'^ '"vention of pri„ti| p^d^s^-:,^^^.^thirt^^^^ three n.i,ii.„ nineteeVtL.'u:a:'d':.^dte?^"^^ ^^'^' ' Sicily, which is 10950 feft high? '" Mount Etna.f How;^k!!;f^^^;-|-^-A.erica.is.48.feethiJ 's 17000 feet high ? ^ ^^^" M^""t Ararat, wh!3 • ^h? highest Jand in North An, • which IS 15000 feet hH;i"^7H^.^^^ Mount feet higher than Mount W.^'? """"""tain is 871J St ' Pete'r '''' " ^"""^ ^^-i^-^^^^ ^^" "^''! '57 feef high^e^tha^n S^"!^' ^^'"^^^ '^ 45o feet hie?, i, what is theLight^o7th'e'fe:Sf ^-^-i. « !• A man wiJIed to his son«, « niore than he willed foi. 7^7496, which was "RosJ he will to his dlS^?'^ ^-ghters : how^Sj?d| • A man owninc o-*.^ Jan.es has .:;i ^^^.^.^^^--r J^J-i. hav^™^^? 1 1 I I * ,1 f.9. MULTIPLICATIOX. ♦7 MULTIPLICATION. 19. The numbers that were added together in the examples in Addition were nearly always different. We now come to a short method of adding together numbers that are the same. £X. 3+3 + 3 + 3:^,2. In this example we have the number 3 taken 4 times, giving 12 as the sum ; but instead of finding this sum by the usual process of addition, we obtain the same by saying 4 times 3 are 12. Again, 5+5+5 + 5 + 5 + 5 = 3,), which result is the same as saying 5 taken 6 times gives 30, or 6 times 5 are 30. 70. This produces a Table, which may be obtained by ordi- nary Addition, for or 71. I + r = 2 2 + 2 = 4 3 + 3 = 6 4 + 4 = 8 5 4- 5 = 10 6 + 6 z=: 12 7 + 7 = 14 S + S = 16 9 4- 9 =: 18 10 + 10 = 20 ir + II = 22 12 + 12 r:^ 24 Again » I 4- I 4- I = 2 + 2 4- 2 = twice I ire 2 << 2 3 4 6 << 4 8 i< 5 10 << 6 12 11 7 8 H 16 << 9 18 i( 10 20 i< II 22 C4 12 24 3 or three times i are 3 6 " " 2 '• 6 In the same way four times 6 will be found to be 24; five times 7 will be 3^, etc. These results v/ill now be pl"ced in the form of a Table, called the Multiplication Table, which must be accu- rately memorized by the pupil. ■ 't ' 1 M AmTHMETIO FOB DEomiri^r., M ULTIPLICATION TABLE. 1^1 tI2. Mult the the] The n the The 11 tipl Thefi |73. The s by. by ( is t IS t the due The p 66, ma} tog< 74. The f kno Since rep cen i gb Since 7 di MULTIl'LICATIOii. 40 ]2. Multiplication is, then, a short method of Addition, or the art of repeating one number as many times as there are units in another. The number which is to be added or repeated is called the Multiplicand. The number which shows ihe number of tines the Mul tiplicand is tcj be repeated is called the Multiplier. The final result is called the Product. [73. The sign of this operation is x , and is read multiplied by. Thus, 1 1 x 6 = 66, would be read ;i multiplied by 6 equals 66. Here ii is the multiplicand, for it is the number which is to be repeated six times. 6 IS the multiplier, for it shews the number of times the multiplicand ii is to be repeated. 66 is the pro- duct, for it shows the result of repeating 1 1 six times. The pupil must clearly understand that while the result, 66, is supposed to be remenbered from the Table, it may be obtained by adding, in the same way, six ii's together, that is, II II II II II XI 66 Ans. 74. The following Mental Questions will test the pupil's knowledge of the Multiplication Table : Ex. — What will 8 peaches cost at 6 cents apiece ? Atis. 48 cents. Since i peach costs 6 cents, 8 peaches must cost 6 cents repeated 8 times, or 8 times 6 cents, which will be 48 cents. Ex. 2. — If a barrel of flour costs 7 dollars, what will g barrels cost ? Ajis. 63 dollars. Since 1 barrel costs 7 dollars, g barrels must cost 9 times 7 dollars, that is, 63 dollars. ■■\ 1 ;} «0 ^^ITBM,mc FOR BEOISNana. TheC' " ""''" """"^^ ''"""■""«- """"P'-^'' MuJlfpSUrT^ht '*''!. '''^ '''«'"="' products in ,ll EXEECISE 18. W. Copy on ,our sia.es and M,„„^,foU,„.„^^ 5 are 15, etc. ^ 5 is 5. 3 t.nies 5 are 10, 3 times 3- Repeat from onrp fi ♦ times 6 ,0 o^ce 6 ^ '" '° '""« «. and back from ,o 4- Repeat from once 7 to ,„ ,i„ 5. -peat fro™ once l: :::;:::-:-- ^e same kind le multiphcan] : products in th umbers that ar roducts. f. 6 and 7. • 13 and 9. >wing • 9X 4 9X o lox 5 f2X I : 10X10 : IX 9 r 8x 5 = IX 3 = 2X 2X 5X 'X 5x >x X 2 = 3 - 4 = 4 = 5 = 1 = 7 = X 4 = X 5 = X 10 = "ce 5 to 10 io> 3 times ck from 10 ck. ck. MOLTI PLICATION. 51 6. Repeat from once g t(j lo times 9, and back. 7. Repeat from once 10 to 10 times 10, and back. 8. What two numbers produce by multiplication the fol- lowing numb(irs: 2,8, 12, 40, 60, 72, 12, 11, 22,36,24, 44. 54. 77. 81, 96, 55, 84, 108, 88, 132, 120, 64, 49, 56, 63, etc. ((J) I. A quart of berries makes 2 pints: how many pints in 8 quarts? 2. There arc 12 inches in one foot length of rope: how many inches in 9 such lengths? 3. If 1 lemon costs 7 cents, how many cents must you pay for 4 lemons ? 4. One gallon cor^^ains 4 quarts: how many quarts are there in 11 gallons? 5. One peck contains 8 quarts : how many quarts are there in 7 pecks ? 6. What would 12 pears cost at 7 cents apiece ? 7. How much would 5 bags of meal cost at 9 dollars a bag? 8. At 10 cents a pint, how much would 8 pints of cherries cost ? 9. At II cents a pound, how much would 7 pounds of sugar cost ? 10. In one week there are 7 days : how many days are there in 1 1 weeks ? 11. At 8 cents a quart, what would be the cost of 7 quarts of berries ? 12. In I florin there are 45 cents : how many cents would there be in 7 florins ? 13. At 3 dollars a bushel, what would 12 bushels of eraoes cost ? ^ *^ 14. There are 4 gills in one pint : how many gills would there be in 12 pints? 15. If a horse runs 12 miles an hour, how far would he run in 5 hours ? •' i 1: I S i ■ ' i M 52 ^niTUMHTIC Foit UKUlSSEns. i6. ^" one yard there aiv, feet- I,. "■ " un^dT^" '^""'^ « PO""''^ ..f ...cu, CO. .. 6 e. worth ? °""""' ^^^at would 9 ounces of blue be ^^" S^inTf^^^^^^^^^ ^- -„y feet would there ' bui^hes'jSt?"^ ^'^^^' ^°^^^ 7 cents, what would y | ^7. At^xo cents. g,u, How ™ch would ..,uis Of „,,„ ^'- .t^^n^Stte "^ * ''-"^ ■• "- -n, pec. „e ''• b4i"o?'L^ets^'' ™-^ "-"^ ".us, ,ou pa. fo. , 30- There is a e-ard^n rwf ,,• in a row: 'o^^^an^y ^^h^arelh^e"'^-^' T^^ ^^ ^"^^P^ 31. Each step in a flight of! '" '^'^ ^"^^^^ ? how n.a„y inches ^ill you a'sceVd fn'?" "^^^^^ ^'^^ : 32. If you work eleven exaLl , '" ''"P^^ you work in six day' p^^^ ^^^^^ Jay. how many will 33. Hown.anyareseventin.es seven pounds? '" tht'^in%ix^^:::,tp^-"--^ ^--anydaysare I I MirijTinuoA Tiny. 58 'any incJies in 35. What will be tho cost of seven pounds of raisin?; at nine cents a poiind ? 36. A slieet of paper can be folded so as to make fonr leaves: how many leaves will eleven such sheets make ? 37. What is the cost of eight pounds of soap at seven cents a pound? 38. There are twelve inches in a foot : how many inches are there in twelve feet ? 39. How many inches in eight feet ? 40. If one barrel of flour costs nine dollars, how much will eight barrels cost ? 41. If one shot weighs six pounds, what is the weight of seven such shot? 42. What is the weight of eight packages of coffee, if each weighs five pounds? 43. If one pound of rice costs twelve cents, what is the cost of seven packages of a pound each ? 44. What is the cost of twelve pair of boots at six dollars a pair? 45. I bought eleven pounds of glue at eight cents a pound what was the cost ? 46. William had six cents ; his sister gave him three more, and his mother gave him seven times as many as he then had : how many did his mother give him ? 75. The pupil is now supposed to be quite familiar and ready with the Multiplication Table, and we will therefore go on with the different cases that occur in multiplication. First we will multiply any number by a number of ono figure, or by any number fr.-m i to 12. Jix. — Multiply 3781 by 7. Multiplicand. 37^ ^ Multiplier 7 ■ » i i Product 26467 AnTTrnrrrrn ran nEom^Ena. ^"J 5 tens; place the 6 tens nHl '^'''- '' ^ ^""^'«^1 "rry the 5 hundreds. 7 ■ ^^ '1 tlT^V ^'"^''' ^'^1 drcds, which, with thn c I 1^ hundreds are 49hun. 54 hundred, or tlutan'dsTnlTf "f ^/-'ed! mi;; 4 hundreds in its own ntre /n V"'^''"^" = P'^^^*''^ sands. 7 times 3 thou arfds a;e " th.'""^ '}'' ^ tho„. 5 thousands we carried mak'oA.."'^"^"' ^"Jthe placed on the left. ''^^ ^^ thousands, wJiich is 76. This may be dnno r« ".e woJ^d. u-il?: ™-^ ;«"y hy ..eglec.,„g f,„ „„ „■„, Thus : 7 times i are >? u„t j 56. put down the gL ujr'! ^'l^ 7' 7 times 8 arc 5 carried) maKe 54!put'd:n^n' hrrand?" ^^i ^^^ 7 tm.es 3 are .,. and the 5 (carded/make'^s"^ *'^ ^ •' ihe pupil should be satisfied th«^ fh have been obtained b/Add/ttVihur^ ''''''' ^°"^'' 3781 3781 3781 3781 3781 3781 3781 26467 not exceed ij, we hav< n! RULE FOR MULTIPLICATION. Pro^ucr-jusr as in Adman ^^"'"' '^ ««>'' '<> th, next (a) I. 2. 3- 4- •7- MOLTlI'LlCATIOlt. H '^ 37fiT, and ts' place, n in tlie usual 5 hundred P'ace, and a''e49h„„. rned, make ." place tlif t'le 5 thoii. rfs, and the s, which is "r ^he time mes 8 arc 'e 49, and J ry the 5 ; 5. 5lllt COUJd ve hai 4;' t/ie ''roduct, 'le next EXERCISE 14. 2. 3- 4- 5- 6. 7- 8. 9- 10. II. 12. «3- 14. «5- 16. 'T- IS. 19. 20. 21. 22. 23- 24. 2j- 26. 27. 28. 29. 3°- 31- M\iltiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply 123 by 134 by 223 by 246 by 278 by 495 by 1312 by 2172 by 3629 by 3785 by 2 4006 by 2 4308 by 2 142034 by 2 1706324 by 2 3614503 by 2 462178 by 2 1203062 by 3 21607835 by 3 93420 by 3 705086 by 3 1039246 by 4 217906 by 4 509367 by 4 567239 by 5 6146802 by 6 4601792 by 5 962078 by 6 729360 by 7 4286072 by 7 237000 by 7 23416 by 2 32- 33- 34- 35- 36. 37. 38. 39- 40. 41. 42. 43- 44. 45- 46. 47- 48. 49- 50. 5^- 52- 53- 54. 55- 56. 57- 58. 59- 60. 61. 62. (65) 3546 II Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiply Multiplv Multiply 45613 by 4. 34520 by 3. 56042 by 5. 21264 by 2. ^.: i53 Sy 6. 32305 b> 5. 5 7. "02 by '. :'o*.i5 by i. gy: Gb, 6. »2i37 by 7- 23460 by 6. 68913 by 3. 57802 by 2. 62819 by 5. 93856 by 6. 28475 by 4. 39586 by 5. 40697 by 6. 17364 by 3. 51708 by 4. 5876 by 4. 8546 by 7. 502 by 9. 246025 by 5. 512604 by 8. 648 by 7. 1082 by 9. 5050 by 4. 73046 by 3. 10708 by 2. 980789 by 8. (66) 5354 12 (67) 81897 II (68) 900867 (69) 8602968 II 12 (70) 716914 12 (71) (72) 765439 8419829 II II I; V ill ■11 I 56 (a) ARITHMETIC FOR BEOUfNERS. (78) 666666 lo 85 86, 87. 88. 89. 90. 91- 92. 93- 94- 95- 7- 5- 83- 82386 X 84- 357 X . 8645 X 8. 2079 X 9 8S42X 4, 3749 X 7. 13146X 9. 876 X 10. 2345x12. 998 XII. 8134X 12. 7312X II. 8183x12. (^)i. (74) 3443 II (79) 9999999 12 96. 1 8889x12. 97- 18476x11. 98. 437958 X 7. 99- 760281x11. 100. 194514X 8. loi. 426847x12. H'2. 859170 X 9. 103. 482403 X 7. ^04. 715736x11. 105- 54S069X 8. ro6. 871392x12. 107. 204625 X 9. ^^^8. 537958 X 7. (76) 77777 II (81) 575757 12 0^77) 888880 12 (82) 48484^ II 109. 960281 XI I. "°- 593SHX 8. 111. 926847x12 112. 760281 X 9. 113. 104748 X 7. 114. 327071x11. 115- 650304 X 8. 116. 382637x12. 117- 6^6960 X 9 nS. 438082 X 7. 119- 871406 X IT. 120. 763867x12. he^e'cdve t'?f?em ""^ '' *^ ^P'-= ^ how „uch did r n '' 't' '"' °' "* ""'^'^^ ""'■ ^' *6a barrel > 3. What ,s .he COS. of .gS; acres of ground, a, ' an 4- 5- Wha. is .i,e cos. of 4786 barrels of flour, a, $9 a barrel ? 'here^'Utr.''" '"'° '"' ^ ''" "-y f-' arc 6. In I mile there are 1760 vaMc . u . therein 5 miles? '7^°^^^^^' "-^ many yards are 7- 1^9 men can sow a farm in ih ^^ • 1 can one man do the same? ^"' '" ^«^"^any days 8. If 6 men can build t wall i„ , j days can one manl^dd'tt "anit^'aTp '" '"^ ""^"^ ^" f,lf l^"'^'i^' V^F^'- ^'ll feed one h how many bushels for the same time ? ^,;i 1, ■^'■se 18 month! wul be necessary to feed 8 horse 1 1. 12. 13- 14. 15- 16. 17- 18. 19. 20. 21. 22. 23- 24. 25- 26. 2?. MULTIPLlCATlOlf. 57 (77) 8888^0 12 (82) 48484S II i028lXll. I35HX 8. 6847x12. 0281 X 9. 4748 X 7. 7071x11. 5304 X 8. 2637x12. J960X 9. io82x 7. 406 X IT. 867X12. luch did barrel .? \ j t $9 an I barrel ? feet are rds art- ay days '^ many lonths: horses TO. I bought 245 cords of maple, at $7 a cord : how much did the whole cost me ? 11. A dealer sold 8 animals, at $253 apiece : how many dollars did he receive for them ? 12. A girl bought 189 yards of ribbon at 6 cents per yard : how much did it cost her ? 13. What is the cost of 2988 boxes of figs, at $3 a box? 14. If a steamer can go 395 miles in one day, how far can she go in 9 days at the same rate? 15. There are 10 companies in the Queen's Own Rifles, Toronto, each having 42 men : how many men are there in the regiment ? 16. In one mile there are 5280 feet: how many feet are there in 4 miles ? 17. If a mill turns out 9757 yards of carpet in one week, how many yards could it produce in 5 weeks ? 18. If a ship can carry 7856 barrels of ore, how many bar- rels could be carried in 6 ships ? 19. If a waggon can carry 5837 shingles, how many shingles could be carried in 7 waggons ? 20. What would 8 miles of pavement cost, at $3489 per mile ? 21. Nine men built a vessel, each one putting in $8457: what was the cost of the vessel ? 22. If a barge can carry 19857 pounds, how many pounds could 4 such vessels carry ? 23. There are 63360 inches in a mile : how many inches are there in 5 miles? 24. How many miles would a yacht sail in going around the earth 6 times, the earth being 24855 miles in cir- cumference ? 25. What would be the cost of constructing 7 miles of embankment, at $35248 per mile? 26. If II clergymen are paid $2212 dollars apiece in. Toronto, what do they receive in all ? 27. What is paid to 12 teachers in Hamilton, at 1 / of $862 each ? rate 58 m W ; I ARITHMETIC FOR BBGINSERS. ''■ ?n ;' Ssf " "" '^^ f-'^ "- "-any fee. ,„ ,h.r. ''• thei?;;'!™';?;?" '^^ ^""^^ "- '---y y^ds are "■ Hidt'reteS; ^Ll'?^ ^"'^"^ ""« "-y dollars 33. W,,a. is the cos, of ,786 boxes of grapes, at $. a box > much did lie pay for them all ? apiece : how ''■ mXT:LI'^'jJ%Z'%l°^t97^pi^e.. how much "■ 'hVre^bei'^^^5".\Xl,s'r^'=^ ""^ -ny quarts wi„ ''■ n't:;%;^e?od:rfhere^y»,' -;na„d= ^^^ '"■ pore'^rtLSn^'r""^ ■" ^^^'^ ^""'»«'.at .. '"■ IVJTLtl'rsr ''' ''''■■ ■>«- ">-y days are when repeated ,49 tfmes, yet we m"v 'jjf .^ b=com. convenience, look upon V as ^h. ,i^^;. T- ""^ ,"^''« >>' duct being dollars, multipUer, the pre i " '— K."..:I,SS;;;~ ' •»« .. ...» ., MULTIPLICATION. f» et are there ^ yards are meal, ho;v iny dollars Its a yard : $2 a box ? ir can she see : how low much uarts will tid: how gs. at 12 days are < of land ' become 5 sake c{ the pro- lich the Ex. I.— Multiply 1396 by 364, that is, find what 1396 becomes when repeated 364 times. 1396 364 1396 X 4= 5584 1396X 60= 83760 1396x300=: 418800 Or, 5584 8376 4188 508144 508144 The number 364 = 3 hundreds + 6 tens + 4. units, hence the multiplicand is to be repeated 300 times, and 60 times, and 4 times. If we then take 4 times the multiplicand, and 60 times the multiplicand, and 300 times the multiplicand, these results added together must give 364 times the muhiplicand. By the previous case, 4 times 1396 gives 5584. This we put down as usual. Again, 1396 multiplied by 6 tens is the same as 6 tens by 1396 (Art. 77), and this we find to be 8376 tens, which result in the addition must (since it is tens) be put one place to the left of the last result, 5584. Again, 1396 multiplied by 3 hundreds is the same as 3 hundreds by 1396, and this is 4188 hundreds, which must therefore in the addition be put one place to the left of the tens' result, 8376. Having placed these three results ready for addition, nothing remains but to add them together in the usual way. We thus find that 1396x364=508144. In the same way we proceed with any number of figures in the multiplier. £x. 2. — Multiply 872 by 307. 872 307 872 V 7= 61 872 X = 872x300 = 261600 04 00 Or, 6104 267704 26160 267704 flO ■ililTHMETIC FOR BBOltfNKRS. Here 7 times 872 gives 6104. xhere are no tens in the multiplier iience we might have filled the usual line with noughts, but one nought is enough to keep the next H'sult m its proper plac. 3 times 872 gives 2616 and this being hundreds, it must be put one place t(i the left of the nought, v.hich makes the tens' place. A jd as before, and we find 872x307 = 267704. £x. 3.— Multiply 371 by 2100. 371 2100 37100 742 82. vSu >y m M 79. 779100 In this example there are no units or tens in the multi- plier, therefore the first result, 371, must be placed to represent hundreds, that is, thre^ planes t- the left of the units' place. The next result, being thousands VIZ 742 thousands, it will be put one place to the left of the last result. Add together as usual. We see from this that any number may be multiplied fL'u' '°°v.V'?fu ^^'^•' ^^ ^^^'"^ ^' 2' 3. etc., noughts to the right of the number to be multiplied. Thus, 389 X 100 = 38900. 40 X 1000=40000. 80. As the^e are 100 cents in a dollar, this principle is very useful in expressing any number of dollars as cents. £x. I.— How many cents are there in $84 ? There will be 84 times as many cents in 84 dollars as there are cents in i dollar, that is, 84 x 100 = 8400 cents • ^/' I'^T^^"''^ ^'"'"y "^^"^^ ^^^ there in $64.52 ; that IS, 64 dollars and 52 cents .? '^ ^ :>''> in-t $64 = 6400 cents. 5400 cents + 52 cents = 6452 cents. 81. Hence, to express any number of dollars and cents as rrilcViJ^'^'ir "?^^' '" ,'^"'''^^^ *^^ P°^"t which sepa- rates the dollars from the cents, and the result will be the required number of cents. ^.v. 3.— $106.97 = 10697 cents. 83. Thi o 84. An U: g< si m ia) 4( MULTIPLICATION. 61 ;ens in the usual line keep the ^ives26i6, ; place ti) i' place. he multi- placed to be left of ousands, o the left lultiplied noughts e IS very cents. )llars as 30 cents. 52 ; that 82. Suppose we have to multiply any number by 16. We know that 8x2 = 16, and therefore to repeat the num- *ber 16 times would amount to the same as repeating it 8 times and then re})eating this result 2 times ; or, since 2x8=16, it would be the same to repeat the number 2 times and then repeat that result 8 times. Again, 4x4=16. We may therefore vepeat the num- ber 4 times, and this r' suit again 4 times; each of these methods would give the same product. Ex. — Multiply 3G2 by 44. 362 83. 15928 Since 44=11x4, we multiply 362 by 11, which gives 3982, and then multiply 3982 by 4, giving 15928 as tlie nnal product. The numbers 8 and 2, or 4 and 4, are called the factors of 16. (See Art. 74.) The factors of 12 a-re 2, 6. 48 are 12, 4. 120 are 12, 10. 84. A result in Multiplication m-ay be proved to be correct by using the multiplicand as the multiplier, which should give the same product, if worked corr.ectly. EXERCISE 16. ;ents as :h sepa- will be tS* As many as possible of the following questions should be worked by factors as well as by the ordinary methixi, (I) (a) 4624 35 (2) 3846 39 (3) 8462 47 14) 7846 147 ea AUnUUHTIC FOR BBOINHBRH. (^) -tJj I 11; ft 11 (5) 3976 J83 (9) 2526 136 (13) 67«54 10234 (17) 39^5 733 (21) 57423 159 (25) 274 167 (29) 3759 3757 (33) 657 408 (.37) 9008 784 (41) 7058 6007 (6) 2243 144 (10) 52365 543 (14^ 6503455 234 (s8) 9S7 89X (22) 194 H (26) 43326 96 (30) 8643 923 (34) 6258 346 (38) 3207 2345 (42) 35768 3456 (7) 763521 433 (n; 3678543 4567 (15) 985io2;5 35789 (19) 7415 387 (23) 3678543 4567 (27) 999 999 (31) 3976 948 (35) 5679 507 (39) 6579 3506 (43) 726 27 (8, ^ (4 1283 \ ('») 36^ •* H4 • it- — • (12) \ i. 7678. ;i? It 1 46' 7615 n -^ (16) (5: :■ 3764 1 5 568 287 (20) 8097 (^) Mult 869 I. 2. I (24) 437 ^ 356 1 3- 4. 10 1 5- 630 1 6. 4 1 7 36 (28) 1 8. 71 841 f 9- 88- 841 ' a 10. 68. ■ 11-753' (32) f 12. 1' 907 I 13- 4. 740 I J 14. 68. — 15' 7( (36) 7856 1 16. 7I 17. 70^ 658 18. 4; 19. 7c 20. ] (40) 21. IC 8579 22. 4078 23- / w 24. t (44) i 25- 7 4628 \ 26. 3S 554 1 27. 67 — 1 28. 74 MULTI PLICA TlOtI . (>3 (8) J2S3 « 144 (12) jOjH: -{It (16) 568 287 (20) 8097 869 (24) 437 356 (28) 841 841 (32) 907 740 (36) 7856 658 (40) 8579 4078 (44) 4628 554 (^) (45) 3648 30 (46) 4275 54 i id) Multiply 1. 74 by 2. roooo by 3. 4698 by 4. looooo by 5. 6307918 by 6. 44670 by 7 367950 by 8. 78609 by 9. 887002 by 10. 684207 by 11. 7532100 by 12. 13- H- 15' 16. 17- 18. 19. 20. 21. 22. 23- 24. 25. 17565 by 43450 by 685900 by 76980 by 78600 by 708060 by 43800 by 70800 by loii by 10009 by 386 by 7815 by 6188 by 7289 by 10. 869. 1000. 76984. 20790. 145- 406. 903- 7006. 4861. 1800. 1700. 190. 16000. 1400. 490. 38506. 69870. 754- 869. 99. 98 97. 999- 26. 38751 by 998. 27. 67583 by 996. 28. 74189 by 995. (58) 2146 179 (56) 81650 789 29. 2572 by 94. 30. 40306 by 127. 31. 86072 by 208. 32. 48746 by 316. 33- 30975 by 507. 34. 6408 by 325. 35- 703475 by 386. 36. 370607 by 4071. 37. 600326 by 2645. 38. 730096 by 5006. 3,9. 2407068 by 3406. 40. 408091 by 2407. 41. 73069 by 46035. 42. 4372 by 128. 43- 3065 by 84. 44. 36204 by 414. 45- 4008 by 3724. 46. 47672 by 234. 47. 302076 by 603. 48. 73008 by 2036. 49. 430605 by 4005. 50. 290361 by 30406. 51. 2784 by 216. 52. 68470 by 435. 53. 3060724 by 2406. 54. 130065 by 8042. 55. 98070 by 12094. 56. 6789 by 2345 64 AJUTII.XfKTir jyjft «AVV/.V.VA7(:J. Ih (r) X. Multiply seven thousand six hundred and one b} seven. 2. Multiply thirteen hundred and eighty-four by eleven and twelve in succession. 3. Multiply tof^ether two, three, four, fivQ, six, seven, eight, nine, and ten. 4. Find the product of three hundred and forty-seven and five hundred and eighty three. 5. Find the product of twelve thousand and three and three thousand and twelve. 6. How much is twelve times four times three thousand four hundred and seven ? 7. Multiply together three thousand three hundred, three thousand and thirty, and three thousand and tliree. 8. Find the square of six hundred and seventy-nine. rS' (The square of a number is that number mul- tipliied by itself.) 9. Find the square of two thousand seven hundred and forty-se\''en. 10. Find the square of seventeen hundred. M. Multiply three millions seventeen thousand and ninety by four thousand and eighty-four. 12. Multiply the square of two hundred and thirty-nine by eleven, 13. Find the product of one thousand three hundred and fifty-six, five hundred and seventy-eight, and two hun- dred and«fifty. 14. Multiply the square of seventeen by the square of nineteen. 15. Multiply six thousand and ninety-seven by nine hun- dred and eight. 16. Multiply fifty-four thousand and forty-nine by six thousand and seventy-five. 17. The two factors of a certain number are 656 and 907: what is the number? 18. Multiply thirty-seven thousand and twentv-eisht by 508. ■ • ^ ^ I ^5 ). Tl sa 2C . W sai 21 .On j is 1 P 22 .Ml 1 hu 1 23 . Ml 1 eig 1 24 Ho L $4: 1 ^^ Ho P 26 A. ■ ma 1 ^^ A V ma m 28. Ho i 29. Ho in ^ i - Ao moi i StO( " i • f 1 ' 1 -!■ 66 8. 9- JO. 1 1. 12. 13 14, 15. 16. !?■ 18. 19. 20. 21, 22. 23 24. ^RiTrntETlc FOR Bi.>;iy\i.:rf!. ,> /Z."'^" "^'^ ^ '"'^^ ^^*' ^^'"'^ ''^ ^J' J'-iVs. how long will It take one man alone to do it? ' ** What will be the com of bnikhnji a hn. cf tel,..a-,„h 274 "iilcs long, at $967 a mile ? '^ ' " If 1049 ponnds of tobnccc. can be raised from an acre of land, how many pounds will 386 acres pn dncc? ayiK73^'a^;ir^^^^^^"^'^'''^^79mnesofr^ cont'-unin^ 7l^Z '""? ^'"^"P '" ^^'' ^^^^^•■^' ^^^'> bal. entire crop /^ ^ - '"'^^^ ''^''^ *^^"' weight of th. What is the value of 108 buildings, at $,89^ each ? What is the cost of 257 yoke of oxen, at $175 a yoke - What is the cost of 428 lots, at $284 each ? In I ream of paper there are 480 sheets: how many sheets are there in 217 reams? ^ If a cotton mill manufacture 658 yrrds of loth in day. how many yards can u nake!.! 309 days 5 ' ?o^^.inJ/rytd1?^°^'^" '" ''' P*^^-' -^^ P-^ ^:Hnin:co^^^'"^--"^^^--^--nit Light travels 192000 miles in i second- how .ar will It travel m 494 seconds ? A drover bought 685 oxen, at $10^ api the cost of all of them? at was A mercliant bought 25 pieces of broadcloth, each piece Cv,ntaining 48 yards at 12. Multiply 12 by 5, subtract 40, add 5, multiply by 2, subtract 25, add 5, multiply by 3, add 7 : rebult ? 13. Take 12 from 48, add 6, take 7. add d, take 3, add 7 take 9, add II take 4, add 3, take 5, add 9, uke 10 add 7, take 8, add 9, add 3, take 5 : result ? 14. From 16 subtract 9, multiply by 3, subtract 7, add 4 multiply by 6, subtract 7, add 9, sulUiact 8 : result ? 15. Add6 to 18, subtract 9, multiply by 4, subtract 2<;. multiply by 2, subtract 40, multiply l>y 7 ; result ? 16. From 19 subtract 8, multiply by 6, subtract 11, add 7 subtract 20, add 8, multiply by 3, add 9: result ? 17. Multiply 7 by 6, subtract 12, add 4, subtract 14, mul- tiply by 6 subtract 20, multiply by 3, subtract 72, add 18. Add n to 29, multiply by 2, subtract 16, add 6, multi- ply by 10 : result ? 19. Take 19 from 39 multiply by 5, subtract 50, add 10, multiply by 3. subtract mo: result? 20. To 23 add 7 multiply by 3, ndd 10, subtract 50, mul- tiply by 2, subtract 100, multiply y 6 : result ? Ens. , subtract g, add 5, ', subtract 6, add 1 1 : subtract 9, subtract tract 5, subtract 20, 12, subtract 7, sub- result ? I JO, take 3, add 4, ? by 2, subtract 15, t 5 : rcsuli? subtra.