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^:? /jt^ ^^ y- * ^ ^i ^ /^^y^ 
 
 
 / 
 
 / 
 
 
 / 
 
 PI 
 
 ^^ 
 
 IN] 
 
 ■ft 
 
 BV Ti 
 
f<-i 
 
 / 
 
 »i4. 
 
 NEW 
 
 PRACTICAL AMTIIMETTC ; 
 
 INf WHICri 
 
 THE SCIENCE AND ITS APPLICATIONS 
 
 ARE SIMPLIFIED BV 
 
 INDUCTION AND ANALYSIS. 
 
 rnEPAunD to acchjmpanv t:- mathematical series of 
 
 BENJAMIN GREENLEAF, A.M. 
 
 BY THE EDITOR OF '' NEIV ELEMENTARY ALGEBRA,'' " XEIV 
 HIGHER ALGEBRA^' ETC., IN THE SERIES, 
 
 
 HALIFAX, N.S. 
 
 A. Sc W. M A C K I N L A Y. 
 
 1872. 
 
4i 
 
 L 
 
 I 
 
4i 
 
 INTRODUCTION. 
 
 I 
 
 Culture is progressive in its nature. Higher, still hialuT, is 
 the true educational spirit. * 
 
 Advance in methods of instruction makes new, improved 
 text-l,ook,s a necessity; and, to subserve wants apparently not 
 heretofore fuliy luovided for, this work has been carefullv pre- 
 pared. " '■ 
 
 Without being in any respect redunrkmt, it is intended to 
 be complete m details and com])rehensive in scope ; 
 
 _ Combining with processes the most scientific the -rcatest 
 simplicity; * 
 
 Developing principles by inductive methods, deducin'^ ruh-s 
 Irom rational solutions, and encouraging self-reliance and 
 originality by numerous exercises in analysis; 
 
 Making written arithmetic in all its steps intellectual ; and 
 
 Keeping prominently in view tlr: practical uses of numbers 
 by various ai)pli cations of a business character. 
 
 While it avoids obsolete or useless mafrial, it properly 
 ti-eats new topics requiring attenti(.n, such as the Metric 
 System of Weights and Measures, Annual Interest, Internal 
 Kevenue, d'c.; and 
 
 Enforces thorough educational results, by ordcu-ly arran-e- 
 ment^ of subjects, and by systematic review questions ami 
 exercises. 
 
 The prominence given in this book to the enunciation of 
 i-iinciph.s, will, It is believed, commend itself to the enli-ht- 
 
 M 
 
 I 
 
IV 
 
 INTRODUCTTON. 
 
 f'no(l educator, sinco, witliout a knoM^edgo of tliose principk s, 
 tlio art of using numbers becomei^iTiere mechanical ciphering. 
 
 Multiplication and Division of Decimal Fractions have been 
 much simplified by assimilating their processes to those of like 
 cases in Common Fractions, and by making the corresponding 
 rules substantially the same. 
 
 By treating of Fractions before Compound Denominate 
 Numbers, the Keduction of the latter is made more thorough, 
 and a number of special rules is avoided. 
 
 Many rules of limited application are also dispensed with, 
 by analysing single examples, of some anomalous kind, as a 
 guide to the solution of all others of its class. 
 
 Tlie examples have been selected with special reference to 
 their adaptation to the present wants of active life. 
 
 Grateful expression of indebtedness is due to Hon. John A. 
 Kasson of Iowa, to Prof. H. A. Newton of Yale College, and 
 to many others, for favours received while preparing this 
 volume. 
 
 PERMANENT EDITION. 
 
 The generous favour extended to tliisjaook, and its wide introduc- 
 tion into the best schools, have led to the sale of several large 
 editions ^a a few months. Encouraged by this marked appreciation, 
 the work has been critically re-examined, and put in a permanent 
 form. 
 
 H. B. Maglathlin. 
 
 Kingston, Mass., Alarch 1867. 
 
it 
 
 CONTENTS. 
 
 DtFINITION'S . 
 
 XoTATlON AND NUMERATI'JX 
 
 Addition- 
 
 SubTKACTION . 
 
 Kkview J'Jxkhcises . 
 
 MULTtPHCATlUN 
 
 PA OK 
 
 7 
 
 8 
 17 
 25 
 ii-2 
 33 
 
 Diviaio.v 
 
 PA OK 
 
 . 42 
 
 HeVIKW EXERCISK3 . 
 
 5i 
 
 Gl^NEIlAL PkINCU'LKS 
 
 50 
 
 Formulas 
 
 57 
 
 Kkvii;w ExEucisrs . 
 
 53 
 
 AunuMETiCAL Analy.sis 
 
 t)0 
 
 i 
 
 United States Money 
 
 Coins 
 
 Exchange of Commodities . 
 
 Factoring 
 Exact Divisions 
 Prime Numbers 
 Factoring op Nhmbers 
 
 Common Fractions , 
 Common Denominator 
 Kelations of Numbers 
 
 G3 
 72 
 
 78 
 79 
 80 
 81 
 
 Bills and Invoices 
 Accounts 
 Ledger Columns 
 
 Multiplication by Factors 
 Division by Factors 
 Greatest Commi^n Divisor 
 Least Common Multii'le 
 
 94 I Exercises in Analysis 
 102 Decimal Fractions . 
 120 j ExERcfsEs IN Analysis 
 
 73 
 76 
 
 77 
 
 83 
 
 84 
 87 
 91 
 
 12t 
 127 
 142 
 
 -*(? 
 
 Weights and Measures 
 Miscellaneous Measures 
 Metric System 
 Comparative Table . 
 Conversion of Prices 
 
 144 
 153 
 155 
 159 
 161 
 
 Denominate Numbers 
 
 PiEDUCTION 
 
 Longitude and Time 
 
 Practice 
 
 Review Exercises . 
 
 102 
 1G3 
 182 
 184 
 
 18G 
 
 ,5- 
 
VI 
 
 Pr.RCrNTAOK . 
 
 C'oMMIHSIOX AND I'lUOKEnAOK 
 
 Insijuanck 
 Profit and T^oss 
 
 llKVIKW ICXF.KCISEH 
 SlMIT.K InTKUDST 
 Phksknt WollTlI 
 I'.ANK DlSCOUNl' 
 
 CONTENT.-*. 
 
 188 
 
 197 
 198 
 202 
 203 
 214 
 21G 
 
 Annual iNTiRrsT 
 Partial I'aymknt.s . 
 
 MkRCANTILK Rt'LK . 
 
 United Statkh PiI^lk 
 
 CONNKCTICL'T JtULK . 
 VkrMONT llULK 
 COMTOUND InTIOUEST . 
 
 Review Exercises . 
 
 PAflB 
 
 2i9 
 220 
 220 
 222 
 224 
 225 
 22() 
 229 
 
 V 
 
 I 
 
 1 ■ 
 
 I 
 
 I 
 
 Ratio and Proportion 
 Simple Proportion . 
 Compound Proportion 
 pARTNi:i{.snip 
 Equation of Payments 
 Averaging op Accounts 
 Settlement of Accounts 
 Interest Method 
 Taxes 
 
 230 
 
 Duties 
 
 . 252 
 
 233 
 
 Customs 
 
 252 
 
 236 
 
 Stocks 
 
 . 253 
 
 238 
 
 Exchange 
 
 255 
 
 242 
 
 Course of Exchange 
 
 255 
 
 245 
 
 Inland Exchange . 
 
 250 
 
 248 
 
 Foi.EiuN Exchange . 
 
 258 
 
 248 
 
 Review Exercises . 
 
 2()1 
 
 250 : 
 
 Exercises in Analysis 
 
 202 
 
 Involution . 
 
 Evolution . 
 
 Square Hoot 
 
 Cube Root . 
 
 Mensuration 
 
 Triangles 
 
 Quadrilaterals 
 
 Areas 
 
 Circles 
 
 Prisms and Cylinders 
 
 266 
 268 
 268 
 274 
 2S0 
 282 
 284 
 2S4 
 287 
 289 
 
 Pyramids and Cones 
 
 Spheres 
 
 Similar Fku-hes 
 
 Medial Proportion . 
 
 Arithmetical Series 
 
 Geometrical Series . 
 
 Annuities 
 
 h view Exercises . 
 
 Exercises in Analysis 
 
 Miscellaneous Exercises 
 
 290 
 2ii3 
 294 
 298 
 ;503 
 305 
 .'508 
 310 
 312 
 317 
 
 Appendix 
 
 321 
 
rRACTrCAL ARITHMETIC. 
 
 'i 
 
 J 
 f 
 
 1 
 
 J 
 
 DEFINITIONS. 
 
 1. A Unit is a .single thing, or one. Tims, 
 A dollar is a unit, an api)Ie is a unit, (V^c. 
 
 2. Quantity is anything that can ho measurod. or com- 
 puted. Thus, 
 
 Distance is quantity, since it can be measured, so as to be 
 named miles, rods, c^-c. 
 
 3. A Number is a unit, or a collection of units of the same 
 kind. 
 
 One, two, three, four, c^-c, which show lir.wmany units tliero 
 are of any quantity, express numbers. 
 
 4. Tlie Unit of a Number is one of tliat number. Thus 
 One dollar is the unit of three dollars, and one the unit of 
 
 five. 
 
 5. Like Numbers are such as have the same unit. Thus, 
 Five dollars and seven dollars are like numbers, 
 
 6. A Concrete Number is a number in which some kind of 
 unit IS named. Thus, 
 
 Two hooks, five days, seven dollars, are concrete numbers. 
 
 Wljat is a Unit? A Quantity? A Number? The Unit of a Number ' 
 Like Numbers ? A Concrete Number ? ' 
 
 I 
 
8 
 
 PRACTICAL AIIITIIMKTIC. 
 
 7. An Abstract Number is a iiuiul)L'r in ^vlli^;h no particular 
 kind of unit is nanu;(l. Thus, 
 
 Two, five, in whicli no particular unit is named, are abstract 
 numbers. 
 
 8. An Operation is a process performed "svith numbers. 
 
 9. An Answer is tiio result of a correct oi)eration. 
 
 10. A Solution is an explanation of the operation. 
 
 11. A Rule is a direction for perfornung an operation. 
 
 12. An Example is an application of a rule. 
 
 13. An Exercise is a lesson for practice. 
 
 14. Arithmetic is the science of numbers and the art of 
 usini' them. 
 
 15. Practical Arithmetic treats of the methods of applying 
 numbers to practical or business purposes. 
 
 t 
 
 I 
 
 i 
 I 
 
 NOTATION AND NUMERATION. 
 
 16. Notation is the method of writing numbers. 
 
 17. Numeration is the method of reading numbers. 
 
 18. Figures are certain marks or characters used to express 
 numbers. 
 
 The method of expressing numbers by figures is called the 
 Arabic, because it was used by the Arabs. 
 
 Ten different figures are used in expressing numbers. 
 
 What is an Abstract Number? "VVliat is an Operation? An Answer? 
 A Solution? A Rule? An Example? An Exercise? Arithmetic? 
 Practical Arithmetic? Notation? Numeration? What are Figures? 
 What is the nietln-a of expressing numbers by ligures called? Why so 
 cidled? 
 

 
 NOTATION AND NUMERATION*. 
 
 9 
 
 N ini''^. 
 
 or 
 
 Fl);iir''t) 
 
 Fi>;iiri's 
 
 NamoH. 
 
 or 
 
 Flsurc^ 
 
 Fl I'll re* ■ 
 
 viiluu (k'UDlcil. 
 
 ttH iirliitcd. 
 
 ai written. 
 
 valiit! (Icii 
 
 oted. 
 
 us iniiituil. 
 
 a.s writt-a. I 
 
 Ciplior 
 
 
 
 
 
 
 Five, 
 
 
 5 
 
 f 
 
 Oil.', 
 
 
 1 
 
 / 
 
 Six, 
 
 
 6 
 
 ( 
 
 Two, 
 
 
 2 
 
 3 
 
 Sloven, 
 
 
 7 
 
 7 
 
 Tliive, 
 
 
 3 
 
 <3 
 
 Eiglit, 
 
 
 8 
 
 S 
 
 Four, 
 
 
 4 
 
 4 
 
 Nino, 
 
 
 9 
 
 J? 1 
 
 19. TIic fi.Ljuros 1, 2, 3, 4, 5, G, 7, 8, 9, arc callod sir/nijicioif 
 figures, because each sitjnijies, or stands for, the nuniijer whifh 
 its name denotes. 
 
 The figure 0. or ciplier, is sonuitimes called ::ero, or nitin/hf, 
 because, when used alone, it stands for 7W mimhcr. Tims, 
 
 dollars means no dollars. 
 
 No number higher tiian nine can be expressed by a single 
 figure, but by combining figures all other numbers may be 
 denoted. 
 
 UNITS, TENS, HUNDREDS. 
 
 20. In naming numbers, nine units and one more are re- 
 garded as forming a single group, or collection, called Ten. 
 
 One ten and one are called ELEVEN ; one ten and two, 
 twi:lve; one ten and three, thirteen; one ten and four, 
 FOUIITEEN, Sec, "teen" meaning "(uid ten." 
 
 Two tens are called twenty ; t^ ree tens, TlllliTY ; four tens, 
 FORTY, &c., "/?/" meaning " tens.'^ 
 
 21. To express ten, twenty, thirty &c., we write 1, 2, .3, 
 &c., denoting the number of tens, to the left of 0. Thus, 
 
 Ten, 10 Forty, 40 Seventy, 70 
 
 Twenty, 20 I Fifty, 50 Eighty, 80 
 
 Thirty, 30 Sixty, GO Ninety, 90 
 
 Why are ' 2, 3, &;c., called signiticivnt figures? What is the cipher some- 
 times called How liigi numbers can be expressed by a siiiglo tigure? 
 Ho'^' may all umbers be it-noted? What name is given to nine and one 
 more ? To ten i one, &c. "' How do we express ten, twenty, thirty, (Sec? 
 
 I 
 
10 
 
 PRACTICAL AKITIIMETIC. 
 
 II 
 
 wlicre denotes the absence o^ ones, or u'iits of the first order, 
 ixnd makes other figures express tens, or units of the second 
 order. 
 
 22. To express the whole niimhero intermediate between 
 ten and twenty, twenty and thirty, thirty and forty, c^'c., ^\■v- 
 Avrite the figures denoting the nuiu'Der of tens, to tlie left of 1, 
 2, o, iScc, expressing the ones, or units of the first order. Thus, 
 
 Eleven, 
 
 Twelve, 
 
 Thirteen, 
 
 Fourteen, 
 
 Fifteen, 
 
 and so on. 
 
 11 Sixteen, 
 
 12 Seventeen, 
 
 13 Eigliteen, 
 
 14 1 Nineteen, 
 
 15 I Twenty-one, 
 
 IG Twenty-two, 
 
 17 Twenty-three, 
 
 18 Twenty-four, 
 
 19 Twenty-five, 
 21 Twenty-six. 
 
 •)0 
 
 2;> 
 
 2t 
 25 
 2G 
 
 23. Ten tens are called OxE Hundred, which forms a unit 
 of the third order, and is written 100. 
 
 Therefore, to express one hundred, two hundred, three hun- 
 dred, &c., we write the figure denoting their number with two 
 ci])hers at the right. Thus, 
 
 Two hundred, 200 
 
 Tlu-ce hundred, 300 
 
 Four hundred, 400 
 
 Five hundred, 500 
 
 Six hundred, GOO 
 
 Seven hundred, 700 
 
 Eight hundred, 800 
 
 aNine hundred, 000 
 
 24. In expressing a whole number by three figures, we 
 j)lace the figure denoting the Jnindreds in the third, the figure 
 denoting the tens in the second, and the figure denoting the 
 units in the Jirst place from the right. Thus, 
 
 Two hundred and sixty-three, or 2 hundreds, G tens, and 3 
 units, is v/ritten 2C3. 
 
 FIa'c hundred and seven, or 5 hundreds, tens, and 7 units, 
 is written 507. 
 
 How do we express numbers between ten nnd twenty ? AVhat are ten 
 tensciillccl? AVhat forms a unit of the third order? In writing a number 
 expressed by three figures, how are the fi^jures placed? 
 
NOTATION AND NUMERATION. 
 
 11 
 
 Exercises. 
 
 Write in figures arranged in colunin.^ : 
 
 1. The numbers between forty- five and si.vty-tlireo. 
 
 1'. The numbers between ninety-one and one Imndred. 
 
 3. One liundred and one, one Imndred and eleven. 
 
 4. Eiglity-eiglit, eiglit, eiglit hundred and eighty. 
 
 5. Thirteen, thirty-one, three liundred and on(!. 
 
 6. Six liundred and five, five hundred and sixty-six. 
 
 7. Eleven, seventy-seven, seven lunidred and eleven. 
 
 n 
 
 THOUSANDS. 
 
 25. Ten hundreds are called One Thousand, whicli f.^rms 
 a unit of the /r;,7r//i order, and is written 1000. 
 
 Therefore, to express one thousand, two thousand, three 
 thousand, vV-c., we write the figures denoting their number 
 with three ciphers at the right. Thus, we write. 
 
 Two thousand, 
 Three thousand. 
 Four thousand, 
 Five thousand, 
 
 2000 
 3000 
 4000 
 5000 
 
 Six thousand, HOOO 
 
 Seven thousand, 7000 
 
 Kight thousand, 8000 
 
 Nine thousand, 0000 
 
 26. Ten thousands are called One Tkx-Tikjusand, which 
 forms a unit of the Jiffh order, and is written 10000. Also, 
 
 Two ten-thousands, or twenty thousand, is written 20000. 
 Thrje ten-thousands, or thirty thousand, is written 30000. 
 and so on. 
 
 27. Ten ten-thousands are called One IIUNDiiEU-TlioU- 
 SAND, whicli forms a unit of the si.dh order, and is written 
 1 00000. Also, 
 
 Two hundred thousand is written 200000; three hundred 
 thousand is written 300000 ; and so on. 
 
 A\h.at are to,, Uu,vl,c,Is called? What u„it does one thousand f.vrtn ' 
 ITowaro one thousand, two tho.isand, &c., Avritten ? AVhafc a.e ton-thou- 
 sands caod? What unit do ten-thousands form? What are ten ten-thou- 
 sands called ? What unit do ten ten-thousands form ? 
 
 !■■ I. 
 
12 
 
 PRACTICAL AIIITIIMETIC. 
 
 28. The first six ouders of units, beginning with the 
 units, are named : UNITS, tens, hundreds, THOUSANDS, tefi- 
 ihousands, hundred-thousands. 
 
 The oiiDEiis OF FIGURES are the positions they occupy with 
 reference to each other, when written side hy side. 
 
 Each figure expresses an unvarying number cf units, and the 
 order of the figure determines the size or name of the units. 
 
 For convenience, in reading numbers expressed by figures, 
 their orders are separated into groups, of thre^ figures each, 
 called PERIODS. Each period takes its name from its right 
 hand order. '!Tius, we have, 
 
 Second Period. 
 
 I'llOUSANDS. 
 
 ^ 
 
 First Period. 
 
 UNJTS. 
 
 . . ^ ^ ,m 
 
 CO 
 
 O 
 
 -(J 
 
 I 
 
 a 
 
 § 
 
 O 
 
 
 
 09 
 
 t 
 o 
 
 03 
 
 a 
 
 3 
 
 H 
 7 
 
 ■0 
 - J 
 
 4 
 
 where the figures express 3 hundred-thousands ten-thou- 
 sands G thousands 5 hundreds 7 tens 4 units, or three hundred 
 six thousand live hundred and seventy-four. 
 
 29. In general, in writing a number by figures, we write 
 each of the figures in the order of its units, and note the ab- 
 sence of a significant figure, in an order, by a cipher. Thus, 
 
 Five thousand and twenty, or 5 thousands liundreds 2 
 tens units, is written 5,020. 
 
 Name the first six orders of units. What .ire orders of figures? To wliat 
 do they correspond? How are orders of ligures separated for convenience iu 
 reading? rroin what does each period take its name ? Name the first two 
 l)eriods. How do we write figured in expressing numbers? 
 
 Ij 
 
I) 
 
 1 
 
 V 
 
 NOTATION AND NUMERATION. 
 
 13 
 
 Exercises. 
 
 Write in figures and read : 
 
 1. Tliree units of the fourth order, Avitli no units of tlie third 
 order, two units of the second, and one unit of the first order. 
 
 Ans. 3,021 ; read, three tliousand and twenty-one. 
 
 2. 5 hundred- tliousands 3 ten-thousands and 4 thousands G 
 Inmdreds and 4 tens. 
 
 Ans. 534,640; read, five hundred thirty-four thousand six 
 hundred and forty. 
 
 3. 415 in the second period, and 405 in the first period. 
 Ans. 415,405 ; read, four liundred and fifteen thousand four 
 
 luindred and five. 
 
 4. 207 in the second period and iri each order of the first 
 period. 
 
 5. 38 tliousands 3 hundreds 5 tens and 2 units. 
 
 G. G hundred-thousands 5 ten-thousands with 7 units of 
 the first order. 
 
 30. The Arabic system of notation is based upon the fol- 
 lowing 
 
 GENERAL PRINCIPLES. 
 
 1. Numbers may be cxjmssed by n-rltlng firjures so as to denote 
 their orders of units. 
 
 2. Ciphers ivrittcn zvith other firjures denote or mark the orders 
 in which units are omitted. 
 
 3. Ten units of any lower order are always equal to one of the 
 next higher. 
 
 That is, ten units make one ten, ten tens make one hundred, 
 ten hundreds make one thousand, and so on. Hence, 
 
 4. Each removal of a figure an order towards the left, makes 
 the value expressed tetifold. 
 
 Give the first general principle. The second. The third. The fourth. 
 
14 
 
 PRACTICAL ARITIIMKTIC. 
 
 vSCALE OF NUMBRRS. 
 
 31. A Scale of NumToers is tlie niimhor or numbers express- 
 ing tlio law* of relation between their different units. 
 
 32. In numbers, where ten units of any lower order always 
 make one of the next higher, the scale is ten, and uniform. 
 
 For this reason, the system of numbers in general use has 
 been called from decern, tlie Laiin for tai, the Decimal System 
 OF Numbers. 
 
 33. The common, or French method of Numeration, is 
 exhibited in the followincc 
 
 Numeration Table. 
 
 f th IVriod. 5th Period. 4tl» Period. ;;d Period. 2d Period. l.st Period. 
 
 QUAUUILI-IO.N'd. TRILLIONS. UILLIONS. MILLIONS. TlIOUbANDS. U.MTS. 
 
 !m 
 
 "^ . • 13 
 
 —-• V) -• m t^ 
 
 ^ « s § a u 
 
 r;._-i.,— I. _ , , — 1./ c^- 
 
 ^ "C "A -^ "A ^ a C! g -^ ?k 'f' rA 
 
 •Ti -;, '^ '~Z' '^ vj '~::! -r-t .rzi;—, „'} '~^ ^ ^"^ 
 
 ■ ^^^ ^H - .1^ ^H •....' , I ^i^ ^^ . II rf >-H (^ , I ^-"H ijj ■ __r 
 
 ^ . Hr-t I' 'rHl.ZH'-''lJlH'-« I --'^a?-*-J 
 
 -4_1 -|_l -l-i -»-J 4-1 -l-J -4-J -.J -i-J *-> -1-) +J 4-J -Ui _J f;-\ ^1 '^ 
 
 CO t- wi O 'ti Cti CI .— I O C5 CO I:- O O ^ • « I— I 
 
 4 3 1, 5 G i>, 7 9 3, 1 5 4, 2 5, G 3 8; 
 
 where the figures express four hundred thirty-one QUADRIL- 
 LIOXS, five hundred sixty-two trillions, seven hundred 
 ninety-three EILLIONS, one hundred fifty-four millions, two 
 hundred five THOUSANDS, six hundred and thirty-eight. 
 
 A dot ( . ), called tlie Decimal Point, is u.sed to nuuk the units' 
 place, by being written at the right of the units' figure. Thu.s, 
 
 8. id read eight imits. 
 
 What is a Scale of Numbers? "What is the scale of numbers in which ten 
 units of a lower order always make one of the next higher? Ecginning at 
 the right, name the periods in the table. Tlie orders. The value exi)re3sed. 
 
 1.- 
 
 a 
 
NOTATION AND NUMERATION. 
 
 15 
 
 The periods above Quadrillions, in their onhir, are Qiiintil- 
 lioiis, Sextillioiis, Septiilions, Octillions, ^suuillions, Decillions, 
 Undecillions, Duodecillions, Tredccillions, Qiuituordecillions, 
 Qiiindecillions, Sexdecillions, Septendccillions, Octodccillions, 
 Novendecillions, Vigintillions, t^'c. 
 
 34. To read numbers expressed by figures. 
 
 1. Let it be required to read 7G0325 1. 
 
 Pointing oil" tlu; given figures into periods, beginning at the uiiit.s' 
 place, we have 7,69;3,2.')4. 
 
 The third j)eriod is 7 millions, the second period is G93 tJioum)i(/.'i, 
 the iirst period i.s 254 units ; therulore, the whole rea<ls, seven million 
 six hundred ninety -three thousand two hundred and lii'ty-i'our. 
 Hence, the 
 
 liULE. Bajiniilng at the wilL< place, point ojf the (jiirn expres- 
 sion into as mani/ j^eriods as possible, of three Jl'j'i res each. 
 
 Then Icfjin at the left and read each period, giciii'j after each, 
 excepting the la.<t, the name of the period. 
 
 The name ol" units' period is not given in reading figures, since it 
 is readily understood , and the decimal point, when not written, is 
 also understood. 
 
 M 
 
 3. 
 
 4. 
 5. 
 
 G. 
 7. 
 8. 
 9. 
 10. 
 
 
 Examples. 
 
 
 
 d read the following : 
 
 
 
 G04 
 
 11. 
 
 80G00 
 
 20. 
 
 99S081 
 
 98 
 
 12. 
 
 51155 
 
 21. 
 
 191G7700 
 
 704 
 
 13. 
 
 99o(;G 
 
 22. 
 
 13245G7 
 
 1132 
 
 14. 
 
 80080 
 
 23. 
 
 82000310 
 
 5005 
 
 15. 
 
 110011 
 
 24. 
 
 17G34042 
 
 7230 
 
 IG. 
 
 33333 
 
 25. 
 
 400400G1 
 
 990 
 
 17. 
 
 520404 
 
 2G. 
 
 789231G 
 
 GGOG 
 
 18. 
 
 70O77G 
 
 27. 
 
 43448801 
 
 7711 
 
 19. 
 
 144189 
 
 28. 
 
 50040030 
 
 29. 
 
 2001002003004 i 30. 570080099044273G 
 
 Nume the periods in their order above Quadrilliuiis. Repeat the liule. 
 Why is not the name of units' period given in reading figures? 
 
16 
 
 PRACTICAL ARITHMETIC. 
 
 35. To write numbers in figures. 
 
 1. Let it be required to write in figures, seven million six 
 Imiidred ninety-tliree thousand two hundred and four. 
 
 Writiiinr tlie 7 millions as the only order of the third period, the 
 (;i)3 thousands as the orders of the second period, and the 204 as the 
 orders of the first period, Ave have 7,693,204, Hence, the following 
 
 EuLE. Beginning vith the highest period to he exj)ressecl, write 
 the figures belonging to each 2^eriod, in their orders, observing to 
 mark the omission of any order of units iciih a cipher. 
 
 Ans. 87. 
 Ans. 301. 
 
 Examples. 
 
 AVrito in figures the following numbers : 
 
 2. 8 tens and 7 units. 
 
 ;i 3 units of the third order and 1 of the first. 
 
 4. One hundred and twenty-five. 
 
 5. 7 hundreds, nine ^-^ns, and 6 units. 
 C. 8 units of the second order and of the first. 
 
 7. Nine hundred and ninety-seven. 
 
 8. Five thousand and sixty-two. 
 
 9. Fifty-five thousand and five hundred. 
 
 10. One hundred six thousand. 
 
 11. 90 thousands and nine hundreds. 
 
 12. 100 thousands 7 hundreds G tens and 4 units. 
 
 13. One hundred thousand four hundred and fifteen. 
 
 1 4. Thirty-six thousand and forty-six. 
 
 15. One million one hundred thousand and uue hundred. 
 
 16. One hundred fifty-one millions. 
 
 17. 3 billions, 4 millions, 14 chousands, 4 hundreds, 5 tens, 
 and 6 units. Ans. 3,004,014,456. 
 
 18. Sixteen trillions seven hundred forty-one billions two 
 hundred twenty-three millions one hundred seventy-eight 
 thousand. 
 
 Ans. 90,900. ^ 
 
 
 Repeat the Eule. 
 
ADDIT10,\. |-r 
 
 ADDITION. 
 
 36 1. Aitluir lin« i books and his sister lias 3; how mnnv 
 iiave both togetlicr ? 
 
 SoLUTrox. Tha^ have, tocjethcr, as r.rnnj InoU a. are equal fo 4 
 ^^>ohand 3 booh, ,cJdch are 7 hooU Therefore thn, Loth together hocc 
 i books. ■ u ■ 
 
 2. raid 8 cents for a pencil and 2 cents tbr a pen-lioLhr ■ 
 how much did I pay for the whok^ ? 
 
 3. How many are 3 and G ? 5 and 4 ? 7 and r. ? 
 
 4. John lias 2 apples, Edward 7, and Henry U; iiow many 
 iiave they togetlier ? 
 
 5. JIow many are 3 and 1 and 8 ? 2 and and G ? 
 
 0. In a yard are 10 peadi trees, 5 apph, trees, and 4 ].hini 
 trees; liow many trees are there in all ? 
 The preceding operations are called Addition. Hence, 
 
 37. Addition is the process of finding a number equal to 
 two or more given numbers of the same kind. 
 
 The Sum or Amount is the result of the addition • and 
 c(mtams as many miits as there are in -dl tlie numbers 
 added. 
 
 38. A Sign is a mark used to denote an operation to bo 
 performed, or to shorteii an expression. 
 
 The Sign of Addition is an erect cross, +, called ph... 
 Jims J + 2, read three plus tico, denotes that 3 and 2 are to 
 be added. 
 
 The Sign of Equality is two short parallel horizontal lines 
 and is read cqnah, or are equal to. Tiius, 3 + 2 = 5 is read' 
 three plus two are equal to five. ' ' ' 
 
 tic^l^^Xt^,:;;;^™""-™"'^ ^«-^ T„o si,n or ..„. 
 
 1 
 
 . 
 
 B 
 
IS 
 
 PRACTICAL ARITHMETIC. 
 
 ] 
 
 Addition Table. 
 
 1 
 
 WW 
 
 1 
 
 
 and 
 
 3 
 
 and 
 
 4 and 
 
 5 
 
 and 
 
 1 
 
 are 
 
 Mi 
 
 1 
 
 are 
 
 3 
 
 1 
 
 are 4 
 
 1 
 
 arc 5 
 
 1 
 
 arc G 
 
 
 5) 
 
 3 
 
 2 
 
 )j 
 
 4 
 
 
 
 „ 5 
 
 
 
 ,3 C 
 
 2 
 
 5, 7 
 
 3 
 
 >> 
 
 4 
 
 3 
 
 >» 
 
 5 
 
 3 
 
 3, 
 
 3 
 
 33 < 
 
 3 
 
 5, 8 
 
 4 
 
 >J 
 
 5 
 
 4 
 
 j> 
 
 G 
 
 4 
 
 3, 7 
 
 4 
 
 3, 8 
 
 4 
 
 „ 
 
 5 
 
 J5 
 
 G 
 
 5 
 
 ?j 
 
 7 
 
 5 
 
 33 8 
 
 5 
 
 35 
 
 f) 
 
 3, 10 
 
 
 
 J) 
 
 7 
 
 G 
 
 35 
 
 8 
 
 G 
 
 „ 
 
 G 
 
 3, 10 
 
 G 
 
 3, 11 
 
 7 
 
 J» 
 
 8 
 
 7 
 
 33 
 
 9 
 
 7 
 
 „ 10 
 
 7 
 
 3, 11 
 
 
 33 12 
 
 8 
 
 5) 
 
 
 
 8 
 
 39 
 
 10 
 
 8 
 
 33 11 
 
 8 
 
 ,3 12 
 
 0. 
 
 3, 13 
 
 9 
 
 M 
 
 10 
 
 9 
 
 39 
 
 11 
 
 9 
 
 ,5 12 
 
 9 
 
 3, 13 
 
 9 
 
 33 14 
 
 10 
 
 >5 
 
 11 
 
 10 
 
 39 
 
 12 
 
 10 
 
 „ 13 
 
 10 
 
 33 14 
 
 10 
 
 9, 15 
 
 
 
 an 
 
 1 
 
 7 
 
 and 
 
 8 
 
 and 
 
 9 
 
 and 
 
 10 and 
 
 1 
 
 arc 
 
 7 
 
 1 
 
 arc 
 
 8 
 
 1 
 
 are 9 
 
 1 
 
 are 10 
 
 1 
 
 are 11 
 
 9 
 
 95 
 
 S 
 
 
 
 33 
 
 9 
 
 2 
 
 55 10 
 
 2 
 
 33 11 
 
 2 
 
 ,9 12 
 
 3 
 
 3) 
 
 
 
 3 
 
 33 
 
 10 
 
 3 
 
 3, 11 
 
 3 
 
 3, 12 
 
 3 
 
 3, 13 
 
 4 
 
 J> 
 
 10 
 
 4 
 
 33 
 
 11 
 
 4 
 
 ,5 13 
 
 4 
 
 35 13 
 
 4 
 
 ,3 14 
 
 5 
 
 M 
 
 11 
 
 5 
 
 33 
 
 12 
 
 5 
 
 33 13 
 
 5 
 
 „ 14 
 
 5 
 
 „ 15 
 
 G 
 
 5J 
 
 12 
 
 G 
 
 93 
 
 13 
 
 G 
 
 ,3 14 
 
 G 
 
 33 15 
 
 G 
 
 33 16 
 
 
 J> 
 
 13 
 
 7 
 
 33 
 
 14 
 
 7 
 
 ,3 15 
 
 7 
 
 ,3 16 
 
 7 
 
 „ 17 
 
 8 
 
 3J 
 
 14 
 
 8 
 
 53 
 
 15 
 
 8 
 
 „ 10 
 
 8 
 
 33 17 
 
 8 
 
 33 18 
 
 1) 
 
 3) 
 
 15 
 
 9 
 
 59 
 
 IG 
 
 9 
 
 33 17 
 
 9 
 
 „ 18 
 
 9 
 
 „ 10 
 
 10 
 
 >' 
 
 IG 
 
 10 
 
 99 
 
 17 
 
 10 
 
 „ 18 
 
 10 
 
 9, 10 
 
 10 
 
 ,3 20 
 
 39. The process of Addition is based upon the following 
 
 PRINCIPLES. 
 
 1. Like mimhers, and units of the same order, alone, can be 
 added. Tims, 
 
 Dollars and dollars can be added, but not dollars and days ; also, 
 units and units, tens and tens ; but not units and tens. 
 
 Hepeat the column 1 aud 1. 2 and 1. 3 and 1. 4 and 1, &c. What is 
 the first Principle ? 
 
ADDITION. 
 
 10 
 
 ml 
 
 
 e 
 
 
 
 > 
 
 7 
 
 y 
 
 8 
 
 ) 
 
 9 
 
 J 
 
 10 
 
 > 
 
 11 
 
 » 
 
 12 
 
 J> 
 
 13 
 
 V 
 
 U 
 
 ?» 
 
 15 
 
 and 
 
 Lie 
 
 11 
 
 55 
 
 12 
 
 55 
 
 13 
 
 5> 
 
 14 
 
 >» 
 
 15 
 
 )> 
 
 16 
 
 5) 
 
 17 
 
 5> 
 
 18 
 
 55 
 
 19 
 
 55 
 
 20 
 
 Img 
 
 can be 
 
 h ; also, 
 
 What is 
 
 2. The sum of iico or mnrc numhtrs is the same in vludcvcr 
 order they are whkd. Thus, 
 
 TIk' sum (>r 2, .*>, and 3 is 10, and the sum of 5, 3, and 2, or of 3, 2, 
 
 and r>, is 10. 
 
 3. Tlie sunt and the nunihers added iimst he like 7U(rnhers, 
 Thus, 
 
 The sum of 4 dollars and G dollars is 10 dollars, not 10 pounds. 
 
 40. To add numbers. 
 
 1. Let it be required to add 230, 541, and 102. 
 
 For convenit ce we write the given nnmT)ers so 
 that all the figures of the same order stand in the 
 same column, and begin with units to add. 
 
 2, 1, and 6 units are 9 units, Avhich Ave Avrite. 
 0, 4, and 3 tens are 7 tens, which we write. 1, 5, 
 and 2 hundreds are 8 hundreds, which we write. 
 
 Therefore, the sum is 8 hundreds, 7 tens, and 
 9 units, or 879. 
 
 OPERATIOX. 
 
 230 
 541 
 102 
 
 Sum, 879 
 
 2. Let it be required to find the sum of 505, 301, and 723. 
 
 3G1 
 723 
 
 Sum, 107 9 
 
 OPERATION. For convenience we write, as before, the figures 
 595 of the same order in the same column, and begin 
 
 with units to add. 
 
 3, 1, and 5 mats are 9 writs, which we write. 
 2, 6, and 9 tens are 17 tens, or 1 liundred and 7 
 tens ; we write the 7 tens and add the 1 hundred in 
 with hundreds. 
 
 1, 7, 3, and 5 hundreds are 16 hundreds, or 1 thousand and 6 hun- 
 dreds, which we write. 
 
 Therefore the sum is 1 thousand, 6 luuidreds, 7 tens, and 9 units, 
 or 1679. 
 
 In practice it is sufficient to name only results. Thus, in the 
 operation, we may say : three, four, nine, — write 9 ; two, eight, 
 seventeen, — write 7 and add 1 with next column ; eight, elewn, 
 sixteen, — writs 16 ; answer, 1679. 
 
 What is the second Principle ? The third ? 
 
20 
 
 nUCTICAL AmTTTMF.Tir. 
 
 KULE. JFriJe the uninl/n:^ to he (uhkd so that Jt'jurcs of the 
 aaiite order shull stand in the same coliunn. 
 
 Begin at the right, add the iiumbers expressed hij the Jignres of 
 each column seiwratelji, and icrite the sum underneath, if less than 
 ten of the order added. 
 
 If, however, the sinn is ten or more, write the right-hand Jigure 
 miderneath, and add the nnmher expressed hij the filJu r Jigwre or 
 figures ivith the nvinhers of the next eolumn. 
 
 JFrite the ichole sum of the I'M column. 
 
 PllooF. A(M tlie numbers <a second time, in tlio same man- 
 ner as at the first, except in an opposite <lirection, and, if tlic 
 result agrees witli that iirst obtained, the work is supposed to 
 be correct. Or, 
 
 Separate the given numbers into parts; a(hl (>acli of tlie 
 parts, and tlien add tlieir sums; and, if the result agrees with 
 that first obtained, the Avork is supposed to be correct. 
 
 The test, by either proof, consists in doing the work twice, in a 
 different manner. 
 
 i 
 
 (3.) 
 
 IC 
 
 812 
 
 407 
 
 Sum, 1235 
 
 Examples. 
 
 (4.) (5.) 
 191 
 
 98 4161 
 803 
 
 1092 
 
 Proof, 1235 
 
 (G.) (7.) 
 
 100 1071 
 
 G73 341 
 
 207 5001 
 
 1092 
 
 JO^ 
 
 ^— "- 
 
 .,l/ 
 
 Proof. 
 
 
 3140 
 
 7301 
 
 
 5908 
 
 
 
 5020 
 
 10928 
 
 Sum, 
 
 18229 
 
 18229 
 
 (8.) 
 
 (9.) 
 
 (10.) 
 
 233 
 
 814 
 
 4444 
 
 511 
 
 816 
 
 1234 
 
 179 
 
 31 
 
 5561 
 
 Repeat the Rule. What is the first Proof ? The second Proof ? 
 
ADDITluN. 
 
 L'l 
 
 s of the 
 
 (jnrcs of 
 CSS than 
 
 idjhjnre 
 fujure oy 
 
 me 111 fin - 
 
 1, if till' 
 posed to 
 
 1 of the 
 CCS with 
 
 I, 
 
 wicf, in a 
 
 oof. 
 
 301 
 
 928 
 
 229 
 
 ).) 
 44 
 
 ^ 
 
 (11.) 
 
 Dollar.s, 
 
 1234 
 olO 
 
 89 
 
 (12.) 
 
 (13.) 
 
 J^/:/^ 
 
 J 
 
 i'0Utl(i.-i, 
 
 Var.lrt. 
 
 (Ijii 
 
 1 780 
 
 31,") 
 
 314 
 
 708 
 
 510 
 
 190 
 
 1710 
 
 J(^9 
 
 fJu 
 
 (11.) 
 
 Mow. 
 
 lln 
 517 
 819 
 110 
 
 (15.) 
 
 1191 
 
 818 
 110 
 
 (US 
 
 /6;(. VlS^ 
 
 in. What is the sum of 42.3, 507, and 385 } An^^. 1375. 
 
 17. What is the .sum of 98, 483, 950, and 85? Ans. 1022. 
 
 18. What is the sum of 15, 003, 1145, and 0342?r. < jC^ 
 
 19. What is the .sum of 370 + 493 + 102 + 315U'^/£^J-^ 
 
 20. What is the sum of 4753 + 0378 + 9257 + 2890? r•i:J^;.,^^ 
 
 21. 974 + 05 + 370 + 487 + 598 + 88 --^ what?--^^;3^ 
 
 22. 1000 + 100 4- 10000 + lODOOO + 11-^ what? ^- /// / / / 
 
 23. 7805 + 3580 + 4321 + 8570 - how many ? 
 
 An^. 24318. 
 
 24. 8 + 105 + 1000 + 810501 + 111 + 3 + 4400 - how 
 many? An>^. 810134. 
 
 25. Find tlie sum of 4080, 715, 1303 1, 29, and 5. Ans. 18403. 
 20. Find the sum of 0037, 2480, 2051, and 333. Am. 11501. 
 
 27. Find the sum of 17890 + 570937 + 781947 + 9078F/,^.fV^' 
 
 28. Find the sum of 10304 + 119 + 18913 + 300204;^.^'^/^/^^ 
 
 29. Ilowmany are 127 + 040 + 300 + 29? Ans, 1102. 
 
 30. Howmany are 014034 + 783420 + 10310 ?^>i^^/ V^ 
 
 31. How many are 3010 + 250 + 40400 + 18 + 3101 + 9?=^^^'^ 
 
 32. How many are 850 + 9193 + 8705 + 4287 + 0090 + 
 9185 + 979 \ Ans. 39901. 
 
 33. 1031 + 125 + 9 + 041 + 10 + 449 + 0O72 = what? 
 
 34. Add five hundred sixtj'-seven thousand three hundred 
 and seven, eighty-eight tliousand and eight, nine hundred and 
 sixty-two, and nineteen. Ans. G50290. 
 
 t 
 
 1 1 
 
 Review Question?!. What is .a Unit?(l) A Quantity? (2) A Nuiu- 
 bei- ? (3) Tiie Unit of a Number ? (4) Like Numbers ? (5) 
 
.10 
 
 PIl ACTI C A L A K IT FI M I:T IC. 
 
 APPLICATIONS. 
 
 1. A f;irmor sold a horse for LMO dollars, a yoke of oxon for 
 175 dollars, sorn(> cows for .'{24 dollars, and some hay for GO 
 dollars; how much did he g«'t for the whole ? ^-Ins. i^O'J dollars. 
 
 Solution. 1/ he sold a horso for 250 doUurs, a yoke of oxen for 175 
 dollars, some coirs for 324 dollarti, and some haij for Go doUarii^lu. 
 received for the whole the sum of 2r)0 + 17r> + '.Vl\ + OO dollars, or 
 SOJ) dollars, Tlitrtfore he receioedfor the whole 80!) dollars. 
 
 2. A merchant bought molasses for 1430 dollars, and sold 
 it at a gain of Ol'O dollars ; for what sum did he sell it 1 
 
 Alls. 2050 dollars. 
 
 3. A man commenced trade with three thousand five hun- 
 dred and twenty-five dollars ; after trading for some time, he 
 had gained three hundred and nineteen dollars ; how much 
 had ho then? Ans. 3844 dollars. 
 
 4. Going out to collect money, I had in my pocket 1500 
 dollars, and received from one per.son 17 dollars, from another 
 132 dollars, from another 527 dollars, and from a fourth as 
 much as I started out with; how many dollars had I then l-=^t^^ 
 
 5. Bought a carriage for 1G3 dollars, a pair of horses for 
 2G0 dollars, and a harness for 84 dollars ; how much was paid 
 for the whole 1 Ans. 507 dollars. 
 
 G. Figures were used by the Arabs in the year 890, and 
 decimal fractions were invented 574 years later ; in what year 
 were they invented ] Ans. 14G4. 
 
 7. In one book are 513 pages, in another 144, and in 
 another as many as in the other two; how many pages in the 
 three books? Ans. 1314 pages. 
 
 8. A merchant bought beef for G44 dollars, pork for 450 
 dollars, and fish for 226 dollars ; for how much must the whole 
 be sold that the gain may be 240 dollars? ^-//^j-^^ 
 
 Keview Questions. "What is an Operation ? (8) An Answer ? (9) A 
 Solution ? (10) A Rule ? (11) An Example ? (12) An Exercise ? (18) 
 
ADDITION. 
 
 '2) 
 
 oxon for 
 
 y for GO I 
 
 I dollars. 
 
 n far 175 
 h)lf(irs, he 
 'oUiD's, or 
 
 iiul sold 
 
 ) dollars. 
 
 five liuii- 
 tiine, he 
 iW much 
 [ dollars. 
 
 iet 1500 
 [ another 
 oiirth as 
 
 )rses for 
 as paid 
 lollars. 
 
 90, and 
 lat year 
 s. UG4. 
 
 and in 
 
 ;s in the 
 4 pages. 
 
 for 450 
 e whole 
 
 ■^ 
 
 0. A hiitcluT has live fat oxen; the first wei;4h.s ll'Jl 
 jioiinds, the second 1l'.*)5 pounds, tlie third I3l)0 pounds, the 
 fourth ItL'O pounds, and the litlh l.")i'5 pounds; what is the 
 weiuht of the whole nuiuber'? .-l /ts. (IG04 i)ou!n!ls. 
 
 10. Purchased a farm for I'JOiJO dollars, paid for repair of 
 fences 820 dollars, for having a ])arn l)uilt 51G dollars, and sold 
 
 it so as to gain 032 dollars; how much was obtained for itJ<^'j^//-jj 
 
 11. A farmer has live stock as follows: on one farm 5 
 liorses, 20 cows, aiul 110 sheep ; on a second farm IG oxen and 
 204 sheep ; and on a third farm 2 horses, 12 cows, and 8 calves. 
 How many head has he in all / Ans. 41. '3 head. 
 
 12. An army was furnisheil at one time; with 1G500 rations, 
 at another time with G30G0 rations, and at a third time with 
 72545 rations; how many rations were furnished altogether >;^> 4';? >i>: 
 
 13. If the Atlantic slo])e contains 0G757G S(pKire miles, and 
 the Mississippi Valley exceeds the Atlantic slo})(; by 2G0535 
 scpuire miles, what is the area of the ]\Iississii)[)i Valley? 
 
 Ah.i. 1237111 sipiare miles. 
 
 14. If the area of Maine is 30000 square miles, of New 
 Hampshire 9280, of Vermont 905G, (jf ^Massachusetts 7800, of 
 lihode Island 130G, of Connecticut 4G74, and the area of Mis- 
 souri 5204 square miles greater than that of the six States 
 named, what is the area of Missouri] ylns. 07380 square miles. 
 
 15. Boui^dit a lot of irround for 075 dollars ; erected a house 
 upon the same, at the cost for carpenters' work 2540 <lollars, 
 masons' work G37 dollars^ painters' work 242 dollars, and for 
 grading the lot 293 dollars; what was the cost of the whole 3^/^j^r' ^ 
 
 10. If the area of Illinois is 55405 sqiuire miles, of Wiscon- 
 sin 53924, of Minnesota 83000, and of Iowa 50914, what is the 
 area of all these States? Ajis. 243243 square miles. 
 
 17. Bought wheat for 0500 dollars^ corn for 10500 dollars, 
 and oats for 1002 dollars, and sold the wheat at a profit of 050 
 
 I 
 
 M 
 
 ■?(9) A 
 3) 
 
 Review Questions. 
 tic? (15) 
 
 What is Aritlimetic ? (14) Practical Arithine- 
 
rf" »■ 
 
 24 
 
 PRACTICAL ARITHMETIC. 
 
 dollars, tlio corn at a profit of 1050 dollars, and tho oats at cost ; 
 what sum was received for the whole 1 Am. 197G2 dollars. 
 
 18. If all your d(.'ljts to different persons are as follows : 
 25,50 dollars, 1200 dollars, 5 dollars, 537 dollars, 495 dollars, 
 and 5730 dollars, how much do you owe? Ans. 10523 dollars. 
 
 19. A certain year Ohio produced 59078G95 bushels of corn, 
 Indiana 52964303 bushels, Kentucky 58072591 bushels, and 
 Tennessee 52270223 bushels; what was the amount produced 
 by all these States ? Ans. 222991872 bushels. 
 
 20. According to Weimer, Europe contains 3807195 square 
 miles, Asia 17805140, Africa 11047428, America 13542400, 
 and Oceixnia 3347840 ; what does tliis make the extent of land 
 on the surface of the globe? Ans. 50150009 square miles. 
 
 21. Bought five loads of hay, which weighed as follows: 
 2500 pounds, 2304 pounds, 3150 pounds, 1905 pounds, and 
 1831 pounds; what was the weight of the whole?// ;-"/'/ /'^ 
 
 22. The skull has 8 bones, the face 14, the ear 4, the tongue 
 1, the teeth 32, the trunk 53, the shoulders 4, the arms 0, the 
 wrists 10, the hands 38, legs 8, ankles 14, and feet 38; re- 
 quired the number in the whole body? ^' <','' '. » ,^, 
 
 23. The number of regular soldiers furnished by each of 
 the States in the Kevolution was as follows : 
 
 1 
 
 New Hampshire, 
 
 12497 
 
 Delaware, 
 
 2386 
 
 ]\Iassachu.'^etts, 
 
 67907 
 
 JVIaryland, 
 
 13912 
 
 Rhode Island, 
 
 5908 
 
 Virginia, 
 
 20078 
 
 Connecticut, 
 
 31939 
 
 North Carolina, 
 
 7203 
 
 New York, 
 
 17781 
 
 South Carolina, 
 
 0417 
 
 New Jersey, 
 
 10720 
 
 Georgia, 
 
 2079 
 
 Pennsylvania, 
 
 25078 
 
 
 
 what was the whole 
 
 number fur 
 
 nished l)v them? Ans 
 
 .231771. 
 
 Review Questions. "What is Not;ttioii? (IG) Numeration? (17) "NVliat 
 are figures? (18) "What is the method of expressing numbers by figures 
 ealletl? (18) What is meant by orders of figures? (28) To what do thi'y 
 correspond? (28) How many orders are taken fur a period? (2S) 
 
SUBTRACTION. 
 
 25 
 
 at cost ; 
 dollars. 
 
 follows : 
 dollars, 
 dollars, 
 
 of corn, 
 ^els, and 
 roduced 
 bushels. 
 
 ) square 
 542400, 
 i of land 
 :e miles. 
 
 follows ; 
 ids, and 
 
 3 tonirue 
 s G, the 
 38; re- 
 each of 
 
 238G 
 3912 
 GG78 
 72G3 
 
 417 
 
 G79 
 
 131771. 
 
 What 
 
 figures 
 
 llo thry 
 
 i 
 
 SUBTRACTION. 
 
 41. 1. j\Iary had G cents and gave away 2 cents; how 
 many had she left \ 
 
 Solution. >S7te hcul h:ft as many us are cqval to (5 ciuf.i less -2 
 cents, and 2 cents from G coits leave 4 cents. Tlierefore she liad 4 
 coits left. 
 
 2. James had 5 apples and gave his brother 3 of them ; how 
 many had he left ? 
 
 3. A hawk having taken 4 c' ickens from a brood of i), iiow 
 many remain 1 
 
 4. How many does 4 from 8 leave 1 7 from 1 ? 
 
 5. How many does G from 1) leave ? 4 from 11 ? 2 from 7 ? 
 5 from 12? 
 
 G. Henry had 10 cents and spent 5 cents ; how much had 
 he left? 
 
 7. John had 13 marbles and lost 8; liow many had he left ? 
 
 8. How many must be added to 8 to make 1 3 .' 
 
 The preceding operations are called SuBTilACTiox. Hence, 
 
 42. Subtraction is the process of finding the difference 
 between two given numbers of the same kind. 
 
 The Subtrahend is the number subtracted. 
 
 The I\IlNUEND is the number subtracted from. 
 
 The DlEFEUENX'E, or Kemaindeu, is the result of the sub- 
 traction. 
 
 When the two given numl)ers are equal, either may be taken 
 as the minuend, and the difference is 0. 
 
 43. The Sign of Subtraction is a short horizontal line, 
 -, called minus. Thus, G — 4, read six minus four, denotes 
 that 4 is to be subtracted from G. 
 
 What is Subtraction? Tiie Suhtrahend? The Minuend? The DiiTer- 
 ence, or Remainder? What is the ditrerenoe wheu the uunueud and sublni- 
 heud are equal? The Si^-u of SJuljtractiiai? 
 
 ii 
 
2G 
 
 PRACTICAL AEITilMETIC. 
 
 Subtraction Table. 
 
 1 from 
 
 2 from 
 
 3 from 
 
 4 from 
 
 fr(jm 
 
 ] 
 
 1 leaves 
 
 2 leaves 
 
 3 leaves 
 
 4 leaves 
 
 5 leaves 
 
 
 2 „ 1 
 
 3 „ 1 
 
 4 „ 1 
 
 5 „ 1 
 
 G „ 1 
 
 ^ 
 
 3 „ 2 
 
 ! 4 „ 2 
 
 5 „ 2 
 
 G „ 2 
 
 7 . 2 
 
 
 4 „ 3 
 
 „ 3 
 
 G „ 3 
 
 7 „ 3 
 
 8 „ 3 
 
 
 5 „ i 
 
 G . 4 
 
 7 „ 4 
 
 8 „ 4 
 
 9 „ 4 
 
 
 G „ 5 
 
 7 „ 5 
 
 8 „ 5 
 
 9 „ 5 
 
 10 „ ri 
 
 
 7 „ G 
 
 8 „ G 
 
 9 „ G 
 
 10 „ G 
 
 11 „ G 
 
 
 8 . 7 
 
 D „ 7 
 
 10 „ 7 
 
 11 V 7 
 
 12 „ 7 
 
 
 9 „ 8 
 
 10 „ 8 
 
 11 „ 8 
 
 12 „ 8 
 
 13 „ 8 
 
 
 10 „ 9 
 
 11 „ 9 
 
 12 „ 9 
 
 13 „ 9 
 
 14 „ 9 
 
 
 11 „ 10 
 
 12 „ 10 
 
 13 „ 10 1 U „ 10 
 
 15 ., 10 
 
 
 6 from 
 
 7 from 
 
 8 from 
 
 9 from 
 
 10 from 
 
 
 6 leaves 
 
 7 leaves 
 
 8 leaves 
 
 9 leaves 
 
 10 leaves 
 
 
 7 „ 1 
 
 8 „ 1 
 
 9 „ 1 
 
 10 „ 1 
 
 11 „ 1 
 
 
 8 „ 2 
 
 9 „ 2 
 
 10 „ 2 
 
 11 „ 2 
 
 12 „ 2 
 
 
 9 „ 3 
 
 10 „ 3 
 
 11 „ 3 
 
 12 „ 3 
 
 13 „ 3 
 
 
 10 „ 4 
 
 11 „ 4 
 
 12 „ 4 
 
 13 „ 4 
 
 14 „ 4 
 
 ,1 
 
 11 „ 5 
 
 12 „ 5 
 
 13 „ 5 
 
 14 „ 5 
 
 15 „ 5 
 
 : 
 
 12 „ 6 
 
 13 „ 6 
 
 14 „ G 
 
 15 „ G 
 
 IG „ G 
 
 
 13 „ 7 
 
 14 „ 7 
 
 15 „ 7 
 
 IG „ 7 
 
 17 „ 7 
 
 % 
 
 14 „ 8 
 
 15 „ 8 
 
 IG „ 8 
 
 17 „ 8 
 
 18 „ 8 
 
 , 
 
 1^^ . 9 
 
 IG „ 9 
 
 17 „ 9 
 
 18 „ 9 
 
 19 „ 9 
 
 1 
 
 IG „ 10 ! 
 
 17 „ 10 
 
 18 „ 10 i 19 „ 10 j 
 
 20 „ 10 
 
 1 
 
 44. The p 
 
 recess of Siil 
 
 )traction is h 
 
 ased upon tl 
 
 le following 
 
 1 
 
 PEINCIPLES. 
 
