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S. HALL, QUEEN STREET. •RS7. Education Office, Province of New Brunswick. _ Tlie Board of Education has prescribed "Hall's Eenicntary Arithn.ctic' as a text book for use i^ the fechools of this Province. WILLIAM CROCKET, ^^'^V Superiutendent of Education. Entered according to Act of Parliament of Canada, in the year 1887, ByM. 8. Hall, In ll.c (ffcc cf tit M iiiati if ^£,i:\iiltiic iX Cltiva. PIIEFACE. dck. "Hall's 3e in the ion. ada, ill lltva, The subject matter of this Elementary Arithmetic lias been arranged on the following plan :— Arithmetic of whole numbers, Arithmetic of Fractions, Business Arithmetic. In dealing with each topic the inductive method has been followed as far as possible. Oral exercises, developing the idea of the operation to be taught, precede the rule, thus leading the ])upil to frame a rule for himself. When the rule has thus been reached, numerous and varied problems are furnished for exercise. Review Exercises, which niay be used as Examination Papers, are appended to such chapters as seemed to require it. It is believed that the exercises are more practical and better adapted to the wants of our scIkjoIs than those of any other elementary Avork of a similar character. See Review Exercises throughout the book. The Chapters under the head of Business Arithmetic have been specially framed to meet the requirements of the Schools of this Province and to f urnisli some prepara- tion for actual business life. See Cliapters 8 and !J. While the Unitary method has not been made the prominent feature of the book, its importfince and utility have not been overlooked, but it was not deemed advisable to omit the subject of Proportion, whicli, in itself, supplies a most valuable means of mental discipline. A Chapter on the Metric System and its applications has been inserted. Fredericton, N. B. I ELEMENTARY ARITHMEi'IC. CHAPTER 1. r llcliuitious. 1 . A Unit is a single thing, or a group of single things considered as one whole ; as one book, one pen, one ton, cne hundred. 2. A Number is a unit or collection of units j as ono desk, threo desks. 3. Numbers are expressed by certain Signs or FigureSi each of which has a value of its own ; as one, two. three, four, five. six, seveyi, eight, nine. 1 2 3 4 6 6 7 8 9 Another figure 0, called the cipher, or zero, or naught, has no value of its own. 4. The science by means of which we reason about nninbers expressed in figures is called Arithmetic. 5. From the principles of tliis science we draw certain EULES for performing operations on numbers, so that Arithmetic is both a Scibnck and an Art. As a Science it has to do witli wh;it we Icnuw about numbers ; as an Art, with what we can do with ihtm. 6. A Concrete Number is a number applied to objects that arc named ; as 3 bof/s, 5 yards^ 10 horses. T. An Abstract Number is a number not applied to any object ; as 4, 7, 9. 8, Numbei's Hit> ot the same kind when they ha^e tho same uut ; as 2 trees, (J ttets, 8 Ines. TJiP. .*iiAiJic sv.srn.'j. Exor<'i»iC 1. 1. Write the figures coiTespoiiflinrr to three, eight, siXy five, nine, seven, ami tlie words corres]>ondiiig to the figures 1, 6, 5, 2, 4, 9. 2. How many units in S pounds, 3 tons, 5 apples, *l slates i 3. What is tlie unit in 4 oranges, six marbles, cents ? 4. Of tlie following numbers which are abstract, and which concrete : o gjillons, 5, 9, bus)iels, 4 men, 1, 8 dollars. 1 acre ! 6. Select the numbers of the same kind in the following : 6 miles, 2 boys, l"» marbles, 9 miles, 4 d«»llais, 7 marbles, 6, 4 boys, 8, 1 dollar, 3 miles, 6 dollars. NOTATION AND NUMERATION*. 9. Notation is the art of writing in ligures or letters a number ex})ressed in words. 10. Kumeration means the reading in words of aciv number e\i)ressed in figures or letters. 11. T'.vo systems of notation are in common use, the ^jabic which uses tigures, and the Iloman which uses letters, for the expression of numbers. TIio ArnUic Sjslc^iii. 12. This system C)f nLitation represent'^, numbers by means of the iiiuv significant rig- ires 1, 2. o. 4. 5, G. 7, 0, 9 and the cijihor 0, or by c. i-lining them ia various ways. 13. Its basis is the iiumber 10 regarded ;\.s a grouj> made Tip of ten uni; . 'i\-u groups of tcii maicc one hundred, 100, which may be c<»ns.dered as one huuilrcil unit^. as ton gi'oups of ten unit'<, or as a single grouit «.)i one liundred units. 14. Starting with one unit, the Arabic System builds units into groups i^i ten, tens into grou]<^ «>f one hundred^ Inmdreds into thousands, thousands int<> tei:- of thousands, T.riis of thousands 111^:0 hvnd/eds of thousands, huuureus of thousands iiit. millions, and so on. 15. Eacli wlioic number less than 10 is expressed by ita own .simiiticaur U'lrure. and every tigure wlu-thtT bijiiiticant or not, wriiTiT, to the right of r-uother signiticant figure, incrcasi'i; liu' \.duc of the tirst written ten-fold. Thus 1 alone v .-.lu- 1 unit If 2 is written ^ifte 1 we li.we 12, in TBDB ARABIC SYSTEM. which number the value of the 1 is increased ten-fold by moving it one place fartlier to the left. 12 may be read, one ten, two units or 12 units. Itt. All numbers between ten and one hundred may be- similarly written. Suppose, for example, we wish to express the number fifty-six by means of figures. Let us first consider how we would write fifty. If we write 5 wo express 5 units. If now we wish to write fifty, we move the significant figure 5 one place to the left by annexing a cipher. We have thus increased its value ten-fold so that it is now Jice tens or 50. But we were required to write not fifty but fifty-six. If instead of the cipher we write the figure after the 5, we increase the value of the 5 ten-fold as before, but our expression will now be read, instead of 5 tens, units, 5 tens, G units or 5G, which ia tlie number we wislied to express. In tlie same way all numbers between ten and on© hundred may be represented. Exercise 2. Write in figures : 1. Six, eight, nine, seven. 2. Forty-two, seventy-three, sixty -nine, thirty, eighteen, eightv. o. Twelve, fifty-seven, t\\Tnty, ninety-one. 4. Eiovon. eighty-six, nineteen, thirty-seven. 5. A\ rite as a single number three tens and seven units ; seven te;is ; one ten eight units ; eight tens four units ; two tens nine units, five tens. AVritc in woixls the numbers represented by the following figures : 6. 7. 8. •>. :o. 17. All In umbers between one hundred and one thousand may be represented by extending the principles contained in Art, 15 to tlie writing of hundreds, tens and units instead of tens and units. Suppose we wish to represent the number six hundred and seventy-two. Six hundred may be expressed by first writing the figure 6 and increasing 6. 8, IG, 34, 18, 75, 69. 7. 90, 30, 50, 42, 68, 11. 8. <>, 39, 9G, 85, 22, 15. •>. 13, 72, 5G, 33, 61, 10. 10. 43, 19, 78, 52, 26, 87. THE AJULBIO SYSTEM. It* ralue one hundred-fold by annexing two ciphers, thm 600. Now if instead of two ciphers we annex to the 6, the figures by which we would express seventy-two, w« have 672 instead of GOO. This number is therefor* made up of six groups of one hundred each, seven groupi •f ten eiich, and two units, making in all (>72 units. Take another case. Su]>pose we wish to express two hundred and nine units. We write the figure 2, move it to the place of hundreds by annexing two ciphers. Now, iince we are required std by six figures, such as 1234.50, and divide it into periods of three figures each : thus, 123 ! 450. TJie highest place held by a significant figure is the sijcth from the right or the place of hundreds of thousands, thus : 1 i ■ I THE ARABIC SYHTEM. T r — - — 2nd PERIOD. I8t. PERIOD , • 1 \ Hundreds of Thonaandu 1 ir 1 Tons of Thouaauda 2 ThourtandH 3 lluiidrods 4 Tens 5 Units 6 \ Tliousanda. " ^ Units, 19. Tlie same piincij)]o may 1)0 extended to numbora •oiisistiiig of liny nuuibor of ri;4urua, thus : Period V. Trillions, -T Ilnndrods of Trillions, oo 'i'ens of Trillioii.3, w Trillions. «< «• r a Hundreds of Billions, IV. Billions J to Tonn of Billions, [ ^ Bllli"n=^. ( -1 riiiu.lruds of IMillions, HI. Millions co Tons of IMillion.s, M 41 II. Thousands. . . . o Mjllions. ri^ Hundreds of Thousands, 0( 'J'eiis of Tliousailus, cr. Thousands. f w iluudreda, I. Units - 10 Tens, [ M Units. 20. The names c;iven to the periods in order above Trillions are : Quadrillions, Quinfcillions, Sextillions, t^'c, but ordinaiy arithmetic has little to du with such large numbers. Express in figures the followijig numbers : 1. Four thousand two h uTKJied iuid lifty-six : six tlujusand three hundred and seventy-three ; three thouf.and four hundred and sixty. 2. Two thousand and t-.vo ; eight thousand nino hundred and nineteen ;' seven tlu^up.aud and ooe. 3. Twenty-eight thcu.sand and twelve ; two bundled and tw-Q thousand two hundi-ed and S\o ; one hundred and nine thousand and six. 10 HOyiAIf NOTATIOir 4. Six millions, thirty-eighfc thousand and sixteen ? fourteen millions, eight thousand and ten ; three hundred, and seven millions, two hundretl and five thousand and eighteen ; nine hundred and eighty-seven millions, three liundred and sixty-eight t'liousand, four hundred and thirty-eigh^ 5. Thirteen billions, seven hundred and forty-eight millions, live liuadred and sixty-nine thousand, eight liundred and fifty-seven ; twenty billions, two hundred thousand and twenty. 0. Ninety-three trillions, six billions, fourteen millions and eighty-six ; sixty-sLx trillions, sixty-six millions, sixty-six thousand and sixty-six ; seven hundred and sixty-eight trillions, nine hundred and lifty-tliree million's, four hundred and seventy-two thousand, one hundred and ■izteen. 7. (^ne liundred and one billions, eleven millions, tsn thousand and one ; sixty billions, and fc vty-eight hundred ; eighty trillions, eight thousand and eight. AVrite in worda tlie followhig numbers : 8. SXIIG : :il0103 ; 0913864. 9. iOlOOO : 129G0018 ; SlOlO.'i. 10. • 5000S(>3040 : 3930l>410805. EOMAN NOTATION. 21. Tlio Roman Notation expresses numbers by meana of seven lebr,ers. Thus : Letters,— I V X L C D M Vah.ies, — one. Jive, ten, fifty, one five cnt hundred, hundred- (houmnd. |H|; . I>y coHibining these le tters, the Romans foruie dtha ^^H follOAA ing table ol notation : Hb 1 u . . XI 21 .. XXI 150 .. CL HHi . . 11 12 .. XII 30 .. XXX 200 .. GO ^H^i .. lu i:i .. XIU 40 .. XL 530 .. D sS' 4 . . IV 14 .. XJv 41 .. XLIV (joa .. DO vs . . V 15 .. XV 59 .. L 1009 .. M ■H . . Yl I'o .. XVI ; (io . . LX 1500 >. MD ■D .. vi; 17 .. V1JIJ8 .. XV n 71) .. J/XX . . LXXX .. XVI II 80 ^■1 .. IX 19 .. XIX 90 .. XC I^B 10 .. A 20 .. XX J 00 .. C I j ROMAN NOTATION. 11 T j 23. From an examination of the above table it will bo seen : 1st, When a cliaractor is foUowed hy one of eqnal or less value, the whole expression is e.iual co the sum of the values of the single el)aracters ; thus, 11 stindy lo- 1] ; HI for 3 ; VI for G ; XXX hn- ;;!). 2nd. When a character i,s preceded >> - o^io of Jess value, the whole expression is e.jual to the differ jiicc jf felio values- of the single cliaracters : thus, IV st.uuis for 5 leas 1, or4 : IX for 10 less 1 , or 9 ; X U for 40 ; XO for 90. Ji4. To write any number in Roman Numerals. Suppose we are required to express the number 1880 in Roman numerals. This number in the Arabic System represent* one thousand, eight hundreds, eight tens and six units ; thus s Arabic. Roman. 1000 M 800 Drno 80 TXXX VI Writing from loft to rigjit wq. obtain MDCCCLXXXVI. Hence to cxjtress any number in Iloaian numerals: Separate the giVv^n iiuiubrr into its different parts and write down one part before proce;dinx to another, beginning always at the left hand, sias. W-vite i.n Roman numei.tls : 1. 24; 9; 48; 19; 37; 41. 2. 394 ; 200 ; 199 ; 537 ; 685 ; 490, 3. m)- 094; 330; 004; 339; 119. 4. 1703; 1784; 1825; 1854; 1807. Write ill Arabic nuriierals : 5. XfdX; IXX-X: XOIX; X^.V^II. C. CCXLilX; DiXXX'V : DCCC)»V. 7. MDOIV ; Mi^UCClXXF ; MLIX. 1. Wliat is a unit I A number i Au ■ibsbract number? A concrete number i 2. What is the difFerence between the Roman and Arabic eystums of notation ? 12 ADDITION. 3. Write in Roman and Arabic numerals and in words the numbers between oO and 50. 4. AVrite tlie largest and the smallest number which can be expressed by the IJgures 3, 0, 4, 5, 2, and by the letters I, V, X, L, using each ligure or letter oriCC. 5. Write and i'e;id the nmnber 35* »2 in all tlie ways you «an. 2 i 1 7' 8 9 AB>i>I i50\. Additaou TabSe. ] 2 3 4 5 C 7 8 9 1 1 2 1 o 1 1 1 1 1 1 1 1 4 5 7 8 9 10 1 3 4 6 7 8 9 2 2 » > ' r 2 " 4~ 2 5 2 2 2 2 4 2 10 ' 2 "11 i> t ■ i) 1 »j • > 4 8 9 • > <> .J * 1 • ) •> "4 5 <; f i 8 ' " 9 10' 11 ~^12~ ^'f i«v. ^«ftr '{)" 1 .^r 9* 4 4 4 4 4 4 4 4 4 4 4 ."> (i" < I..' 9 10 7 12 8 13 9" 1 2 - <> o 4 6 (i 5 5 5 5 5 6 6 5 6 5 " 5 'G ">7 8 9 10"' 11 12 13 14 l' 2 3 4 5 C 7 8 9 (J C 6 C 6 6 14 6 15 f) ^7 • 8 9 10 n 12 13 1 *> • > 4 o 7 8 9 7 7 7 7 7 7 13 7 14 7 15 7 16" 9 7 10 11 12 * > 4 5 7 8 8 8 8 8 8 8 8 a 8 8 8 9 10 11 12 13 14 15 U 9 I 9 2 9 o O 9 4 9 o 9 10 17 7 8 9 9 9 9 9 9 10 11 12 13 14 15 16 17 18 4. A.Dr»ITTr>N. la 4 1 i Oral Exercise 19. 1. How man}' are .iikI 8 ? 17 and 8 ? 2. What is tfio mm of S and 7 ? 18 and 7 ? 34 and 7 ? 3. Wliat is the sum of C and 9 ? 10 and !) ? 40 and 9 ? 4. What is tJic sum of 4 and 3 and 7 and 5 and and 8 ? 6. How many are aj.ples and 3 ai)ples and 4 apples and 5 ap]>]os a!id aj)ph38 v 0. Ijow many are 5 pens and 4 pens and 7 pens and b pens and 3 pens and 8 pons f 7. How many are 15 and 8? 23 and 8? 25 and 8? 35 and 8 ? 42 and 8 ? 19 and 8 ? 8. A boy paid 7 cents for apples and 9 cents for cakes: how much did he pay for the whole V 9. John gave 12 'cents for a ball, 8 cents for a top. 6 cents for a bat : what sum did he pay for the Avhole ? 10. Sarah worked 7 sums on Monday, 8 on Tuesday, 9 on Wednesday, 5 on Thursday, (> on Friday, and 4 oa Saturday : hov/ many did she work in the entire week ? 11. in a yard are 10 peach trees, 5 apple trees, and 4 plum trees : how many trees are there in all s 12. In one ])asture are 10 sheep, in another 7, and in & third 8 : how many are t.jero in all 2 X3. Gave iO cents for a slate, cents for a ruler and 5 cents for a sponge : how many cents were given for the whole '{ 14. Lucy bought pins for 8 cents, some thread for 12 cents and some tape for 5 cents : how much did they all cost ? ^ The preceding operations are called addition. Hence, 25. Addition Is the process of finding the sum of two or more numDei-s. 26. The numbers to be added are called the Addends. 27. The Sign of Addition is an erect cross + , called plus. Thus, 3 + 2, read three plus two, denotes that 3 and 2 are to be added. 28. The Sign^ol Equality is =, and is read equals or equa. ..o. Thus, 54-4=^9 is iidiid^ five plus four equals niiie. 29. To add numbers. Suppose it is required to find the sum of 5, 21, 30, 2 and 41, To add these numbers it is 14 ADDITION. convenient to write them in vertical lines, placing units under vni'ts, tens undev tenn, thus : — Tens, i Unik-!. a 2 1 3 4 1 S 9 Bo,i;Mininads ? 7. 1020 logs on one landing, 21C0 on another, 3400 on another, l2J 6 on another and 2102 on another. Ho\y many logs on the five landings ? 8. 712 bushels of oats in one car. 630 in another and 534 in another. How many in all ? 9. 520 people in one cliurch, J235 in another, 1012 in another and 1100 in another. How many in all tho churches ? 10. In an election one candidate receives 3126 rotes, another 2630 and another 2010 votes. How many votes are cast ? In the preceding exercise the sum of the numbera in any column does not exceed 9, and the figure representing the t i t ADDmON. 15 \ i bXw^if "^ numbers in each column is written inunodiately «fo"^'lw ^?^^'i'''"r7^«^'^ in additiou, liowever, ^ve liavc con- stantly to deal witli colunms of fi .;uies the sum of which exceeds 0. We shall now consider how the sum of u cokunns should be set d(nvn. Example. — What is the sum of 10, 3042, 8 003 and 2809 ? Write the numbers to bo added, keei^in-' units under unit?, tens under tens, S:c., tlius : — jQ Bo'^in at liic units' column and add ; thus 3042 ?' '-' -;^' -- -^- ~« i^ = 'if^^ 'H-e e-iuul to 2 g tens .-ind ,S units. Write tlie ,S units under f^Q^ tlie units column, and c;iirv tiTc 2 tens to the 2SG1) If ^''' ^-'^'^'nim : thus, 2 (carnc>l), .s, 1 J, 18, l!). -i^Ue-.s equal 1 liundred an. 1 !) tens. Write C898 ^!^^ 'V^^"^ '•■^'^^^^" tbe tens' CMlimiU autl carry tiie 1 hundred to tlie huudivds' cu^uun • tliu,'3, 1 (carried), 9, IS. IS hundreds are enual to 1 thousand and 8 hundreds. Write the 8 hundreds under the hundreds' column and carry the 1 thousand to the thousands' column ; chus, 1 (carried), 3, 0. \\'iite the G thousand under the thousands' column. Wo have thus C898 as the sum. ' ' 30. Pkoof. — The readiest method of province tlie cor- rectness of the work is as follows : Begin at the top and add each column downwards ; if the results for each column are the same, the work is probably correct. Note.— The teacher may adopt (jtlior modes of proof when he wishes to provide iulditioiud e.^ercises for the pupil at seat. »I. Fnmi the examples and illustrations given abovo we deduce the following general RULE.— Write the addends under one another so that units come under umts, tens under tens, hundreds under hundreds. &c. Begin at the right and add each column separately, set- ting do_wn the sum underneath il' it does not exceed nine It" how- ever the sum of any column does exceed nine, set down the right-hand figure underneath, and cu- -y the other or others to the next Under the last column set uown its entire sum. 32. TiiO process of Additiou in basyd upon the foUowinff FumciPLES ; — ® 1. Numbers of tlie same khid and unita of the same 16 ADDITION, orcTcr alcno can be added. Thus : Dollars can be added to dollars and cents to cents ; alsc units and units, cens and t^ns ; but not units and tens 2 The sum of two ov more numbers is the same ir wbitever order they are added. Thus : ;) + 4 + 5-^12 ano 4 + 5-1-3 = 12 and 5 + 3 + 4-12. 3 Tliosmn and the addends must be numbers of the same kincl. Tlius ; The sum of 8 cents and 7 cents ls 15 cents, not 15 dollars. 0) 54 pounds. 118 ♦* 96 *< 5 « Exercise 7. 255 dollars. 9 " 362 " 457 i( (3) 67 trees. 211) " 642 " 108 '^ (4) 649 yarda. 2068 '' 119 ** 24 " (5) 78 63 49 52 (6) 81 97 63 45 (7) 32 18 98 75 (8) 93 69 45 86 (9) 16 209 36 584 (10) 352 168 29 375 (11) 463 369 548 276 (12) 981 426 549 387 (13) 654 18 968 594 (14) 3164 286 54 5287 (15) 2369 5498 2697 8364 (16) 3196 428 4937 li564 (17) 355 486 289 192 669 (18) 598 273 694 329 683 (19) 3608 4327 9130 5968 4326 (20) 43695 3-i96 827 51426 18 (21) 36540 59186 23574 ]K693 47189 5 added 1 vinits, «ime ir - 12 and 3 of th€ its i& IS (4) yard*. (« u (10) 352 16B 29 375 (16) 3196 428 4937 1^64 (21) 16540 )9186 i3574 IKfi93 17189 AtMriON. (22) (23) (24) (25) 76134 11214 36982 23689 586i/ 39686 54368 41285 439 543:j9 « 32856 58327 65497 83754 49 75438 1389 91276 t 31825 36984 05 43572 59]8 53962 (26) (27) (28) 316895 432968 978296 432687 58420 3950 34 219864 639 20865 328968 386542 5380 523692 5189 416 732986 369584 429 431095 (30) 449 32] 564327 (29) (31) ( 1 (33) (34' 6 7 2 9 22 47 3 3 8 8 36 99 9 9 9 4 55 53 8 8 7 3 93 95 7 5 5 9 48 52 5 4 4 2 39 38 4 9 1 8 52 29 3 6 3 6 76 , 16 2 3 8 4 54 2T 6 7 6 7 18 52 (35) (36) (37) 316 269 63d 48 388 08 29 493 416 118 268 29 U4 145 13842 4 LJ2G 308 4653 458 418 216 19 630 39 162 492 40529 33 184 5643 IT 18 ADDITION. (38) Pounds. 9 8 2 6 6 8 4 3 9 7 2 6 (30) Yards. 4 3 9 2 8 8 2 6 8 4 1 9 (40) Cents 29 38 56 45 94 37 63 41 53 18 62 29 (41) Dollars. 59 18 26 45 30 87 42 49 62 83 49 56 (42) Bushels. 84 26 43 94 32 65 19 42 63 82 41 75 is the sum of— 3164-4189 + 346S4 + 20. 4320 + 236 + 9 + 56895. 3698 + 43126 + 34219 + 396. 54 + 639+ 18G90 + 64826. 9867 + 325 + 1 1 286 + .'.9685 + iSi 3943+ 1685 + 93^^6 + 41286 + 40. 8254 + 82 1 + 7G390 + 182 + 9. 438269 + 578 + 92816 + 301. 2(;43 + ] 3'.)86 + 489 + 3684r9. 4832 + 39685 + 43260 + 593. 1^ What 0) (2) (3) (4) (5) (6) (7) (8) (0) (10) C'aiia«tiaii iVioney. S3. The sign 8 stands for dollars. Thus $8 is read eight dollai's. The letters eta. stand for cents. Thus 25 eta. ia read twenty-five cents. Dollars and cents, wnen written togetner, are separated by a point. Thus ^12. 37 is read twelve dollars and thirty- seven cents. When the number of cents is less than ten, the firsb place to the right of the point is occupied by a cipher. Thus, $8. 06 is read eight dollars and six cents. Head the following :■ $4.75 #15.07 $27.30 $237.43 $375.04 ^607.75 $1003.37 $9io.oe .^«-***-,*,-,:^1w«^' ADDTTION. 19 1^ Write in figures : 1. Six doliiirs aiKl sovciity-livo cents ; twenty-six dollars uiid ! iiii^y-Huvcn cents. 2. One hundred and twenty-three doHurs and twenty -tive cents ; seven I'ollajs and four cents. 3. Five hundred dollars ; five hundred and seven dollars and f(Air cents. 4. One thousand dollars ; thirteen hundred dollars and fifty-five cents. 5. Five thousand and thirty-seven dolliirs and seventy- six cents ; two thousand three hundred and twenty-fiva dollars. iVt, To add sums of Canadian money, dollars must be placed under dollars, and cents under cents, so that the poirits stand in a straight Une ■ thus — 0) (-)k Ir, ('^) $ 53.45 ^630.07 236.07 126.20 7.50 40.03 7.GL 208.04 500.70 Add as if there were no [loints, and in the result place a point between the second and third figures from the right. Hence : 35. To change any number of cents to dollars, place a point between the second and third figures from the right. The figures to tho left of the point denote dollars, to the right, cents. 1. Find the sum of $207.43, $4.75, ^375.04. 2. Add $375.04, ^5007.75, $1W3.37, ^910.0(5, $15.07. 3. Bought a coat for ^12.50, a vest for $0; a pair ot shoes for $2.25, and a neck-tie for $0.45. How much did the whole cost ? 4. Lemont & Sons sold a bedroom set for $127.65, a book case for $45, and three easy-chairs for $14.75 each. How much did they receive ior all ? a. tons of hard coal tor $6.25 a ton. How much did the whole cost. so CINAPIAN MONET, Exercise 9. Practical Problems. 1. A boy gave 25 cents for a top, 17 cents for a ball, and 28 cents for a knife. How much did they all cost ? 2. A little girl paid 40 cents for a doll, 30 cents for a picture book, and 15 cents for pencils. How much did she ipend ? 3. 54 scholars in one school, 38 in another, 46 m another, and 35 in another. How many in the four schools ? 4. One boy makes a collection of 103 postii^e stamps, another of 239, and another of 370. How many have all ? 6. There are 158 hills in one row of potatoes, 00 hills in another, 249 in another, and 387 in another. How many in the four ? 6. A lumberman hauls 458 logs to one binding, 1092 to another, 710 to another, and 2251 to anotlier. How many logs has he ? 7. A fruit groAver has 2(>4 apple trees in one orchard, 196 in another, and in a third he has as many as in both the others. How many trees in the three ? 8. A farmer raises 48 bushels of wheat, 20 bushels more buckwheat than wheat, and 247 bushels more oats than buckwheat. How many bushels of grain in all ? 9. A saw mill cuts 25000 feet of boards a day. How many feet will it cut in six days ? 10. A merchant owes to one party $426.17, to another $284,35, and to a third $530.86. How much does he owe in all ? 11. The population of Fredericton is 6218, of Portland 16226, of Moncton 6032, and of St. John 26127. How many people in the four towns ? 12. In the year 1882 New Brunswick bold to other countries lumber worth $4724422.26, fish worth $753251, minerals worth $140008, animals and their produce worth $320426.39, farm produce worth $256994, manufactured goods worth $365746. What was the value of them ftUV 1? V, lO. jri. illaXx UUUj^ilt iX iiOISe lOi §^li/v. j O, and. $276.80. He sold them so as to gain $70.25. he receive for them 'i wagon for What did SUHTR ACTION. St SITBTRIC'TIOBT. T.lc.s, ;ind Fannie 9 : how many more apples lias Jane than Fannie ? 9. Subtract G from '.) ; 4 from 11 ; 2 from 7 ; 5 from 12 ; 8 from 13. 10. Jolui had 13 niaibics and lost 8 : how many had he left? What number mu.st be added to 8 to make 13? 11. Ennna worked 7 <|uostions in Arithmetic each day of the week, while Maggie worked 3 questions on Monday, 5 on Tuesday, 10 on Wednesday, G on Thursday, 3 on Friday, anil 8 on Saturday. How many more questions did Emma work during the week tlian Maggie ? 12. A man went to market with 25 dollars. He paid 7 dollars for geese, 3 dollars for potatoes, 4 dollars for fruit, and 1 dollar for tisli : how much money had he left? The preceding operations are called Subtraction. Hence, 36. Subtraction is the process of finding the difference between tv.'o numl>ers. 37. The Minuend is the number to be subtracted from . :iH. The Subtrahend is the number to be subtracted. 39. The Ditterence or Remainder is the result obtained by subtraction. / 22 SUUTUACTIO.^. 40. The Siffii of Siihtraction is— , and is calletl minus. When pl.'ict'd hetween two numbers, it indicates that tlie one after it is to be taken from tlie one before it. Thus, C — 4, read six mimis foiir^ means that 4 is t 7694 58 586213 (22) 439782 419380 (23) (24) C35847 382910 523410 382800 (25) 409287 210053 i (20) 059 48 (31 ) (32) (33) (34) (30) (27) 3942 721 084 90 5820 (i3t>8 980432 (28) 802394 12J4 — 432 — 23 — 214 _ 398 —974122 (29) 52S039 10029 (3r.) (37) (38) (39) (40; 1885 1807 03948 70584 59820 (30) 038426 102 — 1703 — 1004 -52720 — 350 — 9820 Exorcise 11. How many has How many has l»racli<*al I'robleiiis. 1. A farmer has 50 sheep, and £:ells 23. 2. A boy has 48 cents, and spends 25. lie l^ffc ' . , ^,^. i^ . 