- 1 8, add 5, nultij-ly [jy 2, sub- d 5, multiply by 2, dd 7 : rebult ? d d, take 3, add 7, 5, add 9, take 10, result ? subtract 7, add 4, l)tract 8 : result ? by 4, subtract 25. by 7 : result ? subtract n, add 7, dd 9 : result ? , subtract 14, mul- 3, subtract 12, add ;t 16, add 6, multi- ibtract 50, add 10, Mirr/ri PLICATION. 71 \ 21. Add 7 to 9, subtract 6, multiply by 4, subtract 20, add 7, subtract 5, multiply by 2, subtract 8, add 5 : result? 22. Subtract 8 fn.in 17, multiply by 5, subtract 15, multi- ply by 20, subtract 30, add 9, subtract 9 : result ? 23. To the product of 8 and 8 add 6, subtract 30, atld 2, subtract 12, multiply by 3, subtract 4, add G: result ? 24. To ig add 11, subtract 15, nuiUiply by 4, subtract 12, multiply by 2, add g, subtract 5, multiply by 11 : result ? 25. Subtract 9 from 21, add 8, subtract 6, add 11, multiply by 4, subtract 7, add 9, subtract 8 : result ? 26. To the protluct of 9 and 6 add 6, subtract 12, subtract 18, multiply by 2, subtract 20, add 5, multiply by 2, add 10, multiply by 3, add 15, subtract 7, add 6: result ? 27. Fro!n 23 subtract 8, multiply by 2, multiply by 4, sub- tract 20, subtract 21, add 6, multiply by 2, subtract 20, multiply by 3, add 8, subtract 7, add 9 : result ? 28. To 31 add 12, subtract 10, add 6, subtract 7, subtract 8, subtract 4, add 9, subtract 3, add 6, add 8, add 10, subtract 5, add 8, subtract 2, add 6, subtract 7, add 4, subtract 6 ; result ? 29. From 63 subtnct 7, add 3, add 6, add 12, subtract 4, add 10, subtract 5, add 6, subtract 7, add 3, subtract 6, add 9, subtract 8, add 6, subtract 4, add 3, subtract 2, add 7 : result ? 30. Add 7 to 9, subtract 8, add 20, add 14, add 30, subtract 4, add 5, subtract 6, add 7, subtract 8, add 9, subtract 10, add 4, add 5, add 8, subtract 7, add 4, subtract 5 : result ? subtract 50, uuil- 6 : result ? 72 AltrrUMETIC FOB BEaiNNEHa DIVISION. 86. We have seen that 3 dollars repeated 5 ti.nes are ic dollars. Now ot us see how often v/e can take 3 dl lars from 15 dollars. ^ .4 15 dollars 3 12 dollars 3 = ist remainder. 9 " 3 = 2nd «« 6 •• 3 •:3rd '• 3 " 3 =4th •« u f ^ J I 'I O <• =:5th •« Thus, 3 dollars nia)- bo taken 5 times away from ic dol- lars, that is, just as many times as it was before repeated in order to produce 15 dc lars. This fact is expressed by saying that 3 is contained in 15, 5 times. In the same manner it may be shewrj 56 ' "^timeT"*^'"'''' '"'''' ^ *""''" * ^""^ ^ contained in 87. Again, since 3 dollars can be taken 5 times from 15 dollars, this is but another way of saying that $15 can be divided into 5 parts, each part being 3 dollars. In the same way, if 20 units be divided into 4 parts of the same size, each will he 5 units, and since, in repeating the parts in the multiplication, they were, of necessity, the same size, so in this mnr^^cc «r<» ,.,;ii always suppose the parts to be the same size, or of the same value. 88. 89. Wl: tl tl Th] fc n The is The is The ta The tw be on Reac be 45 pa wh 90. Divii plii pre pre Ever wil Di^ by DIVISION, n When we wish to divide 32 into 4 parts, it is understood that they shall be of the same size, viz : 8 units in this case. 88. This operation, then, is called Division, which is there- fore the method of finding the number of times one number is contained in another. The number which contains, or is divided by the other is known as the Dividend. The number which is to be divided into the Dividend is called the Divisor. The number which shows now often the Divisor is con- tained in the Dividend is termed the Quotient. 89. The sign for this operation is •*-, placed between the two numbers, and shews tliat the number coming before it, viz., the Dividen-^, is to be divided by the one coming after it, viz. : the Divisor, thus : 45-*-9 = 5, Reads, 45 divided by 9 equals 5, and means that 9 may be taken from 45, 5 times ; or, that 9 is contained in 45, 5 times; or, that if 45 be divided into 9 equal parts, each part is 5. tS' The pupil should remember that it is the Divisor which always follows the sign of Division. 90. Division will easily be seen to be the converse of Multi- plication, for from 7 and 4 we obtained 28 by the latter process, while from 7 and 28 we obtain 4 by the former process. Every result, then, in the Multiplication Table (Art, 71) will also furnish us with a corresponding result in the Division Table. This is, in fact, the very work done by the pupil in Ex. 13 («). 74 ARITIIMKTIC FOR BEGINNERS. 91. The following Ta'jlo can be seen at once to agree with the Table in Art. 71. DIVISION TABLE. I -i- 1 BS I -: -4- J = J 3 -- 3 — I i 4-^4=1 2 H- I = 2 4 -*- 2 = 2 6 ^ 3 = 2 8 -i- 4 = 2 3 H- I = 3 6 -i- 2 = 3 9-^3 = 3 12 -f- 4 = 3 4 -«- I = 4 8-^2 = 4 12 -J- 3 = 4 16 - -4 = 4 5 -*- I = 5 10 -f- 2 = 5 '5 + 3 = 5 20 - -4 = 5 6 -^ I = 6 12 -f- 2 = 6 18 ^- 3 = 6 I 24- -4 = 6 7 -*- I = 7 14 -f- 2 = 7 21 -J- 3 = 7 28 - -4 = 7 8 -h I = 8 16 -f- 2 = 8 24 -»- 3 = 8 32 - 4 -= 8 9 -*- 1 = 9 18 -*- 2 = 9 27 -*■ 3 = 9 36 + 4 = 9 5 -*- 5 = I 6 -H 6 = I 7'^7 = 1 8 -f. 8 = I 10 -t- 5 = 2 12 -i- 6 = 2 14-1-7 = 2 16 -T- 8 = 2 15 + 5 = 3 18 -1- 6 = 3 21 -»- 7 = 3 24 -f- 8 = 3 1 20 + 5 = 4 24 -^ 6 = 4 28 -i- 7 = 4 32 -f- 8 = 4 25 -i- 5 = 5 30 -»- 6 = S 35 ■*■ 7 = 5 40 -*- 8 = 5 30 -*- 5 = 6 36 -*- 6 = 6 42 + 7 = 6 48 -^ 8 = 6 35 -*- 5 = 7 42 -f- 6 = 7 49 -^ 7 ^7 ' 56 + 8 = 7 40-5-5 = 8 48 -i- 6 = 8 56 + 7 = 8 ' 64 -*• 8 = 8 45 -^ 5 9 -J- 9 = 9 54 -*• 6 = 9 63 + 7 = 9 72 + 8 = 9 =: I 10 -♦- 10 = I II -i- 1 1 = I 12 -*- 12 = I 18 -i- 9 =: 2 20 -J- 10 = 2 22 -4- II = 2 24 -5- 12 = 2 27+9 = 3 30 -*- ID : = 3 33 -*- 11 = 3 36 -5- 12 = 3 36 -9 = 4 40 -^ lO = 4 44 ■«- H ■-= 4 48 + 12 = 4 45 •+ 9 = 5 50 -^ 10 = 5 55 -1- 11 = 5 60 -«- 12 = 5 54 ■*- 9 = 6 00 -i- 10 = 6 66 -*- II = 6 72 + 12 = 6 63+9 = 7 70 -t- 10 = 7 77+11 = 7 84 -»- 12 = ■: 72+9 = 8 80 -*- 10 = 8 88 -*■ II = 8 96 -*• 12 = 8 |8i +9 ssz 9 90 -J- 10 = 9 99 -^ II = 108 H- 12 =: 1 = r = 3 = 4 = 5 = 6 = 7 = 8 = 9 = 4 I = 5 = 6 = 7 = 8 = 9 — 2 ' = 3 = 4 1^ — 3 [2 = 6 12 ■ / 12 = iJ 2 =. 9 EXERCISE 17. Mental Exercises in Division. 76 1. How many 4's are in 12? in 16 ? in 48 ? in 24 ? in 36 ? in 28 ? 2. How many loads of 5 tons each are there in 40 tons ? in 60 tons? in 35 tons ? in 15 tons? 3. How many times can 7 be taken from 14 ? from 42 ? from 63 ? from 84 ? 4. Divide by 3, from 3 into 3 to 3 into 27. by 5, from 5 into 15 fco 5 intf) 45. by 7, from 7 into 42 to 7 into 84. % 8, from 8 into 24 to 8 into 88. by g, from g into 108 to g into 27. by 10, from 10 into 10 to 10 into 70. by 12, from 12 into 132 to 12 into 36. 5. What is the quotient in, 45-^9' 36-^4» 72-^- H, 56-r- 7, io.S-f-i2, 8i-f-g, 72--6, 54-H 6, 132-M1, 64-=- 8, 16^8, 42-4-6, 80-r-io, 27 i6-;-2, 14H-2, 60-5- 5. 35 6. If a box holds 4 pounds of sugar, how many such boxes will he required to hold 36 pounds ? 28 pounds ? 16 pounds ? 44 pounds ? 7. 3/) is how many times y ? 4? 6 ? 12? 8. 32 is how many times 4 ? 8? g. 24 is how many tinier 2 ? 3? 4? 6? 8? 12? 10. 48 is how many times 4 ? G ? 8 ? 12 ? 11. From a pile of 60 bricks, how many loads of 12 bricks may be taken away ? 12. If $56 be equally distributed among 7 men, how many dollars does each man receive ? 13. When apples are 3 cents each., how many can I buy for 24 cents ? (In other words, how many times must three cents be repeated to give 24 cents, ; the answer will be 8. 9. 5. 35-^ 7, 84H-I2, 77-7» 30->-3. 28-4, 63-^7- 70 AmTHMETW FUR UEOtXNEtlS. !,■■■- I could thus buy 8 apples. This must be correct f -r each apple costing 3 cents, 8 apples must cost 8 times 3 cents, that is, 24 cents.) 14. If a man travel 6 miles an hour, how long will it take hm) to travel 54 miles ? 15. How many tubs containing 9 gallons each can be filit-d trom a hogshead containing 63 gallons ? 16. If a man drive 8 miles an hour, in what time will he drive 56 miles ? 17. A farmer bought some lambs for $60, paying $c a liead : how many lambs did he buy ? ^^' ^^ K^ t ^^^^^''' '" ^^^^* ^'"»e will a man earn $^16? $54? $72? $81? $108? *^ ■ 19. If 7 barrels of sugar cost $63, what will i barrel cost ? 20. If 6 kegs of powder cost $72, what will i keg cost ? 21. If a man travel 48 miles in 4 days, how far does he travel in i day? 22. What will be the cost of i ton of coal, if 8 tons cosr 23. If you divide $84 among 7 children, how many dollars A'lll each ch'ild have ? 24. If a man build 72 feet of fencing in 8 days, how many feet can he build in i day ? ^ 25. If 9 dozens offish cost 108 cents, what is the cost of i dozen ? 27. How many lots of 5 acres each are in 20 acres ? 28. How many barrels, each holding 3 bushels, will be required for 18 bushels of onions? For 21 bushels ? 29. How many times can 6 yards of canvas be cut off from a piece containing 30 yards ? 30. How many times can 6 cents be taken from 24 cents? 31. Distribute $28 equally among 7 people: how many dollars will each receive ? 32. W 33. W 34. W 35- A in 56. If pe 37. If nij 38. If fai 39. A: cla 40. A] vva 41. Ai wh cei 42. A 1 frie •(3- If : mil 44. A r loti 92. In all hav nun the I hav clas If the; but the clas for 1 3rrect,f >r ;t 8 times ill it take n be filled e will he ing $5 a irn $36? rel cost ? cost ? does he :ons cosr y dollars >w many cost of I For $84 ? 3? will be shels ? off from |. cents? V many DIVISION. 77 32. What is one of 4 equal parts of 40 ? Of 36 ? Of 48 ? 33. What is one of 6 equal parts of 30 ? Of 42 ? Of 48 ? 34. What is one of 7 equal parts of 56 pounds? 35. A teacher having 66 maps, distributed them equally in a class of 11 pupils: how many did each get ? 36. If g6 pounds of bread are divided equally among 12 persons, how many pounds will each receive? 37. If 88 dollars are divided equally among 8 persons, how many dollars will each have? 38. If 120 barrels of flour are divided equally among 12 families, how much flour will each receive ? 39. A master having 108 pupils, divided tiiem into 9 equal classes: how many were in each class ? 40. A picnic party of 11 persons spent $132 : how much was that apiece? 41. A party of 10 persons found a purse containing $100, which they shared equally : how much did each re- ceive? 42. A lad having $96, wishes to divide it equally among 8 friends : how much can he give to each ? 43. If you pay 84 cents for a horse and waggon to go 7 miles, how much is that a mile? 44. A man having 120 feet of land, divided it into 6 equal lots ; how many feet were th'ere in each lot ? 92. In all the previous examples of Division the pupil must have noticed that the divisor was cotitamed an exact number < f times in the dividend. This is sot always the case, for example : I have 22 pears, and give 4 pears to each boy in the class : how many boys were tliere ? If there had been 5 boys, I would require only 20 pears; but if there had Lv-h n 6 b>.ys I must liavc; 24 pears : so the only thing I can do is to pive the 5 boys in the class 4 pears each, and keep tlm iher 2 that remain for myself. I 78 An/TiiMhTic ton nmiysEJts. 93. This number, 2 (in the rase bef-)ie us), is called tlift Remainder, and may l)e said to he that which is Icit after the divisor has been taken as many times a- possible from the dividend. If the remainder be first taken fnmi the dividend, tlic result must contain the divisor exactly. Thus : 4 may be subtracted from, or contained in, 30, 7 times, hut there will be 2 left, and we see t-liat if the 2 be taken from the 30, the result, 28, will ccntain 4 exactly. Ex. I. — Divide 77 by 8. Since 8 times 9 are 72, the quotient must be 9 ; and siiuc 72 is less than 77 by 5, tl:en 5 must te the remaindei. Ex. 2.— What must be taken from 49 that it may contain 9 five times ? The number that contains 9 five times we know to be 45, and since this number is 4 less *han 49,4 must be the required result. EXERCISE 18. 13* Mental Examples on the Remainder. Give the quotient and remainder, if any, in 18-j- 4, 21-*- 5, 62H- 7, 41-^-11, 39-}. 6, 71 90+12, 80+ 7, 23-^12, 62+ 9, 73^ 8, 45-r 9 83+10, 31+ 3, 42+ 6, 70+ 8, 75+ 9, ,20-rii, 140+12, 93+10. 48+12, 79+ 7, 80+12, 1..C+11. What number must be divided by 6 to give a remain der 3 and quotient 5 ? Ife times 5, or 30, be divided by 6, the quotient will be 5 exactly, with no remainder. Hence, if there is to he a remainder of 3, the number must be 33. Proof —6 is contained in 33, 5 times, and 3 over for a r(;mainder. J. What number must be divided by 5 to give Oiiotient *7-. remaintler 2? QiK.tient 8, remainder 4? Quotient 11, remainder 3? 2. S. 4. To giv must bi by 12? 5. A man much h 6. From c. allowed 7. A. recei men, ai among 8. If I had each : I: 9 Countin and 4 O' 10. How m same nu 11. From T ride 7 11 be at th( 12. How ms they be 1 13. There ar 3 buys 1 each row 14. John has and kee brothers 15. F(nir quc many ga 31 quarts gallon me 16. If lam:- p in front A.'s door far will tl 17- Find the Divisc <( Divisioy 7f> give a quotient 7 and remainder 6, what number must he divided by 7 ? by 8 ? by 9? by i„? ly 12? 5. A man had $1 by 1 1 ? ^O'), and p;ave $9 apiece to H boys : how mucli had lie left? 6. From 93 Dushcis of oats, how many horses can be allowed 10 biisliels each, and what wouhl |,e left ? 7. A, receives what IS left after di\id in men, and B. receives what is left alter d ^ $100 among 8 ividing !ii;i2o than A. ? mong II men : how much more does B. get ...c... .x - 8. If I had 7 more apples, 1 cMild give 8 boys n aprles each: how many have 1 ? ^i ^jj^ies 9 Counting lus marbles by sevens, Joseph had n lots and 4 over : how many had he ? 10. How many could be put in each lot, to have the same number of marbles m each ? 11. From Toronto to Hamilton is 40 miles; a man can nde 7 miles an hour : how far from Hamilton will he be at the end of 5 hours ? 12. How many will be left over from 93 bank notes if tliey be tied in packages of 8 ? of 10? of „ ? T^^'l 13. There are 75 boys in the class, and 6 rows of seats- ,f each mw'/' '" ''''"'^' ''"'" '"""^ '"^^^ ^'^ '^'^'^ i" 14. John has 47 plums, and gives 5 to each of his brothers and keeps the smallest share himself: how manv brothers had John ? what was his % # ^/7 Photographic Sciences Corporation 23 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) 873-4503 V ■O' <^ 'C^ >^ # ^^' t/j 82 m li ARITHMETIC POR B .GINHERS. then be a remainder, place it before the next figure in the divider d. Proceed as before, and carry the remainder to the next J ure in the dividend, but if the divisor xvill not divide, write a nought below in the quotient, and carry thefigi/re or figures to the next one in the dividend. Divide these as before, and so on till all the figures in the dividend are taken in. Place the remainder, if any, to the right of the quotient. Ex. — Divide 116501 by 12. 12)116501 9708-s Here 12 will not divide the first figure i or the two first figures II, therefore we say 12 is contained in ii6, 9 tunes and 8 over ; put the 8 with the 5, and say 12 IS contained in 85, 7 times and 1 over; take the i with the o; but 12 is not contained in 10 ; therefore, put down a nought in the quotient, and take in the next figure i with the 10, and then say 12 is contained in 101, 8 times and 5 over. This 5 is the remainder, and it must be always less than the divisor. Proof .—9708 X 1 2 = 1 1 6496 116496+ 5=116501 That IS, multiply the quotient by the divisor, and to the result add the remainder, if any. This will give the dividend, if the work be correct. 06. Where the process is thus carried on mentally, and the quotient only set down, it is called Short Division. EXERCISE 19. Divide : I. 624 by 2. 2 862 by 2. 3. 684 by 2. 4. 396 by 3. 5. 693 by 3. 6 848 by 4. 7- 484 by 4. 8. 884 by 4. 9- 555 by 3. io. 8642 by 2. II. 3693 by 3. 12. 9306 by 3. 13- H- 15- 16. 17- 18. 19. 20. 21. 22. 746 by 2. 368 by 2. 459 by 3. 756 by 3. 928 by 4, 568 by 4. 655 by 5- 605 by 5. 9246 by 2. 136 by 4. 23- 24. 25- 26. 27. 28. 29. 30- 31. 32. 33- 176 by 4. 215 by 5. 252 by 6. 364 by 7. 434 by 7 336 by 8 568 by 8. 736 by 8. 378 by 9. 459 by 9 8128 by 4 Ai Igure in the inder to the not dhiide, the figure or de these as nd are taken 'he quotient. m VISION. 5 two first :d in ii6, nd say 12 ake the i therefore, ike in the contained jmainder, nd to the give the , and the >ivision. 176 by 4. 215 by 5. J52 by 6. J64 by 7. ^34 by 7. 136 by 8 i68 by 8. '36 by 8. 178 by q. 59 by 9 28 by 4, 34' 35- 3fi- 37- 38 39 40. 55 Divide : 6126 by 3. 5255 by 5. 6312 by 6. 8432 by 8. 756 by 3. 978 by 2. 872 by 4. Uii 48. 49. 50- 51. 52 53- 54. 7535 by 5. 9372 by 2. 6185 by 5. 8491 by 7, 9656 by 8. 9981 by 9. 32568 by 3. 56. 57- 972 by 3. 896 by 4. 67s by 5. 775 by 5. .- 735 by 7. 46. 8208 by 4. 47. 6075 by 3. I ^^. 32305 „y , Twenty.seven thousand five hundred and twelve by Thirty.two thousand four hundred and ninety-six by six Fourteen million eight hundred and sixty-five thousand nine hundred and thirty-two by two. t'lo^sand 58. Thirty-six thousand nine hundredand forty-fivebv nine 59. Seventy-two thousand three hundred ^nlfJriXlhy 60. Forty-five million eight hundred and twentv-ei^ht thousand nine hundred and twenty-seven by nine ^ n™5- P"P'^ ^^°"^^ P'°^^ t^e answers to each of th^ precedmg questions, instead of referring to the In" swers in the book. fixing 10 tne an- EXERCISE 20. Find the quotient and remainder, if any, in each of Hi« foUowing questions, proving each result • ^'^ 4)3654 5)7^84 . 3)i4i 7)40505 (5) 9)476589 (6) " 12)987654 ")334S23 9 10, II. 12. 13- H- 15- 16. 17- iS. 2718065 7893201 5013487 3920384 8372146 4365984-^ 9 453678+11 ^96583-^12, 5703214-*- 7- 6183420+ 8. 8. 9- 6. 7- 8. 19. 3706823+ 9. 20. 6175802-*-! I. 21. 8i6o937.f.i2. 22. 5117284-5- 7. 23. 4465037+ 8. 24. 7600356-^ g. 2 ^-'' ' — ■ 20. 4301 765-+ 27. 7400804 28. 4230569-f-li -1 t ■ 8. 9- (8) 8)639724 29- 78i4873-,.ri. 30. 7o762i7-(-i2. 31- 1275923-1-11. 32. 11330434^-12. 33- 41241154.^11. 34- 2314205-1-12. Z3- 3274604-f-ii. 36. 46;702i-»-i2. n- 857>,ig8-m. 38. 589i27o-<-i2. 1 'I i 1 , 1 f ■' ' 1 i 1 ' 1 t i 1 84 ARITHMETtC h\)ti IlKGI S S Eli.r 39. How many times may 3 be taken iroin 27021 ? 40. How often is 4 contained in 28032 ? 41. The divisor is 5, the dividend is 33515 : find the quo- tient. 42. How many 7's in 44268 ? 43. How many times must 6 be taken from 49392 to leave no remainder ? 44. How many 8's in 44248? 45. How many times 9 is 37845? 46. How many times 7 is 42924 ? 97. To divide by 10, 100, 1000, etc. The number 3766 may be read 376 tens and 6 units. Thus we see that if one figure be cut off the right of a number, the remaining figures shew the number of tens there are in it : as 376 in the case above. In the same manner, 3766 may be read 37 hundreds and 66. Thus, if two figures be cut off the right of ^ num- ber, the remaining figures shew the number hun- dreds in it : 37 in this case. 98. Therefore we see that to divide by 10, 100, 1000, loooo, etc., we need only cut off one, two, three, lour, etc., figures from the right of the dividend, the quotient will be the remaining figures, and the figures cut off will be the remainder. (Compare Art. 79.) Ex. — Divide 87631 by 1000. Cut off three figures from the right. The remaining figures, 87, will be the quotient, and the figures 631, that were cut off, will be the remainder, 99. This principle is very useful in the matter of dollars and cents, for, as there are 100 cents in every dollar, to bring any number of cents to dollars we need only cut off the last two figures as above, and the remain- ing number will be the required dollars, and the figures cut off will be the number of cents left over. 2. 3- 4- 5- 6. DIVISION. -fi'j:.— 86342 cents will be the same as 863 dollars and 42 cents ; or, 86342 cents = $863.42, The dot . being placed to separate the dollars from the cents. (Connipare Art. 81.) EXERCISE 21. I. 2. 3- 4- 5- 6. 7- 16. 17- 18. 19. Divide : 7316 by 10. 83174 by 10. 6192 by 100. 73001 by 1000. 97312 by I 0000. 83916 by 100. 513712 by 100. 8. 712934 by looooo. 9. 392 by 100. 10. 37214 by 1000. 11. 74321 by 1000. 12. 30600 by 100. 13. 3ooo;)oo by 100. 14. 6060600 by looooo. Express 10862 cents in dollars, etc. Express 312 cents in dollars, etc. How many dollars will be the same sum as 461000 cents ? How many cents will be left if 861070 cents be ex- changed for one-dollar bills.? How many dollar bills will be obtained ? 20. How many dollars would buy as much land as 7^10700 cents? ' EXERCISE 22. I. If 2 waggons of equal size carry 4896 bricka, how many bricks will one waggon carry.? a. If 2 houses are bought for 47054 dollars, how much is one of them worth ? 3. If 3 mines cost 156378 dollars, how much does one mine cost ? 4. If 3 times a certain price is price ? 101612, what is the \ ' 1 1 f se ARITHMETIC FOIl BBOIJfXERS. 'Bi ■: h ' 5. A grant of 60148 acres is to be divided among 4 per- sons: what is each one's share ? 6. Divide $10632475 among 5 colleges : what will be the share of each ? 7. Eight men have an equal interest in 2681 12 acres of land : how much has each ? 8. If 9 square feet make one square yard, how many square yards are in 26002197 square feet ? 9. In a market garden containing 8 acres there are 42336 hills of potatoes : how many hills are there in one acre? 10. Add the quotient of 36140292 divided by 9 to the quotient of 31623424 divided by 8 11. Divide 163207431 by 3 times 3. 12. A man died having an estate of 146329 dollars ; his widow received 23193 dollars, and the remainder was divided equally among four hospitals : how much did each hospital receive ? 13. I have 327 lemons, and sell 311 : how many remain ? how much shall I receive at 8 cents each for those I have sold ? 14. At 2 cents each, how many apples can I buy for $43.44 ? The same money will buy how many toys. at 3 cents each ? How many tarts, at 4 cents each ? 15. At 2 dollars a day, how many men can I hire for 346 dollars? For 496 dollars? For 3176 dollars ? 16. At 3 cents a spool, how many spools of thread can I buy for $3.84 ? For $5.73 ? For $49.62 ? 17. There are 4 pecks in a bushel: how many pecks are there in 3844 bushels? In 7688 bushels? In 15376 bushels ? 18. There are 4 quarts in a gallon : how many gallons are there in 132 quarts ? In 396 quarts ? In 792 quarts ? 19. How many pounds of sugar, at 9 cents a pound can I buy for $36.90? For $73,44? 20. At 4 dollars each, how many tickets can be bought for 64 dollars? For 192 dollars ? For 1 152 dollars ? 23- 24. 27. 35- 36. 37- Di visioy. 87 21. There are 3 feet in i yard: how many feet are there in 27 yards? In 16 yards? In 29 yards ? 22 Three feet make i yard : liow many yards are there in 69 feet ? In 276 feet ? In 828 feet ? If 4 pumpkins weigh 108 pounds, how much will i of them weigh ? If 3 weigh 108 pounds, what will i weigh? If 4 iron rods of equal length measure 44 feet, what is the length of each? If they measure 132 feet, what is the length of each ? If 3 bushels of turnips will fill one barrel, how many barrels will 255 bushels fill ? 26. A man bought a lot for 3792 dollars, which was 3 times as much as his house cost him : how much did his house cost him ? 23 24 25 27. 28 Four asylums are to share eaually 7248 dollars: now much does each receive ? How many barrels of meal, at 5 dollars a barrel, car be bought for 3575 dollars ? 29. At 4 dollars each, how many hats can be bought for 796 dollars ? 30. At 1 1 dollars a barrel, how mary barrels of vinegar can be bought for 1749 dollars ? 31. There are 7 days in one week, how many weeks are in 365 days (one year) ? 32. If 9 acres of land cost 1125 dollars, what will i acre cost? 33. If 6 cows cost 1272 dollars, what will i cow cost ? 34. If a horse travels 693 miles in 7 days how far does he travel in i day ? 35. 1704 acres of land are to be divided equally among 8 charities : how many acres v^ill each receive ? 36. If 9 mules sell for 1359 dollars, what will be the sum received for each ? 37. A man bought 12 tons of hay for 180 dollars: how much did he pay a ton ? I I' :' i I i !■ J; ( If 88 ARITHMETIO FOR BEOUTNERS. i 38. 39- 40. 41. 42. 43- 44- 45- 46. 47- 48. 49. 50. 