 1. Like numbers, and units of the same order, alone, can be 
 subtracted one from the other. Thus, 
 
 Dollars can be suhtracted from dollars, but not dollars from days ; 
 also, units from units ; but not units from tens, or tens from units. 
 
 
 Repeat the column 1 from 1. 2 from 2. 3 from 3. i from 4, &c. What 
 is the first Priucii)le ? 
 
SUBTRACTION. 
 
 27 
 
 from 
 
 
 eaves 
 
 
 
 »> 
 
 1 
 
 ;» 
 
 .1 
 
 J? 
 
 3 
 
 5> 
 
 4 
 
 ;j 
 
 
 
 r) 
 
 f) 
 
 >» 
 
 7 
 
 j> 
 
 8 
 
 5' 
 
 9 
 
 •» 
 
 10 
 
 from 
 
 eaves 
 
 11 
 
 1 
 
 5 J 
 
 2 
 
 >? 
 
 3 
 
 ?> 
 
 4 
 
 J> 
 
 5 
 
 J5 
 
 G 
 
 
 7 
 
 5) 
 
 ?1 
 
 8 
 
 5J 
 
 9 
 
 !» 
 
 10 
 
 ■ . ,. J 
 
 lowing 
 
 cYm be 
 
 nil (liiys ; 
 units. 
 
 OPERATION. 
 
 Minuend, 958 
 Subtrahend, 325 
 
 Difference, G33 
 
 2. The difference, minuend and subtrahend, must be lih numlcr^. 
 Thus, 
 
 The difference between 6 dollars and 4 dollars is 2 dollars, and not 
 2 yards. 
 
 3. The difference and subtrahend, taken together, must equal (he 
 minuend. Thus, 
 
 The difference, 5, between 13 and 8, added to 8, equals 13. 
 
 45. To subtract one number from another. 
 
 1. Let it be required to subtract 325 from 958. 
 
 For convenience we wTite the subtrahend 
 under the minuend, so that figures of the same 
 order stand in the same column, and begin at 
 the right to subtract. 
 
 5 units from 8 miits leave 3 units, which we 
 write. 
 
 2 tens from 5 tens leave 3 tens, which we write. 
 
 3 hundreds from 9 hundreds leave G hundreds, which we write. 
 Therefore the difference is 6 hundreds, 3 tens, and 3 units, or 033. 
 
 2. Let it be required to find the difference betvv^een 052 and 
 
 423. 
 
 For convenience we write, as before, figures 
 of the same order in the same column, and begin 
 with units to subtract. 
 
 We cannot take 3 units from 2 units ; but we 
 
 Giui take 1 ten from the 5 tens, leaving 4 tens ; 
 
 and the 1 ten taken is 10 units, which added 
 
 to the 2 units make 12 units ; 3 units from 12 units leave 9 units, 
 
 which we write. 
 
 2 tens from 4 tens leave 2 tens, which we write. 
 
 4 hundreds from 6 hundreds leave 2 hundreds, which we write. 
 Therefore the difference is 2 hundreds, 2 tens, and 9 units, or 229. 
 
 The operation may be performed another way : 
 As the 3 units cannot be taken from 2 units, A\e add 10 units to tlu' 
 2 units ; and 3 units from 12 units leave 9 units, which we write. 
 
 I 
 
 OPERATION. 
 
 ^linuend, G52 
 Subtrahend, 423 
 
 Difference, 229 
 
 r ' 
 
 c. What 
 
 What is the second Principle ? The third ? 
 
l> 
 
 i; 
 
 2S 
 
 PRACTICAL ARITHMETIC. 
 
 To Lalance the 10 units fiddoil to the 2 units, we add 1 ten to the 
 2 ttjiis, in acc()r(huice with the principli* that if any two numhcrs are 
 I'luallij increased their difference remains the same; and say, 3 tens 
 from 5 tens have 2 tens, which we writi'. 
 
 4 hundreds IVoni G liundreds L-ave 2 hundrt'd.s, which we write. 
 
 Therel'ure the dillcrence is 220. 
 
 In practice, a hrief exphination is sufficient. Thus, we may say : 
 .'} t'roia 2, inipossiVile ; hut 3 from 12 h'aves 9, M'liich we write ; 2 
 from 4 h'aves 2, which we write ; 4 from leaves 2, which we write. 
 Answer, 229. 
 
 IIULE. IFriie the less numhcr wider the greater, so that figures 
 of the same order shall stand in the same column. 
 
 Begin icith units, suUraet the numher expressed hit each figure 
 of the suhtrahend from the numher expressed hy the figure ahove it, 
 arid urite the difference underneath. 
 
 If the upper figure expresses a less numher than the lower, con- 
 ceive that numher increased hy ten, suhtract, and write the differ- 
 ence, and considering either the numher e?:iiressed hj the next upper 
 jigurc one less or that expressed hy the ne?:t lower figure one 
 GREATER, proceed as heforc. 
 
 Proof. Add the difference to the subtrahend, and, if the 
 work IS correct, the sum will equal the minuend. Or, 
 
 Subtract the difference from the minuend, and, if the work 
 is correct, what is left Avill equal the subtrahend. 
 
 Minuend, 
 Subtrahend, 
 
 Examples. 
 
 (3.) (4.) 
 
 40G 395 
 
 154 288 
 
 Difference, 252 
 
 lu; 
 
 GOOO 
 1234 
 
 47GG 
 
 Proof, 40G 288 6000 
 
 In example 5, m'c cannot subtract 4 iniits from no units, and there 
 
 Repeat the Rule. "What is the Proof ? 
 
SUBTHACTION. 
 
 29 
 
 I 
 
 arc no tens and no hundreds ; so that we cannot take one of the tens 
 and call it 10 nnits, or one of the hundreds and call it 9 tens and 10 
 units ; hut ■\ve can take one of the G thousands, leavini,' 5 thousands, 
 and call it 9 hundreds, 9 tens, and 10 unit.^. Than we can take 4 
 units from the 10 units, 3 tens from the 9 tens, &c. 
 
 (0.) 
 From 023 
 Subtract 515 
 
 (7.) 
 700 
 4G5 
 
 (8.) 
 1120 
 744 
 
 (0.) 
 4789 
 1987 
 
 I 
 
 (10.) (11.) 
 
 Tons. Sheep. 
 
 From 912 lOGO 
 
 Subtract 453 343 
 
 (12.) 
 
 Yards. 
 
 23G0 
 1272 
 
 (13.) 
 
 809 
 
 87 
 
 there 
 
 14. 
 15. 
 
 U'). 
 17. 
 18. 
 
 10. 
 20. 
 21. 
 
 9-> 
 
 23. 
 
 24. 
 25. 
 2G. 
 27. 
 28. 
 29. 
 30. 
 what 
 
 From 854 take 578. ^-his. 276. 
 
 From 1799 take 1732. Ans.^ 07. 
 
 Find the ditference between 8G41 and 1904. -^ .:y,.Ji/J '/ 
 From 5490 subtract 1492. Jiis. 4004. 
 
 From 1584 subtract 920. Ans. GG4. 
 
 From 5672 subtract 2356. Ans. 3310. 
 
 From 74700 subtract 39817. Ans. 34943. 
 
 52305 - 15423 - how many? Ans. 3G942. 
 
 78507 - 32782 = how many? 
 9730214 - 8878940 = how many? At 
 
 ^Yhat is the difference between 900000 and 123454 ?rj ^j/^t 
 How much larger is 38007 than 3807 1 Ans. 34740. 
 How much smaller is 34730 than 38007?^.;;'/,; 
 
 u^i no. u\jo-x^. 
 
 Ans: 85^268. 
 
 y-. 
 
 •^iV 
 
 ^. 4. 
 /I 
 
 How much must be taken from 2483 to leave 391 ?- 
 How much must be added to 2082 to make 2483 1 . 
 From 7630005 take 3270006. Ans. 4359999^ 
 
 The larger of two numbers is 10040 and the less 9535; 
 is their difference? Ans. 1105. 
 
 .^•^ 
 
 ^ 
 
 Rf,view Questions. Upon what Ge ^ral Principles is the Arabic Nota- 
 tion l)asca? (30) What is a Scale of T: umbers? (SI) What is the Scale 
 in numbers expressed according to the Arabic Notation? 
 
t 
 
 30 
 
 PRACTICAL ARIinixETIC. 
 
 31. Subtract two tliousHud one liundred and nineteen from 
 five thousand two hundred and twelve. ylns. 3093. 
 
 32. If one be taken from one hundred thousand, what will 
 remain? Ans. 99909. 
 
 33. AVhat is tlie difference between nine units and ninety- 
 nine miUions? - 9 '^'/;f":/V7i 
 
 34. From four hundred fifty thousand and ninety-four take 
 ninety-nine thousand nine liundred and nine. 
 
 35. From 35 billions G3 millions and 9 thousand take 7 
 billions 103 millions and 9. Ans. 279G0008991. 
 
 1^ 
 
 I 
 
 APPLICATIONS. 
 
 1. A farmer raised 420 bushels of wheat, and sold 198 
 
 bushels; how many bushels had he ^^^'^'^--'^J^i . ^{h' J ,:,:>' ^ . 
 
 Solution. If Jic raised 420 bushels and sold 198 bushels, he must 
 have left as many bushels as the difference betiocen 420 and 198 bushels, 
 or 222 bushels. Therefore he had 222 bushels left. 
 
 2. A boy had 520 apples, and gave away 411 of them; how 
 many had he left? Ans. 115 apples. 
 
 3. Albert Smith borrowed 1090 dollars, and soon after paid 
 909 dollars ; how much of the sum borrowed did he then owe? 
 
 Ans. 181 dollars. 
 
 4. How many years have elapsed since the first planting of 
 cotton in this country in 1709? 
 
 How many years from the beginning of the revolutionary 
 
 5. 
 
 4f 
 
 war in 1775 to the beginning of the late Avar in IPGl ? 
 
 G. A merchant sold for 3097 dollars goods which cost him 
 2807 dollars; how much did he gain?;-:, a" j 
 
 7. A man owns property to the amount of five thousand 
 eight hundred and twenty-five dollars, and owes one thousand 
 three hundred and forty dollars ; how much will he be worth 
 when his debts are paid ? Ans. 4485 dollars. 
 
 Review Questions. Give the names of the orders in the Numeratiou 
 Table. (33) What is the Rule for reading numbers ? (34) 
 
 l4< 
 
SUBTRACTION. 
 
 31 
 
 IS 
 
 
 liars. 
 
 :atiou 
 
 J!i. Donglit a ship for 42G.30 dollars aiid sold it for 49000 
 dollars; what did I gain] Am. 03.30 dollars. 
 
 0. A ircntlemaii urave 124G2 dollars for a house and sonic 
 land; tlie house alone was worth O'h.") dollars; what was the 
 value of the land? Anx. 3087 dollars. 
 
 10. x\ lumberman, having G50000 feet of boards, sold 1G2372 
 feet of them ; how many feet then remained ? Am. 487 G28 feet. 
 
 11. Tli(> battle of Gettysburg, in 18G3, was 48 years after the 
 battle of New Orleans ; in what year was the latter? J i'//^ 
 
 12. A man having 100000 dollars, gave away 3G5 dollars; 
 how much had he left ? Am. 99G3.J dollars. 
 
 13. A merchant owns property to the amount of 455G3 
 dollars, and owes 2120'J dollars; how much is he worth more 
 than he owes ? ---0 ^ ^S 6'^^ 
 
 14. If two candidates for office received in tiie aggrcgat(? 
 73402 votes, and the successful one had 45309 votes, how 
 many did the other have ? A]is. 28153 votes. 
 
 15. Illinois contains 55405 square miles and Iowa 50914 
 square miles ; how many more square miles does the one con- 
 tain than the other ? rrt r~^^ J /a.^ , 
 
 10. Mount Sorata, in Soutn America, is 25380 feet high, 
 and 1914G feet higher than Mount Washington in New Hamp- 
 shire ; how high is Mount Washington ] Ans. 0234 feet. 
 
 17. Oirard College, in Philadelphia, is said to have cost 
 1422800 dollars, and Trinity Church, in New York City, 338000 
 dollars ; how much more did the one cost than the other^j^/Jjf ^ 
 
 18. If the value of the annual products of the industry of 
 Massachusetts is 200000000 dollars, and that of Pennsylvania 
 is 285500000 dollars, how much do the products of the one 
 State exceed those of the other? Am. 19500000 dollars. 
 
 19. If the population of Ohio was 45305 in 1800, and 
 2339502 in 1800 ; how much was the increase ? Am. 2294137. 
 
 Review Questions. What is Addition? (37) What principles are 
 to be observed in Addition ? (39) 
 
 I ! 
 
32 
 
 rr.ACTICAL ARITUMl'TIC. 
 
 !! ! 
 
 REVIEW EXERCISES. 
 
 1. 573 + G tliousaiid + G million + 5079G;1 + 1215 r. how 
 many? ylm^. G5157iSl. 
 
 2. If the minuend is eight millions six Inmdred seventy-three 
 thousand four Inmdred and one, and the subtrahend six million 
 seven hundred twenty thousand seven hundred and thirty, 
 what is tl'e difference? yitts, 1952G71. 
 
 3. If the larger of two numhers is 100101 and their dilTer- 
 onco 9902, v/hat is the smaller number? Ans. 90199. 
 
 4. A man owning 4G05 acres of land, gave to one of his sons 
 1 420 acres, a id to another 1280 acres ; how many acres had he 
 remaining ? /y^*^" ♦.^ J t^, , 
 
 Solution. If he [lavc to one son 1420 acres, and to another 1280 
 iicrcfi, he must have (jiven to loth the sum of 1420 acres and 1280 acres, 
 or 2700 acres. 
 
 If he had 4605 acres and gave away 2700 acres, he must have had 
 remaining the difference between 4C05 acres and 2700 acres, or 1905 
 acres. Therefore, he had remaininrj 10O5 acres. 
 
 5. A grain dealer bought GOOO bushels of wheat ; he after- 
 wards soil to one man 1575 bushels, and to another 3GO0 
 bushels ; how many bushels remain unsold? •=: ^^^j4*. 'y^^'/t/ 
 
 G. A man died leaving 24000 dollars, of which he cave his 
 wife 8000 dollars, one daughter 3500 dollars, another 4500 
 dollars, and the residue to his son ; what was the son's portion ?.,\p^; 
 
 7. Mr Jones had in a bank 1G830 dollars, drew out 94G0 
 dollars, and afterwards put in 2000 dollars ; how much had he 
 then in the bank? Ans. 9370 dollars. 
 
 8. A farmer had a horse worth 275 dollars, and exchanged 
 it for a yoke of oxen and two cows ; the oxen he sold for 125 
 dollars, one of the cows for 75 dollars, and the other for 58 
 dollars \ how much did he lose by the trade ? Ans. 17 dollars. 
 
 
 IvEViEW Questions. AVhat is Subtraction? (42) "What principles are 
 to be observed in Subtraction? (44) 
 
MULTIPLICATION. 
 
 33 
 
 = how 
 
 ir)78i. 
 
 '-tliiee 
 iiillioii 
 thirty, 
 32071. 
 
 (lilTer- 
 )019'J. 
 
 s sons 
 lad he 
 
 •• 1280 
 ) acres, 
 
 ce had 
 r 1905 
 
 after- 
 3000 
 
 VQ his 
 4500 
 tioii^,j^^,| 
 
 94g6' 
 
 id he 
 
 lars. 
 
 nged 
 
 12.5 
 r 58 
 
 lars. 
 
 s are 
 
 MULTIPLICATION. 
 
 46. 1. How many dollars will G tons of coal cost, at 7 dol- 
 lars a ton ? 
 
 Solution, k^ince 1 ton of coal costs 7 doUari^, (5 tons ?/??<.sV cnat (1 
 times 7 dollars, which are 42 dollars. Therefore (5 tons of coal, at 7 
 dollars a tan, will cost 42 dollarii. 
 
 2. How many cents will buy 5 pencils, at 8 cents each \ 
 
 3. When berries are 8 cents a quart, how much nuist bo 
 paid for 4 quarts ? 
 
 4. If a boy can walk 3 miles in an hour, how many miles 
 ca i he walk in 5 hours? 
 
 5. A farmer had 10 cows in each of 3 pastures ; how many 
 had he in all of then 1 
 
 6. If 1 horse will eat 4 tons of Iiay in a given time, how 
 many tons will 7 horses eat in the same time ? 
 
 The preceding operations are called Multiplication. 
 Hence, 
 
 47. Multiplication is the process of finding the result of 
 taking one of two given numbers as many times as there are 
 units in the other. 
 
 The Multiplicand is the number to be taken. 
 
 The Multiplier is the number denoting how many times 
 the multiplicand is to be taken. 
 
 The Product is the result of the multiplication. 
 
 The Factors of the Product are the multiplicand and 
 multiplier. 
 
 48. The Sign of Multiplication is an inclined cross, 
 X , read muUij^lied hy. Thus, 5 x 4 is read, 5 multiplied 
 
 by 4. 
 
 What is Multiplication? The Multiplicand? The Multii.lier? The 
 Product ? The Factors of the Product ? The Sign of Multiplication ? 
 
 C 
 
 ii 
 
I 
 
 I: 
 
 fill 
 
 
 I' 
 
 h, 
 
 iii 
 
 34 
 
 PRACTICAL ARITHMETIC. 
 
 Multiplication Table. 
 
 Onco 
 
 2 times 
 
 3 times 
 
 4 times 
 
 5 times 
 
 G times 
 
 1 is 
 
 1 
 
 lare 2 
 
 lare 3 
 
 1 are 4 
 
 1 arc 5 
 
 1 are G 
 
 
 
 2 
 
 2 „ 4 
 
 2 „ G 
 
 2 „ 8 
 
 2 „ 10 
 
 2 „ 12 
 
 
 3 
 
 3 „ G 
 
 3 „ 9 
 
 3 „ 12 
 
 3 „ 15 
 
 3 „ 18 
 
 4 „ 
 
 4 
 
 4 „ 8 
 
 4 „ 12 
 
 4 „ IG 
 
 4 „ 20 
 
 4 „ 24 
 
 5 „ 
 
 5 
 
 5 „ 10 
 
 5 „ 15 
 
 5 „ 20 
 
 5 „ 25 
 
 5 „ 30 
 
 c „ 
 
 G 
 
 G „ 12 
 
 G „ 18 
 
 6 „ 24 
 
 G „ 30 
 
 6 „ 36 
 
 7 „ 
 
 7 
 
 7 „ 14 
 
 7 „ 21 
 
 7 „ 28 
 
 7 „ 35 
 
 7 „ 42 
 
 8 „ 
 
 8 
 
 8 „ IG 
 
 8 „ 24 
 
 8 „ 32 
 
 8 „ 40 
 
 8 „ 48 
 
 „ 
 
 9 
 
 9 „ 18 
 
 9 „ 27 
 
 9 „ 30 
 
 „ 45 
 
 9 „ 54 
 
 10 „ 
 
 10 
 
 10 „ 20 
 
 10 „ 30 
 
 10 „ 40 
 
 10 „ 50 
 
 10 „ GO 
 
 11 „ 
 
 11 
 
 11 *>'' 
 
 11 „ 33 
 
 11 ,, 44 
 
 11 „ 55 
 
 11 „ GO 
 
 12 „ 
 
 7 tin 
 
 12 
 
 1 •J V *^ '^ 
 
 12 „ 36 
 
 12 „ 48 
 
 12 „ GO 
 
 19 TO 
 
 les 
 
 8 times 
 
 9 times 
 
 10 times 
 
 11 times 
 
 12 times 
 
 1 are 
 
 7 
 
 1 are 8 
 
 1 are 9 
 
 1 are 10 
 
 1 are 1 1 
 
 lare 12 
 
 
 U 
 
 2 „ 16 
 
 2 „ 18 
 
 2 „ 20 
 
 2 22 
 
 2 „ 24 
 
 3 „ 
 
 21 
 
 3 „ 24 
 
 3 „ 27 
 
 3 „ 30 
 
 3 „ 33 
 
 3 „ 36 
 
 4 „ 
 
 28 
 
 4 „ 32 
 
 4 „ 30 
 
 4 „ 40 
 
 4 „ 44 
 
 4„ 48 
 
 5 „ 
 
 3o 
 
 5 „ 40 
 
 5„ 45 
 
 5 „ 50 
 
 5 „ 55 
 
 5 „ 60 
 
 6 „ 
 
 42 
 
 G „ 48 
 
 6 „ 54 
 
 6 „ GO 
 
 6 „ 66 
 
 6 „ 72 
 
 
 49 
 
 7 „ 5G 
 
 7 „ 63 
 
 7 „ 70 
 
 7 „ 77 
 
 7 „ 84 
 
 8 „ 
 
 5G 
 
 8 „ G4 
 
 8 „ 72 
 
 8 „ 80 
 
 8 „ 88 
 
 8 „ 96 
 
 „ 
 
 03 
 
 9 „ 72 
 
 9 „ 81 
 
 9 „ 90 
 
 9 „ 99 
 
 9 ,,108 
 
 10 „ 
 
 70 
 
 10 „ 80 
 
 10 „ 90 
 
 10 ,,100 
 
 10 ,,110 
 
 10 ,,120 
 
 11 „ 
 
 77 
 
 11 „ 88 
 
 11 „ 99 
 
 11 ,,110 
 
 11 ,,121 
 
 11 ,,132 
 
 12 „ 
 
 84 
 
 12 „ 90 ,12 .,108 
 
 12 ,,120 
 
 12 ,,132 
 
 12 ,,144 
 
 Any number of times* is 0, and times any number is 0. 
 Thus, 
 Ox 1 = 0, X 2 = 0, &c.; 1 X = 0, 2 X = 0, &c. 
 
 49. The process of Multiplication is based upon the fol- 
 lowing 
 
 'S 
 
 Repeat the column ouce 1 ia 1. 2 times 1 are 2. 3 times 1 are 3, &c. 
 What ia any number of times ? times any number ? 
 
 iiii 1 ' 
 
 —^w 
 
MULTIPLICATION. 
 
 35 
 
 }) 
 
 18 
 
 J> 
 
 24 
 
 M 
 
 30 
 
 J> 
 
 3G 
 
 >} 
 
 42 
 
 >> 
 
 48 
 
 M 
 
 54 
 
 >) 
 
 GO 
 
 J 
 
 6G 
 
 » 
 
 72 
 
 (3 
 
 PRINCIPLES. 
 
 1. The product and muUij^dicand must he like numhtrs. 
 Tlius, 
 
 4 times G men. are 24 mcii. 3 times 7 cents are 21 a'7if«. 
 
 2. The muJt'n)licr viud alwajjs he regarded as an abstract 
 number. Thus, 
 
 In fiiKliiij^' tlic cost of G tons of coal at 7 dollars a ton, the 7 dullurs 
 are Uikun G times, ami not multiplied by G tons. 
 
 3. The product of two or more factors is the same in what- 
 ever order they are taken. Thus, 
 
 The i)r()duct of G x 3, or 3 x G, is 18, and the product of 5 x 3 x "2, 
 or 2 X 3 X 5, or 3 X 5 X 2, is 30. 
 
 50. For tlio definition of Multiplication, it follows tliat 
 
 Multiplication, when the si::e or value of a single tiling, or 
 unit, is given, enables us to find the size or value of any uunx- 
 ber of things of the same kind. 
 
 51. To multiply one number by another. 
 
 ^ 1. Let it be required to multiply 5G4 by 7. 
 
 OPERATION. For convenience we write the multiplier 
 
 ^lultiplicand, 564 under the units in the multiidicaiid, and begin 
 
 Multiplier, 7 with units to multiply. 
 
 7 times 4 units are 28 units, which erpial 2 
 
 Product 3948 ^^^^ ^^*^ ® units ; we write the 8 units, and 
 
 reserve the 2 tens to add to the next product. 
 
 7 times 6 tens are 42 tens, which with the 2 tens added are 44 tens, 
 or 4 hundreds and 4 tens ; we wTite the 4 tens, and reserve the 4 
 hundreds to add to the next product. 
 
 7 times 5 hundreds are 35 hundreds, which with the 4 hundreds 
 added are 39 hundreds, or 3 thousands and 9 hundreds ; which we 
 write. 
 
 Therefore the product is 3 thousand 9 hundred and 48, or 3948. 
 
 What is the first Principle ? The second? The third? "V^Tiat Joes JIol- 
 tixilication enable us to find ? 
 
 ii 
 
 
 «MJi* 
 
36 
 
 PRACTICAL ARITHMETIC. 
 
 
 'it 
 
 :1i 
 
 ly 
 
 In practice the name of the ordor of units may l)e omittcMl. Tims, 
 in the operation we can way : 7 times 4 are 28 ; we write the 8, and 
 uilil tile 2 to tht! next product ; 7 tinuis (5 are 42, and 2 are 44 ; we 
 write; 4, uiid a(M 4 to the next pro<luct ; 7 times 5 are 35, and 4 are 
 3S) ; answer, 3U4H. 
 
 2. Let it be requirod to find tlio product of 730 ])y 20G. 
 
 oPKRATiox. I'^'^r convenience we write the multii»lii'r 
 
 Multiplicuud 73G ^^^''l^'r the multiplicand, so that tij,'ures of the 
 Multiplier 2()G ^'''•'"^ order stand in thu same column ; and 
 
 midtiplyinj,' by the units, as in the preceding 
 
 Partial f 44 IG operation, we ohtmn 4416. 
 Products 14720 There being tens, we ^v^ite a cipher in 
 
 the order of tens underneath, and pass to the 
 
 Product, LJIGIG hundreds' figure of the nuiltiplier. 
 
 2 hundreds are 2 hundred units, and 2 hun- 
 dred times f! units are 12 hundreds, or 1 thousand and 2 hundreds. 
 We write the 2 hundreds, and reserve the 1 thousand to add to the 
 next product. 
 
 2 hundred times 3 tens are (5 hundred tens or G thousands, and the 
 1 thousand added are 7 thousands, which we write. 
 
 2 hundred times 7 hundreds are 14 hundred-hundreds, etpial 1 
 hundred-thousand and 4 ten-thousands, which we write, and obtain 
 1472 hundreds, or, with the cipher on the right, 14720 tens. 
 
 Adding the two partial products we have for the entire product 
 151G1G. 
 Therefore the product of 73G by 206 is 151616. 
 
 Rule. JFrite the rmiUipUer under the 7mtlttplicand, so thai 
 tinits may stand under units, tens under tens, &c. 
 
 If the multiplier contains but one order of units, heglnning at 
 the right, multiply each order of the multiplicand by it, vritiiig 
 the right-hand figure of eu'Jb product underneath adding the 
 numbers expressed by the other figures, if any, to the next product, 
 observing to write all the figures of the last product. 
 
 If the multiplier contains more than one order of units, multiply 
 by each of the orders, successively, writing the rightliand figure of 
 
 Review Questions. What la a Sign? (38) The Sign of Aiklition? (38) 
 Of E.iuality ? (38) Of Subtraction ? (43) Of Multiplication ? (48) 
 
MULTIPLICATrON. 37 
 
 each partial pmhid mider the order used. The sum of the partial 
 products icill he the entire product. 
 
 PaooF. Multiply the multiplior by the niultipli.-aud ; iui.l 
 »f the i.r.xluct is the same as that first obtained, th(! work is 
 supposed to be correct. Or, 
 
 Separate the iiiultipli(>r into parts, and make each of them 
 a niulDpher, and if the .^um of tlie pnxhicts eciuuls the lirst 
 product, the work is supposed to be correct. 
 
 Exercises. 
 3. Find the product of 78 by 37. 
 
 OPERATION. 
 
 78 
 37 
 
 Proof. 
 37 
 78 
 
 540 - 78 X 7 
 234 .. 78 X 30 
 
 290 =- 37 X 8 
 259 c:. 37 X 70 
 
 
 
 Ans. 2880 = 78 x 37 
 4. Find the product of 310 by 13 
 
 OPERATION. 
 310 
 
 13 
 
 2880 = 37 x 78 
 
 948 = 310 X 3 
 310 ..: 310 X 10 
 
 Ans. 4108 = 310 x 13 
 
 (5.) 
 Multiply 410 
 By ' 4 
 
 
 / 310 
 
 
 
 11 
 
 
 Proof. 
 
 ] 310 - 
 j 310 .. 
 
 310 X 1 
 310 X 10 
 
 
 f 032 = 
 
 310 X 2 
 
 
 ^ 4108 = 
 
 310 X 13 
 
 (0.) 
 3102 
 
 1091 
 
 (8-) 
 4407 
 
 8 
 
 
 9 
 
 Product, 1004 24810 
 
 7037 
 
 40203 
 
 What is the llule ? The Proof ? 
 
f 
 
 I 
 
 t 
 
 1^ 
 
 I i 
 
 it 
 
 Ni 
 
 33 
 
 PRACTICAL ARITHMETIC. 
 
 (9.) 
 
 Multiply 196 
 By 6 
 
 (10.) 
 7781 
 
 (11.) 
 44G5 
 
 (12.) 
 5532 
 
 /ryi s?f6 6 /^j-6' ^^ 
 
 Multiply 
 13. 130Gby 3. A7is. 3918. 
 U. 971 by 5. Ans. 4855. 
 
 15. 4137 by 6. 
 
 OY^ 
 
 *M 
 
 ■>^1^ 
 
 ♦^ i. 
 
 16. 46002 by 4. ^W5. 184008. 
 
 17. 1190 by 7. ^ws. 8330. 
 
 25. 2538 by 11. Ans. 270 i 8. 
 
 26. 1223 by 8. Ans. 9784. 
 
 27. 67812 by 8. X.oVj^'^/^ 
 
 28. 12091 by 7. ^w.9. 84637. 
 
 29. 7090 by 12. Ans. 85080. 
 
 30. 5009 by 17. ^^.^^Vo'i 
 
 31. 387 by 22. Ans. 8114. 
 
 32. 664 by 19. 
 
 33. 315 by 23. 
 
 34. 1782 by 61 
 
 35. 909 by 29. 
 
 36. 81201 by73.^/z5.5927673. 
 
 Ans. 3546244. 
 Ans. 4832878. 
 
 Ans. 12616. 
 
 y^ws. 108602. 
 Ans. 26361. 
 
 18. 60441 by 5. ^^,>,,i^V 
 
 19. 9943 by 2. Ans. 19886. 
 
 20. 6453 by 5. Ans. 32265. 
 
 21. 978609 by l..>^n,f>?'i;;-' 
 
 22. 1706 by 11. Ans. 18766. 
 
 23. 19774 by 10.^715.197740. 
 
 24. 8320 by 13. Ans. 108160. 
 
 37. 75452 x 47 =^ how many? 
 
 38. 54302 X 89 - liow many ? 
 
 39. 784 X 203 = how many ? 
 
 40. AVhat is the product of 137 by 35 1 
 
 41. AYhat is the product of 567 by 108 ? 
 
 42. What is the product of 5, 25, and 37 ? 
 
 43. What is the product of 3, 17, and IIU 
 
 44. Hov/ many are 1234 times 7013? 
 
 45. Multiply 486 by 259. 
 
 46. Multiply 34618 by 259. 
 
 47. Multiply 80704 by 432. 
 
 48. Multiply thirty-one thousand three hundred and elevv^ii 
 by one thousand two hundred and thirteen. rijPj?^ 60{i^^ 
 
 49. Multiply ninety-three thousand one hundred and eighty- 
 ►.ix by four thousand four hundred and fifty-five. 
 
 Ans. 415143630. 
 
 Rfview Questions. "\Miat is the answer called in Addition? (37) In 
 Subtraction ? (42) In Multiplication ? (47) 
 
 Ans. 159152. 
 
 Ans. 4795. 
 
 Ans. 61236. 
 
 Ans. 4625. 
 
 Am. 8654042. 
 
 Ans. 125874. 
 
 Ans. 8966062. 
 
 ^W5. 34864128. 
 
MULTIPLICATION. 
 
 39 
 
 !0. 
 
 In 
 
 I 
 
 i 
 
 52. "WhoD there are ciphers between significant fii^urcs in 
 the muItipUer, the operation may be shortened hij passinj oi\r 
 each of the multiplier. 
 
 50. Multiply 4230 by 2007. 
 
 OPERATIONS. 
 
 4236 \ / 423G 
 
 2007 1 ( 2007 
 
 29052 
 847200 
 
 8501G52 
 
 or, <' 29G52 
 j 8472 
 
 \ 8501G52 
 
 51. Multiply 15G07 by 3094. 
 
 52. Multiply 60121 by 3108. 
 
 Ans. 48288058. 
 
 53. AVhon the multiplier consists of two significant figures, 
 ■with \jV without intervening ciphers, and begins or ends with 
 1, we may consider the multiplicand as a product ly the 1, and 
 write the other partial product as many orders to the right or left 
 as is required by the midtiplier. 
 
 53. Multiply 251 by 31. 54. Multiply 1235 by 1004. 
 
 OPERATION. 
 
 1235 - 1235 X 1000 
 4940 - 1235 X 4 
 
 ^^^ 
 
 OPERATION. 
 
 
 251 
 
 = 251 
 
 X 
 
 1 
 
 753 
 
 - 251 
 
 - 251 
 
 X 
 X 
 
 30 
 
 7781 
 
 31 
 
 1239940 - 1235 x 1004 
 
 In example 56, the one partial product is units and the otlier tens, 
 and in example 57, the one partial product is thousands and the other 
 units ; and they are so Avritten that, in each case, the sum of (ho 
 partial products may be the required product. 
 
 55. Multiply 3403 by 501. 
 
 Ans. 1704903. 
 
 How may you multiply when theie are ciphers hetween the signiticiint 
 figures of the multiiJier? When either of the two significiuit figures of the 
 multiplier is 1 ? 
 
 \ ( 
 
 ! ; »' 
 
40 
 
 PRACTICAL ARITUMETIC. 
 
 if 
 
 u 
 
 m 
 
 r)G. Multiply 5121 by 1002. Ans. 5131242. 
 
 57. Multiply 61303 by 701. Ans. 42073-403. 
 
 54. "When tlic multiplier is 10, 100, 1000, &c., the p-odud 
 maij he obtained at once, hjj annexing to the midtijjUcand as many 
 ciphers as there are in the multi2)Uer, and regarding the decimal 
 point as removed an equal number of places to the right. 
 
 For the value expressed by fij^ures is made tenfold by each removal 
 of tliem an order to the left. (Art. 30.) Thus, 2 x 10 -^ 20, 2 x IdU 
 = 200, &c. 
 
 58. Multiply G19 by 100. Ans. G1900. 
 
 59. Multiply 11G44 by 1000. Ans. 11G44000. 
 GO. 45G8 X 1000000 - how many? Ans. 45G8000000. 
 
 55. When there are ciphers on the right of either or both 
 of the factors, ice may multiply icithoni r<'ference to them, and 
 annex to the product as many cijjhers as there are on the right of 
 both factors. 
 
 61. Find the product of 2050 by 1300. -i^ £,^'J^^ 
 
 OPERATIONS. 
 
 2050 \ f 2050 
 
 1300 
 
 1300 
 
 G 15000 
 2050 
 
 26G5000 
 
 or. 
 
 615 
 
 205 
 
 2GG5000 
 
 The second operation is evidently the same as the first, except that 
 the ciphers on the right are not written until the partial products are 
 added. 
 
 .62. Multiply 485 by 240. Ans. 116400. 
 
 63. Multiply 36500 by 730. Ans. 26645000. 
 
 64. Multiply six hundred seventy-four thousand and two 
 hundred by two thousand one hundred and four. 
 
 Ans. 1418516800. 
 
 "When the multiplier is 10, 100, 1000, &c. ? How do you multiply when 
 there are cii>lier3 at the riyht of either or both of the factors ? 
 
 -.-«i 
 
MULTIPLICATION. 
 
 41 
 
 0. 
 
 i 
 
 ire 
 
 A 
 
 
 ■1 
 
 )0. 
 
 ,i 
 
 
 
 )0. 
 
 1 
 
 ft'O 
 
 1 
 
 APPLICATIONS. 
 
 1. If a man travel 212 miles a week, how far will he travel 
 in 02 weeks ] =r//^ /^ ^ ^^^^ 
 
 Solution. If a man travel 212 miles in a week, he will travel in 
 52 weeks 52 ti)nes 212 miles, or 11024 miles. Thcrrforc if a man 
 travel 212 miles in a week, he will travel in 52 iceeks 11024 miles. 
 
 2. If an orchard containing 313 trees prodncc 15 l)usliels of 
 apples to a tree, how many bushels is the produce of the whole 
 orchard? Ans. 4695. 
 
 3. How many lights of glass in 18 windows, if each window 
 contains 24 ? ^ ^ J ^^^ ^,?,-/ 4 * 
 
 4. How many bushels of Corn will grow on IGO acres, at the 
 average rate of 45 bushels to an acre ? ^1/^.$. 7200 bushels. 
 
 5. What cost 24G3 barrels of flour at 9 dollars a barrel %^'-^^W' i) 
 
 G. If a man can earn 83 dollars in one month, how many 
 can he earn in 12 months ? Ans. 99G dollars. 
 
 7. What will be the cost of a farm containing G84 acres, at 
 57 dollars per acre ? Ans. 38988 dollars. 
 
 8. A certain field has 625 hills of potatoes, and eacli hill 
 will average 8 potatoes ; how many potatoes at that rate in the 
 field? -^ 6^0 06 4x.^U{^X. , 
 
 9. If 17 men can do a piece of work in 91 days, how long 
 will it take one man alone to do it ? Ans. 1547 days. 
 
 10. An army consists of 6 brigades, each brigade of 4 
 regiments, and each regiment of G13 men, rank and file ; 
 required the number of men in the army ? Ans. 14712 men. 
 
 11. If a saw-mill can produce 4360 feet of boards a day, how 
 many can it produce in 106 days ? Ans. 462160 feet. 
 
 12. If the earth moves around the sun at the rate of 68000 
 miles an hour, how far will it move in 365 days of 24 hours 
 each ? Ans. 595680000 miles. 
 
 Kkview Questions. What is the Rule in Addition? (40) In Subtrac- 
 tion? (45) In Multiplication? (51) 
 
 ll 
 
 'V 
 
 m 
 
 
 
 
 
 ■ 
 
 
 
 
 ( 
 
 \ 
 
 
 
 
 i 
 
 i 
 
 i 
 
 1 
 
 ! ' 
 
f I 
 
 PRACTICAL AEITHMETIC. 
 
 I' .'I 
 
 il 
 
 ii 
 
 DIVISION. 
 
 56. 1. If 4 cents will buy one orange, how many oranges 
 will 12 cents buy? 
 
 Solution. If 4 cents will buy one orange, 12 cents will hui/ as 
 many oranges as 4 cents are contained times in 12 cents, or 3, There- 
 fore if 4 cents will buy one orange, 12 cents will buy 3 oranges. 
 
 2. At 7 cents a pound, how many pounds of rice can be 
 bought for 2 1 cents ? 
 
 3. If 2 boys share equally 8 marbles, how many will each 
 have 1 
 
 Solution. If 2 hoys share equally 8 marbles, each will have 1 of 
 the 2 erpuil parts of 8 marbles ; 1 of the 2 equal parts of 8 marbles is 4 
 marbles. Therefore if 2 boys share equally 8 marbles^ each will have 
 4 marbles. 
 
 One of the two equal parts of a number is called one luilf 
 of ' he number ; one of the three equal parts, one third of the 
 number ; one of the four equal parts, one fourth of the number ; 
 one of the five equal parts, one fifth of the number ; one of the 
 ten equal parts, one terith of the number ; one of the hundred 
 equal parts, one hundredth of the number, and so on. Hence, 
 
 A li'hole number has two halves, three thirds, four fourths, 
 five fifths, ten tenths^ one hundred hundredths, &c. 
 
 4. When 4 apples cost 16 cents, what is the cost of one 
 apple 1 
 
 Solution. TVlien the cost of 4 apples is 16 cents, the cost of one 
 apple is one fourth o/ 16 cents, or 4 cents. Therefore ivhen 4 applti 
 cost 16 ce?its, the cost of one apple is 4 ce^its. 
 
 5. If 20 dollars are shared equally between 5 boys, how 
 many does each boy receive 1 
 
 The preceding operations illustrate what is called Division. 
 Hence, 
 
 What is one of two equal parts of a number called ? One of three equal 
 jmrts ? One of four equal parts ? 
 
 I 
 
 I 
 
DIVISION. 
 
 43 
 
 I one 
 
 how 
 
 SION. 
 
 equal 
 
 57. Division is the process of finding how many times one 
 number is contained in another ; or of finding one of the equal 
 parts of a number. 
 
 The Dividend is the number to be divided. 
 
 The DmsoR is the number by which we di\'ide. 
 
 The Quotient is the result or number obtained by the 
 division. 
 
 The Eemainder is that part of the dividend which is left 
 after finding the exact v/hole number of the quotient. Thus, 
 
 3 is contained in 7, 2 times and 1 as a remainder. 
 
 The division is said to be exact, when there is no remainder. 
 
 58. The Sign of Division is a short horizontal line with 
 a dot above and below it, -f , read, divided hij. Thus, 
 
 6 -^ 2 is read, six divided by two. 
 
 Sometimes, in place of the dots, the number divided is writ- 
 ten above the line, and the number which divides it is written 
 below. Thus, 
 
 4 is read six divided by two. 
 
 Division, also, may be indicated by a curv^ed line, ), the 
 divisor being written before, and the dividend after it. Thus, 
 
 2) 6 indicates that 6 is to be divided by 2, and is so read. 
 
 59. One or more equal parts of a unit are called fractions, 
 to distinguish them from unbroken or whole numbers, which 
 are called integers, or integral immhcrs. 
 
 \, read one divided by two, or one half, 
 ^, read one divided by three, or one third, 
 f, read two divided by three, or two thirds, ike, 
 are fractional expressions. 
 
 What is Division? Tlie Dividend ? The Divisor? Tlie Quotient ? The 
 Remainder? AVhen is the division said to be exact ? "What are the Siyns of 
 Division ? What are one or more equal parts of a unit called ? 
 
 k 
 
 w 
 
 I 'I 
 
4.1. ^S.-'^.'-ll*!,* « 
 
 44 
 
 PRACTICAL ARITHMETIC. 
 
 Division Table. 
 
 
 1 in 
 
 
 2 in 
 
 
 3 in 
 
 
 4 in 
 
 5 in 
 
 1, 
 
 1 time 
 
 2, 
 
 1 time 
 
 3, 
 
 1 time 
 
 4, 
 
 1 time 
 
 5, 1 time 
 
 2, 
 
 2 Limes 
 
 4, 
 
 2 times 
 
 G, 
 
 2 times 
 
 8, 
 
 2 times 
 
 10, 2 times 
 
 3, 
 
 3 y 
 
 G, 
 
 3 „ 
 
 9, 
 
 3 „ 
 
 12, 
 
 3 „ 
 
 15, 3 „ 
 
 4, 
 
 4 „ 
 
 8, 
 
 4 „ 
 
 12, 
 
 4 „ 
 
 IG, 
 
 4 „ 
 
 20, 4 „ 
 
 5, 
 
 T) „ 
 
 10, 
 
 5 „ 
 
 15, 
 
 r> „ 
 
 20, 
 
 T) „ 
 
 ", ri i 
 
 G, 
 
 c „ 
 
 12, 
 
 G „ 
 
 18, 
 
 G „ 
 
 24, 
 
 G „ 
 
 30, G „ 
 
 7, 
 
 7 „ 
 
 14, 
 
 7 „ 
 
 21, 
 
 7 „ 
 
 28 
 
 7 „ 
 
 35, 7 „ 
 
 8, 
 
 8 „ 
 
 IG, 
 
 8 „ 
 
 24, 
 
 8 „ 
 
 32, 
 
 8 „ 
 
 40, 8 „ 
 
 0, 
 
 „ 
 
 18, 
 
 9 „ 
 
 97 
 
 9 „ 
 
 3G 
 
 9 „ 
 
 45, 9 „ 
 
 10 
 
 10 „ 
 
 20, 
 
 10 „ 
 
 30, 
 
 10,^ 
 
 40 
 
 10 „ 
 
 50, 10 „ 
 
 
 G ill 
 
 
 7 in 
 
 
 Sin 
 
 
 9 in 
 
 " lOin" 
 
 G 
 
 , 1 time 
 
 7, 
 
 1 time 
 
 8, 
 
 1 time 
 
 9 
 
 1 time 
 
 10, Itime 
 
 12, 
 
 2 times 
 
 14, 
 
 2 times 
 
 IG, 
 
 2 times 
 
 18 
 
 2 times 
 
 20, 2 times 
 
 18 
 
 3 „ 
 
 21, 
 
 3 „ 
 
 24, 
 
 3 „ 
 
 27 
 
 3 „ 
 
 30, 3 „ 
 
 24 
 
 4 „ 
 
 28, 
 
 4 „ 
 
 32, 
 
 4 „ 
 
 3G 
 
 . 4 „ 
 
 40, 4 „ 
 
 30 
 
 5 „ 
 
 35, 
 
 5 „ 
 
 40, 
 
 5 „ 
 
 45 
 
 , 5 „ 
 
 50, 5 „ 
 
 3G 
 
 c „ 
 
 42, 
 
 G „ 
 
 48, 
 
 G „ 
 
 54 
 
 . G „ 
 
 GO, G „ 
 
 42 
 
 7 „ 
 
 49, 
 
 7 „ 
 
 5G, 
 
 7 „ 
 
 G3 
 
 . 7 „ 
 
 70, 7 „ 
 
 48 
 
 8 „ 
 
 5C, 
 
 8 „ 
 
 G4, 
 
 8 „ 
 
 72 
 
 , 8 „ 
 
 80, 8 „ 
 
 54 
 
 , 9 „ 
 
 G3, 
 
 9 „ 
 
 72, 
 
 9 „ 
 
 81 
 
 . 9 „ 
 
 90, 9 „ 
 
 GO, 
 
 10 „ 
 
 70, 
 
 10 „ 
 
 80, 
 
 10 „ 
 
 90 
 
 ,10 „ 
 
 100,10 „ 
 
 60. The process of Division is based upon the followin 
 
 Principles. 
 
 1. The quotient loill he an abstract number, tvhen the divisor and 
 dividend are like numbers. 
 
 For the quotient will denote how many times the divisor is con- 
 tained in tlie dividend. 
 
 2. The divisor must he regarded as an abstract number, when 
 the dividend is concrete, and the divisor not a like number. 
 
 Repeat the line 1 in 1. 2 in 2. 3 in 3. 4 in 4. 5 in 5, &c. "What is the 
 first Principle ? The second ? 
 
DIVISION. 
 
 45 
 
 1 a7ul 
 
 •011- 
 
 I'Jwb 
 
 tho 
 
 ! I 
 
 For the divisor mupt then denote the number of equal pa;'<s into 
 wliicli the dividend is to be divided. Hence, 
 
 3. // fhe divisor and dividend are not like nuwhcrs, the quotient 
 and dividend u'ill he like nmnbers. 
 
 For, the quotient will denote one of the equal imrts into which 
 tlie dividend is divided. 
 
 4. Tlie remainder and dividend must ahrays be like numbers. 
 For the remainder is evidently a part of the dividend. 
 
 5. Division may be regarded as the reverse of multiplica- 
 tion. 
 
 For the dividend corresponds to the product, and the divisor 
 and quotient to the two factors. 
 
 61. From the foregoing principles, it follows that there may 
 be two kinds of division : 
 
 First kind — when the size or redue of the equal parts ot 
 a quantity is given, to find their number ; and 
 
 Second kind — when the number of equal parts of a quantity 
 is given, to find their i. :c or value. 
 
 CASE I. 
 
 62. To divide l)y Short Division. 
 
 1. Let it be required to divide 1702 by 7. 
 
 For convenience we WTite the 
 
 OPERATION. 
 
 Divisor, 7)1702 Dividend; 
 
 243} Quotient. 
 
 divisor at the left of the dividend, 
 and begin at the left to divide. 
 
 7 is contained in 1 thousand no 
 thousand times ; therefore, there 
 
 will be no thousands in the quotient. Try 17 hundreds ; 7 is con- 
 tahied in 17 hundreds, 2 hundreds times, and 3 hundreds, ecpial 
 to 30 tens, remaining. We write the two hundreds, and add the 
 30 tens to the tens, making 30 tens. 
 
 What is the third Principle ? The fourth ? The fifth ? Name the two 
 kinds of Division. 
 
 i; 
 
 i 
 
 n 
 
 i \ \ 
 
4G 
 
 PRACTICAL ARITHMETIC. 
 
 7 in 30 tf!n=5, 4 tens times, and 2 tens, equal to 20 units, remain- 
 ing. We write the 4 tens, and add the 20 units to tlie 2 units, 
 making 22 units. 
 
 7 in 22 units, 3 units times, and one unit remaining. We write 
 tlie 3 units, and indicating the division of the 1 unit, we aimex the 
 fractional expression, ^ unit, to the integral part of the quotient. 
 
 Therefore 1702 divided hy 7 is equal to 243}, read two hundred 
 forty-three and one seventh. 
 
 The solution by the second kind of Division ifl : 
 
 One seventh of 17 hundreds is 2 hundreds, and a remainder of 3 
 liundreds, e{iual to 30 tens. We write the 2 hundreds, and add 
 the 30 tens to the tens, making 30 tens. 
 
 One seventh of 30 tens is 4 tens, and a remainder of i tens, Ci^iuil 
 to 20 units. We write the 4 tens, and add the 20 units to the 2 
 units, making 22 units. 
 
 One seventh of 22 imits is 3 units, and a remainder of 1 unit 
 We write the three units, and indicating a seventh of the 1 unit, 
 we annex the expression |, to the integral part of the quotient 
 
 Therefore 1702 divided by 7 is 243f 
 
 2. Let it be required to obtain the quotient of 7G3 divided 
 
 by 7. 
 
 OrERATION. 
 
 Divisor, 7 ) 7G3 Dividend. 
 
 109 Quotient. 
 
 For convenience we write the 
 divisor, and begin to divide, as 
 in the preceding operation. 
 
 7 ill 7 hundreds, 1 hundreds 
 time, which we write. 
 7 in 6 tens tens times, and 6 tens, equal to 60 units, remain- 
 ing. We write the tens, and add the 60 units to the 3 units, 
 making 63 units. 
 
 7 in 63 units, 9 units times, which we write. 
 In practice we may say, 7 in 7, 1, which we write ; 7 in 6, 0, 
 which we WTite ; prefix the 6 to the figure 3 ; 7 in 63, 9, which 
 we write ; answer, 109. 
 
 When, as in the above operations, the dividing is performed 
 mentally, except in writing the quotient figures, the process is called 
 Short Division. 
 
 Give the solution by the first kind of Division. The second. When is the 
 process called Short Division ? 
 
 1( 
 
 1] 
 
DIVISION. 
 
 47 
 
 i 
 
 1 
 
 EULE. JFrite the divmr at the left of tlic div-ulcrui. 
 
 Ik' 'J in at the left, divide the numbers expressed hy each figure 
 of the dividend hi/ the divisor, and icrite the result beneath. 
 
 If ther^ he a remainder after any division, regard it as prefixed 
 to the figure of the ni'xt lower order, and divide as before. 
 
 If any partial dividend he less than the divisor, write a cipher 
 in iJte quotient, and prefix sudi dividend to the next figure, if 
 any, for a new dividend. 
 
 If tlicre he a final remainder, write it, with the divisor beneath, 
 after the integral part of tlie qmtient. 
 
 Proof. Multiply the integral number of the quotient by 
 the divisor, and to the product add the remainder, if any ; 
 and the result will equal the dividend, if the work is ri^dit. 
 
 ie, as 
 
 6,0, 
 ^vhich 
 
 )rmed 
 Icalled 
 
 N 
 
 
 Examples. 
 
 
 (3.) 
 
 4 ) 8852 
 
 5 ) 90G51 
 
 7 ) 220523 
 
 Am. 2213 
 4 
 
 181301 
 5 
 
 31503^- 
 7 
 
 roof. 8852 
 
 90051 
 
 220523 
 
 2 ) lOOG 
 
 (7.) 
 8 ) 17620 
 
 (8.) 
 3) 01273 
 
 IS 
 
 the 
 
 Divide 
 
 9. 1896 by 4. Ans. 474. ' 14. 85765 by 3. Ans. 28588.^. 
 
 10. 6819 by 5. Ans. 1303i. 1 15. 65548 by 9. Ans. 7283^! 
 
 11. 60104by2. ,y;^i3^>^'?^,|l6. 578096 by a<. . .^f/.^-/ £ 
 
 12. 321001by7. ^7W. 45857f.|17. 161413by 6. ^/w. 26902}. 
 
 13. 447078 by 8.Ji^ t'/iiM^^- ^080706 by 5.Jfn, /// 6 /f /jf. 
 
 I 
 
 ■; i 
 
 !'. 
 
 Eepoat the Kule. What ia the Proof ? 
 
48 
 
 I'RACTICAL ARITHMETIC. 
 
 Rofjuircfl 
 
 19. One tliinlof 189. ybis. 03. 
 
 20. Onti fifth of 1700. 
 
 A?is. 358. 
 
 23. 870G21-r2 = howmaiiy? 
 
 24. How many are -iii-Jii^ 1 
 
 21. Oneci-litli of 9872.r/^9^ 
 
 22. One ninth of 8011. 
 
 Ans. 800 J. 
 
 Ans. 435310.}. 
 Ahs. 48803}. 
 
 APPLICATIONS. 
 
 1. How many cords of wood can be bouglit for 10G5 
 dc^llars, at 5 dollars a cord 1 = ^9^ ^-'^-^ 
 
 Solution. Since the cost of 1 cord is 5 clollirs, ns many cords can' 
 he bovijJit for 19G5 dollars as 5 dollars arc cart tended tlmci> in 1905 
 dollars, or 393. Thenfvre, there can he hoiight 393 cords of wood for 
 1(JG5 dollars, at 5 dollars a cord, 
 
 2. If 4 hushcls of wheat make 1 barrel of Hour, how many 
 barrels will 9G50 bushels make 1 — J- 
 
 CopTii 
 
 3. At 7 cents a pound, how many pounds orrice can be 
 bought for 3G3 cents ? Ans. 51'-' pounds, 
 
 4. If 9 horses cost 2025 dollars, how much must 1 horse 
 
 Solution. 7/" 9 horses cost 2025 dollars, 1 horse must cost one ninth 
 of 2025 dollars, or 225 dollars, lliercfore, if 9 Jiorses cost 2025 
 dollars, one horse must cost 225 dollars. 
 
 5. When 31 G8 dollars are paid for 6 bales of cloth, how 
 much is paid for 1 bale ? Ans. 528 dollars. 
 
 G. IIow many cords of wood, at 5 dollars a cord, can be 
 bought for 19G5 dollars? Ans. 393 cords. 
 
 7. Wlien a carpenter is paid 581 dollars for 7 months' 
 labour, how much is that a month ?->> j ^) 
 
 8. 7 times a certain number is equal to 22134 ; what is the 
 number? Ans. 31 G2. 
 
 I 
 
 Review Questions. What is a Concrete Number? (G) An Abstract 
 Number ? (7) 
 
DIVISION. 
 
 40 
 
 )25 
 
 US' 
 
 he 
 
 
 I 
 
 CASK II. 
 
 63. To divide by Long Division. 
 
 1. Let it bo required to divide 31531 by IT). 
 
 OI'KIl.VTION. 
 
 Dividend. 
 Divisor. 15 ) 3-1531 ( 230i\V Quotient. 
 
 30 
 
 45 
 
 45 
 
 31 
 30 
 
 1 Iicmaindcr. 
 
 l''(ir c'onvoiiiciifi' we 
 Avrite the divisor at the 
 It'l'l and tlu' ([iintii'iit ut 
 the lii^lit of llic divi- 
 (lend, and ln-^qii to di- 
 vide as in Slioit Divi- 
 sion. 
 
 15 is contuinc'l in '.i 
 ten -thousands (> tcn- 
 thciusaml limes ; theiv- 
 f'oir, there v.'ill lie u 
 ti II - thousands in the 
 '[Uotient. 'J'ake34 thou- 
 sands ; 15 is contained in 34 thousands, 2 th<»usand times ; \ve write 
 the 2 thousands in the quotient. 1") x 2 tliou^ands = ."lO thousands, 
 which, suhtracted from 34 thousands, leaves 4 thousands = 4() liun- 
 dreds. Achlin,^; the 5 hundreds, we have 45 hundretls. 
 
 15 in 45 hundreds, .3 InuKh'eds times ; we write tlie 3 liundreds iu 
 
 tlie quotient, 15 x 3 liundreds — . 45 hundreds, which suV)tracted 
 
 from 45 hundreds, leaves nothinc^. Adding the 3 tens, we have 3 tens. 
 
 15 in 3 tens, tens times ; we write tens in the quotient. Adling 
 
 to the 3 tens, which e(|ual 30 units, the 1 units, we Iiave 31 units. 
 