1 3 A little '^irl goes to a store with 86 cents m Jier purse. She buys a yard uf cotton for 14 cents, a pair of shppers for (0 cents. How much m«jney has she left after paymg for them ? ., i iqe 4. There are 208 bricks in a pile, and a mason uses IdD of tiicm. How many are left? 5. Bought a watcli for 158 dollars and sold it tor loJ dollars. ^How much do 1 lose ? ^ , i * There are 170 rows of potatoes in a held. A man hoes 30 rows one day, 42 rows the next day, and 48 rows the next. How many rows must he hoe on the fourth day to finisli them ? 7 One candidate receives 3098 votes at an election, another receives 2014 votes. How many more votes doea the first receive than the second ? 8. There are 805 apples on a tree, and 320 are picked from it. How many apples are left? n A i.,^V.^Y. rviorfhoTif V^uvs a raft, of loers lOI loZ^O dollars and sells it for 15875 dollars. How much does he gain ? ri ' (• 24 SUBTRACTION. In the foregoing exerciscsi the figures in the subtrahend have ahvays been lesfsthan tlie corresponding figures in the minuend. But, in example?^ such as the following, this ia not the case, and the operation needs further exnlanation. Example. 70.3 37S -^ From 7<-J3 subtract 385. Begin at the riglit. Since we cannot take 5 units from 3 units, we borrow 1 ten from the G tens, and add it to the 3 units, which makes 13 imits. Then 5 units from 13 units leave 8 units, v.liicli wo set doAvn under the units' column. Having borrowed 1 ten from tlie tens we have only 5 tens. left. As we cannot take 8 tens from 5 tens, we borrow 1 hundred from the 7 hundreds, and add it to the 5 tens, whicli makes 15 tens ; 8 tens from 15 tens lea^'o 7 tens, which we set down under the tens' column. Havinjr borrowed 1 hundred from the 7 huu ixeds we have only 6 hundreds left, and 3 hundreds from (5 hundreds leave 3 hundreds, whicli we set down under tlie hundreds" column : we have thus 378 for the remainder. 42. From the examples and illustrations given above we obt^iin the followinof : RULE.— Write the sul)tralie:aci under tlie minuend so that units come under units, tens under tens, &c. Begin at the right, and subtract each figure of the sutatraliend from the corres- ponding figure of the minuend, and set down the remainder underneath. If any figure of the subtrahend is greater than the correaponding figure of the minuend, add 10 to the latter and subtract ; then re2ard tne next figure of the minuend as diminished by 1, and proceed as before. 43. Proot. — Add tlie remainder to the su]»trahend ; if the sum is equal to the minuend, the w jrk is torroct. Note. — As an additional exercise for the pupil he should be frequently required to subtract the remainder from the minuend. 44. The process of Subtraction is based upon the following Principles : — 1. Only n.uml)crs of the same kind anjs 34 J dollars for school purjjoses during a year. At the end of the year there is a balance on han 1 of '18 dollars. How nuich money was expended? 5. Tlie population of !St. John is 26127, of Portland 15226. How nuicli does the po}>ulation '^f the two cities togetlier exceed that <,»f Halifax, which is 362}0 ? 6. The expense of running a mill for a year is 3861 dollars, and the receipts from the mill are 7280 dollars. What is the profit i 7. A farmer raises 1C6) bushels of potatoes. After selling a certain number of bushels, lie has 578 bushels left. How many did he sell ? 8. The Riyer 8t. J(.»hn Wiis first seen by white men in the year 16 ;4. Hetween 601: and 468 will make 1020 i 6. In] 881 the population . .What is his profit, if he sells tho goods in Fredericton for $8375.56 ? >IULTIPI.I€.^TION. ExAMrLE 1. — What will 5 apples cost at 2 cents each ? If 1 apple cost 2 cents, 5 apples will cost 2 + 2 + 2 + 2 + 2 cents, or 5 times 2 cents, or 10 cents. 2, If I pay 4 cents for a pencil, what should I pay for 3 pencils 1 If T pay 4 cents for 1 pencil, for 3 pencils I will pay 4 +-4 + 4 or three times 4 cents, which is 12 cents ,rtife8#iPSft MULTirLlCATlON. 29 3. If 1 pair of boots costs $3, what will 2 pairs cost ? 5 pairs ? 4 pairs ? 6 pairs ? 8 pairs ? 4. If 1 silk liat cost $5, what will 2 silk hats cost ? 3 hats ? 4 hats ? 5 hats ? 7 hats ? 6, What will 2 oranges cost at 6 cents each ? 3 oranges ? 6 oranges ? 7 oranges ? 9 oranges ? 6. If there are 7 days in 1 week, how many days in two weeks? 3 weeks ? 5 weeks? 8 weeks? 7. If 1 barrel of flour cost $8, what will 6 barrels cost? 8. Tf a boy walk 4 miles in a day. how many miles can he walk in 7 days? 8 days? 9 days ? 9. What will 6 barrels of apples cost at $3 a barrel ? 8 barrels ? 10 barrels ? 10. What is the price of 12 articles at $4 each ? 9 articles? 1 1 articles ? 11. What will 4 barrels of flour cost at 9 barrels ? 12 barrels ? a barrel ? The process employed in the preceding operations ia called multiplication ; hence, 45. Multiplication is the process of finding what a number will amount to, when repeated a given number of times. >- i 46. The Multiplicand is the number to be repeated. 47. The Multiplier is t.io number which danotes how many times the multiplicand is to be repeated. 48. The Product is the result of this process of repetition. 40. The Sign of Multiplication is an inclined cross x . When placed between two numbers, it signities that they are to be multiplied together. Thus 7x8, read 7 fiiultiplitd by 8, means that" 7 is to be multiplied by 8. # 30 nULTIPLli'ATlO.N. ifliilsiplicatiou Table. Twico 3 time !3 i 4 times. ; 5 times 1 times ; 7 times 1 are 2 1 are 3 1 are 4 1 are 5 1 are 6 lare 7! 2 '• 4! 2 " 6 2 " 8 2 ' 10 2 t i 12 2-14 3 " 6: 3 " 9 3 " 12 3 ' 15 3 IS 3 " 21 4 " 8| 4 " 12 4 " 16 4 •' 20 4 24 4 " 2S; 5 " 10 5 " 15 5 " 20 5 •' 25 5 30 5 " 35 6 " 12 6 " 18 6 " 24 6 " 30 6 36 6 " 42 7 " 14 7 " 21 7 " 28 7 " 35 7 42 7 " 49, 8 '« 16 8 " 24 8 " 32 8 " 40 8 48 8 " 56 9 '• 18 9 " 27 9 " 36 9 " 45 9 51' 9 " 6i 10 *« 20 10 " 3d 10 " 40 10 " 50 10 6010 " 70 11 " 22 11 " 33 11 " 44 11 " 55 11 66 11 " 77i 12 '* 2412 " 36 12 " 48 12 10 times " 60 12 " 11 times 7212 " 84^ 8 times ; 9 times . 12 times 1 are 8; 1 are 9 1 are 10 1 are 11 1 are lo 2 " 16 2 " 18 20 22 2 " 24 3 " 24' .'3 " 27 3 " 30 3 " 33 3 " 36! 4 " 32 4 " 36 4 " 40 4 " 44 4 " . 48| 5 " 40 5 •' 45 5 " 50 5 " 55 5 " 60, 6 •' 48 6 '• 51 6 " CO 6 " 66 6 " 72! 7 " m 7 " 63 7 " 70 7 " 77: 7 '' 84i 8 '* 64 8" 72 8 " 80 8 " 8!;$ 8 " 96 9 " ".2 9 " 81 9 " 90 9 " 99 9 " 108 10 " 80 10 " 90 10 " 100 10 " 110 10 " 120 11 '♦ 88 11 '* 99 11 " 110 11 " 121 11 " 132 12 " 96 12 " 108 12 " 120 12 '' 132 12 *' 144 50. The numbers which produce or make another number, are called factors of the latter numbers, thus 2 and 3 are factors of 6 ; 9 and 6 are factors of 54 ; 8 and 9 are factors of 72. 51. When a number is produced by the multiplication of two equal factors, it is called a square number ; thus, 4, 9, 16 are square numbers, since they are equal respectively to 2 x 2, 3 x 3, 4 x 4. Oral Exercise. 1. What two numbers multiplied together make 6 ; 8 ; 9 ; 12 ; 16 ; 18 ; 21 ; 27 ; 36 ; 42 ; 48 ; 64 ; 72 ? 2. What three numbers make 8 ; 18 ; 27 ; 56 ; CU MULTIPLICATION. 31 1 3. What are the factors of 45 ; 63 ; 10 ; 132 ? 4. Name all the square numbers in the multiplication table, and give their factors. 5. What will three books cost at 8 cents each ? 5 yards at 7 cents ? 6 pounds at 9 cents,? 7 quarts at 12 cents? 6. What cost 9 barrels of flour at $7 a birrol ? fi tons of hay at $ ' 2 a ton ? 8 cords of wood at ^5 a cord ? 12 pounds of sugar at 12 cents a pound ? 7. 4 rows of seats in a school-room, and 12 seats in a row. How many seats ? 3 r 12 14. 415 X 25. 396854 x 6 = 15. 29f] X 6 - 26. 100,S(;5 X 9 =1 IG. 1028 X 8 = 27. 3685 ;42 x 11 = 17. 3694 X 7 = 21 4932S6) X 8 =1 18. 5')G4 X 11 = 29. 1369482 x 7 = 19. 36902 X 4 = 30. 4060082 x 5 =, 20. 5i:-J27 X 8 = 3i. 3542^16 X 6 == 21. 93684 X 10 = 3.>. 9684327 x 12 = 22. 80039 X 12 = 33. J 587643 x 8 = 23. 4G0387 x 3 = 34. 521(847 x 11 => 24. 62085 X 9 = 35. 6506508 x 10 = If si m tl; It: MULTIPLICATION. 33 Exercise 17. Practicail Trobleuis. 1. II acres at ,f345 per acre 7 2. 9 horses !i?175 each ? 3. 8 cabinet organs at Jji^lSO each f 4. 12 mowers at .*j'07.50 each ? 5. 5 threshing niaehhies at .';475? 6. 9 farms witli 628 acres in each ? 7. 12 car hxuls of oats witli 816 bushels in each ? 8. 7 rafts of logs witli 8692 h)gB in eacli ? 9. 5 ships carry 1256869 feet of deals each from St. John lo Liverpool. How many feet in ali ? 10. 3520 sleepers for 1 mile of railroad. How many sleepers will be rerjuired for 7 miles ? 54. To multiply by using the factors ot tne multiplier. Example.— What will 24 ploughs cost at .532 each ? In this example we are ropiivod to talce ^'32 24 times. Now as 24 =-2 X 1 2 or 3 x 8 or 4x6, it is evident tiiat if we multiply 32 by any set of factors which will province 24, the result nui-^.C be the same as if we had multiplied by 24 Itself ; thus. it 32 e>4 t 768 32 2 64 12 768 3f 96 8 768 32 4 128 6 763 Kxei'cise IS. Multiply, using factors : — M i 7. 1. 672 X 18 2. 763 X 27 3. 9;;3 X 45 4. 1003 X 54 5. 3263 X 63 6. 8095 X 72 7. 1)086 X 84 8. 3u59 X 96 9. 5 76 X 103 10. 7854 X 132 f I 84 MULTIPLICATION, 11. If 1 hogshead of sugar cost $148, what will 18 ll! nUoadof hay weighs 185C poumli?, what will 25 Buch loads weigh ? , » . . i i. -n oo 13. If 1 bushel of oats is worth 54 cents, what will 63 bushels be worth ? , 14. If 1 piece of cotton contain 08 yards, how many yaids will there be in 9o pieces ? , , 15. If 1 acre of land produce 298 bushels of potatoes, how many bushels will J(> acres produce ( 10. If 1 pair of hordes can draw 3040 pounds, how many pounds can 32 pairn of liorses draw ? 17. If 1 cord of wood is worth ^3.00, what will a pile contahiin!? 5(» cords be worth ? 18. If i passenj^er car seat 120 persons, how many persons wilJ a train of 15 cars seat? 19. If a man saves J?350 each year, how much money will he save in 18 years ? .,, 20. If it costs ,^^8204 a nnle to build a railroad, wliat w.U a line 03 miles long co«t i CASE II. 55. When the multiplier exceeds 12. Before beginning this case the pui>il should be shown that annexing a cipher to anumber multiplies that i|^^i'jyer by 10 ; annexing two ciphers multiplies the number by 100, &c. Thus, if is annexed to 24, the result is 240 - 2^ x 10. Example.— Multiply 730 by 20. 730 20 736 X 26 ( 736 X 6 = 4410 1st partial product. = I 730 X 20 = 14720 2nd " 19130 whole product. For convenience we write the multiplicand and multiplier as in case i. We first multiply each figure ot the multiplicand by the units' figure of the multiplier as m the same case ; thus, times 736 are 4410, the 1st partial product. Next multiply by the 2 tens or 20 : 20 times 736 = 2 times 736 x 10 or 1472 x 10 = 14720, the 2iid partia.1 product. This process is the sanie as multiplying ^ WaMNMH >mmm MULTIPLICATION. 35 ! ^a the multiplicand by 2 and annexinjj a cipher to the product. Adding the partial products we obtain 19130 for the whole product. In practice the cipher in the 2nd partial product IS omitted, and tlie first figure of the product is set down under the tens' place of the 1st partial product. The same principle is observed when the multiplier consists of three or more figures. Note. — Tf there are ciphers between the significant figures of the niultii>Uer, pass over each of the multiplier and multiply by the signiiicant figures, only remembering to put the results in their proper places. 56. From the example and illustration given above, we obtain tlic following RULE.— Write the multiplier under the multiplicand so that units stand under units, tens under tens, Sec. Begin at the Tight hand and multiply the multiplicand by each significant figui-e in the multiplier successively, and set down the right- hand figure of each partial product under the figure of the multiplier used. Add the partial products to obtain the whole product. 57. Proof.— A convenient and ready method of proof at this stage is by casting out the 7iine3 from the multiplicand, multiplier and product in the following way. Take the example worked in case ii. above, and proceed thus: 7 and 3 are 10; cast out 9 and I remains;! (remaining) and are 7. 7 is therefore the remainder after casting tlie nines out of tlie multiplicand. Proceed in the same way with the rigiires in tlie multijilier. Thus, 2 and 6 are 8 ; as 8 is less than 9, the remainder is regarded as 8. Set down tlie remainder of the multiplicand and multiplier thus: 2 Mii'tij)lyiiig these tv/o remainders together as 7x8 indicated, the product is 5(5. Cast out the 2 nines from this product. Thus .5 and G are 11 ; cast out 9 and 2 remains. Set down this 2 above the sign of multiplication as indicated. Next proceed to casb tlie nines out of the product. 1 and 1 are 2 (9 of course not included) and 3 are 5 and 6 are 11. Cast out 9 and 2 remains. Place the 2 under the sign of multiplication as indicated above. If the remainder, after casting the nines out of the products is the same as the rem.aindfr .-vfter casting out the nines from the multiplicand and multiplier, as shown in this operation, the work may be regarded as correct. 86 MULTIPLICATION. The process of multlproition is based upon the following RA Principles. 1. The product and multiplicand must be numbers of the same kind. Thus, 3 time. 7 books are 21 books. 5 times 8 men are 40 men. i,„f„„,fc 2 The multiplier is always regai.led as an abstiact number Thus, in tiuding the cost of barrels of Hour at 5 dollars a ban-el, the 5 dollars are taken tnues, and nub multiplied by C barrels. Exercise 10. Multiply and prooe :— 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 6i8 763 8t)9 657 257(1 81.-6 321)7 0254 3026 7 3 JO 95ti7 5482 2198 36'j4 59C8 7329 9i;18 36S0 X X X X X X X X X X X X X X X X X X X X 56 95 147 345 048 73f3 504 625 568 849 972 4(37 3i.5 82G 600 3270 5483 4376 3275 4396 21. 22. 23. 24. 25. 20. 27. 2S. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 5843 X 6;.74 X 49()H5 65847 80.542 .'30485 29087 96438 73654 62946 648 7060 9186 2990 59 20000 X 7l60 X 91600 X 38001 X 46000 X X X X X X X X X X X X X X 0549 .",862 2:64 5472 0387 9084 16854 26598 58021 43987 400 269 4500 3U12 6000 92a 7S0 4700, 500 59000 Exercise 30. Practical ProbleniH. 1 The pay roll of a factory amounts to $2645 per week. What wiU it amount to in 1 year of 52 weeks ? 2. A barrel of flour weighs r.J6 pounds. What wiU »» barrels of flour weigh ? , . . • j tt^« 3 A mill can grind 46 bushels of grain m a day. UOW many bushels will it grind in 8 weeks of 6 days eaehT •.r 4 MULTIPLICATION. 9! 93 4. If there are 35 telegraph polos in a mile, how many would there be between St. John and Halifax, a distance of 276 miles? 5. What will a ho^'shead of molasses containing 126 gallons be worth at 52 cents a gallon ? 6. If 1 acre of land is worth |108, what will 240 acres be vorth ? 7. From 1 acre a farmer raises 28 bushels of wheat. How many bushels ought he to raise at the same rate from 13 acres '< 8. If it Uikcr, 38,) poumls of hay to feed 20 head of cattle for 1 day, how many pounds will tliey eat in 180 days ? 9. A book in liine voUnnes contains 476 pages in each volume, and 80 lines in each page. How many lines in the book ? 10. A ship crossing the Atlantic steams at the rate of 12 miles an hour for 8 days of 2 I hours. Hovr many miles has she sailed i 11. A train runs at the rate of 3) miles an hour for 4 days. How many miles has it cles be taken from 20 apples ? From 32 apples ? Fr^ni 48 apples ? 12. If a yard of ribljon ct^st 4 cents, how many yards can be bought for 30 cents ? For 44 cents ? & CASE I. 65. When the divisor does not exceed 12. Example 1. — How many times is 6 contained in 87G? Divisor. D'vidend. Q lotient. 6) 870 (140 For convenience, we set aown 6 tlie divisor fit the left of the — — dividend, and beQ:in at the 27 24 left to divi 'e. 6 is cortained in 8 hundreds, 1 hundred • times and a remainder. Set 30 down the 1 hundred in the 30 quotient, and multiply the divisor by the 1 hundred. This gives 6 hundreds, which we set down under the hundreds of the dividend. Subtract the 6 hundreds from th© 8 hundreds and the remainder is 2 hundreds or 20 tens. c S 5 DIVISION. 43 Add the 7 tens of the dividend to this remainder and set down the 27 ten?. 6 is contained in 27 tens, 4 tens times.. Set down the 4 tens in the quotie^nt, and multiply the divisor by the 4, This gives 24 tens, which we subtract from the 27 tens. The remainder is 3 tens or 30 units. Add the 6 units of the dividend to this remainder and set down the 36 rnts. 6 is contained in 36 units, 6 units times. Set down the 6 units in the quotient, and multiply the divisor by the 0. This gives 30 units, which we subtract as before, and there is no remainder. The quotient is therefore 146. The process above described, in which each step is written out in full, is called Long Division. When the divisor does not exceed 12, ho\revor, it is usual to shorten the work as follows : — 6)376 We set down the divisor to ^^-^ left of the dividend and begin to divide ^t the left. 6 is 146 contained in 8, once and 2 over. Set do vn tho 1 below the line and put the 2 before the 7. 6 is contained in 27, 4 times and 3 over. Set down the 4 below the line and place the 3 before the 6, 6 is contained in 36, 6 times. This shortened process is called Short Div'sion. Example 2. — How n my times is 7 contained in 756? 7)756 Proceed as in the last example. 7 is contained in 7, once. Set down the 1 below 108 the line. 7 is contained in 5, no times. Set down the cipher below the line, and place the 5 be.ore the 6. 7 is contahied in 56, 8 times. The quotient 38 thus 108. Example 3.— How many times is 5 contained in 6819 ? 5)6819 Here 5 is contained in 0, once and 1 over. Set -— - down 1 below the line, and place 1 before 1363i the 8. 5 is contained in 18, 3 times and 3 oyer. Set down 3 and place 3 before 1. 5 is contained in 31, 6 times and 1 over. Set down 6 and place 1 before 9. o IS cuuta nod in HK 3 times and 4 over. Sot down the 3, and indicate the division of the 4, thus, f. This is annexed to 1363, thus, 1363|. f 1 5 f: 5, i ■ 44 mvisiON. 66. From the examples and illustrations given aboTo, we obtain the following RULE.— Set down the divisor at the left of the dividend. Begin at the left-hand figure of the dividend and divide each figure in succession by the divisor, settin'^ down the result directly under the figure divided. If there is a re:-' •' >r after dividing any figure, prefix it to the next figure of . Mend, and divide the number thus formed as before. len the diviflor is not contained once in any figure of the dividend, write a cipher under that figure; then, regarding this figure as a remainder, prefix it to the next figure of the dividend, and divide as before. If there is a remainder after dividing the last figure, place it over the divisor at the right hand of the quotient. 1 Exercise 22. "Work both by long and short division :— 0) 3,432( (5) 7)756( (9) 2)708( (13 3,657( (17- 9j972( Work the following exercise hy Short Division :— Exercise 23. (2) 6)640( (3) 4)792( (4) 6;864( (6) 3)G12( (7) 8)864( (8) 5)720( (10) 1)321( (11) 5)865( 4^944( (14) 7)'^S3( (15) 8;976( (16) 7)588( (18) 6,366( (19) 8)512( (20) 9)G57( ''1^ 2)048 (3) 4)570 (3) 8.936 i "" Divisioir 5. 535 - - 5 18. 4793 - r 8 6. 984 - - 6 19. 2014 - r 9 7. 525 - r 7 20. 3167 - ~ 3 8. 696 - r 4 21. 36086 - f 4 9. 832 - r 8 22. 353287 - ^ 9 10. 504 - - 9 23. 50:i4 - ^ 12 11. 432 - - 6 24. 4538 -^ - 7 12. 216 .\ - 9 25. 532H43 -^ - 11 13. 766 ^ - 3 26. 98267 -^ - 5 14. 9S7 - - 7 27. 2( 006 -1 - 8 15. 500 H - 4 28. 120354 ~ - 9 1«. 304 -1 - 8 29. 17786 -i - 3 17. 638 H - 11 30. 561385 -J - 12 45 Exercise HI. Practical Problems. 1. At 7 cents each, how many pencils can be boudifc for $2.52 ? 2. At $3 a pair, how many pairs of boots can I bi>y for $o57 ? ' 3. There are 7 days in a week. How many weeks in 364 days? 4. If a ton of hay cost {|i)ll, how many tons vrould $583 buy ? 5. If a steamship runs 14 miles an hour, how long would it take her to run 2002 miles ? 6. If the fare on a railroad is 3 cents a mile, how far can you travel for $'3. 60? 7. A train runs 196 miles in 7 hours. How many miles an hour is that i 8. A fortune of $89650 is divided equally among 5 children. How much does each receive ? 9. What number multiplied by 8 will produce 1296 ? 10. If the quotient is 12, and the dividend 5032, what is the divisor ? 11. If the divisor is 16, and the quotient 36, what is the dividend 7 1*^. If the quotient is 16, the divisor 8, and the remainder 5, what is the dividend? i! f 46 DIVISION. 'M 67. To divide by using the factors of the divisor. Example.— Divide 3758 by 28, using the factors 4 and 7. 4)3758 7)939 and 2 units over = 2 '* I four over = 4 134 Remainder = 6 First, dividing by 4, the quotient is 939 with a remainder of 2. This is 2 units. Dividing this quotient by 7, we obtain 134 for quotient, with a remainder of 1 This is 1 group of 4 units. The true remainder is, therefore, 4 + 2 units or units. Hence, The true remainder is fr.und by multii)lying the firstJ divisor and second remainder together, and adding the first remainder, if any, to the product. Note.— The teacher may al.-o ilhistrate the process of finding the true remainder by assuming the dividend 3758 to be pens, or pencils, etc., to be divided into parcels, each containing 4 pens, etc., and these parcels to be divided into larger parcels, each containing 7. Exercise Divide, using factors : — 1. 2. 3. 4. 5. 6. 13. John lays out $2450 m hogshead. How many hogsheads does he buy 1 14. How many bushels of rye are there in 938074 pounds, one bushel of rye Aveigliing 56 lbs ? 15. A bushel of v.dieat Aveighs 60 pounds. How many bushels are tliere in 71496 pounds ? 16. If 48 coAvs cost .$1780, Avhat is the price of 1 cow ? 17. If 21 bushels of peas Aveigh 1260 pounds, what is the weight of 1 bushel ? 18. Divide ,S0400 equally among 25 persons. 19. If 40 barrels of Hour cost #.325, what is the cost of 1 barrel? 20. A man buys 28 barrels of apples for $60. How much does he pay for 1 barrel ? 672 by 18 763 " 27 962 " 45 1256 " 54 2678 " 63 3872 " 60 lavs out 25. 7. 8695 by 72 S- 9384 " 84 9. 12456 " 96 10. 15684 c. 72 11. 27305 " 108 12. 28046 " 132 sugar, pa_ yinrj $84 per DIVI«nd [)artial dividend. 1 Rem. 19 is contained in 25 once. Set d(.\vn 1 in the (luotient, and midtiply the divisor by 1. Subtract the product 19 from 25. To the remahider o, bring down tlie next figure of the dividend, (3. 19 is contained in 66, 3 times. Multiplying and subtracting as bei'» )re, Ave have a remainder of 9. Bring down 0. 19 is contained in 9 ;, 5 times. Multiplying and subtracting as before, we have a remainder of 1, which we, write above the divisor, thus j^^. Tlie entire quotient therefore, is 2135j\-. ■RxAMPLE 2.— Divide 90325 by 48. 48) 96325 (2006^1 96 325 288 48 is contained in 9(1, 2 time? ; set down 2 in the quotient : 48 x 2 = 9G, which subtracted from 96 leaves 0. Bring down 3 ; 48 is contained in 3, times. Set ,__.__ down a cipher in the quotient, 37 Remainder, and bririgdown 2; 48 is contained ni 32, tnnes. Set down another ciplier in the quotient, and bring down 5 ; 48 is contained in 325, 6 times ; 48 x 6 = 288, which subtracted from 325 leaves 37. Write this remainder above the divisor, thus, 15". The entire quotient, therefore, is 20062^-. 1. r« 43 MrT>^iOM. 69. From the examples and illurttnitiuns given above, we obtain the following : — Rule.— Set down the divisor to the left of the dividend, as In Case 1. Find how many times the divisor Is contained in tho fewest figiires of the dividend that will contain it, and set down the figure in the quotient to the rij^ht of the dividend. Multiply the divisor ty this figure ; subtract the product from the part of the dividend used, and to the remainder bring down the next figure of the dividend. Divide as before, till all the figures of the dlvideiid have been used. If any partial dividend will not contain the divisor, set down a cipher in the quotient ; then, bring down the next figure of the dividend, and divide as before. When there is a final remainder, write it over the divisor and annex it to the quotient. 70. Proof. — Multiply the divisor by the quotient and add the remainder, if any, to the product. The result will equal the dividend, if the work is correct. 71. From what has been said, it will be seen that, in eveiy example in Division, we have to deal with four terms, viz : — Divisor, Dividend, Quotient and Remainder. If any three of these terms be given, we can always find the fourth. mi r\ Dividend — Remainder Thus ; Quotient «=» * — Divisor Divisor, Dividend — Remainder^ Quotient. Dividend = Quotient x Divisor + Remainder. Remainder = Dividend — (Quotient x Divisor. ) Note.— 71 may be omitted, at the discretion of the teacher, till a later stage. 72. The following principles should also be explained and illustrated by examples : — 1. Multiplying or dividing both Divisor and Dividend by the same number does not alter the Quotient. 2. Multiplying the Dividend or dividing the Divisor by any number, multiplies the Quotient by that number. 3. Dividing the Dividend or multiplying the Divisor by «ny number, divides the Quotient by that number. PIVISIOW. EiEcrcise 26. 1. 2. 3. 4. 6. «. 7. 8. 9. 10. 856 916 7216 2304 .0640 7280 81034 50103 2182li.7'). How much has ho left iu tha bank ? 2. A butcher buys 12 head of cattle for ii;'45 each, tho Fame day 153 shoe[> at ^3. 'iO each. He suIIh the cattle for S!)00, and tlio bhecp for i;,CI0. Doo3 he gain or lose on tho whole tran.iactlcn, a.td how jiiuch ? 3. What niup.her divided by 1() will give a quotient of 35 ? 4. Add together 36, 208, 23u0. From the sum subtract 1008, multiply the remaiiider by 36, and divide the product by 144. 5. A merchant l)uys 100 Ijarrols of koroseno oil. Tho oil alone weighs 28000 pounds, Sui)])osiitg a gallon of oil to weigh 7 pounds, what is it worth at 10 cents a gallon? j 6. Tv.'o men, A and !