51 52 53' 54' for $2.75 : liow much did he JO' A boy sold II rabbits receive apiece ? A girl spent $3.54 for buttons, giving 3 cents apiece for them : how many did she buy? X. is worth 15795 dollars, which is 5 times as much as Y. is worth, ami Y. is worth 3 times as much as Z. : how much are Y. and Z. each worth ? A.'s land cost 2358 dollars, which is 3 times as much as the building of the house erst : what was the cost of the building? A butcher bought 12 oxen for 1764 dollars : what was the average cost of each ? A cooper worked 12 months for 216 dollars: how- much did he receive a month ? If 4 yards of tweed will make a coat, how many coats could be made out of 1876 yards ? A grocer spent 3661 dollars in sugar at 7 dollars per barrel : how many barrels did he buy ? How many barrels of cider at 5 dollars a barrel could be bought for 2235 dollars ? There are 4 weeks in i month : how many months are there in 5764 weeks ? A grocer spent $12.75 for baskets at 5 cents apiece: how many did he buy ? A school-house was built jointly by 7 gentlemen at an expense of 2625 dollars : what sum did each subscribe ? In I bushel there are 4 pecks : how many bushels are in 1176 pecks ? A mill worth 43652 dollars was owned by 7 men in equal shares : what was the value of a share ? If a train in 8 days runs 2896 miles, what would be the average run in i day ? A patent valued at 38125 dollars was owned in equal shares by 5 men : how much did each man own ? At 6 dollars a gallon, how many gallons of wine could be bought for 2274 dollars ? From the sun to the earth is about 92000000 miles; light travels this distance in about 8 minutes : how many miles does light travel in a minute ? 100 Drvrsioy. e» :h did he its apiece smuch as ich as Z.: as much s the Cost what was ars ; how any coats hilars per rel could onths are s apiece : len at an ibscribe? shels are ' men in .vould be in equal ivn? ne could 3o miles; es : how 100 We now come to those cases in Division, in which the dividend and divisor may be any numbers. £x.. I. — Divide 18763 by 16. 16 ) 18763 ( 1000 16000 16 ) 2700 ( 1600 100 16 ) II60 ( 1 120 70 16) 43 ( 32 2 II 1 172 — II The dividend is 18 thousand, 7 hundred, 6 tens, and 3 units. 16 is contained in 18 thousand i thousand times, leaving a remainder of 2 thousand, which with the 7 hundred make 27 hundred. 16 is contained in 27 hundred i hundred times, leaving a remainder of 11 hundred, which with the 6 tens make 116 tens. 16 is contained in 116 tens 7 times, leaving a remainder of 4 tens, which with the 3 units make 43 units. 16 is contained in 43 units 2 times, leaving a final re- mainder of II units. The whole quotient is therefore i thousand, i hundred, 7 tens, and 2 units, or 1172, and the remainder 11. The noughts to the right, expressing the thousands, hundreds, etc., are omitted in practice, because the place of the figures shews their value. Ex. 2. — Divide 588491 by 83. 83 ) 588491 ( 7090 21 581 749 747 21 n I 90 i!if ;'. ! ARITHMBTIC FOR HKOryyERS. The least number of figures on the left of the dividend that will contain 83 is 5,8,8, that is, 588 thousand, and tins contains 83,7 times. 83x7 gives 581 ; or, in full 83x7000 gives 581000. Subtracting the 581 from cSs' we find the remainder to be 7. Bring down the m xt ngure, 4, in the dividend, and we see that 74 will not contain 83; or, in other words, 74 hundreds will con- tain 83 no hundred times, and this no hundreds must be expressed in the quotient by the nought. Since 74 win not contain 83, we bring down another hgure, 9. 749 contains 83, 9 times, with a remainder 2. Bring down the next figure, r, and then, since 21 will contain 83 no times, we place a nought in the quotient and call the 21 our final remainder. The following proof shews the correctness of the result : 83x7 thousands = 58iooo 83 X o hundreds = o 83x9 tens = 7470 83x0 units = Remainder 588470 21 01. Dividend . . . 588491 When the divisor is greater than 12, and the different products are expressed, the Drocess is called Long Division. ° 102. We then have the following RULE FOR LONG DIVISION. Write the divisor and dividend as before, leaving a place on the right for the quotient. Find hmv 7nany times the divisor is contained in the fewest number of figures on the left of the dividend Place this as the first figure of the quotient. Multiply the divisor by it, and subtract the product from thcs' fS^ites at the left of the dividend. Attach or bringdown, to the difference, the next figure to the right in the dividend. 5 dividend isand.and or, in full, from 588, 1 the next 4 will not will con- reds mnst 1 another lainder 2. :e 21 will : quotient e result : different d Long place on he fewest 'e this as 'om these n to the Dl VISION. M If the number thus formed will contain the divisor, place tha number of times as the next figure in the qujtient, and proceed as before ; but if it does not contain the divisor, place a nought in the quotient, then bringdown the next figure from the dividend. Proceed tn the same manner until all the figures in the dividend ,.vv brought down. The number that is then left is the final remainder. Proof — 7he same as in Short Division. IS" In finding the quotient figure, the pupil will he ^^sisted by seeing how many times the first figure of the divistjr is ccjutained in the first figure, or, if neces- sary, the first two figures of the dividend ; an allow- ance being made for the carrying figure. IS" If any of the remainders (before bringing down a new figure) be equal to or greater than the divisor. It shows that the previous quotient figure is too small, and must be increased. rS* If any of the products of the divisor by a quotient figure be greater than the number above it, it shows that the quotient figure is too great, and must be diminished. K5* After the first quotient figure is obtained, there nriust be as many figures written in the quotient as there are figures brought down from the dividend. Divide EXERCISE 23. I. 588 2- 759 3. 864 4. 882 5. 2996 6. 3042 7- 3995 8. 5832 9- 5103 10. 7524 11. 5448 12. 5668 13. 7099 by 28 by 33 by 36 by 42 by 14 by 13 by 17 by 18 by 21 by 22 by 24 by 26 by 31 15- 16. 17- 18. 19. 20. 21 22. 23- 24. 25- 26. 8995 by 9576 by 9315 by 27775 by 43692 by 82242 by 88641 by 76875 by 35784 by 30618 by 38232 by 146448 by 199864 by 35 42 45 23 27- 475524 by 612 28. 1445204 by 802 29. 1760225 by 905 30. 3156584 by 722 33 I 31- 5173302 by 834 54! 32. 5926431 by 643 63 j 33- 3214664 by 566 75 34- 6923471 by 555 84 I 35. 14293624 by 675 126 36. 56243121 by 686 236 37. 692348726 by 897 324 38. 496839715 by 1047 301 39. 786935846 by 31 18 i! ! i 1 i " V 4 03 ARITHMETIC FOR BEGtSNERS I IdW many tiir.cs can $86 be taken from $17354 ? 41 ■ $52 from $7012 ? 42. if! 1 7 from $13354? 43- $G2 from $3406 ? 44- $73 f''""i $45078 ? 45- $51 from $60702 ? 46. $55 from $13415? 47. How many 73's in 1731195? 48. 49. SO. 51- 52. 53. 54. 55. 5^. 57. 46's in 761 3 1 702 ? " 381's in 13261467 ? " 937's in 13189212 ? " 754's in 762294 ? " 112'sin 51867? " 999's in 7281711 ? " ^5's in 33490 ? " SSG'sin 3931476? " 2624's in 73484248 ? " 736"s in 863256 ? Find the quotients ?nd also the remainders, if any re sultmg from the following divisors and dividends :— 58. 3076 59. 269181 60. 6739549 61. 62. 63. 64. 65. 66. 67- 68. 6g. 70. 71- 72. 2012 2309 3605 808 9101 7305 6635 7239 3827 5943 73421 «5043 and and and and and and and and and and and and and and and 11214887. 1246038849. 2331883954. S659110. 7861 1003. 4843167. 9863701. 4816657. 71810282755. 33216694340. 28956427101. 2 1 99898 1 3 74. 165x8324782. 472698568233. 1172481547818. 73- 74- Di by Di hu 75- Di tw 76. Di sar thi 77- Di an( 78. Di^ tW( thr 79. Di^ nir for 80. Di. dre hui 103. 16, Th The! wh 24. an( giv Now 4t or This wh DIVISION. 93 75 76 73. Divide forty thousand t\v(j Imndred and seventy-eight by seventy-five. 74. Divide seven liundred and sixty-five thonsand four hundred and thirty-one by ninety-six. Divide three hundred thousand four hundred and twenty-eight by three hundred and twenty-four. Divide forty-three million two hundred and ten thou- sand and forty by one thousand two hundred and thirty-six. 77. Divide fifty-six million thirty thousand one hundred and sixty-nine by two thousand a^d four. Divide one hundred and nine million four hundred and twenty-six thousand and fifty-one by seven thousand three hundred and fifteen. Divide four billion two hundred and eighty milnon nine hundred and sixty thousand three hundred and forty-two by fifteen thousand and three. 80. Divide thirty-one billion eighty-two million six hun- dred thousand five hundred and seventy-eight by four hundred and seven thousand ind fifty-three. 78 79. 103. It was shewn in Art. 83 that instead of multiplying by 16, for example, we could use its factors, 4 and 4 The same principle holds true in Division. £x. 1. — Divide 8413 by 24. 3)8413 8)2804-1 350-4 The factors of 24 are 3 and 8, those being the numbers which, when multiplied together, produce or make up 24, We therefore divide by 3, which gives 2804 and I over, and then divide the quotient by 8, which gives 350 and 4 over Now, the 2804 represents that number of 3's ; hence the ' ' " ~ lust mean four. q's. or 12. This 12 with' the i left which is the exact over remainde by at first makes in all 13, il ^ ARITHMETIC FOR BEOltiHERS. Proof : 350 x 8 = 2800 2800 X 3 = 8400 8400 + 13 = 8413 £x. 2, — Divide 427 11 by 99. 9)42711 11)4745-6 43^-4 4x9 = 3*^. 36 + 6=42 ^m<7/>;fl'^. The factors of 99 are 9 and 11. The final quotient is 43'- The last remainder 4 means four 9's, or 36, and this with the first remamdere gives us the true remainder, 42. 104. We thus have the following RULE FOR DIVIDING BY FACTORS. Fincf the factors of the divisor. Divide the dividend, as usual, by one of them, and then this quotient by the other. This result will be the true quotient. To find the true remainder, jnultiply the last remainder, if a?iy ■ by the first divisor, and to tfie product add the first remainder if any. The result will be the required remainder. This principle enables us to divide more easily by any number endmg in noughts : for example, 800. The factors of this number are 8 and 100, so we divide by the 100 first and then by the 8, and find the true remainder in the usual way. Ex, 1. — Divide 97643 by gooo. 1000)97643 9)97-643 105 106, .-^i 10—7 7 X 1 000 + 643 = 7643 Remainder. Tins might have been done more rapidly thus 9)97/643 10-7643 bi vistos. 95 For we can divide by the looo by merely cutting off the three figures to the right (Art. 79), then divide the remaining figures by the 9, and to the remainder, 7, attach the figures cut off, making in all 7643. £x. 2. — Divide 976341 by 3700. 37)9763'4i(263 74 236 222 143 III 32 3241 Remainder. First cut off the 41 to the right 01 the dividend, then to the remainder 32 attach the two figures 41, and we have the full remainder, 3241. EXERCISE 24. Divide : I. 436899 by 14. 20. 8349 2. 300527 by 18. 21. 7630 3- 83076 II by 16. 22. 7491 4- 439205 by 21. 23- $860000 5- 4031729 by 24. 24. 312946 6. 843043 by 25. 25. 36972 7- 7390478 by 28. 26. 131111 8. 736255 by 42. 27. 23218 9- 6310972 by 49. 28. $22120 10. 5084263 by 35. 29. 40220 II. 5083753 by 48. 30- 131127 12. 6230749 by 56. 31- 89952 13- 4003767 by 36. 32. 73*^6597 H- 5726009 by 44. 33- 4590000 15- $19866 by 77. 34- $13834500 16. 8514 by 99. 35- 115/9112 17- 15336 by 72- 36. 3678900 18. $93312 by 108. 37- 796532 1 9. 4361 by 10. 38. 461:12 by by by by by by by by by by by by 100. 100. 1000. 100. I 0000. 10. 400. 60. 70. 1900. 1 2000. 500. 30- by 306000. by 120300. by 890000. by 326100. by 230. by 8000. II ^crrj- — • -. 96 ARJTIJMJCTIC FOll UEOiySl ..s. How many dollars are there u 41. 7200 cents? 42. 36000 cents ? 39. 4600 cents? 40. loooooo cents ? Express as dollars and cents, 43- 846 cents. I 43. 81243 cents. 44. 750062 cents. I 46. 157 cents x 307 EXERCISE 25. '■ pay a'iecef * '^ ^''"^' ^°' ^'^^^ ' ^°''' "'"''^^ "^'^ '^'^^ 2. A man paid $1400 for oxen : at the rate of $56 each how many did he buy? ^^ ' ^' ^^I'ge? ''''" ' ^^"'^^ °^ "^^""^^^ '°'^' ^^37 barrels cost '• wm bcY; ea'^h^iSflf r '^ '^ ^°^""^^= ^°^ "^-y P^^-^ '• he ta^E^^g^o f33;tflLT'" " ^ '^^ ^ '°" ^°"^ ^'^^ 6. The wages of Jones for 17 months come to $595 ; how much was he paid a month ? ^^ 7. The cost of 97 sheep was $388 : what did each cost ' 8. If^9_^5 lots of land cost $22515, what is that for one 9. My agent sends me from Montreal 4368 hams, bein-- 13 tmies too many : how many did I require ? " ^°' L^ladrboTdo^'^ ^"^^^""^^" X5 weeks, how many ''■ i^n'days!'" '^ ''""'' '" '^'^ ^^^ '' ^^P^^^^ 66360 hours 12. In the workhouse there are 72 men whose ages amount to 5976 years : what is the average age of each ? '^" ISf ^??'^^' '^ "f P^"ters for $3555 for the season wnat do 1 pay each "^■■" - 14. There is a new m lan oon every 28 days : how man moons will there be in 108192 days ? V new Dirisio.y. 97 107 ch did he 556 each, rrels cost iny pages long will 595 : how Lch cost ? for one ns, being owmany 60 hours i amount :h? season : any new 15. How many battalions can be formed out of 32340 soldiers, givmg 420 men to each battalion ? 16. How many lo-cent pieces would make up !?64.6o ? 17. How many 25-cent pieces would ])ay a debt of $468 ? 18. A wealthy merchant distributed to 9S0 p)i)r people, in an equal proportion, 876432 pounds cjf Hour : what would each receive, and how mucii would be left ? 19. If 63 gallons make a hogshead, how many hogsheads will there be in 1449 gallons? 20. How much would be left from $^449 after 116 men had been paid 21 dollars each? 21. An excursion boat can carry 105 people : how many trips must it run to take 2486 people, and how many go on the last trip ? 22. The total outfit of a regiment of cavalry 1200 strong cost $236400 : what was the cost of each man's outfit .'' 23. How many miles of road, at $26000 a mile, can be built for $11050000? 24. To give 236979, by what must I multiply 1809? 25. In an engagement, 4376 soldiers use 205672 cartridges • how many is that for each man ? 26. How many feet are there in a mile, if 42 miles contain 221760 feet ? 27. Of what number is 158 both divisor and quotient? 28. How many bales of cotton, each weighing 427 pounds, are there in a crop of 468419 pounds? 29. A moulder has 17385 pounds of metal : find the least number of pounds he must buy in order to cast cannon- balls each weigiiing 68 pounds? 30. How many could he then cast ? 31. Divide one billion by 256. 32. The quotient is 345, the dividend is 273240 : find the divisor. f * 'r I ! I 98 ARITHMETIC FOIi BEOlNNEnS. EXAMPLES ON ALL PREVIOUS PRINCIPLES. £x. I.— If 5 apples cost lo cents, what must i oav for 8 apples ? If 5 apples cost lo cents, i apple must cost 2 cents, and 8 apples would cost 8 times 2 cents, or i6 cents. A//S. i6 cents. £x. 2.— How many pears at 3 cents apiece ought I to receive in exchange for 12 apples at 2 cents apiece? In order to make the bargain even, the value of all tlir pears must be the same as the value of all the apples, which is 24 cents. How many pears, then, must there be to amount to 24 cents, at 3 cents apiece? A/is. 8 pears. tS* On account of the pears being worth more apiece than the apples, there must be a less number of pears than apples. £x. 3. — If 3 men can build a house in 21 days, how long must 7 men be employed to do the same work ? If 3 men take 21 days, one man would require 3 times as long as 3 men, that is, 63 days. One man doing the work in 63 days, 7 men would need only g days. Aus. 9 days. US' The pupil must be taught very carefully to distin- guish between Ex. i and Ex. 3. It is quite natural to reason thus: If 3 men take 21 days, i man would take 7 days ; in fact, this is the very mistake the pupil will be apt to make. £x. 4. — A boy bought the same number of oranges as lemons, paying 5 cents each for oranges and 7 cents each for lemons : how many would he get for 84 cents ? If the boy had only 5 cents and 7 cents, that is 12 cents, he could only buy i orange and I lemon. Hence, for every 12 cents he owns he could buy one of each, and as he owns 84 cents, or 7 times 12 cents, he could buy 7 oranges and 7 lemons. Ans. 7 of each. ■Ex. 5— I gave 11 peaches to each of 8 boys, and kept 5 myself: how many had I at first ? DTVISIOS. «9 «V ! T' ^'k"^ ^ '^'^"'^ "^^^ S t''"*^ " peaches, or 88 peaches ; but as I want 5 myself I must have 88 + c. or 93. to begin with. j^ts. 93 peached In this problem the 93 is the dividend, the 8 is the divi- spp' Ihi. f'^"?*!>"*V^"^. 5 the remainder. Hence we H^Jic A ^""^.^^^ dividend, 93, we multiply the mlind'er^" ^^^tient together, and add in tlie re- .n^"^' ^'""i^ '"''" ^V^^ ^^^"' °^ '5« acres at 80 dollars an acre ; he pays $5000 down, and the rest in 8 equal yearly payments : what does he pay each year? ^^na «! °^ ^^A ^^™ J' ^^" ^ 150= $12000. After pay. i.?.. ^.°T ^"W" Inhere will be left $ 12000- $5000 L m^,T.K° ^/^'l '".^ ^^"^' payments. Each payment must therefore be $7000-^8 = $875. Ans.%^Ts. I. 2. EXERCISE 26. tr^nflTAf. 1^-53^ *\^^' ^•^"' ^^3289 to hisdaugh- ofhfsprfplrtV?''" "^P'^"="^^* -- ^h« --o-t L^chTfin^^hyw^bSelS^^^ ^- l'^a\V4re'acT:^^d^ ^'^^ ^ ^^ 4. Divide the product of 204 and 238, by their difference. ^" hnw ^^"""^'^fu ""^ ^'^^f "'^" ^'■^ f°'-'"«d i" two rows: row i^H^^ ""^ ^!,'" "^"^ ^^^^ ^ "^^^^ '"^"y '■" each row If they were drawn up in four rows.? How many II in SIX rows ? -' 6. From what number must 72 be taken to leave a re- mainder equal to 3 times 45 ? f ! . ! I 100 ahitiimetic fou nKfUXNEns. II. 12. 13- g. A man bought 47 feet of laiul : for 25 f(;(t he paid $41 a foot, for the rest $45 a foot : how much did all cost? ID. Add three hundred and sixty-two thousand four Inni- dred and nine to eight hundred and sevni thousand nine hundred and eighty-four, and divide their sum by eight. A man left $14389 to be divided thus : to his widow $5000, to his son $4000, to each of fcnir servants $100. and the rest to be equally divided among his three- daughters : what will each of the daughters receive? There are 24 sheets of paper in a quire: how many sheets in 3 dozen packets, each containing 5 quires ? A farmer bought 3 horses and 4 mules for $1122 ; the mules cost $144 each : what did each of the horses cost ? 14. A merchant bought 13 bales of cloth, each bale con- taining 27 pieces, and each piece measuring 34 yards. what would be the value of the whole at 17 cents per yard ? 15. If 36 men can cut a road in 77 days, how many men can do the same in 2i days? 16. How many yards of velvet at 7 dollars a yard, 8 dol- lars a yard, and 9 dollars a yard, the same quantity of each, can a dealer buy for i8oo dollars ? 17. What number added to the product of 327 and 8j will give 30000 ? 18. When a man's property was divided, his son received $5148, and the rest was divided among 11 churches, giving each $936 : what was the property worth ? ig. Divide the sum of 5168 and 5206 by three times their diiTerence. 20. How many weeks will it take a man to build 17 wall.' of 154 feet each, if he build 22 feet a week ? What must be multiplied by 327 to give 23642 1 gallons? 21. 22. A. had 75 cows, B. 90 oxen ; each sold his cattle for $2250: how much per head did A. receive more than B.? otrisios. 101 23. I bought 16 pieces of print of 33 yards eacli at lo cents a yard, and paid for them with tea at 80 cents a pound : how much tea was given ? 24 The quotient is 345, the dividetid 273240 : what is the divisor ? 25. The divisor is 213, the quotient 437, and the remainder 196 : what is the dividend? 26. A. has 2280 dollars to layout for horses and oxen, and wislies to purchase the same number of each: if he pays 5J65 a head for horses and $30 for oxen, how many of each can he buy ? 27. I bought some books for $3.57, and sold them at 20 cents apiece, losing 17 cents: how many books were there ? 28. A vessel saus 5712 miles in 48 days : how many miles does she go in a day ? how many in 5 days ? A man's salary is $3150 a year ; his expenses are $2817 a year : how much can he save in 6 years ? A.'s income is 5 times B.'s, B.'s income is 3 times C.'s, and C.'s income is $1325 : find the incomes of all to- gether. 20 30 31- 32. {^°^v m^"y cases, each containing 6 dozen books, can be filled from 18 parcels, each containing 3124 ? If 27 clerks receive $3888 for 16 days' work, how much a day was that for each man ? 33. What number is that to which if 17 be added the result is five times 384 ? 34. A man's income is 398 dollars a year : if he spend each year 256 dollars, how much will he save in 12 years ? 35. A man had an income of $3742 a vear (52 weeks) ; he spent 1 1500, gave to the hospitar$37o, and saved the rest : hov/ much did he save per week ? 36. If 563 be multiplied by a certain number, and 1043 be added, the result is 23000 : find the number. 37. For 21 pigs and 43 calves a farmer received .$401 ; the ~" "Twere sold at " ' each pig each : what was the price ol > \ \> ■ \ 1 f, ; if ■ ^ ■ '^■ 1" t \ ! 102 4"- 41- 42. 43. 44- 45- 46. 47- 48. 49- 51. 52. 53- 54- AlilTIJMl.Tjr roil ILGINNERS. Find a niimher. such that if th*. sum of 89 and 2::G be subtracted from it, the remauuler is 12 times 399^ A man bought wheat at 47 cents a bushel, and s.ld it What number must be multiplied by 37 to make the product equal to the sum of 1998 andV996 ? f i^!unS"'^' °^ '"" "''' ^'^•'" •• ''^''' '''' '^' P"^^ -f If II men can sod an acre of ground in 12 davs how sTddlng?" "'" ' """ '''' '^ '^ ''^ ^""^ '"^"""'" ll-nl* *u"^'^- "^""^"^ P^'^y ^'^'' 24 yards of cloth what will be the price of 56 yards of the same kind ? fflo^Vof ?33 cos''? '°"^'* '^^ ^^^-5- = '-- --h WiU It tfe"same ^L?^^'' f "". ^' ^°''^^^* ^"^'^ ^3. how many, at tne same rate, can be bought for $51 ? ^ %'c7nTsf:yTr ?' '^"^'^ '°^ ^^ ^-^^' ^- --y will boy's'^in "hoLsr" ^^" ' ''^>'^ ^'^ ^^ "^^^ ^^^ - ^ Jn 6 dlyr?'"^ "^'^^ ''" -^ ^^>'' ^""^ ^^ '""^h ^« « boys '^U'ir.rwTek^f ^ ^^" ^^ '^^'^ earn as much as 3 hort'nTS^^yfp^^ -•" ° "'-- - ---h as :8 Divide the sum ot 1692 and 1786 by their difference valued" aT$i!r^fHT uf' "^^5 each for a horse wmea at ^1^^, and the balance n hats at ftj ani^r^^. how many hats did he receive ? ^^ apiece : tJ'lff- '^'^^ ^•' ^^'''"^ 305 paintings at $45 each, and receiving 77 reapers at $181 each : whith owes the other, and how much ? $89648 is 8 tim.es as much as I paid lor a house • but /t^worth ? ""'' '^'" *^' ^""^^ "^^ --^^^ = -hi; wl's DI VISIOS. 103 ^^" s.I'd'ihemy r t''t f '*"'""''"' ^^ ^5 a barrel, and barrel ? *^^ ' '"""'' ''"' ^^"'"^^ ^" ^^^^» 56. By selling 31 i,.ts f„r $3100 I lose $155: for what should 1 sell 16 lots to gain 1^597 ? ^^' hoirr'' ^''^'■'^' ';''^'y '^'^'^^ ^9237; he spent $136 on house repairs ; for hired men he paid 4 times as wh.^^h^''.^'".^r^55= ^'"^ f"^ «th^^ expensts $1902 what has he left to put by yearly? . *^y"^ • ^^' J>"f* 25 sacks of flour for $125: what must I sell them for per sack to gain $75 ? 59. What will be the gain on each sack in the last ques- ^"" fe"^i!,°i^*^^,^^'"^""mber of plover, snipe, and quail rnnt! ^"^S":!.^^ ^^'T"" ^* '^ Cents, the snipe at 37 e^c^did^he'senr"^ ^' '' ^^"^^ ^^^^ = ^^^^ "^^"^ '^ ^'' 3!!" *?°"f "^ ^^"^v^y -hecks are to be marked by 3 ?^X' ^'^^ marks 2MO an hour ; the second and third each mark 150 an hour: how long will they take to mark the whole, all working together ? 62. If 59 articles cost me ^43.07, how much must I sell 23 ot them for to gain $1.83 on those sold ? 63. A man earns $50 a month, but it costs him $:5o a month to live: hc.w many months will he take to save enough to purchase 48 acres of land at $1 o an acre ? 64. I sold 28 horses at $122 each ; then bought 224 sheep at $12 each, 8 cows at $60 each, and spent the re mainder in calves at $8 each : how many calves did I 65. If it costs $56 for bricks to build a cistern, when bricks are worth $8 a thousand, wiiat will it cost for bricks to Duild It, when they are worth $10 a thou- sand r 66. If 5 barrels of cider are w'^.' multiply-by 9, divide by 7, multiply by 8, add 5 divide by II multiply by 6, add 21, divide by 9, add 20, divide by 3, multiply by c, add is di- vide by 10, multiply by 8, divide by 6 : result ? 20. From 63 take 9, add 16, divide by 10, add 41, subtract 20, divide by 4, add 93, subtract 17. add 2. divide hv 5, multiply by 3 subtract 8, add 27, divide by 7, sub- tract 10, multiply by 13: result? CUAPTRR II. 'ifi!f^.') FACToiima. 108. Tliere are two ways of making up the number 6, eitliLr by adding 4 and 2, or by multiplying 3 and 2, As in Art. 83, where the number 6, is made up by niui tiplying 3 and 2, each ol these numbers is calk-d a Factor of 6. Each of them is also an Exact Divi sor of 6. 109. A Factor of a number may therefore be said to be an Exact Divisor of the immber. It is very desirable that the pupil should be able to tell the different Divisors of any number. Ex, I.— Find all the Divisors of 18. 18 = 9x2, or 3x6, Hence the Divisors are g, 6, 3, 2. The numbers 2, 3, 5, 7, u, etc., have no exact Divisors or Factors, and all such numbers are called Prime Numbers. Since the number 3 divides botn 6 and 9, 3 ir ;iaid to be a Common Divisor of 5 and 9. So 5 is a Corn 'no;5 Divisor of 15 and 20; 4 is a Common Divisor ot \ 12, and 16. A Common Divisor is any numoer that will exactly divide two or more numbers. 110. 111. Ex. Ti'iere.fore Find a Common Divisor of 16, 20, 24. i6 = 2x 8. 20 = 2x10. '.4— 2X 12, 2 is a Common Divisor. (106} factohiso. 107 Again 16=4x4, -J" = 4x5. 24 4x6, ThtTcfoie 4 is also a Common Divisor. Wt thus ste that there may I,e more than one Common Divisor to two or more numbtis ; and, since 4 is the gr.aterof the two Divisors, it is called the Greatest Common Divisor of 16, 20, and 24. 