 15 in 31 units, 2 units times ; we write tlie 2 units in the quotient. 
 
 15 X 2 units = 30 units, which, sul)tracted from 31 units, leaves 1 
 
 unit as a remainder. Indicating the division of the 1 unit, we annex 
 
 the fractional expression, J-- unit, to the integral part of the (|uotient. 
 
 Therefore 34531 divided by 15 is equal to 2302J.. 
 
 In practice we may say : 15 iu 34, 2 times ; Avrite 2 in the quotient ; 
 15 X 2 = 30, which from 34 leaves 4. Bring down 5 ; 15 in 45, 3 
 times ; write 3 in the quotient ; 15 x 3 = 45, which from 45 leaves 0. 
 Bring down 3 ; 15 in 3, times ; write in the quotient. Bring 
 down 1 ; 15 in 31, 2 times ; 15 x 2 - 30, which irom 31 leaves 1. 
 Answer, 2302yV. 
 
 11 
 
 Explain tlie operati(tu. 
 
50 
 
 PRACTICAL ARITHMETIC. 
 
 Wlion, as a1)ovo, tlu; wr»rk of dividing U mostly written out, tlie 
 process is culled Lono Divisiox. 
 
 KL'LK. jrrlfi' the dtL'isor at the left of the dividend. 
 
 JU'ijiii (it (he left, divide the number exj^ressed hy the fewest 
 figures of the dividend that will contain the divisor, and write the 
 quotient at the rir/ht of the dividend. 
 
 MuUiphi the divisor by this quotient ; suhtraet the inoduet from 
 the imrt 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 dividend have been 
 used. 
 
 If there be a final remainder, tvrite it, icith the divisor beneath, 
 after the integral imrt of the quotient. 
 
 PliooF. The same as in Short Division. 
 
 Examples. 
 
 2. IIow many times is 24 contained in 781G] 
 
 OPERATION. rilOOF. 
 
 24 ) 7S1G ( 325 J J Quotient. ' 325 
 
 72 ' 24 
 
 Gl 
 48 
 
 136 
 120 
 
 IG Eemainder. 
 Divide 
 
 3. 8G87 by 7. Ans. 1241. 
 
 4. IGlSbyO. y/jr.rjft 
 
 5. 15702 by U. Ans. 1427/x. 
 G. 25G20by 12. Ans. 2135. 
 
 1300 
 G50 
 
 7800 
 
 IG Remainder. 
 
 781G 
 
 7. 8G000byG3.^/i5. 13G51J. 
 
 8. 18570 by U^^u,.6y/^^ 
 
 9. 5783 by 108. Ans. 5^^. 
 10. 98701 by 75..f^.../j3/^j^_ 
 
 "When is the process called Long Division ? Repeat the Rule. What is 
 the Proof? 
 
DIVISION. 
 
 13 
 
 i 
 
 11. 1735 1 l)y 8G. An^. 'JOi;^,";. 13. 01028 by 100.= dT x^ 6^^-, ^"^^ 
 
 12. 7012 by 52. Ans. \U\i. 14. 071032 by 3(M.'-j^6 
 15. 13351: f 17 =- how many / 
 1<5. 340o -f G2 ^^ how many? 
 
 7 
 
 >/^ 
 
 /^s•. z."^.) 
 
 y//i5. 54; 
 
 I r 
 
 ? 
 
 T^' 
 
 17. 10000 
 
 liow many ? 
 
 18. 10004 4- 110 how many J 
 10. 45078 -^ 73 - how many I 
 
 .^ i/i'f 
 
 Am. CI 7 
 
 Tiff' 
 
 ^ns. llOOI'f.^ 
 Ans, 243 J^. 
 
 / 
 
 Ans. 38G;^i'^ 
 
 Ans. 513i".'V 
 
 Ans. 7108. 
 
 Ans. 308. \. 
 
 ^715. 10002. 
 
 y^w5. 38^111-!}. 
 
 20. null -f 222 = how many] 
 
 21. G0702 -f 51 - how many J 
 
 22. 13415 -r 55 - how many? 
 
 23. AVliat is the value of ';•:!•;"] 
 
 24. Divide 23218 by CO. 
 
 25. Divide G3125 by 123. 
 2G. Divide 15547G8 by 21G. 
 
 27. Divide 200204 by 81. 
 
 28. Divide 100000 ])y 102. 
 20. Divide 400G0 by 1023. 
 
 30. Divide 8317 by 27. 
 
 31. Divide G421284by Gt2. 
 
 32. Divide 120345 by 31 o2. 
 
 33. Divide six millions three hundred forty-six thousand 
 two hundred and sixty-nine by one thousand two hundred and 
 sixty-nine. Ans. 5001. 
 
 34. Divide two million nine hundred fifty-three thousand 
 and seventy-nine by one thousand seven hundred and twenty- 
 eiglit. Ans. 1708}!! §;^. 
 
 64. When the divisor is 10, 100, 1000, Szc, the quotient may 
 he obtained, at once, by removing the decimal ])oint in the dividend 
 as many jplaces to the left as there are ciphers in the divisor. 
 
 For, since the value denoted by figures is multiplied by 10 by re- 
 moving the decimal point one place to the right, by 100 by removing 
 it two places, «&;c. (Art. 54) ; and as division is the reverse of multi- 
 plication (Art. 60); removing the decimal point in the dividend one 
 place to the left divides it by 10, two places divides it by lOO, &c. 
 
 Review Questions. What is Multiplication ? (47) Division ? (57) How 
 may the quotient be obtained when the divisor is 10, 100, 1000, kc. ? 
 
 P 
 
 W 
 
 ^ 
 
52 
 
 PRACTICAL ARITEMETIC. 
 
 The integnil part of the quotient will be on the left of tlie 
 •Iccimal point, and the remainder ic'dl he the part on the right of 
 the iioint. Tims, 
 
 1243 -r 10 - 124.3 - 124-i\, read, one hundred twenty- 
 four units and three tenths; 1243 ^ 100 - 12.43 = \^-y\% 
 read twelve units and forty-three hundredths ; 5004 -f 1000 
 ^ 5.004 = 5j-j/jj-g, read, five units and four thousandths, &c. 
 That is, 
 
 IVte first order at the right of the decimal ])oint expresses tenths; 
 the second, hundredths ; the third, thousandths, &c. 
 
 Divide 
 
 3.5. 1403 by 10. Ans. 14G/^j. 39. G0013 by 1000 
 3G. 0700 by 100. ^,^. (^7 Jus. QO.Hj^. 
 
 37. lG301byl00.y//^5. lG3iJ^. 40. 33444 by 10000. /?.. J ^i^>^.^ 
 
 38, 857G1 by 1000./4?z.j.^6'^ -H- 80000 by 1000. Ans. M^^^ 
 
 loco 
 65. When the divisor has any number of significant figures 
 
 with ciphers on the right, the work may he ahridged hj cuttiiKj 
 off the ciphers at the rigid of the divisor and an equal numhcr (f 
 figures from the right of the dividend, and then dividing the re- 
 maining ptart of the dividend hij the remaining part of the divider, 
 and, if there he a remainder, prefixing it to the figures that ircre 
 cut off from the dividend for the entire remainder. 
 
 42. Find how many times 1 700 is contained in 39792. 
 
 OPERATION. 
 
 17,00 ) 397|92 ( 23iVinr, ^^^s. 
 34 
 
 57 
 51 
 
 092 
 
 17 liundrods is contained in 397 
 hundreds, 23 time.?, with a remain- 
 der of 6 hundreds, which, with 92, 
 make? the entire remainder COO + 
 92, or 692. Therefore 1700 is con- 
 tained in 39792, 23^",^^ times. 
 
 What will mark the parts of the quotient? What will the part on the 
 right of the point be? How may the work be abridged when the divisor lias 
 iiny niuubcr of significant figures with ciphers on the right? lu the opera- 
 tion, what is the remainder found by using the lumdrtds in dividing ? Wliat 
 IS taken to form the entire remainder ? 
 
 fi 
 
DIVISION. 
 
 4-3. How many times 70 in 22120 ] 
 44. How many times 1)00 in 820800 I 
 4."). How many times 1900 in 40220? 
 40. How many times IGOO in 137000 ? 
 
 47. Divide 89952 by 500. 
 
 48. Divide 1.31127 by 12000. 
 
 49. Divide 4590000 by 30G00O. 
 
 50. Divide 13834500 by 120300. 
 
 51. Divide 803402 bv 400000. 
 
 52. Divide 11570112 by 890000. 
 
 53. Divide 3G7S900 by 320100. 
 
 53 
 
 Alts. 3i0. 
 Ans. ll'dlltr 
 
 /. 
 
 
 Ans. 15. 
 Ans. 115. 
 
 /hi<i 1 1 "J ■'""' 
 
 APPLICATION'S. 
 
 1. James Erown bond)t a farm of 387 acres for 8514 dul- 
 
 O 
 
 lars ; how much did it cost him per acre ? Ans. 22 dollars. 
 
 2. An army of 9870 men lost one fifteenth of its wliolt/ 
 miiiil)er in storming: a fort ; how nianv were lost ? 
 
 .-tn^. 058 men. 
 
 3. If a field of 109 acres produces 3379 l)iisi;els of wheat, 
 how much is the yield per acre ] ylns. 31 bushels, 
 
 4. If a field vieldini? at tlie rate of 31 busluls to tin; acit' 
 produces 3379 bushels of wheat, how many acres are there in 
 the field? -/^9d-f.. 
 
 5. How long should 17 men subsist upon a supply of pro- 
 visions which would sullice for one man 0205 dayslrr: ^ (nA ^tP^<'^}t 
 
 0. A teamster removed 31340 oricks at 20 loads; how many 
 did he remove at a load ? Ans. 1507 bricks. 
 
 7. If the valuation of a certain town of 7500 iidiabitants 
 is 2025000 dollars, what is the average to each individuals 
 
 Ans. 350 dollars. 
 
 8. How many years must a person labour to accumulate 
 23100 dollars, by .saving 1100 dollars a year] ^J 7>>t'^ 
 
 1 ) '■ 
 
 I 
 
 Ir 
 
 ! ; 
 
54 
 
 PRACTICAL ARITHMETIC. 
 
 9. If light moves at the rate of 192000 miles a second, 
 how many seconds is it in coming from the moon to the earth, 
 the distance being 240000 miles ? Ans. ly^^T/^J'^f^ seconds. 
 
 10. Certain States and the District of Columbia furnished 
 in the late war 2G53062 soldiers; how many days would it 
 take a person to count them at the rate of 15000 a day? 
 
 Ans. miUU-d^ys. 
 
 \ 
 
 REVIEW EXERCISES. 
 
 1. If the multiplicand is 15G07 and the multiplier 3094, 
 what is the product? Ans. 48288058. 
 
 2. If the dividend is 48288058 and the divisor 3094, what 
 is the quotient? -I^CsG^/ 
 
 3. If two men start from the same point and travel in 
 opposite directions, the one at the rate of 42 miles and the 
 other 45 miles a day, how far apart will they be at the end 
 of 12 days ?tr-/^ ^^ ^ c}>'C<^^ 
 
 Solution. Ij one man travel at the rate of 42 miles a day, he will 
 travel in 12 days 12 tiines 42 miles, or 504 miles. 
 
 If the other man travel at the rate of 45 miles a day, he will travel in 
 12 days 12 times 45 miles, or 540 miles. 
 
 If the one travel 504 miles and the other 540 miles, and in opposite 
 directions, they must he apart, the sum of 504 miles and 540 miles, or 
 1044 miles. 
 
 Tlierefore if two men start from the same point and travel in opposite 
 directions, the one at the rate of 42 miles and the other 45 miles a day, 
 they will he 1044 miles apart at the end o/ 12 days. 
 
 4. If two men start from the same point and travel in the 
 same direction, the one at the rate of 512 miles and the other 
 540 miles a week, how far apart will they be at the end of 8 
 weeks i^^-^/^^^^, 
 
 Review Questions. What is the Rule for Short Division? (G2) Wliat 
 is the Rule for Long Division ? (G3) 
 
I i 
 
 DIVISION. 
 
 55 
 
 [ 
 
 ! i 
 
 i 
 
 5. I have two farms; the first contains IGO acres, worth 80 
 dollars an acre, and the second 220 acres, worth 65 dolhirs an 
 acre ; how much are both worth ? Aus. 27100 dollars. 
 
 6. What will be the cost of 103 barrels of flour at 7 dollars 
 a barrel? Ans. 721 dollars. 
 
 7. A merchant bought 30 hogsheads of molasses at 45 dol- 
 lars each, and paid 800 dollars down, and gave his note for the 
 balance ; for what amount was the note 'hr 7*^ • 'A*^ 
 
 8. If a man sell 90 acres /i land at 38 dollars an acre, and 
 divide the money equally among his 5 children, what is each 
 child's share %-JIQ ^ ^j- 
 
 Solution. If a man sell 90 acres of land at 38 dollars an acre, he 
 will receive for it 90 times 38 dollars, or 3420 dollars. 
 
 If he divide 3420 dollars equally among his 5 children, each child 
 will receive as a share one fifth of 3420 dollars, or 684 dollars. 
 
 Therefore if a man sell 90 acres of land at 38 dollars, and divide 
 the money equally among his 5 children, each child's share is 684 
 dollars. 
 
 9. If a man having 5500 dollars to invest should purchase 
 15 United States bonds, at 105 dollars each, how many shares 
 of railroad stock, at 157 dollars each, could he purchase with 
 the balance? Aiis. 25 shares. 
 
 10. William Miller bought some land for 18050 dollars. 
 He sold 50 acres of it for GO dollars an acre, and then found 
 that the remainder cost him 50 dollars an acre ; how many 
 acres were there of the remainder? Ans. 301 acres, 
 
 11. Bought 5 cows at 50 dollars each, and 7 horses at half 
 the price each of the entire cost of the cows ; how much was 
 the cost of both ? Ans. 1125 dollars. 
 
 12. Smith has 108 acres of land, Johnson 4 times as much 
 and 35 acres, and AVade 3 times as much as both of them less 
 1200 acres ; how many acres in all have they ? ^^.i/^^c^o *^- 
 
 • Ik 
 
 Review Questions. What is the Proof in Addition? (40) In Sub- 
 traction? (45) In Multiplication? (51) In Division? (G2) AVliat is the 
 answer in MultiplicaticM called ? (47) In Division ? (57) 
 
 
5G 
 
 PUACTICAL ARITHMETIC. 
 
 GENERAL PRINCIPLES AND APPLI- 
 CATIONS. 
 
 66. Tlie Fundamental Operations or Processes of Arith- 
 inotic, or those upon which all others depend, are based upon 
 Notation, and are 
 
 ADDITION, SUBTRACTION, ]\IULTIPLICATIOX, 
 
 AND DIVISION. 
 
 67. The Signs used to indicate processes, or to abbreviate 
 expressions, are called 
 
 Symbols. 
 
 + , read plus, or added to. i = , read equals, or equal to. 
 - , read minus, or less. | . *., read therefore, hence. 
 
 X , read multii)lied by. j *. *, read since, because. 
 
 -=- , read divided by. ! ( ), parenthesis. 
 
 68. Numbers in a parenthesis, or under a vinculum, , 
 are to be regarded as all subject to the same operation. Thus, 
 
 lG-(3 X 2), denotes that the lyrodiict of 3 multiplied by 2 
 is to be subtracted from IG. 
 
 IG - (5 + 3) X 5, denotes 5 times the difference between 16 
 and the sum of 5 added to 3. 
 
 Exercises. 
 
 1. What is the value of (31 x 6) - 8G ? 
 
 Solution. 31 x 6 equals 186, and 186 - 86 equals 100. TJtcvc- 
 fore 31 multipUed by 6, in imrentliesis, less 86, equals 100. 
 
 2. (18 4- 6) + 13 - how many ? Ans. 16. 
 
 3. (G x 6) -f (4 X 3) - how many ] Ans. 3. 
 
 4. (8 + 2 X 5) - 20 = how many ? Ans. 30. 
 
 70) + 18 
 
 5 (ii!^ 
 
 12 
 
 how many ? 
 
 Am. 9. 
 
 6. (3 + 9) X (13 - 5 X 2) = how many? 
 
 Upon what are the Fuiulamental Operations of Arithmetic based? Name 
 thv^ni. What are Sj'mbols of Operation ? How are numbers in a parenthesis, 
 or under a vinculum, to be regarded ? 
 
GENERAL PRINCirLES. 
 
 .57 
 
 Formulas. 
 
 69. An Arithmetical Formula is an arithmetical expression 
 of a general rule. 
 
 70. The following formulas, which include the fundamental 
 operations of Arithmetic, follow from the preceding definitions, 
 princi])les, and illustrations : 
 
 1. The Suu = all the jyarts added. 
 
 2. The Difference = the Minuend - the Subtrahend. 
 
 3. The Minuend = the Subtrahend + the iJifference. 
 
 4. The Subtrahend =- the M'umicnd - the Difference. 
 
 5. The Product = the Multiplicand ;< the MultijjUer. 
 C. The Multiplicand = the Product - the Multiplier. 
 
 7. The Multiplier = the Product -^ the Midtiplicand. 
 
 8. The Quotient = the Dividend -r the Divisor. 
 
 9. The Dividend = the Quotient x the Divisor. 
 10. The Divisor = the Dividend ■- the Quotient 
 
 These ten Formulas, from their general nature and impor- 
 tance, may be regarded as Fundamental. 
 
 The ninth and tenth formulas are general, as will hereafter 
 appear ; but, when there is a Pmmaindcr to be considered, they 
 may for present applications be given thus : 
 
 11. The Dividend = (//ic integral part of the Quotient y, the 
 Divisor) x the Remainder; 
 
 12. The Divisor = (^/ic Dividend - the Remainder) -^ the in- 
 tegral part of the Quotient. 
 
 "What is au Arithmetical FonnuLi? To what is tlie sum equal? Tlie 
 Difference? Tlie Minuend? The Subtrahend ? The Product? The Multi- 
 plicand ? The Multiplier ? The Quotient? The Dividend? The Divisor? 
 "Which of the formulas may be regarded as fundamental? "Wlien there is a 
 Renjoiader to be regarded, how may the ninth and tenth formulas be given? 
 
58 
 
 rRACTICAL ARITHMETIC. 
 
 The sixth and seventh formiilcas furnish rehable methods of 
 proving Multiplication by Division. Thus, the multiplication 
 is proved, when, 
 
 13. The Product -f the Multiplier = the 3Iitltiplicand ; or, 
 
 14. The Product -f the Midtijjlicand = the MidtipUer. 
 
 Some Principles of Division. 
 
 71. Multijylying the dividend, or dividing the divisor, by any 
 number, multiplies the quotient by the same number. Thus, 
 
 16 
 
 16 
 
 f-" = 4;and^";" = 8,or-^ = 8. 
 
 4 4 4 -i- 
 
 72. Dividing the dividend, or multiplying the divisor, by any 
 number, divides the quotient by the same number. Thus, 
 
 18 = 4; and liil^ 
 4 4 
 
 2, or -i^- = 2. 
 ' 4x2 
 
 73. Dividing or midtiplying both the dividend and divisor by 
 the same number loill not change the quotient. Thus, 
 
 « = 4; and li±| = 4, or llx-| = 4. 
 
 4 ' 4-7-2 ' 4x2 
 
 KEVIEW EXERCISES. 
 
 1. If the items of a certain debt are 12 dollars, 106 dollars, 
 and 112 dollars, what is its entire sum 1 ^^ 3o 
 
 Solution. Since the sum is equal to all the parts added, if the items 
 of a certain debt are 12 dollars, 106 dollars, and 112 dollars, its entire 
 sum must be equal to 12 dollars + 106 dollars +112 dollars, or 230 
 dollars. 
 
 hi ^ < 
 
 Which formulas furnish a proof of Multiplication ? What is the first Prin- 
 ciple of Division ? The illustration ? The second Principle? The illustra- 
 tion ? The third Principle ? The illustration ? 
 
GENERAL PRINCIPLES. 
 
 50 
 
 2. The larger of two numbers is 1G85, and the smaller is 
 768 ; what is their difference] 9^ y 
 
 3. The difference between two numbers is HG and the less 
 is 407 ; what is the larger ? fp J 3 
 
 4. If 1406 be the difference between two numbers, and the 
 greater number is 4879, what is the less number ? - .^ ^/'/ ,^ 
 
 5. The two factors of a number are 27 and 36 ; what is the 
 number? Ans. 972. 
 
 6. The product of A's and B's ages is 3200 years, and B's 
 age is 50 years ; what is the age of A ? = ^ J^/UM^oA^ 
 
 7. The product of two numbers is 515522, and the larger 
 number is 1601 ; what is the smaller ? Am. 322. 
 
 8. If 168465 pounds of beef be divided equally between 
 11231 soldiers, how many pounds will each receive? 
 
 Ans. 15 pounds. 
 
 9. If a quantity of beef was divided equally between 11231 
 soldiers, and each received for his share 15 pounds, what quan 
 tity was divided % I Co?H-G 6 
 
 10. The product of two numbers is 890368, and the larger 
 number is 3478 ; what is the smaller ? Ans. 256. 
 
 11. 168465 pounds of beef having been equally divided 
 between a number of soldiers, each received as his share 15 
 pounds ; what was the number of soldiers % J J !j J/ 
 
 12. If the integral part of a quotient be 23, the remainder 
 692, and the dividend 39792, what is the divisor? 
 
 Ans. 1700. 
 
 13. John Reed says his number of sheep is such that if he 
 should separate the flock so as to put 115 in each of 5 fields, 
 there would be 4 sheep remaining. How many sheep has he ? 
 
 Ans. 579 sheep. 
 
 14. A farmer having 579 sheep, on putting an equal number 
 of them into each of five fields, had 4 remaining. How many 
 did he put into each of the fields ? -. / / A ' 
 
 V. 
 
 i r ;i, 
 
 i 
 i 
 
 fSfT 
 
 wm 
 
GO 
 
 PRACTICAL ARITIUIETIC. 
 
 ^ 
 
 ARITHMETICAL ANALYSIS. 
 
 74. The process of Golving (juestions ])y considering tlieir 
 elements, or l>y coni}).'iring their niimhers, and reasoning from 
 them to a required number, is called Anahjsls. Hence, 
 
 75. Arithmetical Analysis is the process of solving a 
 question by considering its elementary parts, and obtaining a 
 lequired result by a course of reasoning. 
 
 In analysing a simple question, the reasoning is usually from 
 a given number to a unit of the same name, and then from this 
 unit to the required number. 
 
 
 I! 
 
 Exercises. 
 
 1. If 7 acres of land produce 525 bushels of ^ wheat, how 
 many bushels will 11 acres produce ? ' (^ >( 6 //'y/u/j. 
 
 Analysis. If 7 acres of land produce 525 bushels of wheat, 1 acre 
 v'ill produce one seventh of 525 husliels, or 75 hufhcls, and 11 aa'es will 
 produce 11 tiiJies 75 bushels, or 825 bushels. Therefore, (&c. 
 
 2. If 11 acres of land produce 825 bushels .of corn, how 
 many bushels will 7 acres produce 1 ^ !). {) A^jA:\ t. 
 
 3. When 120 barrels of flour cost 1560 dollars, what will 
 139 barrels cost? Ans. 1807 dollars. 
 
 4. If 139 barrels of flour cost 1807 dollars, what will 120 
 barrels cost ? ^ / A '/./> 
 
 5. If a man can travel 2440 miles in 4 weeks, how many 
 miles can he travel in 9 weeks ? Ans. 5490 miles. 
 
 6. If 20 men can do a piece of work in 11 days, how many 
 
 days will it take 17 men to do it ? • - JO. /<^- 
 
 . ^) 
 
 Analysis. -7/20 men can do a piece of ivork m 11 days, it will take 
 
 1 man 20 times 11 days, or 220 days, to do it, and 17 men 1 seventeenth 
 
 of 220 days, or 12|^ days. Tlie^'efore, d'c. 
 
 What is Aritlimetieal Analysis ? 
 
 iM 
 
ARITiniFTrrCAL ANALYSIS. 
 
 CI 
 
 7. Wlion 13 men can hoe a field in 21 clays, in wliat time 
 can 39 men hoe it? .I^.s'. 7 davs. 
 
 8. If 10 cords of wood will pay for 57 yards of cloth at '1 
 dollars a yard, what is the wood a cord ? Anst. G dollars. 
 
 \). When 540 tons of coal will pay for 30 tons of iron, at IH) 
 dollars a ton, M'liat is the coal a ton? Ans. 5 dollars. 
 
 10. If 14 horses can be bought for 2240 dollars, how many 
 can be bought for 3040 dollars ? - /O /, ^ j^^ 
 
 Analysis. If 14 horses can he hov;/ht for 2240 dollars, 1 horse can 
 he hov.ijht for 1 fourteenth of 2240 dollars, or IGO dollars; and if 1 
 horse can he bought for IGO dollars, as many horses can he homjht f>r 
 3040 dollars as IGO dollars are contained times in 3040 dollars, or J!) 
 /torses. Therefore, djc. 
 
 11. If 19 horses can be bought for 3040 dollars, how many 
 can be bought for 2240 dollars? - />^/. /: y, ;^^^ 
 
 12. When 23 men earn 1380 dollars in a month, how many 
 men will earn 1980 dollars in the same time? Ans. 33 men. 
 
 13. If 31 horses will consume 93 tons of hay in a certain 
 time, how many horses will consume 87 tons in the same time? 
 
 Ans. 29 horses. 
 
 14. How many hogsheads of molasses can be bought for 
 17G70 dollars, if 16 hogsheads can be bought for 912 dollars? 
 
 Ans. 310 hogsheads. 
 
 15.* Daniel White bought an equal numlier of cows and 
 oxen for 2880 dollars, each cow at 45 dollars, and each ox at 75 
 dollars ; how many of each did he buy ? - ^Ji // / f^-xr ^ 
 
 Analysis. Since 1 coin cost 45 dollars and 1 ox 75 dollars, 1 cow 
 and 1 ox together were hought for 45 dollars plus 75 dollars, or 120 
 dollars ; and since the numher of each icas equal, he hought as many of 
 each as 120 dollars are contained times in 2880 dollars, or 24. There- 
 fore, (&C. 
 
 Review Questions. AVliich ave the Funtlanientiil Operations of Arith- 
 metic ? (66) What are called Symbols of Optratiou ? (67) AYliat is an Arith- 
 metical Formula ? (69) How many of them are called Fundamental ? (70) 
 
 * Optional. 
 
 ni, 
 
 M^ 
 
 
G2 
 
 rilACTICAL ARITHMETIC. 
 
 t< 
 
 I 
 
 IG.* I liave 11900 dollars in National currency, consisting of 
 an equal number of 5 dollar, 10 dollar, and 20 dollar bills ; how 
 many have I of each ? Ans. 340. 
 
 17. Frank Homer bought a farm for 11200 dollars, giving 
 CO dollars an acre for one half of it and 80 dollars an acre for 
 the other half; how many acres in each half? Ans. 80 acres. 
 
 18. A farmer sold an equal number of colts at 90 dollars 
 each, and horses at 175 dollars each, for 3145 dollars; how 
 much did he get for each kind ? 
 
 Ans. Colts, 1170 dollars; horses, 2275 dollars. 
 
 19. If A and B together have 1625 dollars, and B 825 
 dollars more than A, how much is the moi].ey of each 1 — - - — 
 
 -- '^bluTiON. If A and B together have 1625 dollars, and B 825 
 dollars more than A, 1625 dollars less 825 dollars, or 800 dollars, must 
 he twice A's money. 
 
 If 800 dollars he twice A*s money, one half of 800 dollars, or 400 
 dollars, is A^s money ; and, since B has 825 dollars more than A, B^s 
 money is 400 dollars 'plus 825 dollars, or 1225 dollars. 
 
 Therefore, <0c. 
 
 20. At a certain election A and B were candidates; the 
 whole number of votes cast for them was 5963, of which A had 
 321 more than B; Iioav many votes did each receive? 
 
 Ans. A 3142; B 2821. 
 
 21. Two men having met on a journey, found that they had 
 travelled 1200 miles, and that one had travelled 360 miles more 
 than the other ; what distance had each travelled 1 ,. - 
 
 22. Albert Smith gave his three sons 1350 dollars, Samuel 
 receiving 50 dollars more than Edmund, and Ernest 150 more 
 than Samuel; how much did each receive? 
 
 Ans. Edmund 366 § dollars, Samuel 41 6| dollars, and Ernest 
 5G6|- dollars. 
 
 Review Questions. How is the sum of two or more numbers found ? (37) 
 How is the diflference of two numbers found ? (42) 
 
 * This page is optional. 
 
 ■■■ 
 
UNITED STATES MONEY. 
 
 63 
 
 UNITED STATES MONEY. 
 
 76. United States Money is the legal currency of the United 
 States. 
 
 The name of its different orders, or denominations, begin- 
 ning with mills, is shown in the following 
 
 Table. 
 
 10 mills make 1 cent, marked c. 
 
 10 cents „ 1 dime, „ d. 
 
 10 dimes ,, 1 dollar, „ 
 
 10 dollars „ 1 eagle, ,, 
 
 8 
 E. 
 
 COINS. 
 
 77. Coins are pieces of metal converted into money by 
 legal stamping. 
 
 The Coins of the United States are of gold, silver, nickel, 
 and bronze, as follows : 
 
 I 
 
 ) i- <>, 
 
 GOLD. 
 
 
 
 SILVER. 
 
 Double eagle, 
 
 value $20 
 
 Dollar, 
 
 value 81. 
 
 Eagle, 
 
 >j 
 
 10 
 
 Half-dollar, 
 
 „ 50 c. 
 
 Half- eagle, 
 
 M 
 
 5 
 
 Quarter-dollar, 
 
 „ 25 c. 
 
 Three dollar, 
 
 >) 
 
 3 
 
 Dime, 
 
 „ 10 c. 
 
 Quarter-eagle, 
 
 )> 
 
 H 
 
 Half-dime, 
 
 „ 5 c. 
 
 Dollar, 
 
 )) 
 
 1 
 
 Three-cent, 
 
 „ 3 c. 
 
 NICKEL. 
 
 Three-cent, value 3 c. | Five-cent, value 5 c. 
 
 t ri 
 
 i ; 
 
 i 
 
 ■- 
 
 BRONZE. 
 
 Two-cent, value 2 c. | Cent, value 1 c. 
 
 What is United States Money? Repeat the Table, "\Miat are Coins? 
 Name the gold coins. The silver coins. The nickel coins. The bronze. 
 
 MMai 
 
Ci 
 
 niACTICAL ARITHMKTIC. 
 
 i 
 
 Tlic St'tnchtrd o{ ^'old cdiii of tlu; Uiiitcil Stutcs i.s jiarts pure 
 iii('l;il, and 1 ])iiit silver and copper; and (»!' silvi-r cioin, is !> paits 
 yww nil till, and 1 part copper. 
 
 The hronzi; cdin is !)5 parts copj)er, and 5 ])arts tin and /inc. 
 
 Tim (lolliir mark, §, may he considered as the letter U written ujion 
 an S, denoting,' U.S., the initials c*' United States. 
 
 NOTATIOX AND NUMERATION. 
 
 78. The Dollar is tho unit of United States Money. 
 
 Ill accounts, eagles fyo written as icm^ of dollars, and indi- 
 cated by the tlollar mark (^) ; and tiiey are usually read as a 
 number of dollars. Thus, 
 
 5 eagles arc written 8^)0, and r'^ad fifty dollars. 
 
 Dimes are written as tenths, cents as hundredths^ and mills as 
 thoumndths of a dollar, and ei)arate(l from dollars by tho 
 decimal ])oint (.). Dimes are nsuall} read as a number of 
 cents, and mills are sometin)es read as parts of a cent. Thus, 
 
 3 eagles, 3 dc liars., 3 dimes, 3 cents, and 3 mills are written, 
 
 0-:3,333, 
 
 and read thirty-three dollars, [Idrty-tluee cents, and three mills; 
 and, 
 
 4 eagles, 2 dimes, and 5 mills may be written, 
 
 840.20^, 
 and read forty dollars, twenty and one half cents. 
 
 [ 
 
 What is tlie stiUidard of gold coin ? Of silver coin ? Of what is the uronze 
 coin comi)osed? What may the dt)llar inaik Vo considered'; Whivt is the 
 unit of United States Money? How are eagles writt'^:; and usually road ? 
 How are dimes, cents, and mills written How are dimes usually read ? Mills 
 sometimes ? ' 
 
 im^i 
 
UNITED STATES MONEY. 
 
 Cf) 
 
 79. In Uniteil States Money, 10 of a lower denomination 
 make 1 of tlio next liiglier, according to tlie Decimal Syjitem 
 (Alt. :)-2). Hence, 
 
 United States Money mnij he written, tided, subtracted, nudti- 
 ;plicd, and divided, hj imcedinrj rules. 
 
 
 4. $21.00 
 
 5. $37.70 
 G. $93,703 
 
 7. 8108.12^ 
 
 8. $1813.19 
 
 9. $35,005 
 
 Ans. $42.18. 
 
 Exercises. 
 
 Copy and read : 
 
 1. $0.15 
 
 2. $9.87 
 
 3. $8,025 
 
 Write in figures : 
 
 10. Forty-two dollars, eighteen cents. 
 
 11. Thirteen dollars, twelve and a half cents. 
 
 A71S. $13.12J. 
 
 12. Seventy-three dollars, eleven cents. ^ji^H' / d' U 
 
 13. Sixty-six dollars, six dimes. Ans. $00.00. 
 
 14. Five eagles, five cents. Ans. $50.05. 
 
 15. One hundred nine dollars, nine cents. >7 /o.)'^//^^'/7y 
 
 16. Six hundred dollars, ten cents, five mills. 
 
 Am. $000,105. 
 
 17. One thousand three hundred and fifty dollars. // 1 ;i{j t ' 
 
 18. Five hundred dollars, eighty-seven cents, five mills. ,>!^/^'/'^/»'^y 
 
 EEDUCTION. 
 
 80. Reduction is the process of changing one number into 
 another of a different denomination, but of equal value. 
 
 The reduction is called descending, when the change is into 
 a number of a lower denomination; and is called ascending, 
 when the change is into a number of a higher denomination. 
 
 How may United States Money be -written, added, &c.? "What is lie- 
 duction ? 
 
 E 
 
60 
 
 PRACTICAL ARITUMETIC. 
 
 II ' 
 
 s 
 
 CASE I. 
 
 81. To reduce a number to another of a lower denomina- 
 tion. 
 
 1. In 59 cents how many mills ? 
 
 Solution. Si7ice in 1 cejit there arc 10 mills, in 59 cents there must 
 he 51) tunes 10 mills, or 590 mills. 
 
 2. In 8 dollars how many cents? 
 
 Solution. Shice in 1 dollar there are 100 cents, in 8 dollars there 
 imi6t oe 8 times 100 cents, or 800 cents. 
 
 3. In 8 dollars liow many mills ? 
 
 Solution. Since in 1 dollar there are 100 cf?2?s a)i(Z in each cent 10 
 17H7/5, t/icre must he in 1 dollar 100 ^i/«c5 10 7?u7/i.', or iOOO ?/u7^s, ontZ 
 I?. '' dollars 8 /mcs 1000 mills, or 8000 ??u7/s. 
 
 KuLE. To reduce cents to mills, mullijdij hjj 10 ; dollars to 
 cents, mnltiphj by 100 ; oi' dollars to mills, riiultij'^y hy 1000. 
 
 When there is no decimal point in the nuniher, the multiplication 
 may he ]>c rformed hy anne^dng ciphers {Ai*'.. 54), omitting the dollar 
 mark, \i nsed. 
 
 Wlien there is a decimal point in the numhor, dollars and cents may 
 he reduced to cents, and dollars, cents, and mills to mills, hy simply 
 removing the dollar mark and the decimal point. For, multi|'lying hy 
 10, 100, &c., has tlie same efTeci as removing the decimal point ix>: many 
 jilaces to the right as there are ciphers in the multiplier (Art 3(.^), 
 
 Examples. 
 
 4. In $102 how many mills? 
 
 5, In J; 1.0 2 how many cents? 
 G. In ^ol how many cents? 
 
 7. In $0,008 how many mills ? 
 
 8. In 87 cents how many mills? 
 
 9. In $;^0.03 hov/ many mills ? 
 10. Ttoduce $100.90 to cents. 
 
 Ans. 102000 mills. 
 Ans. 102 cents. 
 
 Am. 0008 mills. 
 Ans. 870 mills. 
 
 Ans. 10090 cents. 
 
 IIow many mills make 1 cent? How many cents make 1 dollar? How 
 many n-.ills make 1 dollar ? Repeat tlie llulo. "When there is no decimal 
 point in the number, how may the multiidicatiou be performed ? How when 
 there is a decimal point in the nurabor ? 
 
UNITED STATES MONEY. 
 
 67 
 
 I 
 
 I 
 
 si 
 
 CASE 11. 
 
 82. To reduce a number to another of a higher denomi- 
 nation. 
 
 1. In 590 mills how nicany cents? 
 
 Solution. Since in 10 milh there is I cent, there must he in 590 
 mills as many cents as 10 mills are contained times in 590 mills, or 59 
 cents. 
 
 2. In 800 cents how many dollars ? 
 
 Solution. Since in 100 cents there is 1 dollar, in 800 cents there 
 are as many dollars as 100 cents are contained times in 800 cents, 
 or $8. 
 
 3. In 8000 mills liow many dollars ? 
 
 Solution. Since in 1000 mills there is 1 dollar, in 8000 mills there 
 are as many dollars as 1000 mills are contained times in 8000 mills, 
 or $8. 
 
 ItULE. To reduce inills to cents, divide by \0 ; cents to dol- 
 lars, divide by 100 ; or mills to dollars, divide by 1000. 
 
 Cents or mills may be reduced directly to dollars by placing the 
 decimal point as numy places to the left as there are cipliers in the 
 divisor (Art. 04). 
 
 Examples. 
 
 4. In G91000 mills how many dollars? 
 
 5. In 102 cents how many dollars? 
 G. In 3100 cents how many dollars? 
 
 7. In 970 mills how many cents ? 
 
 8. In 3G030 mills how many dollars ? 
 
 9. Eeduce 875 mills to cents. 
 
 10. Reduce 1G090 cents to dollars. 
 
 11. Keduce 18734 mills to dollars. 
 
 12. Reduce 18734 mills to cents. 
 
 Ans. $691. 
 
 Ans. $31. 
 Ans. 97 cents. 
 
 Ans, 87^ cents. 
 Ans. 81 GO. 90. 
 
 Ans. 1873/o- cents. 
 
 Repeat the Rule. How may cents or milla bo reduced directly to dol« 
 lass? 
 
 ii 
 
 't 
 
 ft 
 
 \\ 
 
 ff 
 
 <mm 
 
G8 
 
 PRACTICAL ARITHMETIC. 
 
 ADDITION. 
 83. 1. Add $21.38, $1.82, and $31,015. 
 
 OPERATION. 
 
 $21.38 
 
 1.82 
 31.015 
 
 For convenience we Avrite the numbers so that 
 figures of the Scame order stand in the same column, 
 and begin at the right to add, observing to place 
 the decimal point between dollars and cents. 
 
 $54,215 
 
 2. Add $1G4.00, $5G1.45, $902.03, $31.01, and $.95. 
 
 Ans. $1GG0.04. 
 
 3. Add $80,375, $103.1G, $1500.81, and $100.87^. ■ ^'-j^^ 
 
 4. Find the sum of $3004.05, $10091.03, $00i, and 
 111.621 Ans. $13808.301 
 
 5. $809 + $310.05 + $1701.12 + $10003.14 = how many ?a; v ,^3..' 
 0. $10800.171- + $9850.37 + $0345 + $910032 + $82.12.} 
 
 how much ? Ans. $943115.07. 
 
 84. 
 
 SUBTRACTION. 
 
 1. Subtract $109,021 from %dQ3.03. ''}-//:^'^/^ ^ 
 
 OPERATION. 
 
 $903,030 
 109.025 
 
 For convenience wc write the subtrahend under 
 the minuend, so that figures of the same order 
 stand in the same column, and ])egin at the riglit 
 to sul)traot, ol)serving to place the decimal point 
 between dollars and cents. 
 
 Here, in tlie subtrahend wc write the h cent as 5 
 
 $854,005 
 
 mills, and in the minuend note the place of mills by 0. 
 
 Ans. $715995. 
 
 2. From $90,175 take $19.18. 
 
 3. From $40.95 take $31,008. 
 
 4. From $130 take $1.03. Aiis. $134.57. 
 
 5. From one thousand dollars take three cents and one mill. 
 
 Ans. $999,909. 
 
 How do you write tlio numbers to be addod in Addition of United States 
 Money ? Where do we begin to add ? "Where do wo place the decimal 
 point? How are the numbers written for subtracting? Where do we 
 begin to subtract ? How is the h cent written ? Why is an annexed to 
 the miuucud? 
 
 'I 
 
 9 
 
 J 
 
 ^1 
 
 m 
 
UNITED STATES MONEY. 
 
 69 
 
 6. From 13000 dollars, 93 cents, and 4 mills, take 1300 dol- 
 lars, 9 cents, ami 5 mills. Ans. $11700.839. 
 
 MULTirLICATIOX. 
 85. 1. Multiply $513,125 by 7. 
 
 OPERATION. 
 
 §513.125 
 7 
 
 We begin at the right to niiiltiply, observing 
 to place the decimal point between dollars and 
 (ti "^ •■" 1 87 ^ cents in the product. 
 
 2. Multiply $30G.07 by 13. 
 
 3. Multiply $305,155 by 15. 
 
 4. Multiply $97.38 by 19. 
 
 5. Multiply $39.07 by 100. 
 0. Multiply 10 dollars 25 cents by 27. 
 7. Multiply one hundred thirteen dollars, five cents, and five 
 
 mills, by one hundred and five. Ans. $11870.775. 
 
 Ans. $3978.91. 
 Ans. $5477.325. 
 
 Ans. $3907. 
 
 
 I 
 
 i 
 
 DIVISION. 
 86. 1. Divide 053 dollars by 4. 
 
 4) $053 
 
 $103} 
 
 OrERATIONS. 
 
 or, 
 
 4 ) $053.00 
 
 03.25 
 
 Dividing the dollars, we have a remainder, \vhich we write as a 
 part of a dollar, in the first operation. 
 
 In the second oi)('ration, the division is extended to cents by su])- 
 plying the two ph. > .^ of cents by ciphers in the dividexid. The two 
 answers, although a little dill'erent in form, express the .same value, 
 since \ of a dollar and 25 cents are ecpial. 
 
 In like manner, if it had been recpured, the division could have 
 been extended to nulls, by supplying an additional cii)her. 
 
 How do we begin to multiply? "Where do we place the deciniul point 
 ill the result ? "When there is a remainder, after dividing dollars, how may 
 the division be extended to cents? To mills ? 
 
70 
 
 PRACTICAL ARITHMETIC. 
 
 2. Divide $254.04 by 12. 
 
 3. Divide $562 by 5. 
 
 4. Divide $181,125 by 9. 
 
 5. Divide $07.44 by 1G24. 
 
 6. Divide $61,205 by 8. 
 
 Ans. $21.17. 
 Ans. $112.40. 
 
 ' Ans. $.0G. 
 Ans. $7.650§. 
 
 Here, there is a remainder after dividing mills, wliicli is expressed 
 in the answer ; but in such cases it is customary to omit the re- 
 mainder, and to add + , to indicate that the division is not exact, 
 thus, $7,650 +. 
 
 7. Divide one thousand dollars and sixty cents by one hun- 
 dred and ninety- nine. -'.}; -^'tt £ ^K'rj 
 
 87. When divisor and dividend both express sums of money, 
 but of different denominations, they must be brought to the 
 isame denomination before dividins;. 
 
 8. How many times is $1.25 contained in $40. 
 
 OPERATION. 
 
 125 ) 4000 ( 32 
 ' 375 
 
 250 
 250 
 
 In $1.25 there are 125 cents ; in $40 there 
 are 4000 cents. 
 
 125 cents are contained in 4000 cents 32 
 times. 
 
 Ans. 3816. 
 
 9. Divide $954 by 25 cents. 
 
 10. Divide $141 by 75 cents. y^^^,^ y // 
 
 11. Divide $444 by 370 mills. Ans. 1200. 
 
 12. How many times seven mills in twenty-nine dollars? 
 
 Ans. 4142^. 
 
 13. How many times twelve cents and five mills in three 
 hundretl and seventy dollars 1 Ans. 2960. 
 
 14. How many times one dollar and twenty-five cents in 
 thirteen eagles seven dollars and five dimes? Ans. 110. 
 
 What is customary when there is a remainder after dividing mills ? What 
 must bo done before dividing when the divisor and dividend are suras of 
 money, but of different deuoniinatious ? To what must dollars be reduced 
 iu order to divide them by cents ? 
 
 i 
 
75.6 
 
 UNITED STATES MONEY. 
 
 APPLICATIONS. 
 
 71 
 
 ^ 
 
 1. Bought of C. Washbunic, flour for $13.25, sugar $3.75, 
 tea $9.27, pepper $.17, starch $.121., ^^j^j kerosene $1.87.V; 
 ■what was the whole amount 1 Ans. $28.41-. 
 
 2. Sold a horse for $300, a carriage for $375.50, and a 
 saddle for $15.75 ; how much was received for the whole '^//y/lJ^' 
 
 3. Bought goods for $0G35, but the same receiving injury, I 
 was content to sell them at a loss of $3G7.87.\; what did I 
 receive for them 1 Ans. $9207.1 2, \. 
 
 4. Nathan Soule bought a house for $G1G7, and sold it for 
 $5375.75 ; how much was his loss l// ^ J I ' X 6' 
 
 5. I paid for a horse $375, for a yoke of oxen $2G3, for a 
 cow $75.50, for a yoke $7.37-^-, and sold the whole at a profit 
 of $13.12^ ; how much did I get for them % Ans. $734. 
 
 G. A young lady went " a shopping." She purchased a silk 
 dress for $43, some velvet for $9.75, a shawl for $25, a bonnet 
 for $19.87, and some pins for $.15. If she started with a 
 hundred-dollar bill, how much change should she bring back 1 
 
 Ans. $2.23. 
 
 7. If a barrel of flour is worth $13. G5, how much are 110 
 barrels worth ? A ns. $1501. 5 0. 
 
 8. When flour is $13.G5 a barrel, how many barrels can be 
 bought for $1501.50 ? ///' 4 . ' - / » . 
 
 9. When coal is $4.25 a ton, what will 1000 tons cost? 
 
 Ans. $4250. 
 
 10. When 1000 tons of coal can be bought for $4250, what 
 is the price of one ton '^- jJ-, I ' ^ K 
 
 11. If a farm of 47 acres can be bought for $1774.25, what 
 is the price of one acre ? '/' j /^ 
 
 12. What cost 31G bushels of wheat at $1.63 a bushel ? 
 
 Ans. $515.08. 
 
 13. What cost 316 bales of cotton, at $260.50 a bale '^^^f !l:iJf* 
 
 Review Questions. "What is a Rule? (11) AVliat is an Arithmetical 
 Furmula? (G9) Arithmetical Ainilysia ? (74) 
 
72 
 
 PRACTICAL ARITHMETIC. 
 
 14. If 519 huslicls of potatoes cost $194,025, what will one 
 bushel cost ? Ans. $.37^. 
 
 15. Bought 65 yards of cloth at 27 cents a yard, and 15 
 yards more at $0.50 a yard ; what did the whole amount to*? 
 
 Ans. $115.05. 
 
 16. By receiving $3.25 a day, and paying out for expenses 
 50 cents a day, in how many days will a man earn $825 1 
 
 Ans. 300 days. 
 
 17. If 200 pounds of pork cost $25, how much is it a 
 pound 1 / // ^ r* . 
 
 18. At $.125 a pound, how many pounds of pork can he 
 bought for $25 1 — X CC //{j. 
 
 19. When 5 bales of cotton, weighing each 312 pounds, can 
 be bought for $491,40, what is the price a pound ? Ans. $.315. 
 
 20. Tobey Wasteful spends each day for beer 15 cents, for 
 cigars 12j cents, and for oysters 25 cents; at that rate, how 
 much does he spend in 305 days? Ans. $191. 02J. 
 
 EXCHANGE OF COMMODITIES. 
 
 88. Exchange of Commodities is trade or traffic, by passing 
 goods or wares from one party to another for an equivalent, 
 in goods or money. 
 
 Exercises. 
 
 1. If I bought an arithmetic for 03 cents, a reader for 
 $1.25, a slate for 38 cents, a globe for $15.50, and gave in 
 payment a ten-dollar and two five-dollar bills, how much 
 change should I receive ? tl '!) * J^ Jj- 
 
 Solution. Ij I hoiight an arithmetic for G3 cents, a reader for 
 $1.25, a slate for 38 cents, and a globe for $15.50, / bought in amount, 
 $.(53 + $1.25 + $.38 + $15.50, or $17.70. 
 
 If I gave in 'payment a ten-dollar and two five-dollar bills, I gave 
 $10 4- $5 + $5, or $20 ; and should receive in change the difference 
 between '?.<) and $17.70, or $2.24. Therefore, (&c. 
 
 What is Exchange of Commodities? 
 
 '-^n 
 
UNITED STATES MONEY. 
 
 73 
 
 2. A farmer buys a bill of goods amounting to 8-35.25, and 
 pays down $37.50, and agrees to furnish wheat for the balance 
 at the rate of 81-75 a bushel ; how many bushels must he fur- 
 nish ? A US. 113 bushels. 
 
 3. I have 12 cords of wood, worth $8 a cord, and 17 hundred 
 rails, worth $G a hundred ; if I should exchange them for a 
 carriage, worth $200, how much should I gain by the opera- 
 
 tion? ^Jl^tl/) 
 
 4. Joseph Bryant exchanged 150 spelling-books at 25 cents 
 each, for arithmetics at 50 cents each ; how many arithmetics 
 did he receive? Ans. 75 arithmetics. 
 
 5. If I give GOO pairs of shoes, at $1.25 a pair, for 100 pairs 
 of boots, what are the boots a pair ? •a( '/'(f/'^ 
 
 G. Joseph Howland had 300 tons of coal, at $7.25 a ton, fur 
 which Frank Dunmore gave $1515 in cash, and the balance in 
 wood at $4.40 per cord; how many cords were required ? 
 
 Ans. 150 cords. 
 
 7. If I should exchange 50 bushels of corn, at$.G5 a bushel, 
 for 120 yards of muslin at 15 cents a yard, how much balance 
 would there be in my favour? Aui^. $14.50. 
 
 8. IIow many pounds of butter, at $.28 a pound, must be^ / 
 given for 14 yards of gingham at 32 cents a yard? ~r- I ^.^'/tj 
 
 9. How many barrels of flour, at $9.50 a barrel, can be ex- 
 changed for 475 pounds of cotton at 30 cents a pound, and 7G 
 gallons of molasses at 50 cents a gallon ? Ans. 19 barrels. 
 
 i; 
 
 BILLS AND INVOICES. 
 
 89. A Bill is a written statement of merchandise bought or 
 sold, or of services rendered. 
 
 90. An Invoice is a bill of merchandise sent by the seller to 
 the purchaser. 
 
 Review Questions. What is United States Money ? (76) Coins ? (77) 
 The Unit of United States Money? (70) 
 What is a Bill ? An Invoice ? 
 
 ' ' 
 
74 
 
 PRACTICAL ARITHMETIC. 
 
 To make out a bill or invoice, we find the cost of Ccach of 
 the items, and the amount of the whole. 
 
 91. A bill is receipted when its payment is acknowledged 
 in writing, by the party in Avliose favour it is, or by some one 
 authorised to sign for him. 
 
 The first bill m the following Exercises is receipted by A. 
 C. Lombard ^ Co., and the third bill, by Robert T. Gould, by 
 George Boyd, who is authorised to sign for Gould. 
 
 Exercises. 
 
 Reckon and find the amount of the following bills, or 
 invoices : — 
 
 (1.) 
 
 w/KW 9^oi/f, U6af^ /^ /cS'^^^ 
 
 ^00 /if. ^a, a S .60 < ' ^ '^i^O'OC 
 
 /SO . ,%.ai, . ./^ • • ' • }lh^^ 
 
 6'o ^ ^^o 
 
 ^'SJS na^. ^(:o/iJJedj „ 
 
 /O ^/ &'r/oui. 
 
 ,4^ 
 
 .(fo 
 
 /^.60 
 
 %<3C0.40 
 
 Mifcei'vea c^ui/men/. 
 
 kS^. ^. ^amfaic/^ ^o. 
 
 How do we make out a Bill or Invoice? When is a bill receipted ? "WHiat 
 does the "a " in the bill mean? Ans. At. 
 
UNITED STATES MONEY. 
 
 (D 
 
 -ch of 
 
 i Olio 
 
 '7 A. 
 
 or 
 
 (oa. 
 
 
 it 
 
 ! 
 
 (2.) 
 
 (J/itcrrnfC, c/ctff/ 5, /SCO. 
 
 SCO ^ tl^/ica/, ,» 
 
 230 . ^rt/j, m 
 
 /SO ^ ^ai/^y, 
 SO ^/(^ 2^utu}, ecc/ia, , 
 
 .... Jbo-oo 
 .40 I to ■ r t 
 
 .d'o -- (Jo ■ /' /• 
 
 
 ^ eccivec/ <^ay.mcn/^ 
 
 (3.) 
 
 ^■ 
 
 -^//w yu.c. 
 
 ^2S c/ay,' lOoJ. on ce/'^ui. a %4.2S j f (fj ' y. ^j' 
 
 <S0 , ouot/j on -wactj M 
 
 ., QQ <. WUX o^' ^^/*/i>.en/cce, 
 „ 3000 <§^i<c^ 
 „ /S /ond %y/oftej * 
 
 % 
 
 4.35 lVj,^'0 
 
 /o.co lt>0-O0 
 
 WT^ 
 
 J ^ 
 
 / 6y,' / 
 
 ^o<fei/^y^ou/a\ 
 cO>/ ^coiye cyJo//a. 
 
 How ia the 3a bill receipted? What docs the " Dr." mean ? Ans. Debtor. 
 
 J 
 
76 
 
 PRACTICAL AHITIIMETIC, 
 
 ii 
 
 ACCOUNTS. 
 
 92. All Account is a written stcatement of debt and credit 
 between two parties. 
 
 The party owing is the Debtor, and the party owed is the 
 Creditor. 
 
 93. In tlie settk^ment of an account, it is required to find 
 th(! diffenince due, or balance. 
 
 94. A I)UK-1>1LL is a written acknowledgment that a cer- 
 tain amount is due, and is often given in making settlements, 
 when it is not convenient to make immediate payment. 
 
 
 Exercises. 
 
 Reckon and find the balance of the folio wim; accounts : — 
 
 (1.) 
 
 Philadelphia, Dec. 12, 18GG. 
 Mr George W. Grimshaw, 
 
 To C' imings, Simons, & Co., Dr. 
 
 June 10. For 31 yd. Mishn, a $.15 
 
 „ 13. „ 20 ., Flannel, „ .50 
 
 Aug. 15. „ IG „ Broadcloth, „ 5.50 
 
 Oct. 31. „ 33 „ Gingham, „ .45 $117.50 
 
 Cr. 
 Sept. 25. By Cash, 8100.00 
 
 „ 17. „ Merchandise, 13.50 113.50 
 
 $4.00 
 
 Balance due C. S. & Co., 
 
 Keceived Payment, 
 
 Cummings, Simons, &, Co. 
 
 2. Baltimore, Nov. 16, 186G. James McClintock owed An- 
 drew Saulsbury for 110 bu. of corn, at 75 cents a buslel, 
 
 What is an Account ? WJiat party is the Debtor ? The Creditor ? What 
 is required in the settlement of an account ? What is a Due-Bill ? 
 
UNITED STATES MONEY. 
 
 < t 
 
 bouglit Oct. 1 ; 3 bbl. of flour, at $7.50 a l)arrel, bouglit Oct. 
 7 ; ami G2 bu. of oats, at 43 cents a l)usliol, btmglit Nov. fy ; 
 and Mr Saulsl)ury owed him f(jr G thousand of extra shingles, 
 at .SG a thousand, delivered Oct. 5 ; for cash i?<JO, paid Nov. 1 ; 
 for bill of labour, amounting to .$8.6G, rendered Nov. 10. The 
 account was settled Nov. IG by giving due-bill, amounting to 
 827, for the balance duo A. S. Make out the account, ami 
 settle it in your own name for Andrew Saulsbury. 
 
 LEDGER COLUMNS. 
 
 95. A Ledger is the book in which merchants enter a sum- 
 mary of accounts. 
 
 The items thus brought together often make long columns. 
 
 Accountants, by practice, are enabled to add rapit-lly the 
 numljers of tw(j or more columns at one operation. 
 
 
 Exercises. 
 
 
 (1-) 
 
 (2.) 
 
 (3.) 
 
 (^.) 
 