>, buy a farm containing 162 acres, worth $12 an acre. They agn^c to divide it .so that A shall have 20 acres mf)re than B. How much must each pay? 7. A lady j)urchase3 at a store 17 yards of flannel at 18 cents a yard, 1 2 };irds of cotton at 1 2 cents a yard, 3 pair^ of gloves at $1 2 ) a pair, and a ] a-r of boots at $2.75. What change docs she receive out of ir,=2)? 8. A farmer hires 3 men at $1.2) per day in tho haying season, ^low much must he pay them for 3 weeks' work, supposirg they were prevented from working during 5 da}S, owing to rainy weather? 9. What does it cost to k-ee}> a span of horses for 200 days, if each consumes daily 20 pounds of hay worth $3 per ton of 2000 pounds, and a bushel of oats every 4 days worth 40 cents a lushol? 10. A lumberman hauls during the winter 1560 pine logs, and 16S0 spruce logs. It t^ikes 8 pine logs to make A thousand feet of lumber, and 1 2 spruce logs for th« siima quantity. What are the logs worth on the landing, at C'5.50pcr thousand for the pine, and $4,20 per iliOTJ^and for the spruce? 'a i..i 02 DIVISION. 11 . A tram, m liose usual rate is 30 miles an hour, sUrta on time to make a run of 120 miles. After runnin« 3 hours, an accident occurs and the traia proceeds at half itu usual rate. How late will the train be 'i J 2. A merchant bought 5C0 barrels of a|>nles nt $32 fop every 8 barre s and sold them at $30 for cveij 6 barrels ; how much did he gain ? * 13. A man has a quaf.t-*)y of coal for wh.ich he paid f2r)0 By letailmg it at $6.25 a ton, ho gained $20 ; how uuu.y tons had he / ^ 14 Tf ^-4050 is paid for 30 barrcit of apples, each eontainmg 3 bushels, how much are they a bushel i 15 A man has 16 cords of wood, worth $3.50 a cord, an*J 24 tons of hard coal worth $(}.25 a ton. If he slua-ld v:^^^:^ '^^ * ^"■^'^^^^' ^^'''' ^^^^' ^-- --^ m A fiirmor buys goods to the anu^unt of $235 25 and pays down $37.50. He a.^rces to f.rnish >vLat for tho 17. Define Mr.Itiplicatlon and Division. Show that the product of two numlers is the same in whatever order the operatK.u is performed. r?r.'>?'''ff '''''' ^7"" '^""^^^^ '^f ^^>ich the product is ifr ' *^'«P'*^^t.^'^ number is 875 i lind the sum and ditFerence of the numbers. WKnf^^'^^'f ^^'^ ^T^V f^^^tor,' « product,' and 'quotient.' ^^^^^"^ "^'''^'"' '^ '' '''^' ^-^ ^^- -- I-^-t 20. The Quotient arisinor from ih(^ ,],'vJ^;,,„ ..e nao^ r^ , certain number is 1 7, and the remainder is 373. Find tL» divisor. - - J FRIME NUMBERS. CHAPTER II. 5^ PRIWE ATlJ.nBERS. 1. What numl)er is contained an exact number of timei In 1 ? in 2 ? in 3 ? in 5 ? . ?■ o^y^^i number is contained an exact number of time* m / « m 11 ? in 13 ? 3. What number is contained an exact number of timei in 17 ? in 19 ? in 23 ? Each number given above is exactly divisible only by itseh or unity, and is called a Prime Number. Hence, 7H. A Prime Number is a number which can be exactly divided only by unity or by itself. 74. A Composite Number is a number that can be exactly divided by other numbers besides itself and unity. Thus, 4, 0, 8 are composite numbers. T5. The Prime Factors of a number are the prime numbers, whicli, when multiplied t«)irether, will produce that number. Thus, 3 and 3 are the prime factors of 9. Oral Exi^rciiios. 1. Make a list of prime numbers up to 20. 2. What are the prime factoid of 8 ? 10? 12? 16? 3. What prime factor is found in both 15 and 27 ? in 9 and 21 ? 4. What are the prime factors of 32 ? 3G ? 45 ? 56 ? 5. What are the prime factors of 33 ^ 54? 75? C. What })rime factor is fuuud in both 25 and 45 ? in 21 and 40 ? 76. To resolve any number into its prime factora. Example. — Find tho nrime factors pri 84. 54 PRIME >iUMB»iRS. lif 2)84 Dividing 84 by 2, a prime factor, we obtain 42. 2)42* Dividing 42 by 2 again we obtain 21» ^.— Dividing 21 by 3; a prime factor, we obtain LJL 7. As 7 is a prime number, the division 7 cannot be carried further. Hence the prime factors of 84 are 2, 2, 3, and 7 ; that is, 84-^2 x 2 x 3 x 7. 77. From tlie example and illustration given above, we obtain the following RULE— Divide the given number by the least prima number that is an exact divisor; divide the quotient in the same manner, and so continue tha division until the quotient is a. prime number. The several divisors and the la^t quotient are the required prime factors. Exereiise 20. Find the i)rime factors of :— 1. 35 f). 75 11. 145 10. 1008 2. 42 7. 91> 12. 216 17. 1728 3. 48 8. 105 13. 275 18. 1920 4. 64 9. 120 14. 3')0 19. 2240 5. 70 10. 102 15. 040 20. 3310 Caucellatioii. 78. Since dividing both dividend and divisor by the same number will not cliange the quotent it is often possible to shorten the work of division by rejecting equal factors common to both dividend and divisor. This process is called CcjiceUatiod from the rejected factors being usually croased or cdiicdled. ExAxMPLE 1.— Divide 90 by 1(). $ X X ^ Indicating the division and = factoring, the dividend becomes ^ X ^ 8x0x2, and the divisor 8x2. Cancelling tlie factors 8 and 2, common to both dividend and divisor, the quotient is 6. Example 2. — Divide 84 by 30 . 84 ^)^x7 7 Resolve 84 into 12 x 7, and 30 ~~^ = '" "=-" — = -3 into 12 X 3. Cancelling 12, the 30 Itx^ 3 factor connnon to both dividend and divisor, the quotient is 7-f3 or 1:"3. 90 16 i PRIME NUMBERS. Exercise 3q, 1. Divide by cancelling common factors, 126 by 27 ; 540 2. Divide 87 by 15 ; 132 by 16 ; 175 by 45. 3. Divide 26 X 4 X 10 by 13 X 2 X '15. 4. Divide 18 x 42 x 5 by 7 x 9 x 20. 6. Divide 63 X 48 X CO by 15 x 21 x 16 6. Divide 64 x 81 x 25 by 24 x 27. 7. Multiply 8 times 66 by 5 time;5 18, and divide tin product by 33 times 72. 8. Miiltiply together 10, 5, 4, and divide the product hm 8 times 20. ^ 9 How many pounds of butter, at 16 cents a pound, will pay for 18 yards of cloth at 48 cts. a yard ? 10. How many barrels of flour, at $12 a "barrel, arewortk as much as 10 cords of wood, at 83 a cord ? il. At 14 cents a pound, how much sugar can be bought for 2 cords of wood, at 84 a cord ? 12. How many bushels of potatoes, at 60 cents a bushel, must be given in exchange for 4 calico dresses, each containing 12 yards, at 30 cents a yard ? 13. A farmer gave 3 firkins of batter, each weighin>'^ 58 pounds, in exchange for 300 pounds of beef, at 14 cents a pound. What was tlie butter worth ? GREATEST rOWnO\ WEAStRE. 1. What numb.^r will exactly diviile 8 and i- It will be seen tJiat 2 is a nuni})er and 4 is a numbc* that will exactly divide 8 and 12. 2. Name a number that is an exact divisor of 5 and 29t 3. What numbers are exactly containod in 16 and 32^ 4. What numbers will exactly divide 36 and GO ? 6. Name two numbers tliat are exact divisors of 24 and 56. Each of these exact divisors is called a measure of tho numbers. If a number is an exact divisor of two or more numbers, it is said to be a Common Measure of these num^bers. Hence, 79. A Common Measure of two or more numbers is a number that will exactly divide each of them. Thus 4 is* common measure of 8 and 12. =' i i I I ss • i •t i: - * I > " ^S- OREATEST COBOfON MEASURE. If this Common Measure is the largest number that is an Jtact divisor of two or more numbers, it is called the Greatest Common Measure of those numbers. Hence, 80. The Greatest Common Measure of two or more ■umbers is the largest number that will exactljr divide each « theni. Thus 4 is not only a Common Measure of 8 and «: as above, but their Greatest Common Measure. Ihe Greatest Common Measure is usually written G C. M 91. To find the G. C. M. of two or more numbers. Suppose we are required to find the G. C. M. of 4 and 8 Ifc IS seen that 4 is their G. C. M. Again, suppose we are required to find tlie G. C. M of » and 30. It is plain that 10 is their G. C. M. It is also plain that 8 is the G. C. M. of 16, 32 and 40. ^ Now in the first case 4, which is the G. C. M. of 4 and 8 w equal to the produot of the prime factors 2 and 2 A^ain' 10, the G. C. M. of 20 and 30, is equal to the product of the prime factors 2 and 5. In like manner 8, the G C M ^16, 32, and 40, is equal to the product of the prime factors 2, 2 and 2. Hence the G. C. M. of two or more ■umbers is equal to the product of the prime factors •ommon to those numbers. In exhibiting the work the following form may be used : 2)16, .32, 40 Taking out all the prime facrors common to' 1 2) 8,_10, 20 t'^e given numbers, we find them to be 2, 2)4, 8, 10 ^^"'^,?- Hence their product 8 is the 2 4 5 ^- ^- M of 16, 32, and 40. Example — Find the G. G M. of 36, 48, and 108. 2)36,84, ^08 2)18^42^54 ^)_9, 21, 27 3, 7, 9 Here, proceeding as before, we take out the prime factors common to all tlie given numbers, and find tliom to be 2, 2 and 3 Their product 12 is, therefore, the G. C M of 3G, 84 and 108. Sa. From the examples and illustrations given above, ▼e obtain the following ig 'S^Z%l^t i^l^T%'T"" '^^^^'^ *' an the numbers GREATEST COMMON MEASURE. 67 Exercise 31 Pind the G . C. M . of: 1. 18, 42, 72. 7. 2. 26, 54, 96. 8. 3. 48, 80, 120. 9. 4. 25, 75, 100. 10. 5. 36, 54, 90. 11. 6. 27, 63, 117. 12. 19, 81. 30, 70, 68, 112, 95, 135, 7o, 15, 102, 196, 380. 540. 105. 210. 238. 272 83. When the numbers are large, the following method will be found most convenient, but its explanation should be deferred till a later stage . Example . — Fii.d the G. C 247)323(1 247 76)247(3 228 19)76(4 76 measure required. M. of 247 and 323. Here we divide the greater number, 323, by the less, 247, and obtain a remainder, 76^ which we now make a divisor, and 247, the former divisor, tlie dividend,' and so on. When the remainder, 19, is used as a divisor, it leaves no remainder, and is therefore the greatest common 84. From the example and illustration given above, we obtain the following . RULE— Divide the greater number by the less and the divisor by the remainder, and so on, until there is no remainder; the last divisor will be the greatest common measure required. 85. To find the greatest common measure of three or more numbers. RULE.— Find the G C. M. of any two of them, and then of this divisor and a third number, and so on, until all the numbers have been taken. TLe last divisor is the G. C. M. of all the given numbers. Exerci!»e 33. C. M. of : 1. 2. 3. 4. 6. 6. Find the ( 36 and 48 96 169 136 384 T. u (( u a a 90. 144. 128. 866. 455. 1290. 7. 632 and 1274. 8. 1908 " 2736. 9. 804 " 2240. 10. 2145 " 1157. 11". 465, 1365, and 215. 12. 744, 492, '' 1044. f, ii ; • i I 58 OKEATEST COMMON MEASUlt**. lo. I have r /oils of carpet, 56, 72, and 112 yards long respectively. What is the length of the longest equal pieces into wliich the o rolls may be cut ? 14. Two streams are 1520 and 1280 feet wide respectively. What is tlie length of tlie longe.st e«iiial spans by which they may be bridged, and the number of spans in each? 15. A teamster agrees to haul 132 barrels of potatoes on Monday, 84 barrels on Wednesday, and 108 barrels on Thursday : v/hat is the largest number he can carry at a load, and yet liave the same number in each ? 10. One pile of boards is 22 feet long, another 17, and a third 15 ; what is the length of the longest equal pieces into wliich each of the boards may be sawn? 17. A farmer has 1008 bushels of wheat, 1152 bushels of oats, and 720 bushels of barley. Find the capacity of the largest bins of equal size that will exactly contain the whole without mixing. 18. I ha\'e a lot wliose sides measure respectivclv 42 feet, 84 feet, 112 feet, and 120 feet. I wish to enclose it with boards havnig the greatest possible uniform length ; what w.dl be the length of each board 2 LEAST CO^inoIV MULTIPLE. 1. What number is exactly divisible by 3 ? by 5 ? by 7 ? 2. Name two numbers that exactly contain 4, 8, 6, 12. 3. Name a number that is exactly divisible bv 9, bv 11, by 13. J' ' J' » 4. What number is tliree times 4 ? three times 5 ? five times 7 ? 5. Name three numbers that are exactlv divisible bv 3 ? by 4 ? by 5 ?. " ^ Numbers that are exactly divisible by other numbers are called multlpks uf those numbers. Hence, 86. A Multiple of a number is a number that is es;act^y divisible by that number. 87. A Coin moil Multiple of two or more numbers Is a number tliat is exactly divisible by those numbers. Thus, C and 9 are common multiples of :>. -m:^ LEAST COMMON MULTIPLE. 6»' 88. The Least Common Multiple (L. C. M.) of two or more numbers is the least number that is exactly divisible by those numbers. Thus, 24 is tlie least common multiple of 4, 8, and 12. 1 89. To find the L. C. M. of two or more given numbers. ExAMPLii 1.— Find the L. C. iSh ot ft, 12 i5, and 28. 2)8,JL2^5,J8. 2)X_C, 15^]4- 3)2,3,15, 7. 2. 1, rK 7." Set down the numbers in a lino. Wo see tliat 2 is a factor of 8, 1 2, and £8. Dividing tlies-e numbers by 2, we write the 2x2x3x2x5x7 = 810L.C.M. cjuotfents with 15, the undivided number, in tlio second lino. Seeing that 2 is a factor of 4, (i, and 14, we divide as before, and obtain the third line of numbers. Dividing by 3, avo obtain the fourth line of nunib-rs, wliieh are prime to each other. The divisors 2, 2, and 3 are tlie j)rime fact >rs tliat are common to two or more of the nuDibers, and the (quotients 2, 5, and 7 are the factors not counnon. Then tlie product of tlie divisors 2. 2, 3, and the (paotients 2, 5, 7, or 840, ia tlie L. C. M. of 8, 12, 15, and 28. Example 2. —Find tlie L. C -.^f. of 6, 8, 12, 24, 3G. 2)C, 8, 12, 24,30. 2) ;;] Il2^. 3) ^C, J. 2; 3: Hero, after arranging the given numbers as before, we strike out i'>, 8, and 12, since they are exactly contained in 24, another of the given numbers. 2x2x3x2x3=72L. C M. .._ __. Then, proceeding as in the last example, we have for divisors 2, 2, and 3, wliicli multiplied by 2 and 3 the ouotients in the last line, gives 72, the L. C. M. required. 90. From the examples and illustrations given above, we obt;iin the foUowhiir: — 1 •I ' f if :^ eh JiEAST COMMON MULTIPLE. •«^^fr"? down the given numbers In a Une. and strike out 5i2i^* *" exactly contained in any of the others. Then dMde by any prime factor that Is contained in any two or more Of them setting dowii the quotients and the undivided numbers 111 the line below. Proceed In the same way with the second Une of numbers and so on until a Une of numbers that are prime to each other is obtained. Find the product of the divisors uwd and the numbers in the last line ; this wUl give the least •commonmuUiple of the given numbers. Exercise 33. Pind the L. C. M. of : 1- 18, 25, 40. 2. 32, 3. 7, 4. 25, ^, 6, 8, 12, 50, 7, 45, 64. JO. •75. bO. 6. 12, 18, 54, 80. 7. 30, 54, 75, 120. 8. 84, 1008. 9. 21, 5r,, 84, 104. 10. 256, 309, 75. 13. 14. \l' I' A ^'n P' n ^^' ^2, 32 and 99. 12. 2, 4, 6, 8, 10, 12, 16, 18 and 20. ^. 6, 9, 12, 48, 21, 24 and 16. 6, 10, 14, 18, 22, 28, and 32. 15. Abutcher can buy calves for $3 each sheep for $4, each andoxenfor$75each. What is the least sum of money for which he can buy an exact number of calves, sheep or oxen, and how many of each can he buy with it ? 16. A trader can buy apples at |3.50 per barrel, or potatoes at $1.25 per barrel. He can make 50 cents a barrel on the apples and 20 cents a barrel on the potatoes. What IS the smallest sum for which he can purch.-'.se an exact number of barrels of each ? J'^^'jy^^^ vessel holds 18 gallons, another 25 gallons, and a third 32 gallons. What is the smallest number of gallons which can be exactly measured with each of them ? 18. W^hat is the smallest number of pounds which can Bxactly be weighed with a 5 pound, an 8 pound, a 14 pound or a 24 pound weight ? - ABACTIONS. 61 CHAPTER III. FRi* quotient, the remainder with the divisor underneath for do- nominator. 1. 31. 6. 13H. 11. 12511. 2. 4?. 7. 3511. 12. 221/,. a 6|. 8. 7lf. 13. 115}^. 4. 12|. 9. 68||. 14. 128-}|. & 13^ 10. 1 721 f. 1 casi 1^ J HI. lOU?. Ffl ACTIONS. C5 01< it, Id ExerciNe S6, Reduce tlic following fractions to whole or mixed numbers. 1. 2. 3. 4. 5. 6. 107. To rodiicea fractirm to its lowest terms. A fraction is in its lotvcst terms, when the numerator and denominator are i)rinie to each otlier, or have no common divisor. ExAMPLK.- IUhIuco thefi'action ^'IJ to its lowest terms. Dividing Jxith terms of a DO — 2 15 fraction by tliesame num1)er .-[ .7 = 01 '"^'^^^ does not "alter the value: *--^- ^l Y- 7. 2 sa 5.-. • 13. Wt^. Ail. « 8. 14. •ViS^. ■J* 10* 9. W- 15. ^^. 10. W- 16. ^4^;^ w. 11. r>i.B 17. ^2^(5^. 1 • 12. CASE IV. 18. H .1 *•« %> i '-i A • 15-^3 21^3 5 ■7 hence we divide both terms of 'III by 2, both terms of the result l^i by ;>. Since i has its numerator and de- nominatoipr/wetoeaeh other, it is in its loivest terms. We might have foui d t' e d. C. M. of the numerator and denominator and divided both terms by it at once. Hence, 108. RULE.- Divide both terms by a common divisor and the result ? gain by a common divisor and so on till the numera- tor and denominator are prime to each other, or divide both terms of the fraction by their G C. M. Exercise 37. Reduce to thtii' lowest terms. 1. 2. 3. 4. $. 6. i§. 7. \l 8. 5^ 9. loo 10. ft ^ f ' 11. aao 12. 1.03 8G4' 13. 14. J.}^72 'JOIO' 15. £5000 10. ■2 5 a 2 o450' 1.(132 2 y . (3 • I0A9. -644 j 9 5 ^* 'f'illttt^rtl^fc'i'hi'^W'^ri^aMiWiViil - -, ' ^g-^i nje^aa 63 FRACTIONS. CASE V 109. 10 reduce a compo„„,l' fraction to a simple one no:ti.^r:^-Sn.;:z:;tT -^:'-''- -^ "«■ Oral Ex<»reisi»s, dow" ""^'^ ''^ of 8 apples? of 12 oranges ? of 16 2. ^«^vmuchis^uf8?ofl2?<,f KUofoo? eijlt^;;;""' " ^ '' ' ""^^^^^ ^ ^^ i2 tlmteenths? of 16 I of 8 ai)ples =.4 apples ; therefore I of § ^ 4 nmtli^ (^ \ 4. How mucli IS | of GO ? # ? ^«. ? 11 fa J ' "*^^^ ^^--^ ^ 4ofC0^15;tIiereforeill5'x3U45' o. How much is 1 of G ? of }^ ? of 1 1 y of -i ?.>f -^1 1 ■ t- ^^^"^"^^i-fofi.aof^lrayi'ifi f ^^^ 7. How niucli is .1 of ^ 2 , ^^ 1 5 • 7 or v^-^. =/;. tlierefore 1 of 1 = J. of ■• - t 8- Whati«.of.?iof^?i;f ,j':^M"ofM y. A person buvs ^ nf n 1. 1 / \ i- 7 or ^ ? «eighb J„. i „nt"^u mu^i'lL't 'ir™' ^^"^'"^ • Example l.-Re.luce | of | to a .i,„,,le fraction. * - if, i of fl = ^4^ ; therefore f of ^ = 2 x *~ = ^-^ 2 ^,S'^.:'3tr "sx«;- ;^!;:- j-^SL the 110, RULE-Multiply tog-ether •> i fh. " new numerator and 111 the deromin.fnr.'''''?'®'"^*^^^ ^^^a nominator. "enommacors ror a new de- befo?JapT,bif tl™nt" '""'' ^ ^•"'™^'^ "> "«P™Per fractions, Ex.i,,PLE 2.-2|of5of31 = V "f f of .i=.^'i«=48i Compound fractions niav often 1).. ,■, ,1, ' i i " factors common to a n.Jr:l^l^ ^::I^^'^<^S Example 3. i.ofiofH^''^^l_2xftx3 4 9 10 i'xVxo'x'y- 3x3 9 ^ rr =7 =14. FEACTIONS. 6r of 16 Exercise 38. Express as simple fractions : — 1. 2. 3. 4. 5. 6. 7. 8, 9. 10. & of -% I of f of 4. I of I of 3|. I of § of i\j. ^ of i of 3i. f of|of7|. 3iof Uof 3§ «' & of S nf 7 nf S oi Y or 3 01 Tj. f of i^ of 9 of 01 -^ of ^'- of ^ of -^- 11. -A-of2i of f of Jjj. 12. j% of vfy of f of 7. 13. fy of i§ of 51- of i. 14. 3;^ of 2| of § of tV 15. 21 of Vj- of 4^ of #.. 16. I of 6^ of \l of ijr. U of 2| of 3| of 4i. 4l)f 12^of j^of |of f. 1 -S- of y of 9i^ ( ' ■'- of -^ ^g OI 7 OI ^0 < .^ )I oY" 17 18 19 20. i of ^of 3? of] 4 of 51. 21. What part of a ship is | of f of it? 22. A man has | of a load of hay, and gives his brother J of it : how much does he give his brother ? 23. If [ ov/n 4 of tho steamer ''Acadia," and sell j of my share, lunv mncli f>f the whole steamer do I sell ? 24. Two b(»ys have between them ^- of a dolhir, and spend I of it for peanuts ; how much do they spend? 25. B has f? of a ton of hard coal, which is §as much asO has : how much has C ? ' m CASE VI. 111. To reduce fractions to equivalent fractions having the least common denominator. 112. Fractions have a Common Denominator wnen they have the same number for a denominator. 118. A common denominator of several fractions is a Common JNIultu'LE of their denommators. Hence, The Least Common Denominator of several fractions is the Least Common Multiple of their denominators. Example. — I^educei, §, £ to equivalent fractions with a common denominator. Multiplying the terms of a fraction by the same number does not alter its value ; therefore : — FRACTIONS. h'i 1x3x4 5x3x4 X i ~ M It a _ 3x^5 _ * ~ 4 x 4 X 5~ ~~ 2^4^ 3x4x5 4 5 f Exercise 39. de^u :L^ '^'^^^^' ^-^ti-s having a common 1. 2. 3. 4. 5 5> 1 65 J' a* o a 4» f. 1 9* 7 Example.— Reduce 5. 6. 7. 8. with least coinmordenLLklr?'- 3 o 7? 5 S> 3 .4_ h f. »' 11' H' J', 3 L> 5> 11' IJ? to e(iiiivalent fractions f = SxC) 4x6 _7x3 8 x"3 ,_ J_x2 ^~ 2x12 24: T -_ 8 — 2* 1'4 lii The least common e q.u.tiont^'fi i, to each the other fractions'"™ "' "'"' '"^"'■"' ••'-' -'"'••"•')' with other;'?h'e"oll™ in:.!"'"""'"""' "^ "»' !»'- 'o each wi"b^.l.fL^e!stSnU'5fLfom^;.^' the denominators : this denominator and mSS each S^^^^^ H^'^i?' *^^« ^^^ ^^<^^ Obtained. The produclf ^iirbV^h^n^^S^^^^^^ *^^« frii^S;t,:StrR!S.n^?S^^1.K'"'"^^d t*' ^proper the rule can be a^i^plLd ' "''^''''^ **^ ''^•^"- ^^^'^^^^ terms before f ^ on. FB ACTIONS. 69 a other, as lenomina- fcion, and ator. common Pract ions r is the iiomina- ice24 is 'tors, it minator ^. C. M. ch is 4, ito each ■ly with 'jO each rs; this by each >at thus iiproper !S before Exercise 40. Beduce to equivalent fractions with the least common denominator : — 1. 2. 3. 4. 5. 6. 7. 8. 2 a 5 35 4» ^' 9» ¥T» t* 3 _5_ 11 7» 14» 28* 6 JB^ 14 _ 5_ 0* 18* 4 _3^ 7 55 2(J' on* i^ 9. 10. 11. 12. 13. 14. 15. IG. 91 qi Kl 5 ii^, O.J, Of;, Q- Ih 2^, 3i, 5^. 6' f?' 9' in- S _T_ .fi. 11. 5' 105 255 :jO» _3_ fl JL la 155 45 10> 2o» 2 7 Ilia ;J5 85 125 20" 1 2 A 2. 35 55 e5 8' A ;i -1. 1 _i_ 95 35 45 05 I'Z 116. To compare fractions with respect to magnitude. This means simply to find out which is the greatest of a given number of fractions, which the next greatest, and so on. We must bring them to a common denominator. They may then be compared like other numbers. Example. —Compare the values of f, g, |, i^^r, ^-'}. Beducing them to a common denominator we see that the fractions arranged in order of magnitude, are : ih 1?, ^f5 U^ ti, or, I, i I It, 1^. Exercise 41. Compare the value of : 1. f 5 h tV ^• 2. i,f,f. 3. ^ of I, I of ?. 6' h A> l's5 ^V 3 27 J»^ _7_ 8 5 .')2 5 1^5 lIT' 7. ■?of|of4,i2rof fofS. 8. 9. 10 Si A3 s5 .:aL •»-^« 75 215 4 25 14* _5_ JL iO §1 125 1^5 215 tO' 5 11 _4_ _7_. 65 245 125 15* I*ind the greatest and least of the foilnving fractions ;— * i m II H i ■#. P i. I fV JlDDITION Of rRACnONS. Addition of Fractionii, CASE L 117. To add proper fractions. Oral Exercises. 1. Add together 3 pencils, 4 pencils, 6 pencils. 2. Add together 4 apples, 5 apples, 7 apples. 3. Add together 4 ninths, 5 nintlis, 7 ninths. 4. What is the sum of i, fy, ^-, ^- ? in^h"^? 3"/;,^fV^^^"r^^^^^^ -^^^^ ^t--^^-r, how 6. John bought a slate for 1 1, a geography for $a and an arithmetic for $f . How much moneydid 1^ spei^ hiS TTnw^ "T f \'^- *i^^ Y'^ ^^'^^ ^^ ^^^1 <^o A, and I of it to B. Mow much of his load did he sell ? ' ^ i = llr, and ^ = J^ : therefore ,^ + V 2:. = J, ». It 1 have to pay } of a dollar for eggs, i of a dollar £ pT; in' aT? ^ "' ' ''""■ '"' ^"''^^' ^^^^^ ^^^^^*^^ ^' I ^"v1 i a^nd^i''"^ ^^"^ '''"' ""^ - ^''^ ^ ' "^^ ^ ^^^^^ i J of i and i 5 of Example.— Add together |, J, J, J. We first reduce the given fractions to equivalent fractions having a common denominator, so that they may express L/etl cr^^Tn^ rf'- 7"^' '^T '^^^'^ ^'^'^ numerators iogethei as m wholo numbers, and place the sum over the common denominator. Hence, the followm- •- 118. RULE.— Reduce the fractions to eriiiivnipnt-. f^a«ti««- Sfr I^^rf''^^""''" denominator: add the num"era'tor"s to-ethw ADDITION OF rEACTIONS* n how and nail? toB. ollar have hof 6. A, if, land*.' 6. f, ifandH- 7. f, iHandfl. 8. i, i, I and {^ Exercise 4;S» Add togetner : — 1. I, ^V and j\. 2. §, I and f . 3. ^, § and f . 4. -I, ^, I and -5^, r 119. To add mixed numbers. Example.— Add together 3^, 6J, 2|. 3 + 0-f-2-ll, and IH- 1^ = 12^. Hence, the following : — 120. RULE.— Add the sum of the whole numbers to tho sum Of the fractional par .s. Exercise 43. Add togetlier the followmg : — • 1. 7i 13.1 ^^nd 5§. I 7. 2. 5|, G| and 4L 8. 3. 5fr, 6and72.^ 9. 4. 2^,, 3-5 and 4^. I 10. 6. 17i, 2|and3. | 11. 6. 91, 12.1. j^nd m. I 12. 51, 31, 4f and 6J. ^, l/o, 10| and 5. 2|, 31, 4| and 6. 1^, i,-,^gand2TV 6|, I of 71 and 8iV 1^, lA, f., 2Hand3ft^ '8 litii I j^ubtractioii oT Fraclions. :ion8 )resa itora •tho thei mofi CASE I. 1!21. To subtract one fraction from another. Orail Exercises. 1. What is the difference between 7 apples and 4 apples j 2. What is tlie ditterence between 7 ninths and 4 nintha . 3. Tom has ij of a dollar, Will, f of a dollar ; how much enore has Tom than Will ? 72 SUBTRACTION OF FRACTIOXS. ^ = Ai 5 = f:.t therefore -/,- _ -3._ = j>_ how ,!;^.i;Ce f ;^t7 """^ ^^ ^ ^^^^^'§ ^-^ ^-«^. JanclTr* '' *^'^ ^^iff^rence between i and |? landp and -M '""^ '" ^^'^ difference between i and ,^,- ? | and 8 ? ^ Example. -Find the di.m-rence between " and 4 A 4 _ 2 S ') 5 ~ ;io 21 = a.- 21 ^ J_ orfir tTsnK^"' '" ''■'l^«,''"'l«'from sixtl,.,. Since* ^lumueis must be of the same name or ki'i i w^ rvn^o*. dttmili';? '""''r,'" """™1-" "»e3 Law; 'aTonTon' Exercise 44, Find the difference between : 2. 3. 4. 6. 6. ^ and I. 1^5 and 5%. I and |. f and J. if and i|. 8. 9. 10. 11. 12. 1% '"iiid tV i and J. 2^0 and /^. f f and ^. i^o^j and f, f § and f. CASE II. »ked nu^mbr™:' '"'' *'''" ""' °' ^'^ "^« ^'■^^«oii» are SUBTRACTION OF FRACTIONS. 73 and luch use. H n ) in iwo ust ion hs, ra- ExAMPLE 1.— Find the difference between 5| and 3J. =¥ = 2| Or, 52-3H5a-3-H2f. Either reduce the mixed numbers to improper fractions and apply the rule, or subtract the whole numbers and fractions separately and unite the results. Example 2 . -From 7 take f . 7 - 1 = ^^^ - f = '/ = 6i. Example 3. --From 5§ take 21. r.M _ 91—4^1 r> _ 4 r? _ 5ft _ ^0 _ i2_7. — "8" _ b " .'