112. The Greatest Common Divisor (G.C.D.) is the greatest number that will exactly divrdetwoor more numbers. i5"x— Find the Greatest Com. Div. of 18, 24, 30 We see, by iiispection, that 2. 3. and 6 are the'onlv Common Divisors of 18. 2+, and 30, therefore the w, C D. IS 6. K3- The Divisors, Common Divisors, and Greatest Common Divisors .should, if possible, be found bv inspection. ' 113. The following will be found a simple method of finding the G. C. D. of any numbers: Take 16, 24, and 50. The G. C. D. cannot be pfreater than 16, and must be some divisor of 16. The greatest divisor of 16 is 8 but this will not divide 50. The next divisor of 16 is 4, but this will not divide 50. The next divisor is 2 and since this also divides 24 and 50. it must be the y t. G. D. 11'. This gives us the following RULE FOR FINDING THE GREATEST COMMON DIVISOR. Take the least of the ^iven numbers ami try its divisors in order, beginfiing with the greatest. The fist one that wi. I divide each of the other numher' w^'f be the required Greatest Common Divisor. Ex.— 20, 24, and 28. 108 ARITHMETIC FOR BEQINNERS. The divisors of 20 are 10, 5, 4. and 2. The first (;iie that divides both 24 and 28 is 4. Hence 4 is their G.C.D. EXERCISE 28. t3* These questions should be solved mentally. 1. What numbers will exactly divide 12? 48? 56? 81? 2. Find the exact divisors of 21 ; 32 ; 49 ; 42 ; 36. 3. What numbers under 50 are exactly divisible by 2? 3? 4? 5? 6? 7? 8? 9? 10? II? 12? 4. What numbers between 50 and 121 have for a factor 5? 7? 9? 12? Write down the other factor in each case. * 5. Write down the simplest or prime factors of 64, 54, 78, 120, 145, 152, 99, 117, 189. 6. What numbers less than 150 are divisible by both 3 and 4? 4 and 5? 5 and 6 ? 3 and 8 ? 7. Name the three least numbers that exactly contain both 3 and 5 ; 2 and 5 ; 2 and 3 ; 3 and 4 ; 4 and 5, 8. Write down in order the prime numbers less tlian 50; between 50 and 100. 9. What three prime numbers will divide 42 ? 30 ? 105 ? 10. Find the common divisors of 24 and 30 ; of 27 and 36; of 15 and 45; of 36 and 64; of 72 and 80; of 90 and 120. 11. Name all the common divisors of 12, 18 and 20; of 24, 40 and 60 ; of 36, 48 and 72 ; of 24, 36, 60, 72. 12. Write down the G. C. D. in each part of questions 10 and II. 13. What is the G. C. D. of 16, 24, and 36 ? of 9, 27, and 33 ? cf 15, 35, and 50 ? of 18, 32, and 60? FACTOnrXG. 109 115. When the given numbers are large, the G. C. D. can not always be found by inspection. The following method is then adopted : £'jc.— Find the G. C. D. of 697 and 820. 697)820(1 697 123)697(5 615 82)123(1 82 41)82(2 82 Divide the less into the greater numl>er — the remainder is 123, which we divide into the first divisor 697. This leaves a remainder 82, which we divide into the previous divisor 123, leaving a remainder 41. The number 41 is divided into the previous divisor 82, and since it is contained exactly, 41 is the G. C. D. tS' In finding the G. C D., if the last divisor is i, the given numbers are ' J to be prime to one another. 116. From the above we have the following RULE FOR FINDING THE GREATEST COMMON DIVISOR. Divide the less into the greater of the given numbers, then divide the remainder then obtained into the previous divisor, and so on, until an exact divisor is obtained. This exact divisor will be the G. C. D. required. XS" If there be three or more numbers, find the G. C. D. of any two of Lhem. Then find tlse G. C. D. of this result and a third number and soon. The final result will be the G. C. D. required. Ex. — Find the 0. C. D. of 585, 765, and 285. no ARITHMETia FOB BEOINNEnS. The G. C. D. of Hence 15 is tlie The G. C. D. of 585 and 765 is 45. 45 and the third number 2S5 i. 15 G. C. D. of the given numbers. tS*- The pupil should prove the truth of the result, in other words, see that the G. C. D. obtained \vill exactly divide each of the given numbers. Thus: 585-*-i5 = 39- 765-»-i5 = 5i- 285+15 = 19. I 2. 3. 4 5' 6. 7- 8. 10. II. 12. EXERCISE 29. Find the greatest common divisor of 161 Find the greatest common divisor of 592 Find the greatest common divisor of 2013 Find the greatest common divisor of 576 Find the greatest common divisor of 592 Find the greatest common divisor of 1369 Find the greatest common divisor of 1866 Find the greatest common divisor of 1029 Find the greatest common divisor of 992, 672. Find the greatest common divisor of 867, 714. Find the greatest common divisor of 1134, 630. What is the length of the longest pole measure 84 feet, 56 feet and 70 feet ? and 115. and 332. and 1220. and 960. and 1225. and 703. and 1492. and 1 197. 352 and 1088 and 1386 and that will 117. Wherever we have a Divisor we must have a Divi- dend. The object of the previous exercise was to find the Divisor. We shall now proceed to find the Dividend, of which certain numbers are given as Divisors, When we speak of a Divisor or a Divi- dend, we always refrr to an Exact Divisor and an Exact Dividend. Since 3x4= 1 2, 12 contains both 3 and 4, and is there- fore an Exact Dividend of 3, and also of 4. 119. A N 120. T t3 121. Tl Tc Tl FACTORING. I'll 118. An Exact Dividend of a given niimlicr is therefore a number which will contain the given number with- out any remainder. An Exact Dividend is also called a Multiple. Since 15 contains 3 and also 5 exactly, 15 is a dividend of 3 and also of 5, and is called a Common Divi- dend of 3 and 5. 119. A Common Dividend is a number tliat contain s two or more numbers exyctlv. £^x. — 24 is a Comiion I)i\itlend of 2 and 4. a Common Dividend of 2, 3. ana 3. 18 is Now, 36 is a Common Dividend of 2, 3, and 4 ; and so likewise is 24, and also 1 2. And since of the common dividends 36, 24, and 12, 12 is the least, it is called the Least Common Dividend of 2, 3, and 4, 120. The Least Common Dividend (L. C. D.) of two or morenumbersis the least dividend th.it will contain each of the nuinl)ers exactly. ^•*- — The L. C. D. of 2, 3, and 8 is 24, oecause 24 is the least dividend that will contain 2, 3, or 8. The L. C. D. of 4, 5, 12 is 60. IS" All Dividends, Common Dividends, and Least Common Dividends should be found, if possibl. , by inspection. 121. The following will be f )und a good method of findiii"- the L. C. D. mentally : Take 5, 8, and 12. The L. C. D. cannt^t be 12, because 12 aoes not con- tain either 5 or 8. The next number that contains 12 is 24, but this does not contain 5, although it con- tains 8. Then we try 3 times 12, 4 times 12, etc., until we come to 9 times 12, or io8. None of these will answer, but the next one, 10 times 12, or 120, contains both 5, 8, and 12, and must be the L. C. D. *^^ ARITHMETIC FOn BEOIXHEnS. 122. Hence we have the following RULE FOR FINDING THE LEAST COMMON DIVIDEND. Take the greatest of the given numbers, and try its dividends tn order, l>egtnning zaith the least. The first one that i.ill contain each of the other numbers zvill be the required Least Common Dividend. Ex.~V\\\A the L. C. D. of lo, 24, and 30. The successive dividends of 30 are 60, 90, and lac and since 120 is the first one that will contain 10 and 24, the L. C. D. must be 120. iiXERCISE 30. IS" These questions should be solved mentally. T. What numbers below 50 are dividends of 2> oi x^ of 4? of 5? of 6? of 7? of 8.? of 9? of 10.? of II' of 12? 3. What numbers between 50 and 145 exactly contain 7? 9? II? 12.? 4. Of what two prime numbers is 12 a common divi dend ? 5. Of what numbers is 12 a common dividend? 24 .-' 30 I.S 6. Write m order each number below 100 that is a common dividend of 2 and 3 ; of 3 and 7 • of 2 ^ and 4 : of 4, 5 and 6 ; of 2, 5, and 7 ; of 2, 3! +,' and 5 ; of 3 and 8 ; of 8 and 10. 7. Write down the L. C. D. in each part of question 6. 8. Find the L. C. D. of 8 and 12 ; of 9 and 12 ; of 2, 4, 5,and6; of2 3,and 10; of3,4,and8; of 4,5, and 8 ; of 6, 3, and 8 ; of 2, 5, 8, and 10 ; of 3. 12, and 4 ; ot 6, 18, and 9 ; of 4, 12, and 16 ; of 8, 10, and i- fACTO/USO. an 123. no con. h"'" ^'^'V^""^ °* 3. 4. 12, 15. we need d vVfri ^ '^' '^' ,^^'^"'^ ^"y ""'"^^^ that is a dividend of 15 must be a dividend of 3 ; and any number that is a dividend of 12 must be a dividend *^i 4* "Xavs str^"^ '^ ^' ^' ^- -f-ny numbers we may 124. We thus have the following RULE FOR FINDING THE LEAST COM- MON DIVIDEND OF SEVERAL NUM- BERS. Place the given numbers in a line, and first strike out any one of them that is a divisor of any othtr. Then begin, vith the lo7vest divisor, 2, and divide by it as ojttn as tt ts contained in any two of the numbers, bringing down any numbers that are not divisible. ^ Proceed thus with 3, 5, 7, etc., always striking out in any tZtlne '' ^^""^ '" "" '^'''""' '^ "''^ '*^'''' "'''"^''' '" Finally multiply together the different divisors and all th, numbers in the last line. The product thus obtained will be the required L. C. D. Ex.~~Fm^ the Least Common Dividend of 4 T 10, 12, 30,43, 75, I, JO. ^' ' 3 ) $, '" ~^ 5) 45' 75. roo 45. 75. H 3. 5, L.C.D.-2X 2x3x5x3x3-900. Ans. i , ' i 1 • 114 ARITHMETIC FOR DE0INNER3. ^and"f ^^t ""'"^ers in a lipe, we first stride out 4 and 6, s nee each is a divisor of 12. Then strike out 10, since it I's a divisor of 30. (Art. 123 ) ^ow divide by 2, and we obtain the quotients as fe 1^' :f' "^ ^*^^^^ -^ '5' ^— ^ "-- ^Slifo^ta'fne'd i^^V^" ^'"^^ °"* 3 and .5. for ^?,r;K'"r^ ^ "^'^ "° ^°"^^^ ^^v^'^e any two numbers in the line, wo trjr 3, and then 5. Finally, multiply together the four divisors and the last quotients, and we obtain the L. C. D. as above. 125. Tofind the L. c. D of two large numbers, we first find the Greatest Common Divisor. '^^.n"^'^''' u^ *,^^\^- ^•'*®- '"*° ^^'^^^ o^ the numbers, and multiply the quotient by the other number. The product will be the required least common divi- dend. Ex.—T'mA the L. C.^D. of 970 and 1261. lUeir greatest common divisor will be found to be 97. Then 970 -*- 97 ss 10. 1261 X 10 « i26io,which is the required L.C.D. Proof: Z2610 -»- 970 Z2610 -I- 1261 13. 10. 126. If there be several large numbers, find the L. C. D of any two, then find the L. C. D. of this result and a third number, and so on. The final result will be the required L. C. D. As in the case of the Greatest Common Divisor the pupil should prove the truth of the result by' finding, as in Art. 125, if the L. C. D. will contai'n each of the given numbers exactly. FACTOR! so. EXERCISE 81, 118 Find the Least Common Dividend: '• Of 5. IS. 9. 6 and 3. 2. Of 4, 5, 10, 8, i8 and 15. 3. Of 12, 36, 25, 60, 35 and 72. 4- Of 63, 81, 14, 54, 27 and g. 5- Of 7. 72, 84, 42, 12 and 6. 6. Of 72, 36, 180, 24, 18, 9 and ISO. 7. Of 90, 10, 64, 70, 45, 8 and 32. 8. Of 40. 24. 8, 32, 20, 16 and 10. 9- Of 29, 144, 216, 180, 90 and 252. 10. Of 60, 78, 42, 96, 56, 48 and 39. 11. Of 2041 and 8476. 12. Of 812 and 336. 13. Of 7056 and 7392. 14. Of 7212, 9015 and 24040. 15. Of 7218, 6015, and 5213. 16. Of 2712, 816, 54, and 15. 17- Of 250, 360, 49, and 700. 18. Of 32, 44, 52> 13, 65, and 48. 19 Of 76, 748, 448, 152, and 38. 20. What is the smallest quantity of barley that can be car ed away in either 20, 25, 30, 35, or 40 bushel carts, and how many loads would there be of each ? 21. Find the least amount of money that can be paid by either 2, 3, 4 5, i„, 20, 50, or 100 dollar bills and how many of each kind would be required ? CHAPTER IIT. FRACTIONS. 127. We have spoken of one dollar, one pound, one pint, one f( ot, etc., but have not mentioned any smaller part than one of each. We must now see how we can express any part of a dollar, a pound, a pint or a foot. ' r » If we divide a foot, for instance, into two equal parts, jach part is called a half, and written ^the figure 2 saowing that the unit (in this case a foot) is divided into two equal parts, and the figure 1 show- ing that we have taken one of these parts. In the same way, if a foot be divided into 3, 4 ,5, 6, etc., equal parts, each part will be called a third,' fourth, fifth, sixth, etc., respectively, and be repre- sented by ^, i, X, I, etc. It will also be seen that to make up the whole, or the unit, we must take Two-halves, or Three-thirds, or Four-fourths, or Five-fifths, etc.j etc. This will be easily seen from the following figure, where the unit is taken as a foot in length. £110] Onr unit. Twn-li.-iKcs. Three-thirds. 3 Four-fourths. Five-fifths. Six-sixtlis. Sevcn-scvcinhs. 3 Eight-eighths. Nine-ninths. Tcn-tonths. Eleven-eleventh'-,. Twelve-twelfths. We liave taken the unit to be one foot, and since therr are twelve inches in a foot, eacl) part of the lower line must represent a.i inch in length. 128. The fbllowing result will then be seen by using a straight-edge or measure : ^ ^ The whole equals 12 inches. One-half " 6 inches. One-third " 4 inches. One-fourth " 3 inches. One-sixth " 2 inches. Thtis we see that we find one-half of any number by Hence, j^ of 40 cents = 10 cents, i of 81 apples = g apples. tV of 30 days = 2 days. 129. Again, referring to our figure, we see tliat two-tliirds nL'fj •"? ?, '''■' ^ \"^''"'' ^'^''^^ '^' t^^-*^^ ^s long as one-third ; threc-sixths are 6 inches, or three times as much as one-sixth ; seven-twelfths are 7 inches or 7 times as much as one-twelfth. In the same manner, ,V '^f S44 must be 5 times ^\ of U that IS 5 times §4, wliich is equal to $20. So, f of 63 = 4 times I of 63 = 28. \^ of 26 = ro times J^ of 26 4 ^11 nt ARlTnMETIC FOR DEOOfyERS. 130. The symbols ^, ^, -J, etc.. are called Fractions, and represent one or more of the equal parts of the whole or unit. 131. The lower nuii.oer is called the Denominator, he- cause it points out or shows the number of equal parts into which the unit is divided, or, in other wc.rds, it shows the size of the parts. 132. The upprr number is called the Numerator, for it tells the number of parts taken. Thus -jBj- is a Fraction, and represents nine of the eqn.il parts of the unit. 1 1 is the Denominator, and shows the size of the parts to be elevenths. 9 is the Nume rator, for it tells that the number of parts taken is nine. EXERCISE 32. These questions should be ::olved mentsUy. Read the following fractions, namin;,' thedenomina tor and numerator to each : .j., |, ^, ^, ^, .j, •! TT> TT' TT» e' Ta' TS' Tcf> Ta' T?T' Write the following fractions in figures Five-ninths Three-sevenths Two-fifths. Four-sevenths. Five-sixths. Three-eighths. Four-ninths. Seven-ninths. Seven-tenths. Five-sevenths. Eight-ninths. Nine-tenths. One-twelfth. Five-elevenths. Nine-fourteenths. Eleven -twelfths. Eight-fifteenths. Seven -twentieths. Eight-thirteenths. Four-twentieths. Eleven-nineteenths. Sixteen-twenty-thirds. Three-fourteenths. Four-fortieths. One-seventieth. Seventy-ninetieths. When anything is divided into seven equal parts, what is one part called ? Three parts ? Five parts ? What is one of the eleven equal parts of anything called ? Seven of the twelve equal parts ? Nine of the ten equal parts? Fourteen of the fifteen equal parts? Eighteen of the tv/enty equal parts ? FRACTION. 110 for it What is I of 8? TT(>f 55? i8? 5. What is meant by one-ninth of a quantity of apples ? Seven-elcvrnthsdf a heap of oats ? Tei-twclfths ut a distance ? Four-twentieths of the vahie of a ves- sel ? I''ive-sevenths of a man's property? 6. How many sixths in the whole of an estate ? Tenths in one-half of an apple ? Quarters in one-half of a yard ? Sixths in one-third of a pound ? Eighths in one-quarter? Sixteenths in one-eighth ? :of 12? I of 25? f of 63? i t.f _ of 48 ? ^'^ of 1 20 ? I of 24 ? J of 45 ? T*r of ^8 ? ^ of 21 ? *g of 39 ? I of 27 pounds ? ■| of 48 ounces ? ♦ of i ounce, or 20 pennyweights ? •f of 42 inches ? ^^-j- of 77 acres ? -i-^ of 66 yards ? I of $84 ? 8. How much of anything will be left if | be taken away ? If -^^^ be taken away ? If -J- be taken away ? If y\ be talen away? U j\ of it be lost ? If ♦- of it be given away? If ^^ of it be sold? If f^^ of i' be lost ? 9. How much of anything must be taken from it to leave |of it? -f\of it? T-Lof it? T^ofit? -fofit? \l of it? T^ofit? xiofit? 10. What part of my farm may I sell to have | of it left ? f of it ? t5j. of it ? A| of it ? ^ of it ? ^-^ of it ? -j-\ofit? iofit? 11. A farm contains 2G0 acres: hcnv many acres in | ol it ? in I'tj of it ? in -\ of it ? l of it ? 12. If -I of a vessel be worth $60, what will | be worth ? 13. If I of my property be 100 acres, find the number of acres in ^ of it. 14. If f of a number be 16, what will | of it be ? 15. If ^ of a bushel of oats be worth 72 cents, what would A be worth ? What would a bushel be worth ? What would -rV t)f a bushel be w^orth ? ^\ of a bushel ? y^, of a bushel ? ifi. How old is a boy -f of whose age is just 6 years? 17. If ■*• of a pound of tea be worth 72 cents, find the price of a p;innd. 1 f{ t 1 1 1 120 AltlTIIMKTIC h'On BFJilSSEKS. I8. What is cofic. worth a pound whn. 40 cents pay l"t I fif a pound ? * ■' pay loi -♦- of It. What must F yt pay ? 'Ill 133. W« have seen that J. |, . .3. ^^ of the unit, in Art. 127, \ab the same in each case, namely, 6 of the , ^ equa parts or njclies. This shows th^t a fracti, n may be expressed in many different forms, and st be the same in value. In other words, if we c' then'' n I 7"' '''■ '^'^ ^''''l ^^^' '""^^ '^^^ "'-- of them, and if we incivasc the size of the parts W(> nnis take fewer of them. Again, if we inc ease t h • number ol parts taken we mu\t decrease tidrsiz and If we decrease the lunuber of parts taken we must increase their size. ''TermsXl^;ac?ion. ^^--'-^^ -e called the ^'po2u^iln;$i;f"' "' ''^'•"^ the ^,llowing in. 134. If ike iwo terms of any fraction be m„ltip!i,',t bv t/u number, the value of the fraction is not change/. Again, since ^^ = |, | = i etc.. we deduce the folio important Pruiciple: 135. If the two terms of any fraction be divided by the same ma; - ber, the value of the fraction is not changed. Ex. i.-Express | as a fraction, having ,. for its denominator, or change | to fifteenths. sani, winy Toobtain 15 from 5. we must multiply c by , ^^a ^,.„_, both terms ot a fraction must be n^ul^ij^ i^d ^ ?h same number to have no effect on its value, w. U . multiply the nuinera^ ' • ' " fract Th ion \%. ♦ —12 4 by 3. Th IS gives us the us ■* = TT- % r/?/lCT/().V5. t»l cents pays could only Ex. 2.— Chanfje {;] to ciRhlhs, that is, cxpirss | it, in Art. > of tlie li a fraction , and still if we <\v e more of parts we :rease the heir size, taken we illed the winjT nil. t/ie savh ollowinsj 'tne nui: 5 for its nd sfnc'^ d by th< ,ve mus! us tllu uving its denominator 8. as a fraction h To obtain 8 from i6 we divide i6 by 2, hence W(- must also divide lo by 2, to give a new numerator. Am. \. 136. When tlie fraction \l is clianf^ed to ^, the fraction « IS said t a fraction having its denominator 8 ; 12 , 20 ; 40 ; 32 ; 60. 2. Express | ot a yard as twelfths ; as sixteenths ; as thirtieths. 122 ARITETMETIC FOR BEOiyNERS. ■ i-P II i (p) 3. How many twentieths of a dollar must I exchange for a hall ? for a quarter? for four- fifths ? for seven- tenths ? 4. Change to thirty-fifths, to fortieths, to thirty-sixths, to forty-fifths. Reduce the following fractions to their lowest terms • 6. 12, 8 T J5 8 J5 ± 6 J.* 1 6> 9 » 3 2 ' 2iy' 1 s T^> 7 2 "STS"' T 1 ' 1 2 T2' A B TTJ- *^ a" 3" 6 s 7 o" 1 o Tooiy- 7- 8. 9- 10. 3D TT' 2i T*' 3 2 B 1 T5T> 7 8 Te7> 7 o 6"T5» .9ft ir52» f f> ToT- s 7 So- 10 2^ Tt *■ n I txt;* .s n ST- 11. What fraction with lowest terms will express the same as -^W ? 42 9 i50 ? Shew that the fractions f^, |i, in value. and l^are the same 143. 4 139. Hitherto we have spoken of such parts of the unit or whole as |, -f, |, ^-^, etc., that is, a part less than a whole in each case. We now come to notice such fractions as |, y, \\ x^. etc. Now I means five-fifths and one-fifth, that is the wkole and one-fifth besides, which is written li and read one and one-fifth. So \p means seven-sevenths and three-sevenths, or if V = ^^ eighths and 5 e'ghths = 2|. 140. Such fractions as |, ^, ■^\, etc., in which the nume- rator is less than the denominator, are called Proper Fractions. 141. Such fractions as |, V", V» etc., in which the nume- rator is equal to or f^^reater than the denominator are called Improper Fractions. 142. The expressions 2^, 6|, 8^, etc., are called Mixed Fractions, being made up of a whole numoer and a fraction. («) ;t I exchange s ? for seven- FRACTION &. 128 owest terms : o n T» g ToT- To- 10 2. T* + • n I Ta"y* s n express the re the same of the unit irt less than J 10 21 J2 f T ' T > off- is the wkole lA and read en-sevenths fhths and 5 the nume- are called the nume- minator are lied Mixed unioer and 143. We see by Art. 139 that everj' improper f'-action may be brought to a mixed fraction by means of the fol- lowing RULE. Divide the numeral or of the improper fraction by the denonii- natot , a)id the (jttoticni will be the whole number. Place the remainder, if any, over the denominator, to form the other pari of the mixed fraction, ^.r.— Reduce ^A to a mixed fraction. Dividing 23 by 6 we obtain 3 for the quotient and 5 for the remainder. Hence 3A will be the required mixed fraction. In the mixed fraction 3^ tlie 3 is the same as 18 sixths, which with the other 5 sixths make 23 sixths, y. 144. Hence, to reduce a mixed fraction to an improper fraction we have the following RULE. Multiply the whole number by the denominator, to the pro- duct add the numerator , and under the sum write tJie de- nominator. Ex. — Reduce jf tu an improper fraction. Multiplying 7 by 5, we have 35 ; adding 3 to this pro- duct giveo 38 ; therefore the required fraction is V' EXERCISE ;i4. [a) Express the following improper fractions as mixed fractions, or whole numbers : I. 2. 3- u «_> 3 • V- 3 I 1_9 6 0' t, 6. V- 73 1 • a • 8. »/• 9. V- 10. V. II. ^. 12. 14. 15- 16. 17- i8. 19. 20. 21. 22. 6J1 s • 9 » T • I n a • 1 -i S • l_* a • i__» ♦ • 2 2 23- 24. 25- 26. 27. 28. 29. 30. 31- 32. '^, 33- 9 ■ 3_9 4 • *_? 5 • '_3 a • 8 * 12' 1 8 aT • 8_* 9 ■ S9 l_0 34 35 36 37 3« 3Q 40 41 42 43 44 V- V- 3_0 6 • 3 2 •g- • 1 18 8 • 1 2.9 9 . I +0 3 7 9 T 5 • 1 0.5.8 I I o. *T V. 124 D . fi ,i F; (■ ib) ARITIIMETIC FOR BKGryXKRS. 1. H- ' 12, 9h 2. 9i- 13- I of. 3- 6-/. 14. 8f. 4- I ox. 15- 7^- 5. 6. 9i- 31- 16. 17- 5t\ I of. ?• 4.V- 18. I2i I If 8. 3k. ig. 9- Sit- , 20. 9S-- lo. 7i 1 21. 7t\ II. 8|. ■ 22. 8f. 7-5-. / 8' (c) 23- 24. 25- 26. 27. Hi 28. i6f. 2g. 22|. 3^. 32f 31- 45^. 32. 23f. 33- 5f 8' 9f. lOA. I2f 34- 35- 36. 37- 3«- 39- 40. 41. 57 iV '24i 342^ 200i. I256f. 42. 4091/-^ of a day! 5' 6. 7- 8. 9- ro. II. 12. 13- 14. 15' . How many whole days are there in In y> of a day ? . How many whole ounces are there in •-" of an ounce? In VV ounces? In ^ of an ounce? . Take away as many dollars as you can from V of a do ar and what will be left? Fron^ lp/ .7 ! dollar ? - 1 1 '--^ '^- ^:ZTT^ ir^'p ^ wi '■''"'^" ^ ^""^^ ^p^-^" out of left } ^""'^ °^ ^ '^""^'" w°"ld be Change 5^ to fourths. Change 15 to fifths. Reduce 13^ to sixths. Reduce 8 ■^\ to elevenths. Change 3I to sevenths. To fourteenths. Change 5^ to ninths. To eighteenths. Change 6j\ to twelfths. To twenty-fourths Change 8 j\ to fourteenths. To forty-seconds. Reduce 9/^ to twentieths. To sixtieths. How many eighths of a pound are there in 2^" pounds? In. 30^ pounds? In loo^ pounds? " How many more quarters of a yard are there in Ssi yards than m 32j3-ards? ^' I' n ACT loss. 120 145. If we take the two fractions j?,^ '^'I'l ^. and wish to know the greater of the two, it niigiit not be known from a glance at the fractions. We will therefore find a method of comparing fractions, that is, find- ing out the order in which they stand according to their value. We know that ^ = i±. Also, 4 = ii Now we have the two fractions |^ and ||, in each of which the size of the parts of the unit is the same, that is, twenty-fourths; but in the first there are 14 and in the second 15 of these twenty-fourths taken. Hence we see at a glance that J a is greater than |^ by ^, that is, | is greater than -J^ by JU. Thus, a. boy who received | of a lot of apples would have more than a boy who obtained -^ of the lot. 146. In order to compare the two fractions, they were brought to other fractions, having the same denomi- nator, 24. This is the method we were seeking, and applies to any number of fractions. KS* This same denominator, it will be easily noticed, is the L. CD. of the given denominators. 147. We thus have the following RULE FOR COMPARING FRACTIONS. Find the L. C. D. of all the given doiominators. Bring each fraction to another having the L. C. D. for a new denominator. The value of the fractions will then depend entirely on the value of the new tiumerators. IS* Mixed fractions must be first brought to improper fractions. ^ ^j:.