 .S14.5G 
 
 $321.00 
 
 $264.10 
 
 .$310.00 
 
 31.00 
 
 123.54 
 
 100.78 
 
 152.70 
 
 78.00 
 
 70.25 
 
 88.90 
 
 31. G2 
 
 19.15 
 
 34.80 
 
 142.33 
 
 79.08 
 
 2G.05 
 
 11.19 
 
 6G.08 
 
 10.56 
 
 17.04 
 
 13.33 
 
 72.39 
 
 180.61 
 
 3.36 
 
 55.44 
 
 67.02 
 
 282.39 
 
 13.54 
 
 72.69 
 
 30.50 
 
 67.31 
 
 14.38 
 
 2.18 
 
 50.22 
 
 55.25 
 
 21.00 
 
 13.44 
 
 17.82 
 
 178.90 
 
 6.47 
 
 7.56 
 
 11.12 
 
 300.00 
 
 2.03 
 
 9.00 
 
 44.5G 
 
 65.88 
 
 1.12 
 
 10.13 
 
 14.23 
 
 13.44 
 
 $247.70 
 
 /M-6>' 
 
 ^m^s 
 
 #;^;//^ 
 
 What is a Lodger ; How are accountants enabled to add rapidly ; 
 
li 
 
 II' 
 
 a. 
 
 7S 
 
 PRACTICAL ARITnMETIC. 
 
 
 }h)To, with Exercise 1, wc may illustrate the manner of addiii;^ 
 tlic liiiiiilxTs of two coliiiiiiia at mio o])(>rati(>n, tliu^ : takiiii,' fiivi tens 
 and tlu-ii units, 12 i)lus 3 = 15, ].lus 40 - n"), plus 7 = (52, t>Ius 
 30 = 02, plus 8 = 100, ])lus no = loO, plus 4 = 1.04, jHus 30 ^ 184, 
 I)lu.s () = 190, i)lus 4 = 104, plus 5 = 11)0, jilus 10 = 200, plus 5 = 
 214, plus no = 2(!4, ])lus (\ -= 270 cents, or $2.70 ; wo write the 7o 
 cents, and add tin; $2 with the dollars. 
 
 2 plus 1 = 3, i)lus 2 = .'), plus 0=11, ]dus 20 = 31, ])lu3 1 = 32„ 
 plus 10 = 42, ])lus 4 = 4(5, ]dus 10 = 50, jdus 3 = GO, plus 3 = G2 
 ]>lus 10 = 72, 1.1 us 7 = 70. plus 20 = 90, plus G = 105, plus 10 = 
 11.5, plus = 124, plus 70 = 104, plus 8 = 202, jdus 30 = 232, plus 
 1 = 2:}3, ])lus 10 = 243, plus 4 = 247 dollars, which we write. An- 
 swer, $247.70. 
 
 The process may he ahridp^ed hy simply namin[^ results : 15 ; 55, 
 02; 02, 100; 150, 154; 184, 190; 194 ; 199 ; 209,214; 2G4, 270; 
 we write the 70. 
 
 3 ; 5 ; 11 ; 31, 32 ; 42, 4G ; 5G, 59 ; G2 ; 72, 79 ; 99, 105 ; 115, 
 124; 194, 202; 232, 233; 243, 247; we write the 247. Answer, 
 |i247.70. 
 
 FACTORING. 
 
 96. An Integer, or Integral Number, is a number that con- 
 tains the unit 1 an exact number of times. 
 
 97. An Exact Divisor, or Measure, of a number, is any 
 number that gives an integer for a quotient. Thus, 
 
 2, 4, and 8 arc exact divisors of 16. 
 
 98. An Integral Factor of a number is any integer which 
 is an exact divisor of the number. 
 
 99. A Prime Number is a number that has no integral 
 factor, except itself and 1. Thus, 
 
 1, 2, 3, 5, 7, and 11 are prime numbers. 
 
 -100. A Composite Number is a number that has other in- 
 tegral factors besides itself and 1. Thus, 
 
 4, 6, 8, 9, 10, and 12 are composite numbers. 
 
 Illustrate, by Exorcise 1, the process of adding two columns at one opera- 
 tion. What is an Integer? An Exact Dinsor? A Factor? A Prime 
 Number? A Composite Number ? 
 
FACTORING. 
 
 79 
 
 101. A Prime Factor is a factor that is a primo number. 
 
 102. A Composito Number is equal to tho product of all 
 its prime factors. Thus, 
 
 8 = 2x2x2, and 12 = 2x2x3. 
 
 103. Tlu! number of times a number is taken as a factor, is 
 sometimes denoted by a small figure, called an exponent, written 
 at the right of the figures expressing the factor, and above the 
 line. Thus, 
 
 2« = 2 X 2 X 2, and 18 = 3' x 2. 
 
 104. Two or more numl)ers are said to bo prime vrith rcsjiect 
 to each other, when they have no common integral factor except 
 1. Thus, 
 
 4 and 1) are prime with respect to each other. 
 
 105. Factoring is the process of iinding the factors of a 
 composite number. 
 
 EXACT DIVISOIiS. 
 
 106. Two is an exact divisor of every even number, but of 
 no odd number. Thus, 
 
 2 is an exact divisor of 2, 4, G, &c., but not of 3, 5, 7, &c. 
 
 107. Three is an exact divisor of any number, wlien it is 
 
 an exact divisor (.^. the sum of the units expressed by its 
 
 figures. Thus, 
 
 ?. In .in {^act divisor of 54G. 
 
 108. Four is nr ex.ici di v^isor of any number, when it is an 
 exact divisoi o^'' t-- tf»n^; ;,' i units. Thus, 
 
 4 ia ui> -".vact divisor of 572 ^ 1928. 
 
 109. Five is an exact divisor of any number whose unit 
 figure is or 0. Thus, 
 
 5 is an exact divisor of 15, 20, 25, 30. 
 
 A Prime Factor? To what in .a composite number eciual? "Wlien are two 
 or more numberH said to bo i)riMio to eacli other? "What is factoriug? Of 
 what numbers is two au exact o-vijior ? Three ? Four? Five ? 
 
 ■JL^. 
 

 80 
 
 PRACTICAL ARITHMETIC. 
 
 
 110. Six is an exact divisor of any even number of wlilcli 3 
 is an exact divisor. Thus, 
 
 G is an exact divisor of 12, 18, 21, 30. 
 
 111. Nine is an exact divisor of any number, wlicn it is an 
 exact divisor of the sum of the units expressed by its figures. 
 Thus, . • 
 
 9 is an exact divisor of 7542, as 9 is an exact divisor of 7 + 
 5 + 4 + 2-18. 
 
 rPJME NUMBERS. 
 
 il2. No direct method of detecting prime numbers has 
 been discovered. 
 
 After 2, there can be no even number prime, since 2 is an 
 exact divisor of every even number (Art. lOG). 
 
 In a series of odd numbers beginning with 1, it has been 
 found by trial that every third number after the i)rime 3 lias 
 3 as a factor, every JiJ'th number after the prime 5 has 5 as a 
 factor, and so on. 
 
 113, Hence, we ^ave a practical method of detecting prime 
 iuind)ers by sifting out those that are not prime, as follows : 
 
 JTrite the odd numbers fiom 1 to any desirable limit. 
 
 Bcfjin ivith the Jirst 2>rime number after 2, which is 3, and mark 
 every third number after the 3 by icritiwj 3 over it, every fifth 
 nwnher after the 5 by ivritinrj 5 over it, every seventh nwnber after 
 the 7 by uritiny 7 ov'r it, and so on. 
 
 'Then all that remain are imme numbers; and those marked 
 arc composite, v:ith factors over them. Thus, 
 
 Q OK 3 7 5 
 
 1, 3, 6, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 
 
 3 3,11 5,7 3,13 3,5 7 3,17 
 
 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 
 
 511319 37613 3 "3 8 5 
 
 53, 55, 57, 59, 61, 03, 05, 07, 09, 71, 73, 75, &c. 
 
 Six? Nine? "What is said with regard to detecting prime niimbera? 
 What i.s the only even prime uumbcr? dive the practical method of detect- 
 ing prime numbers. 
 
 I 
 
 ^mmm 
 
 
FACTORING. 
 
 81 
 
 m 
 
 114. Prime numbers to 1000 are included in the 
 Table of Prime Numbers. 
 
 follo"\ving 
 
 1 
 
 r>d 
 
 139 
 
 233 337 439 557 
 
 G53 7G9 
 
 883 
 
 
 
 Gl 
 
 149 
 
 239 317 443i5G3 
 
 G59 773 
 
 887 
 
 o 
 
 G7 
 
 151 
 
 241 : 349 449 ' 5G9 
 
 GGl 787 
 
 907 
 
 5 
 
 71 
 
 157 
 
 251 '353:457 
 
 I 
 
 571 
 
 G73 797 
 
 911 
 
 
 73 
 
 1G3 
 
 257 359 4G1 | 577 
 
 07 7 ' 809 
 
 919 
 
 11 
 
 79 
 
 1G7 
 
 2G3 3G7 4G3'587 
 
 G83 811 
 
 929 
 
 13 
 
 S3 
 
 173 
 
 2G9 373 4G7 593 
 
 G91 821 
 
 937 
 
 17 
 
 SO 
 
 179 
 
 271 379 479 
 
 599 
 
 701 823 
 
 941 
 
 19 
 
 97 
 
 181 
 
 277 3S3 487 
 
 GOl 
 
 709 827 
 
 947 
 
 23 
 
 101 
 
 191 
 
 281 389 491 
 
 G07 
 
 719:829 
 
 953 
 
 29 
 
 103 
 
 193 
 
 283 :VJ7 499 
 
 G13 
 
 727 1 839 
 
 9G7 
 
 31 
 
 107 
 
 197 
 
 293 401 503 
 
 G17 
 
 733 853 
 
 971 
 
 37 
 
 109 
 
 199 
 
 307^409 509 
 
 G19 
 
 739 857 
 
 977 
 
 41 
 
 113 
 
 211 
 
 311 419 521 
 
 G31 
 
 743 i 859 
 
 983 
 
 43 
 
 127 
 
 223 
 
 313 421 I523 
 
 1 
 
 GH 
 
 751 8G3 
 
 991 
 
 47 
 
 131 
 
 007 
 
 WW 1 
 
 317 431 
 
 541 
 
 G13 
 
 757 877 
 
 997 
 
 53 
 
 137 
 
 M w t/ 
 
 331 433 
 
 
 G47 
 
 7G1 881 
 
 1009 
 
 Every prime number, except 2 and 5, has 1, 3, 
 iUi unit liL'^ure. 
 
 7, or 9 for 
 
 FACTOPJXa OF XU:^IBEKS. 
 
 115. To resolve or separate a number into its prime 
 factors. 
 
 1. Kcsolve or separate 84 into its prime factors. 
 
 OPERATIONS. 
 
 2 
 
 84 
 
 
 84 = 2 X 42 
 
 
 2 
 
 42 
 
 or, 
 
 42 =: 2 X 21 
 
 
 3 
 
 21 
 
 7 
 
 
 21 = 3 X 7 
 
 
 Ans. 
 
 84 = 
 
 2x2x3x7 = 
 
 = 9- X 3 X 7. 
 
 ■\Vhat has every prime number, except 2 :md 5, fur its unit fi^nirc? 
 
 ,i 
 
82 
 
 PRACTICAL ARITHMETIC. 
 
 ill 
 
 it' 
 
 By trial, wc find that 84 is coniposetl (;f two factor,?, 2 and 42 ; of 
 whicli 2 is prime and 42 is conipositu. 
 
 The 42 ^vo find composed of two factors, 2 and 21 ; of which 2 is 
 prime and 21 is composite. 
 
 The 21 is composed of two factors, 3 and 7, both prime. 
 
 Therefore, the prime factors of 84 are 2, 2, 3, and 7, and may be 
 written 2-, 3, and 7. 
 
 KULE. Diiidc the, given number hij any imme number (jreatcr 
 ihiin J, Unit is an exact divisor, and the quotient, if comjwsitc, in 
 the same inanner ; and thus continue until the quotient is prime. 
 The divisors and the last quotient ivill be the priiue factors re- 
 quired. 
 
 Proof. The product of the prime factors will equal the 
 given number, if the work is right (Art. 102). 
 
 Since 1 is a factor of every number, it is not commonly specified 
 as such. 
 
 1^ 
 
 Examples. 
 
 "What are the prime factors 
 
 2. Of 252? 
 
 3. Of 1011? Ans. 3,337. 
 
 4. Of 0381? 
 
 5. Of 1000? Ans. 2», 5». 
 G. Of 113G8? Ans. 'I\ 1\ 29. 
 
 7. Of 0109? 
 
 8. Of 707 ? Ans. 7, 101. 
 
 9. Of 7854? 
 
 10. Of 4350? A n.^. 2,3, b"; 20. 
 
 11. Of 32320? Ans. 2% 5, 101. 
 
 116. To find the prime factors common to two or more 
 numbers, we may 
 
 Take out the common factors, by dividing the given numbers hij 
 any inime number greater than 1, that is an exact divisor of them 
 allf and treat the quotltnts in the same manner, until quotients shall 
 he obtained prime to each other. 
 
 12. What are the prime factors commc i to 28 and 50 ? 
 
 What is the Kule? The Pruof ? How may the i)rime f.actors common to 
 two or more numbers .-a iound? 
 
FACTORING. 
 
 83 
 
 OPERATION. 
 
 I 
 I 
 
 a 
 
 ■ 
 
 
 28, 50 
 "7714 
 
 By ti'iiil wc find 2 to Le an exatf 
 divisor of all tlio nuiiiljers, and v;e 
 tlu'i't'f'ore take it as one of lli » com- 
 mon factors, and liave left of 28 the 
 factor 14, and of 50 the factor 28. 
 "* We find 2 to he an exact divisor 
 
 yins. 2, 2, and 7, or 2 and 7. ^f tj,^, f.^^tors 14 and 28, and we 
 
 take it as another common factor of tlie L,dven nunihers, and have left 
 of 28 the factor 7, and of 50 the factor 14. 
 
 We find 7 is an exact divisor of tlie factors 7 and 14, and we take; 
 it as a common factor of the ^dven nnndjer, and we have left only a 
 factor 2 of 50, and which cannot be common to 28, Therefore, 2, 2, 
 and 7 are the conunon prime factors of 28 and 50. 
 
 What are the prime factors common, 
 
 13, To 45 and 75 ? 
 
 U, To 99, 105, and 330? 
 
 15. To 00, L>10, and 390? 
 
 Ans. 3 and 11. 
 Alls. 2, 3, and 5. 
 
 MULTIPLICATION PA" FACTOPS. 
 
 117. 1. Multiidy 452 by 35, using factors. 
 OPERATION. 35 is efj^nal to 7 times 5 ; lience 35 times 452 i 
 
 452 
 
 7 
 
 3104 
 5 
 
 Ans. 15820 
 
 efjnal to 5 times 7 times 452. 
 
 7 times 452 is eijual to 3104, and 5 times 7 
 times 452, or 5 times 3104, is erpuil to 15820. 
 
 Therefore 452 multiplied by 35 is erpial to 
 15820. 
 
 Had 35 contained any other convenient set ot 
 factors, they could hav e been used in like manner. 
 
 Rule. Separate the multiplier info convenient factors. Midtijdij 
 the raultiplicandhy one of these factors, and the yroduct hy another, 
 and so on, until all the factors have been used. The last iwodac.t 
 will he the one required. 
 
 Repeat the Kulo, 
 
84 
 
 PRACTICAL ARITHMETIC. 
 
 Examples. 
 
 ]\rultipl3', using factors, 
 
 Ans. 39G0. 
 
 2. 105 by 24. 
 
 :}. 3405 by 56. 
 
 4. 4031 by G3. Aiif<. 253953. 
 
 Ans. 21523. 
 
 Ans. G1020. 
 
 5. 876 by 28. 
 G. 11350 by 81. 
 7. 1130 by 54. 
 8. Multiply 30451 by 70, or by 10 x 7. Ans. 2131570. 
 
 Hore we annex a cipher to the multiplicand for the product 
 of it by 10 (Art. 54), and then multiply by the 7 ; that is, vc 
 annex the cipher to the multiplicand, and multiphi hy the numhcr 
 expressed hy the significant figure. 
 
 U. Multiply 400G7 ])y 50, or i)y 10 x 5. Ans. 2003350. 
 
 DIVISION BY FACTORS. 
 
 118. 1. Divide 15820 by 35, using factors. 
 
 Since 35 times a number is equal to 7 times 
 5 times th'3 number (Art. 117), oi.e tliirty-lil'th 
 oi' the number must eijual one seventh of (Jiie 
 iil'lli of tlie number. 
 
 One iifth of 15820 is 31G4, and one seventh 
 of one fifth of 15820, or one seventh of 31 04, 
 is 452. 
 Therefore 15820 divided by 35 is equal to 452. 
 
 Had the divisor contained any other con^'^enient set of factors than 
 f) and 7, they could luive been used ■with like result. 
 
 2. Divide G103 by 15, using factors. 
 
 OPERATION. 
 
 5 ) 15820 
 
 7 ) 31G4 
 
 452 
 
 OPERATION. 
 
 3 ) G103 
 
 Dividing by the factor 
 3, we ()l)tain 2034 threes, 
 and a remainder of 1 iinit. 
 
 5 ) 2034, Rem. 1 unit = 1 Dividing l)y the factor 
 
 400, 
 
 J) 
 
 4 threes = 12 tl 
 
 we obtain 406 (live times 
 
 Ans. 40GI4. Trucremainder= 13 
 
 1 5 
 
 irees, or fifteens), and a 
 remainder of 4 threes. 1 
 u)iit-¥-^ threes, or 13^ equal 
 the entire remainder. 
 
 d 
 
 % 
 
 "NVhat is 35 times ta uumber equal to ? 
 
 
I 
 
 I 
 
 1 
 
 1% 
 
 
 It 
 
 FACTORING. 
 
 So 
 
 JU'LE. Separate the divisor into convenient fa rfnrs. 
 
 Divide the dividend hi/ one of these faetors, and the quotient h)f 
 another, and so on, until all the faetors have hecn used. The last 
 quotient vill he the one required. 
 
 Should there be one or more remainder:^, multiphj each remainder 
 hi) the divisors, if any, preceding the one that produced it, and the 
 sum of the products plus the remainder left hij the first division, if 
 aiiij, irill he the true remainder. 
 
 Examples. 
 
 Divide, using tactors, 
 
 a. 31344 by 24. Ans. 130G 
 
 4. 24o28 by 28. 
 
 5. 100(380 bv 50. Ans. 3405 
 
 G. 37u0 l)y 25. Ans. 150^ a. 
 
 7. 200510 l)y 03. 
 
 8. 2010G by 42. Ans. 028^':]. 
 
 0. Divide 134G07 by 700 - 100 x 7. 
 10. Divide 37G875 l)y 315 = 5 x 7 x 9. Ans. 1190.^/ 5. 
 
 119. Since dividing botli the dividend and divisor by the 
 same number will not change tlie quotient (Art, 73), it is often 
 possible to shorten arithmetical operations hy rejecting equal 
 factors from loth dividend and divisor, and using onhj the regain- 
 ing factors. 
 
 Tile process has been called CANCELLATION, from the re- 
 jected factors being usually noted by being crossed or cancelled. 
 
 11. Divide 3 times 00 by 54. 
 
 90 X 3 
 
 54 
 
 OPERATION. 
 
 ).^ X 5 X 
 
 1 
 
 5, Ans. 
 
 ludioatiiiLf tlu; division 
 (Art. 58), and factoring, 
 the divitlond becomes 18 x 
 6x3. and ihc divisor 18 
 
 X 3. 
 
 CunccllinL' the farlors 
 18 and 3, common to both dividend and diviaor, we have {, ur 5, 
 which is the quotient. 
 
 Kopent tlio llulo. How is it often possible to shorten arithmetical opera- 
 tions? AVliiit ia the process of rejecting factors called? 
 
 i 
 
86 
 
 riJACTICAL ARITHMETIC. 
 
 U 
 
 OPERATION. 
 
 ^ 3 
 = 3, Ans. 
 
 1 
 
 12. Divido IG times 21 l)y 8 times U. 
 
 Cancelling' tlio factor 8, common to Ixttli 
 divisor and dividend, we have left in the 
 dividend 2, in ])lace of in ; cancelling,' the 
 factor 2, common to boti divisor and divi- 
 dend, we have left in the divisor 7 in place 
 of 14 ; and cancollinj^ the factor 7, common 
 to the dividend and divisor, we have left 
 ^, or .3, which is the (juotient. 
 AVhen any factor is cancelled, 1 is imderstood to remain, and need 
 
 l>e written only when the last of all other factors in the dividend or 
 
 divisor is cancelled. 
 
 13. Divide 9 times 40 by 15 times 24. Ans. 1. 
 
 14. Divide 7.5 x 25 x 7 by 50 x 3. Ans. 87i. 
 
 15. (3G X G3 X 12) -r (54 x 40 x 10) = how many? 
 
 IG. (510 X G3 X 4) -f (G80 x 84) - liowmany? ylns. 2J. 
 17. Divide 400 x 189 x 33 x 5 by 320 x 12G x 11 x 5. 
 
 S 
 
 f 
 
 Am 
 
 
 ArPLICATIONS. 
 
 1. How many horses, worth ^132 each, must be given for 
 147G sheep, worth ^11 each? 
 
 If one sheep is wortli $11, 1476 
 sheep must he worth $11 x 147G, 
 and as many horses, worth each $1 .32, 
 must be given for $11 x 1470, as 
 $132 are contained times in $11 x 
 147(5, or 123. 
 
 OPERATION. 
 
 n X 147G 
 
 12 
 
 - 123, Ans. 
 
 2. How many pounds of butter, at 35 cents a pouiul, c^n be 
 bought for 105 yards of muslin, at 21 cents a yard? 
 
 Ans. G3 pounds. 
 
 3. At 14 cents a pound, how much sugar can be bought for 
 2 cords of w^ood, at i|j;5.G0 a cord ? ^^ /.• .u><^ 
 
 When a factor is cancelled, what is understood to remain? AVhon onlv 
 need the 1 be writtci'? 
 
 ^r — ^^ 
 
 -.-^mi-- 
 
FACTORING. 
 
 87 
 
 ./ 
 
 4. How many loads of hay, of 18 Imndivds cacli, at 7r> coni^ 
 a liuiidred, will pay for 1G2 bushels of oats, at 50 cont>; a 
 buslu'l i Alls. G l<»ads. 
 
 T). William ^lar.sh sold 3G0 pounds of hov^, at W cents ;i 
 ])ound, for 3 firkins of butter, each weiL;liing 50 pounds. H(»\y 
 much was the butter u pound? ylns. 30 cents. 
 
 G. How many days must a carpenter Wi»rk, at 82.50 a day, 
 to pay for the services of a farmer for 10 days at 81.50 a day \ J^'/ 
 
 7. Sold 1G4 dozen school readers, at ^0 a dozen, and le- 
 ceived in paynKMit ([uarto dictionaries, at $12 apiece. How 
 many dictionaries did I receive? Auff. 123 dictionaries. 
 
 8. When 8i0.50 is pai'^ for 30 barrels of apples, each con- 
 taining 3 busliels, how much are they a bushel ? yin.<. 8.4-5. 
 
 9. How many bales of goods, each bale containing GO pieces, 
 and each piece 49 yards, worth 75 cents a yard, nuist be givt n 
 fur 80 government bonds, worth 8110.25 each. Ans. 4 bales. 
 
 ! \ 
 
 GREATEST COMMON DIVISOR. 
 
 120. A Common Divisor of two or more numbers is any 
 exact divisor (Art. 97) of each of them. Tims, 
 
 2, 3, and G are ccmmion divisors of G and 12. 
 
 121. The Greatest Common Divisor of two or more num- 
 bers is the greatest exact divisor of each of them. Thus, 
 
 4 is the greatest common divisor of 8 and 12, 
 
 But 4 is equal to the product of 2 and 2, the oidy common 
 prime factors of 8 and 12 (Art. IIG). Hence the principh-, 
 
 The Greatest Common Dkmr of tiro m' more numbers is equal 
 to the jyroiluet of all their common prime factovb. 
 
 Review Questions. What i.s an Exact Divisor? (07) A Factor? (98) A 
 Prime Number? (90) A ComiKisite Number? (100)— What is a Common 
 Divisor of two or more numbors? What is the Greate.st Common Divisor of 
 two or luore numbers ? Tlie rrinciplc ? 
 
 !l 
 
.^s 
 
 PRACTICAL ARirnMnirc. 
 
 Ijl 
 m 
 
 i.l 
 
 fi 
 
 122. To find the Greatest Common Divisor of two or more 
 Numbers. 
 
 1. ^VlKlt is tlic greatest common divisor of 8, 12, and 20? 
 
 OPERATIONS. 
 
 o 
 
 8, 12, 20 8 - 2 X 2 X 2 
 
 - t, ~G , 10 or, 12 = 2 X 2 X 3 
 
 2, 3, 5 20 = 
 
 2 X 2, or 4, Ans. 
 
 2 X 2 X .5 
 
 Taking,' out tlio cominon piime factors of the givt'ii mim1>er.s (Art. 
 1 lO), we ii)i(l tlieni to be 2 and 2 ; and tlierefure the j^ieatest coiiiuiuii 
 divi.sor of 8, 12, niul 2(», is 2x2, or 4. 
 
 In the second operation we resolve the given nundx-rs into their 
 I'rinie factors (Art. 115), and liiid the coninion prime factors to be 2 
 and 2, and take their prodnet, witli the same result as before. 
 
 RuLK. Find the common prune factors vj the numbers, and 
 take their product. Or, 
 
 Resolve the numbers into their prime factors, and take the pro- 
 duct of those ichich are common. 
 
 Numbers j^'ime Avith respect to each other (Art. 104), havin;^' no 
 cumniun factor, excejit 1, are said to have no common di\isur. 
 
 Examples. 
 
 "What is the greatest common divisor 
 
 2. Of 45, 72, and 8U Ans.d. 
 
 3. Of GG and 105? ^J^^xr-i!^ 
 
 4. Of 720and9G0? Ans. 210. 
 
 5. Of 54 and 258 ? Ans. G. 
 G. Of 323 and 425? ^^ , ; > 
 7. Of 30, 110, 140, and G80^ 
 
 123. Another metliod, and often the most convenient, is 
 based upon the principle, that 
 
 The greatest common divisor of two numbers is likewise the 
 greatest common divisor of the smaller ond of the remainder after 
 division. 
 
 8. Find the greatest common divisor of 247 and 3 
 
 33. /;? 
 
 Repeat the Rule. "When are two or mure numbeis sukl to have no Uiiu* 
 moQ divisor? What ia thi principle of another nielhoilV 
 
 ii 
 
 t 
 
 - k* e ■ w Nwqw mw < w **'!^' * 
 
FACTORING. 
 
 80 
 
 OPERATIONS. 
 
 k 
 
 
 217)323(1 
 
 247 
 
 70)217(3 
 228 
 
 or, 
 
 323 
 217 X 1 = 247 
 
 228 =: 3 X 70 
 
 r.) X 4 - 70 
 
 10) 70 (1 
 70 
 
 A US. 10. 
 
 Since 247 is tho ^rcatci^t divisor of itself, if it i.s an cxiict divi.sor of 
 323, it will lie the greatest common divisor of 247 and 1323. It is not 
 an exact divisor of 323, for there is a remainder, 7(>. 
 
 If 7(), whicdi is the .greatest divisor of itself, is an exact divisor of 
 247, it nnist he the greatest common divisor of 7<! and 217, and like- 
 wise of 247 and 323. It is not an exact divisor of 247, lor theri; is a 
 remainder, 19. 
 
 If 11), which is the greatest divisor of itself, is an exact divisor of 
 70, it is the greatest common divisor of 10 and 70, likewist; of 70 and 
 247, and of 247 and 323. It is found to "he an exact divisor of 70, 
 and therefore 19 is the greatest common divisor of 247 and 32;'. 
 
 The two operations are exactly the same, except that the second is 
 Avritten in a more compact form than the first. 
 
 124. IIcucc, to find the greatest common divisor of two 
 numbers, wo may 
 
 Divide the (jreatcr numher hy the less, and the divisor by the 
 remainder, and so on, imtil there is no remainder ; the last divisor 
 will be the qreatest conunov divisor. 
 
 9. Find the greatest common divisor of 308 and 500. 
 
 Ans. 2d. 
 
 10. Find the greatest common divisor of 3252 and 42 18. ^ > >. / j^, 
 
 11. What is the i^reatest common divisor of 21 15 and 
 
 34711 
 
 Ans. 30. 
 
 Expliiin the (>i>oriitioii8. How may wo fiud the greatest coininon divisor of 
 twu muiiboia? 
 
 ^^ 
 
00 
 
 I ft 
 
 I^KAcnCAL ARlTllMKTrc. 
 
 have been U,ln,. "' '""' '" <"'> '"'I'l uU the m„„lj, 
 
 •-•'^""Itl,c...eatetcommo„,Iivi,„,„f,,g., ,,, ,, 
 
 *^-, / 41, and 1044. 
 
 44,0 ; «^^-^'"' <'«'"mon divisor of 1 ,.,, 30U4, n„,I 
 
 Aiu^. 442. 
 
 APPLICATIONS. 
 
 J- Tlioro is a cortaiu iiel,I 78s r„ 7 , 
 "■'"'t i.s the length of tl 1 °"" •■""' ^oC ro.Is „-i,le ■ 
 
 ---.othitsien;,;, Sis ;''•'" ""^' ^>-'" --i; 
 
 -■ ^VJint must be the vd.IH, f '^'"- * ''"''«• 
 
 ":« '-t being ,5 feet th inc lTr!''"° '" '' ""■» ™™'«, 
 "'"'« ' - .:^ A/ W,. ' ''"'' -^"^ "'<^ «'"-J :.'l feet 
 
 3 T 1 
 
 .second 450, and thelhi'rV'^o'^-' "'" ''''" ^"'•■"■">ng Sr.J the 
 
 *'■« '-.-. ,K,..h,e i £,::-; 'f ^ «'-ou,d dh.ide it' i^ : 
 '-'■' "0'^ --y acres .oUuK: ::r,';i:7^ -^^^^^ ^" 
 
 4. A farmer lias 7009 i i , ^^^''^' "^ ''^^^^s. 
 
 '^"«l'olsofwi„terwL t L;::',^ f ^"""'^^ ^vheat, 1,50 
 «'- capacity of the l^st tinfof I'f t "' '"■'^^- R<^1"' ^ 
 conf,a,„ the whole, without SL. ^" '"" ''''' ^"" "iutiy 
 5- A has 6079, B $5001 n l^n . ^^^ ''"*'"■'«• 
 
 '■' -' f"^ ^I.co„, a; tlj S;, r; .^f ^^f ^ fe, agree to lay 
 each to exactly i„.est his Cey h ' "f '''"' ^"" •■•"»- 
 
 j!!!::;!:^:^^^^ ^^-^ ^^-''' »^ o oc. sheep. 
 
 How may we finri +ii« , — 
 
 t™.iven:„„te:;'"^«™'«'™»-„aiW„,e„ the. are .e,e .,„„ 
 
FACTORINO. 
 
 91 
 
 LEAST COMMON MrLTirLR 
 
 126. A Multiple of n mmilicr is any nuiiiber Avliioli lias that 
 iiuiubcr as an exact divisor. TIhh, 
 
 8 and I'J are niulti[>U'.s of 4. 
 
 127. A Common Multiple of two os- nion? nundn'rs is any 
 mnnl)L'r Avliicli lias each of those numbers as an exact divisor. 
 Tims, 
 
 21- and 48 are common nmltiplcs of 4, S, and 12. 
 TIic, Least Common Multiple of two or more numbers is 
 tlie least number Avhicli has eacli of those nundjers as an exact 
 divisor. Thus, 
 
 24 is the least common nndti[)le of 4, S, an<l 12. 
 r>ut 24 contains only the prime factors, 2, 2, 2, a'ld .'>, which 
 alone are reciuired to produce 4, 8, and 12. Hence, tiu- prin- 
 ciples, 
 
 1. The least common mitUiph' of tu-o or more inimh.r.i />• a 
 munher containinrj all the prime facidrs of each numbrr, and no nUicr^. 
 
 2. The least common multiple of tie o or more numhirs irhich are 
 prime to each other, is equal to the product of the nuudnr.^. 
 
 128. To find the Least Common Multiple of two or more 
 numbers. 
 
 1. What is the least common midtiple of n, 1 |, anil 10 ? 
 
 OPERATION. Since a multijik; of 4!) must cmi- 
 
 G - 2 X 3 tain 4f), it nm^t contain all its jtrinie 
 
 14 = 2x7 factors ; we take the factors 7x7. 
 
 /^^ - '] x 1 These are all the prime factnr< 
 
 7x7x2x3 = 294, Ans. ''^ ^'^ '"^'^ ^^^ '''^^^'I't •^, whi.h we 
 
 take, and have 7x7x2. 
 
 These are all the prime factors of 49, 14, and G, except 3, Avliich we 
 take, and have 7x7x2x3. 
 
 These are all the prime factors of each of the given nnml>trs, and 
 no others; therefore tlieir product, or 2U4, is the least cumiuon 
 multi[)le required. 
 
 What is a IMultiple of a nuinltcr? A Common Multinle of two or more 
 mimliers? The Least Common Mulliple ? The first principle ? The second '.' 
 ■\Vhat must a multiple of 49 contain? 
 
!MAGE EVALUATION 
 TEST TARGET (MT-3) 
 
 7 
 
 /. 
 
 
 
 <t^ \" 
 
 y. 
 
 i/x 
 
 v.. 
 
 1.0 
 
 I.I 
 
 1.25 
 
 '- illM 
 
 • 5 "'"= 
 
 Z 1^ 
 
 M 
 
 [2.2 
 
 [20 
 
 1.8 
 
 JA 111.6 
 
 <^ 
 
 ^ 
 
 /i 
 
 'c^ 
 
 ^ ^ 
 
 >^^ 
 
 '//J 
 
 M J^'} 
 
 
 »■>» ^'•z 
 
 ^<-^ 
 
 
 /^ 
 
 '>. 
 
 
 #■ 
 
 Photographic 
 
 Sciences 
 Corporation 
 
 ^ 
 
 iV 
 
 iV 
 
 ^\\^ 
 
 
 6^ 
 
 23 WEST MAIN STREET 
 
 WEBSTER, N.Y. 14580 
 
 (716) 873-4503 
 
 % 
 
 ^^ 
 
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 '•«^* 
 
 
92 
 
 PRACTICAL AUITHMETIO. 
 
 ■ . ^ : 
 
 liULE. Piesohe the numlers into their iirime factors, and taJ:e 
 the product of all the different j^rinie factors, using each the greatest 
 nnraher of times it occurs in any one of the given numbers. 
 
 '■ I 
 
 Examples. 
 
 Find the least common multiple 
 
 o 
 O. 
 
 Of 30 and 55. Ans. 330. 
 Of 9, 12, and42.y>oJ7yf, 
 
 5. Of 25, 60, 84, and 15. 
 
 Ans. 2100. 
 
 4. Of 3, 5, 7, and 21. Ans. 105. , G. Of 3G, 56, 75, and 72. 
 
 7. Of 81, 27, 45, and-18.---^^//^ Ans. 12G00. 
 
 8. What is the least common multiple of 5, 19, and 2) 1 
 (Art. 127, prin. 2.) Ans. 1995. 
 
 129. To find the least common mulciple of two or more 
 numbers, we may, often most conveniently, 
 
 Divide hy any 2mme minder greater than 1, that is an exact 
 divisor of two or more of the given numbers. 
 
 Divide the quotients and undivided numbers, if any, in like 
 manner, and so continue, until there is no exact divisor, greater than 
 1, of any tico of them. Tahe the product of the divisors and the 
 final quotients, for the least common multijde of the numbers required. 
 
 9. What is tlie least common multiple of 18, 20, and 30? 
 
 OrERATION. 
 
 2 i 18, 20, 30 
 
 3 9, 10, 15 
 
 3, 10, 5 
 
 3, 2. 
 X 5 X 3 X 2 
 
 180, Ans. 
 
 Taking out the prime factor 
 2, common to the given num- 
 bers, we have left of 18 the 
 factor 9, of 20 the factor 10, 
 and of 30 the factor 15. 
 
 Taking out the factor 3, com- 
 mon to 9 and 15, we have left 
 
 of 18 the factor 3, of 20 the factor 10, and of 30 the factor 5. 
 
 Taking out the factor 5, common to 10 and 5, we have left of 18 
 the factor 3, of 20 the factor 2, and of 30 no factor, and these factors 
 are prime. 
 
 "What is the Rule? How may the least common multiple of two or more 
 numbers be often most conveniently found? 
 
 i 
 
FACTORING. 
 
 93 
 
 Therefore 2, 3, 5, 3, and 2, are all the prime factors of the LjivfU 
 niiiuhers ; hence their product, or 180, is the least common multiple 
 rejpiired. 
 
 10. What is the least common multiple of 15, 35, IG, and 
 5G? Ans. 1G80. 
 
 11. Find the least common multiple of 39, 2G, G5, and 15 1^ :i^ 
 
 12. Find the least common multiple of 12, 3G, GO, and 72 ? ^ (rC 
 
 OPERATION. Here, since a multiple of 72 must he 
 
 12 I 3/^, ^0, GO, 72 a multiple of its exact divisors, 12 and 
 
 3C, we cancel those numbers. 
 
 Again, since 12, althouuh not prime, 
 IS an exact divisor of all the rema'ainj^j 
 immhers, all its factors must be factors of the multiple, and taking it 
 out, we have left of 60 the factor 5, and of 72 the factor 6, which fac- 
 tors are prime to each other ; therefore 12 x 5 x G = 3fJ0, is the 
 multiple required. 
 
 13. Find the least common multiple of 35, 105, 210, and 7-50 Uj'y'/^r^ 
 1-4. What is the least conr; on multiple of 54, 378, 48G, and 
 
 540? Ans. 34020. 
 
 12 
 
 X 5 X 
 
 5, G 
 G = 3G0, Ans. 
 
 {■% 
 
 i II 
 
 APPLICATIOXS. 
 
 1. What is the least sum of money for which I could pur- 
 chase a number of sheep at $3, 84, $5, or $G each, and just 
 expend the whole ] Ans. $G0. 
 
 2. Four men start at the same time and place to walk 
 around a race-course, in the same direction. A can go round 
 in 10 minutes, B in 12 minutes, C in 8 minutes, and D in 18 
 minutes ; in what time will they all be again together at the 
 pohit of starting? — - :i (r^^ i^^o--. >^{/ .{^u-^j ■ 
 
 3. What is the smallest sum of money for which John Fuller 
 can hire a number of men for one month, at either $12, $18, 
 $30, or $3G each, and what will be the number of men that 
 can be employed at each rate ? 
 
 . Ans. $180; 15 men at $12, 10 men at $18, G men at $30, 
 or 5 men at $3G. 
 
 Why, iu the operation, can wo cancel tho 12 and uG ? Why can we take 
 out 12 from the remaiuiug uumbera us a factor of the reciuired multiple ? 
 
94 
 
 PRACTICAL AIIIT1I3IETIC. 
 
 ,1 
 
 1: i 
 
 b \ 
 
 f I 
 
 COMMON FRACTIONS. 
 
 130. If a unit, as one inch, be divided into two equal parts, 
 one of these parts is one half. Thus, 
 
 1 of 1 inch = r-»;i;i'ii'i-';i'Pi;:g i 
 
 If a unit be divided into three equal parts, one of these parts 
 is one third; two of them tico thirds, &c. Thus, 
 
 I of 1 inch— |]J^ '' i ' iii:iMi i I , § of 1 iiich=^S^^^^3 , 
 
 &c. 
 
 If a unit be divided into four equal parts, one of these p?rts 
 is one fourth; two of theTn two fourths, &c. Thus, 
 
 lof 1 inrh^^ ET^IMZl-j L— L -^.of 1 inch=[:: n' n '; |iM : !:3l | u . 
 
 &c. 
 
 In like manner, if a unit be divided into five equal parts, 
 the parts are fifths ; if into six equal parts, sixths ; and so on. 
 Such parts of a unit are called fractions. Hence, 
 
 131. A Fraction is a part of a unit, consisting of one or 
 more of the equal parts, which compose the unit. 
 
 The Unit of a fraction is the unit or thing divided. 
 A Fractional Unit is one of the equal parts into which 
 the unit of a fraction is divided. 
 
 132. The Denominator of a fraction is the number which 
 shows into how many equal parts the unit is divided. Thus, 
 
 Three is the denominator of two thirds. 
 
 133. The Numerator of a fraction is the number which 
 shows how many of the equal parts of the unit are taken. Thus, 
 
 Two is the numerator of two thirds. 
 
 NOTATION AND NUMERATION. 
 
 134. A Common Fraction is a fraction expressed by writing 
 the numerator above, and the denominator below, a dividing 
 line ; as. 
 
 What id a fraction ? The Unit of a fraction ? A Fractional Unit? The 
 Denominator of a Fraction ? The Numerator ? How is a Common Fraction 
 expressed ? 
 
COMMON FRACTIONS. 
 
 95 
 
 fcs, 
 ts 
 
 I 
 :s 
 
 One half, w 
 
 rittcn 
 
 1 
 
 One third, 
 
 
 1 
 
 If 
 
 Two third ^^, 
 
 
 2 
 
 One fourtli, 
 
 
 1 
 T 
 
 Two fourths, 
 
 
 <> 
 
 Three fourths. 
 
 
 4 
 
 One fifth, 
 
 
 1. 
 5 
 
 Two fifths, 
 
 written 
 
 
 Three fifths, 
 
 
 .•i 
 
 "6 
 
 Four fifths, 
 
 
 4 
 
 One seventh. 
 
 
 1 
 
 T 
 
 Three eighths. 
 
 
 3 
 
 Five ninths. 
 
 
 6 
 ¥ 
 
 Seven tenths. 
 
 
 T 
 Iff 
 
 135. A Proper Fraction is one whose numerator is less 
 than the denominator ; as f, -y-. 
 
 136. An Improper Fraction is one whose numerator is 
 not less than the denominator ; as f , |. 
 
 137. A Mixed Number is a whole number with a fraction ; 
 as 3| ; read three and one fourth. 
 
 138. The Terms of a fraction are the numerator and deno- 
 minator. 
 
 The numerator expresses the mimher of fractional units in 
 the fraction, and the denominator their name or dtmomlnation. 
 
 139. A whole number may be expressed in a fractional form 
 by writing 1 under it for a denominator. Thus, 
 
 2 may be written 
 
 
 and read, 
 
 2 ones. 
 
 3 
 
 1' 
 
 >) 
 
 3 ones. 
 
 7 
 
 7 
 1» 
 
 j> 
 
 7 ones. 
 
 
 
 10 
 1 ' 
 
 >> 
 
 10 ones. 
 
 140. A fractional expression may be explained by showing 
 what is denoted by its terms. 
 
 Exercises. 
 
 1. Eead and explain f . 
 
 Solution. Three fifths ; the denominator 5 denoting that the name 
 or denomination of fractional units expressed is fifths, and the numera- 
 tor 3 that three of those units are taken. 
 
 Wlipt is a Proper Fraction? An Improper Fraction? A Mixed Xumber? 
 ^\^^at are tlie Terms of a fraction ? What do they express ? How may a 
 whola number be expr-'ssed in a fractional form? How may a fractional 
 expression be explained ? 
 
'I ^-. 
 
 90 
 
 PRACTICAL ARITHMETIC. 
 
 Read and explain, 
 
 2 "^ 
 
 1 3 
 
 C. f . 
 7 -^-X 
 
 8 "5 
 "• 5 5* 
 
 19 1 flfl 
 
 fi .'5 4 
 
 13. 
 
 iox7bir* 
 
 14. Express by figures seven ninths. 
 
 SoLUTiCN. i^ince the name or kind of fractional nnits to be ex- 
 pressed is ninths, we write 9 as the denominator, and since the mimher 
 of these units to he taken is seven,we write 7 as the numerator, and have 
 as the required expression 1^. 
 
 Express by figures, 
 
 Ans. 4. 
 
 1 T). G ninths. 
 
 10. 17 twenty-fifths. 
 
 1 7. Two fiftieths. 
 
 18. Eleven twenty-seconds." 
 
 1 9. 28 thirty-firsts 
 
 sr 
 
 21. One one hundred fifteenths. / ~ 
 
 22. 98 three hundredths. ::= 111 
 
 23. 51 six hundred fortieths, iy ^^^y^? 
 
 24. Sixteen thousandths. /^ ^^ 
 
 25. 107 nine hundred twenty*-* 
 
 20. 19 sixty-seconds. - 'I ninths. , u. , ^. 
 
 20. Seventy-two thousand twenty-firsts. ^ ^"fVA 
 
 141. A fraction may be regarded as indicated division (Art, 
 58), the numerator answering to the dividend, and the de- 
 nominator to the divisor. Hence, the 
 
 
 GENERAL PEINCIPLES. 
 
 1. The value cxrircsscd hy a fraction is the quotient of the 
 numerator divided hy the denominator. Thus, 
 
 f of 1 inch = h ii '"i i ii"ii ii i"i 'i' iii ' inii I I _, 1 of 2 inches = 
 t>'''iF''ii^P'n"i I I I I I zn . By Art. 7 1-3, it follows, that 
 
 2. Multijjlyiug the iiumerator or dividing the denominator 
 multl])lies the fraction. 
 
 3. Dividing the numerator or multli)hjing the denominator 
 divides the fraction. 
 
 How may a fraction be regarded ? What is the value expressed by a frac- 
 tion ? How is the value of a fraction affected by multiplying its numerator 
 or by dividing its denominator ? What effect has dividing the numerator or 
 multiplying the denominator of a fraction ? 
 
 ^\>Mm 
 
COMMOX FRACTIONS. 
 
 97 
 
 4. MallliAijin'j or dividing loth terms of a fraction hij the same 
 uiuiiher docs not change its value. 
 
 Mental Exercises. 
 
 1. If a unit be divided into 4 equal parts, what will be tlie 
 name or denomination of the parts ? 
 
 Solution. If a unit he dioided into 4 equal 2>arts, each of tltc c^xnt 
 ■jiiirts will he \, Therefore the name or denomination of the j^arts will 
 hefourt/is. 
 
 2. If a unit be divided into 5 equal parts, what will l)e the 
 name or denomination of the parts 1 If into 9 equal parts ] 
 13? 171 62? 92? 150? 
 
 3. If a unit be divided into G3 equal parts, what will 1 of 
 the equal parts be called? 3 of the equal parts? 9? 19? 
 25? 41? G2? 
 
 4. If a unit be divided into 19 equal parts, what is 1 of tlie 
 fractional units called? 7 of the fractional units? 11 ? 10 ? 
 17? 
 
 5. How much is i of 12? iof:2? i? J? -^^1 
 G. How much is J of 8 ? Ofl2? OflG? Of20? 
 
 7. How much is I of 18? 
 
 Solution. Since 1 third of 18 is 6, 2 thirds of 18 are 2 times 6, or 
 12. Therefore f of 18 is 12. 
 
 8. How much is f of 24 ? Of 36 ? Of 40? Of 48? 
 I How much is f of 21 ? Of 28? Of 35 ? Of 49 ? 
 
 10. How much is |- of 18 ? Of 36 ? Of 45 ? Of 72 ? 
 
 11. Which is the greater, J of 12 or J of 12 ? 
 
 Solution. Since ^ of 12 is 6 and ^ of 12 is 4, ^ of 12 is greater 
 than ^ of 12 hy the difference hetween 6 and 4, or 2. Therefore h of 12 
 is greater than ^ of 12 by 2. 
 
 12. Which is the greater, J of 30 or i of 30 ? J of 24 or 
 J of 24 ? 
 
 "What effect has multiplying or dividing both terms of a fraction? "Which 
 is the greater, a fourth or a half of anything? A third or a tenth? 
 
 G 
 
98 
 
 PRACTICAL ARITHMETIC. 
 
 13. Which is the greater, J of 72 or V of 72 ? }. of 100 or 
 
 I 
 
 ' of 100? 
 
 14. Which is the greater, § of 15 or f of 15 ? ^ of 42 or 
 ^- of 42 ] 
 
 15. Ho\7 many fourths in a half? 
 
 Solution. Since there are 4 fourths in 1 imit, in 1 half of 1 «njf 
 tJiere is o)ie half of 4 fuiirths, or 2 fourths. Therefore in 1 half there 
 are 2 fourths. 
 
 16. How many sixths in a third ? How many tv/elfths ? 
 
 17. How many eighths in a fourth? How many six- 
 teenths ? 
 
 18. How many tenths in a fifth ? How many twentieths ? 
 
 19. How many thirds in ^ ? 
 
 Solution. Since in 1 unit there are 6 sixths, in 1 third of 1 unit 
 there is 1 third of Q sivths, or 2 sixths, and in 4 sixths as many thirds 
 as 2 sixths are contained times in 4 sixths, which are 2. Therefore in 
 ^ there are 2 thirds. 
 
 20. How many halves in | ? In ^ ? In |- ? 
 
 21. How many fourths in 1 1 Fifths in -^^J 1 Sixths in ^%. 
 
 22. How many sevenths in -J-2 ^ Tenths in t/^j ? 
 
 23. What part of 7 is 2 ? 
 
 Solution. Since I is I seventh of 7, 2 is 2 tivies 1 seventh of 7, or 2 
 sevenths of 7. Therejore 2 i5 f of 7. 
 
 24. What part of 9 is 5 ? 7 ? 2 ? 8 ? 
 
 25. What part of 11 is 4 ? 6 ? 9 ? 10 ? 
 
 26. What part of 17 is 7 ? 11? 13 ? 16 ? 
 
 27. What part of 19 is 8 ? 10 ? 14 ? 16 ? 
 
 28. 8 is 1 fifth of what number ? 
 
 Solution. If 1 fifth of some number is 8, 5 fifths, or the number 
 itself, is 5 times 8, or 40. 
 
 29. 7 is i of what number 1 i^ of what number 1 
 
 7 is ^ of what number 1 
 30. 10 is ^ of what number ? 
 
 J of what number ? 
 
 I f 
 
 Review Questions. What is a Fraction? (131) How is a Common 
 Fraction expressed? (134) What are the Terms of a fraction? (138) How 
 may a fraction be cxi^lained ? (140) 
 
 . 
 
COMMON F11ACTI0N3. 
 
 99 
 
 31. 12 is I of wliat numbor I '- of what number ? 
 
 32. 1 1 is l- of what num'ocr ? I of wliat number ? 
 
 33. 8 is 5 of what numljor ? 
 
 Solution. If '^ of some number is 8, 1 t/iird of that mimhcr is h 
 of 8, or 4 ; and 3 thirds of that ntviwer are three times 4, or 12. 
 
 o- 
 
 4. IG is 4 of what number ? ^ of what number? 
 
 35. 18 is I of what number ? j^- of wliat number ? 
 3G. 21 is I of what number 1 | of what number 1 
 
 37. 30 is yY of what number 1 'I of what number ? 
 
 38. 40 is ~j of what number 1 y. of what number ? 
 
 REDUCTION. 
 
 142. Reduction of Fractions is the process of changing tLc>ir 
 name or denomination, without changing the valuo expressed. 
 
 . 
 
 CASE I. 
 
 143., To reduce a fraction to its lowest or smallest terms. 
 
 A fraction is expressed in its lowest, or smallest, terms, when 
 the numerator and denominator are prime to each other, or 
 have no common divisor. 
 
 1. Reduce if to its smallest terms. - ^ 
 
 Since in 1 unit there are 12 twelfths, in 1 sixth of 1 unit there is ^ 
 of 12 twelfths, or 2 twelfths, and in 10 twelfths tliere are as many sixths 
 as 2 twelfths are contained times in 10 iwelftJhs, which are 5. Now, 
 in |- the terms are prime to each other. Therefore |§ in its smallest 
 
 terms is ^. 
 
 2. Reduce -^J to its smallest terms. 
 
 3. Reduce ^ to its smallest terms. 
 
 OPERATIONS. 
 
 18^^^x0x33 
 30 
 
 18 
 or — 
 30 
 
 J 
 18 -=- 6 
 
 3 
 
 5* 
 
 ^ X X 5 5' " 30 30 -^ 6 
 
 Since dividing both terms of a fraction by the same number does 
 not change the value e.^pressed (Art. 141), we divide both terms by 
 
 What is Reduction of rractioiis? When is a fraction expressed in its 
 lowest or smallest terms ? 
 
1 i 
 
 t ? 
 
 100 
 
 PRACTICAL AniTIIMETlC. 
 
 tile prime factors common to them, by cancelling, and obtain J?, who.«e 
 term.s have no common divisor. Therefore ^§ in its smallest terms 
 
 IS 
 
 In the second operation, we divide both terms of Iq by their 
 greatest common divisor, with the same result, ^. 
 
 liULE. Cancel in the numerator and denominator all the factors 
 common to both. Or, 
 
 rivide both terms of the fraction by their greatest commun 
 divisor. 
 
 Examples. 
 
 Kcduce to their smallest terms : 
 
 *• 5(r' 
 
 7 1 1: V (I 
 '• a 0-4* 
 
 Ans. -J. 
 
 Ans. f J. 
 Ans. \. 
 
 8. 
 
 27 
 iZ' 
 
 9 ^1" 
 
 Ans. %. 
 ¥Tif Ans. -^. 
 
 11 1200 Ayto i 
 
 10 1 7rt 
 
 ^-'' iwa* 
 
 IT 2 7 () 
 
 14. 1^- 
 
 2^ 987 6 
 
 1 I 
 
 
 Ans 
 
 
 
 IG. Exprosis in its simplest form 114 divided by 285. Ans. f. 
 17. Express in its simplest form 1980 divided by 31 G8. 
 
 Ans. J. 
 
 CASE II. 
 
 144. To reduce an improper fraction to an equivalent whole 
 or mixed number. 
 
 1. Eeduce ^ to an equivalent whole or mixed number. Z>o 
 
 Since in 5 fifths there is 1 unit, in \^ fifths there are as many units 
 as 5 fifths are contained times in 19 fifths^ which are 3f . Therefore 
 y is iual to 3i. 
 
 2. Reduce "-^ to an equivalent whole or mixed number. -=^ ^hZ) 
 
 3. Reduce %^ to an equivalent whole or mixed number.^ 4 i^ 
 
 41 
 
 OPERATION. 
 
 g = 41 -^ 8 = 5J, Ans. 
 Rule. Divide the numerator by the denominator. 
 
 "What is the Hule for reducing fractions to their lowest or smallest terms ? 
 What is the Rule for reducing an improper fraction to a whole or mixed 
 number? 
 
 I 
 

 a 
 
 COMMON FRACTION'S. 
 
 Examples. 
 
 lui 
 
 Kecluco to an rquivalont whole or mixed number, 
 
 Jns. 13^. 
 
 Ans. 8j\. 
 Ans. 17. 
 8. ^\^. Ans. 183,V 
 
 A las 
 *• 
 
 12 
 
 fi. 1*1 
 
 1 7 
 
 7. -Ll« 
 
 1 O.T 
 J. ~^. 
 
 10 2 tt !t 
 11. .12 10 
 
 13. :♦"•! 
 
 Am. 22. 
 
 I • 1 
 
 Ans. 21. 
 
 1. lieauce 3] to an equivalent improper fraction 
 Smco in 1 there arc 4 /„«,.,,„,, ;„ 3 »,,„„ „, 3 ti,,,,. 4 >,«/,. 
 
 2. Eeduee 4 J to an equivalent improper fraction. - ^ 'A 
 
 3. Reduce 5| to an equivale.it improper fraction. ^ 4 ', 
 
 OPERATION. • - 7" 
 
 O-rr = 
 
 5x3 
 
 + - 
 
 '6 
 
 1.'5 +2 17 . 
 
 = -J, Ans. 
 
 3 
 
 /«.»«, fo & ,..«,/„rf ,„,, tu. nnnuratar, aU unU the JL 
 the denommatcr. 
 
 Examples. 
 
 lieduce to equivalent improper factions, 
 
 8. 13J4. 
 q '?7i.'5 
 
 4. 15f-. 
 
 5. 121 J. 
 
 6. 13^. 
 7 20n-2j 
 
 Ans. 
 
 Ans. 
 
 Ans. ^ 
 
 10027 
 
 5 
 
 12. In 115JI how many fifteenth.? 1 
 
 13. In 7191^ how many twelfths ? 
 
 10. 7251. 
 
 11. 128it. 
 
 Ans. 2JL'' 
 
 1 7 • 
 
 Ans. iA^* 
 Ans. 
 
 2 ,S 1 .-.A J 
 
 1 « " 
 
 Ans. L7_3« 
 
 1 5 ' 
 
 Ans. 5iLlL' 
 1 J • 
 
 i^l^ "" ''°''' '" '■='"™« "■^^'' "-"'"' '0 """^^^^I^^i^Lroper 
 
102 
 
 PRACTICAL ARITHMETIC. 
 