^ ~ H ■ Or, 5^U:2?. = 5^-2i, that is o^.^ -3.i: = 2|. AVe ciinudt take ^- from p\ We must, therefore, add 1 to botli minuend and subtrahend the result i.* 2^ as in the first process This uives 5^^ - 3^ and Exerffsc 45. Find the value of : — 1. •-'4 ~ ^i' 7. 1 ^ 109 2. o'i - 2t. 8. 41 01 *-2 4 ~^n;- 3. r. or, 9. 94. .1 01 L. 4. 10:^-^1. 10. i^-ioif 5. 93 -.^. 11. 9:^-3^- 6. 5i - 2i 12. m - 3^. Exercise 16. e rrsic'tical Pioblesns. 1. The sum of two numbers is 13 g, and the less is Gj : what is the greater ? 2. A farmer sold 3f> tons of hay to one person, 2| tons to another, and li tons to a tliird ; what was the whole quantity sold ?, 3. I liave bought three lots of coal, weighing respectively 1|, ^*r;, and 9.;'^ tons ; how much is tliere in all ^ "4, "i boui-hta handkerchief for Ss, a vest for $2^, a pair of gloves for $^\, and a hat for |3;| ; how much did the whole cost? HI ^iii li ,7i SUBTRACTIOX OF FRACTIONS. \^ ^' ^^:°,"^''\Piec,-of cotton confcaiii;na2()'| yards" 12? varda. Trere so d ; how much remained in the plLT ^ ^ ^ L • u- '^ ^^'^ ^''^''''' ^^^G'g'»fc «f 3 tirlcins of butter JionlloTout^f containing 25| gallons of kerosene, 7} gallons le.^K out, hovsr much remains in it ? * J. li i have 10} bantMs of potatoes, and sell l'>^ birrolc how many barrels sliall I liave left i - '' rsi tile' U J;^^' '*'^'' ; '^'' "'1 ^' ^^^'^^ ^'2|, the second ^-»3, the third ^^o^.,, Iiow mucli are they all wortli? o^er1;f"^":™^:^;^^^"''^^' one of po^ acres, and the „ jvi -.' •^^^^''"'; lie i^^iVL'S his edest son 1051^ a('ro« youngest t-cu recui.ve for liis simro ? a dktancToftr'!"! "■' "xl'' ^'■"V,' F'-odorlctou t„ St. John a aiocance ol iH imios. Ho iva ked (ju tlio first ,Uv 9^11 ..nles on tl.e .ccond .'81 miles, oa to aurd%^'Z&^ how far has lie yet to Hulk l " ' rcth^:;^^^?;'!'^!,;; ?"'' '^'•"' «'-'« ^^ «5 be taken the and ,wSti::£r.tr'^'"^' "^ "'^ f™=«-- - *. 1. 1. 1. >>JL|!S^tl:tr'^^l-Bl;^^^Cpndthores. *-y. 4 oJ: an estate beloiurs h. a -' ,.f ,> fTn i 2u remainder which is wo • h ^500 U C VV, f V,'"''^ f^ of the wlio].. o f ,f 11 ^"^'^^' '^^^ ^- ^Viiat is the value ga^"c '.°.y V,>' "' ^''fVl.''?'l,l'^ft?»i "lore than I -■^ui :._^ i at Hr^t i ik.^.,™ MULTIPLICATION OF PRACTIOlirS. 70 IfluUiplicatioGi ot Fractioos. CASE I. 131. To multiply a fraction by a wholo number. Oral Exercises. 1. What is the product of 4 apples multiplied by 3? 2. What is the product of 5 dollars multiplied by V ? 3. What is the product of 5 seven*-hs multiplied by 7 f 5 sevenths multiplied by 7—35 seventlis {'^A==b.) 4. ITow much will 4 pairs of chickens cost at $1 a pairt 5. If a slate cost $1, how much will a dozen cost ? G. If 1 yard of cotton cost "*:' V, how much will 15 yards cost ■? 7. If a man can saw ff of a cord of wood in one day, hoTT many cords could he saw in 1'2 davs ? 8. If a barrel of pork cost .$ls|, whatv/ill 3 barrels cost? Note. — Multiply the wlv-^* mntiber aii'l the fr.ictions separately, and add the product. ' 9. A barrel of a])ples costs $3f; ; what will G barrels cosb at the same rate ? 10. How niuch isn times 5? 5^? B? _ii_v ii ? r^t 11. What is 8 thues 'i ? 3.'? ^i ? i<| ? i^i? " 12. How much is 12 times 2% ] ip. G;] ? i'o|? U\lt Example 1. — Multiply ^ by 3. 4 apples multiplied by 3 = 12 apples. Then 4 sevenths (-') multiplied by 3^12 sevenths Q^.) Orfx3 = -\^ 4x3 / ill' wl Example 2. —Multiply 1*5 by 3. tS^^~ 15 15-5 ^^ 15 ^'^~^KJ^ 4x,' 15^3" 3 times 4 fifteenths, or ^^ is {-§ which reduced is |. Or, Since dividing the denominator multiplies the fraction* 4 3 times j*^ is y^"^ ~ t- H^^^ce, the following : 76 In ^ii MtrLTlPLlCATION OF FRACTIONS. caJ^^'e doL^'^i^oufa r'l'LiiSL "^^ °^ ^^«^ this the Whole niimber "^^^^^^^^^^r* ^^^ide the denominator iby Multiply : — Exercise 47 1, 3. 4. 6. n by 3. 5 A- by 11. " by 3. /■? by 45. U by 25. *? by 21. i^/-j by 18. 6. 7. 8. T^i's by 25. I 9. iVa by 10. I 10. -'"-bvlS bisi^?'" '°^' '^« ^-^'"-"^ of -r„/:^ /of a dollars weeks^ofllaya'S;;,!'?" '""''" "'" " ^'«»e-mason earn in 6 CASE II. 126. To multiply a whole number by a fraction. Oral Exercises. J ofa Wrd'c'oL^ P"*"'""' '"'" ^ <=^"*'' ho" ">"<=!> wm n>onthV ''™ **' ^ ™™"'' ''-^ »"* 'io I earn in J of a 3. If a ton of coal cost «8, how much will | of a ton cost 1 m, , 1 ton costs $8. and I u u 3*;;$o^|g^ ..Sxs^ 4. If a ear load of wood cost $24, wh-^t mil § of it cost ? Bl t S MULTIPLICATION OF FRACTIONS. 77 ExAMPLi:. — Multiply 10 l.y J. 10 X I- —f- j(, 1 fdurth of 10 is 4, and 3 fourths of K) are 3 times 4, or 12. Again 3 times 10 is 10x3, but the inultii»lier is iKjt 3 but 3 fourths, therefore the product istuily a fourth as large, and J of 10 X 3 is — T~"^*^' !-• Hence, tlie f(dlowing : — 127. RULE. Multiply the whole number by th3 numerator and divide the product by the denominator. j^QTf.;, — When the denominator is contained an even imtuber of times in the given nmu1)er, it is better in practice to first divide by it, and then multii)ly the quotient by the numerator. Kxercise 48. Multii»ly : — 1. 112 by tV. 2. 03 1)V I 3. 72 by :^. 4. 305 by ^J^. Example. Mult iply 33 by 3; . 33x17 5. 0. 7. 8. 45 by l^. 093 by I .105 by ..*.j. 1250 by I*. 33x3i=33x^f= o 5 :112l Note. — When the multii>her is a mixed num' it is blotter to reduce it to an improp.-r fraction, and then ai)i)ly the rule. Of course, we may also multiply by the whole number and fraction separately, and tli n add the products. Thus ; 33 x 3 =91), and 33 x } -«V^=13i, andl>; 131-1124. ultiply : — 9. 12 by 9^}, 11. 530 by on. 10. 43 by 8;|. 12. 150 by 10r;>. 13. If a car-load of coal is woi-th f IS, how mui.di are 12^ times that ([lumtity v-'''vt]i ^ 14. Boiiglit a barrel of i)ork for i<20 : how much at the Bame rate shall I pay for 10-^ barrels ,' ^ i 78 MULTIPLTCATIOfi OP FHACTIONS. l\ I CASE III. 128. To multii>ly a fraction by a fraction. Example.— Multiply | by i. -I X 5 = Y- But this result is 7 times too ?reat as ? is nofi to be multij)lie«l hv 5 butbv i r.f r, ., ^^ r ' * "°* i I oy o out by 1 of 5; the product therefore III 3x5 x7. must be only 1 seventh as large, and } of J,^ = iii= L_ ,„ will easily be soon tlvif w^ k, i.. ,. , '"'* ^^7. reduce ,.,ixed .."ulKo r,ipi;™.''';Si;"n^': '"'■''°"'' '""' '™^ Exercise 19. denm^naloS!"""^ ""^ ^^'*°'' ''""'^^^ to numerators and Find the value of :— 2. a 4. 6. 6. 7. 4 ^ li 1 • ■%*- y J>_ V 9 2 lU -^ 16 X ^l^'j. 2Ax2lx^. 191.x lj5. 8. 9. 10. 11. 12. 13. 14. |x^x7J. 2^ of 3f X 4| of 1|. i of I X o^ of 3. 7^xJ^of*of iV ix/,-xvx-i4j-x3^. 5|x}fxiax8. 16* A^^n ^ "^ -,«^ ^ ^-^-^l of cloth at'|2^^ Inl ? flour? ' ^^""^ '^ ^^'^^^^^ ^'^^^ «^^--^J I P'-^y for lo/barrels of agi;;r^iS[;^f!^'-^^^ 18. If a man mows 1 of 102 „ei/ „f "'!7t , how many acres wouki he moVln 2| weelsT" ' ""' ^^' DIVISION OF FRACTIONS. 71^ Dividion <»! Fractions. CASE I. 130. To divide a fraction by a whole nnrnDer. Oral ExerciHOj«. 1. If ('» pencils are divided by ri, what is the quotient ? 2. If sevenths are divided by 2, what is the (luotient^ 3. Divide t' l)y 2 ; f, by 3 ; h'i by 4 ; ^f by 8. 4. If 4 cliickens cost '~^ of a dollar, how much will 1 chicken cost ? Cost of 4 chi( kens = $f i. Then 1 chicken will cost ^ of 5. If 5 slates cost | of a dollar, how much will 1 slate cost? Ayl3. «afh ;" ^^-''^ "'■' i'"'' '"'■ ^ «'"■•'■ ^^ '"'t i^ the price of CASE II. Example 1 — Divide 1 bv " •> Jo' i~-2 = .-1 ^»H i« '^'^ ^'* ^'^ ''^^'^^^^^^ ^^y 2 but by i 0x2 of 2. ihe quotient, therefore, n.ust be 3 times as large, or -,3-^^ ^ dividend multiplied by divisor inverted. Example 2. —Divide 8 by f . Here 8 = -f and f -f 2 = -^- ^^'t 8 is not to be /.f 9 Ti ^ ^- . , ^^^' divided by 2 but bv I ih _^£ ''T ""'' "'"''■^"'■<^' ""'^' '« 3 times a.s lar^^J ■'•<3"' I ^T ■= '-'''■'^''-'"'' '""'tiplied by uivis(;r inverted. DIVISION OP FRACTIONS, 81 Example 3.— Divide 3| by 4§. "We must first reduce the mixed numbers to improper fractions. Thus, ^^ -r V = ¥ x i\ = IS- Hence, th© following general : — 18». RULE— Reduce whole and mixed numbers to Improper fractions ; invert tlie divisor, and proceed as in multiplication. Exercise 51. Divide :— 1. 5 by 7^. 7. 4| by 2f. 2. 49by2|. 8. fbyli- 3. 30 by J, 9. f of Jj by 2f.. 4. 10 by -,Ijj. 10. 6B by 4|. 5. lG|by]24. 1 11. (?of^)byaofJof5.y 6. §by|. 1 12. (4iof3i)by(2iof6i). i. ii' 13. At 20 cents a dozen, how many dozens of eggs can b^^ bought for 8G§ cents ? 14. If 4 pounds of sugar cost | of a dollar, how much will A pouj;d cost ? 15. If $125f are divided equally among 16 men, how much will cacli man receive? 10. The product of two numbers is 36, and one oi them is bf^ ; what is the other ? 17. How many bushels of coin, at foofa, dollar a bushel,, must be given for 1| of a ton of coal, at $4| a ton ?. 18. If the divisor is j^^, and the quotient 8J, what is the^ dividend ? 19. At the rate of 17^ miles an hour, how long will ift take a train of cars to run 486}^ miles ? 20. A cask of cider contains 52% gallons. If J of § of 6 ij'allons are drawn off every day, in how many weeks will the whole cask be emptied? 21. In 3 1 hours a train travels 87 J 3 niiles ; what is its rate per hour ? 22. Bv what must 21 be uiultiplied to make it equal to4»»„? 10 6 62 COMPLEX FRACTIONS. Complex Fractions. 134. To reduce a Complex Fraction to a Simple one. 135. A Complex Fraction ia a fraction in which either uie immei^tor or denominator, or both, are fraSns , L V ^' el' Example.— Reduce i- to a simple fraction. »ro'ri!I''*'"7"'?f " "'" 'l« '""""•■'tora of tile paifckl fractions are removed the traction ,vill be in a simple form. Wo t; ^cirr ', ''"'"'""'i','^''' V nutltiplyiug the fraction 67 such a numbjr as will contain botli of them. This number m clearly their L. C. M. Thus, luul Zi 1. tlo terms of the faction by 28, the L. C. M. of iand 7, wo have -^ = 4x28' 51 1:0 = i.'ff. Hence, the following : th5?^r M^n?Vh??^*'^''' ^°*^* *'^^^ 0^ ^^'^ complex fraction by the L. c. M, of ths denominators of the partial fractions. N"oTii.-If the numerator or deiiomiuatur, or both consist of a Example. ^J - ;:■■ ^ ^ 1:0 -10(7 = 5'(J- ExercLse 53. 1 1 I of ^ § of ^' U of 2} n <'f 3f 1 2 11 o -. • -'•TO- it 16., ^-"'-i . 17 '^1-2^ 18. 2^+T^. 19. ?i±i!. 20. ^"".11, COMPLEX FRACTIONS. 83 Example 1. -Simplify i of J - f of ^V + '^ of 1 | Q-. The pupils should be taught that the operations indicated by " of," X , and ~ must be parformed before adding or subtracting ; thus : — /4 1\ /2 0\ , rn of 2T^— ^_ '! 1 SI. — \J of ^) - (,-- of j;:;^(.-. «f 17) - .-. 17 + .,-, - 34-304-81 85 >5.> 1. Example 2. —Simplify f + ii of f ^ - i -r H I x A- 4 V;^ 11/ V4 3^ ^i^ n^ 3 4. _t 4 "^ 11: — 1 1 It will be ob-eryed that exprei^sions connected l\v th-^. signs "of," X, and -f , have been enclosed in brackets, which at once removes all ambiguity, as the brachts indicate that the numbers included within tliemaie to be treated as one number. ExAMj^i-E o. — Find the value of 04 - ,. oi T - H-(V^^A) 2H + f\j + 4^of5. 1 .'. 4 LLT _n_H 1 91 1 4_ •■'> -j_ni .if ri^ 1 0ii..J_ Ji-_L.:M"5 (!Lil>_^_•)_-t.-•blJL L-'li> iOii 100 ;m mid "~ 1554 -J ■' '.» iii ExAML'LE 4. — Find the value of ,1, of 1 + ^ 3 1 Begin with the lowest fraction and consider /j the numerator and 3 + 1 as the denominator ; tlius, J, of 1 + ^ = ,', of 1-f-^ ^-^, of X + J,=^^,oi ^tl = £, + 4 Jj ^ + 5 + ¥ ExAMiLE 5. — Simidifv the cxpre«sjic:n 11 1, 2^3^ ^4^ 84 COMPLEX FRACTIONS. ; u KHi=-^±i±^ 12 12) ~ -I f- — 126 + 90+70 ^ H 4i = ^+?+e = 2 , 315- J. J. _ -« 4 I. 22 This question has been solved by the principles of Division, but the rule could have been readily applied. £x«r€ijse 53. Simplify : — 1. 3| of 51 of 1-1 of ^. 2. (A + ^)-(3-i)xa + i). 3. 4. L+ i 1 + 1 -^■"l-r [ i + f of 5^ ^ X j f + 2 5. 1 + I of oj X f + | + 3|. S+3i ?^ ( h Q^ T4 . q N. B. —When a number is placed before 6.^/3 r H \ • iX a bracket, with no sign after it, it is im - Vsl ^ / ^ ^ ^* *^® number is to be multiolied by the contents of the bracket. 7. 1| + 2i + 5^ +31. 8. (fi + t of 7*) + A- 9- /T + fof7i + A. 41 -I off 10. ii + 2 5 11 1 4. 1 O- 1 2 + ;^ + 8 3 i 1 3 12. (i + 1) X (1| + 2|) X (2jij-U) X (3Jy— •?). 3-(iof^) 1 1^' 31 14. 2J + "31 + Jl_. 4' COMPLEX FRACTIONS. 85 i ''-•^* 17, ftof*|-«" 4t 4- .:^g 4— ;Ji' llij 18. \-A + 5 vt- 1^' i on - ^ of ^ + f of U?. 5^5^ 1^ of 4;^ ^•(Haf^)-MOi "^ fi'^'lai of Si- Review Exercises. I. 1. Define a fraction. How man;; kinds of Vulgar Fractions are there ? Give an exam})le of each kind. 2. When i.^ a f racti >n said to be in its lowest terms ? Reduce the fractions |)!],l '^, and -^i^;.;^^^- to their lov-est terms. 3. What is meant by the symbol \i 1 Show that multiplying or dividing both of its terms by the sam« number does not alter its value. 4. How are fractions compared in respect of magnitude ? Which is the greatest and which i , the least of tha lOUOWxag .y»Hl';i7) 1:J- 5. Reduce to equivalent fractions having lowest common denominator : j"y, I V, I g, ?:1. C. Why is it necessary hi adding or subtracting fractiona to reduce the fractions to a conunon denominatoi ? K II. 1. In reducing a fraction to its lowest terms, what princiiili is involved ? Reduce 'iiiS^I to its lowesi- terms. 2. State and illustrate the rule for the multiplication of one fraction by anoti-^', 3. Which is the g.c .c-r, f of 2f or I of IJ, and by how- much ? .AC, 1'?. V 4 Simplify Vyx "u / "^ ''^' 5. What number subtracted from 41| leaves 19| ? 6. If § of a r^\vp be worth 137-10, what is the value of the whole ? i.L II iH 86 COMPLEX FRACTIONS. i I' I ''1 I W i 111. 1. Prove by means of an example the rule for the division or tractions. ■ coft?"^' ^ ^^ ^ ^°"^ "^^ ^^''^^ *^^^^ ^'^^^' ^"^^^^^ ^^^^^ » ^^ 2 *^^^^3 4. A man has f of an estate, he ofives his son i of his share ; what portion of the estate lias lie then left ?" 5. A person owes a dolhir to each of G creditors ; to one he pays | of his debt, to aiiotlier |, t<. another 4, and to another ij} ; Avhat will lie still be oAWng altoc^ethei^? ( \^^'' ''.'''"'^^ ^ "^^ '^ steanibor.t and sold | of his share lor ^h4i)0 ■ what was tlie wliole steamboat wortli? IV. 1. In reducing fractions to a common denonnnator, what principle is involved ? Explain the terms numerator and aencmimator of a fraction. » 2. Add together the greatest and least of tlie fraction;? ?, (?, -, .J, ._,,'y : ami subtract tins sum from the sum of the other two fi;ict:ous. .n1* I! ^ l^!}^^^ 2 ^i my money, then I of wliat remains whLe^dl^b^l^i^f ^^^^^ --^^^^^ 5 Divide the sum of |, i and ^ by the difference between "§ anti ^. thfnrf^t^ together 1|, 2f, and 3i ; multiply this sum by the product of tliese fractions ; subtract from the result tha H DECIMALS. 87 CHAPTER IV. DE€I.nAI.S. « «. 137. If a unit be divided into ten equal parts, each of these parts will be 1 tenth. If each tenth be divided into ten equal parts, each part will be ^Q of ji(j, or 1 hundredtli. If each hundredth be divided into ten equal parts, each part will be ^^g of j^^, or 1 thousandth. In like manner, we may, by dividing by ten, continue to obtain fractions, the value of each of which is one tenth of the fraction preceding it. 138. A Decimal Fraction is a fraction which has 10 or some power of 10 for its deneminator : as ^^^j, iVo- A power of any number is tlie product obtained by multiplying the number by itself one or inore times. Thus 100 is the second ];ower of 10 = 10 x 10. 1000'' " third '' ^'10 = 10x10x10. 10,000" " fourth " " 10 = 10x10x10x10. Notation and NumeFation. 1JI9. In writing a decimal fraction, the denominator is commo)ily nnt expressed but understood. A decimal fraction is expressed by writing the numerator with the decimal p(.)int (.) before it. A decimal fraction, for brevity, is usually called a decimal. 140. The first place to the right of the decimal point kl that of tenths. Thu«, sJ'tt is written . 1 and read 1 tenth. " ,2 " " 2 tenths. .3 " " 3 tenths. -2^ " r* 88 JDECIMiXS. of wSf/'Thut *' """ °f«-deoi„>al point i. that Tk is written .01 and read one hundredth. ^p[[ !! '^^ " " ^^^ hundredths, du " " .03 " »« three hundredths And so on. T6^(5 is written .001 and read one thousandth '^"^""ll " •?^? " " two thousandths. IWD (( .003 " " three thouaandtlis. thTmtht^'i^lT '?"^ rig'^ti^ H'afc of ten-thousaudthi; me ntth that of hundred-tliousaudtha ; and so on. numeSn "'* °''''"*''^ ^^^'^"^ "* notation^nd 141. A mixed number is a whol^» nnmT^^T. o«/i j • Sr e'sTf ^^ ^t'^ *"f ^^-> Po-U^^e^f tS^ tW' hundredth?.'Kd aT^"'' "■'"' '" ^«^'' l^, and sixty! «i?.^7?!';!r:7ofa:f:?i:,tf"it"°'^^'"'''^»'-'' •70 = ,% = iV •700 = -,Vo«a=/,. „„f r'Tif '^l''*'' °" "'" '«^f' ''a" 1 ', 7^^' '^^•' ^^^ numerator and c;^u,uh^"tf ^;l;^\i J^^j;!^^^^ ^^-^^mg the as there are ciphers m thV i ? ^ ^^ ^^^^^y %ures 4 .56 -infS 1 ?-. ^'^^ denommator. For examnlo *ioooo = -l Woo and Tn ',t= -O'^'I «^-iampio Exercise 55» Express in vulgar fractions : 1. -8. 2. -0. 3. 03. 4. -25. 5. -875. 0. -784. 7. -1235. 8. -426. 9. '00342. 10. -OGOo. 11. -0212. 12. -4005. 13. 34-625. 14. -04735. 15. 125-03125. 16. 36034G, i te di 1)( 111 DECIMALS. 91 Express as dc-ciUials : 17. ,V -*• 100(7* 18. w,,. 10 - ■ o'} ■ 1 o:i I 14'.^ ' 20. ,j,„. • >j. ".r; 7 1 ■ "•'-^*-"* I odou J 147. Ada tugethcr looi-rouo (il-OLO .\ 23 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) 872-4503 4> ) ^ /WJ3 Ul ^ 92 DECIMALS. 1^. 7 084 + 32o0 + 18 + 00275 + 342-S7^ 4. i«.ok 14. 54 ^3-208 ^ 147-9142 -^ -00875 ' T.^l'f' hundredth; ^en S3" ^Z^^' ?^\!r^^-^ aandths ; four and four thousandth^" ^^^^^^'^'^^^^^ ^^ou- ■tt>. A merchant lias 3 pieces of oUn. respectively 18-375 yards, 29-^35 vards and sV «Sf '^'T^ how many yards are there m the SeVeees t ^^^^"^^^•■ 149. Subtraction. Example. -.From 17 015 take 5 -50304. 17 01500 4-5G304 12-4519^> or 17-015 4-5;C0t 12 45196 iJh. unit tntftlX' ir'". *''f """"^-J- P'--g thia example as twJ 1 ^ *' ^"^^'^ hundredths, .fee , In than in the mtuuend 2 ^"T'' ^«''?'' '" "'« ^^btrahend of decimals/ bv a' ne'Z TY" ^""' '° ''"^ "*"'« """'ber be annexed before per onuinff' "\ T ™PP'«« *em to not alter the v Tue^ff f h " * «'e subtraction . This does in whole number/Ind ,'V'''^!',''- ^^.'^ «•«" subtmct as remainder directW uniXf" "■'*' 'l""'™*' P"'"' '" 'he Hence, the folS-- "'' '" "" ""'"^^'^ ''''''^«- that^^«?e"^'f Th?i'^\^',e\?mf bSIT.''^ '?' »'»»«»«. »• column. Subtra'-t aTin wh«ii ^i^°® ^^^" stand in the same From Take Exercise 57. (1) (2) 20 34 40-68 13-5C 27-12 (3) 2-8703 •49 (4) 52-07 34-71365. .DECIMALS. 93 1 + 37-1. 03462- + 18 25. 6 + -3. ^enty-six "ee thou- leasuring 25yarda; placing &c. In ;rahend lumber hem to lis does 3ract as in the above. Slid, so te same Iscimal iths. >o. < 5. 6. 7. 8. From 18-27 take 7*165. 3-692 " 1-4714, 59 " 58-999. 1 " -91842. 9. 204 1 10. 93-46 11. 17-36 12. 983-9007 take u u [37 0054. 84 03071. 9-01843. 655-97385. Find the value of 13. 8- -000008 ; 3578-6129-31-398. 14. 400 -5201 + 005634 - 287 '1894 15. 2-001 + 5301- 24-56783- -007. 16. (273-29- 41-804)-(8-164 + 35-6789 + 40-36). ■|7. From fifteen hundredths take fifteen thousandths. 18. From a cask of vinegar containing 24*35 gallons, 13*725 gallons were drawn off : how many gallons were left in the cask ? 19. A farmer owned a piece of land containing 345 acres, but gave 125*36 acres to his son, and 126*35 acres to hi* daughter ; how many acres had he left ? 20. Add together the sum and difference of sixty- eight hundredths, and thirty-seven hundred-thousandths. AiaUipllcatioii • 151. To multiply decimals . Example 1.— Multiply -9 by '7. *9 = ^% and 7 - /g; therefore *9 x *7 = fa x /^ = jVa = "63 * Example 2.— Multiply 436 by 07. , •436 = ^(^% and -07 = 11/(1; therefore -433 X -07=: Mo X ih = tMUu = -03052. Example 3.— Multiply 3-43 by 2-7. 3-43 = m and 2-7 = fj ; ' Therefore, 343 x 27 = ^^ x fH?^Si= 9-231. Hence, the following : — 152. RULE.— Maltlply as in wliole numjara, a^idl from tha right hand of tha product mark off as mxiy ft jard3 for djsimila as there are decimal places in both factors. Note.— If there are noi a^ raxny figure? in the proiu^t as there are decimal places in b^th factors, supply the deliciency by pra- fixing ciphers. 4 1 m 94 ri if. I 'ECIMALS. Exercise 58, Multiply (I) 7-36 4-38 4(>-«7 3 46 Multiply : 5. 15-43 by 3 104. 6- 7-42 '' 3 56. 7. -2481 '^ '105. 8. -1234 " -0046. (3) 475-32 •0035 9. 61 -70 by 0071. 10. 0716 " 1-32(3. 11. '0032 " 23-45. 12. 1234 " 1234. 35. If ^ o;d oft^^^^^^^^ ^'5-70 a yard ? n-^any bushels Twhit'nuZl '^^ ^'"'^"^' ^^ ""^'^^^^ ^^ow wood? ^'^""^ "'"^^ be gi^^en for 16-34 cords of i^d^^'^:;[^^^ ^,--Ie measures 3-14159 times 1; t!d 11^- 'i--^s^^^^^^^ — ^~ and -isf '^^^ P^*^^"^^ ^^ the sum and difference of 1 -34 ' ISa. To multiply by 10, ICO, 1000, &o. Example. -^Multiply 6 234 by 10, 100, 10000. 6-234 10 62-340" 6-234 100 '623^00" 6-234 10000 _ "62340-000" Frnn fl O_>Ji4U-000 as many places a=, f'\«^ '? V"^ "gl't m the product Hence, thj folloX» I'" "" "^''^^^ ''» «>« nmLpiJer. If required. *^"''*'* ^" tne multlpUer, aimexlng clpherl DECIMALS. 95 d? Iiow s of lies nee •34* Jell Exercise 59. 1. Multiply 131-634 by 10, by 100, by 1000. 2. Multiply 3478 -9 by 10000. 3. Multiply one thousandth by one thousand. 4. Multiply 67-8456 by 100, by 1000, by lOOODO. Division. 155. Case i. — When the divisor is a whole number. Example 1.— Divide 43399 2 by 856. 850 ) 43399 2 ( 50*7 We divide as in ordinary Division, 4280 and place the decimal point in the when all the fiijures 5992 6992 quotient, ill the inte<^ral part of the dividend have been brought down. Example 2.— -0196174 -^ 241. 241) -0196174 ( -0000314 1928 33~7 241 964 9G4 le Jfc Here tliere are no integers in the dividend : we must there- fore mark down the decimal point and take in the next figure, which is 0. The divisor will not go into this, so we put a in the quotient and take in the next figure which is 1. In this way we get four ciphers in the quotient before any significant figure. Examples.— 12-6025 -^ 355. 355 ) 12 -6025 ( -0355 Here the integral part of the dividend will not contain the divisor: we must therefore mark down the decimal point and take in the next figure which is 6. The divisor will not go into 126, so we put a in the quotient, and take in the next figure. The divisor is now con- The rest of the work proceeds as in 1775 1775 tained in 1260. ordinary division. 'I |l' ! 96 DECIMALS. 1008 Xo 420 336 «40 840 then divide a. i„ Ex. 2, C.«rrfcXS:r " ^^ ^y-^^o^n^i^to?^^^^ It so point Of tne dividend as Sy piaces^o th^nSir" *^^ ^«°i°^al decimal figrures in thP rtr^nL^ ^^^ **^® "S^^* as there were deeimaJclpheSi^necelir^^''«;>,^'^'^*"g^ ^^^ tWs puiJJJJ Ji Whole Smnbers! aTdThen i?th?'2^^ T^«° ^l^^e as 156. CASEiL-Whenthedivisorisnotawholenumbet Example l.-Divide 8 01504 by .099. / | ' 8 0154 -f -009 = £015 04 ~ 99. ' 99)8«a(M(60 96 We ,ethedivi.orawLe„u™„e. ^95fi f^f. Tf f '*^ *^^"^"^^ point beyond 891 Ivefh.^l'^-^^f"^"^- ^e must then _oyi mov e the decimal point of thedividend 594 Jts many places to the right as t has 594 been moved in the diviso'r. We then divide as in the previous case Example 2. —Divide -001029 by 1 -68. L 68) -0010:9 vrr , \ io«r -'020 (.0006,25 Z^7l; l%t,T i:a _i008 point beyond the right hand hgui^. Then move the decimal , point of the dividend as many places to the right as it has been moved in the divisor. This of course multiplies both divisor anddividendbythe same power ot 10 and the quotient conse- ■ quentlyremainsunaltered. We ^E 1. Hence, th^ fr»li..,„;..„ DECIMALS. 97 Exercise 60* Find the value of : — 1. 21.6-r-24. 2. •018-^25. 3. 400-92-:- 2004. 4. 784 02752-^428. 5. 15G2-5f00025. 6 171 -99 ^27 -3. 7. 17 •28-^-0144. 8. 31 -5 -f 0126. 9. 816^-0004. 10. 12.5-=- 0125. 11. 00281-^1 -405. 12. -4543752-^0996. 13. •004095^-273. ?■, 18-9225^4-35. 15. 13 -312053-^501 -69, 10. 31 -5 -r -120. 17. 308-7O49-M75-7. 18. 1882 1096 ^707 -56, 19. 1924-182 H-14-04. 20. 5^-01784. 21. 01255^1004. 22. 1-579197 ^0347. 23. 1 -8019 -f 243 -5. 24. 005808 -r -036. 25. Divide the sum of 47 1265 and 8735 by the differ*, ence between 14*05 and 1-25. 26. What improper fraction is equivalent to the cum ©f 14-5 and -5 divided by their difference ? 27. A young man spent -375 of liis patrimony in dissipa- tion, and -25 of it in bad trades, and then had $1500 left. Wliat was the amount of his patrimony ? 28. What is the value of 62-75 tons of coal, when "7 of ^ ton is worth $Q'QQ^ 1_ lO* 29. Find the value of -tl?5x^"^ 5-t--5 •6-8 30. If I can walk 12 5 miles in one day, how many day* will it take me to walk 102 5 miles ? 158 To divide by 1 followed by ciphers. Divide 146 25 by 10 ; by 100 ; by 1000 ; by 10000. 10)140-25 100 )140-25 1000)140 25 10000)146 25 14-025 1-4025 •14025 -014625 ■MWI i riiMI I M 98 DECIMALS. are ciphers I ^ ^'Z^'^:::^:^^Z^?- ^'>- result wm be thequotienr ^^"^^ ^'^ *^' **^^^«°^' ^^^ tlio Rc'ducflon or Decimals. 160. To reduce a Vulgar Fraction to a Decimal. - Example. -Reduce I to a decimal. 4)3-00 ^ equal 3-4 3 equals 30 tenths or 3-0 • 4^ of 75 ^0 tenths is 7 teutlis or 7 witli 2 f-onfh« ' * i , , ^ - 20 hundredths reniainin. T f 20 1 i' ""^ilf Is 5 hundredtlis or -05. There w'^ ^A luuidredtlis followiiur:-. lutretore ,= ,o. Hence, tho quotient wUl b3 the decimal required. DecimaJs. The Exercise 61. Heduce the following to decimals :— 1. 2. o 5 I ¥• J) 4 «• if. _5 2(J' 7. 8. a 10. 11. 12. _3 4 5* 7 7 1 f5(F' J> iM' '> 7 13. 6 l\ 14. 7|. 15 24 s 10. 17. 18. 170 1 J.)' -l_ 1 L'S* a 1 40- CirculfUiug Becimals. 165. To reduce a circulating decimal to a vulgar fraction. e.\^'Z^lmllfF" ^'^'^T to decimals, sometimes the wMH 1 .. r 1 ^^^"'"^^t^' but the same figure or lir.ures w.i] be repeated continually. "«"»«» DECIMALS. w t Example 1.- P educe § to a decimal I - -66000, &c. Example 2- Reduce -i\ to a decimaL ^^=•2727, &c. Decimals of this kind, in which the Fame fgures are -continually repeated, aie called Circulating or Rej catinor Decimals ; and the part repeated is called the Period or Ilex)etend. 