— Find the greatest and least of the following- I of a pound, ^V "f a pound, x of a pound. TheL.C.D. of theden f of a pound ominators t^ of a poun 12 and 9 is 36. tV of a pound = if of a pound. ^ of a pound = |a of a pound. 'J V26 AniTUMETlC FOR BEOlNyEiiS. Hence we have 24 parts, 15 parts, and 28 parts of a pound. Tlicrefore |^ or f of a pound is the great- est, -I*- or I of a pound is the next, and |i or -^-^ oi' a pound is the least. EXERCISE 35. (a) Arrange the following fractions in their order of value : (i>) h h ± 6 ' S ■ff- 4 "O" To s 5 • i' _7_ 2 O' I R TT- 7 IS- 3. 2" 4 »■ 2 s- 5- 6. 7- 8. 3i. B 1 \ TT' _o_ 1 o* 6. •If ,1 n 2 0" 1 s 1. A. owns I a ton, B. f of a ton, C. ^ of a ton, and D. I of a ton : who has the most, and who the least? 2. A. does I of a day's work, B. f, C. |, and D. ^\: which should draw the most pay? Which the least? 3. John has $3|, James $3^, Henry $5f, Charles $9-1^: arrange their names in the order of their wealth. 4. A man rides 19 quarters of a mile, drives 3|| miles, walks -gSj of a mile, and sails -f of a mile : which was the longest, and which the shortest ? 5. A pole is V f^et high, another 8^^, another f, and another 9|- : arrange them in order of height. 6. Which is the greatest and which the least of the following amounts : $gj, $3^*, $4^, $| ? Arrange them in order of value. it < ADDITION OF FRACTIONS. 148. We now proceed to add fractions when the fractions have denominators that are alike. In this case, since the parts of the unit or whole are the same in each fraction, the sum of the numera- tors will show the number of parts there are alto- gether. If we add | of a pound to -f of a pound, or 2 and 3 of the same si^e parts (sevenths), we must obtain 5 of these sevenths, that is -f of a pound. ADDIlIOy OF FllACrtONS. 127 149. Hence we have the following RULE. ^'Lwf ''^ '^''f'"-'^ """aerators, and under the sum place the given denominator. ^^.— Add together ^V, A. and ^^. "" UoZust t^;r;"^'"'°" '^ -3. so .he requ,red frac In this case, since the size of the parts of the unit or who e IS not the same iu each fraction, the fractions must be brought to other fractions hav-ing tlie sZe toonnnator. Then we proceed as in fhe Irn": 150. This gives the following RULE. ^'tat'oK^ *^" ^'''''^""'' '" '^^"' ^'"""'"^ ^^" '''"" ^'"'omi- Add the new numerators, and under their sum place the new denominator, ^ lO Here ^V .18 Ex. I. — Find the sum of ^ and ^, i- = ii±JL5 _ 13 12 6o ~ 6o tV = I5^> and the sum is |^. IS- If there be any mixed fractions, the whole num- bers are added separately, and then the sum of ^e fractions is added to that sum. Ex. 2.-Add $5J^, $7^, $ioi. 5 4- 7 + lo = 22. lo 8 4 40 $22 + $H = $22i^. 40 I 128 AHITIIMKIIC FfJii nKGrWEIiS 151 SUBTRACTION OF FRACTIONS. In subtracting one fraction from another, the sani,- remarks apply as in the addition cf fractions, except that instead of adding the numerators we subtract them. £^. I. — -i _ a. — » e a ~ ef 12 10 60 ~ 60 Jix. 2— From 23^'^ subtract lyf. In this case we cannot take | f-.m W, that is tV iroi" tV; s*'. as in ordinary subtraction we must use one of the next higher order' thatis.iunit, or-i|. Then|f+^V-if and ,9," from \l leaves J^-^. Again, since we used one ot the 23 units, there are only 22 left, and 17 from 22 leaves 5, giving the final result 5^% or sf. The work might be written in the following form : 23tV I7I Jia 171 =17+ I =17 + A DifFerence = S+A = Sf tS" In giving a result or answer, improper fractions should (unless otherwise stated) be brought to mixed tractions, and all fractions reduced to their lowest terms. EXERCISE 86. KS' These questions should be solved mentally. I. Add together and find the difference between • A andi; ^a-andA; ^andf; | and J ; ^ and i ; $21 and $3i ; | and i ; a of a foot and | of a foot ; 1 of a farm and x of a farm ; /^ and a ; ^V and i ; ^ of a share and a of a share ; ^V and |. Add together f and U > h h 1 1 * of a yard, i of a ^'^'lu, f ui a yard • > • - yan 8> To» a"^' 3. John had | of an apple, and gave away + of tJ apple : what had he left ? IONS. r, the same tioiis, except we subtract that n -fj, tnat is subtractioi), gher order. = H, andJ^ we used one md 17 from g form : ir fractions ht to mixed heir lowest SUBTRACTION OF FRACTIONS. 12» 4. Bought goods for f of a dollar, and sold them for i4 dollars : what did I gain ? ' 5. A boy paid i of a dollar for a book, ^ a dollar for a s end ? ^ paints: how much did he 6. Find the difference between, and also the sum of yand-l; iandl; fand|; a and I ; 3j-and24; 5Tand4i; 3^ and r^ ; 2^ inches and U inches; ci a"d 3|.; 71^ and 3A; 5i yards and 2| yards; 6^ ounces and 4I ounces; 171 cents and i2| cents! doi&r^"^"' ^""^ '^' " °'^"^"' = 7i% ^""^'^ ^"'i si 7. What must I add to $if to make $3^ ? 8. What must I take from $7^ to make $3! ? 9- From what must 3^ be taken to leave 2^ ? '°' ylrds ?''""^'^ ^' ^'^* '^ ^^~ ^'^'^' ^^^" *^^^" f^^'" 6f "' yaTd'??'^^''* '"'''* ^^ ''''* ''^ 51 yards to leave 3I 13- What part of a farm must be sold to leave ^V oHt ? 14. What must be taken from a square rod of land Uo^ yards) to leave 15^ yards ? ^^ * ally. etween : \ "di; $2i foot ; I of U; iofa ird, ^ of a ' f of tlie 152. The pupil will have noticed that in adding such frac- tions as I and I the result, ia is just the sum of the denominators placed over their product. The dif- ference of the fractions would be the difference of their denominators ^placed over their product. For This may be stated as follows : The sum or difference of t7m fractions, having 1 for a nume- rator IS the sum or difference of their denominators, placed over their product. ^ • i< lao A It I Til. ME ri C Foil I! EC I S .\ / ItS. KXKIU'ISK. ;17. I. 2 ?,■ ■\- 5. 6. 7- 8. 4^f-3^ '9l + i3iV Si f 6'. 202— !(,;. rS;-,2|. 22,V+lSV.. 33i + 24i. 10. 4'^J-^r,,v 1 1. 75A-3f>/.- 12. h'o+^'j4 ^m\-2i„ «3 4^':-i5;,- 14. .•3?4 9';2 + 3i2fSf. 15- 3,\ -If If). .V-/yr- 17- 4r^/A+3:. 18. \C). 2i). 21. 22. 23- 24. 25- 26. 27. (1 f 111 28. A lioy had 4 acres; his bnithor ^avc him Dthcr, his sister \ of another, and his fath"er J another: how many liad he then? A merchant had a piece of cahco containintr H" yards, and sold ofi" it 21 J yards; how much \v it left? ' ■ " A man paid $11 for a coat, ."JiJ- for a knife, .$» fo, ,, brush, and $2 f^^r a comb : how much did they ;ill cost ? ^ A man bought 17?. pounds of butter, from which Iir sold 12 J pounds : how much remained ? From a cask containing 42J gallons of molasses, a grocer drew off 174 gallons : how many gallons re- mained? A man bought a horse for ^jt,^, a carriage for $();% a set of harness for $37^^: vvhat was the cost of tin- whole ? A man gave away at different times $^, ,$|., ;?;.,";.. and $-j\: how much did he give away in all? * " • A man started on a journey of 45! miles, and travelled 28J miles : how far had he still to travel ? A merchant bought 272! yards of muslin from one man, 117I yards from another, and 321A yards from a third : how many yards did he buy in all? James has 3 fishing lines ; the first measures 12* feet, the second 14I feet, and the third 15A feet :" how many feet in the 3 lines ? ffURTHACTION OF FH ACTION fi. 181 11 which he 3'. 32. 29. A mr.Tl,a„t h„„,r|,t 3 pieces of calico, the first con- taiiiinf, .5;< yards, the second 22I yards, and the ""•<' 34i \ards: how many yards are there in the x pieces? ^ '^"' have'i'ieft?*'^^^*'''"'' ''""^ ""* ''^^'^'^*^' ^''^ '""'''' '\ •'''fV'^T''^' *"^''»' P'"^l ««i <■*"■ a pocket handker- cluet lisi or a dress hat, $46" for a cloak : how much had she h;ft ? A clerk earned $50^ per month. He paid $2.,J h.r board, ifls^i for washing, and $4* f„r other exp.^nses- How mucJi did he save per month ? 33. What number added to 147* will make 2i6| ? 34. What number added to 307I+210I will make 7003 ? 35. What number must be added to the difference of i«6| and 214^ to make 1042^4 ? 36. What fraction added to the sum of A, » and » will make ||^ ? ^ ^ t*< 37- Bought a quantity of barrel staves for $i6o», and umber for $1 136I : I sold the staves for $205^ a the lumber tor $1240^^ : what was my whole gain ? 38. A man bought a ton of hay for $151, a harn,-l of flour tor ^g^ and a barrel of apples for $1/-. ; what change should be given to him for 3 ten-doilar bills ? 39. What must be taken from 351 to leave 224 ? 40. From what must 24^ be taken to leave 6:5^^? ■J 3 5 41. Bought of Davison, Scott & Co. four cheeses weigh »ng46i 481, 4g-j^'^, and57X pounds respectively: what was their whole weight? ^^' ^ P^i,'?^^has three-eighths of its length painted red two-fifths of ,t white, and the rest of it blue: part of its length is blue? ^nat 183 ARITItMKTir FOn nEGrW'ERI^ ■'11 m 153. 154. MULTIPLICATION OF FRACTIONS. To multiply a fraction by a whole number when thr denominator does not contain the whole number rv actly. Multiply I by 7. Here we have simply to find v. hat x parts become when repeated 7 times, the parts Icing fifths. The result will be 21 parts or fifths, which is written y. Therefore | x 7 = V . When the denominator does contain the whole num- ber exactly. Multiply ^ by 5. Here, instead of repeating (ho number of parts, we will increase their size 5 timrs, that is, make the tenths become halves, since five- tenths make one-half, and the result will be |, taking the same number of parts we had before. Therefore Viy X 5 =f . From the preceding examples we deduce the following RULE. To multiply a fraction by a whole number, divide the denomi- nator by the whole number, if possible, and keep the savu numerator; or multiply the numerator by the whole num- ber, and keep tht same denominator. Ex. \.—^ X 3 = f Ex. 2.— f X 3 = f 155. To multiply a mixed fraction by a whole number. What will 12 pounds of sugar cost at ri| cents a pound ? Or, multiply \\\ cents by 12. The ii| cents may be broken into two parts— II cents and \ cents. We first multiply ^he f by 12. This gives V. which is equal to 71 cents. We write down the a and carry tlie 7 cents to the next order to the left as usual. Then 11x12 = 132; 132 added to the 7 cents makes 139 cents. The result will thus be $i-39f 156. cents 12 i39i Ml LTlll.ICA VIOS OF FUACTIONS. 181 156. From tliis we have tlu: fi)ll()wing RULE. To mitltiply a mixed fraction by a whole tiutnher, multiply the ffiictio/ial fart by the multiplier, and reduce this, if possible, to a mixed fraction. Then multiply the other part of the mixed fraction by tht multiplier, and add to the product the part carried, if any. EXERCISE 88. 13" These questions should be solved mentally. 1. Multiply ^^ of a dollar by 2, 3, 4, 5, 6, 7, 8, 9, 10, II, 12. 2. Multiply T?^ of a pound by 2, 6, 8, 12, 10. 3. If a basket holds ^'5 of a bushel of apples, how much can be put into 3 baskets ? 5 baskets? 7 baskets? 10 baskets ? 4. If each man receive .f 2f, how much will 3 men re- ceive ? 6 men ? 8 men ? 5. At 7.\ cents a yard, find the cost of 3 yards ; 6 yards ; 4 yards. 6. What will \ of a ton of hay cost at $24 a ton ? 7. How many units in 6 times y ? In 8 times y», ? In 11 times -f ? 8. At 6i cents each, what will 12 pencils cost? 9. Find the cost of 7 dozen pens at k a cents a dozen? 10. A man earns %^\ a week : what will he earn in a month (4 weeks) ? 1 1. Find the cost of 10 citrons at 15^ cents each. 12. What will 8 cords of wood cost at $5^.^ a cord ? 13. How far can I travel in 10 hours at the rate mile an hour ? 14. A horse eats 2| bushels of oat much will he eat in two months ? 61 s in one week : how 184 MUTUiitiTu: I'vu uaoiysi-M. i>-^ ' DIVISION OF FRACTIONS. 157. To divide a fraction by a whole mimber wlioii the nu- inerator can be exactly divided by the whole numbn. If 4 men earn | of a dollar, how much is that each ? This may be written : If 4 men earn 8 parts, how much will 1 man earn ? The fourth part of 8 part tliat is, 2 i)arts, or ninths. Therefore $j-=-4 = $|. When the numerator does not exactly contain the whole number. Divide i of an apple into 5 equal parts, that is, divide i by 5. Here, since we cannot divide the 3 parts exactly by 5, we must decrease the size of the parts making each part five times smaller, that is, twen- tieths. 158. Therefore a~-i: =» . This gives us the following RULE. To divide a fraction by a 70/io/e number, divide the tiumerator if possib'e, by the whole number, kee/' >ig the same denomi nator; o-' multiply the denominator :-/, and by the preceding l.rinciple the fifth part of V is \. Hence the result must be $2f. 5 ) I2f Es;:. 2. — Divide 12 remaining. 2f = fore, i2f-=-5 = 2i| _ 1 i 5 ■ 5. i2-f-5 = 2, with 2 y-^5=if There- ^•luMi the nil- bole number. 3 that eacii ? 5 parts, h(i\v rt of 8 parts. contain the at is, divide the 3 parts )f the parts, it is, twen uy with 17. At the rate of 3I miles an hour, how long will I take ro walk 45^ nnles ? 18. A scarf requiresf of a yard of silk : how can be made from 31:! yards of silk? many scarfs DTVISIOy OF FJiACTIOys. 141 Lj of a yard the number pieces. o give 7^ ? be divisor. jy i| of 6 ■ $3171 at ow many of 5. buy with 'ill I take ny scarfs 21 22 ^^* Sf^ !?^^"^ bottles, each holding i^ pints, can be hlled from a barrel of cider containing 6ii pints and how much will be left ? 0.1. 20. A man can run J» of a mile in 5^ minutes, how far can he run per minute ? There are 2i inches in a piece of cloth : how many pieces would be the same length as 123A inches i» If I pay $|v per pound for tea, how much tea can I buy for $9 ? Fur $24 ? For $64 ? 23. If a man spend $4| a month on tobacco, in what time will he waste one week's wages, $27^ ? 24. How long will -H of a cord of wood las't a family "Sing ^\ of i- a cord a day ? ^ 25. How long will a boy take to save $700, if he earns $71 and spends $5^ of it every week? 26. The "Chicora" steamed 156 miles in io4 hours- what IS her speed per minute (60 minutes to the hour) ? 168. Fractions are oftea written in this form ^i. -5 . i of I. i_x.Aili These fractions are called Complex Fractions. 169. A Complex Fraction is therefore a fraction which has a fraction in either its numerator or denomina- tor, or in both. 170. Since we have seen that a fraction means the division of the numerator by the denominator, it follows that the above fractions can be simplified by ordinary division of fractions. Thus: 5i 21 2r 21 8 2| 4 8 4 21 £x. 2.— Simplify ~^^^' 4iofl i^ 3 ■ 27 ^^26~26' 149 ARITHMETIC FOfi REOINNERS. Ex. 3.— Simplify (8i + 3i)-f-7». IS" Since the expression 8|-h3| is enclosed in the Bracket ( ), it must be simplified first, and the result divided by 7|. If the bracket were omitted we should first divide 3^ by 7|, and then add the quotient to 8^, which would give a result quite dif- ferent from the one required. 8i+3i=iii=V Ex. 4— Simplify 2i + 5iX-j9j— 6| of | + ^2r- U 1 :A=2 6^ofi=vxi=V A-WV=fVxV = V IS" The pupil will carefully notice that unless a bracket mterferes.the operations shown by the signs X , of, and -r-, are carried out before any of the others. Thus : In Ex. 4, instead of adding 2| and 51, we mul- tiply 5i l^y 2T. and instead of adding > to ^, we sim- plify eiofi, and-\- »_ 44' A bracket should always be used in any case of doubt. 171. A Bar or Vinculum has the same effect as a brack ,i The expression f -^ of |a has the same meaning as (t-t) ('f if The value of the expression as it new stands is ■^\. If the vinculum or bracket had not been given, the result would be ^' , as the pupil may find. EXERCISE 42. Simplify the following expressions I I. — . s 2. -5. s H' 5- -. IirXli- 7 8 H I^ Of I| 7. ^ ■I- of 60^ X DIVISION OF FilACTfOXf!. lO "fl'-fYT II. Simplify the following expressions : iH- (3-^-f)-2f 20- (i:>-*-2|x5)x3|. 5X|off • «mof^ i of I of 3- 12. 26x^+2i. 13- (4f-2|)X3f 14. 2-f6f-5-^. »5- 3i + (4l-|). 16. 4i-(3l+|-). 17. (i4-8)-^i. 21- (4i + -?H3f-iA)x2 22. 23 ^ of I -I of i 148 \H-f i+i/ U+i kJ" tV ^f^V+Aofi' GENERAL EXAMPLES ON FRACTIONS. Ex. I.— What part of 7 times 4 is one-ninth of 72 ? 7x4 = 28; iof 72 = 8. ^ Now, what part of 28 is 8 ? Since i is ^V of 28, 8 must be ^\ of it, that is f . This question can be written in many wavs. Instead of saying, " What part of 28 is 8 ? " we may say " 8 IS what fraction of 28?" or " Express 8 as a part of 28," or, " What fraction is 8 of 28 ? " Now the answer to each question is -g-V We thus see that we always place in the numerator the quantity which IS to be the part or fraction of the other. £x. 2.-4^ is what part of i8|-? Here we write 4|. in the numerator and i8|. in the denommator, and reduce the fraction thus formed to its lowest terms. I I li- = ^i ^ 92 = i^ i _ 1 18* 5 ■ 5 ~ ""0^ ~4 lU ARlTtlMBTlC FOR BEOIXXEns. ISS"-- The pupil should always prove that ti.e result is correct. Thus: Show that I of r8| is 4^. iofT8|=Iof^i=!_3^ / Hence the answer x is correct. ^^- 3-~-^ of 60 is I of what ? ^'"rv^ill?""'^^'^^'' question now stands: "48is| If 48 is f , fhen I of 48, that is 16, must be 4, and * or the required number, must be 16 x 8= 128 . 16 Proof : | of X^ = 48 £x. 4.— I uf 27 is f of how many times 3 ? Since f of 27 = 18, the question then stands : " 18 is f- of what?' This we know "by Ex, 3 to be 21. Now we must find how many times 3 this 21 is Wc know It is 7 times, which is the required result Proof : 7 x 3 = 21. f of 21 = 18. EXERCISE 48. What must be divided by ii-s^^ to produce 7^ ? To what must ~ + '^ be added to give il + « ? 4i 4 4Vt What part of $51 is $iA ? What part of a day (24 hours) is 5^ hours ? 8f minutes .3 what fraction of an hour (60 mmutes) ? What part of | of a peck is ^ a peck? What part of $n is | of $16 ? 8. f of a peck is | of what ? 9- iV o^ 132 pounds is ^ of what ? 10. 12 is ^ of ^ of what ? ^ ^ 3- 4- 5- 6. 7. ti.e result is Is: "48 is^ i, and |, or 28. 3? ids : «♦ 18 is 21 is. We i result. OEyL'nAL HX.iMPLES ON PHACriONS. I45 ''■thc^n:;p:::jV'^'^^^P-^^-ahorse.w 13. What is t].e cost of a^, loads at $2ao for x6 loads ? '• Jfstf.^lT' " ^^ ^'^ '''' -'^^^^ -^^ parcel : find '^j'T^of|of$47is,of,Vofan.an'srent:findthe 15- A. had $260. and «5npnf ? <• •. A of wh!. b: eartri;^.^„,;fe^';„.^j1^':'- -3, ^-' 10. Of how manv t nies n ;= s ^r ^ ^i ^ 17 Who. T 9 Js I of 20 the five-ninths ? 7. What numbers I of 4* times A of 12? 18. What part of 42 is 3 times I of 30? IQ- ' paid $50 for a wao-pon anH a .>f *u just i of 3 times the cos'^ ^f^^' ^ ''"'* °^ '* "^^^ the harness worth ? "^^ *^" ^^^"^^^ = ^^at was "• whtrdo'rown'r^ ^^^^ ^^^" * ^^ ^ *-- -y -oney. "• f^urthT?"'"^ *^"^^ * "^ $- - * of $25 the three. 6 71? il i o ? mamtes) ? If he^sold ,v of his share, he must have « of h.s share «ofi=i 13 r^ 13 •^Md. Zt"isT^i» ?™d"^,*^ *° "k""^ -'" P"' "e from ^„ whiclVaUh^ugr^o*:,", "'atf.V'"' '^"'5 tmie, for we wnnf f.^ i ^"^"^^^^ causes a waste of the part iSt. ''"°^'' "°* ^^^ P^rt sold, but ". e^f::d''Se\?Lt5o"r.he-; ri' "-= V' " the whole or| = $;7ooo ' ^ = ^"ooo, hence '• via 146 ARITHMETIC FOR IISOINSBRS. 23. A man gave away | of | of | of ^V of his money: what part had he left? ^ 24. I owned * of T^y of y»^ of a business, and sold | of mv share : what part of the business do I now own ? ' 25. A mine is worth $2 jog ; a man owns .,5j. of i „f jt and Jost yV of his share : what part of the mine ha^ he lelt, and what is he now worth ? 26. A man ovyns | of a of ^'^ of an investment ; on sell- ing I of his sliare he finds himself worth $100 less tJian before : what is the value of the whole invest- ment ? 27. B owned (|i- 1) of an estate, and soldi of the estate: what part of the estate, and what part of his former snare does he now own ? 28. A man sold i his load to the first one he met, -'- of the remanider to the next one, \ of the remainder to the next one, and so on : what part of the load had ne lelt after 10 bargains ? ^^' ^^'T^ ^^^^ ^^^^^ ^^ ploughed by A. in 4 days, and by B. m 5 days : in what time could both together do the work ? A. does the whole work in 4 days, therefore .he does i of the work in i day ; also B. does ^ of the work in I day. Both working together would thus do J-4-i = 2% of the work in i day. * " A. and B. do ^W in i day. ^ in 4- day sS " f§ or the whole work in V = 2| days. £x. 8.— Two pipes running together can fill a tank m 20 minutes: one of them could alone fill it in 35 minutes : how long would the other take to fill It? Both pipes fill ^ of the tank in i minute. One pipe fills -jL « «< <. << 7A^ii°*^f Tf fi"^V- 3V = tItt in I minute. . -^s nl^s ^fj of the tank in i minute. " rh " " i minute. " in '• •' 4° =4H minutes. ''■ liran£?"'"''"^^'^"^^= ^"- --h can he 30. A. and B. can together do 4 of a job in 6 days • how liiuch can they do in a day ? ^ " 31. ^- c^" do the work in 8 days, B. in 3 days, and r in 9 days: how long would the three together take ? ''' wiKt^e tt hl^stf U r^"^^ ^" ' '^y- '-- ^-^ '" ^ a^ to Sif tV7.:ioVr '^ ^ '°"- = ^- ^-s -^" ^'^' dokln'^R^?' \^ ^'/"^ '" 5 days ; B. and C. can do It in 8 days : how Jung would A. take to do it ? ^^' ^,;hV"J^ ^' ""r" "^^ ^ P'^^e of work in a week • A of,' c^an t il ''' ^' ^"^ ^' '^ i <>^ ^t : ho.t'uc^h ^^* ^rld1n^£' ''f ^" ^ ^^"'^ ^" 30 minutes, 40 minutes Jake a^l^'""'"' respectively: how long^ will they take, all being opened at the same time ? ^ 39. A pipe can fill a vat with water in 11 minutes • in t vaTbelf/eVl/rr ^""^^^ = ^" ^^^-'"-^ ^ time? ^ '^ ^°*^ ^'^ ^P^"^d at the same wctb men due $850 : wiiat did he owe at first ' CHAPTER IV. CANADIAN MONEY. til a ii I' 172. 173. The Government of every country makes or coins money, which is used by its people in buying and selling. The Canadian Government coins and issues the fol lowing pieces of money: The one cent piece, made of copper; the five cent piece, ten cent piece twenty.five cent piece, and the fifty cent piece, all of which are silver coins. 174. 175. 5 cents, lo cents. 25 cents. 50 cents. The five cent piece The ten cent piece The twenty-five cent piece The fifty cent piece The Dollar, which is equal to 100 cents, is paper money, issued either by the Government or by the Banks of the country, and may be changed into coin at any time by presenting it at the Banks. There are also Two-Dollar Bills, Four-Dollar Bills, Pive-Dollar Bills, Ten-Dollar Bills, Twenty-Dollar Bills, Fifty-Dollar Bills, One Hundred-Dollar Bills. as the Banks or the Government may be pleased to issue. 176. As stated before, the Dollar is represented by the char- acter $. Thus, $45 is read : Forty-five dollars. The cents are often represented by the letter c, thus, 27c. is read: Twenty-seven cents. If dollars and cents be taken together, they are writ- .en tnus: ^,.84,52 is read: i:.":ghty four dollars ana nd fifty-two cents. $7.08 is read : Seven dolla eight cents. rs a [es or coins 1 buying and jsues the ful- piece, made cent piece, cent piece, nts. nts. nts. its. • ts, is paper it or by the ed into coin ks. )ollar Bills, enty-Dollar Dollar Bills, ; pleased to 3y the char- dollars. ter c, thus, y are writ- iollars and iollars and (CANADIAN ::n\rir. 19 75 cents = J .. I^^l .. |^7.y>. 178. The HKKSt necessary requirer-M.t i •„ u ■ the correct addition ,?Tm.; V '"^ business is ^Iollars and cents '"''''" " expressed .-.s /;^.~-Add together $3o.ir «*7i2.4o, $212.04. ^150.75. SSo.73. $30.65 »5o.75 80.73 712.40 212.04 The am„„,„H „rr written as in Simple Addi- tion. 1 i,e sum of tlie cents is 2^7 cents lars and fifty-sevcn cents. \Ve write th„ *.. 86.57 usuil. ' ''"""=■ ■'■"'' P""'^^ t" add as 179. Ca:radiI^"="„;oty is' the'*''"" ^il. S^bTactiun of Care howe":rm;:st't t^e^ thaf 7e'=se1,rr^'rt; d,lre'n«r'""'=" "' "^ """" P'-^ ■'" 'E'lrf ^x— Multiply $241.35 by 8. nio;-. ey $241.35 Wnt ng the numbers as usual, we first .„.i. 8 t.ply the numbe. ..f cents bv 8. which dves «i^i;r^ PI ^f "o"' ""' ' ^""^'■^ and-8ocen s ^ $1930.80 Place the 80 cents in the usual place A^rain Rives io28"doliP'^"^i *'f """^b- of^dollars b'y"' i [iV mam i 111 la- 150 ARITHMETIC FOR BEOINSBRS. 181. To divide a sum of money by any number. 5)1^7^5 Writing the numbers as usual, we first $39.47 divide the number of dollars. Thisdv, oraoocentf Th''"' '"^ " remainder of 2 SaVs'^ make up ^5 J^.^ ^^^ -"*^ '-^-- with the 35 cents 47 cents. ^^ ' ^'''^ ^^^"^ '^''''^^^ by 5 gives Hence the quotient is $39.47. 4x ) 54X2 ( 132 ^x--How often will 41 cents be con- 4x tamed in $54.12? 13 1 This is the same as finding how often ^13 41 cents xs contained in 54x2 cents «^^ '^4ll':nrair3^J^^^^^^^^ ^-^ - '®^' "^tlon'^a^nr^itL^^'ofr' f '^^^^'""' ^^"^^^P^-^" as in whole number/ "''^''" ^'"^^^ ^^ *^«^-- Ts ?ell t^uTiS ^^tats ^mlt;^^^^^ ^^^^^^ ^^^1, EXEECISE 44. "• ^nd-f aft" *^^-'^' *«-^*' «--34. «-.4.. $34... 2. I ga.M to A. p. $3.43. to E. $24.02, to* F $ '^•?3'.toB. $42.23, to C. $14.43, to give in all? 3. AmanowesA.iS27.1S. B «!cfi ,, E. $108.99. F." $6 does he owe; m all ? 24.01 : what did I 3«-J4. D. $45.73, 2.86. and G. $5.09:- h:w"J?ua CANADIAN MONEY. 151 I-I-23 '24.32 n7. ^' S""",^^**"^ J^"- ^°ste^ & Sons: Knives rnnn ^f'^J.