 ^ 1 
 
 I ii 
 
 146. To rofliiro a whole numV)or to an oquivalont improper 
 fraction liaving a given denoniinator. 
 
 MuUiphj the vhole mimhcr hj the given denominator, and take 
 the produrt for the nvmerator of the fraction. 
 
 14. lieduce 17 to fifths. Ans. ^. 
 
 15. Reduce 1 10 to an equivalent fraction -whose denoniinator 
 is 9. '/'/^^ 
 
 IG. Redi)u;e 30G to an equivalent fraction whose denominator 
 
 is 13. 
 
 Am. 
 
 307b 
 i3 
 
 COMMON DENOMINATOR. 
 
 147. Fractions have a Coi^imon Dfcnominator when they 
 have the same number for a denominator. 
 
 148. A common denominator of several fractions is a com- 
 mon multiple of their denominators. Hence, 
 
 The Least Common BE^o^u^iATon of several fractions is the 
 least common multiple of their demminatm's. 
 
 149. To reduce fractions to equivalent fractions having a 
 common denominator. 
 
 1. Reduce -i to fifteenths. 
 
 Since there are 15 fifteenths in 1 unit, in ^ of 1 unit there is ]^ of 
 15 fifteenths, or 5 fifteenths, and in f of 1 unit there are 2 times 5 
 fifteenths, or \{) fifteenths. Therefore § is equal to |^. 
 
 2. Reduce |- to twelfths. To sixteenths. To twentieths. 
 
 3. Reduce -| to twenty-fourths. 
 
 Since the proposed denominator, 24, 
 is 3 times 8, the denoniinator of the 
 given fraction, we nmltiply both terms 
 of the fraction by 3, which will not 
 
 change the value expressed (Art. 141), and have l^. Therefore I is 
 
 eqnal to ^-^. 
 
 How do you reduce a whole number to an equivalent fraction ? When 
 liave fractions a Common Denominator? AVhat is a common deno inator 
 of seveial fractions ? The Least Common Denominator ? Which is expressed 
 in larger terms, 5 or |§? Ans. ||. 
 
 5 
 
 8 
 
 OPERATION. 
 = 8 X 3 = 2i' ^^^' 
 
 " 
 
 • 
 
COMMON FRACTIONS. 
 
 103 
 
 4. ru'tluco IJ and 
 men dciioiaiiiutor. 
 
 Ol'KRATION. 
 
 3 _ 3 X 2 6 
 
 8 ~ « X 
 
 Id 
 
 2 _ ^ ^ 
 2 ~ Iti ( 
 
 10 ; 
 
 Ans. 
 
 to erjuivalciit fractions having a coni- 
 
 Since tlie (Iciioininator, 16, of tint 
 Hccoiul fraction, in 2 times tlio (U'noiiii- 
 nator of tlie first fraction, we nmltijily 
 both turnis of tlie ^ hy 2, causin*,' its 
 (leuoniinatur U- hvcome 1(5, iiiid luive 
 ^"ff and ^ff. Thercfure ^ and ,",^ are 
 (■(inal to ,'{j ami ^\^. 
 
 OPKIIATION. 
 
 
 
 « « 1 1x6 
 
 ^ - - 2 2x0 
 
 6 ' 
 12 
 
 
 ,3 3x3 
 *--x-4 4x3 
 
 
 12 
 
 ../. 
 
 o ^ 1x2 
 
 6 = 2x3 = 0x2 = 
 
 o 
 12. 
 
 
 Least com. niuL - 2 x 2 x 
 
 3 - 
 
 12 
 
 T). Kodnce |, J, and ^ to eqnivahiut fractions having th<' 
 least common denominator. 
 
 We find the least 
 eonnuon multiple of ilie 
 dcnomiiKitor.s to lie 12, 
 which mu8t he the least 
 eonnuon denoniiuat(»r 
 (Art. 148). We then 
 reduce the given frac- 
 tions to e<jiuvalent frac- 
 tions having this de- 
 nominator, by multijtlyiu'^f both terms of each fraction by such a num- 
 ber as will make its denominator become 12, and have y.j, -^.^, and ,'4.. 
 Therefore ^, |, and J are eijual, respectively, to -^.j, ^%, and ^^-. 
 
 Rule. Maltiphj both terms of each fraction hy the denominators 
 of the other fractions. Or, 
 
 Find the least common midlijyle of the denominators for the least 
 co)h. . denominator, and multiply the terms of each fraction hy 
 such a mimher as ivill reduce it to an equivalent fraction ivith that 
 denominator. 
 
 All the fractions should be reduced Lo their smallest terms before 
 finding the least common multiple of their denominators. 
 
 When the number by which to multiply the terms of any fraction is 
 not apparent, it may be determined by dividing the common denomi- 
 nator by the denondnator of the fraction under consideration. 
 
 What is the Rule? AVhen the multiplier of the terms of any fraction is 
 not apparent, how may it be determined ? 
 
'V I :x. 
 
 lot 
 
 PRACTICAL ARITHMETIC. 
 
 i I 
 
 [ I 
 
 Examples. 
 
 Itoducc to equivalent fractions having a common donomi- 
 nator, 
 
 G. 
 
 7. 
 
 1 
 
 and 
 
 T^' 
 
 and J(j. 
 
 An<! Jl- JL 1-0 _7_ 
 
 x^ 
 
 33 _ 
 
 3 0' "5t>' ;>>T' 
 
 lb"' 
 8. f, -j-i and -j^^. Ajis. 
 
 Reduce to equivalent fractions having the least common 
 denominator, 
 
 4 3 r,,Tfl « ^,,„ 20 2 7 ii-l 
 
 *^' 1T> "55 ^""^ 15* ^/t*. -4-5-, -^^, 
 
 10. ^, ^, I, and ^2' ^^^' 
 
 ^ 
 
 4 5' 
 
 3 '.» 
 
 11. 
 
 12. 
 
 ^' "S"' T¥' ^'^^ T(T- 
 
 3 8 7 q«f] 
 
 1 ;•> 
 
 A<n9 C'(^ 4 5 1 r, .'i<» 
 ii/ti. %j-^, Y-r, -f2> Tu" 
 
 ^^"^' l^.c rOJ? jao 
 
 Arta «0 07 3 2 10 ir,r, 
 
 .illlo. irTwTi "^iwu TTiY/Tj •til II • 
 
 
 UFO"? Tioo' ircro"' 3 0" 
 150. If there are whole or mixed numbers with the given 
 
 fractions, they must le reduced to improper fractions, before 
 
 applying the rule. 
 
 13. Reduce \, 3}, C, and |- to equivalent fractions having 
 
 the least common denominator. 
 
 Ans. 
 
 4 
 
 «' 
 
 4S 
 
 "' s* 
 
 14. Reduce 5, 4^^ 3y, and ~ to equivalent fractions^ having 
 the least common denominator. ^=^ (( . ^^\ , LAJL, >^/ 
 
 15. Reduce 1, 4^,, 81-, and 12^- to equiv 
 
 ralent mrctions na 
 
 a common denominator, 
 
 //„o 120 518 102 
 yi/i6. y.j^, y^^, -^y 
 
 vnijx 
 
 \a 7 5 
 
 ADDITION. 
 
 151. Addition of Fractions is the process of finding a 
 number equal to two or more fractions. 
 
 152. Numbers to be added must be of like name or kind. 
 (Art. 39. 1.) Hence, 
 
 Fractions can he added only when they express like fractional 
 units, or have a common denominator. 
 
 1. What is the sum of S, 5, and 2 ? zr= / '■- 
 
 "What must bo done when there are whole or mixed numbers with the 
 given fractions ? Wliat is Addition of Fractions ? When only can fractious 
 be added ? 
 
 I 
 
 II ■•Hiil-ii 
 
COMMON FRACTION'S. 
 
 105 
 
 ? 
 
 /^^ 
 
 Since the fractional units are of like name and kind, the sum of 
 2 s^cvenths, 3 seventJis, and 5 sevenths, is 2 + 3 + 5 sevenths, or ip^ 
 which reduced is 1~. 
 
 2. What is the sum of -1 §, aiidp - / / 
 
 ;'). What is the sum of -^\, H, and ^^j ] _-r' / "^ 
 
 4. What is tlie sum of 4\f, /-, and |^- ? - 
 
 5. Lot it be required to find the sum of J 
 
 f 
 
 J 
 
 u 
 
 
 
 OPERATION. 
 
 
 3 4 
 
 4 + 5 ^ 
 
 15 16 
 ~' 20 "•" 20 
 
 \T^ + IH 
 20 
 
 31 
 
 20 
 
 1^1 Ans. 
 
 Reducin;^' the given fractions to equivalent fractions having,' a 
 common denominator, so that they may express like fractional units 
 (Art. 152), we have 15 tweyiticths and 16 twentieths, which added 
 f,dve 31 twentieths, !5J, or, by reduction, l.J^-. Therefore, the .sum of 'i 
 a-\d * is m. 
 
 Rule. Reduce the fractions, if necessary, to equivalent fnif- 
 tions having a common denominator, and write the sum of thn 
 numerators over the common denominator. 
 
 i\ 
 
 IM 
 
 Examples. 
 
 Find the sum of 
 6. I, f, and -j^. Ans. 1^. 
 
 7 2 6 „„,1 3 1 // / / ? 
 
 Q 7 11 nr,r1 4 ^^10 9 11 
 
 •IJ- 
 
 Ans. 2!!. I. 
 
 10. 
 11. 
 
 19 7 1 17 2. 3 r,,>f] 2 
 
 ^-'' ■§' Taj 18' 2~r> '-^"'^ 2 ; 
 
 1 3 S T,i,| 
 ■2") 4) IT) -"-"^^ 
 
 1 7 11 r,pf] 2 1 ^ fl IP 
 
 TIT' TIT' Til' '^'^^'^^ \\i' - y f- 
 
 Ans. 3i^\. 
 
 „— *• 
 
 ^' T?7' TJ' ^^^ TT" -^^^'5. Sjyy. 
 Q 8 7 4 ^,,,4 ;3 ^„<j Q 109 
 
 1 Q 1 23 2 7 j,„-^ 11 
 
 U. AVhat is the value of ^^ + ^ + \^ + fi ? yi?i5. lOf:-;. 
 15. What is the value of vo + 15 + s + 33 j ^^^^^ 5X 
 
 153. When there are mixed numbers, we may 
 
 Add the fractional and integral imrts separately^ and then add 
 their S2ims. 
 
 IG. What is the sum of 7h, U\, and 5| ? , J^ ^ 
 
 "Why, in the operation of example 5, are the fractions reduced to equiva- 
 lent fractions having a common denominator? What is the Rule? How 
 may we add when there are mixed numbers ? 
 
 t 
 
 1 
 
w 
 
 
 lOG 
 
 PRACTICAL ARITHMETIC. 
 
 1 
 
 OPERATION. 
 
 + 
 
 1 
 
 + 1 = 
 
 -- + 
 
 F 
 
 + S 
 
 7 + 13+5 
 
 ^^ 
 
 What is the sum of 
 
 Ans. lOJf 
 
 18. |,i3-,^,andfJ?-.5'K 
 
 19. 21^, 18-5, 4,. ''-ad 26| ? '^^ 
 
 Ans. 70-^1 
 
 22 
 
 - 11- 
 
 - Ot 
 
 Ans. 
 
 90 *? 1 •'' •'' ^ nnd 1 ^ ^ ? 
 
 Ans. 5M. ^-j-.X 
 21. 17i 10, 141, andl3f?-:^i? 
 
 '''^ ^3^l + ioi,and^? ^"::; 
 
 Ans. W.}. 
 
 I if 
 
 If « 
 
 1.1 • 
 
 ! f 
 
 f :l^ 
 
 SUBTRACTION. 
 
 154. Subtraction of Fractions is tlie process of subtracting 
 one fraction from another. 
 
 155. One number can be taken from another only when of 
 the same name or kind (Art. 44. 1). Hence, 
 
 A fraction can he subtracted from another only ivhen loth 
 express like fractional units, or have a common denominator. 
 
 1. What is the difference between I and -^1 
 
 Since the fractional imits are of like name and kind, the difference be- 
 tween 7 ninths and 4 ninths, is 7 - 4 ninths, or f , which reduced is ^. 
 
 2. What is the difference between il and ^ ? "^Wo^ if 
 
 3. What is the difference between if and y\ ] ^ ^^ ^ 
 
 4. What is the difference between \\ and \\ '{^ 
 
 5. What is the difference between ^ and J 1^=» -iT 
 
 Reducing the given frac- 
 tions to equivalent fractious 
 laving a common denomi- 
 nator, so that they may express like fractional units, we have 7 eighths 
 and 6 eighths, whose difference is 1 eighth, or ^. Therefore the differ- 
 ence between ^ and | is ^. 
 
 What is Subtraction of Fractions ? When only can one fraction be sub- 
 tracted from another ? Why, in the operation, do we reduce the tractions to 
 equivalent fractions having a common denominator ? 
 
 3 
 
 i 
 
 8 
 
 OPERATION. 
 
 { 
 
i 
 
 4 J'^-"^' 
 
 C0M3I0N FRACTIONS. 
 
 107 
 
 Rule. Reduce the fractions, if necessary, to equivalent frac- 
 tions having a common denoirmiator, and write the difference of the 
 numerators over the common denominator. 
 
 Examples. 
 
 What is the difference between, 
 
 K 
 
 6. ^\ and ^V ? 
 
 7. f and -/^ ? 
 
 8. If and } 1 
 
 9. i and ^ 1 
 10. ^V^and^? 
 
 Ans. 
 Ans. 
 Ans. 
 
 11 
 
 _89 
 1 li- 
 
 .17 0^ 
 
 7 o • 
 
 11. J 
 
 and ^ ? 
 
 12. il^and^l? 
 
 13. A^andf ? 
 U. ^»yand-/5? 
 15. i« and^^s? 
 
 Ans. III. 
 Ans. -g^. 
 
 156. When thc^re arc mixed numbers, we may either 
 
 Reduce them to im,proi)cr fractions and apply the rule, or sub- 
 tract the ivhole numbers and fractions separately and unite the 
 results. 
 
 16. What is the difference between 5 J and 31] 
 
 5f 
 
 31 = 
 
 OPERATION. 
 4 fi 2 5 _ 2 1 
 
 2 1, Ans. 
 
 Ur, ov — ""ff — «54- — o-q- — ^isT' -Ans. 
 
 17. What is the difference between 17j^^j and U^^j^l 
 
 Ans. 4./^ 
 
 18. What is the difference between 18^ and ISf ? 
 
 Ans 
 
 19. How much greater is \\^ than \\}y ? Ans. 
 
 20. Take 61 from 14,%.^^^ ^^^^^ 
 
 OPERATION. 
 
 1 
 
 15"- 
 1 
 
 4¥- 
 
 14_s 
 
 61 = lli-^ 
 
 15 
 1 2 
 
 711 
 
 ' I'll 
 
 Ans. 
 
 Or, 14i^ - 61 = 131^ - 6j^ - 7ii Ans. 
 
 Here, iu the second process, as we cannot take [h; from Jjr, we 
 reduce 1 of the 14 units to ticelfths, making i|, and, ailding the i^t, 
 have \l ; taking, then, the 6 units from the remaining 13 units and 
 the y^2 from \^, and, iiniting the results, we have 7^. 
 
 What is the Rule ? When there aie mixed numbers how may we pro- 
 ceed? 
 
108 
 
 PRACTICAL ARITHMETIC. 
 
 21. ^from G5. 
 
 22. IGl from 19. 
 
 5i. 
 
 Alls. 
 Alls. 21 
 
 Subtract, 
 
 23. -,\ from 3. Ans. 2|4. 
 . J|24. 13^ from 42?r. Ans. 26^. 
 
 APPLICATIONS. 
 
 1. A ftirmer sold 3^ tons of hay to one person, 2J to an-/53^2^ 
 other, and If to a third ; what was the whole quantity sold ?/^^ 
 
 Solution. The vhole quantity sold must equal 3i + 2'1 -^ If toii.f, 
 OTy reducing the /^'actions to equivalent fractions having a common de- 
 nominator, 3J§ + 2h^ + ^Ici ^^''•''■5 w7iicA are 8^^ tons. Therefore, (L-c. 
 
 2. Joseph Kirk owns |- of a factory, his brother owns -^^ of 
 it, and his father -g"^ of it ; how much of it do they all own ? 
 
 .Ans. ^-^. 
 
 3. A man has 3 sheep ; tiie first is worth $6^, the second 
 $8y-, and the third $9i ; how many dollars are the whole 
 worth? 5^^^^ 
 
 4. Two boys were talking of their ages ; one said he was 
 9|- years old, the other said he was 4— years older ; what 
 was the asfe of the oldest ? Ans. l^^l years. 
 
 V 
 
 5. If there is a pole standing so that § of it is in the mud, 
 ■?- of it in the water, and the rest above water, how much of it 
 is above water 1 "^ -^ 
 
 Solution. If § of the loole is in the mud and | of it in the icater, 
 there must he in the mud and uuter § + f of i% or \^ + j|y = f j of it ; 
 and as the rest of it is above water, tlicre must he above water ^^ - ^^ 
 = 3p(j of it. Therefore, d'c. 
 
 G. A lady, being asked her age, said her husband was 37^ 
 years old, and she was not so old as her husband by Sy'g- years ; 
 what was her age 'i Ans. 28y^y years. 
 
 7. If I have 16^ barrels of apples and sell 12i barrels, 
 how many barrels shall I have left 1 Ans. 3^ barrels. 
 
 8. A man having two tracts of land, one of 131 1^ acres, and 
 
 Review Questions. What is a Proper Fraction: (135) What is an 
 Improper Fraction? (136) \Vliat is a Mixed Number? (1 !7) The terms of 
 a fraction? (138) 
 

 COMMON FfiACTION\S. 
 
 109 
 
 liad he remaining ? 
 
 the other of 160/^ acres, sold 150„v acres; how many acros 
 
 Ans. 112,'^. 
 
 ]\IULTIPLICATION. 
 157 Multiplicatioii of Fractions is tlie process of multiply- 
 - Avlien one or both of the factors are fractions. 
 
 in- 
 
 CASE I. 
 158. To multiply a fraction by a whole number. 
 
 ]. AViiat is tlie product of -\ multiphed by 3 ? 
 
 The product of 4 multiphed by 3, or 3 times 4 sevenths U \^ 
 sa-enths, or ^^^, wliich reduced is If. ' 
 
 •2. What is the product of |- multiplied by 4 ? 
 3. Let it be required to multiply -Z^- by 3. 
 
 OPERATION. 3 times 4 Jlfieenth^, or 
 
 ^1 X 3 ^ -^ ^ 2 
 ^^ 15 
 
 12 4 , 
 tT = ?, Alls. 
 
 *iV^ ^'^ ih ^vhich reduce. I 
 is i. Or. 
 
 0'-' u^ 
 
 3 - -J— 4 ^ 
 
 15 -- 3 "= 5' ^'^• 
 
 Since dividing' the de- 
 nominator multiphes t^r^ 
 
 fraction(Art. 14i.2),.-itiraes 
 ij*- is — * - , or 4. 
 
 That is, the result is the same M-hether we tak^s'times'the numSer 
 of parts by multiplying the numerator 4 by 3 or the numi r , f 
 re^^ unchanged, .. make their ... ^"l- :;' lis l^J^aS 
 mg tJie denommator 15 by 3. ^ *^ 
 
 rto?i6 W7//io?i/! a remainder. 
 
 Multiply 
 4. If by 8. 
 
 ^- TfVby 11. 
 6. T^V bv 25. 
 
 Examples. 
 
 
 Alls. 
 
 '*■ Twu by 15. Ans. 14X 
 
 a fVbyl20. 
 
 9. J by 3. 
 10. i^ by 20. 
 11- TVby45. 
 12. ^1 by 9. 
 
 Ans. 2]. 
 
 ^/«.?. 21. 
 Anf R ^ 
 _:^^;^;_^^^- i^V by 100. .^;,,. '48^^.' 
 
 WJ.at is Multiplication of Fr'actionsTl^^i^^^^r^ii^^e. 
 
I 
 
 i 
 
 110 
 
 PRACTICAL ARITHMETIC. 
 
 159. When the multiplicand is a mixed number, we may 
 
 Multiply the fractional and integral varts separately, and add 
 the products. Or, 
 
 lieduce the mixed number to an impoper fraction, and then 
 multiply. 
 
 14. Let it be required to multiply 7 J by 5. 
 
 OrERATION. 
 
 I X 5 = «-^-« = 1^. ^ 3} 
 7x5= 35 
 
 7f X 
 Or, 7^ 
 
 5 = 
 
 38f, Ans. 
 
 X 5 
 
 .•31X6 
 
 Multiply 
 15. 5| by 8. 
 IG. 3y«5 by 45. 
 17. 31^^ by GO. 
 
 ip = 382-, Ans. 
 
 Ans. 45J. 
 Ans. 1890. 
 
 Ans. 378f. 
 
 18. 25i by 15. 
 
 19. 19^ by 10. Ans. 198i 
 
 20. lUJby 17. Alls, 1901J 
 
 CASE II. 
 160. To multiply a whole number by a fraction. 
 
 Multiplying a number by a fraction is equivalent to taking the 
 TART of it denoted by the fraction. 
 
 For multiplying lany number, as 6, by h, is taking 1 half of 6 ; by 
 •J is taking 1 third of 6 ; by f is taking 2 thirds of 6, &c. 
 
 1. How many are f times ten, or f of 10 ? 
 1 fifth of 10 is 2, and 2 fifths are 2 times 2, or 4. 
 
 2. How many are f times 14 ] f times 18 ? | times 24 1 
 
 3. Let it be required to multiply IG by f. 
 
 OPERATION. -^ fourth of 16 is 4, and 3 fourths 
 
 4 
 10 
 
 16 
 
 X 5^ = 
 
 X ^ - 
 
 ^ 
 
 of 16 are 3 times 4, or 12. Or, 
 
 3 times 16 is 16 x 3, and, as the 
 multiplier is 3 fourths, the product 
 ~ ' -^""'- must be only a fourth as large ; 
 and ^ of 16 X 3 is 1^, or 12. 
 
 When the multiplicand is a mixed number, how may we proceed? To 
 what is multiplying a number by a fraction equivalent ? 
 
 rg 
 
COMMON FRACTIONS. 
 
 Ill 
 
 add 
 then 
 
 I- 
 
 Rule. Midtijjhj the whole number by the numerator^ and 
 divide the p'oduct by the denominator. 
 
 Multiply 
 
 4. l24by,V 
 ^. G3 by ^L. 
 
 0. 72 by 1 
 
 7. 3G5 by ^^. 
 
 Examples. 
 
 Ans. 22>\. 
 Ans. l^jij. 
 
 Ans. 831-^. 
 
 8. 40by^y_. 
 
 9. G93 by f. 
 
 10. 75 by Jo. 
 
 11. 1000 by" -^V 
 
 Ans. 17. 
 Ans. 385. 
 
 'Ans. 130. 
 
 161. When the multiplier is a mixed number, we may 
 
 Multiply by the fractional and integral parts separately , and add 
 the products. Or, 
 
 Reduce the mixed 7iumber to an improper fraction and then 
 multiply. 
 
 12. Let it be required to multiply 33 by 3 1. 
 
 OPERATION. 
 «?C{ 2 __3 3x2 _ Ofi _ iqi 
 
 33 X 3 - 
 
 99 
 
 I 
 
 33 X 3f = 1121 Ans. 
 
 Or, 33 X 31 = 33 X 1/ = ^^ = 1121 Ans. 
 
 13. What is the value of 4 x f L|. ? Ans. l-j-lV 
 
 14. What is the value of 106 x 31/^? ^'^rh^ 3M f^' 
 
 15. Find the product of 536 niultiplied by ^V^^Ans. SgIbSI. 
 
 CASE III. 
 162. To multiply a fraction by a fraction, y y . 
 
 1. How much is f times |^, or | of ^ ? ^^ oj^ 
 
 1 fourth of I is 5 twenty-eighths (Art. 141), and 3 fourths are 3 
 times 5 twe7it^-eighths, which are 15 twenty-eighths, or ig-.' 
 
 2. How much is | times f ? | of j^^^ ? § of ^^ ? 
 
 3. Let it be required to multiply |- by f .-= 3 
 
 72 
 
 What is the Rule ? How do /ou proceed when the multiplier is a mixed 
 number ? 
 
 -^mammtaaUmmmiM 
 
112 
 
 vr.ACTICAL ARITHMETIC. 
 
 orEllATION. 
 
 3 
 
 5 '^8 = 
 
 ^ X 3 
 2 
 
 5 X 
 
 letr/hthoi'f^ is-J-y, ana 3 n(jhihs 
 'J of j are 3 times .,-^*— , Avliich is ±.li? 
 
 10'"^''*- or, bymlucin-, j^Ij. Or, 
 
 3 times | is i-J-'^, and, as the mul- 
 tij)lier is 3 eiyhths, the iirodiict must 
 he only 1 eiyhth as large, and ^ of ^ is J-^^ (Art. 141), or -'q. 
 
 Rule, Multiphj hij the numerator of the muItijiUcr ami divide 
 the 'product hy its denominator. Or, 
 
 Multiply the mtmerators tog-'Jier far a new numerator, and the 
 denominators for a new denominator. 
 
 This rule is general, and applies in the two preceding cases, since a 
 whole number may he written in a fractional form (Art. 139). 
 
 
 Multiply 
 
 4. I by 1. 
 
 5. 1 by i-«. 
 
 Examples. 
 
 Ans. ^^. 7. ^V by ^V 
 
 Ans. 
 
 •J 5 
 
 8. .'V by A. 
 
 y- T^ by vo. 
 
 Ans. 
 
 Ans. 
 
 ^1 00" 
 
 163. When fractions are connected by the word of, as | of 
 -^, 5 of f o{j\, the expression is called a Comjmmd Fraction. 
 
 A Compound Fraction may be treated its an ctcpression of 
 Midtiptlication (Art. 1G2). 
 
 10. What is the value of f of i« 1 
 
 3 „ IG 16 
 
 4 "^ 21 = "21 " 4 
 
 OPERATION. 
 4 
 3 Id x^ 
 
 7 
 
 1- 
 
 7 
 
 
 
 ^1^ 
 
 4 , 
 
 I-, Ans, 
 
 11. What is the value of -|§ of |f ? 
 
 12. What is the value of f of | of § ? 
 
 1 3. What is the value of i- of § | of |-|- ? 
 
 72 
 TIT' 
 
 Ans 
 Ans. f 
 
 164. When there are mixed numbers among the factors, 
 
 What is the Rule ? What is said of the Rule ? AVhen fractions are con- 
 nected by the word oj\ what is the expression called ? How may a compound 
 fraction be treated ? 
 
COMMON FRACTIONS. 
 
 113 
 
 Heducc the mixed numbers to equivalent im2))'oper fractions, and 
 then mvltij)!//. 
 
 Multiply 
 
 14. Ui by i 
 
 15. 29^byf. 
 IC. 10^ by 115. 
 
 /I us. O-V. 
 ylns. 3G. 
 
 17. l.^^by 1^1. Ans. 3.^. 
 
 18. ^Al>ylA. ytns. 2U. 
 
 19. 27|lby 8^. ./^//x. 240^;!. 
 
 20. What is the value of ^ of ^- of 7i ? 
 
 Ans. 5 ?>:":, 
 
 / 
 
 APPLICATIONS. 
 
 1. John has ^ J of a dollar, and his brother has 15 times as 
 much ; how much has his brother? Ans. !?14|. 
 
 2. What cost 19G bushels of corn, at | of a dollar a busheU/ //^ 
 
 3. At -{jj of a dollar a day, how much can a boy earn in 2G 
 ■ days ?^/^ '"I, 
 
 4. At 80G0 for a tract of land, what cost l^^ of it ? 
 
 Ans. .9544. 
 
 5. At $7 a yard, what cost -J-f of a yard of cloth ? ^(f)^ 
 
 6. At $10f a barrel, what cost 24 barrels of flour ?><^^ /J'^' 
 Solution. At %\^% a barrel, 24 barreb of Jlour ivill cost 24 times 
 
 $1()5, or 24 times $10 + 24 times ^. 
 
 24 times $10 is .$240; 24 times $1, or $^'^^"^,is ^Ui ; and .$240 
 + $15 is $255. Therefore, &c. 
 Y^ 7. At the rate of 23fl miles an hour, how far can a train of 
 cars move in 24 hours? Ans. 5G0t miles. 
 
 8. If 2|g- of a yard of cloth be required to make a coat, 
 how much will be required for 4 coats ? -"'=- f/) h- 
 
 9. If a hat cost \ of %\^\, and a vest % as much, what was 
 the cost of the vest 1 Ans. $3}-|. 
 
 10. William Benton has in his fiirm ^^ of G4 acres, and his 
 son owns f as many acres ; how many does his son own ? 
 
 Ans. IG acres. 
 
 When there are mixed nixmbers among the factors, how do you proceed ? 
 Review Questions. What is the Doiioruiuutur of a Fraction? (1.32) 
 
 The Numerator? (133) 
 flenominator ? (138) 
 
 What does the nuiiierator express? (13S) The 
 
 H 
 
 ■r 
 
 !t^ 
 
lit 
 
 PRACTICAL ARITHMKTIC. 
 
 DIVISION. 
 
 165. Division of Fractions is the procoss of dividing, when 
 the dividend or divisor, or both, are fractions. 
 
 I 
 
 I i 
 
 I 
 
 CASE I. 
 
 166. To divide a fraction by a whole number. 
 
 1. Wliat is the quotient of -}-J divided by f) ? 
 
 The quotient of }? divided by 5, or } of 10 eleventh is 2 elevenths, 
 
 or A. 
 
 2. Wliat is the quotient of §]• divided by 7 1 Of Yi divided 
 by 4? 
 
 .3. Let it be required to divide \'i by 3 
 
 12 
 
 15 
 
 -r 3 
 
 Or, 
 
 OPERATION. 
 12-^-3 
 1.0 
 4 
 
 r4 
 
 - ,., Ans. 
 
 1 r,' 
 
 "" - 3 ^ J^ , = ^, Ans. 
 15 15 X ^'- 
 
 15' 
 
 ^ of 12 fifteenths, or U-tJ^- 
 Or, 
 
 Since multiplying the de- 
 nominator divides the frac- 
 tion (Art. 141. 3),^ of in i'^ 
 
 _!_:-_, which reduced is A. 
 15 X :i' '" 
 
 That is, the result is the same whether we take ^- of the number of 
 parts by dividing tlie numerator 12 by 3, or, the nuuiber of parts 
 remaining unchanged, we make their size l as large by multiplying 
 the denominator 15 by 3. 
 
 EuLE. Divide the numerator hj the whole numher, when it can 
 he done without a remainder. Or, 
 
 Multiiily the denominator hij the whole number. 
 
 -i 
 
 4. 
 5. 
 6. 
 7. 
 
 Divide 
 
 i-l by 8. 
 *o by 20. 
 
 Examples. 
 
 7 
 
 by 29. 
 
 Ans. 
 
 Ans. Y^f^-- 
 Ans. 
 
 W1 
 
 12. What is the value of i^4 -f- 25 ] 
 
 3 1 
 
 f by 7. 
 
 tVit by 48. 
 ^ by 128. 
 
 8. 
 
 9. 
 10. 
 11. xl^by 23. 
 
 Alls. fV 
 Ans. -il^. 
 
 j.'L nS. T'g'. 
 
 Ans. Y^. 
 
 What is Division of Fractions? What effect has multiplying the denomi- 
 nator ? What is the Rule ? 
 
I 
 
 COMMON FRACTIONS. 
 
 U.-i 
 
 167. When the dividciul is a mixed numhor, we mny either 
 Divide the integral part of the mixed numtier, and redneing the 
 remainder to an ijnproj^erfrartinn, divide it. Or, 
 
 Itedure the mixed number to an iinpi'oper fraction, and then 
 divide. 
 
 13. Let it be required to divide 2G;-; by 4. 
 
 OPKRATIOX. 
 
 4)2G=(G 
 Divide 
 
 14. 75j^ by 9. Ans. 8^ 
 
 15. 11^^ by 31. Ans. ,[J,^. 
 
 16. 203^. by GO. .i;... Z^ 
 
 17. 3^^i-by 17. ^«s. ,-j. 
 
 18. 91|-by 15. y/7is. G-V 
 
 19. Sl3{-^hy 100. Ans. 8-,'^;)/^-. 
 
 m 
 
 CASE II. 
 
 168. To divide a whole number by a fraction. 
 
 1. How many times is i contained in 10 ? 
 
 1 is contained in 10, 10 times and since I fifth nmst be contained 
 in any nnml)er 5 times as many times as 1 is contained in it, 1 fifth 
 is contained in 10, 5 times 10 times, or 50 times. Or, 
 
 10 is equal to -^^ . and 1 fifth is contained in 60 fifths, 50 times. 
 
 ' 2. How many times is 1 contained in 6 ? ^ in 16 ? } in 
 lo? 
 
 3. How many times is | contained in 5 ? 
 
 5 is equal to U^ and 2 thirds are contained in 15 thirds, 7^ times. 
 
 4. How many times is | contained in 7 ? f in 3 ? -/^ in 
 6? 
 
 5. Let it be required to find the quotient of 12 divided by ^. 
 
 r:i 
 
 When the diviclenJ is a mixed number, how may we proceed? 
 
IIG 
 
 PRACTICAL ARITHMETIC. 
 
 orERATIOJ 
 
 12<livi<loa Iv 1 is 12 or 12, 
 niwl, sim'e 1 fourth jimst be 
 coiitaiiied in any iiuiiibcr 4 
 times as many timeH as 1 is 
 contained in it, 12 divided by 1 fourth is 4 times Y> or -'-,—, which 
 
 12 -f i = -J- - 48, Am. 
 
 is 48. 
 
 6. Let it be required to find the quotient of IG divided by f. 
 
 1(5 divided by 2 is y, and 
 16 divided by ^ is 5 times \^^ 
 or -^-5~, which reduced is 40. 
 Or, 
 
 Reducing the dividend 16 
 to an equivalent fraction of 
 like name with the divisor, so 
 
 OPERATION. 
 
 8 
 %6 X 5 
 
 16 - I = 
 
 40, Ans. 
 
 Or, IG 
 
 = bo ^ I ^ iO,Ans. 
 
 that they may express like fractional units (Art. 6< . 1), we have 80 
 fifths to be divided by ^fifths, and '2. fifths are cont; iied in 80 fifths, 
 40 times. 
 
 Rule. Divide by the numerator of the divisor, and multi'phj hy 
 its denominator. Or, 
 
 Reduce the given whole number to an equivalent fraction of the 
 same denominator as the divisor, and divide the numerator of the 
 dividend hy the numerator of the divisor. 
 
 When the divisor is a mixed number, it must be reduced to an 
 equivalent improper fraction before dividing. 
 
 7. 
 
 8. 
 
 9. 
 
 10. 
 
 Divide 
 5 by 
 29 by J. 
 10 by ^^. 
 
 Examples. 
 
 7 
 TTT* 
 
 7'' 
 
 Ans. 
 
 97 by 5f. 
 
 Ans 
 Ans 
 
 1000. 
 1710 
 
 11. 49byl«. 
 
 12. 50by i\. 
 
 13. 17G by fj. 
 
 14. 100 by 4^-. 
 
 Ans. 88if. 
 Ans, 3G8: 
 
 Ans. 
 
 90^0 
 
 CASE III. 
 
 169. To divide a fraction by a fraction. 
 
 1. How many times is \ contained in }1 
 
 1 
 
 t 
 
 Why, in the second operation of example 6, is the dividend IG reduced to 
 an equivalent fraction expressing fifths? What is the Rule? When the 
 divisor is a mixed number, what must be done? 
 
I 
 
 COMMON FRACTIONS. 
 
 117 
 
 1 is contiiiiiPtl in J, | of a time, and, since 1 third must bo contained 
 in any number 3 time.s a.s many time.s as 1 is contaiued in it, 1 third is 
 containi'd in ^, 3 times ^, or \ times, whicli is 2^ times. Or, 
 
 ^ is equal to y^ and I to ^j, and 4 twelfths are contained in 
 twelfths 2| times. 
 
 2. How many times is J contained in ^] i i^i 5^ 
 
 3. Let it be required to divide -J by ^. 
 
 4 
 
 OPERATION. 
 
 ^ ^ 3 6 
 
 5 X ;l' 
 
 5 
 
 I divided by 2 is 
 jj', .J, and, as the 
 divisor is 2 thirds, 
 the quotient must 
 be 3 timeh' as hvi'ge, 
 
 or 
 
 
 whi 
 
 eh re- 
 
 Op 4^9-12 . 10_19j^in_11 ^„<; 
 
 duced is 1^. Or, 
 1 13 equal to ]9, and J to [g, and 10 fifteenths are contained in 12 
 fifteenths \\ times. 
 
 EuLE. Divide hij the numerator of the divisor, and midtiphj 
 the quotient hy the denominatur. Or, 
 
 Hcduee the dividend and divisor, if lucessnry, to equivalent frac- 
 tions having a common denominator, and divide the numerator of 
 the dividend hy the numerator of the divisor. 
 
 Tliis rule is general, and applies in the preceding two cases, since u 
 whole number may be written in a fractional form (Art. 139). 
 
 The process of the rule, in brief, consists in invertimj the tei'ms of 
 the divisor, and then finding the jn'oduct of the dividend by the inverted 
 divisor. 
 
 i 
 
 Divide 
 
 3 
 
 4* 
 
 4. A by 
 
 5. « by i|. 
 
 6. t\ by ^%. 
 
 '' 8 "J' IS)* 
 
 .8. jjj by YfTU' 
 
 Examples. 
 
 Ans. f . 
 
 Ans.f-^^. 
 
 ^LnSt i-Tj-jT. 
 Ans. 10. 
 
 9. f } by A4. Ans. i«j. 
 
 10. oV by m- ^ns. ^l 
 iu by f §. ^^/yt^ , /^ 
 l^^yyih' Ans. 61 
 T(J(jVo "^y TUo' Ans. ^^. 
 
 11. 
 12. 
 13. 
 
 How many times is ^ contained in 1? If lis contained once in any number, 
 how many times is \ contained in the same number? Keiieat the Kule. 
 A\Tiat 13 said of the rule ? Thu process of the rule ? 
 
r 
 
 118 
 
 PRACTICAL ARITHMETIC. 
 
 170. When either the dividend or divisor, or both, are 
 7aixed mimb^s, 
 
 Reduce the mixed numbers to equivalent imp'oper fractions, and 
 then divide. 
 
 Divide 
 
 14. 5Gf by §. 
 
 15. I by 51 
 
 A 7lS. Vt. 
 
 Ans. 94^V- I 16. 4f by 5J. 
 
 Ans. l\. I 17. 18§^ hyii. Ans. 33^^- 
 
 171. When a fraction has one or both its terms fractional, 
 
 as ^, a, or ¥, the expression is called a Complex Fraction. 
 '5 s 
 
 A CoraiAex Fraction may be treated as a case in division 
 
 (Art. 58). 
 
 18. What is the value of -^? 
 
 n 
 
 T 
 11 
 
 OPERATION. 
 
 2 
 
 X 11 
 
 22 
 
 = 31, Ans. 
 
 7 • 11 ~ 7x0 ~T 
 
 Here the given fraction is regarded as an expression of division ; 
 the nnmerator ^ answering to the dividend, and the denominator ^ 
 to the divisor, and the value is found by performing the division 
 indicated (Art. 1G9). 
 
 19. iii = what] 
 6 
 
 Ans. j^V- 
 
 20. ^^ = what ? c:>^. - 
 
 ro 
 
 21. 
 
 100 
 
 17 
 
 la 
 
 = what?^w5. llljf 
 
 Ans. y. 
 
 22. ? ^ what ? 
 t 
 
 24. ^\s_^ what? ^«5. 4320 
 
 172 8 
 
 what % 
 
 Ans. \. 
 
 o t a 
 
 25. What is the quotient of ^ divided by ^v? Ans. \\. 
 
 ^^. What is the quotient of | of ^^F divided by | ? 
 
 When either dividend or divisor, or both, are mixed numbers, how do you 
 proceed ? When one or both terms of a fraction are fractional, what is the 
 expression called ? How may it be treated ? 
 
 ,. 
 
COMMON FIl ACTIONS. 
 
 Ill) 
 
 / 
 
 ^ 
 
 1 
 
 ' 
 
 APPLICATIONS. / _ 
 
 1. If 4 pounds of sugar cost | of a dollar, liow much will a 
 pound cost ? y^iis. -^^ of a dollar. 
 
 2. If a liorse consume 1| of a ton of hay in 5 months, how 
 much will he consume in one month? Ans. | of a ton. 
 
 3. If 10 bushels of wheat cost 81 3f, how much is it a 
 bushel? yins. |lf. 
 
 4. If .^250 1 are divided equally among 19 men, how much 
 will each man receive? yins. '*i?13f'/^. 
 
 5. If f of a barrel of flour cost $9, how much will a Nvliole 
 barrel cost ?^/^''^^^ 
 
 Solution. If -i? of a barrel of flour cost $0, ^ of a barrel must 
 cost ^ of $9, or $'3 ; and |, or a wJiole barrel, must cost 5 times §3, ivhich 
 is $15. Therefore, due. 
 
 G.' When $4| is paid for f of a cord of Avood, how much is 
 it a cord ? Ans. $0/^^. 
 
 7. If 104| yards of cloth can be bought for .i?87, how much 
 is the cloth a yard ? Ans. -^ of a dollar. 
 
 8. At f of a dollar a pound, how many pounds of coffije can 
 
 be bought for {^ of a dollar ? ^==^^ ]J 
 
 Solution. At ^ of a dollar a pound, as mayiy pounds of coffee can 
 he houyht for \^ of a dollar as § = -p^ of a dollar are contained times 
 hi \^ of a dollar, ichich are 2i-. Therefoi'e, d-c. 
 
 9. At $95 a barrel, how much flour can be bought for 
 ,$3^ ? Ans. ^ of a barrel. 
 
 10. At I" of a dollar a bushel, hmv man^ bushels of corn can 
 be bought for $87 ?"=--/ ^ ^ %-/iu^LJi^ 
 
 11. At the rate of 17^ miles an hour, how long will it take 
 
 a train of cars to run 180G| miles? 
 
 Ans. 1031 hours. 
 
 Review Questions. What is Reduction of Fractious? (142) When 
 is a fraction expressed in its smallest terms? (143) Wlieu have fractions a 
 Common Denominator? (147) When only can fractions be added? (152) 
 When only can fractions be subtracted? (155) What is niultijiiying a 
 number by a fr.'iction equivalent to? (K!0) 
 
 r 
 

 ff-* 
 
 sm ii-jti ur . 
 
 120 
 
 PRACTICAL ARITHMKTIC. 
 
 RELATIONS OP NUMBERS. 
 
 172. The Relation of Numbers is the number of times one 
 number contains another with which it is compared, or the 
 2)art the latter is of the former. 
 
 173. Numbers can only be compared, and their relations ex- 
 pressed, when they represent units of the same name or kind. 
 
 174. To find the relation of numbers, or the part that one 
 number is of another. 
 
 1. What part of 5 is 3 ? 
 
 1 is |t of 5, and 3 is 3 times 1 of 5, or f of 5. 
 
 2. What part of 9 is 5 ? Of 12 is 4 ? Of 8 is 5 ? 
 
 3. What part of 6 is f? 
 
 1 is I of 6, and ^ of 1 is f of 
 ^, or /vr of 6. 
 
 Or, 6 is ^7^ ; and f is the 
 same part of ^ as 5 is of 42, 
 
 OPERATION. 
 
 7 X(5 
 42 
 
 = A, Ans. 
 
 \JL, V — 7 J 7 
 
 ■1 2 
 
 ~7~ 
 
 ■j'tt, -A flS. 
 
 or x'^T. 
 
 OPERATION. 
 
 3 
 
 20 _ 20^ 
 4 > 4 
 
 4. What part of 2 is 4 ? Of 1 is | ? Of 3 is |? 
 
 5. What part of f is 5 ? 
 
 I is ^ of I, and |-, or 1, 
 is I of |, and 5 is 5 times 
 
 Or 5 - i^ • ^ . 3 _ ^Ji _ f59 ^,,, ^ °^ ^' *'^ ^ °^ t' ^^^^^^^ 
 
 Or, 5 is -^, and ^ is the same part of | as 20 is of 3, or -§, which 
 is 6| times |. 
 
 6. What part of | is 1 ? Of j is 2 ? Of f is 3 ? 
 
 7. What part of f is |? 
 
 ^ is ^ of f , and ^ 
 or 1, is ^ of I, and ^- 
 _ „ is # of 4 of #, or A 
 
 ^^) -g" — i^ij"? 4 — air > s(J ~ ^u — TJj -^^t^'' of I 
 
 OPERATION. 
 5X3 ~ TT' -^'f*' 
 
 What is the Relation of Numbers ? When can numbers be compared and 
 their rehtions exi)ressecl ? 
 
 \ 
 
RELATIONS OF NUMBERS. 
 
 ' 
 
 121 
 
 Or, f is A^ and f is ^%, and ^ is the same part of 15, as 8 is of 15 
 or ^ ; hence, f is j^ of |. 
 
 8. What part of | is i ? Of ^ is | ? Of ^ is ^\ ? 
 
 Rule. Divide the number denoiiwj the imi% by that icith which 
 it IS comimred. 
 
 Examples. 
 
 What part of 
 
 Ans. 
 
 
 Ans 
 
 13. 13is^? 
 
 14. If is 8? 
 
 15. liis 14? 
 
 9. 45 is 9 ? 
 
 10. 16 is 56? 
 
 11. 35 is 21? 
 
 12. 9is^VV? Ans.:^-,^. 
 
 17. What is the relation of -|f to %% ? 
 
 18. What is the relation of 125 to 1000 ? 
 
 Ans. /y. 
 
 Ans. 8^. 
 
 T^ ^» tt f Ans. If. 
 
 16. 30iis5i? Ans.-i-^. 
 
 Ans. As 1 to 1. 
 Ans. ^. 
 
 APPLICATIONS. 
 
 1. A liberty pole, whose height was 75 feet, was broken ofl^ 
 by the wind 45 feet from the ground ; what part of it was left 
 
 standins; ? //•?/". 
 
 ° Ans. "I of it. 
 
 2. If I perform a piece of work in 6 days, what part of it 
 can I perform in 1^ days '^==■'4; 
 
 3. When 24 pairs of bootsTost $156, what will 20 pairs 
 
 OPERATION. 
 26 
 
 20^^. W0X 5 _ 
 24 6' 6 
 
 = $130, Ans. 
 
 When 24 pairs of boots 
 cost $156, 20 pairs, whicli 
 are fQ, or |-, of 24 pairs will 
 cost § of $156, or $130. 
 Therefore, &c. 
 
 4. A and B mowed a field for a certain sum of money A 
 mowing 121 acres, and B 18f acres. What should be A's 
 share of the money, if B's is $30 ? j^g ^20 
 
 5. If a boy has spent | of ^ of his money, what part of -«- 
 of his money has he left 1 ^L ^ "^ 
 
 6. A farmer owns 320 acre^ of land, and his son 2 as many 
 
 "\\liatistheRule? 
 
 H 
 
 4iJ 
 
?^ 
 
 I ! 
 
 1 00 
 
 niACTICAL ARITHMETIC. 
 
 Sliould the son add 40 acres to his land, how would his numbt^r 
 of acres compare with the number his father has % 
 
 Ans. It woul ' be ^ as many. 
 
 jnLTlS, •y-y* 
 
 Ans. |« j. 
 
 Ans. "ooo«. 
 1 «« 
 
 REVIEW EXERCISES. 
 
 1. Reduce jjl'-jf to its smallest terms. 
 
 2. Change |]- to an equivaleno fraction having 07 1 for its 
 denominator. 
 
 3. Change 100}[|[| to an improper fraction. 
 
 4. Reduce 'i, -1, |], and j^^ to equivalent fractions, having 30 
 for their denominator, ^^/-i^^^ f-diJi-., -2^ 
 
 . Tfl 1 11 -^ ^^ ^^-^0 1, 
 
 c5. li I should receive irom one person ^19y^, and from an- 
 other $5 If, and then pay away $G3f , how much should I have 
 remaining ? Ans. %!%%. 
 
 G. Joseph has $13vj^yy, Andrew has $7|- more uiian Joseph, 
 and Henry has as much as both of them ; how many dollars 
 each have Andrew and Henry % 
 
 Ans. Andrew, $20f ; Henry, $33^^ 
 
 7. If I be a minuend, and \ a remainder, what is the sub- 
 trahend ? J*-. 
 
 8. A merchant owned \\ of a ship, and sold f of his share ; 
 what part of the whole ship did he sell ? ' Ans. ~. 
 
 9. If a product be 12jL, and the multiplicand 3|, what is 
 the multiplier % Ans, 3^^. 
 
 10. How many oranges can be bought for 2/^ times ^ of 2 
 cents, when they are 3 of -^^ of 2| cents apiece ? -^^^JC 
 
 11. A has $3240, and B has | as much, lacking $500 ; how 
 much has B ? Ans. $2092. 
 
 12. If there leaks out from a cask containing 31^ gallons of 
 cider, | of a gallon each week, in how many weeks will the 
 whole have leaked out ? Ans. 47j weeks. 
 
 1 
 
 Review Questions. What is the unit of a fraction? (131) AVhat is a 
 fractional unit ? (131) 
 
RELATIONS OF NUMBEIIS. 
 
 123 
 
 13. If the divisor is J and the quotient only | as large, what 
 must the dividend be ? A71S. ■: ^. 
 
 4 «" 
 
 14. AViiab is the value of ^iTh + {H ^ 10) " liV^ ' 
 
 Alls. 311^ 
 
 15. If 3 be added to each term of the fraction §, how much 
 will the value expressed be increased or diminished 1 
 
 An^i. Increased ^. 
 
 16. If 3 be added to each term of the fraction ^, how much 
 will the value expressed be increased or diminished 1 
 
 Ans. Diminished -{•^^. 
 
 17. What number must be multiplied by }J-, that the pro- 
 duct may be 100 ? ' Ans. lll|i?. 
 
 18. If I pay away |- of my money, then ^ of what remains, 
 and then \ of what still remains, Avhat fraction of the whole 
 will be left \ Ans. \. 
 
 19. A farmer, having a flock of 120 sheep, sold 30, and the 
 dogs killed ^ of the remainder. What part of the original 
 inimber then remained 1 Ans. ^. 
 
 20. If 8 horses eat in a certain time l^j- tons Qf hay, how 
 much will one horse eat in the same time ? ^^ -v^ 
 
 21. At |8| a ton, how many tons of coal can be purchased 
 for $41^ ] Am. 5 tons. 
 
 22. Bought 4 casks of molasses, containing 314 gallons each. 
 What part of the whole is 100 gallons 1 Ans. -iff. 
 
 23. A man owning -^^ of a steamboat, said that liis part was 
 10 times as large as that of another man, who owned --^ ; but 
 not being a good arithmetician, he had made a wrong calcula- 
 tion. How many times as large, in fact, was his part ? 
 
 Ans. 8 times. 
 
 Eeview Questions. How may a fractional expression be explained? (140) 
 Hov may a fraction be regarded 'i (141) What is the value expressed by a 
 fract'.on ? (141) How is a fraction reduced to its smallest terms ? (143) How 
 is a mixed number reduced to an equivalent improper fraction 't (145) 
 
 i 
 
 ■• li 
 
 ^' ii 
 
 t ' 
 
r 
 
 124 
 
 PRACTICAL ARITHMETIC. 
 
 EXERCISES IN ANALYSIS.* 
 
 1. If 11 tons of coal cost $99, what will 7;^ tons cost?^^f 
 
 Solution. If 11 tons cost $99, 1 ton loill cost ^ of $99, or $9. // 
 1 toil cost $9, 7^ tons will cost 7^ times $9, or 8C8. Therefore, tC-c. 
 
 2. If 20 barrels of flour cost $210, what will 27 barrels cost 1 
 
 Ans. $283|. 
 
 3. If 27 barrels of flour cost $283i, what will 20 barrels 
 cost? <^^IO^OO 
 
 4. If I of a yard of cloth cost $2.80, what will 5|- yards 
 cost?^^/'>^^ 
 
 Solution. If \ of a yard of cloth cost $2.80, \ of a yard ivill cost 
 I o/$2.80, or $.70, aiid f, or 1 yard, will cost 7 times $.70, or $4.90. 
 
 If 1 yard cost $4.90, 5| yards will cost 5| times $4.90, or $28.42. 
 Therefore, ct-c. 
 
 5. Iff of a pound of tea cost $.60, what will 553^ pounds 
 cost? ^ws. $442.G0. 
 
 G. When | of an acre of land cost $75, what will 74- acres 
 
 cost? ^725. $1560. 
 
 7. If 7|- acres of land cost $1560, what will f of an acre 
 cost ? ^ yA\ d{/ 
 
 8. Ii $2.o0 will buy ^ of a yard of cloth, how many yards 
 will $28.42 buy? ^'#^^' 
 
 Solution. If $2.80 will hxiy ^ of a yard of cloth, \ of $2.80, or 
 $.70, will buy ^ of a yard, and 7 times $.70, or $4.90, will buy f, or 1 
 yard. 
 
 J/" $4.90 ivill buy 1 yard of cloth, as many yards can he bought for 
 $28.42 as $4.90 is contained times hi $28.42, or 5f yards. Tlierefare, &c. 
 
 9. If $7 will buy 5| bushels of rye, how many bushels -will 
 $15 buy ? Ans. llff bushels. 
 
 10. If $5.60 will pay for \ of a ton of coal, what part of a 
 ton will $5.40 purchase ? Ans. | of a ton. 
 
 Review Questions. How are fractions reduced to equivalent fractions 
 
 having a common denominator ? (149) The Rule for Addition of Fractions ? 
 
 (152) 
 
 * Optional. 
 
 ' 
 
RELATIONS OF NUMBERS. 
 
 125 
 
 ' 
 
 « 
 
 11. When 19 J pounds of coffee cost $1U-, how many pounds 
 can be obtained for $2| 'i Ans. 4| pounds. 
 
 12. When $33^ will pay for 4| barrels of flour, how much 
 can be purchased with $27.50 ? Ans. 3} barrels. 
 
 1 3. AVhen 4f tons of hay will suffice for 1 1 horses for a 
 certain time, for how many horses will 71- tons suffice for the 
 same time? Ans. 18 horses. 
 
 14. If A can do a piece of work in 7 days, and B the same 
 work in 5 diiys, in what time can both do it by working 
 together?^ -^^3^ r 
 
 hoLUTiON. If A can do a 'piece of work in 7 days, he can do ^ of 
 it in I day ; and if B can do the same in 5 daj/s^ he can do ^ of it in 1 
 day. 
 
 If A can do ^ of it in 1 dai/, and B i in 1 day, they can, by workinrj 
 together, do \ -^^ \, or ^f , of it in 1 day. 
 
 If by working together they can do ^-| of the work in 1 day, they can 
 do Tj^g- of it m ^ of a day, and ^|, or the ivhole, in 35 times ^ of a day, 
 which is 11 of a day, or 2\^ days. Therefore, d'c. 
 
 15. A man can trench a garden in 13 days, and his son can 
 do the same in 10 days; in what time can both working 
 together do it? Ans. 5|J days. 
 
 1 G. A can mow —^ of a field in a day, and B ~j ; in what 
 time can both, by working together, mow it ? ^ cLO/Ho 
 
 17. A cistern has 3 pipes; the first will fill it in 10 hours, 
 the second in 15 hours, and the third in IG hours. What 
 time will it take them all to fill it ? Ans. 4jy hours. 
 
 18. In an orchard, \ of the trees bear apples, \ peaches, f 
 pears, and the remainder, which is 38, cherries. How many 
 trees are there in the orchard ?^ f D m^-^ t 
 
 Solution. &%nce \ of the trees hear apples, -^ lyeaches, and f pears, 
 i + 5" + f > ^'' tIjVj ^^^^ apples, peaches, and pears, and ][{?, or the 
 v:hole orchard, less ^^^, or -j'Jl , 'imist beir cherries. 
 
 n 
 
 1 m 
 
 H 
 
 Review Questions. The Rule for Subtraction of Fractions? (155) 
 
 General Rule for Multiplication of Fractions? (1G2) Division of Fractions? 
 
 (169) 
 
 * This i^age is optional. 
 
 ..-issk'- 
 
120 
 
 PRACTICAL ARITHMETIC, 
 
 If the 38 trees hearing cJierrien arc -j\,^^, of the orchard, y^^^ is ^ of 
 38 <r(;t'.s, or 2 trees, and if 2 trees are y^^, J[{?, or the whole orchard, 
 must he 105 times 2 trees, or 210 t)'ecs. Therefore, J:c. 
 