16». Tt is usual to express any circulator by writing it •down and placing dots over the Lr»t and last figures of the -part repeated. "When there is only one Mgure repeated ithe dot is plactd over it. Thus, -COOf), &c. , is written '6. ■ -2727, &c., •S777, &c., (( t( •37. 161. To reduce a pure circulating decimal to a fraction. A pure ciroulatinn; decimal is one in which the period "begins immediately after the decimal point. Since }= -lllll, &c., f = '22222, &c., 0=C5555, &c. Again T>'y = J ^11= -010101 &c., hence ■^^= -0^0505 &c.j U = -232323 ttc. In like manner, since 71^=^-^111= 001001 &c., 1 -^5 = -135135 &c. Hence, the followim 'S 165. RULE.— Write the figures that repeat as numerator, and for the denominator as many nines as thv^re are circul tlnff figures. .» laliaWW ^■' 100 DECIMALS* 166. To reduce a mixed circulating decimal to a fractioi*. A mixed circulating decimal is one in which fchfl npr.'^ does not begin imu.ediately after the Zh^llotl ^^ 1. •023=02§ = _2L= ^1 _ 23-2 » 100 ""<^~ "Doo"* 2. -543= •54:i = 5u'^ =6.'}8_ 543-5 10 ^^ um • a yuo •138=-13«-y» = 7 2.-_ l.'iS-l3 *' 100 "<>"- -.loo-- 4. -0443 = •044'^ = 111 ^ 3 '.» _ ^ »-^ - ? t y 1000 '•'""<> ""DouiT- Hence, the followincr • tJ^who?e^d!Sn\?*' wl^^^^^ ^^!?'^ ^^ "^t r^Pe^t. from as many nine^ ?f th«,y l^® remainder as numerator, and nianycfx)hSaa?tb?rat ^'« ^^""''"^ circulating: followed by aa to? o^f ?£ fractioi ^^^'' "°* circulating, as denomina- Exercise 63, Reduce to vulgar fractions :— 1. o 6. •0416. 11. 34i8. 2. •05. 7. •203. 12. 1-283. 3. •520. 8. •01236 13. 11^280 4. •024. 9. •00449. 14. 3 •024. ?. •45. 10. 1-145. 15. £•418. I«S. The Addition or Subtraction of CirouIatiji--■ ^^^ '-^^ »s It S Example 1. Add together -OS, 0432, 2 345 correctly to ^ decimal i>laccs . •3333333 •C4;]2432 2-3404545 2^72203l0' Ana. 2 72203. Example 2. Subtract -2910 from '989583 correctly to .6 decimal places . 9895P33 29i()(;(i7 097V>ioO Alls. -60791. 169. To multiply or divide one circulating decimal by ;anoLhor. Example 1. Mujtiply -3 by -78. ox ^O— yXj, y — y,J— «U. Example 2. Divide '10 by -0027. Hence, the ioUowing :-- 170. RULE.— Raduca the circuIatinT dacimals to Vulgar Tractions. Find the product or quotient as a Vulgar Fraction, «,Ed reduce the result to a decimal. Exercise 6S. Find the value (correct to G places of decimals) of 1. 2-418-fl-lG + 3 609+-7354 + 24042. 2. 234 '0 + 9-928+ -6044 + 450. 3. 51-3b2 + 22-793+ -1824+ •5 + 3-736. 4. G-45--3 and 7-72- 0015. -5. 7-5i-4-5and309--94724. 102 DECIMALS. ■I i. - III I'. 6. 17-2- -MS and 294-5 --372. 7. Multiply 2-;i by 5- by iS'h 8. " y7-2:ihy •2{JHnd22.U by ^m. 9. •* "^-'IJ'yOl-O and 7-72by297. 10. Divide r>4 by 17 and -3 by -09. 11. •• -■'i^'^l>y3(; and 53-4(1 by 7-:i. 12. " -iir by • 148 and 1-18 by 295 i. Review Exercises. J. 2. Find the value of 10''-;-l^^ j. 7 , is i. ^.u i 5. Find tlie value of 3>3x 004 . OOti ' *"° reduce j^ + ^^ ~r2S "' » deciiiiai. 6- (71 of } + J| _ -02) ^ -005. II. . ^ ^''^Vm^^^'iTit'?""^ ^^ ^25 will give the sum of S' T6» 1) ^9o7o and 2 -40 / 2. Prove the rule for multiplication of decimals bv means of the example 432 multiplied by -^^r^ "^^^^ ^^ DECIMALS . 103 3. How do you proceed in the diviRion of decimals when the divisor is not a wholo luniihcr ^ What princii>]e ia involved in tlie operation i Divide 00126 by 2 5 and also 2-5 by 00125. 4. Show that if 1^^,, 2^,, ^l^^, 4^^ be adtfed togetlier, (Ist) as decimals, and (2nd) as fractions, the results coincide. 5. A man walked (50 miles in 4 (hiys ; in each of the firsfi three days he walked an eipial distance, in the tourth day he walked 13-95 miles ; tind the number of miles he walked in each of the first three days. d. Simplify, expressing each result in a fractional and decimal form : — 1. §+ 14 + 1 of 1-0784. 2. -^:~l' III. 1. Define a decimal fraction, and hiking '37 «> as an ex- ample, show from your definition that 375 — {'JJu' 2. Multiply the sum ox* -€489 and 537-0 by their differ- ence, and divide the product by -009. 3. State and exi)lain the rule for reducing a vulgar Taction to a decimal. Express l(0^-}-2|- 3) and ^IH as decimals. 4. The population of a certain town is 2000. Of this '3 are men, '275 women, and the rest children. Find tha number of each class . 5. If a horse eat -0025 of a ton daily, and a cow eat •0715 daily, how long will 307775 tons last 3 horses and 4 cows 'i 6. Find the value of 4-8 x •09-r-OlG, and show which U the greater 11-035 x 0008 or-i9 x 003. IV. *. What is meant by a Circulating Decimal ? State and prove the rule for reducing a pure circulating decimal to ^ fraction. Multiply 5- «i by -4583. mmmmrim ptmrntt^^^ i ? i : 104 DECIMALS. 2. What decimal added to the suii of 1^ « ^r^A is -n make the sum total equal to 3 ? ^^' " '^'"'^ ^^ ^'^^ 3. Reduce '000725 and 31 -2^5 i-n ,.„i r .• How much proi,erty did he leave ? ^^^^' ^ 2 1;. 6 6| .4-f V, 1, What vulgar fraction is equivalent cD the sum of 14-4 hXiZL l'44dxvided by the difference ? * 2, The circumference of a circle = 3'141G timr^« fl.^ r-miigior ^i.oi) per cord and coal at $0.75 per ton ? fi. Find the value c' the expression :- \ {- X J of I of 3 of 20 ) -f|-. *.*-.:^,'5.^'"?'' «"i'.°f.* P"''^e I of its contents. ? of the it'tetT' '■■ '"""" '" """ *"-'° i "■ "'" «"'« did it contain 105 BUSINESS ARITHMETIC. ) CHAPTER V. Rccluetioa. 171. Reduction is the process of changing a number from one denomination to another without altering its value. Reduction is of two kinds, Descending and Ascending. 172. Reduction Descending is the process of chaiiging a number fivjin a higher denomination to a lower ; as shillings to iience. 173. Reduction Ascending is the process of changing a number from .a lower denomination to a higher ; as ponce to filiillings. Britis^i Currency or Sterling Money. 174. Sterling Money is the legal currency of Great Britain : — 4 farthings {far.) = 1 penny or Id. 12 pence = 1 shilling " Is. 20shillhigs = 1 pound " £1. Farthings are generally written as fractions of a penny ; thus, 1 far. — | d. ; 2 far. ^ 2 d. ; 3 far. = | d. £1. sterling — $4,80f Can. Currency ; and i s. = 24^ cts. Oral Exerciises. 1. How many far. in 1 d. ? in 2 d. 'I in 5 d. ? hi 7 d. ? in 9 d. I 2. How many pence in 8 far. 1 in 12 far. ? in 32 far. ? in5()fai.? 3. In 3s. how many pence? in 4 s.? in s. ? in 8 s. ? in 12 s. ? 4. In 36 d. how many shillings? in48d.? in 5Gd.? in72d.? in80d.? ■,«^,««.-— ^,«VB*M-«tt'--r-VW';0 -far. ; 49U0 far. ])lus 2 far. aro f] .a^o. , ■ ^it'f''-^- Therefore in £5 3s. 4 id. there aro 4902 fartlmi'«lir ^ • -XT ^ ^uTK.-When tv.-o nmiiLers arc to be imiJtlphed together it makes no diflerence, as far us the ,,rc,duct is coi c.^rnPd,^. eh of; them IS taken as the multiplicand or umltiplier. Tor c, n e ie ce ^^ therefore we mult.p].y i'^ by 20 and call tlu' product shil ings S B>> with the pence, etc. Jleiice, the following :- '^ RULE. -Multiply tlie number of the higliest denomina- tion given by the number require.1 ot the next JSwer denomination to make one of that\i^her, and to the piSt add the numDer. if any, of the lower ddnomination. ProceecHn this way tiU the whole is reduced to tnere^uired denonUna^io^. CASE II. 176. To perform Reduction Ascending. Example. - Reduce 4f);J2 fartlnngs to £ s. d . 4;49(.2 Sii.co in 4 far. there is Id., in 4002 far I2)1240d. + 2 far. thereare ;is many pence as 4 is contained 2|0}10|3s. +4 d. t;"it3.s m 49(;2, or 1240d. with a remain- TsTs il?'/'^ r'- ^^^^''' ^ ^' 2 far. , or h d. Since iu A x r o \ , , ^^'^' ^^'^'^^ ''^ 1 ^- ' J" ^240 d. there are a* Ans. +o OS. 4^,1. many slnl]ijigsasl2 is contained times iu rpi . . . , , ^. ^2'*^''' "1" i^^3s., with a remainder of 4. This 4 IS 4 d. Since m 20 s. there is £1, in 103 s. there aro as many pounds as 20 is contained times in 103, or £5 with a reriKiinuer < .f 3. This l^ .s 3 ? Thercii'ore in 4962 far. thero are £5 3 s. 4^1, n r s q r b b f; t REDl ON. 107 Inv Henc* , the following ; — P.U'LR.— I -vide the given number bj'^the number of its denomi- nation required to make one of the next higher. Set dotvn th© remainder, if any, and proc33d in th3 same manner with each successivo denomination till the whole is reduced to the re- quired denominatioa. Th3 la^t quotient with the several remainclars aaaex3d, will b3 tlu answer raquLred. Note.— Reduction dc^^ccndiivi, it will bft naticod, is performed by sii>U',(i)sdoii iiia'ti 'liaiUotu, vvliiL* reduction ascciiilin/, which i» but the revers!3 of the former, is! performed Ijy succesnivc divi^sions. Exerttiso 61. TlirouglKntt Ilo'.liiction tlio pupil sliouM work the questions in Reduction Dj^ueiidiiig before proceeding to those in Reduction Ascendin'. 'o* Reduce : — 1. los. lid. to pence. 2. £4(U;s. 8tl. to far. 3. 12M51)far. to shillings 4. £v^'i U)s. 7h(\. to far. 5. 2525(5 pence to pounds 6. £32 12s. lOd. to pence 7. h\ £of> 4s. Od. liow many pence? 8. "£43 12s. how many shillings^ 9. "217s. how many farthings ? 10. " 51)237 far. how many pounds? n. " £481 lOs.U J 1. how manyfar? 12. "18080 jlfar.liowmany pounds? 177. Canadiau :floiiey. 10 mills (m.) = 1 cent (ct.) 10 cents -~ 1 dime (d.) IC dimes - 1 dollar ^ 178. Uuited States ^lotiey. 10 mills = 1 cent(ct.) 10 cents = 1 dime (d.) 10 dimes = 1 dollar ^. 10 dollars = 1 e.igle(E.) L J 103 I' I ! J B 179. RfcitfOx^xxv/N. Long measure. Long measure is used in measuring leiigth or 12 inches (in.) = 1 foot (ft.) 3 feet = 1 yard (yd.) 6^ yards = 1 rod (rd.) 40 rods = 1 furlong (fur.) 8 furlongs = 1 mile (ni.) o miles = 1 league i^l.) 69A miles or 60 ) ^ -, , . 11 1 ^ = 1 decree (deg. or 0.5 geographical miles ) o v o •/ Note, — For measiirijig land, Surveyors use a chain, consisting of 100 links of equal length. This chain is 4 rods or (>r> ft, long. A pace is reckoned at ;i feet, although in i)acing K)ng distances 5 p» -''3 are reckoned as a rod. A hand is 4 inches, and is used in monsuring the height of horses. Sea-depths are measured in fathoms. A fathom is l5 feet. A knot, in sea phrase, answia-s to a (jaojjniphical uv nautical mile. Oral Exereises. 1. How many inches in 3 ft.? m 4 ft ? in 5A ft.? in 9 ft ? in 12 ft.? ^ 2. How many feet in 4 yds.? in 7 yds.? in 11 yds.? in 13 yds.? 3. How many feet in 2 rods? in 3 fathoms ? in 2 chains? 4. In 72 feet how many yds,? in 27 ft.? in 72 in.? in 3 fathoms ? 5. Repeat the table of Long Measure. Example 1, Reduce 25G rodsj Examplk 2. —In 4237 feet ■4 yd. 1 ft. to feet 2)2.5;; rd. 4 yd. 1ft. 1284 128 1412 yd. 3 4237 ffc. how many rods ? 3)-lL'37ft. 5.^1412 yds. 1ft, over. 2 llj28lM~half yards. 25(J rd. 4 yds. over. Ans. 25G rd, 4 yd. 1 ft. d IIEOUOTION. 109 In dividing by 5.V Ave reduce both the divisor and the dividend to hahes ; then performing the division the result is 25() rds. and 8 rem. Now tliis 8 remainder is half yds. %nd equal to 4 whole yds. Exercise 65. Reduce : — 1. ] 24 rd. 4 vd. 2 ft. to inches. 2. 4<5 furlongs to indies. 3. 9728 inches to rods. 4. 89()rds.3yd. 1ft to feet. 5. 751)324 inches to miles. C. 401eagues6fur.2in.toin.. . 80. Niirlacc or Square Measure. A surface is that which has two measurements or dimensions,— 7,f>H;/f/* and InetMh ; as this page, the black- board, the outside of a block. A S(iu;ire is a surface which ha,4 four , ecjual sides arranged as in the annexe 1 , tigure ; hence a i'/ifttrc in-Ji. is a s(iuare j j ) whose sides are each an inch hi length. 'f ^- S(iuare Measure is used in measuring surfaces. 1 inch. /I M J » 1 » 'It 144sqTiaro inches = 1 square foot{sq.ft.) ; 9 square feet - 1 square y.ard(sq. yd. ) h- i 30^ square yards = 1 square rod (sq.rd.) ^ \ 40 square rods =1 rood (r.) ^' Sq. inch. 4 roods MO acrea — I acre (a. ) =r 1 sq\iarenule(sq. m. ) 1 iacli. \\ Oral Exercises. 1. How many sq. ii). in 4 sq. ft.? hi | sq. ft.? in ^^ sq.ft.? 2. How many sq. -v' ■ in \\\.1 in y'o «i-'^ i" t'V ^-^ "^ ^ ^"-^ 3. In 12 sq. yd. h'.y.v many S(|. ft.? in o^ sq. yd.? in 200 sq. in. : 4. Repeat the table of Square Measure. I ! \\ 110 I. EDUCTION, E xercise 66. I:i : 1. 2. Bolidi In 17«>r) sq. I'd. 19 s(i. yd. how many sq. ft. 7 How many s(|. ft. in 5(5 a. 3 roods ? H . . w n ) any ac res i n 21 1 8 1 < i^". ^ yd . ? 4. How many sq. rods in 25G40 feet? J^}' . S*>1««1 or Cubic Pleasure. A Solid or Volume is that which lias three dimensions or measurements, —k'j/,;//7i, breadth and thickness, as a box. or a brick. A Cube is thus a solid body bounded by six squal squares. A Cubic inch is a cube, each sida of which is a eauare iivAi. '■ Solid Measure is used in estimating the contents of lids. 1728 cubic inches - 1 cubic foot (cu. ft.) 27 cubic feet = 1 cubic yard (cu. yd.) 128 cubic feet =-= J cord (cd.) NorK.-A file of wood 8 ft. long, 4 ft. wide and 4 ft. high con- tains a curd. Exorcise 67. 1. In 47i8".7 cu. in. liow many cu. yds.? 2. Reduce cu. yd. 18 cu. ft. to cu." in. o. Jn 14 cords, how many cu. ft.? 4, lieduce 03cu. yd.s. lo'cu. ft. to cu. inches. i^lea^ures «l I'apstcily. The Ca])acity of a vessel is tlie amount which it contains — its contents. IH'i. Llqiii,/ Measure h u.sed in measurin«? liquids, as oil, milk, niolas.ses, S:e. = i > I8». Dnj Measure is used in measuriim '^rain, fruits, routs, A'c. LiQrih MEASTRE. 4,i,nlls (gi,;, . 1 pint (pt.) 2 i>ints = 1 ,pi;,,:-.. (,|t.) 4 quarts -= 1 (,^r;,j.) OOgalofwine) -lli<>^«iiead(hhdjj^ i-'^'^^ -x uusaeUbu.j DRY ME,4SUIIE. 2 pints = 1 (juart. 4 quarts = 1 gallon. 2 gallons = 1 {)eck(|)k ) or REDUOTIOX. Ill Note.— The "Weights and Measiivr? Act" of 1,^7;^ adopted the Imperial gallon of 277.274 cubic inches a.x the standard gallon in 'f Canada. Kef(»re that date the Winchester or Wine orallon of 2:il ' cubic inclieswas in use. This is still the standard liquid gallon in the United States. 5 Imperial or Standard gallons =^0 Winchester gallon!*. By the Act of 1873, a bushel of 'Wheat =60 lbs. Buckwheat -4Sll>s Rye =56 " Corn =50 " . Barley =48 " Data Ciov^er Seed Flax Seed 34 '^ 60 " :50 " Timothy Seed = 48 lbs Beans =60 " Peas =6r " PotatuQS =60 " Oral Exercises. 1. How many gills in 3 p s. ^ mlfjt,? in3(jts.? iiiopts.? 2. How many quarts in 3 gallons ? in 10 gal.? inl peck? rin2pk.? 3. In3gaL 1 qt. how many quarts ? How many pints? 4. How many pints in ^ a gallon? in 1;^ gal.? in 3 gal.? in 3^ qts.? 5. How many times can a cup h( tiding 3 }»iiits be filled .from a vessel containing 3 gal. 2 t. to gills. Change 27 bu. 3 pks, to quarts. 4. H(.)W many gal. of water in 5127 pts.t 6. Change (5 gal. 3 qts. 1 j)t. tu gills. G. Reduce 442 \)t. to bushels. 7. Reduce 5 bu. 3 pk. to pints. 8. In 110 gal. 3 qt. 1 pt. liow many pints ? Measures oT Weight. 'The weifdits u-cd arc of throe varieties : — ■ Avoirdupois^ Ti'oyi '<^^'^^^ Apothecaries" Weight. i 112 I m 1 i I II- r REDUCTION. 1S4. /tToJrdnpois Weight articte7:^'"^'''' ^^^'^''*' '' ""'^"^ '"^ weighing all common 16 drams (dr.) = 1 ounce (oz.) 16 ounces = 1 pound (lb. ) 2o pounds = 1 quarter (or ) 4 quarters = 1 hundred-weight (cwfc.) 20 hundred-weight = 1 ton (t.) '' N0TE.-In Great Britain, 28 lb. =1 quarter ; and 112 lb. =1 cwt. Oral Exercji^e. 1. How many ou9ices in 21b ? in 31b? in 31b 7oz.? in |lb { 28 oz?" ''' ■'' "'"""^ VomuWi in 30 oz.? in 40 oz.? in in 3 ton 7 cvvt*?'^^^'' ^'''''' "^^""^ ^"""^'"^^ ^ '" ^ ^^** ^^^ ^ 8 ctvt^'"' '"'"'^ '"''*'• "' * ""^ "^ *'''' • '" ^ *^''^ ^ ^" ^ ^^^^s « ^p * ''^^•J/^^^ ^"'•^"y ft) ? in 1 ^ cwt. ? in 3^ cwt. ? in 31 1. ? t). liepeat the table of Avoirdupois Weight. Exercise 60. 1. Reduce 36 cwt. 2 qrs. to pounds. 875 ounces t(, pounds. 3r> tons 7 cwt. 151b. to oimceg, lotiolb. to cwt. 3 qr. 15tb 10 oz. to drams. 1728 drams to pounds. 425ib. 3 oz. 5 drs. to drams. 8526720 drams to cwt. 2. 3. 4. 5. 6. 7. 8. (I J 185. Troy weight. jewIiT^ ^^"^'^' ^' ^'^^ ^" ^^^^S^"^S g-lJ> silver, ancj 24 grains (gr. ) = 1 pennyweight (dwt. ) 20 pennywe.glits .- 1 ounce (oz. ) 12 ounces ^ 1 pound (tt>.) ^ li REDUCTION. 113 m non WlJ. in ).? -J Oral Exercises. 1. How many grains in 2 dwt.? in 3 dwt.? in CA dwt ? in5dwt.? 2. Tn aft 8 oz. how many oz.? in 4.^1) ? in 3.1 It) ? in (!'i!],? 3. How many ft in 53 oz. ? in 7'^ oz.? in"l08 oz.? iu 144 oz. ? 4. Repeat the table of Troy Weight. Exorcise 70. 1. Reduce 21ft oz. 14 dwt. to dwt. 2. " 41Uh2oz. 3 dwt. to dwt. .^. " 1011) 5 oz. ()dwt. to grains. 4. In 35210 gruins liow many pouad.s ? 5. " 2150 grains liow many ounces i (i. " 715 oz. how many grains ^ 186. Apollieearies' Weight. Note. — This table is omitted as being- of no practical -value to pupils. 1^7. i^Ioasurc of Time. 60 seconds (sec.) = 1 nunute (min.) t)0 minutes 24 hours 7 daj-s 12 calendar months or ^ 3{)o\ days (nearly^) j 3(j(j days 100 years 1 hour (hr.) = 1 day f da) = 1 v.'eek (wk.) = 1 year (yr.) = 1 leap 3'ear, = 1 century. N(JTE.— When the year is divisible by 4, it is Leap Year. Thui 1876 is a Lenp Year. The ninuber of days in eacli month may be easily re^ membered from the following lines : — " Thirty days hath September, April, June, and November j February twenty-eight alone, ' - All the rest have thirty-one, But leap year coming once in four, Gives to Februaiy one dav more," I 114 REDUCTION. M H 111 counting : — Ot Paper : — 12 units = 1 dozen. 24 sheet.^ -= 1 12 ilo/on= 1 gross. 20(juires = 1 12 gross — 1 great gross. 2 reams — 1 20 units = 1 score. 5 ?»inulle;^ = 1 Ornl KxorcisoH. 1. How many days in 5 weeks? in 7 wk. 4 da.? in lOwk. 3da.? 2. How many minutes in •] hour ? in 1 }, liour ? in .Ti lir, ? in2:>hr.? tS. What montlis liave .30 days eacli ? 'M days each i 4. Repeat the tal)le of Time Measure. Ivwrcisc* 71. 1. Jn 1-5 da. hr. *■) iniii. liow many sec? 2. Row many hours in .■)!»824') sec? 3. Keduce 3(j5 da. <) hr. to min. 4. How many min . in 21 years of .305 da. G lir. each ? 5. How many weeks in 8058451} minutes i C. 1)1 30b30 min. hu\v many days f 188. iMtscellaiieoiis Tables. quiro. ream. bundle. bale. 100 pounds = 1 (piintal of lisli. 10() pouuils = 1. barrel of llour. 200 pounds = 1 barrel of pork. Oral Exci*el!«cs. 1. How many dozen buttons in 2 gross ? in Ih gross ? in 2h gross V 2. How many quires in i ream ? in 2i reams? in 3^ reams '!■ 3. How many sheets in 3 quires? in 2| quires? in GJ quires ? Review Exercises, 1. How many £ in 08520 farthings ? 2. Reduce 85270 grains to pounds Troy. 3. £7 9s. OJd, are how many farthings i 4. Reduce to grains 121b. Ooz. 9 dwt. 16 gr. 5. What is the value of 4 lb, oz. 10 dwt. of gold dust at 85 cts. a dwt, ? Solution. ~4tb. 6 oz. 10 dwt. -1000 dwt. If 1 dwt. is worth 80 cts. 1090 dwt. will be worth 1090' times as iimeh or 85 cts. x 1090- !i^92L)'50. '• REDUCTION. 115 - 0. T boncrht 2 t. 15 cwt. 31ft> of sng^r at 12i cts. a pound. What did it cost ? 7. What will 4 bu. 3 pk. 3 qt. of Pood cost at 20 cts. a quart ? 8. What is the cost of a chain weijjfhincc 3 oz. 2 dwt. afc 70 ctH. a dwt. ^ ^ n. Reduce 150 days 3 hours 30 min. 42 sec. to seconds. 10. Reduce 15778403 socfuids to days. 11. Tn 2 t. 12 cwt. 701b.s. 10 o./. how many ounces ? 12. In 12 211. 7 fur. 28 rdrs. liow many rods ? 13. Jn 58075 ft. how majiy miles? 14. Reduce 1-3I) acre.s 23 s([. rods to square feefc. 15. Reduce Kit. 17 cwt. 1 Gib. to lbs. 10, Reduce 42 a. 98 sq. rd. 7 sq. yd. 2 sq. ft. 40 sq. in. to .sq. inches. 17. In 2588028 drams how many tons ? 18. J^educe 3!)752850 S(j. in. to acres. 19. Find the ditterence between 28 .s'i. ft. and 28 ft. sq. 20. Wm. .Tackson bou«,dit 2 t. 8 cwt. 12 lb. of hay at 1 ^ cfc. per lb. Wliat did it cost him / 21. How many fatlioms in 45038 inches ? 22. Wliat i.s tliecostof 1875 miles of cable at 1 ct. per inch? 23. Reduce the month of February in the year 1884 to seconds. 24. How mucli will be lost on 4 tons 8 lb. spices, bought at 75 cts. a pound and sold at 4^ cts. an oz. ? Conipounil Adililion. ISO. A Compound number is a concrete number of two or more denominations ; thus, £5 3s., and o rods 3 yds. are compound numbers. 190. Compound Addition is the process of finding the sum of two or more compound numbers. Example.— What iy the sum of £10 12s. lOd. : £18 Gs. 7^. £5 33. 5.? The sura of the pence is 22d. = Is. and lOti over. AVe write the lOd. beneath and add the Is. with the shillings The sum of the shil- lings is 22s. =£1 and 2s. over. We write the 2s. beneath and add the £1 with the pounds. The sum of the pounds is £34, which we write beneath and have as the entire sum £34 2s. lOd. Hence, the following:— £ s. d. 10 12 10 18 7 5 3 6 £34 o 10 |! i.' i •- m 110 IIKDUCTION. I. J i1 I i ( < i M RULE.- Set down the addends under one another so that units of the same name or denomination shall be in the same vertical column. Begin at the right-hand side and add the first column ; reduce the sum to the next higher denomination. Set down the re- mainder, if any, under the column added, and carry the quotient to the next column. Proceed thus through all the columns to the last. ExerclSie 73. (1) (-0 (") Jb. 07.. dwt. g^. cwt. lb. oz. bu. pk. (]t 8 IG 7 5 11 G 15 2 3 7 D 12 3 15 9 10 1 2 JO G 15 10 IG 13 10 20 4 G 21 3 4 5 92 20 11 18 4 5 18 «.* s, 7 2 n 81 Alls £1-3 4 h', 4. Add lOu. 35 r. 10 sq. ft.; 3a. 10 r. 15 so. ffc.j IS 11. IGr. 23 sq. ft. 5. Add 12 \vk. 3 d;i. 5 hrs. 20 min. 42 sec. ; 4 da. 12^!, hrs.; 3 wk. 1 da. 10 lirs. 40 niiii. ; IG hr. 3r^ inin. Coiiipouiid .SiibtrttetJoii, 191. Compound Subtraction is tlic process of finding the dillcrence between two conq^ound luiinbers. Exam I'LL.— From £'1S 7s. 4ld. take £3 23. ^^d. £ s, d. Bei/inning witli tlic lowest denomination we have 1 rartliing from 3 farthiiij^'s = 2 f;itthiu-s, which we write beneath. As i*''.L vuniiMt be taken fj'uni Id. we borrow lj-'.,--lLd. from the 7s. leavinj^ Gs. and add to the 4d. wl)ich gives IGd.; then 8d. taken from IGd. leaves 8d. which we write beneath. 2s. from G.s. leaves 43., which we write beiieatli. £3 from £18 leaves £15, which we write beneath. Therefore the difference requir<.'., is X15 'Is 8.',d. llei:ce the following :. — RULE —Write the subtrahend under the minuend, so thu.1 uuita of the same name shall be in the same vertical column. Begin at the right and subtract each denomination of ths subtrahend from the corresponding' denomination of the minu2i.l. If the mumber Oi any denomination in the subtrahend is greater than that of the saAVTc .'enomination in the minuend, increase the upper num))er ;\y ais ma y units oi' tliat denomination as make one of the ne_t h'.-nei', before subtracting. Then consider the number of th 5 -.ioxt higher denomination ot the minuend as diminished by oue. " REDUCTION. 117 '' lb. 8 5 0) oz. dwt. 9 12 2 15 -r. 14 12 Fmcrclse 78. . (2) t, cwt. II). oz. 3 15 18 1 7 (3) bu. j)k, qt. 340 2 4 110 3 G 4 . From 25 in . 150 rd. 12 ft. take 10 m. 120 id. U\ ft 5. From 130 bu. 1 j.k, qt,. take 75 bii. 2] pk. 6. From £10 7«. 4^d, take £2 Os. 9.U. Comiioniiil iniiH:i»li<*a'Joii. 192. Conq)oimd Midtiplication is tlie process of finding Aviiat a compound number will amount to when added to itself or repeated a i^iN-en number of times. Example . - Find the price of 7 tons of iron at £ 12 4s. Gd. per ton . £ 12 s. 4 d ii 7 £S5 11 G 7 times Od. ==42d. =3s. and Gd..over. Write the (id. under tlie pence and add the 3s. to tiie product of the shillings. 7 times 48. = 28s. plus .'Is. carried- 31 s = X'l and lis. Write the lis. under the shillings, and add the £1 to the product of the pounds. 7 times £12 = £,54, plus the £1 carried = £ 5. Ans. £ -.5 lis. Gd. Hence the following : — RULE.— -Set down the multiplier under the right-hand term of the multiplicand. Be^in at the right, and multiply the num- ber ol each denomination in its order, reducing each product to the next higher denomination. Set aown the remainder, if any, under its own denomination, and add the quotient to the next product. Exercise 71. (3) da. hr. min. sec. 3 9 7 17i 7 (1) yd. ft. 6 2 in. 8 7 (2) s. d. 9 5 (4) m. rd. yd, ft. 3 98 4 1 12 Multiply : — 5, 3 oz. 15 dwt. 10 grs. by 10. G. It. 3 cwt. 3qrs. 12 1b. by 7 7. 3 a. 35 sq. rd. by 30. C 118 REDUCTION. If )■ : iO; ! l I I: Mil 1 , i ; 8, 7 bu. 3pk. 6qc. 1 pt. by G. 9. 5 lb. 8 oz. 10 (hvt. by « : by 7 ; by 8. 10. 7 t. ]2 cwt. 10-5 lb, by 8 ; by 9 ; by 7. 11. 4oz. 12dwt. 21grs. by 132. ■oz. dwt. 'jjvs, 4 12 21 1.32 9 5 18^ product by 2 . 7 G 0= '• '' 30. 8 7 12 '^^ '1_1^^- " 132^ 11 38 ( ( 511b. Ooz. 19dwt.l2gTs. 12. 17 a. 3.3 s(i. rd. 4 s(i. yd. 5 sq. ft. by 23. 13. 4 m. 110 rd. 17 ft. by 127 ; by 250. C'oiii3>oiiii 20 53 52 1 12_ "13 13 d. £ s. 1(2 4 d. 1 13 is contained twice in £2S and £2 over. The £2^ 40a. which added to the 13,5.= 53s. 13 is contained 4 times hi 53s. and Is. over. Tlie Is. = 12d., which added to tlie Id. = 13d . 13 is contahied once in 13d . Alls. £2 4s. Id. Hence tlie following : — RULE.— Set the divisor to tho left of the dividend, as in Simple Division Tiien find how often the divisor is contained in the Xiigli°«t denomination of the dividend ; set down this number in the quotient ; multiply as in Simple Division and subtract.^^ If there is a remainder, reduce i& to the next lower uenommation, adding to it the number of that denomination in the dividend, and divida as before. Proceed in this w^y through the whole dividend. RI':DrCTION. no Example 2.— £24 l.'ls. 4d. is to ho put up in envelopes contJiiniU!:^ £'<> 3s, 41 . cncli. How nuiny envelopes will bo required ? £24 3^8. 4.1. 5020 d. £f £1 10s Therefore OOd. = ?^;'fi or ] of £1 lOs, both must be I'oduced to the same denomination. Then since Id. is .^^r^ of £1 lOs., 90d. is 90 times as inuch, or ^%, or ^ cf £1 10s. Hence the followini' : — 4; ■ i I i! 120 KEDUCTION. ! i |i ■ ii '■■1 r 'I i I i ■ if f 1 .1 ^:r :!: ' RULE.— Reduce the givenimrpbers to thelo^vast denomination mentioned in either or them Then write that number which is to be the fraction or the other as numerator, and the other number as denominator. Kverc'iMC 70. 1. I' educe 3 pk. o qt. to tlie fraction of a busliel. 2. AVlint part of a mile is 45 rd. 2 yd. 2 ft. I o. lleduce 3 (it. 1 pt. '6^ '^\. to tlie fraction of a crallon. 