^ '^''"^'' ^34.23 1 Fire-arms Gunpowder, $42.43 ; Sundries, $32.43 : find the toTal 5- I made the following deposits in the Dominion B..-,k • Cn^Tl"' ^"^'W^i ^2'''^ *4°^-2^ ' Silver, $i2;..o.' Gold, $132; Drafts, $301.24: find the tot^l deposit |78.5i, $90.84. $112.79. $29.08. $5.18. $919.03: ^' toT'bit\'?n;^° '? '^°-°"/"' ^7540 more in Hamil- 8. Bought a farm for $3273.08. a house for $1^0:101 horses for $429.17, cows for $273.54, sheep^f,; $290.09, hogs for $447.26, and furnitl?e for $29^8.98 what was the total amount ? v^yo.yo . '■ ?ackfe $88^88^ r"'^' t ^°" '^' ^"^^"^^"^ bill: lackle $88.88; Rope, $99.99; Pulleys, $90.09; VV^ie, $770; Flags, $17.90; Steel, $183 84 • and Cutlery. $611.12 : find the amount of the bill ' 10. After lending A. 60 dollars. B. 139 dollars 44 cents, ij fT 73 ce"ts. D. 78 dollars 17 cents. E f atfr'st ? ^^^•°'' ' ^^^^^f^ $357.28: what sum hrd "■ ^^A n ^'*^^' '^ ^°^^^'' 3 cents. 50 dollars 90 cents cents fi^d' Il'"''.«'^ ^°"^^^ ^4 cents, 77 dollars 25 cents. 83 dollars 68 cents, and 40 dollars 8 cents. 12. From $593.15 take $208.28. '^" stmowel™ ^^°^-°'' ^"^P'y '^'^^.oS: what do I What change do I receive from a hundred dollar bill if I pay for goods worth $67 43 .? Find the difference between nine hundred and three dollars and twenty cents, and $705.82 fhe^'nP w'\^'^^'7.3o, and B. $29120.91 : what is tfie une worth more than the other? H 15 16 I^dfhref." ""?' ''\^V "^'" ''' twenty-seven dollars ana tnree cents each ? 152 i8. 19. 20. 21. 22, 23. 24. 25- 26. 27. 28. 29. 30- 31- 32. 33 34- 35- ARITlliiETIC FOR EEGINNBIiS. What W.I, be ,he cos. of ..g. p„„„d3 „f Ur, J.t. o„":ch''Ca:.s:.^«^L^^.,^e7ofar;i°-'-"""«--- At 32 CIS. a foot, what will 33, feet of rope cost > Messrs. W; A Miirrav Xr r^ ■ , . .0 bales of siIk.Scl1^n°inr4 pt ™o7 ,?"T each, what was the value of^hl^XraVli:^* How often will »85.i4Conlain 99 cents? l buy books at J1.08 each and „=„ « many books do I buv? ' P^'' *533." ; how Divide $18321 by $2.40. mu?rwin°remal?r"^ ''""^"^ among ,9 n,en, ho. A bank pnvs off a debt fif «T^-,^o /■ . . *a5..6'a .„o„.h: h^^'l^'n »H7o85^96^a. tl. rate . 38. 39- ■'iTEItLiyo MOXET. jg, 36. A man having $379, bought 97 lambs at $2 oc each: Ji w much liad he left? '"^^s at 5,2.95 35- Sold J. Cleghorn & Son 37 boxes of figs at $2 7. a box. losing §,27.30: what did they cost n.e at first? Received from A. $19.89, from B. $33.2. f^om C $25.47 more than from A.: Jiow much less w^" received from B. than fr.m C? "' Bought 359 shares at $1.20 a share, and had left $99' 80 : what sum had I at first ? 40. Bought 324 pounds of t 'a for $243 : if I sold it for 15c. a pouncTmore than I gave foi i^ what wic^ whole gain and se ing pn?e per ;ou;Jp '' "'^ 41. A man sold his house for $1567.30 and his land for ^3121.30, and b ...ght buildmg lots at $2, 80 each how many could he buy ? ®^3-oo each : ''' fi've week^'Xt'Tl^r'"*' T"^' ' ^^^^^ ^98-35 in live wecKS . wJ)at d d 1 spend a week ? " ->•> Borrowed from A. $93.86, $46.31, and $101.88 ; from B. $9.08 ; and then paid off my debt to C ,. u was $197.58: what had I left? ^ ^'' '"^"'^ lz::iz:^-r' p'^^^^ --^^ p^y f- 8° horses 45. •^'"^" s°ld 53 bags of flour at$i.o4abag- his neirrh bor sold 13 bags less, but at 15 cts^mr rea bae^ow much more did one get than the other' ^' ^ 43 44 BRITISH OR STERLING MONEY. ^^^" "^tceTsfrfto"^'"^-''^' '°'r^^ "^ -- —try, it is necessary to enquire into that of Great P.r\fLJ ^',?u^ ^'^ '""^'V^ P="" in 21 Shilling (a silver -oin^ which passes m Canada for 24 cents. '' Twenty shillings make the Pound, or gold Sovereig,,. ■>> 154 ARITHMETIC FOB BEOINNEItS. 185. We thus have che following TABLE OF STERLING MONEY. 4 farthings make i penny, d :2 pence « , shilling, s' 2o shillings '< I pound. £. fcisr The farthing being a quarter of a pennv is wnf 186. We see that any number of pounds are brought to f r ^f^!""&s to pence by mult plying by I2 • ner^Z to farthings by multiplying by 4. ^ ^ ^^ ' P^"^^ £x. I.— Express jBS as pence. 8 '^^Zl ^'^ ?°u '^^"'"^^ ^» »^. hence 20 ic *t """^^ ''^ ^ *™«s 20 shillings in ^S, that is 160 shillings. 160 shillings. There are 12 pence in i shilling, hence 12 ^^^'■^r" ^^ '^° t'^^es 12 pence in 160 shillings, that is 1920 pence. 1920 pence. Therefore £8 = 1920 pence. £x. 2.^Express £2 los. ii^d. as farthings. 2 10 i?i "w '^''' Y^4f shillings, as before, 20 ° "* h"t ^e .must add to these the 10 shil- lings, which makes 50 shillings. Again, in 50 shillings there are 600 pence, which when added to the u pence gives 611 pence. ^'?!l^^-' '" ^M P^"^e there are 2444 farthings, which with the ^d., or 2 tarthings, makes 2446 farthings. - _ _ ^ Therefore £2 los. i i|d. = 2446 farthings. 187. In the same wav, any numb«f nf a hirrh^^ j tinn r,-,o„ k„ j- ' ^''v -^-^"'f- • oi a Higher denomiua- 50 12 611 4 2446 STEBLIXO MOSEY. IB5 • '—Change 1200 farthings to pounds. 4)2200 Since 4 farthings nake i penny, 1200 i27i^> [■^■tiu.gs wil make just one-quk^ter o^ 2 '^, l,n? ^'""'"' '^''' '' 300 pence. In Jhe ' 5 n f T ^'? ^^^'^^'"«s. that is 25 shil- of shillings bv 20 tn l"'^ ^\r "^'^'^^ *he number Hence 1200 farthings = £i 5s. £^- 2. - Express 3477 farthings in the higher orders 4)3477 Jr!n "^ *''" farthings by 4. we obtain 12^66^1 ^('g pence and i farthing remaining 2 'ohV-cX ?, ' f'n^' ^'^^ ^^^9 pence by 12, ve find 2 o)7_£_5i_ ILl^":,"^^ ""^ 5 pence oWr, and 3 12 5^ ^"^> dividing the 72 shilhngs by 20 ^^'«hav«3puunJsandi2shiningsover. _ Therefore 3477 farthings = £3 12s. sjd. EXERCISE 45. How many farthings in i4^d. ? !„ 27 j, p How many pence in 468s. ? In £55 ,gs. 7d. ? Express £754 17s. g^d. i,, farthings. How many pounds in 76G0S. ? In ii472od. ? Reduce £15 8s. 7^d. to farthings Change 21368 farthings to pounds, etc. Reduce 854d. to pounds, etc 8. Bring £3 igg. yd. to fartliings. 9- Express 4s. 2fd. in farthings. 10. Reduce £21 os. o4d. to farthings. 11. Express in £ s. d. : I. 2. 3- 4- 5- 6. 7- 156 ARITHMh-rW FOR REGINSERB. 12. How many six-penny pieces in £52S 6s. 6d > 13. How many pounds, etc., in 37,5 three-penny pu ces ' 190. To add together any sums of money. • 4*a., k\ IDS. it4( . ^ *"•• I Hi,.. 4 19s. iifd. s. 8 9 10 ^9 £10 8 % ^^;^ fi'-st place the quantities so that ^ ^1 the same kind are in the same column I Adding the farthings' column, we get o J| farthings, that is 2 pence and t fa, l',^ _1 n)g. We write the i farthing in its 10^ prop. • place, and carry the 2 pence ^ to the ptnce column. ^ the pr„per colu„,„, alfd ca„y1hefpS^«^ '""'" "■poinl!" '"'^^ '° "■= P°-"'^' -l™n, gives ,o Hence the total sum is £,o 8s. loid BTBRLISG MOXEY. EXERCISE 40. Wl Add together the following sums of money i- ^15 los. 9d., £8 9s. 7d.. £i zas. lod., £, x^s. 4d. 2. ^8 gs. 7|d £7 ,.s. 4id., £x .9, mH. ,5s. 8|d. 8. ^,78+5 ,7s. 8d.+f347g ,3s. „d.+6 83 ,« ,d 4. *7358 .3' >d •+*'*'' '"■ ^''■+^^79 .Is. U.t fa33|^„s.8d.+/8ya or•od.-X'^^d!i.t,-^, ■9S. gd. +,,,46 ,,| 8d + Is^f '3'- 5<^-+*8735 ios.4id. " ™-+*«74 13s- 4jd. + «68 191. To subtract one sum of money from another. £'■ 2.-Take ,3 5s. 8Jd. from £5 4s. 6}d ■ I'^^^fTV^" "5™"""== as usual, we say. 3 farthings from 2 farthings we order, that is, i penny or 4 farthiniis which makes 6 farthings Then^ ,' farthings from 6 farthings leaves I £ s. 4 5 d. 8f &i 18 9j les ■IBITmiETW FOB asaiSSSBS. w"hav?f ■,o1fk':'8'"!i"'^ 'f"" 0"e of the 6 pence, cannot. Add , sliillKfr '""" ^ P"'«' ^'"•^l' w« from , r panel leSes gVenc: ""'"• ^''^" « "="" Add f poun7or'i™hf„,^|;;i''»8^--'"^'> «= cannot. La',1/ f """^' 'T ^3 ^Wilings leaves ,8 shillings ^nce the diffiince fs^ .rrSs. Th ""^ ''°""'- -. because ti'i'^tr J :;-;'/d";ri^ier iu own ^132 1 10 88U EJeTe," {^'f '^'^l" ^^'•^- ^^o, thus : * .i-Jt-ven tunes i farthing is n farth 1 1 times 10 shillings \\ V'"^^^?.^"^^ ^^ ^8 pence; pounds is 132 pounfs. '" '^""'"^^^ = ' ' t''»^« i^' The result is £1^2 I in^ 8S11.1 xt 2 pence and 3^fa thLs ^ W. ^Tl'' ^""''^^^^^ ^re ings and car/y the p^'nceTo t"e stn" '^" ^ V-^^^' It 90 pence. A^ain 00 ^1 , P^"^^' making 6 pence. Place th J « • '''>"' ^ shillings and and carry the 7 sM^lf "?, ""^^'^ ^^ own column, it 117 shillings^ But n^.V^^^ '^° '^^^^^'"««' ">al- ts tarne/fn an ?-?hat1's fi^nd^^^^^ ^^^^' ^^^* lo^d. ^^ '^' ""^ 24 times JE3 15s. "J ''e factors of 24 are 3 and 8 Fir.f 7 ^hhiings and 7 pence half-penny" or 2 farthmgs. Next multiply hvl' fa%'l°jri!j'-3^^f-thinVs'ix^^ ^ o .6penc\-&.«--^-.^^^ i'3 15 II £gi 10^ 3 7i 8 !f" / I li ■ I '60 gives Oj peijce, which - r d ii' Hence the amount earned IS i-g, is. KXERCI8E 48. '• ^X '' '"^ ^-"'"^ "f '"- ar,iciss a. f, 6, ,Jcl, 2. One man can earn £q «= ^i^ . earn? *9 «s. 4jd. : what will 4 men times ? ^* '*'^- P'^oduce when repeated 6 5. Find 8 times £18 OS, nd ■ *■ °rh:':^'oie°l:,;h^r' '='"-"'' *■' '5- oid. : wi., 7. Wha. Will :o persons spend a. the rate of .,3 3s. ;Jd. 8. Find the value „f „ s™ng.mach.„esa..,8os.4}d. 9- Multiply i>,o OS. ,,Jd. b. ,2 '" efgif pT„"ce-th''rerfa«i;^,/n-',''"V-"^' -^ subscription,. runt to? "'" ""^ "^ole II. At Id. each, .T-ticles cost £1, „= „,j , Ihey cost at 3s. each? ^ ' ' "*''■• "'''" will ''■ ruirTto'^a^ZS, -,^„7e1/4^- af "' T 13. What amount must b. ij 1 ' * """" each may receive 4132 .,s. jjd. ? ^°"* ^^ "'" ">"' H- A drover makes a nrofif nf L animal he takes to market wit ^'n^^''' "" "■">• d^y when he takes 31 to marStI "'" '"= «='" '" » 'i. Of what amount is f8r OS. 8Jd. the .87th part ,> '^TE/iL/SO MOA'E}' l6. Tim " Dominion • carriM f„ T ,■ ■32 head „f cattle oTlr "'f,''''""'^™"' Q"el,ec 9id. l-er head : Xai d,d th, f "">' '"''' f""- ^^ 7^. '7. Brick is worth U^Y'TZ '""'"'''''''''"' ' -ha, -ill.,,;',?,; ■ ,,^ f ^,"rfck1 f r; ",""""""" ^ requiring ,o thousand? '"''' " ''"ke-huuse "■ 7l.%Te"a1,l'? P^''' f- - <'-» Chris,„as,ee.ea, 195. To^divide any sum of money by a whole num- w:me^rwL*.';^i,i=;;e?te!v,''ri,''r'='H''"i™^'' ^5S. o^d. by II. J^ectuc ? or, divide £812 11 £ s )8l_2_£ 73 17 d. Set down the numbers as usual As ns,n.p]e division. XI is containedl tl L; iPn 'f ^"^' 9 pounds over .^73 and .9 ren^linln^^! ' WrtjT&r'^ ^/ "'^^ '^ place, and carry the £g. "^73 in its proper Now the £9 must bobrou^hf fr. ci„ri- to the^ther .^31^^^^^ Again, II is con'tained in in, ,\'' ^^°+^5=r95. lin-s over. Write t'..^ rZ il^'u^ ^'""^^ ^"^J » s^ii- carry the 8 shilhngs. ' ^''''P^^ P^^", and These 8 sliilh'nfrs are Pn„oi 4- /r are no pencf to L added ? ?'"''' ^"^ ^' '^^'^^ times a.^d 8d. over! Wrfte\Ve 8d"'""'' 'V'^ ' cany the 8d. left over. "^^ ^^ "^'^"^J- and •■'"iigS, ^^»?f,? frPirc"?;"-"' "'" ' f="Wn^.n,al,e33 f' Each woman, .heref„re,wo„,d receive. 73 .7s.8Jd. ■'» 169 Mil Til M uric fou nmiAWEiis. 106. When the dnisor is greater than 12. ^^■•- ''-D'^'''« -^'93 i6s. lo^d. intc, 99 equal parts. ^ s- ence over, ent. Bring (196), and 's, niakipfT k'ide by gg^ or |(i., ta i. y with the ', tlie mul- ling 10 in , and 4, as er. ow many er words, contain we have • Hence o units of STSIiLlSo MOSi:v. ,*^ cc^^ined in 483^8 northings. This will be fbund by 803 ) 4832,^ ( 56 4315 517S Hence there will be 56 persons. 198. From the above we obtair, the followin farthings RULE. division. ^'"" '^^''^'^c 16. Ho^v often is £a8o :5s, o^d. contained in £3397: os. ''' ?pTf £.77r5r?^ °' '" '" '°*'- ^^^' ^^" ^^ "--^^ 18. How many times can £13 os. oid be t-^kf^n r £175 3S. 6fd., and how much win be left ? "" 199. After the Table of Monev fhf. ^^ 4- haps is ...eTaUe^rw^eiJ^%r= j;raS fS' 200 T, "\r '" »- ''■"■eht and sold byZtgt. 200. The smallest we.ght used for this purpose, in practice 901 T, T J"""' ""'"" °' "'"" "^''^ "P =" P™»d!""' ^Ul. 1 he Table in full is as follows : AVOIRDUPOIS WEIGHT. 16 ounces (oz.) make i pound, lb 100 pounds « I hundredweight cwt It is used in weighing all heavy mereliandise such .s iron, groceries, etc. "•"uibe, sucn as It' .. ,sl' TABLE OP WEIGHT. 1G5 Bristol £35 eight ? 1 8s. loM., e carriage 12s. io|d. ^6 ss. ofd. '33971 OS. I be made 'und. cwt. such as 202. 1^ The pupil should be taught to express every order m units of every lower order ; for example I T. = 2o cwt. = 2000 lbs. = 32000 oz. I cwt.= ioolbs.= 1600 oz. . . I lb. = 16 oz. S?asuri.'^^^ ""^^ ^^ ""^^^ ^°' ""''"'y '''^^' ^^'Sht or APOTHECARIES' WEIGHT. 20 grains (gr.) make i scruple, sc. 3 scruples " i dram, dr. 8 drams «' i ounce, oz. ^Ud. The following ,s the table used by jewellers in weigh- ing precious stones, gold, silver, etc., and is called TROY WEIGHT. 24 grains (gr.) make r pennyweight, dwt 20 pennyweights '« 1 ounce, oz. 12 ounces " i pound, lb. r^ The "grain," -ounce" and "pound" are the d?,T '" *u kT ^^'''' tables, that i.s, a grain of go d dust would balance a grain of quinine All examples in the preceding tables are worked just as in the case of sterling money. ^ ^grai^?" ^ ^^'' ^ ^'''^' °^ ^°^^' ^^°^ '"^"y lbs. dwt. 2 s Since there are 12 oz. in i lb., there must 12 be 12 times 2 or 24 oz. in 2 lbs. As there were no ouhces given, we have none to add to the 24 ounces. 485 dwt. Again ; since there are 20 dwt. in i oz. Q04. 24 oz. 20 24 1940 970 1 1 640 grs. there must be 24 times 20, or 480 dwt in 24 ozs. This, wilh the 5 dwt. given, '^^* ^^ i^o fs^vt. in the same manner, 485 dwt. equals 485 times 2S, or 1 1 640 grains. * I I 166 ■^^iT,sM^r,c ^on B^aim^^s. hoJd? 5 ^ats, what mil each vat 22 ^•7ZZliV^{l-' -^r^Z. •°° "'^- -"■ This, f ^-^Si^illncet!"' ' "'- - 48 ounces over. Each va. will thus hold , x , cw, ,. i <:«"■ 22 lbs. 9j oz. TJ, fl ^^XEficlSE 50. -Whatpanofatouis, ;;r '''^•' " - ' 3- What part of a cw.. is "./'"^'-^ « "«• ? 4- What is the differenced? '° '"'• ' " ""S. ? ^'hI:::!::::-:— M3.iver, ;fdr sv '» ^ °?- Wst,?^:- 'lb ^" ^ '^^^ ;terep;?rr^'--ea.,ib.o.s„,lrd 9- ^Viiat isthe cost of * jk r ounce ? Of 4 nf a * ^' °^ nutmegs at r ^ . '0. How „any „„„oes in i a cwt. ? *'° ^" """« ^ ".W.ch.s ..Cheaper on,. .llo„.„„,,,^ t 1 ^ T. 6 cwt. each vat 1 weight. 5. we get ver. '. with the -s 26 cwt. r. This, :)ver. r oz. 'y. 1202.? cwt. ? '.5 ibs. ? penny, ween 1 iof; 2ibs. 02. of f, and ts an nts a nee? TABLE OF WEIGHT. ((t) id. a grain or is. a dwt. ? (^' 25- How many ba^s r.f e i* 3.!b.,a.e^fee^r3^T!'3r,Xf'"« --■ what will be Jeft over ? '^ ''"'• °f silver, aiJd ^\'»crn&^L^e:;a&dSr ^ -^- -'^ 4 cwt. 9dwt...^^r^-^"^--^<'"'-ogrs.by.,b3^. , 3<'- Divide 3 T 9 3'- If one.,^irtee,X if*" 'ctrH^'i'' "?."'■ ^« '"s. what is tlie nnantit, '=7'""' goU coiriage bi. aJi„ weighing 5, ,tt"eac'L;' ^"^ «^°'-^ ■■" ' 74 pSs 32. y^^h^t is tlie total weight of •, djshes, each weig,:::!',^ :.-'-; in ''^'f ^ dozen S„T: "'^'^ weigl„„g,6„z f,H .''^"•: a dozen 33- Add toeethpr a r -ibs.fandV.:?i^^--4oibs.;,of3,ons6cwt 205. 206. 207. tons 14 cwt. sighing 1 x. s- of goods, 2. 16 dwt. ; grs., what ghing 6 T. parcels of ^ere be ? 'g 2 cwt. dwt. c'li/ 'ver, and ^d 4 cwt. X boxes ; 13 grs. bs. 7 oz. e alloy, pieces dozen dozen saJver, 5 cwt. TABLE OF WEIGHT. 169 34. What is the difference between ^ of 4J of 2 lbs. 5 oz. 6 dwt. ; and ^ of 2^ of 6 oz. 10 dwt. 10 grs. ? 35. How often is f of 3 scr. of quinine contained in a package containing 6 lbs. 7 oz. ? 36. What weight is that of which 17 lbs. 2 oz. is J--* 9 of which 5^ cwt. is yO^ ? ' * 37- 01 what weight is 2 o«. 3 dwt. three-seventeenths? 38. What part of 3 cwt. 6 lbs. would just balance 2 lbs. oj oz. ? 39. Add together J of 2 lb., f of 5 oz., 6| dwt., and 3 J grs. 40. By how much does the ^ of 6* tons exceed the i of 13 cwt. 17 lbs.5joz. ? 41. How much weight must be added to 3^^- cwt. to make i ton 2 cwt. 12 lbs.? and what weight taken from 23/^ tons will leave f of gi lbs. ? 42. Take from 5 tons of potatoes its third, its fourth, and Its fifth part ; what part of 171 tons is the re- mainder ? 205. All distances, lengths, breadths or widths, and heights, are expressed in miles, yards, feet, inches, &c. Ihus; Toronto is distant from Montreal 333 miles; a room is 20 feet long ; a piece of cloth is 22 "inches wide ; a flag pole is 80 feet in height. 206. These are included in a table called LINEAR, or LONG MEASURE. 12 inches (in.) make i foot, ft. 3 feet " I yard, yd. 5i yards ♦« i rod, rd. 40 rods '« I furlong, fur. 8 furlongs " i mile, mi. 207. A yard measure might be shown thus : I TIT i I I I I I 13 1 he three iaK^er d!"'=;'""'^ "'r^'Or? • •- '• each. The thirty-si- smaller divisions would each reoresent one inch. ^ J! 1/ \i I \n !f i: I - Vt U I 170 ^JiJnWETW FOU B,:oiys^,, "iUa. CJovh, ribbons etr o common: '""^ ^^rd bemg the most Half of a yard. _,« ; v Quarter o/a yard, r '"?-• E'ffMhofayard, = ', . Sixteenth of a yard.=. ^ .< *®' The sixteenth of a v;ir,i , a "nail." ^^^^^ ^^s formerly known as TheFJemishEhv.asfofav.M " EnghshEJl .. ^^V^'^^'^^ 27 inches. " French EJl u * „ °r 45 inches. T,, * o^ 54 inches. •^^T,e^ fo„owi„, „eas.« a,e u^ed fo. ,,,,,, I Hand =^ ,„ f^ ' Cham =100 Jinks = 66 ft fort 'P'^' °^ ^^ter. surveymg Jands. '" ^°^ "^^asunng roads a„d 8ochams=imile. 209. By Long 210. If a piece of paper were r in j , and I inch brold 7. '"^^ Jong present what is cl'jied T?^^ ''■ Inch of Area oVsirfa^e^^"'^- "^souaT ^"'.^ ^'^^ *^--<-^^ore a square, one inch in Jen^th =.' 1 one inch in breadth. ^ ^"'^ 211. his measure ^S the most 212. SQUARE MEASURE. In the 171 ine same way, a Square i'oot IS a square, each side of which IS one foot, or 12 inches in length, and from the figure we see it must contain square inches. 144 ^nown as hes, les. les. '^ special ^ horses. ^ water, oads and gth and of feet, surface A Square Yard will thus contain S 9 square feet, as seen from the I ^^■ hgure where each of the small [~ squares represents a sq. foot FOOT 213. These are all included in the fol- lowing table, called 214. SQUARE MEASURE. 144 square inches (sq. m.) make i square foot, sq. ft 9 square feet . , ^ 30i square yards . i square rod. q^rd 160 square rods . , ,,,,^ ^^^ ^ 040 acres -^^^- 2 yards long = 6 feet lon^; ^' 6x4 = 24 sq. ft. = its top surface. Now, a 24 cu Therefc] 24 cu feet w J'.'x. Ex 2 11)150^ 4 '0)1 37 8)3H written wit S(/. rds. , 12 30f 366 3 369 y( 9 6 '0)332 '8 ft. JJ 6 O viding the ISJ^The I. What 3 m. ? ». What] in. ? Q 3. What 36 in.? SQUARE MB:: y RE. 178 Now, a slab from the top, i foot thick, would contain 24 cubic feet. Therefore, being 3 feet deep, there would be 3 times 24 cubic feet = 72cubic feet = 2 cu. yds., 18 cu. ft. rS- A cord of wood is 8 feet long, 4 feet high, and 4 leet wide ; or 128 cubic feet. F.x. Express 7544 yards as miles, etc. First, we reduce the yards to rods by dividing 5^, thus : Bring the 5^ yards to half-yards, that is II ; then bring the 7544 yards to half- yards, that is 15088: then 11 half-yards are contained in 15088 lialf-yards, 1371 times and 7 half-yards, or 3^ yards remaining. Proceeding, as usual, we obtain 4 mi 4—2 fur. II rds. 3* yds., which may written with the 3^yds. expressed as 3 yds. i ft. 6 in. Sq. rds. Sq. yds. Sq.ft. 7 Ex 5^)7544 2 11)15088 4'o)i37'i— 7 8)34-11 2 be 12 3oi 366 3 369 yds. 9 A garden is 12 rds. 6 yds. 7 ft. in area. One side is 60 ft. in length ; find the length of the other. Since the area is found by multi- plying the length and breadth together, we shall find the breadth to be 55^-^ ft-, by di- 6 '0)332 '8 ft. viding'the area 3328 sq. ft. by the length, which "is^Go "ft EXERCISE 51. K^The first 11 examples to be solved mentally. 1. What part of a foot in length is 8 in. ? 6 in. ? 4 in ? 3 in. ? 2 in.? t • • What part of a yard of rope is 18 in. ? 2 ft. ? 12 111 ? r. in ? ^ in ? o in ? 2 in. ? 3. in. .'' in. 4 in. J* a in What part 01 a square foot is 72 in. ? 36 in. ? 24 in. ? 12 in. ? 6 in. .? 3 in. ? 48 in.? li 1/ i i: i 174 ^niTUMETIC FOB BEOINNEBS, 5. An acre of farm Jand is wort i %->oc^. , 1 . mf^>r8o.ods.,orod:;%^v::;^cSr^j 2 rods ? I furlong ? , mTle ? *^^^ "="'* "^ 7- A cubic foot of metal is worth «,.toS. , IS the value of i solid inch 9 ill .*7"^J ^vhat cubic yard? ^ ^^'^ mnth part of a 9. A room is 20 yards long and 12 varrk wJ 1 \ many yards in length of carm f /v 5 "" ' ^""■' required for it ? lindts'ccSlt^'^f a y^rd "'^ ''^ "^^L^t^^r^rP^°°-^^^-^^^>^5f;ard.. II. 'VN .-:h is the cheaper ofthe following prices ? \V* 5^. an inch, or 50c. a foot, for lead ninp ? te ;^: ^" '"-J^. or $4 a yardUi c^,P^^^ /y f ^^'^' ^'^ 5^- ^" i"cl^» for lace ?■ \' In/ ^''^V^' ^^ ^ ^"^^' fo^ fenc^^g'?- (^) 6oc. a sq. foot, or $>6 ^^o n e.-, „^^ r (/) 24c. for^8 in., ;r ifa^l^drK^bl?^^^^ " "• al7;!Yr^!ro^:^^ '°^ ^°"^ ^"^ S^-- Measures, 13. Express 213 inches in length as yards, etc X4. How many yards, etc., in 1649 inches of vvire^ ^5' How many inches in 6 rds 4 vds of, '■ . distance? ^ >^*^^* ^ ft. 9 m. of 16. What part of a mile is 2 fur. 36 rds. 2 yds ? '^' lonT "'3^ 1°^^' -'rf ''' ^^^^^ ^" - pile 40 yds iji!^, 2 3.a3. higii and 4 ft. wide? >^^i-^iib. .8. How many fathoms are there in a depth of J of a SQlTARhMEASm "6 19- Find the cost of digmnfr a drain or. f . ^ w.de, a„c, H fee. ^M.^tt^TJV;:^. i^lfi^L'j- in. ; 76 yds. 89 in. ^ ' ^ ^^- J 3i yds. 100 «^^n dividing by ao^. bring both divisor and dividend ..rthn 21. The area of ri board is 21 fp^^t • ifo 1 what is its I.readth ? ' '^' ^^"^^^^ ^^ '« in., ;w.atisth;co^;-if^a;:r/;;/;rLt^t.^^^^^^^ 23- How many suits of clothes mn K« , / pieces of cloth, each containinJ^.ri T'^^ ^'""^' '^ 7} yards to make a suU ? ^ '^ ^'^'•' '^ '^ takes 24. In 987,654,321 inches how many miles, etc ? ''■ ^^::nS:^'^:^^ °^,3 i^^-^^ with a will the line measure the distance? ""'"^ ''"^'^ ''• milel^'efc^r'^''^^ ^^^ ^"^^-' ^-- -any square 27. If a plank be 6f inches wide, what length of ;, •„ give a surface of 2 square feet? ^ "^'^^ 28. A block of marble confflino T-,^/; r anddep.,, are eac^rCVXrii^f /^IT,?^'J ought each of the others to reap J ' " ""'='' '"■ iZ:'Zl\la nialn^te^"--"' ■•" ^ "-' ''■ of\1Z-^t':;\:j°^^ -' -"- '-« -ceed one ''• rod;:?h?s'ec"„7;oJ,'':c'8ls';™'t%f« ==• '33 sq. 69 sq. rods, the fo^th ;, ae ^;„ '• ""^i''''''^ **5ac. fifth 112 ac 08 so rol • if ^^ l^-, ''"'''■ ^"d the ' ac. 98 sq. rods ; how much land do I own? "». IMAGE EVALUATION TEST TARGET (MT-3) /. :/i 1.0 I.I 11.25 ■ 50 ""'^= 2.5 2.0 JA 111 1.6 P^ <^ /: Pnr»ir»rnxiT^nir« Sdences Corporation 4\^ iV *^\ \ c\ -^^ ^^^m. ^ 23 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) 872-4503 p .■; y 176 Mil VUM trie FOli BEGIUXERS. 33- If a mountain be 4 J- miles high, express its aUitu.i ■ asajraction >f the earth's dumete?. which ?sS 34. What length is that of which 2^ yards is ' .^ ? f which 7 feet is ^Vo ? '"'' ' 35. What length In leet would require to be added f. a telegraph wire, which reaches only -t_ of the d.. ance between Toronto and ThornhXl^ to cornn etJ the whole distance, 12 miles? ^mp'ttt. ^^' ^j^3%*P^"*°^ 2 miles would just measure ^ of lo ^^* w ^'m l'v,''^^^u'^.'' '^ "^''^^ «q"^^«' l^ow many acres would there be in Ju of it ? ^ 38. Divide loi square miles of land among 33 persons and give each an equal share ; what wo\ii each 1"! 221. 220. Time is measured by seconds, minutes, hours days weeks, months andyears, as shown b^ the foilowing TABLE OF TIME. 60 seconds (sec.) make i minute, min. 60 minutes - i hour, hr. 24 hours u J j^y^ ^^_ 7 days .. 1 ,,.eei,^ ^j^^ 3^5 days .. j y^ar, yr. '^MaH" Anrif M^' T^^ t"? J^""^^^' February, October \n' ?' ■^""'; J;^'^' ^"^^"^^' September uctober, November and December. April, June September and November have each v> days, and all the others have 31 days excent Fel ruary. which has 28 and somekiesig days^ ' ^TelnVear 'n ^^^^^ ^'^'^'^^ ^^' ^9 ^^y^ ^^ called ^eap Year, and thus has 366 days '^XT888'"^^^^*^^^^^^^^^^^'^y4.such as x86o, ^"'/.f^''^"??"^ ^?"'' ^''"^ ^2 °'^I°^k. noon, on th^ the loth o^T'"^'"'' *° '" o'clock, midnight, on me lotrt ot January next? many acres TABLE OF TIME. ,„ • and 12 hours. ' ' ^^ + io, or 26 days. 62! t^o^f^^^ hours = 624 hours. °24+ 12 = 636 hours. EXEKC'ISE. 52. 1:^ The first seven questions to be solved mental! v I. How many minutes in A an honr9 in 3 !"^"^f^^>'- m i of an hour ? in z^ Lurs'"?^ WJ^ '^"^ " 3. How many days in 2 vvk rt\^ r^( c ^ 1 J ""^i to 20 mm elapses.""' ^*^ "f September, how much time 24. The true year contains 365 days c hrc; .« • 49tV sec. ; how many da^.f etc.^,; iVyL'rsf ""•' ''- di^rdtraVe"ek'^"^ '" ^^^ ^^ ^ -"te, of a ^'' S ihJ^ri^r;:^ ^^^^^^ ^ ^™- i^ r->^ 27. Of what time is 3 min. 10 sec. the seven-fifths ? 28. What time would you have to add to 5 hrs ^- r«in 20 sec. so that it would become | of a day ?* ^ 1888? irjihe 2 sec. ? . 19th ? and 30 min..^ day and a of '>3> to Augufc; 9 to 32 miri. '■ who is idle hrs. 17 min. Jc. ? irs. to leave TABLE OF TIME. 179 r.-i- of '3 '» to 20 mm much time '. 48 min., ars? inute, of a n. is I '.> of ifths ? ^s. 5 min. 39. How lonfr would it take a man to complete a iour allrt^fat r ''^ "^^ "' ^* "^"- in ho\r%Tnd woSdteTni^iit?""