 19. A person having bought a house for a certain sum, found 
 tliat after he had paid one half and one third of the cost, he 
 then owed $400. How much had he paid 1 ^X^O/) 
 
 20. A farmer had | of liis sheep in one field, ^V in a second 
 fiekl, and the remainder, 75, in a yard. How many had he in 
 each of the two fields ? 
 
 Ans. In the first, 135 ; in the second, 150. 
 
 21. A, B, and C purchase a mill ; A pays |, I \, and C 
 $2000 of the cost. What are the sum* paid by A and B ? / / J j-. 
 
 Ans. By A, $M%' ; by B, i^l^i.^. " 
 
 22. A and B together have $13G, and | of A's money is 
 equal to f of B's. How much has each 1 ^^^»CC 
 
 Solution. Since \ of Ah lyioney equals | of B's, ^ of Ah equals 
 h of ^, or f, of B's, and -:|, or the whole of As, equals 3 times § of B's, 
 or ^- of Bs. 
 
 If Ah money equals § of B's, as Bh must equal ^ of itself , A and B 
 together have f -H |, or ^-, of B's money. 
 
 If y of Bs money is $13G, ^ is -^V of $13G, or $8, and §, w A's 
 money, is 9 times $8, or $72, while g-, or B's money, is 8 tirdes $8, or 
 $04. Therefore, &c. 
 
 23. The sum of two numbers is 350, and f of the larger 
 number is |- of the smaller ; what are the numbers ? 
 
 Ans. 200 and 150. 
 
 24. My carriage is worth 2| times as much as my horse, and 
 
 What is the value of each ? 
 
 a horse, chaise, and harness, for 
 The chaise cost \ as much as the horse, and the harness 
 
 o 
 
 $300. 
 ^ as I 
 of each % 
 
 ^ as much as the chaise and horse both. 
 
 AVhat was the cost 
 
 Ans. Horse, $150; chaise, $75 ; harness. 
 
 ;75. 
 
 Review Questions. "What is meant by the Relations of Numbers ? (172) 
 What is the Rule for tintling what part one number is of another? (174) 
 
 * This page is optional. 
 
DKCIMAL FRACTIONS. 
 
 127 
 
 DECIMAL FRACTIONS. 
 
 175. If a unit be cUviJod 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 -jV of j^Q-, or 1 hundredth. 
 
 If each hundredth be divided into ten equal parts, each part 
 will be -Jy of yJ^, or 1 thousandth. 
 
 In like manner, we may, by dividing by ten, continue to 
 obtain fractions, each of whose values is one tenth of the frac- 
 tion preceding it ; such fractions are called Decimal Fractioi/s. 
 Hence, 
 
 176. A Decimal Fraction is a fraction whose unit is divided 
 into tenths, hundredths, thousandths, &c. 
 
 A decimal fraction, for brevity, is usually called a decimal. 
 
 J 
 
 •I 
 
 NOTATION AND NUM JIATION. 
 
 177. Decimal fractions are commonly written without tlu^ 
 denominator, and distinguished from whole numbers by having 
 the decimal point (.) at the left. 
 
 The figures at the right of the point are called decimal 
 figures. 
 
 The first order to the right of the decimal point expresses 
 tenths. Thus, 
 
 j^ may be written .1, and read 1 tenth ; 
 itj- » M .2, „ 2 tenths; 
 
 y^ ,, ,, .6, ,, o tenths ; 
 
 J' 
 
 and so on. 
 
 The second order to the right of the decimal point expresses 
 hundredths. Thus, 
 
 When a unit is divided into ten equal parts, what is each of the parts 
 called ? What is a Decimal Fraction ? What is a decimal fraction usually- 
 called? How are decimal fractions usually written? What doe.-i the first 
 order at the right of the point express? Second? 
 
 It 
 
 » 
 
1*1' 
 
 128 
 
 PBACTICAL ARITHMETIC. 
 
 2 
 
 » 
 
 >J 
 
 3 
 
 » 
 
 » 
 
 M 
 
 jj^ may be wiitten .01, ami read 1 huntlredth ; 
 
 .02, „ 2 liuiulredths ; 
 .03, „ 3 hundredths ; 
 and so on. 
 
 The third order to the right of the decimal point expresses 
 thousandths. Thus, 
 
 ToW ^^y ^^ "written .001, and read 1 thousandth ; 
 ■YQwu » >i '^02, „ 2 thousandths ; 
 
 Txnnr >» »> •^^'^t » ^ thousandths ; 
 
 and so on. 
 
 The fourth order to the right expresses ten-thousandths ; the 
 fifth hundred-thousandths^ and so on. Hence, tlie following 
 
 GENERAL PRINCIPLES. 
 
 1. The value expressed ly decimal figures is determined hij the 
 ]jlace of each ivith reference to the decimal imnt. 
 
 2. The denominator of a decimal is understood to he 1, with as 
 many ciphers annexed as there are orders in the decimal expres- 
 sion. 
 
 3. Ten of any lower order of decinals are always equal to one 
 of the next higher. 
 
 178. A Mixed Number may be a whole number and a 
 decimal expressed together, with the decimal point between 
 them. Thus, 
 
 5.34, read five units and thirty-four hundredths, is a mixed 
 number. 
 
 179. A whole number may be regarded as a decimal by 
 placing the decimal point on the right of tlie order of units; 
 and the expression may be read, according to the decimal 
 
 What does the third order at the right of the point express? Fourth? 
 How is the value expressed by a decimal determined? AVliat is the denomi- 
 nator of a decimal? How many of one decimal order make one of the next 
 higher? How may a mixed number be expressed decimally? How may a 
 wliole number be reirarded as a decimal? 
 
Di;CIMAL FRACTIONS. 
 
 V20 
 
 places niincxcil, as a luiml'cr cf Unths, humlndlh, \c. 
 Thus, 
 
 10 may be written IG.O, IG.OO, S:c., aiul re:i.l 100 ie,iih, 
 IGOO Jium/recWis, »'cc. '' 
 
 180. Tlie orders of decimals are named from llio decimal 
 point to the right, 
 
 Te?iihs, hundredths, thmmndths, ten-flummndlhs, hundred- 
 ihousandihs, miUkniths, ^c, according to tlie following 
 
 Integers. 
 
 Table. 
 
 Decimals. 
 
 il' 
 
 a 
 
 
 
 
 
 
 
 
 
 
 J 
 
 TS 
 
 
 
 
 
 
 
 
 
 
 4^ 
 
 § 
 
 
 
 
 
 
 
 
 
 yj 
 
 
 tn 
 
 •/3 
 
 
 
 
 
 
 
 
 ^^ 
 
 rt 
 
 o 
 
 ^ 
 
 rt 
 
 
 
 
 
 
 
 , 
 
 
 — 
 
 3 
 
 ^ 
 'S 
 
 o 
 
 
 
 
 
 
 
 CO 
 
 
 O 
 t 
 
 £ 
 
 
 C3 
 in 
 
 ^ 
 
 
 
 w 
 
 C-> 
 
 c3 
 
 O 
 
 
 s 
 
 1 
 
 o 
 
 H 
 
 
 • 
 
 «3 
 
 • i-H 
 
 
 
 in 
 
 o 
 
 H 
 
 
 r-i 
 
 3 
 
 4 
 
 6 
 
 7 
 
 8 
 
 2 
 
 5 
 
 
 
 9 
 
 1 
 
 .3 
 
 VJ 
 
 r:; o 
 
 where the mixed number is read, three liundred forty-six 
 thousand seven liundred eighty-two units, and five hundred 
 i^e thousand one hundred thirty-two millionths. 
 
 ill 
 
 ' 4 
 
 CASE I. 
 
 181. To read decimals expressed by figures. - 
 
 1. Let it be required to read the decimal .035. 
 
 Solution. The figures express tenth, 3 hundredths, and 5 t/' 
 sandths; or, since 3 liundredths equal 30 thousandths, and 30 thw.- 
 sandths phis 5 tliousaudths. ecpial 35 thousandths, we have as the 
 value expressed, 3o tliousandths. ' 
 
 Therefore, .035 is read thirty-five thousandths. 
 
 071- 
 OU- 
 
 Name tho orders of decimals. What liumber is expressed in the Table ? 
 
 *ft 
 
130 
 
 PRACTICAL AUITLIMETIC. 
 
 KuLL. Head the dcc'uiud ai a whole number, gh'uig it the name 
 of the rijht-haud order. 
 
 Examples. 
 
 Write and read the fuUowiiii' decimals 
 
 10. .55555 
 
 11. .000G7 
 
 12. .0G3443 
 
 13. .134507 
 
 2. .05 0. .500 
 
 3. .175 7. .7571 
 
 4. .012 8. .991)9 
 
 5. .003 9. .00000 
 
 182. When the decimal is part of a mixed number, we may 
 read the integral i)art as exj)ressin<j units. Tims, 
 
 10.45 may be read, sixteen units, and forty-live hundredths. 
 
 Write and read the followimr mixed numbers : — 
 
 14. 4.5 
 
 15. 30.03 
 
 10. 10.0005 
 17. 100.158G 
 
 18. 50783.56 
 
 19. 50.78356 
 
 CASE II. 
 
 183. To write decimals in figures. 
 
 1. L(;t it be required to express in figures thirteen thou- 
 sandths. 
 
 Solution. Since tliirtecn thousandths equal 1 hundredth and 3 
 thousandths, we write 3 in the order of thousandths, 1 in the order of 
 hundredths, and, as there are no tenths, in the order of tenths ; and, 
 prefixing the decimal point, we have .013. 
 
 Therefore, thirteen thousandths is expressed by .013. 
 
 KuLE. Write the decimxil in the manner of a wlwle number, and 
 l^lace the decimal point so that each figure shall express its p'oper 
 value. 
 
 Write in figures the following : — 
 
 2. Four thousandths. Ans. .004. 
 
 3. One hundred sixty-three ten-thousandths. Jy^, J^i'OOJ 
 
 "What is the Rule for reading decimals? "When the decimal is part of a 
 mixed number, how may the integral part be read ? "What ia the ilule for 
 v/riting decimals ? 
 
DECIMAL FKACTIONS. 
 
 131 
 
 |04. 
 
 4. Five lmn(ln'(l nine units, and live hundredths. 
 
 Am. 509.05. 
 
 5. Xint.'teen units, atid nintitoen thousaiidtli.s. An^. I'J.Ol'J. 
 G. Onr tliousand ei^i^'iiL lnindn'<l ciglit units, and one liun- 
 
 drod sixty-one hundnid-lhousandtiis. Ans. 1808.00101. 
 
 7. Three units, and one hundred two niilHonths. 
 
 Anx, 3.000102. 
 
 8. Fifteen thou.sand nine hundretl eiglity-three tcn-niil- 
 lionths. An^. .0015083. 
 
 9. One million units, and eleven lundved-millionths. 
 
 Jim. 1000000.00000011. 
 Write in the decimal form, 
 
 10 I") '•■' 
 
 Am 
 
 S. D.73. 
 
 11. ToVo'o- ^^'^^' -IC^^- 
 
 12. 17iiia- ^^>^^' 17.041. 
 
 13 1 i*^^ 
 14 
 
 1 tl K <.l I (I 1 
 1 (Ml U • 
 
 in 100 ''"i 
 
 17. 18(551 
 
 \r. 1 (»{) I 1 Q 7)jr; i i .i 
 
 ^•^^ iOoOocCT' ^°' '""^"iTroOo J* 
 
 PwEDUCTION. 
 
 184. Annexmg a cipher to a decimal does not alter the value of 
 
 the deeimal. 
 
 For, the order of the signifu-aut figures of the decimal is not 
 cliaiigcd. Thus, .3, or .30, is the same as yg- 
 
 185. Hence, to change decimals having dilVcrent denomina- 
 tors to equivalent decimals having a common denominator. 
 
 Make each decimal have the same number of decimal places, by 
 annexing ciphers. 
 
 Exercises. 
 
 Change to equivalent fractions having a common denominator, 
 
 1. .8 and .0015. I 3. .9, .00, and 1.G34. 
 
 Ans. .8000 and .0015. Ans. .900, .000, and 1.G34. 
 
 2. .005 and .31. Jfy^.^ooH^^*^^^' I.IG and 17.0016. 
 
 ' Ans. I.IGOO, and 17.001G. 
 
 What effect upon tlie value expressed has the annexing a cipher to a deci- 
 mal? How are decimals changed to equivalent decimals, haviny a commou 
 denominator ? 
 
 1 I 
 
 i| 
 
132 
 
 I 
 
 PKACTICAL ARITHMETIC. 
 
 5. .13, .178, and .33G7. G. 1.5, 10.44, and 1.95656. 
 
 ^^'/3^^^"0^^,'3U)Ans. 1.50000, 10.44000, 1.95656. 
 
 CASE I. 
 186. To reduce a decimal to a common fraction. 
 1. Let it be required to change .35 to a common fraction. 
 
 Removing the decimal point and 
 OPERATION. writing the denominator (Art. 177) 
 
 .35 = -jY^y = ^-^, Ans. wo luu-e ylfjy, which reduced is -J^y 
 
 Therefore, .35 is equal to /y. 
 Rule. Omit the decimal 2mnt, ivrite the p-opcr denominator, 
 and reduce the common fraction to its smallest terms. 
 
 Examples. 
 
 lleduce to common fractions : — 
 
 2. .495. 
 
 3. .0075. 
 
 4. .375. 
 
 yt ns. TT (J (J* 
 An<^ -^ 
 Ans. #. 
 
 5. 12.8. 
 
 6. 68.1875. 
 
 7. 100.00125. 
 
 Am. 124 
 Ans. 68j\. 
 Ajis. J 00^-1^. 
 
 A decimal with a common fraction annexed is called a Cowr 
 
 plcx Decimal ; as, 
 
 .66|, read sixty-six and two thirds hundredths. 
 8. Reduce .43|- to a common fraction. 
 
 OPERATION. 
 
 .43} 
 Reduce to a common fraction, 
 
 43| 
 100 
 
 17£ 
 
 4 
 
 100 
 
 400 ~ 16' ^"^' 
 
 Ans. |. 
 
 9. .831. 
 10. .41f. Ans. j%. 
 
 11. .091. Ans.-^-^. 
 
 15 099 7 -r-r •'// . 
 
 U. .00781 A?is.ji^. 
 14. .01561 Ans. ^V 
 
 CASE II. 
 
 187. To reduce a common fraction to a decimal. 
 
 1. Let it be required to change | to a decimal fraction. 
 
 Wliat is the Hule for reducing a decimal to a common fraction ? What is 
 a decimal with a common fraction annexed colled? 
 
 
DECIMAL FHACTIONS. 
 
 133 
 
 i 
 
 OrERATION. 
 
 4 ) 3.00 
 
 f eqiiais 3 -^ 4. 3 (.'([xiah 30 tentlis, or 3.t> ; | 
 of 30 tentlis is 7 tenths, or .7, with -2 teiilh.s, tM|Uiil 
 
 2(> Imndredth?, rcnuaining. 
 
 .75, Ans. i of 20 Immlredths is hundredths, or .05. 
 Therefore, £ is equal to .75. 
 
 llVLE. Annex clpliers to the numerator, divide hi/ the deno- 
 minator, and point off as manij ])laccs for decimals as there icere 
 ciphers annexed. 
 
 Examples. 
 
 lieduce to decimals : — 
 
 *M • 
 3. 
 4. 
 
 3 
 
 -ibo* 
 
 Tr".')U"' 
 
 Ans. .375. 
 
 
 8(J(J' 
 
 Ans. .00875. 
 
 Ans. .95. 
 
 Ans. .0025. 
 
 Ans. .012. 
 
 8. 
 
 9. 
 
 10. 
 
 I 
 
 l-JS* 
 1 1 
 
 '4 0* 
 
 1 
 
 Ans. .0078125. 
 Ans. .015025. 
 
 Ans. .G25. 
 
 
 
 
 6 4 
 
 188. AVhoii the division does not terminate, it may be carried 
 to a desirable degree of exactness, and the sign ^- annexed to 
 the result to indicate its incompleteness, or the remainder 
 annexed, as a part of a complex decimal. 
 
 11. Picduce yV to a decimal of four places. Ans. .5833 + . 
 
 12. lieduce -^,j to a decimal of four places. Ans. .0404 + . 
 
 13. Reduce -^j to a decimal of six places. Ans. .024390 + . 
 1'. Eeduce -p- to a complex decimal of three decimal places. 
 
 Ans. .11^. 
 15. lieduce ^V to a complex decimal of four decimal jjlaces. 
 
 Ans. .0933 L. 
 IG. lieduce y\ to a complex decimal of six places. 
 
 Ans. .727272-fV 
 
 189. When decimals have figures which continually repeat, 
 as in the answers to the last three exam})les, they are said to be 
 indeterminate, and are called Injinite or Circulating Decimals; 
 
 What is the Kule for reducing a comuiou fraction to an e<iuiva!ent docinial? 
 "When the division does not terminate, how can you i)r()0(M!il ? V>'lien are 
 decimals said to he indeterminate? "What are such decimals called? 
 
 si 
 
' Tirir -[Tr rmmmiiitmmiit^ttmim 
 
 134 
 
 TRACTICAL ARITHMETIC. 
 
 and the ropoating figures, as 1, 3, and 72, in the answers 
 referred to, are called liej^etends. 
 
 A repetend may be distinguished, when of only one figure, 
 by placing a dot over it, and when of more than one figure, by 
 placing a dot over the first and last. Thus, 
 
 .111^ = .1, read repetend one; .0933^- = .OOfi, read nine 
 hundredths and repetend three; and .7'2727'2^ = .7'2, read 
 repetend seventy-two. 
 
 Ever?j repetend is equivalent to a common fraction of which the 
 repetend is the numerator, and the denominator as many 0'5 as 
 there are figures in the repetend. Thus, 
 
 17. Reduce J^ to an equivalent repetend. Ans. .0^. 
 
 18. Eeduce -Jj to an equivalent repetend. Ans. .024:30. 
 
 19. Reduce .02430 to an equivalent common fraction. 
 
 20. Reduce .11115 to an equivalent common fraction. 
 
 /lllQ 1 2 3 s 
 
 
 ADDITION. 
 
 190. Since ten of any order of aecu ..li^. make one of the 
 order next higher (Art. 177. 3), decimals may be added in the 
 same manner as whole numbers. Hence, the 
 
 Rule. JFrite the numbers so that figures of the same order 
 shall stand in the same column; add as in whole numbers, observing 
 to note the decimal in the amount hg the decimal point. 
 
 AVhat are the repeating figures called? How may a I'cpeteiid be dis- 
 tinguished ? To what is every repetend equivalent ? How may decimals be 
 added ? What is the Rule ? 
 
 A _h*t^- 
 
0) 
 
 32.4056 
 245.379 
 
 12.0476 
 9.38 
 459.2375 
 
 IMAL FRACTIONS. 
 
 
 Examples. 
 
 (2.) 
 
 (3.) 
 
 11.275 
 
 3.73737 
 
 .34132 
 
 .873 
 
 .00414 
 
 51.77778 
 
 .0001 
 
 108.2 
 
 23.001 
 
 73.4G313 
 
 135 
 
 758.4497 34.G2156 238.05128 
 
 4. What is the sura of 450, 31.47, 370.004, 1.08, 450, .70, 
 and .05? Ans. 1315.304. 
 
 5. What is the sum of 05.30, 8.125, 983, .405, 7.305, and 
 8.12345? ^^,^J^i^/^'9Sf^d' 
 
 6. Add seventy-three units and twenty-nine hundredths, 
 eighty-seven units and forty-seven thousandths, three thousand 
 and five units and one hundred six ten-thousandths, twenty- 
 eight units and three hundredths, twenty-nine thousand units 
 and five thousandths. Ans. 32193.3820. 
 
 ■ I ■ B 
 
 \ 
 
 l\ 
 
 
 5[ - 
 
 V' 
 
 SUBTEACTIOK 
 
 191. Since ten of any order of decimals make one of the 
 order next higher, decimals may be subtracted in the same 
 manner as whole numbers. Hence, the 
 
 Rule. JFrite the less numher binder the greater, ^o that figures 
 of the same order shall stand in the same column ; subtract as in 
 whole numbers, observing to note the decimal in the difference by the 
 decimal jpoint. 
 
 Examples. 
 
 (1.) (2.) (3.) 
 
 From 6827.4081 2.4181 570.271 
 Take 6018.91 1.2234 89.7107 
 
 Ans. 
 
 808.5581 
 
 1.1947 
 
 480.5543 
 
 t 
 11 
 
 How may decimals be subtracted ? What is the Rule X 
 
ff! 
 
 13G 
 
 PRACTICAL AKITDMETIC. 
 
 Ik' re, ill Example 3, as there are no ten-tliousaiidtlis in tlic minnond 
 to subtract from, avc consider tliat order in the minuend as filled hy 
 0, since annexing a cipher to a decimal does not alter its value 
 
 (Art. 184). 
 
 4. From 9G.71 take 9G.709. Ans. .001. 
 
 5. What is th(3 difference between 107, and .0007 W/^^.J^^^ 
 C. Take eighty-five units and seven hundred thirty- seven 
 
 thousandths from one hundred. Aiis. 14.203. 
 
 7. Take one thousand four units and four millionths from 
 two thousand units and sixteen hundredths. Ans. 90G.15999G. 
 
 MULTIPLICATION. 
 
 192. Each removal of J,he decimal ])oint one place toward the 
 right midtiplics hy 10. 
 
 For, each figure is made by tae removal to denote units of an order 
 next higher ; hence the value expressed is made tenfold (Art. 30). 
 
 Thus, .08 X 10 = .8; .8 X 10 = 8; .30G x 100 = 30.G. 
 
 193. Each removtd of the decimal x^oint one ])lace toicard the 
 left divides hy 10. 
 
 For, each figure is made by the removal to denote units of an order 
 next lower ; hence the value expressed is made one tenth as mucli as 
 it was. 
 
 Thus, 6.5 -r 10 = .C5; M ^ 10 = .0G5 ; 73.4 -r 100 = 
 .734. 
 
 194. To multiply when one or both of the factors are 
 decimals. 
 
 1 Let it be required to multiply .5G7 by 4. 
 
 OPEiiATiON. 4 times 507 thousandths is 22G8 thousandths, 
 .567 which, reduced by dividing the numerator 22G8 by 
 4 the denominator 1000, by pointing off three places 
 from the right, is 2.268. 
 
 Ans 2 268 Therefore, .5G7 muUiidied by 4 is 2.2C8. 
 
 What effect has each removal of the dechunl point one place toward tho 
 light? "NVhy? AVhat effect has each removal of the decimal point one place 
 toward the left ? Why? 
 
DECIMAL FrvACTIONS. 
 
 lo7 
 
 2. Let it be requiretl to multiply 3.1G by .4. 
 
 orKUATiON. 4 times 3.16 is 12.()4 ; but as the inulti[»lior is .4, 
 
 3 1(5 the product must be ouly a tenth us lari^e, and 1 2.(54 
 
 .4 divided by 10, by removing the decimal point ouu 
 
 ^ place to the left (Art. 193), is 1.204. 
 
 Ans. 1.2G4 Therefore, 3.16 multiplied hy .4 is 1.264. 
 
 If we change the given decimals to the form of common fractions, 
 and then multi])ly, ve have 
 
 3.1G X .4 = U^ X 4 
 
 I'A"*- = 1 %4 
 
 1000 i.-^^t. 
 
 100 '^ 10 
 
 In like manner, it may be shown, in every case, that 
 21ie number of decimal ]jlaces in the irroduct is equal to the 
 numher of decimal places in both of the factors. 
 
 EuLE. Midtijdij as in ivhole numbers, and ;point off as mam/ 
 decimal 2)laces in the 2^i'oduct as there are decimal i)laces in the 
 multiplicand and multiplier, supplying the deficiency, if any, hy 
 prefixinrj ciphers. Or, 
 
 If the midtiplier is a decimal, multiply hy its numerator, and 
 divide by its denominator. 
 
 AVhen the multiplier is 10, 100, 1000, &c., the multi})lication may 
 be performed by removing the point to the right as many places as 
 there are ciphers on the right of the multiplier (Art. 1D2). 
 
 11 1 
 
 1 
 
 1 
 
 
 Multiply 
 
 Examples. 
 
 3. 12.375 by 1.25. 
 
 Ans. 15.4G875. 
 
 4. 8.5 by 83.7. Ans. 711.45. 
 
 5. .3785 by .003. 
 
 ^?«5. .0011355. 
 
 G. .125 by .025. ^/i6\. 003125. 
 
 7. 4.3125 by 100. yif«.s.431.25. 
 
 8. 4.3125 by 1000.^^i/^*3' 
 
 9. 4.3125 by 10000. 
 
 Ans. 43125. 
 
 10. Multiply one thousand by fifteen ten-thousandths. ^s=s. I k (j / /), 
 
 11. Multiply one thousand by one thousandth. Ans. 1. 
 
 I ; 
 
 i 1 
 
 To what is the number of decimal places in the product equal? AVhat is 
 the Rule ? Huw may the multiplication be perfo .-med when the multiplier 
 is 10, 100, &c. ? 
 

 138 
 
 PRACTICAL ARITHMliTIC. 
 
 12. ^Multiply two himdred thousand by three tenth3.= ^/'^^^# 
 
 13. Multiply three tenths by three hundredths. Ans. .009. 
 
 14. Multiply one by one hundred-thousandth. y//?.s\ .00001. 
 
 15. Llultiply two hundred forty-one ten-thousandths by one 
 hundred sixty-five thousandths. --=^^?^/;J' ^/^;2 ^^3' 
 
 IG. Multiply two thousand five hundred thirty-four mil- 
 lionths by thice thousand two hundred fifty-six hundred thou- 
 sandths. Ans. .00008250704. 
 
 DIVISIOK 
 
 195. To divide when the divisor, or dividend, or both, are 
 decimals. 
 
 1. Let it be required to divide 2.2G8 by 4. 
 
 OPERATION. 
 
 4 ) 2.2G8 J of 2268 thousandths is 567 thousandths, or .567. 
 
 Therefore, 2.268 divided by 4 is .567. 
 
 .5G7 
 
 2. Let it be required to divide 1.264 by .4. 
 
 1.264 divided by 4 is .316 ; but, as the divisor is 
 .4, the quotient must be ten times as large, and .316 
 multiplied by 10, by removing the decimal point one 
 place to the right (Art. 192), is 3.16. 
 
 Therefore, 1.264 divided by 4 is 3.16. 
 
 3. Let it be required to divide .00115 by .05. 
 
 OPERATION. .00115 divided by 5 is .00023 ; but, as the divisor is 
 
 .05 ) .00115 .05, the quotient must be one hundred times as large, 
 
 and .00023 multiplied by 100, by removing the point 
 
 Ans. .023 two places to the right (Art. 192), is .023. 
 
 If we change the given decimals to the form of common fractions, 
 and then divide, we have 
 
 OPERATION. 
 
 .4 ) 1.264 
 Ans. 3.16 
 
 .00115 -^ .05 = 
 
 115 
 
 100 
 
 115 xJOO _ _23_ _ 
 100000 X 5 ~ 1000 ~ •^^'^• 
 
 100000 
 
 In like manner it may be sho^\^l in every case that 
 
 The number of decimal places in the quotient is equal to the excess of 
 the number of decimal places in the dividend over that in the divisor. 
 
 Explain the operations. 
 
 II 
 
DECIMAL FRACTIONS. 
 
 139 
 
 Rule. If the divisor is a tchole number, divide as in whiic 
 numbers, and point off as many decimal places in the quotient as 
 there are such palaces in the dividend. Or, 
 
 If the divisor is not a ichole mimber, divide hy its numerator, 
 and multiply by its denominator. 
 
 If the divisor is a decimal we may make it a whole number, hy 
 removing its decimal point a suflicient number of places to tlie right, 
 and remove the decimal point in the dividend as many places to the 
 right ; then divide, and point off in the quotient as many decimal 
 places as there are in the changed dividend. 
 
 For, multiplying both divisor and dividend by the same number 
 will not change the value expressed by the quotient (Art. 73). Thus, 
 
 OPERATION. In working the third example, if we multiply the 
 05. ) 00.115 given decimals by 100, by removing the points two 
 
 places to the right, the divisor Ijecomes a whole 
 
 Ans. .023 number, 5, and the dividend 115 thousandths. 
 ^ of 115 thousandths is 23 thousandths, or .023. Therefore, .00115 
 divided by .05 is .023. 
 
 When the divisor is 10, 100, 1000, &c., the division may be per- 
 formed by removing the point to the left as many places as there are 
 ciphers in the divisor (Art. 193). 
 
 Examples. 
 
 Divide 
 
 4. .7935 by 23. Ans. .0345. 
 
 5. .7935 by 2.3. Ans. .345. 
 
 6. .7935 by .23. Ans. 3.45. ^ 
 
 7. 7.935 by .1Z.^/y0 ,5^'d* 
 
 8. 79.35 by .2Z.^/noJ^^* 
 
 9. 793.5 by 2.3. Ans. 345. 
 10. 711.45 by 8.5. Ans.^?,.l. 
 
 To what is the number of decimal places in the quotient equal ? What is 
 the Rule when the divisor is a whole number? "When it is not a wliole 
 number? How may it be made a whole number? How may we divido 
 when the divisor is 10, 100, 1000, &c. ? 
 
 ii 
 
 
 V 
 
 '*t 
 
 11. 00.11355 by .003. 
 
 Ans. .3785. 
 
 12. .05G875byG.5.yi/i5.. 00875. 
 
 13. 987.5 by 100. ^7^,9' f^^' 
 
 14. 987.5 by I000.^y^/K;f . • ^/?5 ' 
 
 15. 987.5 by 10000. 
 
 Ans. .09875. 
 
 IG. What is the quotient of 3G5.8 divided by .002 l^o^^- ifipc 
 
 .. 
 
140 
 
 PRACTICAL ARlTiniETIC. 
 
 ! 
 
 1 
 
 i 
 
 f 
 
 ! 
 i 
 
 ■ 
 
 
 1 f 
 
 Iloro^ liy removing tlie decimal point three places to tlio right, in 
 Lotli the divisor and dividend, we have 
 
 3G5.8 ^ .002 = 3G5800 -^ 2 = 182900, Ant:. 
 Divide 
 
 17. 8.05 l)y .0023. yi»5. 3.500. 19. 17. -IS hy .OUL Ans. 1200. 
 
 18. 2.117 by .0073, ^.^^f 20. 18G.9 by 7.i7G.,^^,^/fp 
 
 196. AYlien there is a remainder after dividing, the division 
 can be continued by annexing decimal ciphers to the dividend. 
 
 If the division does not terminate, it may be carried to a 
 
 desirable degree of exactness, and the sign + annexed to tlie 
 
 result, to indicate its incompleteness. 
 
 I' 
 
 Divide 
 
 21. 2.5 by .32. Ans. 7.8125. 
 
 22. .97 by .8. Ans. 1.2125. 
 
 23. 37.4 by 4.5. Ans. 8.311+. 
 
 24. 7.7byl28.y^/?ii'..06015G25. 
 
 25. 4 by .00255. 
 
 Ans. 15G8.G27 +. 
 
 26. 7.43 by .0079. 
 
 Ans. 940.50G3. 
 
 APPLICATIONS. 
 
 1. A ship sails in four days as Iv^llows : — the first day 
 197.025 miles, the second 211 miles, the third 163.175, and 
 the fourth 150.65 ; how far did it sail in the four^days? 
 
 Ans. 721.85 miles. 
 
 2. The difference between A's money and B's is $7691.55, 
 and A's money, which is the least of the two, is $1006.45 ; 
 required, B's money. Ans. J^8698. 
 
 3. Mr Wade had in his farm 640 acres, but has sold off 
 221.125 acres. How many acres has he left ? 
 
 Ans. 418.875 acres. 
 
 4. What cost 17.75 tons of coal, at $4.54 a ton? 
 
 5. Such a quantity of bread was divided equally among 13 
 sailors, as allowed each sailor 1.230^ j^ounds. How many 
 pounds were divided ?== /c) * uCo/C H^ 
 
 When there is a remaiutler after dividing, how may the division be con- 
 tinued ? How may you proceed when the division does not terminate ? 
 
I 
 
 DECIMAL FRACTION'S. 
 
 HI 
 
 6. What is the cost of 19.95 tons of hay at 820 a ton ? 
 
 Arcs. 8399. 
 
 7. What improper fraction is equivalent to the sum of 14.5 
 and -.5, divided by their dilference? ytns. }'J. 
 
 8. If the length of a year be taken at 305,25 days, instead 
 of 3G5.2422G4, the true length, what will be the error in 400 
 years? An.'^. 3.0944 days. 
 
 9. A man bought a fiirm, consisting of 75.8 acres, at 831.50 
 per acre, and sold it for 82274 ; how much did he lose per 
 acre? Aiis. 81.50. 
 
 10. According to the United States Coast Survey, a meter 
 is 39.3685 inches, and allowhig G33G0 inches in a mile, how 
 many meters are there in a mile ? - j f^ Q -"' « J^^/) f "7- 
 
 11. A young man inherited a sum of money ; after sjiending 
 .375 of it in dissipation, and .25 of it in bad trades, he had 
 81500 left. How much did he inherit? A71S. 84000. 
 
 ■7 12. If a merchant purchases G50 barrels of Hour for 8-1875, 
 and sells it at 88.25 a barrel, how much does he gain on a 
 barrel ? Ans. 8.75. 
 
 13. If a section of land is worth 8C400, what is .875 of it 
 worth. #$^^^7^^ ;P ^=:^ 00. J^ 
 
 14. \yhat part of a section of land worth 8G400 can be 
 purchased for 85G00 ? Ans. .875. 
 
 15. What is the value of GO. 5 tons of coal, when .9 of a ton 
 is worth 8G.GG ? Ans. 8447.70. 
 
 16. If I expend $128,925 for corn at 8.60 a bushel, and barley 
 at 8-75, in equal quantities^ how many bushels of each do I get, 
 and how much money is paid for each kind of the grain ?>,' 
 
 o^ 
 
 ' "^ ^. 
 
 r;i< ' yf'*> 
 
 M- 
 
 .)iM^ 
 
 Review Questions. What is a Decimal Fractio'-? ^i70) How is a ^' 
 decimal usually written? (177) "What is the denominator understood to be? 
 (177) What is the Rule Tor reading decimals expressed. by figures? (181) 
 For writing decimals in figures? (183) AVhat effect has the annexing of 
 a cipher to a decimal? (184) How may decimals be changed to equivalent 
 decimals having a common denominator? (185) 
 
 f 
 
 \ 
 
 II 
 
r — 
 
 !■! 
 
 • 
 
 ..'i 
 
 142 
 
 PRACTICAL ARITHMETIC. 
 
 Exercises in Analysis. 
 
 1. At $4.50 per hundred, what cost 9G34 pounds of fish ? 
 
 Solution. 9034 reduced to hundreds bt/ j)oi7ituuj off two 2)Iaces from 
 tJie ri(//tt is 0(i.li4 hundreds. At .Sl.50 per hundred, 90.34 hundreds 
 of fish must cost 90.34 times $4.50, o/*^433.53. Therefore, d;c. 
 
 2. At $25 per hundred, what cost 820 apple-trees? -^--^.../'/J^ 
 
 3. What is the freight on 5G70 pounds, at $1.20 per 
 hundrea 1 "->/j (^ - *( '^ 
 
 4. What is the cost of 9 GO water-melons, at $12 J per 
 liundred? ' Ans.%120. 
 
 5. At $22.40 per thousand, wliat cost 43750 feet of boards? 
 
 Solution. 43750 reduced to thousands by pointitig off three places 
 from the right is 43.75 thousands. At $22.40 per thousand, 43.75 
 thousands of boards must cost 43.75 times $22.40 or $980. Therefore, 
 d-c. 
 
 G. At $G.50 per thousand, what cost 31G84 bricks? 
 
 Ans. $205. 94G. 
 
 7. How much will it cost to transport 53725 pounds of 
 freight, at $1.14 per thousand ?rr-.'^'*e'/*'^-''^ ' _' ^ 
 
 8. What is the cost of 3G506 feet of timber, at $40 a 
 thousand; 5G80 feet of plank, at $50 a thousand, and 16 
 thousands of shingles, at $5.25? Ans. $1828. 
 
 9. At $6.25 per ton of 2000 pounds, what cost 4480 pounds 
 of coal? -^P/^- 
 
 Solution. 4480 pounds are 4.48 thousand pounds, and since 2 
 thousand pounds make a ton, there are half as many tons as there are 
 thousand pounds, or 2.24 tons. At $6.25 per ton, 2.24 tons of coal will 
 cost %24: times $6.25, or $14. Therefore, dx. , 
 
 10. At $21 per ton, what cost 2560 pounds of haylw''i?(i'"/v^ 
 
 11. At $9.50 per ton, what cost 3248 pounds of plaster ?.;./^/9';<^y,^^ 
 
 12. What is the freight on 96880 pounds of coal at $2.50 a 
 ton? ^ri5. $121.10. 
 
 Review Questions. "What is the Rule for reducing a common fraction 
 to an equivalent decimal ? (187) 
 
DECIMAL FRACTIONS. 
 
 l\S 
 
 per 
 
 70 
 
 13.* A and I> have together a certain sum of money, and A'.s 
 share is to B'd as 2 is to 3. How many hundredths is eacli 
 man's share ] 
 
 Solution. ^Since A's share U to B's as 2 to 3, if the moneji they 
 have toijethcr be divided into 2 + 3, or 5, equal yaHSy 2 of these parts, or 
 '■i of the mo7U'}/, i;i A's share, and 3 of the^e parts, or -.J tf the inoneij, is 
 D^s share. But j; is equal to .40, and 'I to .GO; hence, A's share is .40, 
 and B's .00. Therefore, d-c. 
 
 14. The cost of one liouse is to that of anotlier as 5 to 7 ; how 
 many hundredths of the cost of the two is the cost of each ? 
 
 Alls. .41 §; .58.\. 
 
 15. Three men own a ship together; their parts of it are to 
 each other as 1, 2, and 5 ; what are their shares, expressed in 
 hundr<.'dths ? Ans. .12.V, .25, and .02.}. 
 
 IG. Ijiiilt a house and barn for 82470.10; the cost of the 
 liouse was to that of the barn as -^ to |j ; what was the cost of 
 
 Solution. Since the cost of the house luas to that of the barn as ^ 
 to §, it was as |f to \^, or as 12 to 10. Jf then the mone^j the>/ both cost 
 be divided into 12 + 10, or 22, equal parts, 12 of these parts, or Jlj of 
 $2470.10, ivhich is $1350.00, must be the cost of house, and 10 of these 
 parts, or -|2 o/ $2470. 10, whicJi ts $1125.50, must be the cost of tJie barn. 
 Therefore, dec. 
 
 17. Divide 398.00 into two parts, which shall be to each 
 other as .35 to .05. Ans. 139.51 ; 259.09. 
 
 18. In a certain union school having 475 pupils, the number 
 of boys is to that of girls as 13 to 12 ; what is the numb(T of 
 each? Ans. 247 boys; 228 girls. 
 
 19. If gunpowder is composed of .70 parts of nitre, .14 of 
 charcoal, and .10 of sulphur, how much of each of these will 
 be required for 2000 pounds of powder? 
 
 Ans. nitre, 1520 pounds ; charcoal, 280 pounds ; sulphur, 
 200 pounds. 
 
 Review Questions. "VVliat effect has the removal of the decimal point 
 one place to the right? (1U2) One place to the left? (193) 
 
 * This page is optional. 
 
 mt 
 
VRACTICAL ARlTIIMKTIC. 
 
 WEIGHTS AND MEASURES. 
 
 I i 
 
 ! 
 
 197. Measure is tli.it 1)y Avliicli cxtonsion, capacity, force, 
 duration, or value is ostimatccl or (Ictcirmiuod. 
 
 198. Extension is that Mhich has one or more of the 
 dimensions of length, breadth, and height or depth ; as 
 
 199. A Line, or that ■which lias only length ; 
 
 A Surface, or that which has only length and hroadth ; 
 
 A Solid, Body, or Vcjlnme, or that which has lengtli, 
 breadth, and height or depth. 
 
 The Faces of a solid or volume are its bounding surfaces. 
 
 200. Weight is the measure of the quantity of matter in a 
 body, determined by the force by which it is naturally drawn 
 toward the earth. 
 
 201. A Unit of Mc e is some quantity used as a standard 
 of comparison in measuring a quantity of the same kind. 
 
 COMMON WEIGHTS AND MEASURES. 
 
 202. Common Weights and Measures are those in general 
 use. 
 
 TROY WEIGHT. 
 
 203. Troy Weight is used for weighing gold, silver, and 
 jewels. 
 
 Table. 
 
 24 grains (gr.) are 1 pennyweight, pwt. 
 20 pennyweights, 1 ounce, oz. 
 
 12 ounces, 1 pound, lb. 
 
 A^Tiat is Measure? A Line? A Surface? A Solid, Body, or Volume? 
 Weight ? A Unit of Measure ? The Common Weights and Measures ? For 
 wiiat is Troy Weight used ? Repeat the Table. 
 
 C; 
 
WEIGHTS AND MEASURES. 
 
 145 
 
 and 
 
 ! 
 
 AroTiiECARiEH, ill luixint,' inodiciiu's, use tlu; pound, ounce (5), ainl 
 ymin, of this \vt'i;^flit ; l)iit diviili! tin; (»imc(i into 8 drams (3), fuch 
 »(liial to .'i scruiilt'ii ( fj), each jscruplo bi-iiij,' imjuuI to '20 (/rains. 
 
 A cam^, for ^olil-wt'i;,'lit, i:; 4 ^M-aiiis ; \\>v (liiiiiiinid-wci^ht, is .'3.2 
 ;4 ruins. 
 
 UlllS. 
 
 A jioiind Troy loiilains i'4(> pniiiyweixdits, or HTdO ,mi 
 
 AVOIRDUPOIS WEIGHT. 
 
 204. Avoirdupois Weight is used lor nearly all articles esti- 
 mated by weight, I'xcept gold, silver, and jewels. 
 
 Table. 
 16 drams (dr.) are 1 ounce, oz. 
 
 16 ounces, 1 pound, lb. 
 
 lT) pounds, 1 quarter, <jr. 
 
 4 quarters, or 100 lb., 1 hundredweight, cwt. 
 
 20 hundredweight, 1 ton, T. 
 
 Formerly, 112 pounds, or 4 (puvrters of 28 pounds each, were ri'c- 
 koiied a huiidredwci^lit, and 2240 pounds u ton, now called the long 
 ton. This is now seldom em[)loyed in this country, i!Xce])t at the mines 
 f(jr coal, or at the United States Custom-Houses for goods imported 
 from Great Britain, in which country such weight continues to he used. 
 A pound Avoirdu])ois is e<piivalent to 7000 grains Troy, so that I 14 
 pounds Avoirdupois are equal to 17') pounds Troy. 
 
 LINEAR MEASURE. 
 
 205. Linear or Long Measure is used in estimating di.stances 
 
 and the length of articles. 
 
 12 inches (in.) 
 
 .3 feet, 
 5^ yards, 
 40 rods, 
 
 8 furlongs. 
 
 Table. 
 
 are 1 loot, ft. 
 
 1 yard, yd. 
 
 1 rod, rd. 
 
 1 furlong, fur. 
 
 1 common mile, m. 
 
 How do Apothecaries divilo the ounce? AVhat is a carat? What docs a 
 pound Troy contain ? For what is Avoirdupois "Weight used ? llei)cat the 
 Table. How was a hundredweight formerly reckoned? Wliere is the long 
 ton now used ? To what is a pound Avoirdupois equal ? For what is Linear 
 Measure used ? Repeat the Table. 
 
 K 
 
 I 
 
 I 
 
 it 
 
 i» 
 
146 
 
 PRACTICAL ARITHMETIC. 
 
 1 X 
 
 
 I ! 
 
 Til nioasiirinfT doth and other woven fabrics, the linear yard is 
 divided into halves, quarters, eighths, and sixteenths. Formerly a six- 
 teenth of a yard, or 2j inches, was called a nail. 
 
 An en(/i:ieer's chain, or measurinfj tape, is usually 100 f(!et in length, 
 with each foot divided into tentlis. Surveyors, however, make fre- 
 quent use of Gimter's chain, whicli is 4 rods, or 66 feet, in length, 
 and divided into 100 links of 7.92 inches each. Links are usually 
 expressed as hundredths of a chain. 
 
 4 inches equal 1 hand in measuring the height of horses directly 
 over the fore feet ; 6 feet etpial 1 fathom, in measuring depths at sea ; 
 and 13 common miles equal 1 league on tlie land. 
 
 A geographic or nautical mile is ^q of the length of a degree of 
 latitude. 
 
 The United States Coast Survey employ 6086.34 feet, or 1.1.5 + 
 common miles, as the average length of a nautical mile, and 69.16 
 common miles as the length of a degree < 'longitude on the equator, 
 
 A mile is 320 rods-= 1760 yards = 52^0 feet = 63360 inches. 
 
 SUEFACE MEASUEE. 
 
 206. Surface or Squaic Measure is used in estimating 
 surfaces. 
 
 207. The Area of a figure is its quantity of surface. 
 
 208. An Angle is the difference in di- 
 rection of two straight lines, which meet 
 at a i^oint. Thus, A C D and D G B are 
 angles. 
 
 209. A Square is a surface having four equal sides 
 and four equal angles. 
 
 The area of a square, each of whose sides is 1 foot, is 1 
 square foot. 
 
 How ia the yard divided in measuring cloth ? "What is an engineer's chain, 
 or measuring tape ? Gunter's chain ? A geograpliic or nautical mile ? The 
 length of a nautical mile ? A degree of longitude on the equator ? To what 
 is a mile equal ? For wiiat is Surface Measure used ? What is the Area of 
 a &Qme ? An Angle ? A Square ? 
 
WEIGHTS AND MEASURES. 
 
 147 
 
 of 
 
 lating 
 
 E 
 
 )t, is 1 
 
 r's chain, 
 le? The 
 ITo what 
 Area of 
 
 210. A Rectangle is any surface having four sides and four 
 equal angles. 
 
 1 
 
 
 In the large square are reiDresented 3 rows of small squares, 
 of 3 squares each. If each of these small squares is supposed 
 to be one square foot, then in the large square there evidently 
 are in 'me of the equal rows, 3 times 1 square foot, or 3 square 
 feet ; and in the three rows, 3 times 3 square feet, or 9 square 
 I'eet, or as many as the product of the number expressing the 
 length by that expressing the breadth. Hence, 
 
 The area of a rectangle is equal to the p'oduet of the length by 
 the breadth. 
 
 Table. 
 
 144 square niches (sq. in.) are 1 square foot, sq. ft. 
 
 9 square feet, 1 square yard, sq. yd. 
 
 30^ square yards, 1 square \:.l or perch, P. 
 
 160 square rods or perches, 1 acre, A. 
 
 640 acres, 1 square mile, M. 
 
 In surveying l)y Guiitcr's cluiin, I square chain is 16 square rods, 
 and 10 square chains are 1 acre. 
 
 In measurement of government lands, 640 acres, or 1 square mih', 
 make 1 section of land. 
 
 A rood equals 40 square rods or perches, but this denomination is 
 not now much used. 
 
 An acre is 4840 square yards = 43560 square feet = 6272640 
 square inches. 
 
 What is a Rectangle ? To what is the area of a rectangle equal ? Repeat 
 the Table ? How much is 1 square chain ? 10 square chains ? How many 
 acres in one square mile or section of laud ? What does a rood equal ? An 
 acre? 
 
 Ill- 
 
U8 
 
 m <■ 
 
 'I 11 
 
 : i 
 
 
 1 
 
 PRACTICAL ARITHMETIC. 
 
 SOLID MEASURE. 
 
 211. Solid or Cubi^ Measure is used in measuring bodies, 
 or things having len^' •.. breadth, and height or depth. 
 
 212. The Contents of a solid or volume are the number of 
 times it contains a given unit of measure. 
 
 213. A Cube is a body bounded by six square 
 and equal sides or faces. 
 
 If a cube has each of its equal faces 1 square foot, 
 it is 1 solid or cubic foot. 
 
 214. A Rectangular Solid is a body bounded by rectangular 
 faces. 
 
 In the large cube are represented 3 tiers of small cubes, of 
 3 rows each, and 3 in a row. If each of these small cubes is 
 supposed to be one solid foot, then in the large cube there 
 evidently are in one of the equal tiers 3 times three solid feet, 
 or 9 solid feet, and iii the 3 tiers, 3 times 9 solid feet, or 27 
 solid feet, or as many as the product of the numbers denoting 
 the length, breadth, and height or depth. Hence, * 
 
 The contents of a rectangular solid are equal to the product of the 
 length, breadth, and height or depth. 
 
 Table. 
 
 1728 cubic inches (cu. in.) are 1 cubic foot, cu. ft. 
 
 27 cubic feet, 1 cubic yard, cu. yd, 
 
 128 cubic feet, 1 cord, C. 
 
 For what is Solid or Cubic Measure used ? What is the Contents of a solid 
 or volume? What is a Cube? A Rectangular Solid? To what i& the con- 
 tents of a rectangular solid equal ? Repeat the Table. 
 
WEIGHTS AND MEA.SURES. 
 
 149 
 
 1 cord. 
 
 Ic. f. 
 
 A pile of wood 8 feet long, 4 foot wido,. 
 and 4 foot high, is a cord. A cord foot 
 (c. f.) is ] foot in length of this pile, 
 or 16 cubic feet. 
 
 A 2^€rch of masonry, or of huildlng 
 stono, is 24| cubic feet. 
 
 A ton of timber is commonly estimated at 40 solid foot, but as 
 usually measured is 50 solid feet. 
 
 With transportation companies a ton of freight is quite variable, 
 being for many articles estimated by the space occupied, and for 
 others by weight. 
 
 A solid yard, or 27 solid feet, is equal to 46656 solid inches. 
 
 )tmg 
 
 , solid 
 con- 
 
 LIQUID MEASURE. 
 
 215. Liciuid Measure is used in measuring all kinds of 
 liquids. 
 
 Table. 
 
 4 gills (gi.) are 1 pint, 
 2 pints, 1 quart, 
 
 4 quarts, 1 gallon. 
 
 pt. 
 
 qt. 
 
 gal. 
 
 A barrel (bar.), in some States is 31^ gallons, and in others 28 to 
 32 gallons. 
 
 A hogshead (hhd.), when regarded as a measure, is 63 gallons ; but 
 the term is often applied to large casks of varying capacity. 
 
 Ale, beer, porter, and milk, were formerly sold ])y what was called 
 Beer Measure, of which the gallon contained 282 cubic inches. 
 
 Apothecaries reckon one pint (0) equal to 16 fluid ounces 
 (f. 5) ; 1 fluid ounce equal to 8 fluid drams (f. 5). 
 
 The standard liquid gallon of the United States contains 231 cubic 
 inches, and the Imperial gallon of Great Britain, 277.274 cubic inches. 
 
 A hogshead is 252 quarts, equal 504 pints, or 2016 gills. 
 
 "What is a cord of wood? A ])oioli of masonry? A ton of freight with 
 transportation companies? A solid yard? For what is Liquid Measure 
 used? Eepeat the Tabk\ "What is a barrel? A hogshead? How many 
 cubic inches in a gallon of Beer Measure? In the standard gallon of Liquid 
 Measure ? What docs a hogshead equal ? 
 
 n 
 
 I 
 
 :ii 
 
 i 
 

 150 
 
 PRACTICAL ARITHMETIC. 
 
 I'M' 
 
 I 
 
 1 
 
 ti 
 
 ( I 
 
 DEY MEASURE. 
 
 216. Dry Measure is used in mea'^uring such dry articles as 
 grain, fruit, roots, coal, &c. 
 
 Table. 
 
 2 pints (pt.) are 1 quart, qt. 
 8 quarts, 1 peck, pk, 
 
 4 pecks, 1 bushel, bu. 
 
 The chaldron, a measure of 3C bushels, formerly employed with 
 some kinds of coal, is now seldom used. 
 
 The standard busliel of the United States contains 2150.42 cubic 
 inches ; and the Imperial bushel of Great Britain, 2218.192 culyic 
 inches. 
 
 CIRCULAR MEASURE. 
 
 217. Circular Measure is used for measuring angles, latitude 
 and longitude, difference of direction, &c. 
 
 218. A Circle is a plane figure, bounded by a curved line, 
 all the points of which are equally distant from a point within, 
 called the centre. 
 
 219. The Circumference of a circle 
 is its entire bounding line. 
 
 220. An Arc of a circle is any part 
 of the circumference, as A B, or A B E, 
 
 221. A Diameter of a circle is any 
 straight line drawn through the centre, 
 and terminated by the circumference, 
 as A E. 
 
 222. The circumference of every circle whatever, is supposed 
 to be divided into 360 equal parts, called degrees. 
 
 For what is a Diy Measure used ? Repeat the Table. How many cubic 
 inches does a standard bushel contain ? An Imperial bushel ? For what is 
 Circular Measure used ? What is a Circle ? The Circumference of a circle ? 
 An arc ? A diameter ? 
 
WEIGHTS AND MEASURES. 
 
 151 
 
 IE 
 
 Ised 
 
 lubic 
 it is 
 hole? 
 
 rao" 
 
 A Quadrant is a fourth of a 
 circumference, or an arc of 90 
 degrees, as A B. 
 
 223. Tlie Angle A C B, whose 
 two lines meet at the centre C^ ^s ^so" 
 measured by tlie arc of the circle 
 drawn around that centre, in- 
 cluded between the opening of 
 those lines. Thus, A C B is an angle of 90 degrees. 
 
 A right angle is an angle of 90 degrees. 
 
 Table. 
 
 , 60 seconds Q are 1 minute, ' 
 
 60 minutes, 1 degree, ° 
 
 360 degrees, 1 circum., C. 
 
 A siffn, used only in astronomy, is ^ of the circumference of a 
 circle, or 30°. 
 
 A minute of the earth's circumference is a geographic or nautical 
 mile. 
 
 A degree is y^ of the circumference of any circle, small or large. 
 
 A circumference of any circle has 21600 minutes, or 1296000 
 seconds. 
 
 TIME MEASURE. 
 
 224. Time is used in measuring portions of duration. 
 
 J 
 
 
 
 Table. 
 
 
 60 seconds (sec 
 
 ) 
 
 are 1 minute, 
 
 m. 
 
 60 minutes, 
 
 
 1 hour. 
 
 h. • 
 
 24 hours, 
 
 
 1 day. 
 
 d. 
 
 365 days, 
 
 
 1 common year. 
 
 c. y. 
 
 366 days. 
 
 
 1 leap year. 
 
 1-y- 
 
 Also, 7 days make 1 
 
 u 
 
 eek, 12 calendar months 
 
 1 year, and 
 
 100 years 1 century. 
 
 
 
 
 What is a Quadrant ? A right angle ? Repeat the Table. What is a sign ? 
 A minute ? A degree of the earth's circumference ? What does a circum- 
 ference equal ? What is time ? Repeat the Table. 
 
 il 
 
 \ -tit 
 
 
r. 
 
 152 
 
 PRACTICAL ArvITIIMETTC. 
 
 I '■• 
 
 ( ! 
 
 1 I 
 
 225. Tlio Calendar Months, their names, order, and number 
 of days, are as follow : — 
 
 January, 
 
 1st mo. 
 
 31 days. 
 
 July, 
 
 7th mo 
 
 . 31 days. 
 
 February, 
 
 2d „ 
 
 28 or 29. 
 
 August, 
 
 8th „ 
 
 31 „ 
 
 March, 
 
 3d „ 
 
 31 days. 
 
 September, 
 
 9th „ 
 
 30 „ 
 
 April, 
 
 4th „ 
 
 30 „ 
 
 October, 
 
 10th „ 
 
 31 „ 
 
 May, 
 
 5th „ 
 
 31 „ 
 
 November, 
 
 11th „ 
 
 30 „ 
 
 June, 
 
 6th „ 
 
 30 „ 
 
 December, 
 
 12th „ 
 
 31 „ 
 
 The nr.mber of days contained in each month may be remembered 
 by recollectin^L"" that the months are long and short alternateli/^ with the 
 exception of Au;,aist, wliicli, as well as July, is long. 
 
 226. A true year^ also called a solar or tropical year, is the 
 exact time in whicli the earth makes a revolution around the 
 sun, or 365 d. 5 h. 48 m. 49.7 sec. 
 
 227. The common year of 365 days comes short of the true 
 year 5 h. 48 m. 49.7 sec, or 1 day, lacking only 44 m. 41.2 sec. 
 in 4 years, so that an approximate correction of the calendar 
 can be made by having every fourth year of 366 days. 
 
 But, by making every fourth year a leap year, there will be a 
 gain in the calendar of 18 h. 37 m. 10 sec. in 100 years, or a 
 little over 3 days in 400 years ; hence, a second approxim;ite 
 correction can be made by having only every fourth of the 
 centennial years a leap year. Hence, 
 
 Every year ivhose number can he divided by 4 ivithout a re- 
 mainder, except centennial years, and every centennial year whose 
 number can be divided by 400 ivithout a remainder, is a leap year, 
 and the others common years. 
 