4. AVhat fraction of 5 tons is It. 11 cwt. 1() lbs.? b. Ex]>ress 2 yd. 1 ft. as the fraction of o yds. 2 in. 0. Reduce 3 })k. 7 qt. Ih pt. to the fraction of o bushel. 7. What part of a penny is £4 1,,^? This may be considered a question in reduction ; thus £-.1, y •10 12 _ liiOf] _ 1,] 8. Keduce y.^^ of a week to the fraction of a day. 1>. Reduce yY'^-, of a mile to the fraction of a r(»d. 10. Chani-e tooVioo ^^^ «^ ton to the fraction of a pound. 11. What part of a second is 1 y.j^oj) of a day ? 12. VViiat fraL'tion of a sq. ft. is j = j^l^^j of an acre ? 1 JM>. To express a couipouns.! number as a decimal of a liigher denominati*»n. ExAMPLR.— Reduce 12s. Od. to the decimal of £1. By the previous case reduce to the fraction of £1. Then changt! tills fraction to a decimal. (Art. ) Th.:s 12s. (kl. =. loOd.: £i. - 240d.; £:'J] - £5^ -£-G25. The following method is sometimes move convenient. ExA\ri'LK. — Reduce C*. Od. to the decimal of £1. 12X»,^) pence --= /'^s. ^■= "Os. ^Oj OlT) T herefore Ga" Gd. =6 5,5, ; G.5s. =£".(; = £325. •o25 Hence Gs. Cd. ~£ "325. Hence the foiiowing: : — '/ RULE. -Divid:=i the lowest denomination, annexing ciphers if necessary, by that number which will reduca it to one of the next higher denomiaation. Prefix to this decimal the term, if any, of this higher denomination, and reduce the result to the dscimal of the next higher denomination; and so on, until the wliole is reduced to the reiiuired deiiomiuatioa.. Or, Reduce the givcii number to a fraction of the required de* nomination Then c' a ge this iractioii to a d.ciniaL REDUCTION. 121 a Exorcise 77. AVork tlie questions by both iiictlnMls. 1. Reduce 5 cwt. 2 qr. 15 lb. to the dcciuial of a ton. fur. 30 rds. to tlie di-riiiial nt 1 mile. 13 oz. 8 dr. to the deeiiual of a pound. 15 cwt. 3 qr. 14 11). tw the deciiual of a ton, 4^. ll^d. to the decimal of £> Us. 3 fur, 17 rds. to thedecima! of 2 in. 4 yd. 1 ft. 3 gal. 2 (\t. 1 \)t. of wiuc to the decimal of a h()gshead. 8. " 18 hr 15 m. 50 sec. to fchj decimal of a day. 2. 3. 4. 5. 6. 7. 197. To find tlie value of a fraction of a denominate number. Example. — Find the value of g of £1. £ s. d. Since ^ of £1 is the same a.s J of £o, we 8)5 divide £5 bv X as in tllvision (jf comjtound (TTii C) numbers, and obtain for the result 12. Cd. Hence the followin^j; : — RU:.E -Coasidtr thenumer .tor as S5 many units of the givea denomination, ana divide uy lae d_-.omina,tor. Exercise 7S. Find the value of : — 1. I of a yard. 14. ^\ of a min. ;7. '^ -jf 4 lb. Avoir. S, vr- of 3 cv\"t. ). i of 3 miles. 2. I of a week iO. i of a bu. 3 \}.; of a Ib.Troy'o. I of a.ga,l. 10. What is the value of | of a furlon;.^ I 11. Add together ;i of a day, 5 of an hr., \ of hrs. 12. From I of a mile subtract ^^ of a furioiig. 198. To find the value of a decimal of a denominate number. Example. — How many oz. &c. in "045 lbs. Troy? I f I 1 i ; ill ! > ! ^i 122 •C45 lb. 12 •740 oz- 20 14-800 dvvt. 24 REDUCTION. First multiply by 12 to reduce the given decimal from pounds to ounces. The result is 7 oz. and the decimal '740 of an oz. Then nndtii)ly this decimal by 20 to leduce it to pennyweights. The result is 14 dwt. and tlie decimal •800 of a dwt. Then multiply 19-200 gr. Ans. 7 oz. 14 dwt. 10'2gr. tjiis decimal by 24 to reduce it to f'rains. The result is 11) gr. and tlie decimal ^200 of a gr. llcnce the answer is 7 oz. 14 dwt. 19*2 gr. Hence the following : — RULE,— Multiply the decimal by that number which will re- duce it to the next lower denomination, and point off as in multiplication of decimals. Proceed with t.*ie decimal part of the product in the same way, and continue the operation till the decimal is reducsd to the required denominations. The several whole numbers on the left of the decimal point will be the answer. Note. —When tlierf is n, dnciiual in the hxi^t product, it may bo reduced to a Vnlj.;ar I'raction. Thu.s in tht: example given, we anight hove written 10 i gr. 9. 3. 10, 11. I'- -'8. I 9. •024 ton. •07 of 210 lbs. •21075 ton. Kxercise Find the value of : — 1. -325 of £1. It. • 01 C miles. 2. '125 sfp yd. .5. •7o75 miles, •aolb. Avoir- 6. '070 cu. yd. I wk. + 8 da. + •75 hr. +n min. + '70 da. +35 sec. 0^75 t. +4-7 cwt. + \ cwt. +24 lb. +2J g lb. 12. 4-8 bu. +2i5 bu. +'8125 pk. +3 lai. +i:| pk. +^ bu. Kc'view I^xorc'isies. 1. I bought 3 1b. of sugar for 37 i cts. What is that i)er cwt.'^ 2. If 9 cows eat 2l t. 7 cwt. ? ([r. 12 lb. of hay in a year, how much will one cow eat in the same time ? 3. If 42 acres produce 5t;7 bu. 3 pk. of oats, how much ■will 1 acre produce ? -i If a man travel 21 m. o fur. in a day, how many days will it take him to travel 207 m. 3 fur.? 5. If a i)i!e of wood be 140 ft. long, 3 ft. in. wido, how high mu.st it be to contain 18 cords i f I REDUCTION. 123k 6. S»l)ti'aog° i'\ of a furlonj from ./^ of a mile. Give tlie( answer in yards. 7. If the circumference of a v.lioel ^o 8 ft. 3 in. hoAV many times will it turn in a distance of 184 miles? 8. How many d():':en of tea-spoons, each s)>oon weighing 1 oz. 3 dwt, Clin be made out of 25 lb. 10 oz. 10 dwt. of silver ? 9. A farmer bought 2F?, cords of wockI, at 83.50 per cord ; he sold 13 cords at ^-'l.S'Jiind the remainder at.*?o.20 per cord. Did he gain or Iohu. and h >'x nnicli ? 10. Bought h'^ bushels of salt at 75 ct^. por biuhel, and Sold it at 23 cts. a peck. Wiiat was the gain ? 11. A farmer jturchased 4 acres of wood land at $150 per acre ; lie ])aid for cutting 300 cords t-f wood J?^ per cord ; for hauling tlie same 81.', p;-r cor.l : lie sold the 300 cords of wood at 84 A per cord. Did he gain or lose by the bargain, and how much :" 12. What is the sum of£*;i and Its. 9d. 3 far.? 13. What is the diiierence belween 25 gal. 3 qt. and 14-0241) gal.? 14. A grocer bouglit.15 cwt. 3 qv. 21. lb. of colloe at 80.50 per cwtTand sold it at 12:^ cts. per lb. What did he gain on the w hole ? 15. Bought I of- an acre of land for 8325.. 50, and sold ib at t cts, iter S(p ft. NS'liat was the ;i;ain ov lo^isV 10. A farmer paid 8iO«\5U tor the lease of a farm for one year ; lie sold,]*) tons of liay at ^13 per trm ; 150 bu. of potatoes at o^\ cts. ]>er bushel ; 155 bu. of corn at 75 cts. per bushel ; Ol' bu. of oats at -12 et^. per bu. ; 21^ bbls. of apples at $J .75 per bbl.; and 150] ib, of cheese at cts. per lb. He par. for ex]>eiu. > rds is 10 .^ c u. i -: . ? 21. lleduce 3 j)k. 7 qt. to tlie aedmai of a bushel. 22. 4.', inches is wdirt part of .*-.• miles i 23. How many !?teps, 2 ft. in. in each, will a man take la walking 5^ miles? u Vi ii ■ 1 r.4 124 rp:ductton. ii .,. U. ' :i ^^^^^■^ i i ! 1 ^■r: ! 1 ■mi}. 1 .1 ^^H^' Hf ; , ^^^■! ' j X 24. How many working days of 10 hours each will be gained in 20 years by rising 45 min. earlier 300 days in the year ? 25. How many days old is a boy wliose age is 13 yrs.? 26. How many leap years l)etvveen 1851 and 1885 ? 27. A grocer paid §^7.20 for a barrel of vinegar, and found that it co.st him 3 cents the pint. How many gallons were there in t'he barrel / 28. If $1.85 is paid for j\ of 2 tons of hay, how much is it per cwt. ? 29. A farmer bought 45.^ cords of wood at $2.40 per cord. He sold 18i cords at S2. 85 per cord, and the remainder at $2.90 per cord. What did he gain ? 30. At$tg^ per ton, how many tons can be bought for 1=5331 ? 31 . ^ I exchanged 1 4 -'cords < >( wo( .d at i24 per cord for flour at $65 per bbl. How m.-my bl)ls. of ilour did 1 receive ? 32. A man oAving .«s*233A paid at ditFcrent times $381 ; $29| ; $100=-; and $.5^. If he is relieved from paying 3 of the balance, liow inuch will settle the bill ? 33. H(»w much butter at 23 cts. per pomid will pay for 17f yd. o'' })rint at 13^ cts. per yd. i 84. How many farms of 12 a. 10 rods each can be made from a lot containing 113 a. 145 rods ? 35. When $-';| is paid for i of a cord of wood, how much is it a cord f 30. In what time can a man walk 20 m. 22 r. walking 35 m. an hour ? 37. J. Stewart bought 475 yd. of cotton for $30. He sold g of it at 8 cts. per yd. and ^ of the remahider at 10 cts. per yd. What, is the value of Avliat still remains at 10 cts. per yd. i oS. From a Held there Avas sold to James Parks 5 sq. rd. and to S. Morton a piece 5 rods s«piare. There remains U}^SnM 13G«i;:^;4 sq. rods. (Vixc the breadth of the field sup- posing tlie lengtli to be 47 ,\ roi's. 39. How many pounds of gold are actually as heavy as 12 lb. of iron ? . ^ 40. If 2 of an inch on a map corresponn when the price oione is oiven, or (2) tlie cost of any quantitv of goods ut mixed denominations when the price of a ^ovjUmnt of any denomination is given. 1.1 200. Case I. When the articles are of the samo denomination. Example 1.— Find the cost of 530 tb. of coffee at 37 J. cts. per lb. 25 cts. = 191 cts, = i i;40.75" =:. •' " 25 ct=?. per tb. iof25 0^7i_=__^_^_12^^ ii f 1?6 rRACTICE. 2. 507 " 3. CC4 " 4. 700 5. ^8G 0. 008 7. 049 8. 10(J4 u u Exercise 80. Find tnc price of : — 1. 700 articles at 25 cts. 14. 751K tea at 43"^ eta! at 50 cts. i5. 15-' "atol^cts. at $.1.20. IC. 25(; \i\<. cotton at 18| cus. at $^.1. ;>'.){. 17. 875 III. whoat at $1 .00^. at $o.o[). 18. 7G8lh. cotlec at 25 etn. at $1.87^.. 10. 29711). tea at 50 cts. at $3..37.'>. 20. .304 yds. clotli at $1.20. at.$l.l()T. 21. 201 " " at $1.33^. 0. 10870 yds. at .';i2. 87 -i. 22. 485 ton.s coal .-.t $5.50. 10. 1407 " at$3.02.V. 23. 328 ga). oil at $1.87^ 11. 50 tons at $ ).87A. 24. 147 bbl. meal ac $3.37.V. 12. 75 bl.l. at $ {.03:>. 25. lOor^ yds. clotli at 37jct3. 13. 25 " at .^1 . J{ '^. 20. 107^ cwt. sugar at $0.00. 27. Find the j.rice of 15yd. at $3/20 per yd. ; at $2.10; at $4.33.1 ; at $3,12! ; at $1,001^. 28. Wli.it will 500 i'eut of'jiiniLer cost at $8.50 per thousand ? 20!. Cask ii. When the quantity is of a mixed denomination. Example.- -Find the pi-ice of 14 t. 13 cwt. 2 qrs. 1015. of iron at $80 i)er ton. 9^ 14 X $80:^-: $11 20.00^ price of 14 tons, "^il3x$^4_- 52.00- '' "13 cwt. gJJxSF. 2.00: 10x-04== -40 = $1174.40-- u " 2 (irs. •• 10 1b. "i4t.l3cwt. 2qr. 10!h. $80 X 14 - $1120 cost of 14 ton. Then if 1 ton corst $80, 1 cwt. Avill cost :\-^ of $80 or $S0^20-$4, and $4x13- $52, price of 13 c\\ t. If 1 cwt. cost $4, 1 qr. will cost \ of $4 = $4-f4-$J, and 2 ([ra. will cost .$2, $1-^25 gives "04 r»/-.c.f ,^P 1 H. .,».,] ,rH N, 1 /\ _ ,if\ _.^^l. i« 1 /\ ii rill r. cost of lit), and -04 x 10- '40 cost of 'lO It,. the severttl amounts will be the answer. 11. . .. -,--- jC lie tiUUi \Jl 1 PRACTICE. 127 I t ExcrolvSe SI. 1. What is tlic cost of 7 yd. 3 qr. at 75 cts. per yd.? 2. Find tl)e cost of 15 It). 15 oz . of butter at 25 cts. per tb^ '^. lvt;({uirod tlie price of 5 cwt. 3 qra. 10 !b. of sugar at $9.50 per cwt. 4. IviH^uired tlu.' cost of 15 gal . qt. 1 pt. of molasses at (')S':l cts. })er gal . 5. Wliat will 17 ac. 1 r. 15 rd. of land cost at §25.25 per acre i iu What are 150 bu 3 pk. 7 qt. 1 pt. of n-heat worth at 93| cts. a Ijushel i 7. 20 gal. 2 ([t. 1 pt. of inola.3sesat55cts. per gallon. 8. 24 bu. 2 ])k. 4 (jt. of corn at 83 cts. per bushel. 9. 3 b. 6 cwt. 2qr. of ha^. a^ $12 per ton. 10. 1 t. 5 cwt. 2 qr.- j&lb,'"6f sugarat 11 cts. per ft. 11. 207 cu. yds. l(^i^:s;^ffc. a't'28 cts. jrercu. yard. 12. 3J1 a. 2 r. 20 |i- of land. at $4.55 per rcre. - 13. ^15 gal. 3 (ft. 1 pt, of- molasses at 0-^ I cts. per gallon. ^14. 7 yd. 3 qr. of cloth at 75 cts. per yard . 1-3. 15 lb. 15 n;/. butter ;,t i;5 ct;.. per pound. ll constructing a road 17 mi. 8 fur. 15 rd. long at $4700 per mile ^ 20. AtSi.12:'. I'cr yard vvhat must be jDaid for builtling '^yd. 2;; ft. ofVali i 21. At $1080 per year what is the amount of a salary for 1 yr. 7 m. 22:», da. ^ 22. Fijid the coot of grading 10 mi. G fur. 20 rd. c*f rail- road at ^iJlOi) per mile. 23. Wm. Burns' salary is .f 550 per annum . How mnch does he receive for 8 m. 4 da. ? 11 m. 18 da. ? 7 lu. 12 da. ^ 10 m. 8 da.? 2 yr. 7 ni. 12 da. ? 24. The interest on ^ibC>0 for 1 year is $93. (.0. What is it at the same ratu for 5 m. 20 da . ? 7 m . 14 da. ! 9 m. 25 da. ? 3 yr. m. 15 da. I 8 yr. 8 m. 8 da.? ii i? • i u 123 CHAPTER VI, f P1 W'i '■ 'i ' RATIO A3fD PROPORTION. Ratio. Oral Exercises. 1. James li;i3 18 marbles and William has 3. James has how m;iny times as many as William ? 2. 00 is how many times greater than 10? 48 than G? 18 tJian .'5? 55 than 5 ? 45 than 1) ? o. 28 It), is Jiow many tinjes greater than 7 ft.? 15 oz. han a o'A.! '27 bii. than \) Ini.? 42 cents than 7 cents ? 4. Uow does $12 comi)are with $f2 ^ 72 days with 12 days ? 90 t. with 8 t? 06 ac. with 7 ac,? 202. Ratio is the quotient obtained by dividing ono number l)y anothur oHlmsanie kind. Thus :— 'Hie ratio of 10 to 5 is 10^5 or 2. The ratio of $3 to$5 is ^o-^^o or ■}. 20;$. A ratio therefore represents tlie relative greatness of two numbers ; ;is in Llie lirst example, 10 is shown to bo twice as great as 5 ; and in the second, ^3 to be 'j vi $'). XoTi;:. — Tiie ratio i.s always an i^bstract nuTubor, ah hough the nuiub'ji's cuuii^-u'cd may be concrete. Again, tlie contrasted num- bfTs, when cojicrete. must bo of the same kind. It k manifestly absurd to compare Sr> with 7 acres. 201. Tlie lirst number is called the Antecedent, and the second the Consequent. These form the Terms of the ratio, and taken tt)gether are called a conjdd. "205 . The sign of ratii^, made thus, (:) is placed between the terms : but the ratio between two numbers may be in- dicated l)y writing the antecedent as the numei' n- and the conseiiuent as the denominator of a fraction, ''' is 8:11 and -^^j_ have the same meaning, and either may be read " The ratio of 8 to 11." The sign ; may be considered as an abbreviation of —. ^P\ RATIO AND PROrOllTlOy. 129 Oral ExerriHOfl. 1. What is tlio r-atio of 8 to 2 ? 14 : 7 ? 20 :4 ? 15 : 5 ? 7 : 21? 2 ; lOM? : .".- 7 : 3? J :; : 3^ $5 : <52? 8t1.. : 2U.. ? 18 ft. :3 ft. ? | : i? l:'i'l t :1M, : i ? 2:1 / $13 : $(91 ? i<4 : $25 ? $1;$01? 10 ft, :2 ft. ('. in.? 2. Which is i^roator : — 0:2or8:4? 8:4 or 12:3? y:3(.i(> :2? 3:2 or 4:5? 15 : 3 or 20 : 5 ? jr or ^.^ ? Kxorctse H2, ExA*viPLR 1 . - Whiit is the ratio of 2J to J[? 2| : A = V - ^ = V >^ r = V -^ ^^ Ans. • ExAMi'LK 2.— What is tlio ratio of 3.s. (kl. to £1 1 3s. <)(!. : £1 - 42 : 240 = v^/,j = /j Ans. What is tli'o ratio of : — 4. 2 : 1 ? 1. (m; :11:>? 2. 4S :31>? 6. 5 . 8 / 6. 5V:'2;;j? - 3. 28 :1()2 ^ 7. IT)! : *)' ? o ;? .7,1 o' 9. JB-.Ot.O :.«^24.80 ? '~ 10. iN.p. of Halifax 30000 : pop. of Fredericton 7000.? 31. Top of St. .Fr,hn 2H000 : pop. of Fredericton 7000.? 12. 27 It. 3 in. :10 ft. 7 in.? 13. 8o yds. 3 <|is. : 5 yds, (j in.? 14. 7 Im. 3 pk. :lli oal.? lo. 104 a. : 101 so. lui.? 10. 20 da. 2 hr. :7 hr. 15 m. 200. I'iunMf'LK.s. — Since a ratio, like a fraction, is an indicated division, it lias all the proi)orties of a fraction, I»ence : — i. I. IMultijilyiuL,' or dividing both Antecedent and Con- BC(]uent l)y the same number does not alter tlie value of tlie ratio. il. Multiplying the Antecedent or dividing the Con- scfjuent multiplies the ratio. J II. Dividing the Antecedent or multijdying the Consequent divides the ratio. 8 ^M H J.> 1 } ■( 1 j ^r" ' } ij §^1 13a RATIO AND PROPORTION. Also : — ^^' '^^^ Consequent = Antecedent -f- rntio. V. The Antecedent = Consequent x ratio. Exer«l«<» Hii. 1. Write sevei-jil ratios which are ouch o(juul to : 63-^72'* • '■' ^ = 3, 10 : 1] , 55 : 33, 21 : 105, 54 : 108, 4G : 36, 2. Supply the missing numbers in the followini? :^ 1. 90:24-( ). 2. 6:( )..48. 3. ( ):So = -. 4. 72 :( ) = 18. 6. 8:( )^l «. ( ):5-.9. 7. 72 :( )^G, «. ( ): 102=^17. 9. ( ):9 = 12. 10. 121 :( )..ll. 11. 850 :( )=.8. 12. ( ):100--=17. 15. 1?:( ):^[o. Ki. 2 ft. 3 in. :( )..|. 17. $17..'15:( ) = yt.-. 18. 1 yd. 7 in. : ( ) = ^. 207. A Simple liatlo is tho ratio of two numbers as 20 : 5, ^ : 4. ^ SOS. \ Compoimd Ttatio is the product of two or more simple ratios, as a Compound Fmcfioa is the product of two or more simple fractions. 4:5] Thus the compound ratio 3 : 4 J- = * x 5 x 2 = i" or 18 • 35 0:7] Hence, to change a Compound Ratio to a simple one w© have the following : — RULE.— MultliDly the antecedents together for a new ante- cedent, and tlie oonsequcats for a new consequent, I'xercise 84. Find the value of the following compound ratios : — 4:7) 8 : 12 = ( ;. 2. 7:35^=( ). 6 : Gi 8 : 18 1. 84 : 8 0:;; 10 : 13 3. 30:42 ).-r 84:14 ?:18 ) 4. 2J- : 1.) l^( i 36, [13 ro >. 3 5 PROPORTION. 131 209. A proportion is an equality between twa ratios, and is expressed thus, ( : : ) or by the aiirn of equality( = ). Tlius : 3 : : 14 : 7 is a proportion because tlie value of each ratio is 2. Jt is read, is to 3 as 14 is to 7. a 10. Every proportion, therefore, consists oifonr terrns^ Of these the 1st and 4th are called extremes ; and the 2ndi and 3rd means. Exercise 85. Which of the following are true proportions ? 1. 7:3:: 20: 11. 6. 2.1 : 5 : : 4 : 8. "2 5:7::4:(). a 10 : 13 : : 20 : 36. 4. 5 : 12 : : 25 . CO. 6. 2:1^-5:30. 7. 3.?: 10:: 7:21. 8. X. 1 . . .1.4 9. 15 : 2^ : : 11 : If. ail. PjiiNtiPLE.s.— I. The product of the extremes is equal to the product of the means. Hence : — II. The product of the extremes divided by either mean will give the other mean. III. The product of the means divided by cither extreme will give the other extreme. Proof of I.— In the proportion 4 : 8 : : 3: G the ratios f and ^. are equal. Reducing these fractions to equivalent ones having a common denominator we have — 4 X 3x^ "48 These fractions have like denominators, and their '^"^i^rators, here given in factors, a-re seen to be equal. x>ut 4 and 6 are Lhe extremes^ and 3 and 8 are the means of the proportion. 162 il mm rcOPORTION. Ill Exercise 86. Supply the missing term in. each of the folio wing- proportions ; — ** 1. 15-3 : :90:( ). 2. 33-flO ::7: ( ) a 103-( )::U2:18. 4. 896-112: :( ) : 72 \^5. 84- li: :G():( ). ). n . 1 ir • 3 12. 8 t. : 6. 121- (>|: .g .( '^ 23-05 4-5::7-l:( ). 342il-05::{ ): 100. ■^ ^. a ^ .-, • • 8 ^ k )■ 10, ll-^^21::4:( )■ : : ( - 1^. ( ): : $1«.50:$7.25. 13. ( ):13^ yd.:; ^i).3o;^17.10. 14. f>a. uOrd. :( ) : : ijBlJiJ :^1008. 16. 5 ga]. 'A (jt. : ( )::.§! 7. U) : ^53. Hi. ll(lH.Hhr.;i3d.lOh. ;:, !:$()0.!0 17. 1 cvvt.oOtt;. : ;}r»it) :: ( ) ; ^15.60 18. 84.75 :$l(>0::3(im. :( ). 19 4^ yd. :i3l yd. : : ^9.75 : ( ). 20. ,V : ,-^ : : £51 : ( ). SIMPL.E PROPORTION, ^ 212. Simple Pro\)ortion is an e(]uallty between two simple ratios, and isM* term applied to the moth^jd lued in the solution of problems in which three terms of a propor- tion are given toJi,}id the fourth. 21 a. Tlie first and second of these terms form a ratio, and therefore must be of the same kind. The third will be of the same kind as the answer or fourth term, with which it lorms the second ratio. Example.— If 2 bbls. of flour cost $13, what will 5 bbls. cost ^ Since, in a ratio like things only can be compared, there- fore tlie 2 bbls. must be compared with 5 bbls. We know the cost (^f 2 bbls. while the cost of 5 bbls. is required. Now, it is evident that the ratio of 2 bbls. to 5 bbls. is the same ms (he latio of the cost of 2 bl)ls. (5^1.3) to the cost of 5 bbls. (the answer sought.) Hence : — 2 bbls. : 5 bbls. $13 Answer. X sl3 4> = $32.50. Hence, the following: — T i ^ a i^. r< PROPORTION. \Z'S 214. RULE.- Write for the third term that number which ia of the same Jord as the answer sought. If from the conditions of the question, the answer must be greater xhan the third term wrne the greater of the remaining numbers for the second term ; but if the answer should be smaller, write the smaller of these numbers for the second term ; the remaining number will be the first term. «.!I!?^'K^ !5® l^^^^^ ^d third terms together and divide th» product by the first term ; the quotient will be the fourth term or answer. w«*«** «^,*u». Exercise »7. If 9 bushels of wheat cost $11.50, what will 455 bu. 1- cost ? 2. How many yards of cloth can be bou.crht for $2C4 if $17 IS j)aid for 5 yds.? 3. If 12 o7,. of mustard cost 20 cents, what will 14 lb cost ? 4. How long will it take a man to pay a debt of ^oij if he pays $5.(.;5 in ?>\ months? 5. ] f^^^ of a yard costs $ ./. , what must be paid for t ■{ ya. . 6. $2520 was the sum })aid for 30 acres 3 roods of land • at the .^ame rate what would 21 ao. 20 rds. cost/ ' 7. 4| yds. of cloth cost f 14.50; iind the price of 25,''^ yds. ^ ** 8. If 108| cwt. of cheese cost £09 4s. 8d., what will 54 cwt. 1 qr. cost? 9. If [^l acres of land sell for $103. aO. what will 20 acres 3 roods 18 rods cost at the same rate ? 10. If 37 uc. 3 i\ of land cost $1800, what will 150 acrea cost at the same rate ? 11. WhaL will 403 bu. 1 pk. 2 qt. of apples cost if I bu. cost II. 08| ] ^^ ^ 12. A man received $52.50 for 1^| days' work ; how much sliould he receive for 27.^ days' work ? 13. John paid $7.50 for berries at the rate of $1.20 for 15 qts. ; how many did he buy ? 14. If a man tradf 50 bu. potatoes for 21 bbls. of flour, how many barrels should he revive for 153 bu.? 15. How many lbs. of butter at 23 cts. per lb. will pay for K \ j'ds. of print at 8 J cts. yer yard ? 16. If butter is worth "l8i cts. "per lb., and 42 lbs. of sugar IS exchanged for 23.^ lb. butter, what is. the price of the sugar per lb. I ilil ^ I 134 PROPORTION, I 17. Construct and solye three diflerent"problems from the ioUo^mg sutement : In 11| days, 25 men earn $690?and in 18 days 31 men earn $1339.20. Jq T^Qli""- of ^ats cost $2^, what will 191 bu. cost? In jilt ""^ ''^'''^^ T^ ^^I-^^' ^'^^<^ «^i""'4| tons cost? cost? ' cost $11.37^, what will 12^ cords 21. How many yds. of cloth can be bought for $582 18 if 174 yds. cost $695.53 ? W 9?i'T^'Y T^A. ?f ''^'''^''^^ ^ >^^- ^^^« ^i" it take to line 254 yds. of cloth 1^ yd. wide ? 23. What will be the cost of 375 tons of coal if 14 tons can be bought for $72? ^ Jf ^[f ^"- of .wheat make 9 bbls. of flour, how many DWs. of flour can be made from 185 buf^hels ? 25. How much sugar can be purchased for $195 40 if 5 cwt. 45 lb. cost $82.50 ? .-^^ u o Jf' ■^U,?^^ ^"^ ^^^® of land is worth $42.50, what is the Talue of 24f acres at the same rate ? 27. If one yard of broad-cloth costs $4^, how many yards can be bouglit for $124;] ? ^ 28. When 35| tons of'hay are sold for $880, what will 18$ jtons cost ? • 29. I exchanged 15^ cords of bark at $^ per cord for pour at ^7.m per bbl. How much llour d.d I gj^ !q Vl^- ^f^^- ^'i^'- of h'^y^'i^ be purchased with ^ cords 80 cu. ft. of wood, how much wood can be bouc^ht N^itJi 1 t. 12 cwt. of hay ? ° 31 If i of I of a bbl. of flour cost $3?, what will 1 of A Cf a bl)]. co.st? 3 rr Stiitement : — ■* x ? I X -3 3^ 1 I gi=49 + cts. 32. If # of j^ of -I. of 10 yds. of cloth cost $1.23^, what must be paid for ^ of ,«i of f^ of 15 yds. ? ^^'iF^X^'f H ofg ^f 3 cords of wood cost $I5;V, what ^ .Will ] I of I of i;* of 25 coMs cost ? 34. If i of -i of fe P uirin^ ~ *, ' ^' ?f? .?' « ''f ^^ ^^^^- of clotli cost J of A of jl of ^10, what will I ot ;?^ Yds, cost ? ^ "^ . 3», A merchant fnillDg caii pay but 70 eta. on each doUar ,et liLs ni(lehrediie.s.s. He owes A. '"^ '" " ' ,$1100 ; what will each receive ? 1090. B. $2000, and C. FBOPORTION. 195 36. If the expenses of a family of 3 persons be S86.57 for 4| weeks, what would it be fo.- 183 days ? 37. If 7 schooiiers carry 1,575,000 feet of lumber, how much would 19 carry ? KA?^'u^°^ ™^^^ bushels of wheat at $1.06| will pay for 5000 bu. of corn at $.45| ? * f J ^'^ 39 If a locomotive run 85 miles in 4| hours, in what time can it run 222 miles at the same rate ? 40. If a post 12 ft. high casts a shadow of 7 ft. m length at noon, how high is a steeple tliat casts a shadow of 85 ft at the same time ? . 41 If i5 hhd. of molasses cost $40, what would 3.V hhd. ./ cost ? , - V 42. Find the value of 23 yd 1 ft. of cloth, supposing 4 yd. ^1 in. of the same quality to cost $15. > 43. If 26 yd. of clotli cost 48 shillings, what must it be ( €old at per foot to gam 4 shillings oii the purchase ? / -^ tfi^^^^ circumference of a circle is to its diameter as \. »5.i4it);l. J^^Id, m feet and inches, the circumference of I a circle whose diameter is 22 '5 foet. ^^45. If two numbei-s are to each other as 8 to 12, and the /less is 320, what is tlie greater ? *> ^ 40. If a i)erson walk 390 miles in 14 days of 12 hours each, 1 in how many days of 9 hours each can he walk the same Vdistance ? *.ioL E ^^*i '''^^' ^ "l'"^- ^* ^^- (^^'''''^ weight) of flour cost ^1290.3<, what cost 2 t. 3 cwt. 2 qrs. (long weight) at the aame rate ? r:> ^ o / COMrOUND PliOPOKTION. 5215. A Compound ProporUou is an equality between a compound ratio and a simple ratio. 4' 7 ) Thus,j^^;gQ > : ; 15; 150 is a compound proportion. That is 15 is to some required number (150) in the com- pound ratio of 4: 7, and 14 to 80. 216. Compound Proportion is therefore applied to the solution of such pi'oblems as would rcvjuire two or more satiple proportions. y ?' M .y- 13(i COMrOUND PROFORTIOy, Example. — If 15 men mow 12 acres of j^^rass in 20 dayS| how many acres can 18 men mow in 14 (IH3S ? 15 : 18 ? 