-' '' -^-^^-^by the clock 30. Two men start to walk, the one from Toronto the otuer from Newmarket, a distance of 32 miles a o oi"e; a mi?er^'l ""' ''' ^^^l°^3 milef an hour? h? ^ll£/^r ""''''' at what time by the clock 31. A boy studies on Monday 5 hrs. 30 sec. f of ii of a day on Tuesday, f of 12 hours on WednidaV f of 10 hrs. 20 mm. on Thursday, and i of 8 hrs u, sTd ^? '"'• °" ^('^^y ' how many Lurs would he study during a school week? => vumu ne 222. Goods are bought and sold, not only by weight, but are often measured: as, a pint of beans, 1 gkllon of milk, a bushel of potatoes, a hogshead of wine 223. Liquids such as wine, ale, etc., are measured in a different way from fruits, grain, etc. 224. The latter and such commodities' as are taken up in the hand, are measured by the following ^ TABLE OF DRY MEASURE. 2 pints (pt.) make i quart, qt. 4 quarts - i gallon, gal. 2 gallons " I peck, pk. 4 pecks «. I bushel, bush. IS* The standard measure in Canada is the " Imperial Gallon, containing 277^- cubic inches. ^ '^tlJr ^.^t^^:?'"^^ tJi^ weight of a bushel of different kinds of produce, as follows": ""lerent 1* ■*! I: : I t ,i 180 AhnHMKTIC FUR BB6IS.\BRS Oats, - . Barley, • Buckwheat, Timotliy Seed Flaxseed, - Rye, - 34 'bs. 48 lbs. 4>* ibs. 48 U)s. 5u ibs. 56 lbs. Corn, .56Jbs. VVheat, - . 6oJbs. Beans, . 60 lbs. ijfas. . . . 63 Jbs. Clover Set i gallon, gal. A teh '"^ (^.^K^-) ^^ ^'"« <^on(ains 63 gals A hogshead of beer or ale «• ^. „ A barrel of beer or ale .. ^t ^^''• , 30 gais. KXERCISE 53. IS- The first S questions to be solved mentally " tor; "' "^'^^ '°" "^-^ ^^'-' ^ovv many a h^lf-bushel P Zee-q^ua^Ters of a^^shelt ' ^^^^ ' 3 I paid 80c for a gallon of Fulton & Michie's Hp.. vinegar: how much was that a Quart ? 1^ pint? for6g.!lsP lor half a gallon ?^ "" ^''^^■ *• P°"^!!* ? '^"s'^el of Spanish chestnuts from ri horn & Son for $4 , what was the rh:e for ha f ^1' gallon? for a quart., for 3 bushels J f^r 4 quans . 5. Bought from Farmer Jones, potatoes at 60c a bush el ; in order to gain loc. on every neck T splf u\ must I charge for a bushel ? for 1 of fneck ' f bushel and a quarter ? for half a 'peck ?^ ^°' " 6. What part of a bushel of corn is r, ,>ks ? i o i Ion ? 4 gals.? 8 qts.P 8 pints? ^ ^ ^ ^ S^^' 7. Which lb the cheaper of the following prices : 5 lbs. > Jbs. >Ibs. )Jbs. -lbs. lbs. wing RE. J gals. I- gals. > gals. tally, how many he price of ^f a peck ? :bie's best ? a half- ■om Cleg- ■or half a J- quarts ? c. a bush >ell, what ^ ? for a i a gal. ?s; (^) $1 a bushel, or 25c. a peck? il>) 20c. a qt., or loc'a pin?? U j 5c. a gallon, or 60c. a bushel ? W) ^2 a quart, or 30c. a gill ? l^; 80C. a peck, or 1 2c. a gallon'? '• at raquSfo?nutf '" ^^^^^ ^^"^^ ^^^ ^ ^^-^-ad of '• "rarTels^ofr? '"^^'"^ ^^^"^^ ^"^^ '-f^iiedfrorn 10. % how much does the number of ffals in, of w,ne exceed the number of hoSheads^r' ^''- gallons ? "^Jgsneads in loooo ... How many pints of water in 6 gals. 3 q.s.,p,, .2. How many bushels, etc.. of nuts ,„ X387 ^J, .3. How many hogsheads of wine in 6324 giiis ', H. I bought 4 barrels of spirits each l,„i r ^^^ ^*^- ^ ^■" ■• -'-' "^; I s^en' i.'}"'r*r,fas f! 15. How many gallons of cider at 12^0 a „M obU,„.„ exchange for ,30 bushe&^.f :p^pt:ar3';c! 16. What is the value of a^c horro c u .aining2 bush. . pk^aV/o^a ^ stlT' "^^ ™"- 17. How many loads of apples eirh,^^,.* • • a. 45c. a bush., can bTbrghur,':™"^''* '''-''• .8. What part of 1 gallon is 2 qts. ,} pt > 19. If 376 gals 3 qts. , pt. of mill; ,,g j- .., among 9 chanties, how much will each i^iT?"'' 20. A man bought -^x of a hnc^^oi r them a. roc^ qVart b', '"o^;' "L™ I f "'' T"' quart; how much did he receive .' * ' ° ^ 21. A man boueht ia hno-c ,,f k„ bush. 3pks..'for$t-!^^nVlldTem1,fVox^^ ^ bush. 3 pks. each ; find the price pTr box ' "^ ' 22. What part of 62 gals. ; from a hogshead of wine leave must be taken 42 gals. I pt..? M 182 •i ri •^' 4 1/ ,'. h ■ |5 I 1 J ' i M ■ ' ■ 'I: J:! -*'""'«™ 'OK /!OT;,v.ve/,s. ;3""' b„t,,es „'„„,d^hiJ'-,, -/d W many d,,„, at each meal? ^^y, and given 5 quarts ''' & ro\t^t t?-i:aV^,^^^.%-^^ ^en^ade wheat to make 2 lbs. of flou' ?' ^ '' ^'"^^^^ 3 lbs. of 226. Many articles of merch;in^^ special names, the most^^ n T ^""""r^^' ^"^ ^^Id by the following ^^^^ '"^P^rtant of which compose GENERAL I dozen (doz.) I score I quire I ream I gross I great gross I stone I brl. of flour I " pork or beef TABLE. = 12 articles. = 20 '< = 24 sheets. — 20 quires. = i2 dozen. = 1 2 gross. = ■14 pounds. = 196 " = 200 <« EXERCISE 54. t,J^"'""°""« ■'"-"■'>- should be solved .en- 12. ■|| <"-:yBRALTAnLE. 18ft II +i of a score ? i„ 3 „,,, ,,,-^Vi doztn ? ' "'""" a score of Jemons f of^a^^ss^of ^pelfs^lTc 'ali'f 3- What part of a score is a dozen ' 'ht:?fr?o?:"re'i:;:,^^"'"f-'^%"S^^^ 5.Which,s.hecheaperofthefol,owi„gprices:_ ,'c"'aM°''"' ""^ "°'=- a score ? ^o; /"■"'■ 3°<=. a doze,,.' 50c aqu,re,or2c.asheet.' J a dozen for aoc nr R«,^ „ .ic.each.orloc:a°d'ze„r""' ^ score for a shilling, or i shilling a score? Men-^t^tTfLtef:,?--- "- --=" per gain .50. a dozen ? toTose Vc ^^1 l^l ^P'?« ' '<> score ? to lose roc. a score ' ""'" ' "> Sain aoc. .-, '■ i?g98ooXe;T°'='"*P'" 5 fl'^"^. each con.ain- "• "°nTneSi'o°Te„:r''^°'"^'"'"^ ' «--■ are .here 9- What would 120 eroq.; nf o^ 1 dozen ? ^"^"^^ °^ ^Poo^s cost at 500. a *'• How many barrels in t,<, what woulS the ^tl^cost'" IT^^ ^^^ «-' = »" "• Page™e':chr,d°&de''°'? """^""""^ "^ ^oo into 8 leaves ? ' ' °"e ^heet is folded '" b"arel?^f"^o1kr= " '''"" -'" -Sh as much as a- I(». }! 1 h I rr N 184 • "™ """y reams of „,„„ u-m ''^ '^S'^ ' 2500 c„p„, „f thefiSVa" r"*„^" edition o, each copy cootainine js!^ '''?" deader No , « leaves ? "8 384 pages, ,f „„^ ^^ ^'"''' "' ='«! pe„s, at 6} "• ^he!.fofpa^;e7cr? '"''- -"' 9 reams, a ,ui,es, . 22. How many dozen r.KK dozen? Hnm 1 ^"^ *" market .<. "^^ ^"ey be «--„y acres are. hte1„t2."=;iS,„7 CIFAPTKR V. DECIMALS. 227 1 ti units' place. ''"^' "^''•""^^ *« the right of tt 228. In the number 84vs-267 if the values to its feVincJ^ase Tn/fl^ '^' ^ "'"^s. decrease tenfold for each pL^e "^ ^^°'" *" ^^^ ^ght Thus, 5 means 5 units. j3means3tensofuniis 2 means 2 tenths of a unit o means 6 hundredths of a unit 8 means 8 thousands of units '■ •^^7 = T^+^+^^^^2oo+6o+_7^_267 •lecimals rapidiv, when .h^, ^''""'^i in wri,i„„ way. •nen.heyare read in the iatte? tI85J ■'•. I I8 ,. «>.''« /w /i -sS^v^I,;*! ;;,";"'v./>/«« "- «-'^" "Ww^O-- ' ■'"'*'**"«• n>/«ri to l/„ Ufl.Z Ex, a. — -J! oa ,, jr. •>' T() 000- =0502. orders are added t^itw' ''^^1'"^' ^^^t only R-e another. ^^.ether. or subtracted from one ^^•5. Find the sum of 4.02 •oo7c « 4-02=4._i . . °°75. 16-31, and 4x03,. l6'5I — rfisi ToTJo^ 4+16 + 41 = 61 ^^^'^ T^ + T^Voo+t'oV + ^.^, 222±75±^3^zoo+32o 3695 Hence H '°°°° "Toooo- = '369S. Hence the sum is 61.3695. This may also be worked thus .- 4'02 •0075 16-31 41-032 2 in Art. 228 5 the begi,,. ' the lf.ft jj. onal form /,y merator, and alform in a ^tcimal pari OS there art '0 the ie/t, tf therefore, ition and ke whole pnJy like rom one 41-032. DkVUULS. £x. 6. — Find 4o'935- 187 th< ay also be ^\orked thus :— •042 74 148 •"^>i5i4 235. From this we see that «r «,?///,>/« ^^ ,/ ./ -£•*. 8.— Multiply 31.25 by .0196. 31-25 0196 18750 28 1 25 3^25^ •612500 ing -6125. ^ ^*'""'=^ *'^' ^eav. llx. 9.— Divide -0296 by -08. ■02q6 = — iiu ■' 1000 O » To ()• ^Ti^XJ.^ = VU,= tVo = -37. iS^ Proof: •o8x-37 = ,,8 x '^ - 2o« ^ 188 ^HimMETJC FOR BEGiy.yEns This Example may also be worked thus: ■o8)-o2g6C37 ii. 56 J^iow, since there ir^ /•.. 1 dividend, j;LX^X-is^^''.°^ ^^^^'^-Js i" the '\^ see that tJie mm her of nf ""^ ''''^ ^" ^"^^ quotient always be found by akin^^ j'^'^^" 1^^ quotient can the dn.sor from the nun^b^r " nT^'' ^^ ^^^^^'^ ''" dend and marking off thi^ cHfff ^ '''^^ '" ^^^^ divi- of places in ih^ quotient ^'^"^^"^^ ^^ the number We also notice th^f ^-, • ", ^9a,.stasifS--;tKl,^^^^^^ 236. From this we obtain the following ^ULE FOR DIVISION OF DECIMALS ^iri^fe the numbers as iff/,. ^^M ALS. ^^ein the dividend more thn: • "T^''' 'f^^^^'^s theu ''^^'-;^^ot;.eleft,ifZ7sarr "' ^'^'^or, preji:c,;, %he-f to ttl?4,Vu"°* '^'?^^ ^" ^--J^e by addm. ^ dividend have Xe 01:'?..'^"' ^^''"^y^ "^ake S ;^j:. io.--Div,de 52 3 bv^-tfc ^" '^" ^^^^^°^- The ..dendma^yb^elS 33.3,,^, -^iT-^4:8, Inthed,.,,,.,,^^^^^^^^ ~2i^ «.onn' '\*^^ dividend m!,«f\ ^^^ quotient we 237. In ,. , any cases t!,.^- ■ ■ 5^3000=52.3. often L, roV.!;?„tr"aS?f,r'^ '°"S' -d ve.v •he .,„„.,.„, ,, , ce„at„^'4'eV'orpSs"r °f-^ ''^' naJs in the e quotient, uotient can 'f places ill n the divi- ;he number lings into LS. ^'ii^ Mark '>laces tkew y adding 'ake tile '. ^ave 3 ividend places : ent we which 125 = d very to find s: — Divmos OP DECIMALS. jgg 109)25100000(230275 ~- Now there are 3 places of deci- "i^ls in the divisor, and since we are to have 4 places of decunals in the quotient,there must be 7 places in the divi- dend, or 2-5100000, and tlie operation will be performed as shown, and the -^otient will be 23-0275. EXERCISE 55. TuSfon^"^" ^'^"^^ P^°- ^'- --Its in each (-)^Expr_ess as ordinary fractions in their simplest I. -495; -0075; 12-8; 68-187;; 2- -375; -225; -0068; .3125' {f') 1. I he distance from \ tn n ;. B to C is 13-06 ch., from C to n l"* J^ "=''*'?'• f™» L!;;?SE^p^^-'^™-^-B7sV;:tri5o,^^^^ 3. From -00038 take 36 ten-miUionths •■ 'sw"^' '^ ''''" '-■» 34-<534 acres to leave ■'■ lu^LTlSt °/rct;c:i™"^^3-S55 po„„ds pounds? produce a mixture of 29-796 '• •o;^:6 ;^ile*? ''''^"="« ^^'-- '00 miles and '■ r,:und"d.ir ."VtL^r""^?^' -5 millionths thousandths ? ^ '""■"'^'^"dths, 834 hundred.' ■u ^;f III I 'li I 'hi' IL'L 190 AlilTmtETW fOJl UEGi fi. What is the cost of XSJi/iS. yard? 3r5 yds of cloth, at I3. 9. Since 16.5 feet niak '5 per th lere in 237 rods? e a rod, how many feet are 10. The product of What IS the other ? two numbers IS •( >o48; one is -06. II, 12, Divide Divide each to four pi •04905 by -^.y ^35-05 by -037. 71-142 by -0071. aces of decimals; 15075 by 30-2C. 300-402 by 1 2 -I 4'oo334 by 6-31. 13- Find the quotients each '3412 -^ 8-4706 •°«4i34 * -3243 to five places of decim ah •oooy.joSs + -8 '3497- tion to ,ts equiva/e,u decimTl " ^ "''«" f"'^- '"bysXs'S""'^- ^--b.ai„ed by dividing 3-0 »d the latter b.divid4, .00 by „,.hus:- 25)1.00 239. Hence we spp fj^o*. l-, etc., may be &r V'^ ■ ''"^'' fr»«i™^ as* i . «c., any^ract,™Z"y be^Sed? '° "l -^^ '^^^ ■"raply dividing „,e„L,oS bathed '''"='!"'" "^ ^ oy the denominator. Dl VISION OP DECIMALS. jjj -e^.i.-Express^V as a decimal. i6)5-oooo(-3i25 -_ 4« H^re, the value of the numer- ator IS not altered by adding ciphers to the right of the de- cimal point. It will then be seen that the po- sition of the dedmal point IS iound as in ordinary division ol decimals. £x. 2.— What decimal of a oound ;«= t\.^ a pound ? ^oxxna is the same as /^ of 4^000 Here the division is carried on bv '""9275 the same .«''^^"'^' «? °^ ^ P°""d is The previous result. k ""^^^^ of a pound. ^ "^ ^^''"^^'^ "'ay be proved thus ;— 20 16 40 32 80 80 ' 1 O 11 (I if -:r-=-- -' J 10000" — Tfl. 3a* , 3— Reduce II to a decimal 45J28-ooo)-622 ^ Here the remainder, 10. is contin- Qo ,t\ ^^P^^ted ; then we see that ^ the figure 2 in the quotient nSutf loo also be reneatpd ti,^ r?"^^ Of, K • . "^^P^^S^a, the quotient _9^ being written -62. 10 ^^. 4— Express ||| as a decimal. i65)io3-ooo(-624 990 The remainder 40, repeats, and therefore the figures in the quotient from thi 3 will ^ P^'-iv, and rhe quotient is written in the fonn .624. hi' ' } ■f h f : i 192 ARITHMETIC FOR BEGIifNERS. ^^- X:ri=r,-rais torn th'ff"'r;r« ''-™^'=. - figures i„1he,„ circuia[: or re^t";' ""' ""^ "' ■"-= tn'^'L^re.c''^ -.ri""' ;l-"-d 0"'. be w„t- written ■62424a;, etc *° '5"°""" *°"M be ^Pe'rTh'„^^"''"^'=^*^«e;ure or figures .hat re- •0572 means -057205720572, etc. •62947 means -62947947347, etc. 242. Decimals like -^572, where ail the ii»„r,« -f. .1 po.nt repeat, are called Pure'li^atintoed! Decimals, like -62047. where «;nni^ r.( 4.1. c 243. to vulgar fractions by These decimals are reduced the following RULE. If a Mixed Circulating Decimal, subtract the iart that ^.. Ex. I. •0^7=-A7_ Ex, 2. — •o^7=-?--<-^. ■J' BOO' ^nume'ral^r.''"""'^'^"" 3+- "^ich is placed as Ex. 3. — 5'892. 5892-58 = 5834. ^i^mo^V OF DECIMALS. Thus, the required fraction will be 191 V.V or 5A|*. '''• "^^^^^?^!^^^ worked ., are then performed as in (vf „, . "P^ations £X\ I .—2740 X -243 = ?JMU» V « ♦ ■< _ 2 ;- EXERCISE 56, (-) Express the following fractions .s decimals.- t. 2- tV J 20 4. ^^. T as' '■ '" '"• ^- ^5- ^. 16. ,v. 17. -.. ,8. ::: (fi) Express the following deciin^lc . f lowest terms :- ^ ^ecuuaJs as fractions in their '• 7205. 2. 72, 3. -i 3, ^^ _ ■ 6. .0169. 7. .2045. 8. .1045. Q K>i^.n (') I. Multiply 55-69 by -3. 2. Fi"d the product of 5.4,46i and -z'ai. 3- Divide -082 by -6. 4. Find the quotient of 3,.7,; divided by 3.07:; 5. Divide .;97 into 2-297. e The product of two •245. find tlie other, two decimals »s H,' one of them is .1 ■h :. i CHAPTER \i iU I PERCENTAGE. ^46. The term Per Cent, thus means per hundred and ,= only a short way of writ.n, peAeatuT^.'^^tZ rS- We see that 20 per cent, means 20 on everv hnn dred ; therefore. 20 per cent of anything V h. same as JLo_, or 1 of it. 'iuyuimg, ,s the I^The expression % stands for per cent.; thus ."/ means 5 per cent., or ^"5. ' 5^ ^A-. I. What is 8% of 150 pounds? o lot) — -2 5' 7\ of 150 ]bs.^= 12 pounds. iT.r. 2. What is 120% of 80 yards? /o 100 — 5' -S- of 80 yards = 96 yards rioi] fURCENTAGK. •inac IS, Jie lost 2 per cent. Phoop : 2% = ^=^,^, ,„j ^^ of 400=8. '- hilt IS 8 horses lost. ^"V fhem r dis'cas^ " l"' "' '°" ^^'=''- -" '-' -^o' ' ny uiscase , Jiow many sJieep were left? 5 01 »uo=32o sheep. 047 r^ "''" ''" "'*^' ^■^- 5. page .45. t ;:/:;:^5:! ^^^ ^"^ ^ --- ^^ a peculiar sense Hence 20% .;, 60c. a yard is | of 60c., or 7,0 a yard 4o%^//icxi. apoundisiof Inr^ ^^ /'[^- a }ard. Afr;,;n o f „ """'^»°^^°a.,ori4d.apound. Again a farmer selling a plough pf 00°/ /r.u would just receive 80%, or^lof thf cos^ '"^ ' ""'' Kence 15% ^^$40, would leave 857 or -' r ^f s.. ,, . IS $34. making a loss of $6?'^' ' " ^'^"' '^'^^ 30 %<#8o bushels would hp -70°/ ^.. , r ^ bushels, which is 56 bushels ^ '^ ^" °^ ^° EXEKCISE 57. IS- The r,rs. ,. questions should be solved raentally "• f«Ealt/„;'l-'"S''"""'^ees in their si^pi.s. ^7ii ik 41 3?^- ,'S> "*/• 45%. 15°%. 9oi ^1; US- f./ff 15%. 40%. 1807; V? .??i-^^^"'%'f«n^eis|of it? 1? _,_"? II- .f ■ill . i I ' r I -' -il IfiiA ^^'ITiriETtO Pon DEGIiINEIi\ 3. ^Vhat is the difference between Qn°/ of »>-., r of it ? A of a lot and 60/ of it^?'^ tyli^Z"'"^ f 40% of $800 ? ^ of 80 pksCand 807 J?io pfs? "v of 200 acres and f of 200 acres? ' °° P^^ •'' ^S/,' 4. A man owned 60% ol a farm of 640 acres and sol.l i of It ; l,ow many acres had he Jcft ? ^ 5. A merchant makes $Go on $200; what is his gain '• whtt d.!t U* t, ^"' ^^ ^^- ^-^^- ^'- ^900; '■ ^i^SV ont '^1 r '"^^" '' ^ ^^ ^ ^^- ^^ -^d ^r ^" of whV^V'" ^Ti" ^' ^^^^"" ^"°^^« ^'■^^old so that ? of what they all cost is received for half the goods ? 9. Sold I of a Iihd. of molasses for what the whole cost me ; what was my gain per cent. ? 10. What per cent, is gained by buying oil at So cents a gallon, and selling it at 12 cents a pint ? ''°""*'^ "• c^entTssTs^ysfharrsf" ^'^ ^ '^'^^ ^^^ ^^^^ 13. What is 65% on $145? 40% of $560? 14. What is i2i% ^« i of a shilling? 25°/ of J'- of 1 gallon ? 20% oj H of a yard'? zSo^ i || ff ^ 15- What %^« $9 is $12? $18? $13.50? i6. A merchant gains 25%; what is the gain in a sale of i2^o los. 5d., and what is tJie cost price ? 17. A merchant loses 12^% ; what is the loss in a sale 01 i^72i.yo, and what was the cost price ? 18. What per cent, more than -«- is ^ ? 19- What per cent, less than f^ is ^ ? 20. One-fifth is what per cent, of three-foMrtl«s ? u farm and ■> 'f $Soo :u]([ ! and sold ■ liis gain lias $goo; is sold for so that f e goods? hole cost 'o cents a uch per it cost, II of a n of a I sale of I a sale I'B:'.CSS'TdGE. 1:7 ""'■ Sffi^'^f-T"^ "">' '^'^^°" t'^i-ee years. I sold it f -.50. A few examples will sI,ow these more clearly. ^lertionL"^"'' ": ^''''''^'''' ''^'' ^" '^^ P^^-io"^ ^andTft 7r,'' ,'+ ?• " •"• '""- '^ f^- 7 in. ^v;dc, fntlfe walls" ''^^' "^ '^^^^'"^^"^ square feet are thcrJ 14 ft. " in + lo ft. 7 in. = 25 ft. 6 in. = 25' ft. = length of the two adjacent walls ; 251 X 2 = 51 ft. = length of the four walls ; SI It. X 9 ft. 4 in. = 31 X V = 476 sq. ft. (5). ' £x. 2.— Find the cost of painting the walls in the pre- cedmg example at 42 J-c. a square yard. ^ • 1?« t %T5i~ ^y^t- ^" tl^« ^^"s ; 42.C. =^^3 ; ^.r. 3.-Find the cost of papering a room 40 ft. / in. long, 20 ft. 8 m wide, and 12V feet high, with pape; i yd wide, and i8c. a yard, if the windows, doors. etc., take up i of the walls? ' 40 ft. 7 in. + 20 ft. 8 in. = 61J- ft (i) • 6ii: X 2 =4AA = length round the room : H^ X i2| =xj^ X V - "lumber of sq. ft. in the walls (5). ^ Since i of the surface does not reauire papering, we take I of It (6); ^ a X 7- X ^ X i -= sq. yds. of paper required Now to find the length of the paper we divide its area by its width (4). Thus^lA X V X f X i X f = length of paper re- quired. ^ ^ "^^ ^^^ it^ ^''^bie, multiply its length by the price per ■^F X V X >- X } X } X i8c. = $27.20. MO I ^rUTUMKTlC r^H BEOISSEM. f'«^i^tl:r;S:;;/:;-;-^'«^rnt Steps are .^ *»^*'' i"* hx tins mA " *'^"^- '"'J then simn P^^r foot in length. '^' ''^^"^ ^^«» sold at 40c' i^ in. =: .1. yd . , , _ Therefore tneVengtl?n7u^thp"c" area of the e.,.|. and aoyds. . ^^,,, u:.tj^,,\rx^ ?o^:^i^t ^^^^ kxercisp: sa 1. 3. A block of ice is ^ - ; , ""^ '^^ '9^- W- >=ins .6. cubic feet': Lfe.f.LSes"""' ""■ -'■ 4- The paDerino- of o '^"css, ;5c. a yard /osrs Ix^sT- Tiw ^'^'^ ^° '"• ^^'^^^ •''t he walls and i( the Wth be Tf? '"^'IT ^^'^^ '' " ft., find its height. ^^- ''"'^ t'^e breadth '• f-t r'$°^P^'".';^f ''■-alls of a roo„ a. ,.. , ,, HC a sq. yd. : find the cost ' ' *° ^^ P^ved at 7- It taJies ia6s vHq <^f many inches wide is the paptr^?'- '° '"■ '"Sh : I.ow "fro/^.l;i;t\-^7,%^,---5..,.hepape. ^•^^^^r:'-£^}i-r^'-:ecos.of are worth $L p. o..sa^:V"- ^^ 4| in., if the slates steps are kq.t J then simph. iiicejjing save i'lchcs square II sold at 4(,c. ^fthe end 20 yards fj); ' = $24. ie, and con- tJie carpct- H 19s. 8(1. J> and con- in. wide at re yards in lebrsadtl) ^2c. a sq. > and tile a garden paved at y^s. 5 in. gh : Jiow 'e paper- cost of le sJates CHAPTER viir. BILLS OR ACCOUNTS. 251. When cfoods etc nm c- j 1 1 '• Bill or Acco»n? i Y °"^ P^""" t" anotlier, sent with the 'ooc^ to tir'f" ""^ ^'>' ^'^^ ^^^^'er. an.l "^- ^^^.^:rs:;&=^i^-a^i^^^ Messrs. J.o. M.cno...o ^ ^o!'""""""' ^^'^ =°' '««'• 1881. '^'°''""'"- p ., ,^ J^o t of Edwa rd TIugfies & Sons. "5 ' 10 o Sept. 5 420 yds. Aubusson Axminster © f 5/6 ^'°° " I^i-i'ssels, 5 Frame C. " 3/6 Sept. 12 Soo " u ^ , li.'^' 3/3 ^'' 3 220 " Tapestry, .. ! ^^^ 10; o o 130 I o I 22 ' 18 £373 i 8 n- Since 80c. = lofSr, 75 IDc,. @ 80c. would cost -*- of «7c thatis$75_.^:5 = $^>^°**75. [201] 202 ■«! I i m i. EXERCISE 59. 2. 7 Make out r? f %*«%%'! •,?" i^o^^es G,af isf9 ■ 2ao fcg» K^ ^S.ay^ , 3 jc boxes Tin pi/ 3" ' 4o kegs i ubJish no- Pn T ™^-> vVhitby. biiv r,f n n^r A/T ^ ■' Toronto • lannr^ 17 . ^ °* Canada per M ; 375 Pass-Books" /^f T ^"^^^Pes @ $2 874 5' 'Ljeo. Lewis <;nM + /- @ 9c. ; Jibs M.nt^cP'- M'=Master: 2, lb, c ■• Chas' pin "J^''" ® «' ""'^' '5-' l-fll^c • ^*^ >'<''5- Velvet i i',y*^''"- Lace a $ ' P B r,f ® '°'^- P" *S ° ^" '"'•• "^* ''o-- Scarfs @ 8jc, each ® *'-'5 i'" doz. ; Too Me^'s ^'53 and 1:4. t^^'s ciass of each of the ^am & Co., 220 kegs 40 kegs ^ $4.20. '.Simpson, in Navy " ts "Dark -s " Dark • D. King @ $4-25; Ts Boys' @ $4.50. Canada ? 12.87^ iates @ •2. Maps '• Sugar Cheese at 15c.; Barrie; @$i4 5^ do2. Brush h-65 doz. ; Men's EXAMINATION PAPEKS. ""^--So^fi^Sl^^t^il'Yc^^^^-- ^'-" ^^ the Institutes. ^"^rance to High Schools and Collegiate PAPEB I. \o mSett„*B° '"''"'^"' '^ ""<' ^' » - '° give A "ir^sfb^lS te\t^ *„T «?^ "-" "- -™ai„der "'at is S23 apiece. ^'"''' receives after tliis, Thus B receives $25. "' Falrm^'Cgn?"'"'™ '""^°'-- I™P-P- Fraction, 2. Express vV of «» of sa r denominator. '* ^' ^' ^ ^""^^t^^" 'saving y^ for its 3« A iiouse is worth '^c^p. „i,- i • . ^ times the value of a barn fi'i^fJ ^^^6 more than | of 4. A man sells io|"bs of suJ .". '^' '^^"^ "^^^^ l^-- and gains Z7c. f find I'^/ost ? ' " ^"^^^ ^^ « ^^^- ^- ^^^. What is the frro;,f;J r '^^ ^ugar per lb. eachofthenuSrfxxTil ^ -" --tly divide I sold a farm for 2/°/,^' ^°^;,4°^"' ^"^ ^445? -?Jd it to B for lioS.iS was" 'i?f '"^to A, who J"ni: what did it cost me? 33 J /less than it cost 7. Reduce the fraction ^yy^uL to if = , 8- The sixth nart of 0%""*'" *° '^s lowest terms. A room is 67 ft. lon^. i x f .^f- ' :'^'''' ^'^^ ^^^^^ g^t ? 5. G. A room IS 67 ft. lonir u ff • 1 • ", ^'^^eac doors and windows I'lV,--^ '^ ^c"^'^ 38f ft. wid and ceiling: find J'^ic. per sq. yd. t! le th e cost of le area of the walls pamting the remainder at [:03j J04 PAPElt II. 2. 3- 4- lo. II 12, ^-st 4Sc , what vviJl inx Jbs off' "'^ 4^ ^b^- of yt ius. oi sugajT cost ? I JIj. of coffee = 1^c. = ^ ijib. oftea =tV 9*"' TT^ ' Jh. of su^ar -''a M ^°^ee= 7 of ej.c K^t^.- r '""^^^^°•■"-""-P'■■- $3 a ba^ I jra/n 4- * ,^ ^^S I iose S^o h„f v t , tea .0 ™..-\--J/- i^^^ of «-„e ..., A and B earn $g^o between 1, ^?''"' '"'"'' 'ffcn from iro' b'T'" " ^"d f °l I2I.+ , . The furniture of a LT '' '' '° P™d"ce 3/3^" V ~ '* f ol the furniture co"t?i' """'i? «'°«'°. and ." f . . A ■■ what'd,^ric,f .^f. t"hT' '"^ S'ii'=-d 8. 5 6 t as much I as 8. i'XAMjyATWy PAPEns. PAPEB III. 205 ^x. 3. — After spendincf $60 morp than 1 ,.fu; man ]aad $.ao left : Vhat had he at fi^^f ?"' "^°"^^'' ^ ''it t'eTeto°rtlor' '^^ "°"^^ ^^^ ^^^^^ ^^^ ^-^ Then, i.aving spent | of his money, he must have -. of it Therefore, | of t!ie money = $180. = $60. 1. << V " " =185480. Pj^-">f: i of 480 = 300. io- ^360 = $120, " left. I. 5' G. What IS a pure Circulating Decimal ? Hn,„ ? duce It to its equivalent frfcti'n ? "" "^^ ^^^ '«■ ■t-xpress as a decimal : 34- ^2 lU ol ,.i be multiplied by y, of el,e square of 8f, and 't be added to the product, from what should the result H taken to produce 1-^^? 8. A 81 man worked 6| weeks, and. h than A of his earnin^rs 1 ad S?c 1 h^ 'Pf"* ^'5 "lore week? '^'^"in&s, J^ad )?45 Jeft: what did he earn a 20G Paper iv. '•3-7hrs.,timetogo2i„nJ. Going down th ^ «- - "Hies up. ^^-=-7=3 'jrs, time for, •7+3Xxohrs., whole time. '• Divide (to four ^^l^^^^Z~r~. 3. I save a jl S, a"?'"" " '^"'^^ '" ^ " "'""^ ;-y : What i:f/ra",t'r> "^"^ *^°- "ore .„a„ , g,,, 4* A man paves h;<5 j 5. A stream runs haU o t • of gold ? ■" P^rt of a grain would balancn • , , . , ^ °°oo7j ibs. 'ose ,„ case ol'd"' •"" '-'"'y. -orei^S^f/^^-j^-oe o« Divide c'7n 8. stiJl water : stream a:iU ur, making T, malci "g 075 and Jch, but kv many J gave 'ide at iuare ; n rf)\v going 5ibs. ance Id I gB i C 207 ^-I'/j.r/.v.iy/o.v />Anintf!. TAPER V. -Ex c \ ' n'l.lesan'i.'.rfa',', ™',';» !" '"?"■• «'/"'' »• who goes 8 will be overlalcen ' * "'"" ' ''"" '""g before U (B)\!^'rcE'r '" '"^ '"^^ "-d (A) and .hes^al, hand A ^ A goes 60 min. spaces in 60 n,in. ''6ot:i!'^~'-"--"ti.eL?„f SS min. spaces in 1. 3 4' 8 Sisr.S ■""=' "^ -"'P'^ed b, .0800S to give ,8536.4S,S From the sum of ♦ ]b 4» ^, ference between |. o^. and^fd^wt" ^'^ ^''^- '"^^ *^^^ ^if. Simplify -i. X W 44l_^ 36f S^i 196^ igi ' 7^ • J7™ac":s^;,te'iha7."i,t ,f™ ■" Manitoba; one had ■ ^ of the farn, : ^T^^^ ^i^:^^,^ 70o acres i.orc .b::; '■ re,;a?„d%r;^i,f b?|% ':,f---'=l'>y ? ^'f the s„„, , , find the sum. ^ ""'"" ">"" i°i "I the remainder on'at"ip^ohVo^nS^. Z^T^ ^'"'^ =' ' °'^'°ck p .Z starts from the san e p/ace , .'^^-'''S ^7 mf,es an h , V when and where win he former bi'"' P?V "''^'o^'' '-■«• • Div de Sua hpt,„. /- "^"^ ""- overtaken ? George ^/,f mctTha^^I^e^s^:^?' ^^^^-^-'g-ing ^ore ban the other two together '' ""^ "^'•^>' $So By selhngmy el.th at $1.26 a vard T • than I Jose by selliny seHin.i 800 ya^is at | r.^o ^ ^^rd ? "^^'^^ ^^^^ ^ 6. 208 I ^ I 1 i i PAPER VI. ■howl/ng ii'/n? lf^\3^ yards of fencing i„ 5, ,, require ' ol ^f ,, ^ amount of work r?r . PraperfracLn "■■!?' '' f"™'"' into a nr"^' "'"'^''y <^<"-r«- =■ Multiply ^o^"!,'"; ' +'■ =']■'"• to produce sandths! * "'' '5625 millionths bv ,fi„ . 3- If f of the sum .f ^^ ten-thou- be added to ,h.^':"'ain ""mber and ,-, , -"at is i':zsr' """"'"■ <'- rl' t\:i,rLe ""'• ■t- Divide $6000 between C » ^ r, '''• ^■•Ar.«ir;/dSpr^'"''^^"''°"'^^^^^^^^^^^^^^^^^ "B:;-eigJit tliousand. ('■) ; f.7. 2568,242 1 wo milhon, five hundred and sixtv-eiirht thousand, two hundred and forty-two ^ 19' I'lT^y^t. ^ T "^^^i'""',^*^ht thousand and three. L, l^^f^^', ^'"'^ hundred an J twelve rniHion, three forjy^ete^"' -venty-five thousand, six hundred and =0. 609,003,588. Six hundred and nine million, three thou- sand, five Imndred and eighty-eight ^"rcetliou- 21. 897,856,846. Eight hundred and ninety-seven million an^^for/^sif '"' '^'^^"" ^^^""^^^^^'' ^^'^'^ ^-^ -^' (/'') I. 9. Nine, 200. Two hundred. 