 A common year has 8760 liours, equal 525600 minutes, or 31536000 
 seconds. 
 
 Give the Calendar Months, their names, order, and days. How may the 
 number of days contained in each month be remembered ? What is a true 
 year? What years are leap years, and what years common years ? 
 
WEIOUT3 AND MEASURES. 
 
 153 
 
 MISCELLANEOUS MEASURES. 
 
 228. Counting. 
 12 units make 1 dozen. 
 12 dozen, 1 gross. 
 
 "20 units, 1 score. 
 
 6 scores, 1 hundred. 
 
 229. Paper. 
 
 24 sheets make 1 quire. 
 
 20 quires, 1 ream. 
 
 2 reams, 1 ])undle. 
 
 5 bundles, 1 bale. 
 
 'I 
 
 000 
 
 the 
 
 true 
 
 are 1 bushel. 
 1 bushel. 
 1 bushel. 
 1 bushel. 
 1 bushel. 
 1 busliel. 
 1 quintal. 
 1 cental. 
 1 barrel. 
 1 barrel. 
 
 230. Capacity. 
 
 56 pounds of rye 
 
 5G pounds of coru, 
 
 CO pounds of wheat, 
 
 60 pounds of beans. 
 
 60 pounds of potatoes, 
 
 60 pounds of clover-seed, 
 100 pounds of fish, 
 100 pounds of grain, 
 196 pounds of tiour, 
 200 pounds of beef or pork, 
 
 Meal, either of Indian corn or rye, is usually estimated at 50 
 pounds to a bushel, and wheat bran at 20 poumls to a bushel. 
 
 A quarter of grain, in England, is equal to 8 Imperial bushels, or 
 to 560 pounds. 
 
 Mental Exercises. 
 
 1. How many grains in 2 pennyweights \ In 3 penny- 
 weights I*' In 4 pennyweights ? ' 7 ^ 
 
 2. In 40 pennyweights how many ounces? -^ x>^J'*^*^^ 
 
 3. How many ounces in 3 Troy pounds ?:;^i. In 12 Troy 
 pounds %~/Jj'^ /y*; 
 
 4. How many ounces in 3 avoirdupois pounds ? r-: iX-^ 
 
 5. In 60 hundredweight how many tons %j In 80 hundred- 
 weight ?y- In 100 hundredweight?-:^ q 
 
 6. How many yards in 2 rods ? // In 5 rods %"■ %^ \. ' % 
 
 Repeat the Table of Counting. Of Paper. Of Capacity, 
 estimated to a bushel? "Wheat bran? Quarter of grain ? 
 
 How is meal 
 
 
154 
 
 PRACTICAL ARITHMETIC, 
 
 i 
 
 Solution. Since there arc 5.} yards in 1 rod, there are in 2 rods 2 
 limes 5^ yards; 2 times 5 yards are 10 yards, and 2 times J- « ?/arrf 
 are 1 yard; 10 yards and 1 v/arc^ arc 11 yards. Tlierefore, &c. 
 
 7. In 30 feet how many yards 1/^ In 45 feet 'l/.^In 48 feet lj/p 
 
 8. How many square yards in 2 square rods 1^;- n.n 3 square 
 rods?^^^ 'f. ^ 
 
 9. In 54 cubic feet, liow many cubic yards ? = ^ 
 
 10. In 10 gallons, how many quarts ?f^In 15 gallons? Ip 
 
 11. How many rods in 11 yards ^y^In 22 yards? J-^ 
 
 Solution^ Since there is in 5-J yards, or in 11 half yards, 1 rod^- 
 there are as many rods in 11 yards, or 22 half yai'ds, as 11 half yards 
 are contained times in 22 half yards, which are 2. Therefore, d-c, 
 
 12. How many minutes in 3 degrees ?%In 5 degrees 1 / 6 
 
 13. How many quarts in 7 pecks ?'6'(i^ In 10 pecks ?/3^In 12 
 pecks ?^^ 
 
 14. How many square rods in 2 of an acre l/^i^n | of an 
 acre "^^ In | of an acre?/^^ H/^^- 
 
 Solution. Since in 1 acre there are 160 square rods, in -] of an acre 
 there must he \ of 160 square rods, or 40 square rods ; and if 1 fourth 
 is 40 square rods, 3 fourths must be 3 ^iwies 40 square rods, )r 120 square 
 rods. Therefore, d'c. 
 
 15. How many minutes in | of an hour Ij^ln f of an hour ?^^ 
 In j-^2- of an hour ?5':^ In /^ of an hour ? 'y//; m'); ^^ v/;.7 
 
 16. How many pounds in | of a bushel of wheat ?^In £- of 
 a bushel of corn ?^^In y^^ of a barrel of beef ? ^^ 
 
 17. What part of a furlong is 5 rods ?-;;:0f a pound Troy is 
 8 ounces Ir^Of a cubic yard is 18 cubic feet ? jL 
 
 Solution. Sitice 1 furlong is 40 rods, 5 rods are 4^ = ^ o/ a fur- 
 long. Therefore, &c. 
 
 18. What part of a hogshead are 15 gallons ?^ Of a bushel 
 are 24 quarts ti- Of a day is 18 hours ?j^ 
 
 19. What jfert of an acre is 120 rods ?^- Of a bushel of 
 wheat are 48 pounds ?^0f a barrel of beef are 60 pounds ? 7^ 
 
 Review Questions. What is Arithmetical Analysis ? (75) What is a 
 Rule ? (11) What is a Formula ? (69) What is a Solution ? (10) An Opera- 
 tion ? (8) An Answer? (9) 
 
/ard 
 lare 
 
 •ards 
 
 n 12 
 
 if an 
 
 I acre 
 'ourth 
 quare 
 
 our ij^y^ 
 
 fof 
 
 oy is 
 fur- 
 ishel 
 
 3l of 
 
 ^ )0 
 
 IS a 
 ipera- 
 
 DECIMAL WEIGHTS AND MEASURES. 155 
 
 DECIMAL WEIGHTS AND MEASURES. 
 
 231. The Metric System of weights and measures, autho- 
 rised by Congress, in 18G6, to be used in tlie United States, is 
 formed according to the decimal scale. 
 
 The Higher Denominations of a weight or measure are 
 expressed by prefixing to the name of its principal unit, 
 
 Deka, IIecto, Kilo, IMyria, 
 
 10, 100, 1000, 10000; 
 
 and the Lower Denominations by prefixing 
 
 Deci, Centi, Millt, 
 
 10th, 100th, 1000th. 
 
 MEASURES OF LENGTH. 
 
 232. The Meter, the principal unit for tlic measure of 
 length, is very nearly one ten-millionth of the distance on the 
 earth's surface from the equator to the pole. 
 
 Table. 
 
 10 millimeters (mm.) make 1 centimeter (cm.), equal to .3037 inch. 
 
 10 centimeters, 1 decimeter 
 
 10 decimeters, 1 meter (me.), 
 
 10 meters, 1 dekameter, 
 
 10 dekameters, 1 hectometer, 
 
 10 hectometers, 1 kilometer (km.), 
 
 10 kilometers, 1 myriameter, 
 
 The meter in used as the unit of measure for all common lengths and 
 distances. It is about 3 feet 3 inches, and 3 eighths of an inch in length. 
 
 The kilometer is taken as the unit in measuring long distances, as 
 the length of roads, distances between cities, &c. It is about 200 
 rods, or ;^7 of a mile. 
 
 25 millimeters nearly replace the inch, 3 decimeters the foot, 5 
 meters the rod, and 1600 meters the mile. 
 
 How is the Metric System formed? How are the Higher Denominations 
 of each weight or measure expressed? The Lower Denominations? To 
 what is the Meter equal ? Repeat the Table. About how much is a meter ? 
 
 3.937 inches. 
 39.37 inches. 
 393.7 inches. 
 328 feet 1 inch. 
 3280 feet 10 in. 
 6.2137 miles. 
 
 'hi 
 
 1! 
 
 i 
 
ino 
 
 PRACTICAL ARITHMETIC. 
 
 i 
 
 
 1 
 
 I 
 
 MEASURES OF SURFACE. 
 
 233. Tlie Square Meter, the principal unit for the measure 
 of surface, is the square whose side is one meter. 
 
 Table. 
 
 100 aq. millimeters (mm.-), are 1 sq. centimeter (cm.-), = .00155 sq. in. 
 100 sq. centhneters, 1 .sq. decimeter, .1076 pq. ft. 
 
 100 sq. decimeters, 1 sq. meter (m.-), 1.19(5 Hfj. yd. 
 
 Since the side of a square meter is 1 meter, or 10 decimeters, a 
 square meter is equal to 10 x 10 = 100 square decimeters ; since the 
 i^ide of a square decimeter is 1 decimeter, or 10 centimeters, a square 
 decimeter is equal to 10 x 10 = 100 square centimeters, &c. Hence, 
 
 The scale is 100, and two orders of fi<,'ures nmst be allowed to 
 each denomination. 
 
 234. The Are, the principal unit in measuring land, is a 
 square whose side is ten meters. 
 
 Table. 
 
 100 centiares are 1 are (ar.), equal to 119.6 sq. yd. 
 
 100 ares, 1 hectare (ha.), 2.471 acres. 
 
 A centiare, or square meter, is about \\ square yards, and a hectare 
 about 21 acres. 
 
 40 ares nearly replace an acre of common surface measure. 
 
 I'- 
 
 MEASURES OF VOLUME. 
 
 235. The Cubic Meter, the principal unit for the measure 
 of volume, is the cube whose edge is one meter. 
 
 Table. 
 
 1000 cu. millimeters (mm.'), are 1 cu. centimeter (cm.^), = .061 cu. in. 
 1000 cu. centimeters, 1 cu. decimeter, 61.022 cu. in. 
 
 1000 cu. decimeters, 1 cu. meter (m.-^), 1.308 cu. yd. 
 
 To what is the Square Meter equal? Repeat the Table. What is the 
 Are? Kepeat the Table. How much is a centiare? A hectare? What 
 nearly replaces the acre? What is the Cubic Meter? Reijeat the Table. 
 
DECIMAL WEIGHTS AND MEASUUES. 
 
 15; 
 
 in. 
 u. in. 
 yd. 
 
 Since the e([<fi,v of u cubic lueter is 1 nit-tiT, or 10 docimetei-ri, a 
 cubic meter i.se(iu;il to 1(> x 10 x 10 = looo cubic (IccinictiTs ; since 
 tlie edj^'e of u cubic decimeter is one di-cimcter, or 10 centiiueti-rs, a 
 cubic decimeter is e(£ual to 10 x 10 x 10 = 1000 cubic centimeters, 
 &c. Hence, 
 
 The scule is 1000, and three orders of li-jurcs mu-sl be allowed lo 
 each denomination. 
 
 236. Tliu Liter, the principal unit for li({uid or dry ineafturt'. 
 is a cubic deciinetcr. 
 
 10 milliliters 
 10 centiliters, 
 10 deciliters, 
 10 liters, 
 10 dekaliters, 
 10 hectoliters, 
 
 Table. 
 
 arc 1 cenlilitrr (cl.), ecjual to ..'J.^S fluid oz. 
 1 deciliter, .815 gill. 
 
 1 LITER (It.), 1.05G7 quart?. 
 
 1 dekaliter, 2.0117 gallons. 
 
 1 hcctollkr (hi.), 20.417 gallons. 
 
 1 kiloliter, 201.17 gallons. 
 
 The literh usediu measuring lii[uids, and i« about Ij^g liquid quart. 
 
 The hectoliter is used in measuring grains and like articles, and is 
 2,837 bushels, or about '2^ bushels, or | of a barrel ; a liter is very 
 nearly .U08 of a dry quart. 
 
 4 liters a little more than replace the li<iiud gcdlon^ and 35 liters 
 very nearly the common bushel. 
 
 A milliliier is equal to 1 cubic centimeter ; ,:i ccntUiler is equal to 
 10 cubic centimeters. 
 
 237. The Stere, the }>rincipal unit for ni asuring wood, is a 
 cubic meter, or lOUO liters. 
 
 Table. 
 
 10 decisteres are 1 steiik (st.), equal to 1.308 cubic yards. 
 
 10 steres, 1 dekastere, 13.08 cubic yards. 
 
 36 decisteres, or 3.0 steres, very nearly replace the common cord. 
 
 What is the Liter? Eepeiit the Table. For what is the liter lined? 
 About how much is a liter? For what is a heitolitcr used? Abtjut how 
 much is a hectoliter? AV'hat nearly replaces the iifiuid gallon? The com- 
 mon busliid? AVhat is the Stere? ilei)eat the Table. What very nearly 
 replaces tho conmion cord ? 
 
 ' 1 
 
 AM 
 
 W 
 
 1 
 
 1 
 
 % 
 
 3 
 
 Mii 
 
Ii!< 
 
 ! 
 1 
 
 1 II 
 
 
 IS' . 
 
 X III; 
 
 158 
 
 PRACTKJAL AIIITIIMKTIC. 
 
 10 mill if/rams 
 10 centigrams, 
 1 decigrams, 
 10 grams, 
 10 (l(;kagrams, 
 10 liectograms, 
 10 kilograms, 
 10 myriagrams, 
 10 quintals. 
 
 .1543 
 
 grain. s. 
 
 1.543 
 
 >» 
 
 15.432 
 
 ?» 
 
 .3527 
 
 av. oz. 
 
 3.5274 
 
 >> 
 
 2.2040 
 
 a v. lb. 
 
 22.04(5 
 
 >> 
 
 220. IG 
 
 )' 
 
 2204. G 
 
 
 WEIGHTS. 
 
 238. The Gram, th(5 principal unit of weights, is tho weight, 
 in a vacuuni, of a cubic centiineter of distilled water, at its 
 gi'eatest density. 
 
 Table. 
 
 are 1 centigiam, equal to 
 
 1 decigram, 
 1 (UiAM (gm.), 
 1 dekagram, 
 1 hectogram, 
 1 kihvjram (k.), 
 1 myriagram, 
 1 quintal, 
 1 millier, or tonnean (t.). 
 
 The kilofjram^ or, for brevity, kilo, is the ordinary weight of com- 
 merce. It is iiliout 21 pounds. 
 
 The tonnemi (pronounced tonno), or metric ton, is used in weighing 
 heavy articles, and is about 2200 pounds. 
 
 The gram is used in mixing medicines, weighing letters, gold, 
 iewcl., &c. 28 grains nearly replace an avoirdupoi.s ounce; and ^ 
 kiiu, a little mure tlian a pound. 
 
 239. Li expressing Metric Weights and Measures, by figures, 
 the decimal point, as in United States Money, is placed between 
 the unit, and its subdivisions written as decimal orders. 
 
 One, two, or three orders of figui'es must be allowed to each 
 denomination lower than the unit, according as the scale is 10, 
 100, or 1000. Thus, 
 
 3 kiloliters, 7 hectoliters, 2 dekaliters, 5 liters, 6 centiliters, is 
 written, as liters, 3725.06 It. 
 
 4 cubic meters, 030 cubic centimeters, as cubic meters, 4.00063 m'*. 
 
 24C. The integer of a metrical expression may be read as a 
 number of its primary unit ; and the decimal ])art, if any, as a 
 numl)cr of the lowest denomination denoted. Thus, 
 
 Wliat is the Gran)? Repeat the Table. "Wliat is the kilogram called for 
 brevity? About how much is a kilo? Atonneau? What very nearly re- 
 places the avoirclupt is ounce ? The pound ? How is the decimal point placed ? 
 
 ai: 
 
DKCIMAL WKIOIITS AND MRASirRKS. 
 
 150 
 
 iK 
 
 .lOO.OTfj kilos luiiy be ruud as three huiitlnid sixfy kilos, luid 
 eevetity 'ivo ^'ruius. 
 
 30.15 meters, iw thirty-aix metera, ami lirtecii ceutuuelei-s. 
 
 Comparative Table. 
 
 A meter = 
 A meter = 
 A meter = 
 A kiloiiu^ter = 
 A sq. meter = 
 A s(|. meter = 
 A H(j. meter -- 
 All are - 
 
 A liectiire - 
 A liei-ttuv 
 A liter - = 
 A liter = 
 
 A liter 
 
 A hectoliter = 
 A liter 
 
 A hectoliter = 
 A stere = 
 
 A stere = 
 
 A ijram = 
 
 A kiloi^n'am - 
 A kilogram = 
 A kilogram = 
 A toiineaii = 
 
 PA)M7 inched. 
 
 3.28 feet. 
 l.OiKUJyar.l.s. 
 .02137 mile. 
 lofjO s(|. inches. 
 
 10.70 Mj. feet. 
 
 l.l!)() s(|. yards. 
 
 3.!)r).3 s(|. rods. 
 
 2.471 acn's. 
 .OOIJSO s(|. mile. 
 
 33.81 lluidoz. 
 1.0507 (jiiarts. 
 .20417 gallon. 
 
 2.837 bushels. 
 01.022 cu. inches. 
 
 3.531 cu. feet. 
 
 1.308 cu. yards. 
 
 .2750 cord. 
 15.432 grains. 
 
 35.27 av. ouncies. 
 2.08Tr.])oun(ls. 
 2.2O40av.pounds. 
 1.1023 tons. 
 
 An incii 
 A foot 
 A yard 
 A mile 
 A s({. inch 
 
 A Hi[. f(J0t : 
 
 A s<|, yard 
 A B(i. rod 
 An acre 
 A s([. mile 
 A. Iluid oz. 
 A ([iiart 
 A gallon 
 A bushel 
 A cu. inch 
 A cu. foot : 
 A cu. yard 
 A cord 
 A grain 
 An av. oz. 
 A Troy lb. 
 An av. lb. 
 A ton 
 
 .0254 
 
 .3048 
 
 .!)! U 
 
 1.00!K{ 
 
 .000()452 
 .0!»2(> 
 .8301 
 .252:) 
 .4047 
 25!) 
 .02!)5S 
 .5)405 
 3.780 
 .3524 
 
 : .0103!) 
 .2S32 
 .704(5 
 3.025 
 .0018 
 .0283 
 .373 
 
 : .4530 
 .9071 
 
 nuter. 
 
 nu'ter. 
 
 nu'tcir. 
 
 kilometora. 
 
 .s([. meter. 
 
 s(|. meter. 
 
 s([. meter. 
 
 ari!. 
 
 hectare. 
 
 hectares. 
 
 liter. 
 
 liter. 
 
 liters. 
 
 luictoliter. 
 
 liter. 
 
 hectoliter. 
 
 st(U'e. 
 
 steres. 
 
 gram. 
 
 kilogram. 
 
 kilogram. 
 
 kilogram. 
 
 tonneau. 
 
 These equivalents are only determined approximately ; but are 
 sufliciently e.xact for all ordinary business. 
 
 How many inches make a meter? How many miles a kilometer? Square 
 inches a .square muter? .Scjuare roads an are? .Acres a hectare ? (Quarts a 
 liter? liu-shels a liectolitor? Cords a stere? Grains a gram ? Poiuids 
 Troy a kilo? Avoirdupois pounds a kilo? Tons a tonneau? IMeters an 
 inch? Kilometers a milo? Square meters a s(iuare yard? Ares a square 
 rod? Hectares an acre ? Liters a quart ? Hectoliters a bushel? .Stores a 
 cord? Grams a i,'rain? Kilograms a pouud Troy? Kilograms an avoir- 
 dupois pound? Tonnoaux a ton? 
 
 n 
 
 m 
 
\\f 
 
 l! 
 
 160 
 
 Read 
 
 1. 1686.45 me. 
 
 2. b37A-25 k. 
 :i 634.56 t. 
 
 4. 76.7 St. 
 
 5. 100.56 ar. 
 
 PllA CTICAL AKIT .IMETIC. 
 
 Exercises. 
 
 6. 2.006 m.^ 
 
 7. 517.5 hi. 
 
 8. 3.1605 ha. 
 0. 15.1005 m.'^ 
 
 10. 678.06 It. 
 
 11. Write in ligures thirty-one hectares and fifteen ares, as 
 liectares. 
 
 12. Write in figures seventeen and five tenths steres. 
 
 13. Write in iigares sixteen hundred meters and twenty-five 
 centimeters. 
 
 14. Express by figures one hundred thirty-two cubic meters, 
 and seven thousand three cubic centimeters. 
 
 15. Reduce 8.07018 kilometers to meters. 
 
 Solution. Since in 1 Jcilonictcr there are 1000 meters, in 8.07018 
 kilometers there 'mast be 8.07018 times 1000 meters, or b070.18 inelcrs. 
 
 16. Reduce 16.85 grams to kilograms. 
 
 Solution. Since in looo <jrams the-e is 1 kilogram, there are as 
 mani/ kilof/r'ims in 10.85 </rams as 1000 grams are cuutained times in 
 10.85 (jrams, or .0]()85 kilogram. 
 
 That is, to cluuinc froiu a nuiuber of out.' denoiuiiiation tu an 
 t'quivuleuL miiubei' of jinotht-r denomination of tlic same weight or 
 measure, we may 
 
 liemove the decimal point one or more places to the rigid or left, 
 (/.•>■ the caf<e may require, sujtplijlng ciphers If nec.cssarij. 
 
 17. In 5<10.5 hectares, how many square meters \ 
 
 18. In 5605000 square meters, how many hectares ] 
 
 19. In .0605 liters, how many centiliters ? 
 
 20. In 6.05 centiliters, liow many liters ? 
 
 21. Express .01531 tonneau as kilograms. 
 
 22. Express as millimeters and add, 31.55 meters, 13.005 
 meters, and 75 millimeters. Ans. 4063 nnn. 
 
 23. From 56 hectares take 156 ares. Ans. 54.44 ha. 
 
 How may we change from one deuombiutiou lo aiiutlier of the same wciyht 
 
 or measure ; 
 
DECIMAL WEIGHTS AND MEASURES. 
 
 IGl 
 
 CO.MPAEISON OF UNITS. 
 
 241. The units of the metric and common systems may bo 
 readily compared by means of the Comparative Tabh) (Art. 
 240). 
 
 1. In 125 meters how many feet] 
 
 Solution. Since in 1 meter there are ^.28 f<;et, there will he in 12.') 
 meters 125 times 3.29 feet, or 410 feet. 
 
 Or, since in 1 f(JOt there are .3048 meter, there loill he as many feet 
 in 125 meters, as .3048 meter are contained times in 125 meters, or 41t) 
 feet. 
 
 2. In 150 acres how many hectares? 
 
 3. In. 2.5 hectoliters how many bushels ? Ans. 7.09 + bu. 
 
 4. In 15 cords how many steres? Ans. 54.375 steres. 
 
 5. Keduce 1G.25 kilos to pound' avoirdupois. 
 
 0. How many tonneaux in 100 tons? ylns. 90.71 tonneaux. 
 
 () an 
 it or 
 
 left, 
 
 .005 
 
 lllUl. 
 
 ha. 
 
 leight 
 
 CONVERSION OF PIUCES OK RATES. 
 
 242. When the price or rate per unit is given either in llie 
 common or in the metric system, it may be readily found for 
 a corresponding unit in the other system, since 
 
 The prices or rates must have the same relation to each other as 
 have the magnitudes of the given units. 
 
 1. If the price per kilogram is §.80, what is it per pound ? 
 
 Solution. If *he price j^er kilogram is .$.80, it miost he per 'potiml, 
 which is .4536 times a kilogram, .4536 times 8.80, or .$.36 + . 
 
 2. Tlie price per liter being $1, what is it per quart ? 
 
 ylns. J?.94G + . 
 
 3. The produce per hectare beiiig 10 hectoliters, what is it 
 per acre ? Ans. 4.047 hectoliters. 
 
 4. The produce per acre being 45 bushels, what is it jxt 
 hectare? Ans. 111.195 bushels. 
 
 rl 
 
 How may the units of the metric and commo!i systems bo coinparod ? To 
 what docs the price or rate of units in the ditferent systems correspond? 
 
 i 
 

 i 
 
 
 )i 
 
 162 
 
 PRACTICAL AKITUMKTIC. 
 
 DENOMINATE NUMBERS. 
 
 243. A Denominate Number is a concicte nunil)er ex[)re.^s- 
 iiig weight, measure, or currency. Thus, 
 
 5 rods, 3 pounds, 2 ounces, 9 dollars, are each denominahi 
 numbers. 
 
 A Simple Denominate Nujibkr expresses only units of a 
 single denomination of a weight, measure, ^c. 
 
 A Compound Denominate Number expresses units of t^vo 
 or more denominations of the same kind of weight, mt asure, 
 &c. Thus, 
 
 5 rods 3 yards, 8 pounds G ounces, 7 dollars 3 dimes 2 
 cents, are each compound denominate numbers. 
 
 EEDUCTION. 
 
 244. Reduction of Denominate Numbers is the process of 
 changing their denomination, without changing their value. 
 
 CASE I. 
 
 245. To reduce from a higher to a lower denomination. 
 
 1. Let it be required to reduce G3 11). oz. 10 pwt., to 
 pennyweights. 
 
 OrERATION. 
 
 03 lb. oz. 10 pwt. 
 
 12 
 
 Since in 1 pound there are 12 omiccs, 
 
 in 63 pounds there are 63 times 12 
 ounces, or 756 ounces. 
 
 Since in 1 ounce tlicrcare 20 penny- 
 weights, in 756 ounces there are 756 
 times 20 pennyweights ; and 10 penny- 
 weights added, make 15130 penny- 
 weights. Therefore, &c. 
 
 120 
 63 
 
 75G oz. 
 20 
 
 15120 
 10 
 
 15130 pwt., Ans. 
 
 What is a Denominato Number ? A Simple Denominate Number ? A 
 Counoouud Dcuomiuatc Number ? lleiluetion of Denominate Numliers ? 
 
 
DENOMINATE NUMBEllS. 
 
 163 
 
 RULK. MaUiijhj the ninnbcr of the Jiujlied denomination given 
 by the ninnhrr required of the next lower denomination to make one 
 of that higher, ana lO the jmjduct add the number, if anj/, of the 
 lower denominatio .. 
 
 Proceed in like 'uamier till the u-lwle is reduced to the required 
 denomination. 
 
 Examples. 
 
 IJ educe 
 
 2. 3 T. Id cwt. to pounds. J^^^^ 
 
 3. 2 lb. 8 oz. to drams. (^^^ 
 
 4. 9 miles to rods. 'X/^^O 
 
 5. 98 gill. 1 pt. to pints, y^/i" 
 
 G. .583 A. 130 P. to square rods.;^ '>/.' 
 
 7. 17 cords to cubic inches^ /^^/•^^ 
 
 8. 27 bu. 3 pk. to pints. /> ^6 ^ 
 
 9. 5" G' \h" to seconds. jVJh 
 
 10. In 5 fur. 12 rd. 4 yd. Iiow many feetlj^^ /^ 
 
 11. In 3G5 d. 5 h. 48 m. 50 sec. how many seconds ^i/*^ '6 1;/^^ 
 
 12. In 24 hlid. 18 gal. 2 qt. iiow many pints? I %%^-'¥- 
 
 13. In 17 m. G fur. 22 rd. 4 yd. 2 ft. 7 in. how^many inches ?//''^^'/^' 
 
 14. In 75 bu. 3 pk. 5 qt. how many quarts'? J2./-^/' 
 
 15. In 4 cwt. 99 lb. 10 oz. 12 dr. how many drams? / '-- / /'/ u) 
 10. In 180 degrees how many seconds? (j^ -j^ i!f >y O 
 
 246. The rule r.lso applies when the denominate number is 
 a common fraction or decimal. 
 
 17. lieduce '1 of a gallon to pints. 
 
 OPERATION. 
 
 1 'J 
 
 4 = Y qt. ; V- X 2 = ^ = 3-^ pt., An^.. 
 
 Since in 1 pdlon there are 4 quarts, in ^ of a gallon there are ^ of 
 4 quarts, or \" (quarts. 
 
 Since in 1 quart tliere are 2 pints, in ^ ([uurts there arc \^ of 2 
 pints, or "/ = 3^ pints. 
 
 18. lieduce yjT- of an acre to squaie rods. ~ §0 
 
 19. In y- of a yard how many quarters? -' / ^ 
 
 20. In Ss\ days how many minutes?^ j i fJ^O 
 
 21. What part of a pennyweight is ^^^ of i pound t -* I- 
 1 -_^- 
 
 Whatisthe Rule? 
 
 * For answers, sco corresponding examples in Ciiso 11. 
 
 llj 
 
 !' / 
 
 i'. i 
 
■ill III" 
 
 16i 
 
 PRACTICAL ARITHMETIC. 
 
 22. 
 
 23. 
 
 24. 
 26. 
 
 26. 
 27. 
 28. 
 29. 
 30. 
 
 C 
 
 Express tj|^ of a cwt. in terms of a pound. ^ >. 
 
 Reduce .145 cwt. to ounces. =:iJiL^. I ^//i./^'i^^^'y'^'^ 
 
 OPERATION. 
 
 .145 X 100 - 14.5 lb. ; 14.5 x 16 = 232 oz., Ans. 
 
 In .0003 weeks how many minutes ? ■=\j ' C^ jL ^Ov 
 
 In 6.35 miles how many feet ?- fi^'*^ / 
 
 In .1756 kilometers how many meters? "^ I ^ -^ * ^ y. ^ 
 
 What part of a quart is .0015 of a hogshead? - ' <j I ^ 
 
 How many mills in $15.69 ?- /^''^;/^;; 
 
 How many kilograms in 3.675 tonneaux? -tj 6 7i 
 
 How many square rods in .94375 of an acre 1 i^ 7 yj 
 
 '! 
 
 I 
 
 CASE II. 
 
 247. To reduce from a lower to a higher denomination. 
 1. Let it be required to reduce 15130 pennyweights to 
 pounds. 
 
 OPERATION. 
 
 2!0)1513|0pwt. 
 
 12)756. . lOpwt. 
 
 Since in 20 pennyweights there is 
 1 oimce, in 15130 pennyweights there 
 are as many ounces as 20 penny- 
 weights are contained times in 15130 
 pennyweights, which are 756 times, 
 and a remainder of 10 pennyv/eiLj;]its. 
 
 Since in 12 ounces there is 1 pound, 
 in 756 ounces there are as many 
 
 63 1b. 
 Am. 63 lb. oz. 10 pwt 
 
 poimds as 12 ounces are contained times in 756 ounces, or 63 times. 
 
 Rule. Divide the given number by the number oj its denomi- 
 nation required to make one of the next higher, and reserve the 
 remainder, if any. 
 
 Proceed in like manner ivith the quotient, and so continue till 
 the whole is reduced to the required denomination. 
 
 The number of the required denomination, with the several re- 
 mainders, if any, will be the answer. 
 
 Review Questions. What is a Denominate Number? (243) A Simple 
 Denominate Number? (243) A Compound Denominate Number? (243) A 
 Measure? (197) A Weiyht? (200) Explain the operation. Repeat the 
 Rule. 
 

 DENOMINATE NUMBERS. 
 
 105 
 
 Examples. 
 
 J* Ui/ • 
 
 Tleduce 
 
 2. 7500 pounds to tons.^,.'/0 
 
 3. G40 drams to pounds. ^L"/ 
 
 4. 2880 rods to miles. = Q ),ut^ 
 
 5. 785 pints to gallons. O^^O"^! 
 
 A f 
 
 6. 93410 snuare rods to acros.-)< 
 
 7. 37G0128 cubic in. to cords./; ' 
 
 8. 1770 pints to busliels. -j, /'/^ 3 
 
 9. 18375 seconds to degrees.^' 
 
 10. In 3510 feet how many furlongs ? =0 •/^, ^ta ■ 
 
 OPERATION. 
 
 3 ) 3510 ft. 
 
 ^ 
 
 iy 
 
 
 5^) 1170 yd. 
 
 '> 9 
 
 11 2340hlf. yd. 
 
 40 ) 212. .8hlf. yd. = 4 yd. 
 
 Here, to divide 1170 by 5.^, 
 we first reduce both numbers 
 to halves, by nmltiplying by 
 2, and have 2340 Judves to be 
 divided by II halves, which 
 gives 212, and a renuiituhr 
 of 8 halves - 4. 
 
 5 . . 12 rd. 
 Ans. 5 fur. 12 rd. 4 yd. 
 
 Eeduce^^y^'^^^'"^''^'^^"^^^ 
 ^11. 31550930 seconds to days.lU. 2429 quarts to bushels. 
 
 12. 12244 pints to hogsheads. 
 
 13. 1129171 inches to miles. 
 
 15. 127910 drams to cwt. 
 10. 048000 seconds to deirrers. 
 
 17. Reduce 3f- pints to the fraction of a gallon. 
 
 q:5 
 
 OPERATION. 
 
 -- , — -r -. ^ qt. ;, ^ - 1 - Y gai., Jin.'^. 
 
 24 
 
 Since in 1 quart there are 2 pints, in 3-^ = ?_* pints there are a.s 
 many tpuirts as 2 pints are contaii)ed times in ?^^- pints, which are 1^ 
 times. 
 
 Since in 1 gallon there are 4 quarts, in ^ quarts there areas many 
 gallons as 4 quarts are contained times in ?-, which is j of a time. 
 
 18. Reduce 70 square rods to a fraction of an acre. ~ y^ 
 
 19. In H quarters how many yards? -^ 
 
 llEVIEW Questions. Repeat the Table of Troy Weight. (203) The Talih; 
 of Avoirdupois AVoight. (204) Of Linear Measure. (20.")) Of Surface 
 Measure. (210) "What is a Surface ? (20C) The area of u figure? (207) A 
 Square ? (209) A Rectangle? (210) 
 
 / 
 
 ■■'c 
 
 1 
 
 * 
 
 
 .n 
 
 ,1 1 
 
T~T^^S5^HJ 
 
 1 
 
 ii 
 
 106 
 
 PRACTICAL ArwITDMETIC. 
 
 (, 
 
 Q 
 
 20. In 13020 Tninutes how many days? • / J y 
 
 21. What part of a pound is |- of a pennyweight? T7^^^ 
 
 22. Express -} of a pound in terms of a hundredweight. J^-?- 
 
 23. Reduce 232 ounces to a decimal of a hundredweight. ' J'^S' 
 
 OPERATION. 
 
 232 --10 = 14.5 lb.; 14.5 ^ 100 = .145 cwt., Ans, 
 
 24. In 3,024 minutes how many weeks? * ^c^ C f 
 
 25. In 33528 feet how many miles ? ^ * "^ '^ h /y 
 
 26. In 175.0 meters hoAv many kilometers? * / * h 
 
 27. What part of a hogshead is .378 of a quart ? ; ( ■ ' -' 
 
 28. How many dollars in 15090 mills ? /'' jh"' If^ 
 
 29. How many tcnneaux in 3075 kilograms ? 3 ' iP^ ^ ^ 
 
 30. How many square acres in 151 square rods ? */ ^J / O 
 
 CASE III. 
 
 248. To reduce a fraction of a given denomination to in- 
 tegers of lower denominations. 
 
 1. Let it be required to reduce y- of a pound Troy to equi- 
 valent integers. 
 
 OPERATION. 
 
 1 X 12 = M = 10.^ oz. 
 
 2 X 20 = i« = 13^ pwt. 
 
 24 
 
 21 = 8 ^r 
 
 Alls. 10 oz. 13 pwt. 8 gr. 
 
 Since 1 pound e([uals 12 ounces, 
 § of a pound eqiial.s ;} of 12 ounces, 
 or 101} ounces. 
 
 Since 1 ounce equals 20 penny- 
 weights, § of an ounce equals ;•} of 
 20 pennyweights, or 13^ penny- 
 wc'i<fhts. 
 
 Since 1 pennyweight equals 24 grains, ^ of a pennyweight equak 
 \ of 24 grains, or 8 grains. 
 
 Therefore, § of a pound Troy is equal to 10 oz. 13 pwt. 8 gr. 
 
 IlULE. Multiply the fraction hj/the mimher required of the next 
 lower denominaticn to make one of its denomination, and reduce 
 the result, if possible, to a tvhole or mixed number. 
 
 Proceed in like manner icith the fractional parts, if any, and so 
 continue till reduced as required. 
 
 Explain the operation. Kepeat the Rule. 
 
DENOMINATE NUMBERS. 
 
 1C7 
 
 / 
 
 Examples. )^c ^^^^V^- 
 
 2. IJc.'duce § of a furlong to equivalont intogors.*j''^ ' o >> i) >' ^ 
 
 3. lieduce -./j of a hogshead to oqiiivak^nt intogcre. %j J '• J ' 
 
 4. Express in integers -]" of a week. .-- J*-^*^. /'-•,/ ^ 
 />, Find the vahie of -j^- of a degree. :r // . ^ l^jL 
 G. Find the vahie of l\ of a hiindredweight.r %'^h */S" i'j' 
 7. lieduce .282 of a ton to equivalent integers. ^^ *■ Q. . I ^ ^ 
 
 FIRST OPERATION. (SECOND OPERATION. 
 
 .282 X 20 = 5.G4 cwt. .282 
 
 20 
 .04 X 4 = 2.50 qr. 
 
 .50 X 25 = 14 lb. 
 
 Ans. 5 cwt. 2 qr. 14 lb. 
 
 It is often most convenient to multiply the decimal 
 part, without re-writing it, as in the second operation, lb. i4.000 , . '^j 
 
 8. Reduce .875 of a hogshead to equivalent integers. A' />'-^'"/ 
 1). Eeduce .4705025 of a mile to equivalent integers.^ '.?>^"/" *f 
 
 10. What is the value of .09375 of an acre? "• / b'u^' ^^^ r 
 
 11. Reduce 5.141 tons to equivalent integers.^A'^^ //,' ^'^^' 
 
 12. What is the value of .701 of a day ? =•/ // j ^j'^a Iji' V-' • 
 
 cwt 
 
 5 
 
 040 
 4 
 
 qr 
 
 2 
 
 500 
 
 
 
 
 25 
 
 
 800 
 
 
 11 
 
 20 
 
 I 
 
 mJl 
 
 
 CASE IV. 
 
 249. To reduce integers of lower denominations to a 
 fraction of a higher denomination. 
 
 I. Let it be required to reduce 10 oz. 13 pwt. S gr. to an 
 equivalent fraction of a pound. 
 
 OPERATION. 
 
 8 gr. = -/j- pwt. = A pwt. 
 
 grains 
 
 13Jpwt. = ^pwt. = ^ - 20 ^ ^oz. 
 101 oz. = ^ oz. ^ iif ^ 12 = « lb. 
 
 Since 24 
 ecpial 1 pennyweight, 
 8 grains equal ry^, or 
 ;^, of a pennyweight. 
 
 Since 20 penny- 
 
 Exiilaiii the operation of examjile 1. 
 * For uuswerri, see corresponding' examples in Case IV. 
 
 H 
 
1G3 
 
 TRACTICAL AIIITIIMETIC. 
 
 weights equal 1 ounce, y pennyweights e([ual vrV of ^ , or 3 of an 
 ounce. 
 
 Since 12 ounces equal 1 pound, 'Y ounces equal ^^ of ^j^, or ^ of ii 
 jiound. 
 
 Hulk. Hedvce the mimher of the lowest denomination to a 
 fraction of the denomination next higher, and ivrite it as a frac- 
 tioufd jwj't of the number of that higher denomination. Eeduce 
 the nnraher thus formed in like manner as before, and so contin ne 
 till all the numbers are reduced as required. 
 
 n 
 
 ■;: 
 
 Examples. 
 
 2. Reduce 35 rd. 3 yd. 2 in. to an equivalent fraction of a 
 furlong. :r^ ,^ ■ 
 
 3. Reduce 2 qt. 1 pt. 1 gi. to an equivalent fraction of a 
 hogshead. ^ ^ 
 
 4. Reduce 3 d. 4 h. to an equivalent fraction of a week. '^jT, 
 
 5. Reduce 21' 255^" to an equivalent fraction of a degree, ^.'-j. 
 
 6. Reduce 2 qr, 20 lb. 13 oz. 5 J dr. to an equivalent fraction 
 of a hundredweight. = ^^^' 
 
 7. Reduce. 5 cwt. 2 qr. 14 lb. to an equivalent decimal of a 
 ton. ■=^-*i'<^^-» 
 
 FIRST OPERATION. SECOND OPERATION. 
 
 14 lb. = 14 -r 25 - .56 qr. 
 2.56 qr. = 2.56 -r 4 = .64 cwt. 
 
 25)14.00 lb. 
 
 4 ) 2.56 qr. 
 
 5.64 cwt. = 5.64 -r 20 - .282 T., Ans. 20)5.640 cwt. 
 
 .282 T., Ans. 
 
 The second operation, which is an abridged form of the fir.st, is 
 generally preferred, on t^ccount of its conciseness. 
 
 8, Reduce 55 gal. qt. 1 pt. to an equivalent decimal of a 
 hogshead. ~ * S / ^ 
 
 9. Reduce 3 fur. 32 rd. 8 ft. 3 in. to an equivalent decimal 
 of a mile. ^ '^y^io^^O' 
 
 What is the Kulc? Explain the operation. 
 
DENOMINATE NUMBERS. 
 
 IGO 
 
 10. Reduce 15 square rods to an equivalent decimal of an 
 acre.-.^5?or'>'^" 
 
 11. Express 5 T. 2 cwt. 3 qr, 7 lb. as a mixed decimal of a 
 ton. •" -^ •/ ^/^ 
 
 \'l. What is the value of 18 h. 15 m. 50.4 sec. in the decimal 
 of a day ^ — * ^ ^./ 
 
 250. When it is required to find the part that one conq>ound 
 denominate number is of another (Art. 174), 
 
 lieduce the numbers to the same denomination, aid divide the 
 number denoting the part by that with which it is compared. 
 
 13. What part of 1 lb. 4 oz. 12 pwt. 12 gr. is 2 oz. 15 pwt. 
 10 gr.? An.s. \. 
 
 14. What part of 7 bu. 1 pk. is 2 qt. 1 pt. ? Ans. -^l^. 
 
 15. What part of 3 acres is 1 A. 26 P. ? Ans. 'y^. 
 
 IG. What part of 1 T. 6 cwt. 15 lb. 10 oz. is 10 cwt. 4(; 11). 
 4 oz. ? Ans. r. 
 
 17. What decimal of 148 m. 4 fur. is 18 m. 4 fur. 20 rd. ? 
 
 Ans. .125. 
 
 18. What decimal of 7 w. 4 d. is 2 d. 17 m. ? 
 
 Ans. .0379585 + . 
 
 19. What decimal of 45 T. 15 cwt. 25 lb. is G T. 10 cwt. 
 75 1b.? Ans. .142857 + . 
 
 APPLICATIONS. 
 
 1. If a silver pitcher weighs 2 lb. 3 oz. G invt., how many 
 pennyweights is its weight 1 Ans. 54G pwt. 
 
 2. What is the cost of 3 cwt. G3 lb. of flour at 5 cents a 
 pound? Ans. $18.15. 
 
 3. How many acres in a rectangular field 80 rods long by 
 65 rods wide 1 Ans. 32 A. 80 P. 
 
 How do you iiiid the part that ouo compound denominate nuuxbur is of 
 another ? 
 
 i 
 
 » 
 
170 
 
 I'RACTICAL ARITHMETIC. 
 
 4. IIow m.iny hottlos, holding 1 qt. 1 pt. each, 'will he re- 
 quired to hottlc a liogsliead of wine ? yliis. 1G8 ])ottles. 
 
 5. W'liab will ho the cost of excavating a mass of earth 
 lOO.U fett long, 12.45 feet wi<le, and 10 feet deep, at 20 cents 
 a cubic yard ? y//^^•. 51:92.351 + . 
 
 0. What sum can he .saved in 10 voars hv risini.' 45 minutes 
 earlier, 300 days of each year, if an hour is worth 15 cents? 
 
 y//^s^ 81350. 
 7. A farm consists of exactly 78843G square feet; what is 
 
 its value at -| of a dollar per S(juare rod? 
 
 yln.^. S1810. 
 
 8. How many tons of iron, at 3 cents a pound, ean Ije bought 
 for $353.79 1 Am. 5 T. 17 cwt. 3 qr. IS lb. 
 
 9. What is the value of a range of wood, 100 feet long, 4 
 feet wide, and 12 feet high, at li?5 per cord? =^ , -^ ^ 
 
 10. Which are the leap years between 18G<j and 1873? »*/6^/'^'^ 
 
 11. How many square yards of carpeting will be required to 
 carpet a room 24 feet long and 18 feet wide? yim^. 48 yards. 
 
 12. What is the cost of 2 T. 5 cwt. 2 qrs. 20 lb. of beef, at 
 $0 per hundredweight ? A as. $41 1.30. 
 
 13. If you ar<'! 10 y. 3 w. 18 h. 30 m. old, how many minutes 
 have you lived, allowing that every fourth year was lea}) year? 
 
 .7/^5. 8440710 minutes. 
 
 14. At $3.50 a gallon, how much wine can be bought for 
 $204.25? Ans. 1 hhd. 12 gal 2 qt. 
 
 15. At $9 a hundred .veight, how much beef can be bought 
 for $229.05? Ans. 1 T. 5 cwt. 1 qr. 20 lb. 
 
 10. How many more minutes in March than in February of 
 1 808 ? Ans. 2880 minutes. 
 
 17. When the distance between two places is 31.5 kilo- 
 metres, how many miles are they apart? 
 
 Ans. 19 m. 4 fur. 23 rd. 0.7584 ft. 
 
 Keview Questions. What is a Cube ? (213) A P.cctangvilar Solid? (I'H) 
 Table of Solid Measure ? (214) A cord of wood ? (214) A jierch of masonry ? 
 Table of Liquid Measure? (215) Table of Dry Measure? (21G) 
 
 Zj 
 
 \ 
 
DENOMINATK NUMUEIIS. 
 
 171 
 
 I f 
 
 ro- 
 les, 
 nh 
 iits 
 
 + . 
 
 lt<'.S 
 
 ? 
 
 .50. 
 
 t is 
 10. 
 
 -ht 
 11>. 
 
 ,4 
 
 hmri 
 
 Ito 
 r.ls. 
 
 , at 
 30. 
 
 itcs 
 
 tcs. 
 for 
 qt. 
 
 ;ht 
 11). 
 
 ^ (jf 
 tes. 
 
 ilu- 
 ft. 
 
 L'14) 
 
 i.y? 
 
 
 'S 
 
 IS. IIow many cords of wood in a load iiil('(l in two titers, 
 each t ft'ct wide and G^ foot liigli, tlio length of the wood heiiiL,' 
 4 foot? An>^. 1 C. .T c. it. 
 
 1!). AVhat is the valuo of .0G2S of a C(>niTnon yoar? 
 
 Am. 351 d. in h. 7 m. 40.8 sec. 
 
 20. How many hoct< -liters are 80 1)U. 2 i>k. \ 
 
 Am. 28.;)G82 hectoliters. 
 
 21. If a pound Avoirdupois of opium he divided into dose- 
 of 15 Troy j^rains each, and sold at 20 cents a dose, how i;,ii«h 
 will it amount to? Am. .$93.33 '.• 
 
 22. ^Vhat is the cost of .095 ton of nails, at 8 cents a 
 pound? y/w.9. $111.2(1. 
 
 , 23. What decimal of 4 d. 3 h. is 2 w. G L d. ? Am. 4.909 \ . 
 
 X, 24. If the length of a degree of longitude on the ocpuitor be 
 
 taken at G9l miles, instead of G9.1G, tlu^ true length, how 
 
 much too large will it make the o(piatorial circumference of the 
 
 earth? Ami. 122 m. 3 fur. 8 rd. 
 
 25. If a comet move at the rate of 40' 3(>" per miinite, how 
 
 loniT will it be in moving throuL^h GO" 45'? Ail^. 1 h. 30 m. 
 
 20. There is a house 1 12 feet long, and each of the two sides 
 
 of the roof is 25 feet wide ; how many shingles will it take to 
 
 cover it, ii" it rcf|uire G shingles to cover a s<piare foot? 
 
 Am. 33GU0 shingles. 
 ADDITION. 
 
 251. 1. Let it be required to lind the sum of 5 cwt. 2 (jr. 
 10 lb., 2 cwt. 2 qr. 17 lb., and 10 cwt. 3 qr. 21 lb. 
 
 For convenience, we write tlie. numberH 
 so that units of the same name stand in llu- 
 siime coluiun, and bi'Lfiu at the riirht to add. 
 21, 17, 10, are 4S ; 48 lb. are etiual to 1 
 (|r. 23 lb. ; we write the 23 lb., and aild the 
 I qr. in with quarters. 
 A)\^. 19 23 1, 3, 2, 2, are 8 ; 8 <[r. are 2 cwt. (• (jr. ; 
 
 we write the qr., and add the 2 cwt. in witli hundrodwciglit.s. 
 
 Kkview QuESTU)N.s. Fur what is Circuliir iMoiisurc useil ? (217) Wliat Ih 
 a Circle? (218) The circumference of a circle? {2i:i) An Arc? (220) \ 
 Diaijjeter? (221) Kepuat the TaUe vjf Circuhir Mea^«lu•e. (22.;) 
 
 OPKUATION 
 
 . 
 
 cwt. 
 
 11 r. 
 
 lb. 
 
 5 
 
 •) 
 
 10 
 
 2 
 
 O 
 
 17 
 
 10 
 
 3 
 
 21 
 
 w 
 
 1 
 
 if 
 
 < 
 
172 
 
 I'HACTICAL AKITIIMKTIC. 
 
 i 
 
 2, 10, 2, T), are 10 ; 10 cwt. we write. 
 
 TluTtlorc tilt' sum i('«|uireJ is IJj cwt. qr. 2;J. lb. 
 
 ItUrj':. U'^rite the numhcrs so that vnits of the same name shall 
 i^tavd ill the samr column. 
 
 Bcfjin at the rUjht, add each column, and reduce the sum to the 
 next hl(/her denomination ; write the remainder, if any, and add 
 the quotient vith the next colnnm. 
 
 PuoOF. The same as in siiiipk' iiuinbcrs. 
 
 lb. 
 
 7 
 5 
 
 Examples. 
 (2.) (3.) 
 
 07.. pwt. gr. M. fur. rd. yd. ft. in. 
 
 5 9 78 5 30 4 1 
 
 G G 7 G8 3 10 3 5 
 
 G 8 GO 3 14 2 
 
 5 
 
 7 
 1 
 G 
 
 Sum 21 11 18 
 Proof 21 11 18 
 
 207 
 
 or 
 
 207 
 
 4 15 U 2 
 4 15 4 1 8 
 
 In example 3, the \ yard in tlic sum is equal to 1 ft. G in. ; by 
 80 cou.sideriug it, and adding, a form of answer is obtained without 
 a fraction. 
 
 \\} 
 
 A. 
 
 120 
 
 (4.) 
 
 p. sq. yd. sq. ft. 
 
 131 25 G 
 
 
 d. 
 
 121 
 
 li. m. 
 
 16 45 
 
 sec. 
 
 15 
 
 243 
 
 10 3 5 
 
 
 
 20 35 
 
 5 
 
 G 
 
 UG G 2 
 
 Bq. in. 
 
 108 
 
 17 
 
 8 
 
 G 10 
 
 
 
 
 370 
 
 98 4f 4 
 98 5 1 
 
 12 
 
 Or 370 
 
 147 
 
 19 30 
 
 32 
 
 The second form of the answer to e.xample 4 is obtained by 
 regarding tlie f yd. as equal to 6 sq. ft. 108 sq. in., and adding it to 
 the 4 sq. yd. 4 sq. ft. 
 
 1 
 
 Explain tlie operation. Repeat the Rule. What is the Proof? How 
 may the fraction iu the answer to example 3 be cousidorod. ? 
 
DKNOMINATK NUMBERS. 
 
 173 
 
 .It 
 
 C. Add 310 en. yd. 18 cii. ft. 13 12 cii. in., 718 cu. ytl. 9 en. 
 ft. 710 en. in., l.'JO on. yd. G en. ft. 310 oil in., and 7 cu. }yl. 
 20 en. ft. IG 12 cu. in. /^^ ^r^'^ J jf-ir^ , ^ »/^ 4^r/^., 
 
 7. Add 120 gal. 2 qt. 1 pt., 25,^ gal, 13G gal. 1 (jt. 1 pt., and 
 118 gal. qt. 1 pt. Am. 10 hlid. 3 gal. ([t. 1 pt. 
 
 8. Add 11 Im. 2 pk. 5 qt., 23 U. 3 i)k., 8 Im. 7 (jt., 19 bn. 
 
 1 pt., and .09 bn. 4 qt.--- JJl^ /^//, 3/^^'^^/A //^' 
 
 9. Add G'' 27' 48", 3" 32' 12", 8' 20' 30", l' 39' 31", 9' T)!)' 
 48", and T 4G' 41". Ain^. 37^ 40' 30". 
 
 10. What is the snm of 31 m. 5 fur. 30 rd. 2 yd., 42 ni. 
 G fur. 25 rd. 4 yd., 03 m. fur. 10 rd. 3 yd., 7 fur. 17 rd. .> yd., 
 and 10 m. IGrd. 4 yd.? Am. 154 m. 4 fur. 21 rd. I yd. 1 ft. in. 
 
 11. What is the sum of 4 T. 11 ewt. 2 qr. 22 11). 12 oz., 
 10 ewt. 1 qr. 24 lb. 7 oz., 9 cwt. 11 11). 15 oz., 3 T. ewt. 1 <|r. 
 
 2 lb. 7 oz., 2 T. 9 cwt. 1 lb. G oz. ] 
 
 AiK. 11 T. 12 cwt. 2 i\Y. 12 lb. 15 oz. 
 
 12. Wliat is the sum of 00 y. 90 d. 50 m., y. 7G d. 1 h. 
 57 ni., 3 h. b^ m., G y. 1 d. 2 h. 1 Ans. 72 y. 107 d. 8 h. 45 m. 
 
 252. When one or more of the denominate numbers to be 
 added are fractional, before adding, 
 
 IltducG the fractional numbers to integers of lower denominatiom 
 (Art. 248). 
 
 13. Add .282 of a ton, and | of a hundredweight. 
 
 OPEUATION. 
 
 cwt. qr. lb. 
 
 .282 T. = 5 2 14 
 
 u* 
 
 cwt. 
 
 The value of .282 T. is 5 cwt. 2 qr. 
 o K 14 11)., and of ^ cwt. is 3 c^r. 5 lb. ; and 
 luKling, we have G cwt. 1 (jr. 19 lb. 
 
 Am. 1 19 
 14. Add J of a mile to ^^ of a furlong. Am. G fur. 28 rd. 
 
 In the answer to example 4, liow i.s the second form of answer obtained ? 
 How do you proceed when one or more of the numbers to be added are frac- 
 tional ? 
 
 11 
 
174 
 
 rUACTICAL ARITHMETIC. 
 
 liX Add .G of an acre, .85 of an acre, and 17 A. 32 ^'^f\lS'i.^ 
 10. Add ^ of a week, j-j of a day, and \ of an hour. 
 
 Ans. 3d. 1 h. G ni. 55 ^^^ sec. 
 
 M I 
 
 
 APPLICATIONS. 
 
 1. Bought three load.s of liay ; tlie fir.st weighed 1 T. 14 cwt. 
 
 1 qr. 17 lb., the second 1 T. 2 qr. 17 11)., and the third 1 T. 
 
 2 qr. 10 lb. ; what was the weight of the whole'/ 
 
 Ans. 3T. 15 cwt. 2 qr. 19 11). 
 
 2. A minor has an incrot of silver wei<jrhin2: II lb. 4 oz. 
 IG pwt. 11 gr., another weighing 2 11). 5 oz. G pwt, 14 gr., and 
 a tliird weighing G lb. 7 oz. 14 pwt. 17 gr. ; required the en- 
 tire weight? Ana. 20 lb. 5 oz. 17 pwt. 18 gr. 
 
 3. I have four pieces of roj)*; ; the first is 2 yd. 2 ft. 7 in. 
 long, the second -^\- of a inile, the third -^^ of a furlong, and 
 the fourth 3 yd. 1 ft. 11 in. ; what is the length of the whole I 
 
 Ans. 37 rd. 2 yd. 1 ft. G in. 
 
 4. In one range of wood are 50 c. 104 cu. ft. 172 cu. in., in 
 a second, 30 c. 110 cu. ft. 100 cu. in., in a tl iid, 45 c. 3 cord 
 ft., and in a fourth 9 c. 5G cu. ft. G78 cu. in. ; how much wood 
 in the four ranges ? =-/J6 C - (o H^^^^'O^ • 
 
 5. "What day of the year is the IGth of May in common 
 years? Am. 13Gth. 
 
 G. A ship sailed east from Boston, the first day 4° 45', the 
 second 3" 0' 45", tlu^ third 2° 25' 5", the fourth, 3° 10' 15"; 
 how far then was sh) from the place of departure? 
 
 Ans. \y 21' 5. 
 
 7. Through how many degrees of latitutle does a .ship pass 
 in going from Philad(^lphia, latitude 39° 58' 24" north, to Cai)e 
 of Good Hope, latitude 32' 24' 3" south? Ans. 72" 22' 27". 
 