20 : 14 S 12x18x14 12 : Ans. = 10.,-- acres. Here the question is about the inmiher of acres, and tiie answer sought being in that name we write 12 acres for tlie 3rd term. The answer, ln.v.cver, depends 15x20 on two conditions : (1st) on the number of men, and (2nd> on the number of days they work . "^Ve consider tirst tho number of men, without any reference to tlie second con- dition : and ask, if 15 men can mow 12 acres in a given time will 18 men mow a larger or smaller nuiuber of acres ?' Larger ; hence, as by this the answer will l)e greater than the third term, we write tlie larger nunil)er 18 in th second term and state aa 15 : 18 : : 12 ; ans. Xext, we pro- ceed to the second condition, the nunilter of days, and ask : If 12 acres can be mowed in '/O days, can a larger or smaller number be mowed in 14 day .5? -Smaller; hence we make, the smaller number, 14, tlie second term and write, as; 20 : 14 : : 12 : ans. As in both statement:, tl'e third term is. the same we combine them into the compound statement given. Henci. t!ie following : — RULE -Write, in the third place, that tsrin which is of tfie^ same Irind as the answer. Of tlie other quantities, tak3 each pair of corresponding' terms, and having first reduc3d hoth to tne sam.3 denomination, arrange them as in simple proportion. Then, multiply together the third term and all the second terms, and divide the result by the product cf all tho first terms. The quotient will be the answer, in the same denomination as- the third term. Note. — Before thus multiplying anl dividing, be careful to re* duce, by cancellation, all the terms as much as po:Sv>^ibie. i 1/ Exercise 89^ si.:i 1. If 30 men in 7 days of 9 hours each, reap 21 acres, in how many days of 10 liours each will 22 men reap 35 acres ?' 2: If 5 men spend $37t> in 13 weeks, what will 7 men spend in 52 weeks? 3. How much will $272.50 gain in 4 months if $100 gaim $7 in 1 year f COMPOUND PROPOUTIOX. ■X 4. How many acres will 15 horses plough in 5 days, if 8 horses can plough 11 acres in 7 days 7 5 . Three trains can draw 53 tons of coal in 5 days ; how many tons can 7 teams draw the same disttince in 17 days ? 6. If a cai-penter receives ,f 18 for days' work of 10 l^rs. each, what should he receive fur 23 days' work of hours each ? 7. $35.55 was the sum paid for 12 yds. of silk | yd. wide; what should be paid for 10 yds. of tlie .-jame quality 1 \ yd. wide"? 8. If 3 tt). of yarn will nsa'ce 10y 20. How much will $800 gain in 3 years at 8 per cent, if $500 gain $60 in 2 years at 6 per cent.? 21. If 2 yds. of cloth 1^ yd. wide cost $10.25, what will 13 yd. of like (juality cost, wliich is 1| yd. wide ? 22. If 7 men can mow 84 i acres in 12^ days, working 8 hrs. per day, how many days of 10 hrs." each will 20 men require to mow 254 * acres ? 23. If 100 men by workin;? 6 hr. each day can, in 27 days, dig 18 cellars, each 40 ft. long, 30 ft. wide", and 12 ft. deep, how many cellars, each 24 ft. long, 27 ft. wide, aud 18 ft. deep can 240 men dig in 81 days of 8 hours each ? 24. If it require 1200 yd. of cloth | yd. wide to clpthe 500 men, how many yardi^ ?.yd. wide will it take to clothe 960 men? 25. If 4 men in 2^. days, mow 6;| acres of grass by working 8] hours a day, liow many acres will 1. men mow in 3| days^ "Working hours a day ? 26. If 2i yds. of cloth 1^ yd. wide cost $3.37.\, what will be the cost of "i^S}^ yds. 1^ yds. wide ? 27. If 5 men reap 52*2 acres in days, how many men will reap 417 "6 acres in 12 days ? 28. If 54 men can build a fort in 21^ days, working 12.^ hours per day, in how many days will 75 men do the same, vhen tlicy work but 10.^, hours per dav ? ^ 29. If 24 men dig a trench 33| yd. long, 53- yd. wide, and Sg yds. deep in 189 days, working 14 lujurs each day, how many hours per day nmst 27 men work, to dig a trench '23iyd. long, iq yd. wide, 2^ yd. deep, in 5^ days? Ct / b t r a. . I \ .a" / ±. e the cominissioa or ■brokerage required. Exercise 91. 1. What is the commission on .$1800 at 1^ per cent.? 2. What is the brokerage on $ ;7S.90 at 5 per cent.? 3. An agent collects debts to the amount of $12.")0 ; what is his brokerage at 3 per cent.? 4. A broker purchases stock to the amount of 8-^420 50- what IS his brokerage at 3^ % ? ug a debt of 5.^If my agent charged 5 % for colleeti..^ .. ....... ^, f 107o.40, how much will he have to pay over to mo, and what will his fee be? (). An agent collects debts to the amount of SS75.2o ; what !s his conmiission at 8 %, and what will he have to pay o\'er ? 7. A lawyer collects bills amounthig to $-.<-02 ; what is Ins connnission at 5 %? 221. When the given amount includes the commission ov brokerage. 8. A broker receives STOns w^ith instructions to deduct his brokerage ^ %, and invest tlic balance in Imnk stock ; what sum had he to invest, and what was his brokeraf'e ? 14? PERCENTAGE, ll $7035 ^$1.00J = $7000 to be invested. $703o-{i?7000 =--$35 brokerage. Since tlio brokerage is i % of the sum to bo investeil, the broker receives $1.00^ for each dollar to be invested, and will have for investing as many dollars as $l.QC'^ is contained times in ^7035, or $7000. The diflerence between the sum received and the sum to bo invested, or |3j, must be his brokerage. ' If.' 9. An agent receives $o81.85 to lay out in wheat, deducting his commission at 2^ % ; how much money does he lay out ? 10. I receive 81200 with instructions to deduct my com- mission at 2.^ % and invest the remainder in Hour. What is my commission, and how many bbls. of flour can I purchase at $6 per bbl . ? ISTTEREST. 225. Inb rest is the sum paid for the use of money, J a26. Tlie Principal is the sum lent. 227. The Rate per cent, is the sum paid for the use of each hundred dollars. 228. The Amount is the sum of the principal and interest. 229. To find the interest of a given sum, for a given time, at a given rate per cent. Example 1.— Find the interest on $250 at G per cent, for 1 year. Principal = $250 Since the rate is G %, or '06, the Rate = -06 interest of .$250 for 1 year is "00 of Int. fori yr = $15. 00 $250, or $15. Example 2.— Find the interest on $430.75 for 3 yr», 2 mos. at 7 %. •i' i I INTEREST, 143; f' Principal Rate Int.for lyr. « $430.7o Since the rnfo is 7 /. of -07, "■ '!li .t'"' nitcre.st of $i?.0.7o for 1 vr = $30.1625'«/>7of $4r>0.7r,, or $30.1525.' ^^ «'>A H''' '"^*^''c«t for 1 year is " 904575 *;^-l*>->> f<'r 3 yrs. 2 ni.,.s., or Int.forOJyrs. -- $ VKHOiii .,1 ' — " .r'^* - 'iM'M., or 50254 + l%,-^;':^;i '^ "^>»-^t bo 3/. times % DS'^'r+H V^ n-''' <'^'^')^-582y + . Hence decim^n^w^I'iH^y ^^^^'■'^l^P^^^y the rate, per cent expressed In vfn^. ^' ^ ^^® product by the number expressiag- the time- h,.k'»'i"'''~^''"'''"''''L^^ "^^^'^^^ •'^^' quo.sti..as in Interest mar be- sol\ ed are reserved for an Advanced Arithmetic. Exer<»ise »2 Find the interest of : — 1 . $!){;0.50 for 2 years at 8 per cent. 2. $150.40 for 4 years at 5 per cent. 3. $50.78 for 3 yrs. 11 nios. at 10 per cejit. 4. $800 for yrs. 5 nios. 18 tl . at 8 per cent. Ii(Hluce 5 mos. ISd. to tlie decimal of a year. Thus G yr. 5 mo. 18 d. =G'4» ' yr. 5. $210 for 4years 5 monclis at 7 per cent. 0. $333 for 5 years 2 months 5 day.s at 8 per cenfc. 7. $17<>. 70 for 10 years 8 mouths at 5 per cent. 8. Wliat is the amount of $;;75 for 2 years 9 months at o percent.? 0. What is tlie amount of $570 for 2 years 6 months at 4 per cent. ? 10. AVhat is tlie amount of $84.50 at 7 per cent, for 2 year o mon til s anil 12 days? 11. What is the interest of ,f70.35 for 1 year 8 months and 18 (hiys at per cent.? 12. \VIi;a is the amount of -$80 for 1 yoar 5 .. onths 12 days at G per cent.? f ,i 144 Cn AFTER VITI. h^ i I i i i 1 » i il^ ^ ¥. 230. An Aconu-.it i.s flu- ivcnrd, w'lich a i)crson keeps of gdO'b iM,iiL,'!)t or sold, cash ]>aid ov received, or servicoa retidcrod aiuithcr. A ]»ers()ii wl-.o owes a debt is callod a TkJiior, and the •one to \vl;on» the debt is owed, is a Cfi'ilUor. Name tlie debtor and creditor in each of the foHowing bills. 2»l. A lii1l i.^ a written statement of an account made out bv the ci-edilor for the debtor. A bdl shoiilil also state th'.; names of the debtor and creditor and the tin.o and jdace of the transaction. A bill is rexeipU-d wlu-n the words '' Received i)ayinont," or "Paid'" are written at the bottom and the creditor allixes bis name, as Ex. '1. In bills the following abbreviation.s are often used : — (d)^[x.t. Dr. =- Debtor. Acc't-Accunt. Do. or Ditto -Tlic same. Co.=Con>pany T*er = J5y. Or. = Creditor or Credit. RecM. -Received. Copy and tlnd the amounts and balances of the following bills :— (1.) FiiEnEiiiCTON, N.B.,July 10, 1885. Mr. William Logan. BoLuht of GEO. HATT & SONS. •25 lb. Tea S 45 Sacks Flour @ 40 ti>. iSuojar @ 2 cans peaches ® .*-^ Ill ACCOUNTS AND BILLS. 145 When an account at a store has been nmnin!^ sonic time, the merchant generally makcfi a copy of it for the debtor upon settlement, which he receipts. The following ia a specimoM of (20 Receipted Bill. -_ ^ Saint John, N. B , Oct. 23rd, 1885. Mr James Brown. To SAMUEL L. LONG, Dii. 1885 Sept. Oct. 3 To lolb. Oran. Sugar at .lOcts., 8 10 4 3 " 3'^ Te<' 1*^ Sept. 10 *' ilo a at .00 ct.s., " 1 Bbl. Flour, "18 1b. Butter at 20 ct.s., "125 "Oatmeal at3.U'ts., Cu. By 2 Cord.s Wood at $2.25, " o Bbls. A])ples at $2.50, Oct. 27th, 1885. Reoeivod Payment, SAMIE-L L. LUX(;. $150 105 7,b0 3b0 4 38 450 750 18 73 12! 00 73 Exercise 93. Make out in proper form, bills for the following accounts supplying where neee.s.U cts. ; Jm. tea at 70 cts. ; 2^ ft)! coffee at 4l> J cts. ; and lU!b. dried fish at 10^ cts 9. krs". F. King bought of Landry & Co., 10 yd. calico at 8^ cts. ; 6 spools thread at 6| cts. ; 2.1 yd. sheeting at / , cts. ; 3 yds. cloth at f^l , tri.nntings .^1.25; 4 table-cloths at $1.75 ; and 3 pairs hose -a-z 2^ cts.; and paid on account ^2 00 lb. W. C. Culc sold Peter Grant, 145 lb. butter at 25 cts ; 240 lb. coffee at 141 cts. ; 643 lb. sugar at 9^ cts. ; 847 lb. lard at 15 cts. ; 142 bbls. Hour ut $0.20 ; 324 gal. molasses at 62^ cts.; and received on account $20.00 in cash and 20 tons of hay at ^10.50. 232. Farmers, lumbermen, mechanics, i^c. , as well as merchants and traders should keep an account of their business transactions, and thus have at hand a recora giving an estimate of their property and debts. v 233. The difierent transactions as they occur from day to day should be put down in what is termed a Day Book. The following pages are tjiken from the Day Book oi William Johnston^ a general Country Merchant. ACCOUNTS AND BILLS. •147 Note. — Let the pupil copy them neatly and correctly in an ex- ^ ercise book, or if preferred, on a sheet of paper, the 1st and 2nd pages of which are properly ruled for the Day Book, and the 3rd for the Ledger in the form given on page 149. ] Fredericton, N. B. 1884 Nov. »i •ft tt i< K «( • t 8 11 12 13 15 James Smith, To 51b. Tea at45cts., " " Soda at 8 cts., Dr. $2.25 .48) "Dr. 2 Robert Sharp, To 15 yds. Print at 11 cts. , $1. 65 "3 " Sheetin g at 12 ct8., .30 James Smith, Dr. To 1 bbl. Flour, ^- -- Cr. By 8 bu. Potatoes at 40 cts. , $3. 20 '' 1 bbl. Apples, George Evans, Cr. I 3 By 5 days' work sawing woodatHOets. | $2.01 1 ---- Robert Sharp, By Cash, Samuel Long, To 5 gal. Molasses at 40 cts . , $2. 30 " lOtb. Sugar at 11 cts., 1.10 Cr. By 121b. Butter at 21 cts. , $2. 52 " 2doz. eggs at 15 cts., .30 4 James Smith, Dr. 1 To32yd.GreyCottonat8 cts., $2.56 "14yd.bleachedcottonatl2cts. 1.08 "20 yd. Fl annel at 55 cts., 11.00 Samuel Long, Dr. To 2 bbl. Corn Meal at $3. 25, 86.50 ' ' 50 lb. Oat Meal at 4 cts. , 2. 00 7^ 01 .0 50i 40 15 8 g:4& ! 4|00 2101 24 50! 82 I 148 ACCOUNTS AND BILLS. Fredericton, N. B. Nov. «i #( 17 17 18 George Evans, Br. To 5 lb. Butter at 25 cts. , $1.25 " 3 " Tea at 55 cts., 1.C5 ; James Smith, Dr. To 5 bbls. Flour at $7.25, $36.25 Cr. By Cash, $ 5 27 (1)** Not eat3mo 3., 50J[^ Samuel Long, Cr. By on Ace' I., 2 36 90 25 55 27 00 i 234. Posting to the liedger. In the Day Book the different transactions are recorded as they occur without any regular order of the names ; it therefore follows that the name of any one person may appear several times during a few months or even weeks. The object of the Ledger is to collect these scattered accL ;nts of each person under one liead, and thus to show at a glance his indebtedness . The several debits and credits are placed on one page, suitably ruled for the pur- pose, the former in the left hand columns, the latter in tho rinhi- \Vr\t\n(r fliPHA I'p this wav- into the Ledsfer, is called liosting. - Dr. ACCOUNTS AND BILLS. James Smith. 14V Or. 1884. Nov. 3 " 7 " 13 a 17 Dr. To Goods, •* do. '• do. •'• do. 1884. 1 2 73 Nov. 7 1 7 50 .. 17 1 ilo 24 a 17 36 2o 6172 1 By Goods " Cash * Note® 3ms Robert Sharp. 1884. Nov. 4 To Goods 1 2 01 1 1884. Nov 11 1 1 By Cash Dr. George Evans. 1884. 1884. Nov 17 To Goods 2 2j90 Nov. 8 Dec. 1 *' Balance ilio \ 4 00 i Dec. 1 Dr. Samuel Long. 1884. Nov 12 " 15 To Goods " do. Dec. ^ To Balance 1884. 1 3 40 Nov 12 I'i 8 50 ' " 18 Dec, 1 n 9C j 4 08 By Goods " Cash " Balance 6 45 5 27 50 00 6172 Cr. 01 Cr. By Work 1 4 4 00 00 By Balance 1 10 Cr. 2 5 4 11 82 00 03 90 160 ACCOUNTS AND BILLS. 235. Bnsinesis Forms ■f s I 'W' A reference to the Day Book (p. 147) shows that on Nov. 11th William Johnston received from Robert Sharp the full amount of his indebtedness in Cash. Johnston would, in return, hand Sharp the following : RECEIPT IN FULL. $2.01 Fredericton, N. B, , Nov. 11th, 1884 Received from Mr. Robert Sharp the sum of Two Dollars and one cent, in full of all demands to date. William Johnston. ¥S^ Or, Received, Fredericton, N. B., 11th Nov., 1884, from Mr. Robert Sharp, the sum of Two ^l^ Dollars ($2.01) in full of all demands to date. William Johnston. On Nov. i8, Samuel Long paid part of his account and received from Johnston a RECEIPT on account. $5.00 Fredericton, N, B. Nov. 18, 1884. Koceivcd from Mr. Samuel Long Five Dollars on account. William Johnston. Again, on the 17th James Smith in order to settle hia account, gave Johnston a PROMISSORY note. $50. Fredericton, N. B., Nov. 17th, 1884. Three months after date, for value received, I promise to pay William Johnston or order, Fifty Dollars, with uitercst. James Smith. If no time for payment had been fixed he would have given a I \m 151 I NOTE ON DEMAND. ^50. Fredericton, N. B., Nov. 17th, 1884. On demand, for value received, I promise to pay Williair Johnston or order Fiftv Dollars, with interest.' James Smith. money order. $17^(1^. St. John, N. B., Nov. 15th, 1884. Messrs. E. Forrest & Co. Please pay to Norman Jordan, or order, Seventeen ^i^ Dollars, and charoje to my account. Levi Patton. CHA.: TER IX. J MEASHJREMEJ^fTS A\n ESTIHITES. MeasiireinoiH ot Keelaiisuiar Siiriaces. 2»« . A Eedawjh is any plane figure bounded by four straiglit lines, and having four equal angles ; as the surface of a d'< ■i'^i: "I Find the area of rectangles of the following dimensions :— • 1. 26in.byl8in. 3. 12|ft. square 5. 8-5 yd. by 9 ft. 2. 4 ft. by 20 ft. 4. 45ft.0in.byl2ft.9in. 6. 279 ft. by 180 ft. 7 . How many square yards in the walls of a room 16 ft. long, 12 ft. () in . Avide, and 10 ft. 4 in. high ? 8 . What is the width of a room 24 ft. long, the floor containing 444 sq. ft . ? (See formula 2. ) 9. A room 18 ft. by 16| ft. ; carpet 1 yd. wide at 90 cts. First find the area^ and then find the length of a piece of cari>et of same area that is 1 yd. in width, by formula 3. 10. A room 24 ft. by 36 ft.; carpet 28ni^. wide at|l^. 11. A room 13§ ft. square ; carpet 30 in. wide at $1.84. 12. A hall 42^^ ft. by 8'4 ft. ; carpet \ yd. wide at $2^. 13. Wh : will be the cost of glazing 6 windows, each 8 ft. 4 in. by 3 ft. 6 in. at 90 cts. a sq. ft? 14. What will be the cost of plastering a room 21 ft. 6 in. by 16 ft., and 9 ft. high, at 35 cts. a sq. yd., allowing 225 sq. ft. for doors, windows, t&c? 15 . What will it cost to paint the walls of a room 10 ft. 3 in. high, 36 ft. long by 24 ft. wide at 8 cts. per sq. yd. ? 16. How many rolls of paper, each 9 yd. long and 18 in. wide, will it take to paper the walls of a room which is 20 ft. 6 in. long, i4 ft. 8 in. wide, and 10 ft. high ; deducting 10 in. for base-board and (i in. for cornice ? 17. What will it cost to put on a tin roof 48 ft. by 70 ft. 2 in. at $12.86 per 100 sq. ft.? 18. What will it cost to floor a room 80| ft. wide by 60 ft. 3;«> \r..^n, o4- Oft'^i Aa T%oT. inn an. f f. 2 19. What must I give for #lating a roof 80 ft. long by 47 ft. wide, at ^13.40 per 100 sq. ft.? MBVSUREMENTS AND ESTIMATES. Biii]cler* i ;J iiv m 154 MEASUESMENTS AND ESTIMATES. 9Ie»K'ov^Mnent oT Rectangular Solids. 241. A IvocbMiigulaT' Solid is a body bounded by six rectangular surfaces, called Faces. 243. If the faces of a rectangular solid are eiual, it is called a cube. Example. — Find the number of cubic inches in a rectangular solid 4 inches long, 3 inches wide and 9j inches high. The lower face or base contains 4x3 = 12 sq. in. Hence if the solid is 1 inch high it would contain 12 cu.in. But the solid is 2 inches high, and therefore contains twice 12 cu. in. or 24 cu. in, Hence, To find the cubic contents of a rect. solid, multiply the length, breadth and thickness together. Find the contents of solids of the foliuwinij dimensions : 1. 10 in. by 3 in. by6.Un. 2. 45 ft. by ()ft. by 15 ft. 3. 3Gft. by45ft. by6Ht. 4. 6ft.4in. by4ft. Gin. by3ft. i. 18 ft. by 10 ft. Sin. bvO'Sft. «:< yd. bv4^'ft. byd'25 ft. 4V«1. by G| it. by 10 in.' 12 y ft. by 3| ft. by 3 ft. 9 in. ll'.i 9. What will be the cost of digging a cellar 40 ft. long, 36 ft. wide, and G ft. 6 in. deep at 50 cts. per cu. yd.? 10. How many cords of wood in a pile 44 It. long, 8 ft. wide, and 5 ft. high ? 11. At $3.50 a cord what is the value of a pile of wood 3 ft. long, 8 ft. v/i in. tliick ? 19. How many bricks of average size will bo required to build a house 54 ft. long, 27 ft. wide, and 24 ft. high, the wall being 13 in. thick, allowing 25B sq. ft. for doors and windows ? What will be their cost at $7.50 per thousand? 244. Board Measure. — Lumber or sawed timber as boards, planks, itc., is usually measured by board measure. A board foot is 1 ft. long, 1 ft. wide and 1 in. thick. 12 Buch board feet would therefore make 1 cu. ft. ^ In the lumber business all boards are assumed to be 1 in. thick even when less, and for every additional ^ in. in thickness the piece is increased one-fourth. ,1 Thus :— 1200sq. ft. 1 inch thick or less =- 1200 ft. board measure. 1200sq.ft. llin.thick = 1200xli= InOO '' '' 1200 sq. ft. 2 in. thick = 1200 x 2 = 2400 " " '^ Hence i — To find the contents of aboard plank, &c., multiply the product of the length and breadth each taken in feet by the number denoting the thickness in inches. 20. How many board feet in a plank 14 ft. long, 12 in. wide and 3^ in. thick ? 14 X 1 X 3^ =49 board feet Find the contents of boards measuring '— 21. 10 ft. by 14 in. 24. 22. 18 ft. by lorn. 25. 23. 21ft. by 20 in. 20. Find tJie cost of the following : — 27. 40 boards 14 ft. long, 9 in. wide at $2.75 per 100 ft. 28. 8 planks 12 ft. long, 14 in. wide, and 3 m. thick, at $15 per 1000 ft. ^^^ ,.^. 29. 30 sciuitling 9 ft. long, 4 in. by 3 ui . at |2| per 100 16. 23 ft. by 1 ft. Gin. 21ft. by 10^ in, 10 ft. by 12^ in. 150 MEASUREMENTS AND ESTIMATES. If f 1 1 t II, i i , ! 1 ^f'- I SI f W'^f^ I 30. 350 fence boards 10 ft. long, and 8 in. wide, at$12 per xUUU it . 31. 4 sticks of timber 32 ft. loni,^ 10 in. by 14 in . , at $1.60 l)er 100 ft. 32. How many board feet in G joists 14 ft. long and 4 in, square ? 33. How many feet in 5 boards each 12 ft. long, 10 in. wide, and 1-f in. thick ( Farmers* E.sUmates. ai5. Tlie number of busluls of (jrai7t, held by a granary or bin is equal to the contents of the' bin expressed in cubic feet, multiplied by ^**^j, or more accurately by j%^g. 246. About G bu. of wheat yield one bbl. of flour. 247. The net ircUfht oi fat beeves is about f of the live weight ; of slieep ^^ ; of swine *. 248 . Horses, sheep, and young growing cattle are esti- mated to consume daily about 3 pounds of hay, for each 1001b. of weight; and oxen and cows about 2^ lb. i !12 per 11.60 d 4 in, 10 in. THE METRIC SYHTEM. ranary I cubic le live 3 esti- each 1 The Metric System of Wei;:5ht3 and Measures is based upon the decimal notation, and has for its base a unit called the Metre. The Metre, which is the ten-milli(mtJh part of the dis- tance from the e(iuator to the pole, is the unif of len^'th. It is also the fundamental unit, since from it every other unit of measure or weight is derived ; liencr) the nam© Metric Sij.it',m. The following are the Standard units, from which all the other units of the system are derived . rsiTH. NAMES. PRONUNCIATION. SIfJNIFICATION. Lenfjth.' Metre. mee-ter. Measure. Surface. Are. air. Surface. Vjlnme. Stere. stair. SoHd. Capacity. Litre. liter. l*ound. Weight. Gramme , gram. Smali-weight, 'The Are, which is the unit of layid measure, is a square whose side is ft^ji metres. The Stere, which is the unit of Cubic or Solid Measure, is a cube whose edge is a metre. The Litre, the unit of all mea.mres of capacity, is a cube whose ede-e is the tenth of a metre. The Gramme, the unit of ireiyht, is the weight of a cube of pure water at its s,'reater^t density, wliose edge is the hundredth part of a metre. The n-.imes of the loirer or hiqhcr order.s in the decimal Scale are formed by pretixhij to the several Standard units named above, — 1st, for Lower Denominations, the Latin ordinals :— Deci, signifying one tenth, ^^j, '1. Centi, " one- hundredth, -jj,j, .01. Milli, " one-thousandth, xd'oU' '001. 2nd, For Higher Denominations, the Greek Numerals:— I 158 THE METRIC SYSTEM. ^(i » il' i 1'; |! II Deka, signifying Ten, 10. Hekto, " One Hundred, 100. Kilo, " One Thousand, 1000. Myria, " Ten Thousand, 10000. These terms ])rehxed to the standard unit furnish the key to the whole System, showing at once funo many times greater or less the respective unit is than the Standard unit, Tlie names miU, cent, and dim\ used in Decimal Currency, correspond to 7nilli^ centi, nnd clxi, in the Metric System. Thus, taking the dollar as the Standard unit, a dime might be called a deci-dollar, since it is J^q of a dollar; a cent, a centi-dollar, since it is ^^0 of a dollar 'y and a will, a milli-dollar. In the Metric System, only a few of the denominations are much used. These will be noted in the tables by the difference in type. • lieiigtli Measures. The Metre is the standard unit of length, 39.37079 inches, or 1-09 + yard. Table. and 13 CANADA VALUES. •03937 in. •3937 " 3-937 " 39.37 •''* 32. 81 ft. 19.93 rds. •6214 mi. 'J 6-214 mi. SYMBOLS. 1 Milli- metre, mm. 10 Milli-metres = 1 Centi-metre. cm. 10 Centi-metres = 1 Deoi-metre. dm . 10 Deci-mdtres =1 Metre. m. 10 Metres =1 Deka-metre. Dm. 10 Deka-metres = l Hekto-metre. Hm. 10 Hekto-metres = 1 Kilo-metre. Km. 10 Kilo-metres = 1 Myria-metre. Mm. The Metre is used in measuring cloth and short lenifths and distances, like our yard. The^Kilo-mctre .is generally used in measuring roads, and long distances. It is nearly f of a mile, j HOW TO WRITE AND READ METRIC NUMBERS. '"03j ' As in Decimal Fractions, a Metric Number'.is written with the decimal point separating the unit from its decimal parts ; thus, — jd 3t:t? metric system. 159 i 3Km. 2 Tim. 5 Dm. Cnhu. 5 cm., writton as metres, ia 3259(J5m., and may be read three tlionsaml two Inindrod and Hfty-niue, and .sixty-tivo liuiuiiedtli.s metres ; or, tliroo thouHaiid two luuulred aud htty-aiiie metre.s and sixty-fivo ceiiti-metres ; written as kilometres, is I.J2t">l)l»r) km., and niay be read tliree kilometi-es, two hundred and tifty-iiine nietroa, and sixty-five centimetres. ExercitKO. Read the follow inij : — 1. 8-58m. 203-9 m. 83-405 km. 7o0-053 km. 17 -OOO km. 3-908 km. 9 909 km . 183-93 m . 2. How many metres and hundredths of a juetre are ex- pressed by 13 '12 i\).( 3. How many metres and centimetres does 392-87 m. express ? 4. How many kilometres and thousandths of a kilometre are exi)ressed by 43-38l» km.^ how many kilometres and metres ? 6, Reduce 18-72 m. to centimetres ; 25 -723 km. to metres; 83 km. to centimetres. (j. How many centimetres in 9-52 m.? in3'70km.?jn 18-093 km.? "^ 7. How many Dm. in 20 m.? in 80 m.? in 305 m.? Siurface iflcasures. The Square Metre, or Centaue, is the unit commonly- used in measuring surfaces of small extent, and is l'19o sq. yd., or 10'7b4 sq. ft. The Are is the unit of land measures, and contains 100 sq. metres. An acre is 40 ares nearly. Table. 100 Sq- Milli-metre8(sq,mm.)= 1 Sq. Centimetre. 100 Sq. Centi-metres = 1 S(i. Deci-metre. 100 Sq. Deci-metres = 1 S(i. Metre, or Centare, 100 Centares, or Sq . Metres = 1 Sq. Dekametre, or Are. 100 Ares, or Sq. Dekametre3 = l SqHektometre,orZieft;£a^r6i lOOHektares, or Sq. Hekto-metres = 1 Sq. Kilometre 160 THE METRIC SYSTEM. ! U ! :| ) I Symbols. Sq. cm. 8q. Dm. Sq. m., ca. Sq. dm., a. 87 sq. dm. as .sq. m.; assq. cm. 3. In 1.537 a, how many ca.? how many Ha,? Volume, or Cubic Mea^ure««. The Cubic Metrp: is the ^(nit of ordinary Solid measures. It takes the name of Stere when applied to the measuring of wood, lumber or stone. Table. 1000 Cu. MiJU-motres (cu. inm.) = lCu. Centimetre. 1000 Cu. Centimetres. 1000 Cu. Decimetres SYMBOLS. cu. cm. cu. dm. cu. m. s. = 1 Cu. Decnnetre. = 1 Cu. Metre, or Stere, CANADA VALUES. '061 CU. in. •0353 cu. ft. 35-3100 cu. ft. Also : — ■ lODecisteres (ds,) = l Stere, S. i0 8teres =1 Dekastere, Ds. The cubic centimetre and cubic mifh'metre are used for measuring min-iite bodies. The .Surface and Cubic Measures aie only the squares and cubes of the length measures. 35 -3106 cu. ft. 2 '759 cord THE METRIC SYtiTEM. Kl 111 the first part of this table, 1000 units of a smallor da- nonunatioii make a unit of a ilononHnation luxt larger ; tlio scale is 1C03, aivA three plac^vs of figures inu.st be allowed fo* each denomination. Thus, 37 cu. ni. 702 cu. dm. G03 cu. cm., written as cubic metres, is 37-702i>03 cu. m., which may l)e read 37 cubic metres, and 702003 cubic centimetres ; and, wntt n as cubic decimetres, is 37702 •f)03 cu. dm. 1. Tn 9 cu. dm . how many cubic centimetres ? In 7 cu.m. how many cubic decimetres? 2. Express 0.730-355 cu. dm. as cubic centimetres ; as cubic metres. . j 3. In 8030 cu. mm. how many cubic ccutimetres . in 374 Ds. hov/ many steres ? The Litre is the unit of capacity, for both liquid and dry measures, and is 1 cubic decimetre, or about \S8 qts. Table. symbols, can a i) v valuk ci. dl. 1. Dl. HI. Kl. Ml. •0088 qt. 086 " 88 " 2-2 gal.. 220 220J 10 Millilitres (ml.) - 1 Centilitre. 