60,002. Sixty thousand and two. 700,000,000. Seven hundred million 230,000,060. Twohundredandthirty million, and sixty 6. 81,501,007,012. Eighty-one billion, five hundred and one million, seven thousand and twelve 7. 30,000,000,000,603. Thirty trillion, six hundred and ^' 7;;°'°8o,ooo,ooo,ooo. Seven hundred trillion and eighty '- 'roiii^rrnd Sit' '''°"' ^^^-^^ ''^''-^^ ^-^^-^ 10. 15,000,018,207,000,081. Fifteen quadrillion, eighteen billion, two hundred and seven millic^n and eighty-one. I. 2. 3- 4- 5 (0 I. 60,701,892. Sixty mill sand, eight hundred and ninety-two n, seven hundred and one thou- Ir ''•VSirA'^.9. 3. 60 !;~^iiiiL;;fe' f^j-^^cu .„„ ,,.,„ '.OJO. Six } lundrctl t and one. loiisand. 4. 4Q 000 nrirt r- ^" ^^» moiisanc iion, SIX milJion 190,190 ooi.goo.'^One I i-'^d and ninety-tJirfe b,!- 'nindied and died. n.?:'^/;;r»-^-'..fivehu.fd.... rod. ninety niiJlion, one tl ^" l'nt',?f';68 One hundred loiisand, nine Juui- Jiundred and and sixty-three niiJ] 8. "'•Mwicu and ninetv-fni.r fi, 'vj-v'"^*; niujion s'xty-eight. ^ "' ^'><^"sand, five luindred ion, one and 10 1 1, 9' 5-000,204. p l.nnHr.^ ^^ iuindred and ninetv-r..nr fi,,, 12. 2 iiundred 12,000,012. Twelve miJJi f( 'nnr. ^y-four thousand lou- tnne .007,980,134. Two hill and ei ighty th 'on and twelve, lon.seven million, i'O I- 18,000. 2. 2,060,153. 3' 60,060,060,060. 4* 60,200,500. 5- 402,348,213,020. ^- 78,640,000,006,016. /• 6,542,000,025. 8- 6,542,000,025. W I- Jg. Nineteen, 2. 21. Twenty-one. 3- 10. Ten. 4- 45 • Forty-five. 5. 65. Sixty-five. 6. 64. Sixty-four. 7- 79- Seventy-nine. 8. 85. Eighty-five. 9- no. One hundred and ,7: ;^9. One hundred and 12. II ousand. one hundred and nine hundred tlnrty-four. 10, 9. 402,348,213,020. 5.008,949 II. 200,300,800. ^2- 29,599,000,601. 13- 4.000,558,240,07 ^4- 32,001,343,404 '|- 555.777.669, o. 16. 806 .070,005,206. ten. 00. 5n;;zs^"'"'"^^^-- 4. One hundred and fourt 13- 160. One hundred and ecu. SI xty. and seven AMnrE/is. y-tliree bij- 1(1 rtd. J'illion, one , nine Jmn- 'liJlion.one "fired and Rfty th ou- nir. sand, nine ■ Jnindrcd -four. :o. 70. ? >4 15- 16. 17. iH. 19. 20. 2r. 22. 23- 24. 25- 26. 27. 28. 29. 30. 31- 32. 33- 34. 190. 260. 35- 36. 37. 3«. 39. 41. 42. 43- 44- 45. 46. 47- 48, 49. 50- 51. 52. 53. 54 One lumdred and ninety i wo linndred and sixty 290. Two Juindred and ninety 629. Six hundred and twenty^nine. • « I. Eight hundred and eleven. ■ 950. Nino hundred and fifty 7^6!' Sev^'enV-six"""'' ^"'^ ''""^^^^ ^^^ ^'ftynine. Ifof ?"r.l ""'^"1- ^'^^^^ ^""^Jr^'- 15.749. Go. (<^) I- II loads. 2- 12 cents. 3- 12 (Jays. 4- 14 clieiTies. 5- iC apples. o- 14 slice]). 7- 13 dollars. ^' 14 trees, y. x6 cents. ^ '• 21 jilums. ^i- Mduckt-. (/^) r. 85. 2. 79. 3. 5«- 4. 58. 5. 89. 6. 364. 7. 893. ^- 7795- HXKficiSE 7.-P„«, .,1. *2. 19 cents. 13. 13 books. '4- j6 cents. ^5- 13 chickens. '^'- gappJes. t? Jetters. 12 fwoks. 23 dollars. »^i dollars. 2r) cents. '7 18. 19. 20. 21. II. 12. 13- ^5. 2435- 799. I ) »< ». 678. 4. 568. «• 984811 «• 28172. T. 2598. «• 2584. 9- SS89. lo. 77g,j, 997Q. 6759. 997 8- 8999- - <^f>47. f 't). 49gg, j EXERCISE 8._p,„^ 2,_ 9. 221C8. j 10. 164918. I • ^'37- '' 3161. 21286. 23940. 93545. 1245901, 197351. 22 23- 24. 25- 26. 27- 28. 29. 3<>- 3^ '4- • ^5 cents. ■ 17 l>irds. '7 cents. »5 J)anes. 16. »9 pears. 18. 26 cent.--'. 8898 acres. '7- -6999. 18. 77969. 19- 848699. 20. 148798. 21. 8199398. 22. 899969. 23- 9999999. 24- 3747655. '«. 196395. '■»• ^54970. '°- 2035342. 2- ^384473- 3- 145406. 4- 1670551. 3- ^93372. ^^- 45 23 18 1. 7- 96379092. (J and forty, and forfy- l1 ninety. 'ity-niiie. twenty-five f,''ity-t\v(). cents. )irds. 'ents. )anes. ears. AyiWKR.i. v.i 8 9 lo. 1 1. 12. IJ. 14. i , *, 16. iH. 19. 2 , i739i«J. 1347^^2, 9'i'J754- • '•■^2594'- • '0965339- • '5^4442- ■ '"^594353- 9339 1 9'^- 11178170. 10306156. 10670291. 4289 trees. $2844, 679 .sheej), $9212. 19 '4- 2r, 22. 23- 24- 25 26. 27- 2S. 1'53. *7549 3'>'9. ^'575- •"^3233 1. $11425 J 1K21. 75922. 29- 3394^6 n:en. 30. !*>i 36"? ,. 31- $764-' 32. $10766. 33- I13025. 34- $2ir4.$2935, $10897. 35. $141252. 3^- 37199. 37- 3935' niilr.s. 3«. 154^*91. 39ji374"i7''^- 40. J 3690009 J ), 4'- I. 16665, *• 444-. •• 53328. 42. 375540. 43- 6) diihlia.s. 44. 466 cents. 45. 131 cents. 46. $2538. 47- 96 years. 4S. 140 yards. snt.s. acres. h )8. 98. 9- 99- 55. EXERCISP] 10.— Pago ;;7, 1. 142. 2. 1512. 3. 6251. 4. 5"2i. 5. 1440. 0. 41 1 2. 7. 3201, . 8- 5310. 9. 2411. I '. 2412. II. 1120. '2. 3732. '3- 52221. M- ^52556. 15. 603054. 16. 864201 1 10. 17. 10035174. 18. 5162142. ig. 30640, 2 ). 57172. 21. 43242. 22 . 4421 1 23 . 32134. 24 • 56314. 25 . 33662. 26 24570. 27 22625. 28. ^3432. 29. 43220. 3^'- 12442, 31- 10853. 32. 25468. 33- 31064. 34. 6' 6453. 35- 64124. 36. 37001. 37- 205443. 3«. 601. 39- 35002. 40. 46150. .1 r. '7( >r^ ■* A 1 / P-. 42. 10644. 43" f2 marbles. 44. Ill qnai). 45- $441". 46. 213 busliels. 47. $211. 4^. 230 slieep. 49- $2560. 50. $1404. 51. $2453. 52. 4003 3'ards. 53. 42r3. 202 bushels. 6r. 1 1602 barrels. ^2. 14432 dollars. 63- 56io92dolJars VIH ^ysw£jis. m •*. f (a) I. 83. 2- 445. 3- 221. 4- 10S6. 5- 313. 6. 2384. 7- 3335- «. 24465. 9. 185173. 10. 1147. (^) I. 276. 2. 67. 3- 4004. 4- 664. 5- 33^6. ^- 34943. (*■) I. 409095. 2- 274850. 3- 707970092. 4- 9610. 5- 896. 6- 51306. 7- 60964492. *• 549913964. 9- 46924. ^°- 75757. 11. 56300. 12. 699901, 13- 5200. J4- 9688. ^5- 75999- ('/)t. 64349020. 2. 899450. 3- 1 1600800, 4- 25535001 1. 5- 1215. 6. 99999500. EXERCISE ll_p,., 4,^ 11. ^18. 12. 909. '3- 949. M- 3745. '5- 2757. lo- 41830. ^7. 581949. 18. 495829. '9- 383969. 20. 194959. 7. 5498. ^' 1 757 1. 9. 18654. 10. 23017. '^- 57921. 16. 90014. 17. 145129. 18. 254999. '9- 319527. 20. 663367. 21. 427165. 22. 587979. 23. 758451. 24. 82301 1. 25- 900829. 26. 6898220. 27. 19542. 28. 45961 1. 29- 31229. 30. 698. 7- 478. ^- 459079260. 9- 40020. i<^. 99002992. ir. 615. 12. 776546, 2r. 491978. 22. 261636. 23- 294928. 24. 680929. 25. 281939. 20. 490909. 27- 492929. 28. 391616. 29- 17359. 30- 432099. 12. 19238. '3- 37i86. H' 11T530. '5. 591203. 16. 666667. 31. 996. 32. 4598/2. 33- 44007. 34- 491693. 35- 62796. 36. 49S9050. 37- 1 700261. 38. 6634585, 39' 69994. 40. 8974088. 41. 287949, 42. 852642. 43- 899999. 44- 90100199. ^3- 34740. ^4- 3877. 15' 2092. 16. 401. '7. 4359999. iij. 1105. I. 175 sheep. 2. , 3- $7604. 4- $5210. 5- $9161. 6. 285866. 7- 3602. «. 30. 9. 47. 10. 30. 11. $1462. 12. 128. 13- 4794 miles. 14. 5862 pounds. 15- $221708. 16. 21834 acres. AysWERS. KXEKCISE 12.— Page 17. 48805 barrels 18. 19883. 19- 138094. 20- i3733 feet. 21. $8674. 22. 275 acres. 23- 94760000 miles. $6350. $3P^7- 487628 feet. 1808. $99635- $24354- 28153 votes. 4491 sq. miles. IX 24 25' 26. 27. 28. 29. 30- 31- 4;5. 32. 3571 feet. 33- 1769- 34- 85 3 ears. 35- 664. 36. no years. 37- 2984679. 38. 7925 feet. 39- 7812 feet. 40- 8712 feet. 41- 293 teet. 42. $13667. 43- 6326a'^res. 44- 28278 barrels. 45- John 32; James 90. (<0 I. 246. 2. 268. 3. 446. 4- 492. 5. 556. 6. 990. 7. 2624. ■ 8. 4344. 9- 7258- 10. 7570. 11. 8012. 12. 8616. 13. 284068. 14- 3412648. 15. 7229006. 16. 924356. 17. 3609186. 18. 64823505. 19. 280260. 20. 21 15258. 21. 4156984. 22. 871624. EXERCISE 14— Page 23. 2037468. 24. 2836195. 25. 36880812. 26. 23008960. 27- 5772463. 28. 5105520. 29. 30002504. 30. 1659000. 31. 46832. 32. 182452. 33- 103560. 34. 280210. 35- 42528. 36. 360918. 37' 161525. 33. 1156)4. 39" 400675. 40- 547476. 41. 86499. 42. 140760. 43- 206739. 44- 1 15604. 45- 314095. 46. 563136. 47. 1 13900. 48. 197930. 49- 244182. 50. 52092. 51. 206832. 52. 23504. 53. 59822. 54- 4518. 55' 1230125. 56. 4100832. 57- 4536. 58. 9738. 59. 20200. 60. 219138. 61. 21416. 62. 7846312. 63. 26^43, 64. 37704. 65. 39006. 66. 6424S. 4i 1 1\ (^7- 900867. 6.S. 9909537. ^^9- 1 03 2356 J ( 7"- y6(.;296cS. 7f. 8419829. 72, 92618119. 73- 99249624. 74- 37«73. 75- 16140. 7^' %5547- 77- i"666596. 7«. 6666660. 79- 119999988. 2;^- 44444444. "1- 69()go84_ J2. 5333328. "3- 576702. 84. 17S5. 2- $1104. 3. $178^3. 4- $43074. 5- 26400 feet. 6. S800 3 ards. 7- 162 dajs. S- 894 daj's, 9- 3680 bushels 10. |:7,5. , !!• !1P2024. 12. II34 CtS. 13. $8964. H- 3555 miles. .'l.V.S'iry.;/',^ P5. 69i6(\ 86. 1N71J, ^7- 353^)8. 88. 26243. 89- 1183,4. 9 '. 8760. 9^. 28140. 92. 10978. 93- 97608. 94- 8 '432. I 95- 98196. 96. 226668. I 97- 203236. I S^*^- 3' '65706. I 99. 8363091. I 100. 15561,2. ' ^"i-,5i22i64. I ro2. 7732530. 15. 420 men. 16. 21 120 ie(it, '7- 48785 yds. ^^- 47136 brls. ^9. 40859 shingles. 20. $27912. ( 21. $76113. 22. 79428 pounds, 23- 316800 inches ^4- 149136 miles. ' 25* $246736. 26. I24332. 27. $10344. («) r. 1 6 1840. 2. 149994. 3- 397714. 4- I 153362. 5. 727608. 6. 322992. 7- 334422198. EXEKCISE L5.-Page 61 8. 184752. 9- 343536. 10. 28434195. 11. 16799905881. ^2. 584720181340 13. 694417836 IS2 ' 103. I04. 105. ro6. 107. 108. I09. 33 7682 r. 7873096. 4384552. 104567,4. 1841625. ^0563<9r. ^^'^. 4748112. ^^1. 11122164 , ^'2.6842529.^ I ''3.733236. ' ''4- 359778;. ^15. 5202432. ''^- 4591644. 17. 555264,, 118. 3066574. ''9- 9585466. ^20. 9166404. 28. 36960 {QQi 29. 14080 yds.' 30. 980 pounds. 31. $700. 32. 445 CtS. II- $3572. 34. 3483 miJcs. ^5- $7156. 36. $679. "iT- 33804 quarts. 3». 1760 sq. rods. 39. 394992 pence 40. 4015 days. '4= 1521808704. ;5. 3529163131975. 16. 163016. 17. 2921005. 18. 879417. 19. 2869605. 20. 70362 i 2r. 32Q3 9130257 22. TI058. 23- 167999058S1 24- 1555/2. 25- 4575«- 26. 4159296. 27. 99800 1. 28. 707281. 29. I4I3008I. 3'>- 7977489. 3- 3769248. 32. 671 180. 33- 268056. ('j) I. 740. 2. 8690000. 3. 4698000. 7698400000. 131141615220. 6477150, 278170200. 3^915254- 9. 8;>,)g628o6. ^"- 4793554242. II. 36613538100. 3 161 7000. 73865000. I 13032 1000. ' 1 23 1 680000. I 10040000. I 17- 3469494* 'O. ! 18. 1686562S00. I 19. 4946796000. I 4. 5- 6. 7- 8. 12. 13. 14. 16. (c) I. 53207, 182688. 3628800. 202301. 36153036. 163536. 30026997000. 8. 461041. g. 7546^^09. 10. 2890000. 11. 12321795560. I. 2. 3- 4- 5- 6. 7. AysiVFus 34- 2165268. 35- 2879253. 36. 5169248. 37- 7062272. 3«. 7520415- 39- 23065974. 40. 34985162. 41- 42397406. I 42. 123614208 I 43- 19602. I 44. 2563912. 45. 131328. 20. 762294, 21. 8697821. 22. 38214. 23. 765870. 24. 6o(j236. 25- 7281711. 26. 3867349H. 27. 67312668. 28. 73818055. 29. 241768. 30. 5 1 18862. 31- 17902976. 32. 15403736. 33. 15704325. 34- 2082600. 35. 271541350. .36. 1508741097. 37. 1587862270. 38. 3654860576. 12. 628331. ^3- 195942000. 14- 104329. ^5- 5536076. 16. 328347675. 17- 594992. 18. 18810224. 19- 633259^- 20. 4903524000. 21. 502705700. 22. I41S516S00. 46. 230850. 47- 31812417. 48. 3379446. 49. 2420880. 50. 4040138. 51- 2738352. 52. 384134- 53- 2145594. 54- 146456 1 2. 55. 16003352. 56. 64421850. 39- 819S473608. 40. 982275037. 41- 336373'4i5- 42. 559616. 43- 257460. 44- 14988456. 45- 14925792. ( 46. 1 1 155248. \ 47- 182151828. 48. 148644288. 49- 1724573025. ) 50. 88287 I 65G6. 51- 601344. 52. 29784450. 53- 7364101944. 54. 10459827:5 .. 55- 118605858c. 56. 15920205. 23- 415143630. 24. 500 ; 6. 0. ; 1800; 470,' ; 2205; 872c C; 54010; 8080. 25- 725 cent.s. 26. 90 cent.s. '^7- 975 cents. 28. 60000 cents. 29. 3800 cents. 30. 228240 cents. f'^)-^ 4695 bush. \2l 6H.0 f 2- 432 panes. I'a, ,?f S' ^^^^' h^- 4- -^^^797 dollars. 1 2fi^o«-^°- 5- 1494 dollars, i'f iTlr* •, 6- 38988dollars 1 26 Itf '"'^*'^'- 7- « 135 potatoes.;' 37 tso^'f «• ^547 days. Ls f^^ ^^et. 9. 1264958^ 1^^- 79000 pounds. ,^. ^2. $7640136. ^3- f Jf ^ ^^^eep. I,, 13- 202640 lbs o^ '37970 miles. r2 M. $204^6. • 33. 32166 dollars. If, 15. l4497r l^"^- 29730 dollars, rf rr, r!975. f35. 541112 dollars, ct' /3&. 492729 dol'ars. cfi 5451 dollars. HG89 dollars. '^'- '^^2155^. (^5. 541112 dollars. 55 V7- ;04i6osheets.K7- ^92729 dol'.rs. j|, -.§299720. ^^l^^J^Pts. 57, EXERCISE 10.-P,ge 60. -J. 21. 3. 32. 4- 33- 5- 20. 6. o. 7- 26. 8. 20. 9- r8. 10. 10. 734481 dollars, r. 228984 njen. • 273249 yards. • 980019 pounds. • 372480 plants. ^358112 letter?. 349S60 pcmnds.* 03360 feet. 40824 plums. 89232 rails. 10348 soldiers. 108 miles. ^4712 men. 462160 feet. 595680000 miles 291214 cents. ^ 68400 cents. 1- 312. 2- 431. 3- 342. 4- 132. 5- 231. 6. 212. 7- 121. EXERCISE 19.--p,ge 82. '|- '53. ^6. 252. 17- 232. 18. 142. ^9. 131. 20. 121. 21. 4623. 5451 doJJars. i46«9 dolJars. 734481 dollars. 228984 njen. 273249 yards. 980019 pounds. 372480 plants. ^358112 letterv. 349860 pounds.' 53360 feet. L0824 plums. 9232 rails. 0348 soldiers. 38 miles. 4712 men. 32160 feet. 15680000 mi] 'js •1214 cents. 400 cents. 41. ^ ?7o. )2. roo. 4- 14. Co. B. )• 22. 34. 23. 44- 24- 43- 25. 42. 25. 52. 27. 62. 28. 42. 29- 71. 30. 92. 31- 42. 32. 51. 33- 2032. 34- 2042. I- 913—2. 2. i45i6_4. 3. 2S823— 2. 4. 5786—3. 5- 52954— 3- 6. 82304—6. 7. 30411— 2. 8. 79965—4. 9- 339758—1. 10. 877022—3. 11. 835581-1. 1.2. 560054--6. 13. I 0465 I 8 — 2. H- 485109—3. 15- 4^243—5. 16. 124715—3. ^^.v.s•|^A^^^y. 35- 1051. 36- 1052. 37- 1054. 38. 252. 39- 489. 40. 218. 41. 324. 42. 224. 43- 135- 44. 155. 45- 105. 49- 2052. 47- 2025. EXERCISE 20.— Pago 83. XI ii ^7 18 19 20. 21. 22. 23 814744—6. 772927—4. 41 1869 — 2. 561436—6. 680078—1. 731040 — 4. 5^8129—5. 24. 844484. 25. 568345—5. 537720—5. 822311 — 5. 384597—2. 710443. 589684—9. 1 15993. 26. 27 28, 29, 30. 31- 48. 1507. 49- 4686. 50. 1237. 51' 1213. 52. 1207. 53- 1 1 09. 54- 10856. 55- 3439. 55. 5416. 57- 7432966. 58. 4105. 5J- 14469. 6 >. 5092103. 32. 944202— ]0. 33- 3749195—9- 34- 192850—5. 35- 297691—3. 36. 389751— g. 37- 779108—10. 38. 490939—2. 39- 9007 times. 40. 7008 times. 4^- 6703, 42. 6324. 43- 8232 times. 44- 5531. 45- 4205 times. 46. 6132 times. 1. 731—6. 2. 8317—4. 3- 61—92. 4- 73—1. 5. 9—7312. 6. 839-16. 7- 5137—12. EXERCISE 21.-Page 85. 8. 7—12934. 9. 3—92. 10. 37—214. "• 74—321. 12. 306. 13. 30000. 14. fio — 60600. 15. $108.62. »6. $3.12. 17. $4610. 18. 70 cents. 19. 8610 billr. 20. 73107 dollars. xiv ^ v. n'j-:/:s. ■5 cent?, ^y b'lttons ^59 Y's; 053 ^'s. "^6 dollars. r/ dollars. ^ dollars. '9 coats. 3 barrels. 7 barrels. 41 months. 5 baskets. ) dollars. I- bushels. 6 dollars, miles. 5 dollars, gallons. 30000 miles. 384-2534. 58. 59- 60. Gi. 3645—2867. 46^9. 346. 19214—542. XV O2. 34045—1098. ^3- 64. 1343— 1652. 12207—445. 529—2228. I. 2. 3- 4- 5- 6. 7- 8. 9- lo. ir. 31207— I. 16695—17. 519225— II, 20914— II. 167988—17. 33721—18. 2S3945— 18. 17529—37. 128795—17. 145264—23. 105911—25. 12. 111263 — 21. 13- 111215 — 27. 14- 13^^136—25. ^S. $258. 16. 86. ASSWErvS. Co. 9830291-7000. 67. 5006284. 68. 4000059. 69- 5748362. ^/usu— ,,, 70.2779458-5888.1 7I ;4^5'' ^ij- 71.6438192-73401.1 ^9. 28?So-!;;2 72. .378_6926. I I^. yefei%tl9s: 74. 7973—23. 75- 927—80. 76. 34969— 716. 77 27959—333. EXEKCISE 24._Page 95. 17. 213. 18. $864. 19. 436—1. 20. 83—49. 21. 76—30. 22. 7—491. ^3' $8600. 24. 31—29^6. 25. ^697—2. 327—311. 386-58. 1316. 21—320. 10 — 11,27. 179—452. 26. 27 28. 29. 3 '• 31. 32 33 34. 35- 36. 37- 38. 39. 40. 41. 42. 43- 44- 45- 46. • 243553—7. • '5- ■ $115. 13— 9112. n— 91800. 3463—42. 5—6512. $46. $10000. $72. $360. $8.46. $7500.62. $812.43. $481.99. i^XEiXisK i:;^.-r..c co. ^37- 7- 6. $97- ■ $25. ■ $8. ■ ■ 3-5 P'l^'c.i. 74 ('-ys. $35. . $4- 8. $237. 9- 31^> hams. ^■^- 365 questions. II. 2765 days. I. 2. 3- 4- 5- 6. 7- 12. 83 years. 13- $237. 14. 3864. 15. 77 battalions t6. 646. 17. 1872. 18. 894 pounds ; 312 pounds. 19- 23 hhds. 2.). $13. 21. 24 trips; 71. 22. $197. 23- 425 miles. 24- 131. 25. 47cartri(!."-cs. 26. 5280 feet." 27. 24964. 1097 bales. 23 pounds. 256 balls. 39''625o. 792. 28 ■29. 50. 31. 32. XVI .1 ys ivj-jjts. I- $10394. 2. $825. 3- Houses; $167=: 4. 1428. 5- 354. 177. 118. 6. 207, 7- 3')6. 8. 156 times. 9- $2015. 10. 146299 — I n. $1663. 12. 4320 sheets, 13- $182. 14. $2028.78. ^5' 132 men. 16. 75 yards. 17. 897. 18. $15444. 19- 91. 20. 119 weeks. 21. 723 gallons. 22. $5. 23.^66 pounds. l^XERCLSK iJG.^Pago 'JU. 24- 792. '■5' 9*277. !6. 24. 7- 17 books. 8. 119 miles; 595 miles. 9- $1998. 0. $25175. 1. 781 cases. z. $g. ■• $1704. • $36. • 39- . *i5. • 5133. ■ 36405 bush 162. $3-6o. 33 days. $336. $465.50. 68 bush. 6. 8 oranges. 7- 9 Jiours. ^' 12 days. )• 5 weeks. '• 15 days. • 37- • 217 Jiats. • Aowesli; $212. ■ $11080. - - ' $3' 56. §2277. 57' $6750. 5^' $8. 59. $3. 6>). 15. 61. 20 hours. 62. $18.62. 63- 24 months. 64- 31 calves. 65. $70. 66. 3 tons. 67. 21 yards. EXERCISE 27.-Pago 104. 1. 20. 2. 42. 3- 24. 4. 96. 5- 34. 6. 24. 7- 36. 1. 23. 2. 4. 3. 61. 4- 192. 8. 18. 9- 5^- 10. 64. 11. 21. 12. 10. 13. 3- 14- 4. 15- 20. .16. 10. 17. 4- 18. 17. 19. 8. 20. o. EXERCISE 2a-Page 110. 5- I. 6- 37- 7- 2. 8. 21. 9- 32. 10. 17. ir. 126. J 2. 14 feet. 1. go. 2. 360. 3. 12600. 4- 1134- 5- 504- 6. 360. 7- 20160. «. 480. 9. 438480. KXKUCISK lil.-Pagc 115. 10. 43680. ir. 1330732. 12. 9744. ^3- 155232. 14- 72720. 15- 469170. 16. 4149360. 17- 441000. xvii iS. r)S64o. '9- ^591744- 20. 4200, 210, 168, 140, 120, 105. 300, 150, 100, 75> 60, 30, 15. 6, 3. 21 KXERCISE n:).-Pacro 121. CO 2. __: 12 w ' 1 a"> 10 -"_• -lA 76~' 3 2 1 ro 1. 4 5 » 6 > V 2. i 8 ' a ' a"' 7 ' 1 T' T' 7. _3 1 ;. ('0 I- 4:^ 2. 7. 3- 7- 4- 6A. 5. 3f 6. 5|. 7. 5i. 8- 4|. 9. 6f 10. 7^. II- 7f (<^) I. V- 2. V- 1 B ?o ■J 2 ( _"_ : i (I • 1 + 2 > 2(7> ■^O' *0 ' 3U' 4 J- i-1 J. 1 10' ' To 1 1 0- 1 U ' 1 o O > 9"! 8. ^ 3 ' f > Ti- l.V! 1 7 9- 10. ir. 1 9 > ■i +• 1:XEKCISE a4.-Pafeo IL 12. 15. .4- 16. 21. 17. 3|. 1 8- 7i- 19- 4i- o. 6-1 I- 4f ". 4t'o 6. V. 9. V. 10. y 23. 6. 24. 9i. 25- 9^ 26. ,21. 27- 7- 28. 18. 29- 7^- 30. 7h 31- 81. 32. 5. 33- 4^ ri. V- 12. '?», 14. "LP 1 3 > 3 ' ry- 3 3' 3 ' 1 -t • 3 ' 2 1 "3 • 34 • Ci 35 4|. 36. 6. 37- 4. 38. Hh 39. Hh 40, '7h 41. i6|. 42. 28ii 43. 3HI 44- 61-2-9 47 8 16. «.«. 1 1* 17. 7 6 T • 18. V. 19. ■^§^. 20. 8.0. xv: j ^ysTri:s. 22. y. 23. y. 24- y. 25. V. 26. (^) I. 2. n » 12 da3-s; 5 days. 6 ounces; II ounces; 9 ounces. 28. 29. 4>. 30. *p. 31. 33. V 34. J.1 ) 5,« 8 32. 35. VV'. 36. Vt". 37- H"^. 4- 5- G. 7. 8. 9. 40 hoys; $ « 7 5 s • 7 B^ • t 1 T' t 'i . T > 1 *• EXERCISE C5.-Pa 10. II. 12. 13- 14. '5- r?) I. 1 Ji ■f 2 > Tj. 2. 3 J.a ii o» To' 3- n 6 > 4^ ^0 ' 4. .■1 *5"' li 7 7 T2> — 7_ 1! ♦ 4 0» n "1 J' n '2-5' _2 So"' 2 39. ^V-"- 40. L'yi.i. 41. 9-y-*. 42. tnpnj, u VV; vv 1_» 4 34 • 3_4 .l 4 ij • BUS „ 00 • ifl^; £4_4. jsna Al 4 •* .10 12G. (^) I. C has most; A least. 2. Cmost; D least. 3- Charles; Henry; John; '^' i^i^'"l-^°"&est; Walk, iiig— shortest. James. I: j^K:..'^:'. .Vf v« feet. 1 '-L7_a .iiao ><{ 9.8 17 EXERCISE 37.-Page 130. 1J8 0> Vastf* I /3 3- 2 J-30 3 • ^4f 4 • 4-i. 5 3^ 6. 6^1- "2 H' 7- 4'15-. 8. 36t"4. r. 57tV- 16. 22,%. II. 393V. 1^2. 28iX "oo- 13- 31^1:. H- ^Hh 15. 16. 17. 18. ig. 20. 21. 22. 2-'- 2 S- 8-x_ "1 35- -■''_•_ 2 4 0* :2 3 5i-% acres. 13/0 yards. $i4H. S\ pounds. 23- 24|gaJ]ons. 24. $208^i-. 25- I If 26. i6if miles. 27- 7 1 2i yards. 2S 42, feci. 29. 31- 32. 33- 34. 35- 36. 37- 38. 39 40 41. ^H yards. f96A. f29^ $19,^. ^6 3* 182^. 'OI4H. r49TV. 13t^. 88i. 201H Ihs. 42. Ji 40. 39. \2, a.V5I»'i7JS. 1 • " » I 9 . T » I s • o > ff 1_» 4 3 +J» • 4 B ' a • o' i2(y 5. irW- t; WaJk- U feet. yards. !• Li 1 3" lbs. I. 2. 3- 4- 5- 6. 7- 8. 9. (1 28. 3|. 238^ I'XKIICISE 40.-IVtco lli?. xix i6i cts. 1 1. 30^, fi.36f 40 miles. $224. lu. II. 12. 13- 14. 17. 38^ cords. 18. $31.25. >io/^. 19. 20. 21. 22. 23- 24. 25- 26. if>i5.95. 1953 cords. $10.60^ t 'P5734- 42 knots. EXERCISE 41.-Pago MO. T. 2. 3- 4. 5. 6. 7- 8. 9. 10. T* 3 ?• 4 T' a 3' 2| times 8 times, 6. 3h f times. 1. 2. 3. 4- 5. 6. _9 20' 3 s- ♦ a' t'V. ^2. .,3H yards. 13' 4'J hours. 14. 368 times. lb. 3Q^ pounds. 1 7- 1 2^j hours. 18. 50 scarfs. 19- 44 bottles; TT pint. EXERCISE 42.-Page 142. 20. ^J^ mile. 2^ 55 pieces. 22. 9t''t; 26A ; ^9tv pounds. 23. 6A months. 24. 8 days. 25. 400 weeks. 26. /^ mile. 7- h 11- 7t'o- 12- I3tV "49' I T 8 2. -"J- 504* a T* 9 9' 1 y 7. t'V. 8- U Pk. EXERCISE 43.-Pajro 144 9- 80 lbs. 3- 4. 5. 10. i9|. 11. 1 140. 12. $30. 13. 32 ounces. M- fsmh 15. $308. 16. 3 times. 17. 18. 19. 20. 21. 22. 23- 24. 24. $15. $35- 5 times. 1 2 years. a u 1 XX ;r:ii ■j< $100. 1000. ■5,1' t't- 1 1 S' 25. 26. 27. 28. 29. ^\. 30. J. '• f ^74-55. 2. 1149.35. 3- §'3i5-22. 4- $214.96. 5- $1370.68. 6. $i3i^<.95. 7. f2375.8s. o. vJ,V.VH'/;a'5. HI (lays. 5 (lays. 12.J Jiours. 1 3 i- days, u liay. 6 days. 37- 38. 39- 40. 41. 42. 13-i- mit). 24o- IIIH). $^912.50. Co feet. 150 i'eet. EXERCISE 44.-Pa^rc 150. $6516.03 $1099.52. $830.10. $435-'>9- $384.87. 13- $235-93. H- $32.57- 15- $197.38. 9. 10. II. 12. 16. $9793.61. 17- $8838.81. 18. $28.47. 19- $135- 20. $197.28. 21. $784. 22. $51975. 23.. $1564.50. 24- $1804. 25. $125.12. 26. $25839. 27. 86 times. 28. 864 books. 29- 7630 times. 30. $3.16. 31. i6cts. 32. 581 months. 33- 15 cts. 34. 105 pounds. 35- $35845.35. 30. $92.85. 37' $129.05. 38. $12.12. 39" $530-60. 40- $48.60; goc. 41- 197 lots. 42. $60.33. 43. ^53-55' 44. 32800 pieces. 45. $7-52. EXERCISE 45.-Pa.o 155. I. 58; 1092 far. 2- 56i6d.; i3435d. 3- 724695 far. 4- £3231 ^478. 5. I48i4far. 6. £22 5s. 2d. 7. £3 IIS. 2d. 8. 3820 far. 9. 203 far. 10. 20163 far. II- 1. A'22 3s. id. 2. £i7 6s. 5^d. »• ^39 IIS. 8fd. '- .£3001 los. I ,d. [12. 2 1 133 sixpences. I 13- £46 I2S. 3d. 14- 316 boys ; £3 OS. I id. EXERCISE 4(5. -Pago 157. I 2 3 4. 5. £27 IIS. 6d. £18 17s. 8|d. £63 i6s. 2d. £66 2s. lid. _£92H 83. 9d. I 6. ^4825 5s. g3_,l. 7. £99153 lOS. 2i-d. o. £66807 9s. t^d. 9- £55862 OS. 4d' 10. £'50650 6s. J^d ■5. i3i min. 24^ nun. $1912.50. Co feet. 150 i'ect. ' cts. I nionths. cts. 5 pounds. 5«45-35. 2.85. 29.05. 2.12. }o.6o. *-Co; goc. lots. ■33- ■55- 10 pieces. 12. . 8fd. OS. 1 >d. ^peaces. 3d. > » id. .1 ysn'Kna. KXEIICISK 47.-Pii.'o 158. kil t. £8 17.^5(1. 2. i'9 Hs. lod. 3- -^'94 15s. 5id. I. Cia'igs. lold. 2- -37 13s- 5d" 3. i'e IS. 2fd. 4. ,1'449 13s. gd. 5. i'144 7s. 4d. 6. i'i5cj 15s. 2jd. k. £14 gs. SJd. 2. £154 IS. gd. 3. £149 17s. I id. 4- i'73 17s. 8id. 5. £3 19s. 7irs.42«min.; 17?- min. to 4 RM. 3". 4fVmm. to 2 P M 3i- 43hrs. 47min. oi«ser 32. Saturday.atio.aoX!M: (^) »6grs. • 2of ^ grs. - Jbs. ; '9§U lbs. 2ac. 20 sq. is. 5 sq.ft. rds. 4 sq. l8 sq. in. rds. rds. q rds. min. S sec. > min. 45 min. hrs. o sec. I? sec. 9- to. 12. (13- 14. >5- 1 6. »7- 18. 19. EXERCISE 5a— Pago 180. xxiii 480 dozen, 1091/^. 55 pints. 43 busli, I pk. 3 qts. 3 hhds. 8 gal. 2 qts. i pt. I90.56, ^ ^ 364 gals, l5n.87f 4 loads. 20. I3.80. 21. $1.05. 22. -J. 23' 30 gals. i 24. 141 qts. 25. 36 barrels ; 432 dozen. 26. i69f bush. 27. 922^«j. bush. 28. 171 bush. 3 qts. 4X gals. 3 qts. 1 pt. EXERCISE 5i. 7- 2450 score. 1 8. 62500 boxes, 9. $720, ) ro. 80 brls. ; $47040. ir. 1152 books. 12. 300 stones. 13- 115 brls.; $i8n7.co 14. h , EXERCISE 55. 29. J 5* (/'^l in 20 (") '■ 2%\', xh^; I2i; 68tV I 3 65'046 ch. 549-375 ac. •0003764. 5-6857 ac. 5-937 lbs. 99-96154 miles. •850955- 30. 2045>,- barrels. ■Page 182. 15- 4i6| doz. 16. A • _?- • ♦ 2 ' 10 > T' 17. #13.20. ^8. 83^ reams. 19. $12. 20. $2i3.i6i|. 21. 2970 volumes. 22. 18-5 doz.; $726; 2ac. -Pago 180. J • 7 8 ' T6 ' I. 2. 3- 4- 5- 6. 8H. *0 ' IS' 8. 9- 10. II. 12. 13- $108-675. 391-05 ft. •08 '^5; 3650; 10020. 4-9834; 248266; -6344. •04026; -01274; '00094. (a) I. 2. 3- 4- 5- 6. •625. •0625. •85. •52. •35. 2-625. EXERCISE 56.-Pago 193. 7. '024. 8. -0390625. 9. 24-6. 10. -484375. 11. -1 1584. 12. 1-5008. 13- -027. 14. -083. 15' \i23. 16. -238095. 17. -624. !«• •3iy. XXIV AyswEns. {f>) M' 2. -A, 3- 4- nun iTo w • _8 iT* _« sf 7 5- 6. 13 7 5" y 2 8 • 9_ 8 -*A. lO. A luaTTo' k) I. 18-56. 2. °6. 3. -123. 4. 8. 5- 6. 772. 7-3 EXERCISE 57.— Page 195. 13- $239.25; $336. 14. loid ; ^V gal.; I yd.; $21 15- 33*%; 100%; 50%. 16. A'50 2s. id.; ^6200 8s. 4(1. 17. $103.10; $824.80. 18. 50%. 19. 28%. 20. 261%, 21. $110, 22. 50%, 25%. $30000. - 19845- 26. 60 CtS. 20. .2.87i. 600 lbs. I St year ; 720 lbs. 2nd year. Wife and son, $480 each; daughter, $360. 23 24 25 27. 28. 29. 30- 1. 9 ft. 2. 15^ ft. 3- I ft. 1. $2750.53. 2. $843. 3. £189. 4. $650.10. EXERCISE 58.— Page 200. 4. 60 sq. yds.; 10 ff. 5. 24 ft. 3 in. 6. $166.88. EXERCISE 59.— Page C02. 7. 34 in. 8. 12 ft. 9. $9. 5. $13.84. 6. $832.70. 7. $7981.20. 2, 3 _1 1 1 OOOO' 772. 7-3 ear ; ►'ear. , $480 each ; 5o. 7. 31^ in. 8. 12 ft. 9. $9. o. jiSSWERS. EXAMINATION PAPERS,-Pago 203. XXV o "9 ^' rg 3. $224. 4- loJ'gCts. PAPER 1, 5- 17- 6. $1200. 7. •?^^3- I. 1854. 2. 14839, 3. CXIV; XMCMLXXXIII. 4. 2880. 1. i97"32S. 2. Soo ; 200. 1. '0122. 2. 32|- hrs. 3. I90.70, .]. I68.60. 3- 4- 5. PAPER 11. 5. $2000. 6. 6o-. $2.50. 7 7 gals. TAPER III. Sift. 3i lirs. PAPER IV. 5- 4 his. 6- -432. 7- -f. PAPER V. 8. 288 ac. 9- $3561; $4022. lo. $89.11. 8. A $360; B $480. g. 3 times, 10. $18333^. 11. 282 ac. 12. $550. 6. 21 bris. 7* 7i40* 8. $16. 8. A 315 ac. B 127^ ac. C 127I ac. 1. 606100 gals. 2. I lb. 3 oz. 8 dwt. 2 1 grs. J" 1 !• 1. 4840 sq. yds.; 13' 2- f>-S3055' 3. 196. 4- 3410 ac; 3940 ac. 5- I246. 6. 8 min. to 11 p.m.; 2461 miles. PAPER VI. 4. C $1050; DiR 495°. 5. 2i hrs. 6. $4.80. 7. Henry $31- George $130; Fred $100. 8. $240, 7. A $75; B ^ i 1 I. 8. 16 men. 9. 100 gals.