10 Centilitres = 1 Decilitre. 10 Decilitres =1 Litre. 10 Litres = 1 Dekalitre. 10 Dekal itres =1 Hektol i t-re. 10 Hektolitres = 1 Kil< 'litre. 10 Kilolitres = 1 INlyrialitre. The Litre is used in measuring liquids, &c. , in moderate quantities. IMie Hektolitro is used_in the measurement ot grains, roots, &c. 1. How many centilitre.'] are 7-5 litres? 13-02 litres? 831 dekalitres ? , , 2. In 30G litres how many decilitres? how many centilitres? 3. In 830751 centilitres how many litres i how many hi J 1 • I „ _ 4. Express as litres, 302 cl; 430 di.; 30D1.; 94o HI 162 THE METRIC SYSTEM. Weight Measures. density, and is eqiiul to 15 432 grains «"'""''' Table, lOnSt-r,!! '■»="•> = 1 centigranune.. 10 Ceiitigranmies 10 Decigrammes 10 Grammes 10 Dekagrammes 10 Hektogrammes 10 Kilogrammes 10 Myriagrammes 10 Quintals SYMBOLS. = 1 decigramme. eg. dg. g- Hg. Kg. Mg. Q. T. 1 1 1 1 1 1 = 1 CANADA VALUE. •1543 gr. 1-543 '' 15-43 " •3527 oz. Av. 3-527 " " 2-2046 it,. Av. 22-046 " " 220-40 2204-6 gramme. dekagram me. hektogramme. kiligramme. myriagramme. quintal. ton. u u The Gramme is used iri weighing letters ' crol.l o,. i • mixmg medicines ; the ^.%?a..me is used !n w^^^^^^^^ common articles. ^" weighing Eercise. maiymmil^^S^sT"' ''°" """'' centigramme.? how is f og^?'"* '''^<'""--^' ^^'>' "f ^ ton is a kilogramme? of a Hg. 4. How manv kil<'"^'r^mryir-.=f 1 1 i 1 . J^ilogrammeare expressed by 73 -532%'? ! 163 1 cubic greatest lid in jhing how lany Hg. )f 04 ANSWERS. 1. 3. 5. 6. EXERCISE 1, Page 6. 1 3 : 8 ; 6 ; 5 ; 9 ; 7. One, six, live, two, 1 nr, nine. 2* 8 • 3 ' 5 • 7 . 3. 1 orange, 1 marbL 1 cent. 4.' 1,'a, 9 are abstract ; 1 acre, 3 gallons, uushels, 4 men, 3 dollars are concrete. ^ , i 6 5 miles, 9 miles, 3 miles ; 2 boys ; 4 boys ; 6 marbles, 7 marbles ; 4 dollars, 1 dollar, dollars ; G, 8. EXERCISE 2, Page 7. fi.Q.Q. 7 2. 42; 73; 09; 30; 18; 80. 12r57;'20;91. 4. 11; 85; 19; 37. 37 ; 70 ; 18 ; 84 ; 29 ; 50. Eicdit : s-xteen ; thirty-four ; eighteen ; seventy-five; %^f' "Ninety ; thirty ; fifty ; forty-two ; sixty-eight; eleven. 8. Six ; thirty-nine ; ninety-six ; eighty-tive ; twenty- two ; fifteen. 9. Thirteen ; seventy-two sixty-one ; ten. 10. Forty-three ; nineteen : twenty-six ; eighty-seven. EXERCISE 3, Page' 8. 101 ; 110 ; 111. 706 ; 972 ; 439. 85() ; 330 ; 299. , ^ , , • One hmidred and eighty; two hundred and six; three hundred and thirty-three ; five hundred and sLxtv-eight; one hundred and nineteen. 7 Six hundred and ninety-seven ; nine hundred and * ninety.nine ; five hundred and fifty ; seven hundred and fifty-six. ^ , i i ;i 8. Two hundred and thirty-nine ; five hundred and sixty-one , eight hundred and forty ; six hundred and sixty-three ; one hundred an'l five. , , , ^ 9. Five huhd-ed and eighty-four ; rff.'eu hundred an^ twenty; three hundred and sixty ; two hundred anu twenty-two ; four hundred and tlurty-seven. 1. 3. 5. 6. fifty-six ; thirty-three ; seventy-eight ; fifty-two ; 2. 220 ; 519 ; 666. 4. 365 ; 440 ; 708. ^I]l 104 ¥M ; i AUBWEBS, I. I. 5. 6. 7. 8, ^'7^^^ hundred and ninety-one ; six hundred and fifty- nine ; nine hundred ; three hundred and eighteen Mira hundred and thirty-four. «'«"t^en , nre EXERCISE 4, Page 9. 4256 ; 6373 ; 3i60. 2. 2002 • SOI') • •7nni 28012 ; 202,202 ; lOOOOG. ' ^^ ' ^^^^ ??2?S^ro^^^^^^^?nn^ ; 307205018 ; 987308438. 13< 48^09857 ; 2000i)2000->0 93006014000086; (>()00006 i , uui V , MDCCCLIV ; MDOCCLXVII. >. 49 ; 79 ; 99 ; 48. Q, 049 . kqa . qa^ I 1604 ; 1871 ; 1059. . ^^9^584,804. EXERCISE 7, Page 16. 1 tlnn't- 2. 853 dollars. 8. 936 trees. I 2860 yards. 5. 242. 6. 286 7 2->3 I 293. 9. 845. 10. 924. 11. I056 1^ 4l^' 13. 2234. 14. 8791 ].. 18903 ' r-"" ^^4*-' 17. 1891. IS. 2577. 19. 27449. 20. 99262. 21. 185182! ANSWERS. 165 PAGE 17. 22. 149380. 23. 323831. 24. 161998 25. 289685. 2(5. 298(>787. 27. 1253791. 28. 1904747. ^29- 53- 30. 61. 31. 53. 32. 62. 33. 493 34. o08 35. 1507. 36. 3595. 37. 66011. 38. 64 poimds. 39. 59 yards. 40. 565 cents. 41. 61o dollars. 42. 668 bushels. PAGE 18. 1 37218. 2. 61460. 3. 81439. 5 61206. 0. 56340. 7. 85656. 9. 385577 10. 88370. 4. 84209. 8. 531964. EXERCISE 8, Page 19. 1. $587.22. 2. $7911.29. 3. $21.20 EXERCISE 9, Page 20. 4. $216.90. 5. |35.25. 1 70 cents 2. 91 cents. 3. 173 scholar" 4. 778 8tam])s. 5. 890 hills. 6. 4517 logs. 7. ^20 trej: 8 443 bushels. 9. 1.50000 feet. lO. $1247-i. 11. 52603 people. 12. |6561749-65 13. $502.80. EXERCISE 10, Page 22. 1 25 2 51. 3. 33. 4. 43. 5. 36. 0. 2, 7 466 ' 8. 512. 9. 465. 10. 225. 11, 2600. 12. 244 i3 1220. 14. 111. 15. 43. 16. 15112. 17. 262ft 18. 1043. 19. 3003. 20. 11000. Page 23. 21 183225 22. 20402. 23. 112401. 24. 110- 25 253231 26. (511. 27. 3221. 28. 861100. 29: 512010. 30. 638324. 31. 252 32 73. 33. 5612. 34. 6000. 35. 12310. 36. 122. 37. 263. 38. 11222. 39. 76234. 40. 50000. EXERCISE 11, Page 23. i 33. 2. 23. 3. 12ct3. 4. laSbrirVs. J- f^ 6. 50 rows. 7. 1084 votes. 8. 545 apples. 9. $-6^^. 166 ANSWERS, 1. 32o. 6. lOS. 11. 340. 16. 1515. 20. 997. 24. 17006. 28. 8999. 32. 611. 36. 1982. 40. 357227. EXERCISE 12, Page 25. 2. 103. 7. 155. 12. 95. 17. 1688. 21. 6227. 25. 7497. 29. 10996. 33. 998. 37. 928. 41. 68920. 3. 255. 4. 82. 8. 94. 9. 190. 13. 184. 14. 227. IS. 1802. 22. 998. 26. 4491. 30. 6966998, 34. 1994. 38. 13888. 5. 239. 10. 269. 15. 493. 19. 7176. 23. 5734.5. 27. 49029. 31. 858. 35. 678317. 39. 200, 1. $720. 4. $292. 7. 482 bushels. 1. $186.19. 6. $186.59. 9. $4.37. 3. $1870. 6. $3410. 10. 177 miles. EXERCISE 13, Page 26. 2. 761 acres. 5. 5153. 9. 632 cents. EXERCISE 14, Page 27. 2. $136.69. 3. $227.06 4. $656.07. 6. $13.13. 7. $711.90. 8. $2249.59. 10. S32.38. EXERCISE 15. 1. 309. 2. 1013. 3. 1718. 4. 1392. 5. 5641 6. 9287. 7. 186. 8. 3092. 9. 1098. 10. 289501! REVIEW EXERCISES, Pages 27 and 28. 1. 50 yards. 2. $68.57. 3. 2817 votes. 4. Wand 21 \ !?t-.n J- ^^^^^^^- "• ^^^'^-^^^ ^' He gains $576.' JL $lo50. Jo. EXERCISE 16, Page 32. » hJ,^^' ^- l^'^l- ^- 2792. 4. 4380. 5. 23550. • ?2^n^', n ^- '^^^^^^' ^- ^^2623. 8. 3234 bushels. ?o ^?o2±"^^^'- , ^^- ^^^^^ ^^^'*^- ii • 41643 yards. 12. 12o90 pounds. 13. 6.3912 men. 14/2075. 17. 25858. 15. 1776. 16. 8224 19. 147608. 20. 4:14016. 21. 936840. 22 960468 23. 1408161. 24- -^.')«7 32. 1899140. 33 36. 18560000, 39. 19000500. 1. $137540. EXERCISE 19, Page 36. 2 72485. 3. 127743. 4. 226665. 6.'l89o')36. 7. 4095504. 8. 2060625 10. 3333174. 11. 7153920. 12. 4467789. 14 1815548. 15. 2460204. 16. 19o51168. 18 11793320. 19. 30188950 2l' 38265807. 22. 24616388. 24. 360314784. 25. 552743754. 27. 500344698. 28. 2565057924. 30. 2768805702. 31. 259200. 41337000. 34. 9005880. 3o. 354000. 37. 5227200. 38. 430520000. 40. 2714000.000. EXERCISE 20, Page 36. 2. 19208 pounds. 3. • 2208 bushels. PAGE 37. 4. 9660 poles. 5. $65.52. 6 J259.20. ^^ ^64 bu^^^^^^^^^^ 8. 70680 pounds. 9 . 342720 lines 10. 2304 mdes. 11 OQ>ini..Jlp« 12 ^17'>28. 13. $1101.60. 14. $o6250. Ib! I^i^r 16. 56^20*"rcUcr 168960 feet. 17. 74088 ears. IS. $282.2'l. 19. $175. 108 1 i :\' ANSWKRg. EXERCISE 2!, page 38. 1-4. $85.32 „,»,1, S12.(i0 gain. 14. $ '{■:•.• ■,,u|,r *^ ■*^- RFVIEW EXERCISER, PA,.K;;n. ^iif!*^"" ■, 2. First, *!,•.!. .". Onimr.^S. 4. fVy'^o^'-n-;. 5. $2.54. «. .s.-,.U 7.0. 7. 40t,0 hills. PAGH 40. ll. $1^41. 12 Second crop, $i::J3.07 1,„rt,,,, jo ^^^g 1. 144. 6. 204. 11. 173. 16. 84. EXERCISE 22, Pacje 44. 2. 128. '-' 108. 3. 198. 4. U4. V. iu.-^. 8. 144. 9. ;;.->4 12. 230. 13. 219. 14 li9 17. 108. 18. 61. 19 «>4. 5. 108. 10. 321. 16 122. 20. 73. 1. 324. EXERCISE 23. 2. 144. 3. 173. PAGE 45. 4. 117. 11^72- %^9t V^^^- 1^^- •*• ''''■ 1^-S«- i« ',1' ll' ft ^''- ^'^-'^ J^- ^^1- lo. 125. 16. 38. 17. 58. 18. 599^. 19. 22.3' 20 1055S 21. 9021|. 22. 39254^. ^ 23. 4191;'- 24 S 25. 48440/V. 26. 19653?. 27. 2500 ' 28 133m 29. 5928f.'' 30. 46782^.^ "' ' ^'^'^^^^' EXERCISE 24, Page 45. 1. 36 pencils. 2. 119 pairs. 3. 52 weeks. 4. 53 tons. 5. 143 hours. 6. 190 n^iies 9. 162. /. ~u inijcs. 10. 419,^. 11. 576. 12. 133 ANSWERS. 169 ,t EXERCISE 25, Paok 40. 6 6 J'''" 7.^2o'r'8. lll-> '■•!». I2V»;^/ 10.217^^. 14. 10404 -jMuishek. 15. IIIU;:,;; l)usl.els. J'';^ ^::' V^V' 17. COpomuls. 18. .$25i>. V-l ^H.12.^ 20. ^2.14.i^g. EXERCISE 20, Pace 40. 1 iru6 2 3'>-« '^. il4/*. 4. IH.'','',;. 5. 52r4v, in i'VVi'>y 11 4777''S' 12. 40;j.;;A' ;. 1'3. ' •'•'V:- I'l:'.- EXERCT^^'E 27. 1 1023ya.as,12inehe:,^; ^ ^' t... 802 p.^; ^- ?l':S;na. 7; 125 e:.:.;.anlo. 8. 2000 oa.rel. 9. 45\}f{ys. 10. 14,>:J^]^^, times. EXERCISE 28, Paoe 50. 1 flt^nno 2 <%'n 17 •'^- 5=40.95. 4. $0.80. K dfer>){) 20 ':;'J barrels. <• |^<^--^i3o- 12. 212|''^T pounds. 1«>. 5t-^ ■*•>• -i^- -* iiuu ^ REVIEW EXERCISES, Page 51. 1. $1302.25. 2. Oah>.$147. J;- ?50. ^ If^- 5. $040. 0. A 81092, B ^852. 7. ^U-lo. 8- ^^f'-^O, 9. 72. 10. $1000.50. ^^ 11. Ihour. 12. $50o''f'l3r'44t.>ns. l^^^^^^' 15. .«44gaiii. 10. li:n,ushels._ _ 18. 130. sum, 448dme?ence. 19. 815. 20. oO^^. EXERCISE 29, Page 54. 'k i ir ^: . i 170 ANSWERS. 3. 2^. 9. 54. 12. 2, 2, 2, 3, n, 3. 13. 6, 5, 11. 14. 2, -. 2, 3, 3. 3, 5. 15. 2, 2, 2, 2, 2, 2, 2, r>. 16. 2, 2, 2, 2, 3 3 7. 17. 2, 2, 2, 2, 2, 2, ", ., .j. 18. 2, 2, 2, 2, 2 2, 2 3 5. VJ. 2,2,2,2,2.2, r^, 7. 20. 2,2,829. '''''* EXERCISE 30, Page 55. 1. 4§, 15, 07. 2. 5i, 8^, 3^. 6. 3fi. C. 200. 7. 20. 8. 2. 11. 57}. 12. 24. 13.j;>0otuu. EXERCISE 31, Page 57. 1. 6. 2. 2. 3. 8. 4. 25. 5. 18. 7. 19. 8. 27. 9. 15. 10. 5. 11. 34. EXERCISE 32, Page 57. 1. 18. 2. 43. 3. 32. 4. 1. 5. 5. 7. 2. 8. 3G. 9. 32. 10. 13. 11. 5. Page 53. 13. 8 yards. 14. 80 feet, Id spans, 19 spans. 15. 12 barrels. ' IG. O (J. C. M. 17. 144 feet. 18. 14 feet. EXERCISE 33, Page 00. 1. 1800. 2. 1()00. 3. 70. 4. 225. 5. 000. 6. 2100. 7. 5400. 8. lOuS. 9. 2184. 10. 2301000. 11. 3108. 12. 720. 13. 1008, 14. 110880. 15. $J00. 100 <'alves. 75 sheep. 4 oxen. 10. $17.50. 17. 7200. 18. 840. EXERCISE 34. 4. 3. 10. 4. 6. 9. 12. 4. 6. 48. 12. 12. 1 * _1 .3 a 4 8 10 3 3 6 4 4 7 '.i •J.UOO • i (T* 1 ti » 1 '. • ft S'-iUO iJT 74 /);U4 *'' J4 J J4 5 ,ji • 2. 72 108 344 24Q . M)' 19. ?. 20. 10^ 24. ,\- 5. |. 11- i^iV IG. 3^. 6 32* 21. >: EXERCISE 39. 6. ijg &c. 6. i^r^d Ac 1' roU &<^- 8. ,j-o^, &c. EXERCISE 40- 2. i^*^ i&c. 3. ?J &c. 14. ^M). &c. 15. j\;;, &c. 1^- 1^ *^- 4. 2^^ &c. 9. ^4^ &0. 13. 3^ Au EXERCISE 41. 1- »o' yB' tto* *: 'u .w' ir' ^.-^ 2.feA,j?-,„ ,,3.J.58,.^ ^4 iiwn "I'J' fP.O' n TOO _5;i. ^o. ^y 11. I greatest ; 4 lease. fi 700 805 868 31 10. ill, n^ i^',^'\ 12. i^ greatest ; iJ least- EXERCISE 42. 1. %' 2. 2M- 3- l^]- 4. 2^. 5. r^5- 6. 2H. ! ' 1 f i ) i i 'i I |j .1 11. 1 172 1. 2q. 6. 321. 11. IBiJ. ANdWERS. EXERCISE 43. Q 1. .V. 2. Jl. 7. 1. 2i. 6. 3i ■' 11. 55:^. EXERCISE 44. • 4 5 0» 6- A- 9. e^o" 10. aV 11. ?SS. 12 EXERCISE 45. 2 21^ 21. 4. lOj^s^. 7. 'i^[,._^ 8. i^. 9. 2i:il 12. 10/>j. 5. 92||. 10. 3JJ. EXERCISE 46. 1 ]2i 2. 8,v tons. 3. 11,\, tons. 4. ,fr>3X 5. 14/, yd. «. 72^ tb. 7. Hi. 8. 17Jl*4!l. ?-3fbbl8. 10. $^8r^ II. 1911 Aeik 12. 3;^"miles. 33. ^1011. 14. lO-v 15. :,V 1«. | greatest ;| least.. ir I'A- . ^^' -•• 1'^- ^-iV 20. $473G|f ;^*. 21. ll},yd. 22, $48.V EXERCISE 47. 1- 2|. 2. .^{1. 3. i?^ 4. 14f* 5. 6A. 6-2]. 7.21. 8. 23vJ. 9.19.' 10. 2^; 11. $174. 12. $151, 13. $124^. 14. 419-^ mife'e. 15. §117. 16. 128 rods. EXERCISE 48. 1- ^^' 2. 14. 3. 64. 4. eOvfff. 5. 238. 6 386. 7. 20. 8. 025 ,V 9. 115,^.'^ 10. 379|. 11. 36381. 12. 1629. 13. $594. 14. $332 EXERCISE 49. 1. "■ ^ t' 2. i. 3. f. 4. 13i. 5. 1^, 6. ^ 7. 76. 8. 513. 9. 409^. 10. 49.1 «. 11. 5|. 12. ^,. 13. h 14. 36, 1.5. $m - k;. ^74^ 17. r^vVV 18. 56S Acres. "" ^ "' ! V, 5. 2^. 10. 3ff. At. 5. 92||. 10. 31% 4. r>!f. L7J1 gal. ^^ miles. ; h least. Off ; hh 5. 6^. LO. 2^. ^ niiies. 5. 23f. ). 379|. r. $332. 6. 11. 5|. ANSWERS. 178 • EXERCISE 50. 7. i^r 8. id' 9- 72'^,: 10. ^4iVj. 11- 5/2 rois- 12. Il'oJ. ^ EXERCISE 51. 1. 7^. 2. 21. 3. 34^ 4. 10()0. 5. U. G. J. 13. 4idoz. iV. I/;, "15. $7'4;]. 10. d.";. 17. 8 bu. 18. 7t^ is. 27?.[d l"'s- ^0- i^'^i ^H's- 21. 25]i^ miles. 22. l2i. EXERCISE 52. ). i. 4. ,|^.. 5. 4. 0. 35. 7. 21. 10. 4. li.'li 12. r-,- l'^- l.',>'o. 1. ?. 2. |. 3. 8. If,. 9. 10. 10. -. - , .., ^ , ,. ., 14. 4. 15. 31. 16. ^;>,. 17. II 18. Wil 19. 3f. 20. U, EXERCISE 53. 1. 14JhV 2. ..^. 3. i. 4. 28] ^ 5. lOi^ I 12. Ij,. 13. ,^V\^. 14- 2?; . 15. 14,^13. IG. 1,1,. J 17. ^f 18. U^. V'). 1. 20. 4^^ REVIEW EXERCISES, Page 85. I. 1. Book work. 2. i?., i;2- • 3. Cook work- 4. ^0 greatest, i least. 5. Ul Hjji, jr^C> SilJ- ^- ^^""^ ^^^^• II. 1. T6 4. 2. Book work. 3. i of 1] ; by v^gQ. 4. I. 6. 21|. 6- |y073i- TIL 1. Book work. 2. 90? cts. 3. $0. 4. ,^^. 5. $3. 6. I(f25800. IV. 1. Book work. 2 i 5 5. 14'i G. 121.1. "liO- •>. fi' $^1^^- » 4 R-'fl f ! rn iij 174 ANSWERS. EXERCISE 54. 1. Fifty- six thouaandths. 2. One thousand and three ten -thousandths.* 3. Two tliousand seven hundred and eighty-six ten- thousandths. 4. Three hundred and sixty-four ten- thousandths. 5. One, and fifty-six thousandths 6. Thirty-four, and twenty-seven ten- thousandths. 7. Seven, /^nd two hundred and sixteen thousandths. 8. Sixteen, and forty-seven ten-thousandths. 9. .Five, and six thousand and one ten-thousa\idths. 10. TAventy-six, and four tliousand and sixity-seven hvmdred-tli'Hisandths. 11. One hundred and thirty-four, and six hundred- thousandths. 12. Two hundred and forty- three, and thirty-two thousand and thirty -six Imndred-thousandths. 13. Five tliousand and sixty-seven, and six thousand five hundred and one millionth^. 14. One thousand four hundred and fifty-six, and thirty thousand two hundred and forty-six ten-millionths. 15. Two thousand three Imndred and four, and thirty thousand six hundred and seventy-five millionths. 16. Fourteen thou.s.md eight hundred and ninety -two, and eix millions seven h andred and eighty-four thousand fiv© hundred and two ten-millionths. 17. -8 ; -34; "00^ ; 4 037. 18. 465-0080 ; 4007 086 ; 32 00106. 19. 7 007 ; 852 032675 ; 32 000006. 20 . 456 • 0030246. EXERCISE 55, Page 90. 1. f. 2. « • Si (To (J • "• 12. #o^oV 16. 36;i\j^. 3 .21,3 COO' 13. 17. 3. j^o,' 4. i. 5. 5. 6. ^^. 9. 5^Uo. 10. li^. 11. 2lh^ S4|. 14. lUhh 15. 125;^. •4. 18. 07. 19. -027. 20. 004 ■ 21. -036. 22. •0125. • 23. 01034. 24. 35-7. 25. 127W.36, 26. 342 00106. 27. 1480007 28. 3124-00085 . t ' 'W'ww w^ iW ftlMIWBWipi y HI I p iiiB iiftiyjwwt AIv^SWEES. 175 IX ten- 3. iS. dths. ths. ■y-seven undred- houEand and five d thirty 1 thirty }wo, and and fiv© 030246. 6. 126 1. _5.3 2 5 0lJ* 125;^. 0. •004- 4. 35-7. 480007 1. 82-6469. 4. 1415-514485 EXERCISE 56. 2. 812-7843. 3. 5002 244323. 5. 7233-4084. 6. 10-917. 7. 346 122. 8.347-2589. 9. 114-4662. 10. 1130-9634. 11.295-5405. 12. 707-18032. 15. 37 047. 13. 386-53735. 16. 82 035. EXERCISE 57. 14. 207-89095. 1. 6-78. 5. 11 105. * 9. 167-0946. 13. 7-999992 16. 147-2831. 20. 1-36. 2. 13-50. 3. 2-3806. 6. 2-22(16. 7. -001. 10. 9-42929. 11. 8-34157. 3547-2149. 14. 113 336334. 17. -1:^5. 18. 10-625. 4. 17-35635. 8. -08158. 12. 327-92685. 15. 30-43617. 19. 93-29. EXERCISE 58. i. 0770236. 8. -00056764, 12. 152-2756. 15. 58-3338 bu. 18. S825-50. 1. 32-2368. 2. 161-4782. 3. 1-06362. 5. 47-89472. 6. 26.4152. 7. 0260505. 9. -438496. 10. -0949416. 11. -07504. 13. -0013014. 14. $153-525. 16. 387-87640935 yds. 17. 1 777644. EXERCISE 59. ^ 1. 1316-34 ; 13163-4 ; 131634. 2. 34789000. 3, '1 4. 6784-56 ; 67845 6 ; 6784560. EXERCISE 60. 1. -9. 2. -23. 3. -00072. 4- 1-83184. 5. 0250000^ 6. 490000. 7- 1200. 8 2500. 9- 2040000. 10- 1000. 11. 002. 12. 4-562. 13. 015. 14. 4-35. 16. 2o0. 17- 1'757- 18. 2-66. 20.280-269 + . 21- 0000125. 22-45-51. 24. -163. 25.3-75. 26. ]5. 28- $597-02f 29. 1. 30- 8| days. 15. -0237* 19. 137-05, 23. -0074. 27- $4000- EXERCISE 61. 1. -625. 2. -25. 3. -36. 4- -3125. 5. -95. 6- -09375. 7. -075. 8. -875. 9. -0875. 10. -00625. 11. -078125. 12. -2375. 13. 6-171875. 14. 7'625. 15. 24-04- 16- 1-36. 17- 0078125. IS- -21 la- n W6 ^i: r I fcii' I: I i> ANSWERS^.. EXERCISE 62. ^m. ■^:i- -^^^ -^^'- ^■^^^»- EXEPvCISE 63. 1 oi.q-ixqo 2 700-599955. 3. 78-630G06. i 0^1^!^ 1-08185, 5. 2055555; .08052752. a 16'6737a7 ; 294183183. . H 1QO o,..>..-40 8. 9-928 ; 1300-24 9. 19 814; 2297. 10. 30375; 3'b. 11. 081, 7 2». 12. 2-5227 ;4. REVIEW EXERCISES, Page 102. I. 1 «io7*;s0952 2. 13 0125. ' , • u*. 3 ^^1a-e: bumUed and ninety-seven thousand and eight. andTiur hundred and Hye thousand ^f -- ;;i^^-^^S^; 3970084050 ' 09 ; 39 •700840o( )09. 4 . 2o i bo.-o , i uu 5. -0002; '2607. 6. 432. ^ 1 .03493 2. Book work. 3 . • 0005 ; 2000 . 5 15-35 6.(1) Sifandl 5488(2) i^fi and -7007 111. 2 32112030-7.3431. .3. 3 083 ; 2-87296. 4 0^n?en,550womcn,850chd:heu. O.Oodaya. 6.27. IV. 1. 2-6 2. -72. 4. ^ 6. ei666|. 6. 7910000; 00536 V. 1 li. 2. 3056 miles. 3. ISii ; 4-2142857 ;j|ii5; 5^; -00341. i. $23.80,V &. i^i- ^. ^l^-^^' EXERClSrE 64. 1 md. 2. 44480 far. - 257 « • L^.', ^ ^Pg.o b' 5. £1054,s.8d. 6. 7e34d.. 7. H4.4d. o. 872b. 9 . 104ii) lai" . l•^^ . xob ix o . . 4 U . 12. £1883 7s 8|1. 11, 402287 tar. ''K:m^m ,5. A. 10. i-h- 15. 2Ut- ^8-630006. )8 •052752. U mi; 7-29. and eight, luilliouths. 3225 i 100. 005; 2000. and -7007 lya. 6. 27. 57 ; rib'i,, 0. $10.87^. 4. 81438 far. I. 8. 872 H. 402287 far. ANSWERS , 177 1. 24720 m. 4. 14794 It. 6. 7650722. EXERCISE 65 2. 356400 m. 3. 49 r. 2 ft. 2 in. 5. 11m. 6 far. 29rds. 2|yd. 1ft. EXERCISE 06. 2. 2472030 ft. 3. 437 a. 102 rd. 2. 311040 cu. in. 1 . 4809641 ft. 4. 943q. rd. 4| yd. 8fr.. EXERCISE 67. 1. 10 cu. yd. Sen ft. 93 cu. in. 3. 1792 cu. ft. 4. 2965248 cu. in. EXERCISE 68. ' 1. 96 gaJ. 2 qt. 1 pt. 3 gi. 2. 1156 gills. 3. 888 qts. 4. 640 gal. 3 qt. 1 pt. 5. 220 gi. 6. 6 bu. 3 pk. 1 gal. 1 (^t. 7. .368 pts. 8. 887 pts. EXERCISE 69. 1. 3650ft). 2. 54 ft). 11 oz. 4. 15 cwt. 2 qr. 15 lb. 5. 23200 dr. 7. 108853 ar. 8. 333 cwt- 7 lb- 8 oz. EXERCISE 70. 2. 98683 dwt. 3. 1131440 oz. 6. 6 lb, 12 oz. 1. 5174 dwt. .^. 60144 Ljr. 5. 4 oz. 9 dwt. 20 gr. 4. 6 ft). 1 oz. 7 dwt. 2 gr. 6. 343200 gr. EXERCISE 71. 1. 1318140 sec. 2. 157 h. 50 m. 40 sec. 3, 525960 min. 4. 11045160 min. 5, 850 wk. 7 h. .S6 min. 6. 21 da. 6 h. 30 min. REVIEW EXERCISES, Page 114. 1. £71 7s. 6d. 2. 141b. 9 oz. 13 dwt. 4 grs. 3. 7155 far. 4. 72232 grs. ' 6. |G91-375. 7. $31. 8. $4340. 9 12973002 sec. 10. 182 d. 14 hr. 54 mm. 23 sec. 11. 84330 oz. 12. 4148 rd. 13. 10 mi. 319 rd. 3 yd.2ft.Giu- 14 6540261? sq. ft. 15.33716 1b. 1(). 267302272 sq. in. 17. 5 t. 113 lb, 18. oac. 53 rd. 2\>l yd. 4 ft. 72 in. 19. 756EO.it. 20. $72.18. 21. (Jli31uth(.ma 5 ft. 2 in. 22. .fil88(.)00. 23. 2,505,600 sec. 24. ^240 24. n • ' \ . ,; 1: it 1 i •I t 1 > Is I" \'\ 178 AXSWEKS. EXERCISE 72. 1. 45 lb, 4 oz. 2 dwfc. 10 gr. 2, IIG cwt 2 qr. 11 lb. 4 oz. 3. G bu. 1 pk. 4. 31 ac. 01 r. 48 ft. 5. 16 wk, 2 da. 21 h. 7 inin. 12 sec. EXERCISE 73. 1. 3 lb. Gra. 17 dwt. 2. 8 oz. 10. GO t. IGcwt. 84 lt». ; G3 t. 8cwt. 9i lt>. 8 oz. ; 53 t, 4 cwt. 73 lb. 8 oz. 12. 39o ac. 8 sq. rd. 14 so. yd. 3«> ^.q.in, 13. 552 m. 20 rd. 14 ft. ; 1086 ml 237 rd. 9 ft. G in. EXERCISE 75. 1. 6 bw. 3 pk . 31 qt. 2. 3 Itx. 11 oz. 8 dwt. 4| gr. ; 21b. 7 oz. 12 dwt. 2,f gr. ; 1 lb. 11 oz. 14 dwt. 2i gr. 3, 1 fc. 4 cwt. 4 1b. 15 oz.; 8 cwt. 1 lb. 10|\)/,.; 10 cwt. 30 lb. 11 oz. 4. 55 gal. 2 qt. 1 pt. ; 43 gal. 1 qt. 3?^ri, 5. 2 ac. 37 ,\, rd&. G. 15 bu. 7^ (its. 7. £1 Is 0^ d. 8. 1019 n%^ 9. 15 da. ; 12U da :io-6' 1. S^Obll. 2. 5. A. 6. yi-. O 7 11. §i sec !a 12, I sq. ft. EXERCISE 70. 9. I^rd. 8. ,y>.da 4. 10 7 70 2") Off* 5^0 lt>. EXERCISE 77. 1. -2825 t. 2. •84375 m. 3. -84375 lb. 4. '7945 fc,- 5. -0976307. G. -213799. 7. -00992 hhd. a -760995370 da. 1. 2 ft. 3 ill. 4. 10 sec. EXERCISE 78. *' » 2. 3 da. 2hr. 40min. * 3. 8 oz.'^dwt.T^rs, 5. 2 pk. qt. Iv' pt. 0. .i qt. 1 pt. ». 4 oz. h. 10 o7>. Ft. ir.51min. wfc. 4 trra. .7(it.6[)fc. *z. ; 53 t> 3(> t.q»in. t. 4| gr. ; rt. 2^ gr. 10| (./..; Is 0^ d. 7 70 5^0 lt>. •7945 fc.- qfc. 1 pt. ANSWERS. 179 ^ /% " ion ™ ft 94. Ih 9. 2 m. 3 tur. 8rd. V4'-rd'^" rd. ftM'"- 11. 19 hrs. 52 min. 12. 1 fur. 5fj| rd. EXERCISE 79. 11. 7t. 65 1b. lOoz. 12. 11 bu. 3pk.4qt,I,«5Pfc- EXERCISE 80. 1 $177 25. 2. f^253.50. 3. $796.80 4 $1053'33^^^ R fii^/fW'i fi «iU40 V. $3202-87i 8. $r^41-3ov l\ &.50.'-ll''l'317Wi' ll.|3H5^^ 12^ m>-^^- 25. |396.93i 26. $1510.87^ ^7- $48 ; S^-aO , $6<^ . J4^7^ ; $2> 28. $4.76. h EXERCISE 81. ^ 4. $5-8125. 2. $3-98^^ 3^ ^f^i^^' f $^(.7 $339-2^; $470-ob;^ $i4o9->l- -^^ ^** ^v^, «» ^76-70, $331-50, $813-28. REVIEW EXERCISES, Page 122. 1 ^12.50. 2. 2t.7cwt,2qi-Ulb. ^^'^^^^^ pk- 11 pt .•oio^^fr ff 4 ft. 8^-^-* ri. 6.776 yds. /. 117760 Umea t ^ t . "&.1^5.80 l^. , ^p- $2-805. il'|V4-Vfttir-'l7:-"'-t- i.'. •Sl'.iSac. 19. 2-50227. 5 04 20. ^..- "21. 24. 4t>(Hk. 2&. 1^ oolite • •y6875. " 22. 25. 4746 da- 41:^ AH* 26. y. 23. 12408 steps. 11./:^ V I !!■«' I'll * f' 180 ANSWERS. 31. 5 bbls., $3-89o remaining. S3. 101b. Oi^^oz 30. 5 11.34 in ' ^4- nU' 32. $35-60. |t3-25. 35. sec. 37- $ 10-55 R value of remainder. 38. 29/^ rods^'' y30. 14,-^ lb. 40- Ijy in- 41. $1626. 42. 3i t. 43. £36 Os-Od. 44. 124628033290 m. 33 rd. 5yd.4in. 45. 2 ac. 3 r. 9 p. 21 yd. 6 ft. 78 in ' " 47. 37026 feet. 48. -02083 EXERCISE 82. 49. 138 gal. 46. -21875. 50. 37026 ft. f , lii i i| 7» 2. li^.,. 7. 3. 8. -/^^. ' 13. leei 14. 5^. 11 9. 200.^ 3. \L 4. 4. 0- 51. 15. ,ii,. y; 661^ EXERCISE 83. J 5 1 () 6. 11. 3|. 12 if) 5 • O 7.-} 1. 4. 7. 12. 32. 3332 2. '\. 8. 1734 13 17. $>)5 06|. .14 I'V 3. 56. 4. 4. 9. 108. 10. 14. 11. 15. If I}, 5, 11 0. 45, 11. 107. 18. 5<;]yds. EXERCISE 84 1. 12. 2. e\j. 3. 42. 4 EXERCISE 85. The 4th, 5th, 6th and 7th are true proportions. EXERCISE 86. 16. 3 ft. 4^ in ^. 1. 18. 2. 2.*.5. 6. 4/.. 7. l"38|^f. 11. 1{^. 12. 3173. 13 16. 17 gal. 3iVt qt- " 18. 1212{| miles. 19. 5^. 11. laO 3. 20/f^. 4. 576. 8. 32571 42? . y.L 10 5 yd. 0,V, qr. 14. 32 ac. 77^'i"ds. 16. |50-76}i^t. 17. $>980f. $2925. 20. £10 18s. 6M. EXERCISE 87 1. 531 -381 fLm^^yr o. {Jjo. « Oq , 4. 2yr.10ni.2da. 5. $-763." 6. iM^. ' '• 7. $S4•30•^ 8. '£'.34 12s. 4d. 9. $575-805. 10. .97152 -31] ly. 11. $1533-67ili . 12. $77. 13. 2 bus. 3pks. Igal. 1 qt. 1| pts. 14. 7^^ bbls. 15. 61b.l2..\cz. 16. lOiV^c 18. $11.50. 19 $",3-91'^. lO. $39-40g. 21. 145^, yds- nearly. 22. 608 yds *>i\ ^^j •J HA 37 bb! s . 25. 1? 1^1808 -78^. 27. 25jVry^- •J 1 Kl .S(i IVi 28. $456 131?:) A ..-iii wa-.niii ANSWERS. 181 ' i 2 r- , 11 150 3. $6-35|. 0. 162-10. 10. 350 men. 13. 7^2 pairs. 16. $118-52+. 43 8d. 44. 70ft. 8-232 m. 45. i»u. *"• -^"3 J 47^ ^171 -67 + . % EXERCISE 88. 1 14.7, days. 2. $2105TjO. 7 «54-3U 8. 501b. 0. ^117 ll.^'So bricks. ,,12..,»«^- 14. 866061 yd. if.J ^^''^ iq s32.'"' 20. $192 17 . 200 men . 18 • $1827 • ^y . |^. ^ , oi (ife'77'795 22. lOidays. 23. '57trr ^*-^^^;'/';; ^^' S77ni • 9A $->9 794- 27. 20men. 28. 21 days. 25. 40|Vday8. 2b. *J^ <»+ • ^'• EXERCISE «9. ^ /M- Q Aft 4 11. 5 035. 14. h,-io.hh lb- IIXERCISE 90. . ^o «A 9 ayove. ^^^.^trly'. *;805 ■ 23 to pay < )ver . i - >/'* • '"^ ■ ^- '^ iO $129-26 commission, 195^ bbls nearly. EXERCISE 92 1 dKi-QAft 9 SSSO 08 3, $22-23 4.$413 86|. 1. $la3 68. ^'^^^I^^"^-. A.>|4-272. 8. K^6-87i. 9. leif:'' 10.$98.9iK a. $7-d64. 12. $buyo. 182 ANSWERS, I ' EXERCISE 93. L $S-24i- 2. $1110^^. 3. Balance $3-K 5. $41-89^. 6. $5320-181. 7- 17-77. 9. $15'2GfV 10- $1112-56. EXERCISE 94. 4. $82-85i. 8. $7 -284 th. 36 sq. in. . 3 sq. ffc. 36 sq. in. 2,/^6q. 7. 2 sq 8(1. ft, rd. 3 sq. yd. 1. 3sq. 3. 17sq.yCl. , / 8 sq. ft. 90 sq. in. 5. Miiq. yd, 4 sq. ft. 72 acj. in, 6. 184 sq. rd. 14 sq. yd. . ^. 65 sq. yd. 4 sq. ft. y^ ^ 8. 18,ift, 9. 33 yd. $29*70. 10. 123^ yd. $186. 98^ 11. 24|:ifydv $45-82 noiirly..> 12, 45^ yd. $96-33+ . 13. I157-60. ^ 14. $m3Srfear]y. 15. $10-93J, J6. 15.]^ rolls.ffc 17. $433-125. 18. $186,982. 19. $503^84. EXERCISE 95. 1 . 4486 nearly, 3. 2100|, 2. llf thousand, 58^ lb. nails. 4. 2^62+. EXERCISE 96. 9 6. 380| cu. ft. $173-33|. 12. 90 cords. 15. 10 -98 + cords 18. 22320. 19. 22. 224 bd. ft. 25. 33 bd. ft. 5 01 29. $7.29. ft. 10. I. 195 cu. in. 2. 4050 cu. ft. 3. 390 cu. yd, 4. 851 cu.ft. 5. 2850 cu. ft. 8. 175'f^cu. ft. II. $3-93|; 15-25. 14. 34 cords 58 cu. ft 17. 7 i^y cords . 21. 18| board ft. 24. 34^ bd. ft. 27. $11.5.", 28. . 31. $23-89, 32. 112 bd. ft. 33. 87^ bd. ft 7. 02|cu 13| cords. 13. 145.^ cord, 16. 41 '55 cords. $590.20, 23. 35 bd. ft. 26. 17bd. ft. 30. $15.57. f ' 4. S82-85i. i. $7 -284 (1- ft, d. 3 sq. yd. t. 72 8(1 . in, : sq. ft. y^ i ^d. $186. 96v $96-33+. 5. $10-93J, , $186,982, b. nails. 4. 85?, cu.ft. . 62|"cu. ft. I cords. !. 145,^ cord, 41 '55 cords. $590.20, 3. 35 bd. ft. 6. 17bd. ft. 30. $15.57. ft CONTENTS. CHAPTER I. PAGE DEFINITIONS, Notation and Numeration 5 Addition 12 Addition of Canadian Money 18 Subtraction 21 Canadian Money 26 Multipli caton 28 Canadian Money 38 Division 41 Canadian Money 50 CHAPTER II. PRIME NUMBERS 53 Cancellation 54 Greatest Common Measure 55 Least Common Multiple 58 CHAPTER III. FRACTIONS, Definitions 61 Reductions of Fractions 63 Addition of Fractions 70 Subtraction of Fractions 71 Multiplication of Fractions 75 Division of Fractions 79 Complex Fractions 82 CHAPTER IV. DECIMALS, Definition, Notation and Numeration 87 Addition of Decimals ^^1 Subtraction of Decimals 92 Multiplication of Decimals 93 Division of Decimals 95 Reduction of Decimals 98 Circulating Decimals. • 98 il n ! '1 ^^'^ CONTENTS. CHAPTEfl V. REDUCTION AND TABLES i06 Compound Addition [ ^ xi5 Conii)()und Subtraction . . . .IK} Compound Multiplication 1^7 Compound Division ..!!.. 1 18 Donominafce Fractiims .119 Practice . . 125 CHAPTER VI. RATIO, Definitions 228 Proportion, Detinitiona .131 Simple Propf)rtion 132 Compound Proportion 135 CHAPTER VII. PERCENTAGE, Deliniiions 139 Commission and Brokerage 141 Simple Interest 242 CHAPTER VIII. ACCOUNTS AND BILLS 144 CHAPTER IX. MEASUREMENT OF RECTANGULAR SUR- FACES 151 Builders' Estimates 153 Measurement )f Rectangular Solids .154 Farmer's Estimates " . . I5(j Metric System 157 Answers I(j3 ! s I! it II ffl III liil ^-^~'»« w.>.£;L.itu . ■jiei.iMMix£i^^iVimmjf!.tJma 106 . ......115 IKJ 117 118 119 125 128 131 132 135 139 141 142 144 SUR- 151 153 154 156 157 1()3 i^*itti^fA.^.niMftgnwr.»i»