 Kevikw Questions. Wluit i.s Time? (2'_M) Kepoat the Table of Time. 
 (2*24^ Give the namea of the Calendar Months. (225) "What U the length of 
 a inic year? (22l3) 
 
 i 
 
|A»«*«W*'*" 
 
 DENOMINATE NUMBERS. 
 
 17.-) 
 
 8. My farm coii.«ii.st.s of tlireo fields; the f.rst contain.-s 
 •"'.881l*.") acres, the second 19^ acres, and the third U A. 17 V.; 
 how many acres does it contain f Ans, (JH A. OS P. 
 
 in 
 d 
 
 od 
 
 [iU 
 
 tie 
 
 S3 
 
 >e 
 
 hi 
 
 
 OPERATION. 
 
 
 cwt. 
 
 ■I'- 
 
 lb. 
 
 VJ 
 
 .) 
 
 17 
 
 15 
 
 3 
 
 15 
 
 
 3 
 
 •) 
 
 SUBTRACTIOX. 
 
 253. I. Lot it l)t' ivqnired to find the differcnco between 
 l^J ewt. '2 (jr. 17 lb., and 1-5 cwt, 3 qr. 15 lb. 
 
 For convenience, we write the .sul>- 
 trahond under the minuend, so that 
 units of the saiuo name stand in thi* 
 same coluiun, and begin at tiie rii^'ht t^' 
 subtract. 
 A>i.^. 3 3 '2 15 lb. from 17 lb. leaves 2 lb., whicli 
 
 we wiilc. 
 \Vi; cannot take 3 qr. from 2 (|r., but we can tuke I cwt. fiDiu the 
 li> cwt., having 18 cwt., and the 1 cwt. taken is 4 qr., which ad«letl 
 lo th^ 2 (p'. makes G (^r. ; 3 (^r. from G (|r. leaves 3 (jr., wliich wc write. 
 15 cwt. from 18 cwt. leaves 3 cwt., wliicli we write. 
 Therefore, the dilference re(|uired is 3 cwt. 3 ([r. 2 lb. 
 Instead of making the upper number, 1!) cwt., less l>y 1 cwt., tho 
 n.-sult would have been the same if we liad increased the corresj)oud- 
 ing lower uundjer, 15 cwt., by 1 cwt. (Art. 45.) 
 
 ' ItULK. ITrite the less nuinher n/ukr ilie greater, so that units 
 of the same name shall stand in the same enhnnu. 
 
 Beginning at the right, subtract each dcnomimition of the siib- 
 tnihend from the corresponding denomination of the minuend. 
 
 If the number of any denomination in the subtrahend is great* r 
 than that of the same denomination in the minuend, increase the 
 vjijicr number by as many nnits of that denomination as make 
 one of the next higher, before subtracting ; and consider the num- 
 ber of the next higher denomination of the minuend diminished 
 by one. 
 
 Proof. The same as in subtraction of simple numbers. 
 
 c 
 
 E.xpliiiu the openitiou. licpeut iho llule. 
 
17G 
 
 PRACTICAL AllITIIMETIC. 
 
 ■ 
 
 
 
 Examples. 
 
 
 
 
 
 I 
 
 (2 
 
 •) 
 
 
 
 
 (3.) 
 
 
 
 '' lb. 
 
 oz. 
 
 pwt. 
 
 pr. 
 
 m. 
 
 fur. 
 
 nl. 
 
 y<i. 
 
 ft. 
 
 10 
 
 G 
 
 15 
 
 IG 
 
 IG 
 
 ( 
 
 8 
 
 3 
 
 2 
 
 8 
 
 3 
 
 19 
 
 11 
 
 11 
 
 G 
 
 29 
 
 4 
 
 1 
 
 Ans. 2 
 
 2 
 
 IG 
 
 5 
 
 5 
 
 or 5 
 
 
 
 
 18 
 
 18 
 
 4 
 
 1 
 
 Proof. 10 
 
 G 
 
 15 
 
 IG 
 
 
 
 4J 
 
 ! <1 
 
 
 in. 
 
 4/;FromJ80 A. 40 P., si 
 
 5. Take 13 gal.*3 qt. 1 pt. from 1 lilid. 2 gal. 1 qt. pt. 3 gi. 
 
 subtract 39G A. 13.5 P. 
 
 Am. 383 A. Gf) P. 
 
 G. Take 5 bii. 3 pk. 1 pt. from GO bu. 1 pk. 7 qt. 
 
 Ans. 54 bu. 2 pk. G qt. 1 pt. 
 
 7. Take 310 d. 5 li. 45 m. from one common year-^^"^'-^/^ /^^/^k 
 
 8. Subtract 10 cu. yd. 25 cu. ft. 1014 cu. in. from 23 cu. yd. 
 13 cu. ft. 357 cu. in. Ans. 12 cu. yd. 14 cu. ft. 1071 cu. in. 
 
 254. When one or both of the given denominate numbers 
 are fractional, before subtracting, 
 
 Reduce the fractional number, or numhcrs, to intcr/crs of a lower 
 denomination. 
 
 0. From 1 1 lb. Troy, subtract .17 lb. Troy. 
 
 ^ V 11, 
 . 1 7 lb. 
 
 Ans 
 
 OPKRATION. 
 (iz. pwt 
 
 7 G 
 - 2 
 
 IG 
 
 19.2 
 
 5 
 
 5 
 
 20.8 
 
 Tlie value of |J Ih. Troy i.s 7 oz. 6 
 pwt. IG gr., and of .17 lb. Troy i.s 
 2 oz. pwt. 19.2 i^T. ; and, suhtract- 
 ing, wo have 5 oz. 5 pwt. 20.8 gr. 
 
 Ans. IG lb. 1 oz. 12^ dr. 
 
 / 
 
 10. From r; of a quarter, take [j of a pound. 
 
 Ans. IG . 
 
 1 1. From fl of an acre take y^ of an acre. . ^ -^ -^ ., '' 
 
 12. From |^ of a square yard take ^} of a yard square. _ ^^ ^ ./^/' , 
 
 13. Subtract j\ ^^^ ^ degree from .37 of a degree. 
 
 Alls. 4G; seconds. 
 
 ^ i k 
 
 How is obtiiint'd the second form of answer to example .S ? Wlion one or 
 lioth of the givcu ilouomiiuite nuuibcra are fructioual, how ilo yuu iJiocceU ? 
 
denominate; numbers. 
 
 i: 
 
 in. 
 
 G 
 P. 
 
 
 pt. 
 
 . in. 
 
 bcTS 
 
 loZ. K 
 ty is 
 raot- 
 
 t clr. / 
 
 
 ^■-ipy't^ ' 
 
 >iuls. 
 
 ine or 
 bed? 
 
 255. The lulo applies M finding tlio. clilkivncc between two 
 
 dates. 
 
 14. What time elapsed between j\Iay IG, 1819, and ^larch 
 
 9, 1SG5? 
 
 A\' 
 
 tl 
 
 ilacc iiic nnin'iei'ii o 
 
 .t 11 
 
 ic furiuT 
 
 operation. 
 
 .hit 
 
 uiue mil 
 
 ler those of the hitii', writiiii' tin 
 
 ino. 
 
 1SG5 
 1819 
 
 d. 
 
 9 
 
 UUlUl''^'' O 
 
 r tl 
 
 u; year. 
 
 tl 
 
 u'li 111 on 
 
 111- th 
 
 iiumlxT of iiiouths ami davs w liich have 
 
 4 
 
 IG rl, 
 
 ipsed of the year, aiul, in siibtracliii;. 
 
 A 
 
 ns. 
 
 21 
 
 coviit as man// duijH to a month as are in 
 
 the month next earlier than that named in 
 the later date. 
 Here that month has 28 days. 
 
 It is quite connnon to count every month as containing,' ;i(i days, 
 which, iji this caHu, would have given the ditleii-nce 2 days more, and 
 to that extent wanting in exactness. 
 
 " la. What time elapsed lietween Feb. 11, 1807, and Sept. '">, 
 18GG? A as. 59 y. G m. 'I'l d. 
 
 ^J6. What time elaps(;d between July 4, 1 o'clock, P.M., 
 177G, and January 8, G o'clock, A.M., ISGf)'/ 
 
 Ans. 88 y. G m. 3d. 17 h. 
 Since the hours of the day begin to count from miilnight, (5 o'clock, 
 A.M., is G hours from that time, and 1 o'clock, I'.M., i;} hours. 
 
 '^ 17. What time elapsed between Oct. 14, 1492, and April .'i, 
 
 iSGf) I . ^^^ :/ , •. ^2) ,). ' j' ^\ ■.^, , 
 
 - 18. W^hat time elapsed betwed^n the termination of the 
 American Kevolution, January 20, 1783, and the evacuation 
 of Fort Sumter, April 14, l^Gl. Ans. 78 y. 2 m. 25 d. 
 
 ATMVLICATIOXS. 
 
 1. A forwarding liou.-^e having received 20 T. 2 (p*. 14 lb. 
 of freight has shipped 10 T. 13 cwt. 2 (jr. 14 li». iiy .steamer, 
 and the balance liy sail vessel ; what quantity was sent l»y the 
 latter conveyance? Aa^. 9 T. 7 cwt. 
 
 2. Bought 7 cords of wood, and 2 e. 78 cu. ft. having been 
 
 stolen, how nnich remains \=r-^ La. ~ , ^jo J''- . ■ • J ' 
 
 » 
 
 In subtriicliiig dates, how uiatiy days arc counted to a month ? Fioiii 
 wliat time do the hours of the day bcyin to count ? 
 
 M 
 
 fl 
 
 t* 
 
 
 ! f 
 
I 
 
 t ■ 
 
 I 
 
 i 
 
 n 
 
 178 
 
 PRACTICAL AlilTHMETIC. 
 
 3. There are two cities, 98 m. 5 fur. 3 rd. apart ; how far is 
 a man travelling between these cities from one of them, if he 
 is 12 miles ^^^ furlongs from the other ] 
 
 Alls. 85 m. G fur. 39 rd. 
 
 4. The longitude of Boston is 71° 3' 30" west of Greenwich, 
 and that of San Francisco 122^ 20' 18"; what is the differ- 
 ence? Ans. bV 23' 18". 
 
 5. The latitude of New Orleans is 29° 57' 30" north, and 
 that of Chicago 42" ; what is the difference ? ^J^ "X^'J^ " 
 
 G. How many days arc there from the 5th of January till 
 the 25th of the next June of a common year? rr. / '/ / //^ _y 
 
 Solution. 31 daiis iuJanucuy, less 5 dat/a ~ 26 days of Janyar)/ ; 
 20 days of January + 28 days of February + 31 days of March + 
 30 days of April + 31 days of May + 25 days of June = 171 daijs, 
 Ans. 
 
 7. flow many days from Feb. 22, to the next July Ith, of 
 a leap year? Ans. 133 days. 
 
 8. If a man was born February 29, 1820, how many anni- 
 versaries of his birthday will he have had on February 29, 
 1880? Ans. 15. 
 
 9. If September 21st is Monday, on what day will the 25th 
 of the next December be? Jf ticiiii.- ' 
 
 Solution. The number of days between Sept. 21 and Dee. 25 is 
 9^ days, or 13 weeks 4 days; countinj four days after Mondaij, the 
 2hth of Decembe)' must be Friday. 
 
 10. A farmer has in one crib 325-}^ bushels of corn, in a 
 second 43^- bushels, and in a third 587 bu. 3 pk. 7 qt. Of this 
 he will have to pay for rent 307 bu. 2 pk. 4 qt. ; to use for 
 fatting cattle, 50 bu. 2 pk. 3 qt. ; for fatting hogs, 35 bu. 3 pk. 
 2 qt., and will require for his own use 298 bushels ; how much 
 will he have left to dispose of? Ans. 199 bu. 2 pk. 
 
 g(^ 
 
 Rkview Qukstions. ^." '.vli-it purposes are decimal weiglits ami measures 
 nerally adopted T (yv.?) U[u;u what scale is the Metric System formed? 
 
 W 
 
 (2:J9) 
 
1 
 
 i\V«»,»^- 
 
 DENOMINATE NUMBERS. 
 
 179 
 
 ■^ 
 
 111 a 
 
 f this 
 
 so for 
 
 3 pk. 
 
 much 
 
 2 pk. 
 
 easurcH 
 ormed? 
 
 I 
 
 bu. 
 
 4 
 
 OrKRATION. 
 l-k. qt. 
 
 3 4 
 
 pt, 
 1 
 
 
 
 7 
 
 Ans. 34 
 
 
 
 MULTIPLICATION. 
 
 256. 1. Let It bo required to find the proihich of 4 bu. 3 pk, 
 4 qt. 1 pt. iiiultii)licd by 7. 
 
 For convenience, wehoi^'inwitU the 
 lowest denoniijKition to multiply. 
 
 7 times 1 pt. are 7 pt, which equal 
 2 qt. 1 pt. ; we write the i pt., and 
 reserve the 3 ([t. to add to the pro- 
 duct of the quarts. 
 7 times 4 qt. are 28 ([t., wliich, with the 3 qt. added, are 31 ([t., or 
 3 i>k. 7 qt. ; we write the 7 qt. and reserve the 3 pk. to add to the 
 product of the pecks. 
 
 7 times 1) pk. are 21 pk., whieli, with the 3 pk. added, are 24 pk., 
 or 6 hu. pk. ; we write the pk., and reserve the G bu. to add t(» 
 the product of bushels. 
 
 7 times 4 bu. are 28 1)U., which, witli the G bu. added, are 34 bu., 
 which we write. 
 
 Tlierefore the product reipiired is 134 bu. pk. 7 <[l I pt. 
 
 Rule. Bcfjimiing at the I'njht, multiplij the number of each deno- 
 mination in its order, and reduce each product to the next higher 
 denomination; write the remainder, if any, and add the quotient to 
 the next product. 
 
 PuoOF. The same as iu nuilliplicatiou uf tiimplc numbers. 
 
 Examples. 
 
 cwt. qr. 
 
 3 
 
 (2.) 
 
 11). oz. 
 
 G 13 
 
 dr. 
 15 
 
 8 
 
 
 
 (3.) 
 
 
 
 m. 
 
 fur. 
 
 nl. 
 
 yi. 
 
 ft. 
 
 1 
 
 2 
 
 15 
 
 .J 
 
 1 
 
 9 
 
 6 2 4 15 
 
 4. ]\Iultiply 3 hhd. 57 gal. 3 qt. 1 pt. by 11. 
 
 Ans. 43 hlid. G gal. 2 (^t. 1 pt. 
 
 5. Multiply 3G d. 21 h. 48 m. 56 sec. by (j.^-0^^^l^/cl-^3^%.jr^ 
 G. Multiply HI C. 7 c. ft. 7 cu. ft. by 12. 
 
 Ans. 1343 C. 1 c. ft. 4 cu. ft. 
 7. Multiply 4 bu. 1 pk. 5 qt. 1 pt. by 7. 
 
 Ans. 30 bu. 3 pk. G qt. 1 pt. 
 
 Explain the operation. Repeat the Rule. Wliat is the Proof ? 
 
180 
 
 PRACTICAL ARITHMETIC. 
 
 ( 
 
 8. INFultiply 17 A 71 T. by 72. Ans. 1255 A. 152 P. 
 
 9. Multiply 2 lb. oz. 9 pwt. 22 gr. by 50. 
 
 Ans. 131 lb. 2 oz. 15 pwt. 20 gr. 
 
 APPLICATIONS. 
 
 1. What is the weight of 5 hogsheads of sugar, if each 
 weighs 12 cwt. 1 qr. 23 lb. ? Ans. 3 T. 2 cwt. 1 qr. 15 lb. 
 
 2. What is the weight of 12 spoons, if each weighs 1 oz. 12 
 
 pwt. 20 gr. ? =//^'y/^/^:-/ , 
 
 3. If a car will take on 4 C. 5G cu. ft. of wood, how much 
 will 8 cars take on 1 Ans. 35 C. G-4 cu. ft. 
 
 4. If the daily motion of the moon is 13° 10' 35'^, how much 
 is it in 15 days ? Aiis. 107° 38' 45'^ 
 
 5. If 3 }d. 1^ qr. of cloth are required for one garment, how 
 mucli is required for 14? z^J/.(p ],//'■"'/ ''/'j'/^ 
 
 6. How much molasses in 25jCasks,' if each contains Gl gal. 
 1 qt. 1 pt. 1 Ans. 1534 gal. 1 qt. 1 pt. 
 
 7. If a man travel at the rate of 22 m. 7 fur. 32 rd. 4 yd. 
 per day, how far can he travel in 5G days ? 
 
 Ans. 128G m. 5 fur. 32 rd. 4 yd. 
 
 8. How much land in 9 farms, each containing 74 A. 87 P. 
 4 sq. yd. 1 Ans. G70 A. 144 P. 5 sq. yd. G sq. ft. 108 sq. in. 
 
 
 f 1 
 
 DIVISION. 
 
 257. 1. Let it be required to find the quotient of 34 bu. 
 pk. 7 qt. 1 pt. divided by 7. 
 
 For convenieiico, we begin with the 
 highest denomiiiation to divide. 
 
 Cue seventh of 34 bu. is 4 bu., witli a 
 
 remainder of 6 bu., equal to 24 pecks ; 
 
 we write the 4 bu., and add the 24 
 
 pecks to the pecks in the dividend. 
 
 24 ])k. and pk. are 24 pk. ; and one seventh of 24 [)k. is 3 pk., 
 
 with a remainder of 3 pk., eqital to 24 quarts ; we write the 3 pk., 
 
 and add the 24 quarts to the 7 quarts in the dividend. 
 
 OPERATION. 
 
 bu. pk. qt. 
 
 7 ) 34 7 
 
 pt. 
 1 
 
 1 
 
 Explain the operation. 
 
I 
 
 
 DENOMIN.^TE NUMBERS. 
 
 181 
 
 24 q(. and 7 (|t, are 31 (jt. ; and one i'oventh of 31 qt. is 4 qt. Avitli 
 a renmindcr <>t 3 qt., equal to pints ; we write the 4 qt., and add 
 the 6 pints to the 1 pint of tlie divitUi d. 
 
 6 pt. and 1 pt. are 7 pt., and one seventh of 7 pt. is 1 pt., which 
 we ■write. 
 
 ItULE, Bcginimg at the left, divide the numher of each deno- 
 mination of the divicund in its order; ivrite the quotient^ and 
 reduce the remainder, if any, to the iicxt lower denomination, 
 adding the same to that denomination in the dividend, before divid- 
 ing it. 
 
 Proof. The same as in division of simple numbers. 
 
 HI. 
 
 bu. 
 
 ith a 
 
 cks ; 
 
 le 24 
 
 lud. 
 pk., 
 pk., 
 
 Examples. 
 
 hhd. 
 
 11 )39 
 
 (2.) 
 
 g'll. qt. 
 
 G 2 
 
 pt. 
 1 
 
 m. 
 
 (3.) 
 
 fur, 
 
 31 ) 28G ( 9 
 279 
 
 venient 
 
 oper 
 
 Repciit the Rule? What is the Proof? "Wheu the divisor is large, how 
 is it often couveuient to proceed ? 
 
1/ 
 
 182 
 
 PRACTICAL AHTTIIMRTIC. 
 
 258. When both dividend and divisor arc denominate num- 
 bers, 
 
 llcduce them to cqidralent nnmhers of the same denomination, 
 and proceed as in division of simjde numbers. 
 
 10. How many times 5 pwt. 9 gr. in 9 lb. oz, 3 pwt. 12 
 gr. 1 Ans. 43G. 
 
 11. IIow many times 17 m, 5 fur. 27 rd. in 513 m. 4 fur. 
 23 rd. ? Ans. 20. 
 
 V 
 
 
 ,1 I 
 
 APPLICATIONS. 
 
 ' 1. If 169 gal. 3 qt. 1 pt. be contained in equal quantities in 
 9 casks, how mucli is tliere in each cask ? 
 
 yiw.s\ 18 gal. 3 qt. 1 pt. 
 
 * 2. If 5 hogsheads of sugar weigh 3 T, 2 c)vfc. 1 qr. 15 lb., 
 what is the average weight of each ? rr. V "^. / £ . /'> "^V^^' 
 
 ... 3. If 2 dozen spoons weigh 7 11). G oz. 13 pwt., what is the 
 weight of each spoon ? Ans. 3 oz. 15 pwt. 13 gr. 
 
 ^ 4. When a ship passes over 99° 22' 30'' in 30 days, what is 
 the average progress per day? Ans. 3^ 18' 45". 
 
 • 5. If it take 4 bu. 3 pk. of Avheat to make a barrel of flour, 
 how ninny barrels may be made from 450 bushels? Ans. 96. 
 
 ^ 6. When a railroad train moves 118 m. 4 fur. in 8 hours, 
 
 what is the rate per hour ? Ans. 18 m. 4 fur. 20 rd. 
 
 / 1. If a man travel 12 m. 3 fur. 19 rd. per hour, in what 
 
 time can he travel 174 m. 26 rd. ? Ans. 14 hours. 
 
 8. When the weight of 27 loads of bay is 30 T. 8 cwt. 2 
 
 qr. 23 lb., what is the average weight of a load ? 
 
 yins. 1 T. 2 cwt. 2 qr. 4 J lb. 
 
 LONGITUDE AND TIME. 
 
 259. The earth, by turning upon its axis once in 24 hours, 
 causes }^ of 360°, or 1 5°, of longitude to pass under the sun 
 
 When both the dividend an(i divisor are denominate, how do you proceed f 
 
V 
 
 DENOMINATE NITMBERS. 
 
 183 
 
 in 1 hour, and -^^ of l.V, or IT' lo pass under it in 1 minute 
 of time, and ^'^ of 15', or l.V, lo pass under it in 1 second of 
 time. Hence, tlic following 
 
 Table. 
 
 15" of longitude equal a difference of 1 hour in time. 
 
 15' „ „ ,, 1 miinite ir. time. 
 
 15* „ ,, „ 1 second in time. 
 
 Whence is derived the 
 
 Rule. Divide the difference of longitude of two j^/^/rr'.s' hi/ 1.5, 
 and the quoficnt maired hnxirs^ minutes, and seconds, instead of 
 'lerjrees, minntes, and second-^, will express their difference of time. 
 
 MnUiply the difference of time of two places hy 15, and the pro- 
 duct marked decrees, minutes, and seconds, instead <f hours, minutes, 
 and seconds, will express their difference of lon(/itude. 
 
 Since the earth turns from west to east, time is later to places cantf 
 and earlier to places ^vest of any given meridian. 
 
 lb. 
 
 Examples. 
 
 1. When it is 9 o'clock, A.M., at AVashington, 77° 2' 48" 
 west of Greenwich, what time is it at the latter place 1 
 
 Ans. 2 h. 8 m. Ill sec. v.m. 
 
 2. Tf the time at 8t Louis, longitude m' 15' 10" west, is 40 
 m. 2:- sec. earlier than that of a place east, what is the lomri- 
 tudc of th(; latter })lace I Aus. 80' 14' o4" west. 
 
 3. When it was 10 o'clock, P.M., Dec. Ml, J8(;5, in San 
 Francisco, longitude 48° 20' 40" west of New York, what time 
 was it in New York ? 
 
 Aus. 1 h. 13 m. 47 sec. A.iM., Jajf. 1, 186G. 
 
 4. A captain, at sea, at noon, oliserved that his fli tonometer, 
 set to the time of Greenwich, longitude 0, pointed to 2 h. 45 m. 
 30 sec. ; in what longitude was he i ylns. 41° 22' 30'^ west. 
 
 What (Iocs the earth, hy tnrmiig upon its axis, cause? Repeat the Table. 
 The Rule. At what places is time earlier with reference to any yiven iiieiidiau ? 
 
 11 
 
 J 
 
 11 
 
K 
 
 
 184 
 
 V 
 
 PKACTICA L AHITH M KTIC. 
 
 rUACTICE. 
 
 260. An Aliquot Part of ;i mniibor is an cxart divisor (Ait. 
 97) of it. Tims, 
 
 2, 2h, 3/,, ami 5 arc aliquot parts of 10. 
 
 261. Practice is an abridged method of computation, by 
 means of alitjuot parts. 
 
 Table of Aliquot Parts. 
 
 Of u 8. 
 
 Of a 
 
 Ton. 
 
 Of a Cwt. 
 
 Of an Afve. 
 
 Of a^loiitli. 
 
 Cts. $ 
 
 cwt. 
 
 ton. 
 
 lb. cwt. 
 
 nl. A. 
 
 d. m. 
 
 50 = J 
 
 10 
 
 
 50 = ..V 
 
 80 - }, 
 
 15 = .V 1 
 
 
 5 
 
 _ 1 
 
 2.-) = -} 
 
 40 ■= ] 
 
 10 = . ' 
 
 
 4 
 
 _ 1 
 
 20 = ^ 
 
 .32 ^ -i 
 
 n = 1 1 
 
 IGH = ^ 
 
 01 
 
 _- 1 
 
 m" 
 
 1 01 _ 1 
 
 20 ^ l- 
 
 G --, ! 
 
 ■ 1 
 
 
 2 
 
 _ 1 
 
 10 =-;o 
 
 10 = ,v 
 
 r, .1 
 
 10 -iV 
 
 1 
 
 _ 1 
 
 - 11C) 
 
 5 ^j. 
 
 8-.V 
 
 •^ ""Iff: 
 
 Exercises. 
 
 1. Lot it be required to find the cost of 3G02 bushels of 
 wheat, at $1.87.1 a bushel. 
 
 OPERATION. 
 
 3G32 bu. at 81 per bu. will cost $3G32 
 
 „ „ .50 „ „ \of.<?3n32 - 181G 
 
 „ .25 „ „ .lof.^l81G = 908 
 
 .12.V „ „ ;! of 8908 = 454 
 
 >» 
 
 }j 
 
 
 
 „ $1.87} „ „ 86810, Ans. 
 
 2. Required the cost of 3 T. 5 cwt. 2 qr. of hay, at $24 a ton. 
 
 Since 1 T. 
 3 T. 
 
 5 cwt. 
 
 OPERATION. 
 
 cost 824.00 
 
 will cost 824 X 3 = 872.00 
 
 i of 824 = G.OO 
 
 qr. 
 
 3 T. 5 cwt. 2 qr. 
 
 J) 
 
 ^Vof 80 = _.60 
 
 $78.00, A71S. 
 
 . ll ! 
 
 What is au Alitjuot Part of a number ? What is Practice ? 
 
 I 
 
 u 
 
of 
 
 t 
 
 DKNOMINATK NUMBERS. 
 
 183 
 
 3. "\Vli;it is tlie amount of a sa i y for 1 y. 7 mo. 2'2\ d., at 
 $1080 per ye.-'? 
 
 ¥ov 1 y. 
 
 »♦ 
 
 G mo. 
 
 »» 
 
 1 mo. 
 
 »» 
 
 
 » 
 
 
 OnillATION'. 
 
 it is $10^0 
 
 „ .i of 61080 = 540 
 
 M ;of 8--ilO ^ 
 
 15 .1. „ • (,f .91)0 = 
 7.'. d. ,, }, of .sl'"» - 
 
 90 
 45 
 L'J.50 
 
 ,, 1 y. 7 iiio. liL'i fl. „ 
 
 81777.50 
 
 i1 
 
 - 4. Wliat h tho value of GO A. 120 P., at 880 pe- acre? 
 
 Jns. 81800. 
 
 5. What is the cost of 13 gal. 3 (jt. 1 pt. of molasses, at 
 .8.G0 per gallon ] =■ S f' 'J i^ ' 
 
 G. Ilcquireil the rost of grading 10 ni. G fur. 20 id. of rail- 
 road, at 8G11)0 per mile ? Ans. 87017:5.12^. 
 
 ,. 7. KtMpiired the cost of 2117 yards of cloth, at 37cj cents 
 per yard .? Ans. $7'r3.67L . .^ 
 
 . 8. What cost 120 yards of broa<U'loth, at 83. GG;] per yar<l h^^j^,^ 
 
 9. How much nuist be paid for 24 hu. 2 pk. 1 ([t. of corn, at 
 8.88 per busheU y/;w. 821.G7. 
 
 ,. 10. If a man can walk 18 m. 5 fur. IG rd. in one day, liow 
 far can he walk in 10 d. 8 h. ? Ans. l'J2 m. 7 fur. 32 rd. 
 
 11. At 8240 a year, what is the rent of a house for mo. 
 25 d. ? -- c , ,. ^, ^ 
 
 / 12, What is the cost of excavating 108 cu. yd. 18 cu. ft. of 
 earth, at 42 cents per cu. yd. ? Ans. 845. G 4. 
 
 Rkvikw Qt:r:sTro\a. What is Reduction of Denominate Numbers? (24 1) 
 "What is the Ilulo for reducing from a higlier to a li)wer donomination ? (245) 
 For reducing from a lower to a higlier denomination? (217) For reducing a 
 fraction of a given denomination to integers of loweg uiuominations? (L'48) 
 For reducing integers of lower denoniinatioj^oy) The dittereu' a higher 
 denomination? (249) .ne? (259) n 
 
 I 
 
 \\ 
 
 % 
 
 \ 
 

 ^ 
 
 
 
 (MAGE EVALUATION 
 TEST TARGET (MT-3) 
 
 /, 
 
 ^.^s^'/ J<'' €?_ 
 
 
 7 
 
 ^ .o i,y 
 
 ^^. 
 
 
 f/i 
 
 f/. 
 
 1.0 
 
 I.I 
 
 1.25 
 
 "i^lllM IIIII25 
 
 I" m 
 t 1^ 
 
 1.4 
 
 1.6 
 
 ^^ 
 
 ,P>-. 
 
 Ajt- 
 
 'c^l 
 
 
 
 o 
 
 / 
 
 
 Photographic 
 
 Sciences 
 Corporation 
 
 23 WEST MAIN STREET 
 
 WEBSTER, N.Y. 14580 
 
 (716) 872-4503 
 
 t\ "^ 
 
 r^l, 
 
 ,\ 
 
 ^^ 
 
 ft- 
 
 :\^^ 
 
 \ 
 
 
 \ 
 
 #^ 
 
 '[>".^>^ 
 
 C^.. <^ 
 

 x9 
 
 ^ 
 
186 
 
 PRACTICAL ARITHMETIC. 
 
 i# 
 
 h «: 
 
 REVIEW EXERCISES. 
 
 ''l. What cost 1 T. 5 cwt. 5G lb. of sugar, at 11 cents a 
 
 pound? y^7i.s. $281.10. 
 
 / 2. At 11 cents a pound, how much sugar can he bought for 
 
 / 3. A farmer, having GlO acres in his farm, values one half 
 of it at $20 per acre, and the other half at 20 cents a square 
 rod ; how much is the one half worth more than the other? 
 
 Ans. $3840. 
 / 4. A druggist retailed {| of a pound Troy of a drug, at 2 
 cents a grain ; how much did it amount to 1 ■= ^ I 0.' h-O 
 / 5. How much money can be made on 500 tons of coal 
 bought at $5 per ton of 2240 lb., and sold at the same number 
 of dollars per ton of 2000 lb. ? :=:. "^^ ^? n ,-> , n 
 
 Solution. 2240 Ih. - 2000 lb. = 240 Ih. ; if 240 lb. can he made 
 on 1 ton^ 500 times 240 ih., or 60 tons, can be made on 500 tons ; and 
 00 tons, at $5 per ton, sell for $300, tlie money that can he madt; 
 Therefore, ,&c. / . : ^ ^ , .. V A./r/* ^ - / <L -^7 7 • . 
 
 / 6. How much can be made by buying 10 lb. of rhubarb, at 
 $6.50 per pound avoirdupois, and selling it at the same price 
 per lb. Troy 1 Ans. $1 3.99 + . 
 
 y 7. A labourer being 8 h. 4 m. doing a job, another man 
 agreed to do a like piece of work in ^ of that time ; how long 
 was that? Ans. 6 h. 43 m. 20 sec. 
 
 / 8. AVhen $1.80 is paid for r}^ of 2 tons of hay, how much 
 is it per hundredweight ? Ans. $1.12?,. 
 
 / 9. At $1.50 per yard, in length, what will be the cost of a 
 X carpet, | of a yard wide, that will cover the floor of a room ^ 
 29.5 feet long by 11.25 feet wide ? Ans. $88.50. 
 
 / 10. What must be the length of a rectangular let, that 
 measures 72 feet on the front, to contain 9432 square feet? 
 
 Ans. 131 feet. 
 
 Review QuESXt. 2 .^What is the Eule for Addition of Compound 
 
 Denomiiv •— :!i^ ^~—~^-- Subtraction? (253)? Multiplication? (256) 
 
 Divisir- "What is an Aliquot Part 
 
DENOMINATE NUMBERS. 
 
 187 
 
 ^jj:) 
 
 500. 
 
 uch 
 
 '^^ 
 
 )f a 
 jom 
 [50. 
 
 hat 
 
 lund 
 1256) 
 
 S 
 
 i 
 
 Solution. Since the 9432 rtuiM be the product of the mimher de- 
 noting the front and another denoting tJie length, the latter must he 
 1)432 -^ 72, or 131 feet. 
 
 / 11. How wide must be a rectangular lot, which is 42.4 rods 
 long, to contain exactly an acre ? -— :^ h ^7 i '^} ^ T'^-f- ' ^"^f^*'-^ 
 •- 12. How much will a man's wages amount to, at $25 per 
 month, from April IGth to the last day of the next March ? 
 
 Ans. $287.50. 
 
 1 3. How much money may be made on 2 bushels of chest- ^ 
 nuts, bought at 84.80 a bushel, and retailed at 15 cents a 
 quart, liquid measure? Ans. $1.5G + . 
 
 14. When tea is 75 cents a pound, how much is it per kilo- 
 
 gram?-: ///./; ^:'.'/ ^ 
 
 Solution. Since 1 kilogram is equal to 2,204G Ih., when tea is 75 
 cents a pound., it is 2.2046 times 75 cents, or gl.65 + , |;e7- kilogram. 
 Therefore, dx. 
 
 15. When a field produces 40 hectoliters per hectare, how 
 many bushels is that per acre ? Ans. 45.9 + bushels. 
 
 ^^16. What is the cost of a stone wall 132 feet long, 4 feet 
 high, and 1.5 feet thick, at $2.25 per perch of masonry? 
 
 A71S. $72. 
 
 ^^ 17. A waggon is 8 feet long and 3.75 feet wide 3 how high V 
 must wood be piled upon it to mtike a cord 1 .'—-- J/-- ^ / 
 
 Solution. Since 128, the number of cubic feet in a cord, must be 
 the prod^ict of the numbers denoting the length and undth, and the re- 
 quired height, the latter must be 128 -=- (8 x 3.75), or 4.2G +,feet. 
 
 18. If a pile of wood is 4 feet wide and 6 feet high, how 
 long is it, if it contains 4 C. 6 c. ft. ? »= ;2_ h-^ "^^ • - • - ^. , 
 
 ^ 19. Wliat time is it in 30° west longitude, when it is 1 
 o'clock, A.M., July 4th, in 7° 30' east longitude 1 
 
 \ Ans. lOh o'clock, p.m., July 3d, 
 
 Review Questions. What is the Rule for finding the difference of time 
 corresponding to tiie difference of longitude? (2;")0) The difference of longi- 
 tude corresponding to the difference of time ? (259) 
 
 1 
 
 ] 
 
 -tt 
 
 vf 
 
 t n 
 
 ■ 
 
 ! 
 
 I ■ 
 
 ' 
 
188 
 
 iMi 
 
 PI 
 
 '1 i 
 
 PRACTICAL ARITHMETIC. 
 
 PERCENTAGE. 
 
 262. Per Cent., from per^ by, or on, and centum,, Imndred, 
 moans hj/ or on a hundred. Thus, 
 
 2 per cent, of a quantity, means 2 on a hundred, or 2 hun- 
 dredths of the quantity. 
 
 263. Percentage is the process of computing in hundredths ; 
 and 
 
 The Percentage of a number is so many hundredtJis of it 
 as is indicated by the per cent. 
 
 The Base of percentage is the number upon which it is com- 
 puted. 
 
 The Rate per cent, is the number of hundredths denoted 
 by the per cent. 
 
 264. Since rate per cent, is a number of hundredths, it may 
 be expressed in tiie form of a decimal or of a common fraction. 
 Thus, 
 
 1 per cent, is Avritten .01, or yi^. 
 5 per cent. 
 10 per cent. 
 25 per cent. 
 100 per cent. 
 125 per cent. 
 
 \ per cent, is .00^, or .005 ; |- per cent, is .00], or .0025 ; 
 I- per cent, is .OOi or .00125, Sec. 
 
 The sign % is used, in business, to stand for the word per 
 cent. 
 
 Exercises. 
 
 Express decimally, 
 1. 8 °/. Ans. .08. 
 
 >> 
 
 >> 
 
 5» 
 
 55 
 
 55 
 
 .05, 
 
 55 
 
 5 
 
 100* 
 
 .10, 
 
 55 
 
 ] 
 100* 
 
 .25, 
 
 55 
 
 2 5 
 
 loo"- 
 
 1.00, 
 
 55 
 
 100 
 lOO* 
 
 1.2.5, 
 
 55 
 
 12.'; 
 
 10 0" 
 
 2. 15 
 
 K °/ 
 
 /o' 
 
 Ans. .15. 
 
 3. 121%. Ans. .in. 
 
 4. 1| %. 
 
 Ans. .01^. 
 
 5. i %.- - ' 006\ 9. 45 %.^ • ^6 
 
 6. \:i-*C^0!l6 10. 90%.-=. .y^ 
 
 7. ^^ %.=^^C^3\ 11. 150 %. = I'OO 
 
 8. 7-i\j%. '%0l3\ 12. 271 
 
 /o- 
 
 What does Per Cent, mean? What is Percentase? The Percentage of 
 a number or quantity? The Base? How may per cent, be expressed? 
 What sign for per cent, is used ? 
 
T 
 
 rEUCENTACE. 
 
 189 
 
 i. 
 
 ^6 
 
 .16 
 
 J. 76 
 
 |e of 
 
 lied? 
 
 Express as common fractions in their smallest terms, 
 
 12.^ 
 
 13. 12i%. Solution. 12^ % = ^j^ = ^^ = ^ ^ns. 
 
 IG. 37^ % 
 
 17. IGi %... ^. 
 
 U. 15 %. Ans. ^. 
 
 15. 110 %. ^^«5. l^V 
 
 Express as a per cent., 
 
 20. •§. Solution. -^ = 5.00 ^ 8 = .62^ = G2;^ %, Ans. 
 
 19. m %. ^^ 
 
 21. H. ^7^.^ 125 %. 23. ^. ^ /^ f- I 25. |. -- ' ^^'^jy. 
 
 22. -,-i-V Aus. 7,^- %. 24. 3i.^-- ^^f^l 2G. 5^^ t:-^ '^ 6/ 
 
 CASE I. 
 
 265. To find any per cent, of a number. 
 1. What is 5 per cent, of 3G5 bushels ] 
 
 OPERATION. 
 
 3G5 Since 5 % is .05, 5 % of 365 hu. is .05 of 3G5 bu., 
 
 .05 or 18,25 bu. Tliercfore, &c. 
 
 Ans. 18.25 ha. 
 
 KuLE. MultipJy the given number hy the rate ^er cent., ex- 
 pressed decimally. 
 
 Examples. 
 
 What is 
 
 2. 5 % of 5G3 yards? Ans. 28.15 yd. 6. 13 % of $150.50?= rrH^.\ 
 
 3. 91 % of GOO men ? Ans. 57 men. I 7. 19 % of 81000 % -^ f Cj , • . . * 
 
 4. 15 % of 120 sheep ? Ans. 18 sheep. | 8. -i % of $13G7.50 % j q.^^^^ J 
 
 5. 35 % of 834.G0? Ans. $12.11. i 9. 103 % of 7GOpounds?-.^d^<4 
 
 10. Find 4 per cent, of 3 pk. 2 qt. -,,. ' , , ' '- 
 Solution. 3 pk. 2 qt. = 26 qt. ; 26 ([t. x .04 = 1.04 qt., Ans. 
 
 11. Fhid 45 % of 12 cwt. 2 qr. 20 lb. 
 
 An^. 5 cwt. 2 qr. 21^ lb. 
 
 266. When the number of the per cent, is an aliquot part of 
 100, to find the percentage, we may 
 
 Take the same ixirt of the base that the rate per cent, is of 100. 
 
 ' i 
 
 if 
 
 i 
 
 Explain the Operation. 'What is the llule? "When tlic number per cent. 
 is au aliquot inirfc of 100, liow may the percentage be found ? 
 
190 PRACTICAL ARITHMETIC. . 
 
 / 12. What is 33,^ % of 2880 hogsheads ? -r ^ (/, 0. 'A ^^' ^- 
 
 QOl 
 
 Solution. 33^ % of 2880 hhd. = ^^J, or ^, of 2880 - 9G0 hhd., Ans. 
 
 ^ 13. What is 25 % of $968.50 ? Jw^\ $242.14. 
 
 14. What is 50 % of 4 h. 25 m. 12 sec. ? 
 
 ^i/i.9. 2 h. 12 m. 36 sec. 
 
 / 
 
 » V 
 
 / 
 
 / 
 
 V 
 
 APPLICATIONS. 
 
 1. A farmer raised 3674 bushels of grain, and has sold 05 % 
 of it ; how much has lie left 1 
 
 Solution. If he has sold 65 % of it, he has left the whole, or 100 %, 
 less 65 %, which is 35 %; and 35 % o/ 3674 iws/ic/s w 3674 x .35 = 
 1285.9 bushels, Ans. 
 
 2. If cloth will shrink in sponging 5\ per cent, of the length, 
 what will be the shrinkage of a piece of cloth containing 43 
 yards before sponging 1«=. UJ j^ /£? d' "^-^^^^ 
 
 3. If a man's salary is $2250 a year, and his expenses 87 \ 
 per cent, of that sum, how much can he save yearly ] 
 
 Ans. $281.25. 
 
 4. A speculator bought 3100 barrels of produce, but on 
 examining them, found 15-^ % of the whole worthless ; of the 
 remainder he reserved for his own use 5 %, and sold what was 
 left at $3 per barrel } how much did he get for what he sold 1 
 
 Ans. $7610.07. 
 
 CASE 11. 
 
 267. To find what per cent, one number is of another. 
 
 1. What per cent, is 18.25 of 365 ? 
 
 OlERATION. 
 
 1J:l25 _ 05 or 5 V A^<, ^^"^ ^^"^^ ^' ^'^^ = '^^ °^ ^^^' 
 £65 - •'^'^' ^^ ^ /o' ^^^- and .05 = 5 %, 18.25 is 5 % of 365. 
 
 EuLE. Divide the number denoting the percentage by that denot- 
 ing the base, and extend the division to hundredths. 
 
 Expkin the operation. Eepcat the llule. 
 
PERCENTAGE. 
 
 101 
 
 I 
 
 5 
 
 Examples. 
 
 What per cent, is 
 
 2. 15 lb. of 50 lb.? Ans. 30 %. 7. $28.-17 of .9G57 ?r-'-' ^ J/^j *' 
 
 3. 57 men of 600 men? , 6) ??."*. 8. 199.^ rd. of 2850 rd. ] ^'^ 
 
 4. $7.80 of 81040? Ans. | %.' " Am, 7 %. 
 
 5. $782.80 of $7G0?- -J'CS'^. $235.50 of $1500] 
 
 6. 90.848 of801.G]y/w.<?. Hi %.'! Aiu. 15/^%. 
 
 10. Find what per cent. 1.04 qt. is of 3 pk. 2 qt. ~'C V- .. 
 Solution. 3 pk. 2 c^t. = 2G qt. ; 1.0-i -^ :^G = .04 = 4 %, ^Ih^-. 
 
 11. Find what per cent. 5 cwt. 2 qr. 21:^ lb. is of 12 owt. 2 qr. 
 
 20 1b. '^j-af, 
 
 12. Find what per cent. \ is of .G25? Ans, 20 %. 
 
 i! 
 
 !' X 
 
 !7l 
 
 365, 
 5. 
 
 uot- 
 
 APPLICATIONS. 
 
 1. On a plantation containing 4000 trees, 20 trees died ; 
 what per cent, is that of the whole ? Ans. \ %. 
 
 ^- 2. If 43 yards of cloth shrink in length, by ?. ponging, 2.365 
 yards, what is the shrinkage per cent. ] — '^'^^'^J 
 ■^ 3. A farmer had last year 320 sheep, and this year 344 ; 
 what is the increase per cent. ? 
 
 Solution. // lie, had last year 320 sheep, and this year 344, the 
 increase is 24 sheep ; and 24 is t>%*(j = .07^ = 7^ % o/ uhat he had lust 
 2/ear. 
 
 -.. 4. The population of a certain town was once 5600, but is 
 novv 4802 ; what per cent, is the decrease 1 Ans. 14-^ %. 
 
 ~ 5. Mr Wilson has paid $110 of a demand for $235 ; what 
 per cent, remains unpaid 1 Ans. 53 j^ %. 
 
 ~ G. A farmer gave his son Thomas 100 acres of land, and his 
 son Arthur the same quantity ; but Thomas decreased his 
 50 %, and Arthur increased his 50 % ; what per cent, then was 
 Thomas' land of that of Arthur 1 Ans. 33^ %. 
 
 J ' 1 
 
 Review Questions. What is meant by per cent. ? (262) What is Per- 
 centage? (2G3) Percentage of a r.umber? (2G3) The Base of percentage? 
 (263) The Rate per cent. ? (263) 
 
 I 
 
 n» 
 
 1i 
 
It 
 
 PKACTICAL AHITUMETIC. 
 
 «■ i 
 
 lii i 
 
 Or, 
 
 CASE III. 
 
 268. To find a number when any per cent, of it is given. 
 1. $i50 is 15 per cent, of what number of dollars ? 
 
 Since 15 % of some number is 
 $450, 1 % of it is ^ of $450, or 
 630, and 100 %, or the number 
 itself, is 100 times $30, or $3000. 
 
 Or, shice 15 %, or .15 of the 
 luuuber is $450, that number 
 must be J.^ of $450 = $450 -v- .15, 
 which is $3000. 
 
 OrERATION. 
 
 30 
 
 $,i0O X 100 
 
 1^ 
 
 = $3000, Ans. 
 
 $450 v .15 = $3000, Ans. 
 
 Rule, Divide the given percentiuje hi/ the rate per cent, ex- 
 pressed decinmlly. 
 
 Examples. 
 
 -^•2. Of what number is 9G bushels G %'? Ans. IGOO bushels. 
 , 3. Of what number is 57 men 9^ % ? 7;:^ .-^ // ^ 
 
 4. $375 is 30 % of what number? ' Ans. 1250. 
 
 5. 1.04 qt. is 4 % of what number? Ans. 3 pk. 2 qt. 
 
 6. $235.50 is 15j-^ % of what number? — 4^ / O ■-' ^ 
 
 7. $7.80 is f % of what number? ' ' Ans. $1040. 
 
 8. 7 lb. 11.2 oz. is 5 % of what number? 
 
 Ans. i cwt. 2 qr. 4 lb. 
 269. When the number of the per cent, is an aliquot part 
 of 100, to iind the base wo may 
 
 Consider the percentage the same fraction of the base that the 
 rate per cent, is of 100. 
 
 9. 9G0 hogsheads is 33^ per cent, of what number ? ^' - ' ^ 
 
 Solution. If 3^} %, or J, of some number is 960 hogsheads, f , or 
 that number, must equal 3 times 960, or 2880, hogsheads. 
 
 . 10. $242.14 is 25 % of what number ? r=. V ^s ^' * hG 
 11. 2 h. 12 m. 3G sec. is 50 % of what number? 
 
 Ans. 4 h. 25 m. 12 sec. 
 
 Explain tlie operation. What is the llule ? How may the base be found 
 when the rate per cent, is an aliquot part of 100 ? 
 
 
 
tERCENTAOB. 
 
 1D3 
 
 I 
 
 ^ 
 
 APPLICATIONS. 
 - 1. From a oaslc of mol,mo., thorc loaW out lO.OS ..Mlon, 
 
 -- ^. ihe tax on a liouse is $8, wliicli is » V of ,> . , i .• 
 wliat is its valuation? ^^ ^ '^ its vaJiutiori ; 
 
 " 3- If ^Y'''"' y ''"'""- ^' ^ % «f ^"« in<^ome, lays up .^281 '>5 
 a year, what is liis income ? ^ y >^- ^, , ^ ^ 1 . -m .. j 
 
 - 4- If \7 bu. 2 pk. is 7i % of a farmer's crop of o.,ai„ u,^ 
 much IS the whole of it? ./J233bu 1.7 
 
 ^. In a battle an army had 75 men killed, 93 Wound'^.l^ud 
 11- taken prisoners, and the entire loss was I7J, ^/ ■ what was 
 the number at first ? V \ 
 
 Ans. 1000 men. 
 
 of 
 ^/i6'. $2007. 
 
 A man took from a bank $393, which was 13 ^- ^ 
 what ho had deposited ; how much then remained ? " 
 
 CASE IV. 
 
 l.TfJr!' ^""^ ^ ''''"'^''' ^^^'^ ^' " ^^^^" P^^ <^^^*' ^ore or 
 less than a given number. 
 
 to 38.51'"' ""'""""■' '""■'''"' '^^' '** ^'' '™'- of ''^^'f' i^ "1»»1 
 
 A number increased by 10 % of 
 itself is equal to 110 % of itself.' If 
 110 % of a number U equal to 385, 
 1 % of it is equal to jj- of 385, or 
 3.50, and 100 %, or the number, i:. 
 equal to 100 times 3.50, or 350. 
 
 Or since 110 %, or l.lo, of 'some 
 
 number is equal to 385, that number 
 
 • must be equal to ^x^ of 385 = 385 
 
 ~ 1.10, which is 350. 
 
 Rule. Dkide the gmn nnnhr, acmrding as it ma,j be more 
 
 Mu^n the rcimred number, b,j 1, i,era.ed or dJnisl./Z 
 
 trie rate per cent, exm-fs^sp,!. rUn^r..,.ii., ^ 
 
 OPERATION. 
 
 10 
 
 385 X 100 
 
 ~- = 350, Ans. 
 
 440 
 11 
 
 Or, 
 
 385 - I.IO = 350, Ans. 
 
 the rate per cent, expressed decimally. 
 
 Explain the operation. Repeat the Eule. 
 
 1 I 1 
 
 1^ 
 
 : 
 
 -. ■!?. 
 
 N 
 
~'^°Tlit- lifciJB tlUfl'IM II I 
 
 194 
 
 PPwACTICAL ARITHMKTIC. 
 
 
 h: 
 
 
 i 
 
 Examples. 
 
 .. 2. 4550 is 25 % more than what number? Jiu. 3010. 
 
 3. 7402.5 is 17.>- % more tlian wliat numborl 2^ ^ ^00 ' 
 
 , 4. .$1047.80 is i^ % more than wliat nuniber? Am. $1040. 
 
 5, What number, <limini.slic(l by 14 ^/ of itself, is equal to 
 129?-- /6Y;' 
 
 Solution. 100 % - 14 % = 8G % ; .-J^- ol' 129 = 129 -v- .86 = 
 150, Ansi. 
 
 ^ (j. 534.85 yards are 5 % less than what number? 
 
 Ans. 5G3 yards. 
 . 7. 1738 is 13-^ % less than what number? Am. 2000. 
 
 8. 543 men ai'e 9A % less than what number? -r-. "> / ' ^ru-n, 
 , 9. What fractiov,, diminished by 10 % of itself, is equal 
 to ^4? Ans.-j^. 
 
 Tl\. It is sometimes most convenient to use the given rate 
 \>('X cent, expressed as a common fraction. 
 
 -' 10. What number, increased by 25 % of itself, is equal to 
 750 ? rz=. (^ r 
 
 Solution. 25 % r= -^^^^ = I ; a, or the number, + ]- = ^ ; if ^ 
 = 750, I = 1 of 750, or 150, and ;[ = 150 x 4 = 600, Am. 
 
 . 11. $G20 are 33/3 % less tlian Avliat number? - ^]7>0 
 
 '12. I is IGg % more than what number? 
 
 APPLICATIONS. 
 
 Am. -j^. 
 
 • 1. A candidate for re-election received 3G40 votes, which 
 was 12 y more than he received at his first election ; how 
 many had he at the first ? Am. 3250 votes. 
 
 - 2. A regiment, wliich lost in a severe engagement 31] per 
 cent, of its men, haa 440 men left ; how many had it at first? 
 
 Am. G40 men. 
 
 IvEViEW Questions. How may rate per cent, be expressed ? (264) What 
 is used in business to stand for the word per cent. ? (204) 
 
PErtCKNTAOE. 
 
 195 
 
 .^3, A man spont 00^ % of his innonio, and savod i5 .").") 3. 3 3,^ ; 
 what was his income ? rr^^^ I 1^ C C 
 
 Solution. If he spent 00^ % of /i/.s- income, he siov.d the whole or 
 100 % - G6|| %, vhlrh /,.• '.V.\l '/, or ?, ; if ^-i?>'.\:.\?,\ in .\, ;|, or hU ichole 
 income, must he ^WX^.'.V.)}^ x ;j - $1()()0, Ans. 
 
 — 4. Tho popnlati<in of a certain village is 4059, which is 
 12.^ % more than it was five years ago ; what was it then? 
 
 Anx. 3G08. 
 '- 5. A gentleman comph^ted a journey by steamer in 26 d. 
 10.4 li. which was 30 % longer than would have heen required 
 to have performed it by railroad; how long would it. have 
 required by railroad I Am. 20 d. 8 h. 
 
 2H 
 4 6- 
 
 if:? 
 
 
 
 What 
 
 . 
 
 COMMISSION AND BPtOKERAGE. 
 
 272. A Commission Merchant, or Factor, is a person who 
 buys or sells goods for others, for a percentage. 
 
 273. A Broker is a person who buys or sells stocks, bills of 
 credit, t^'c., for a fee, or a percentage. 
 
 274. Commission is the percentage paid a commission mer- 
 chant, or factor, for the transaction of business. 
 
 275. Brokerage is the percentage paid a brok(}r for the 
 transaction of business. 
 
 The Base is the amount of property sold or purchased, of 
 money invested or collected, cl'c. , 
 
 276. To find the commission or brokerage. 
 
 1, An agent sold property to the amount of Ji?5G50 ; what is 
 his commission, at 3 per cent. 1 
 
 oi'ERATioN. Since tlie commission is 3 %, 
 
 $.3050 X .03 =^ 81G9.50, Ans. or .0,3, on $5050, it must be 
 
 3 hundredths of $55650, or 
 I5G50 X .03, which is 81 GO. 50. Therefore, &c. 
 
 EuLE. Multij)hj the (jivc7i amount hy the rate per cent, expressed 
 decimalhj. 
 
 AVhat is a Commission Mcrcliant, or Factor? A Broker? "What is Com- 
 mission? Brokerage? The Base? Explain the operation. Tlie Rule. 
 
 w 
 
PRACTICAL AUITIIMKTIC. 
 
 \ Examples. 
 
 \^«. An agont collocts debts to the amount of $005 ; wliat is 
 
 his commission at 2 %] ytns. $19.00. 
 
 ^ 3. Wliat is the hrokerago for purcliasing $10000 of gold, 
 
 »-. 4. A comtnission morchant disposes of 718 barrels of flour 
 at $7.13 per barrel; what is his connnission at 4} per cent. ? 
 
 Ans. $217.r)7 + . 
 -' 5. If a broker charges \ % for exclianging $5G70 of cur- 
 rency, how much is the brokerage] Ans. $28.35. 
 •^ 6. If my agent charged 5 % for collecting a debt of $564.40, 
 how much will he have to pay over to me, after deducting his 
 
 fee?r-rz; o'j^(;^'/^ 
 
 Solution. $564.40 x .05 = $28.22 ; $5G4.40 - $28.22 = $53G.18, 
 Ans, 
 
 / 7. What is the fee of a tax collector for collecting $19500, 
 at 1-} %, and how much will he have to pay over ? 
 
 A)xs. Fee, $292.50; pay over, $19207.50. 
 
 277. When the given amount includes the commission or 
 brokerage, 
 
 The given amount divided hy $1, increased hy the commission or 
 brokerage on $1, will denote the sum to he invested. 
 
 The sum to he invested suhtracted from the given amount, uill 
 give the commission or hrolcepage. 
 
 ^ 8. A broker receives $0045, with instructions to deduct his 
 brokerage, f %, and invest the balance in bank stock ; what 
 sum had he to invest, and what was his brokerage ; 
 
 OPERATION. Since the broker- 
 
 $6045 -r $1.00} = $6000, to be invested, age is | % of the 
 $6045 - $6000 = $45, brokerage. sum to be invested, 
 
 the broker receives 
 $1.00| for each dollar to be invested, and will have for investing as 
 many dollars as $1.00| is contained times in $6045, or $6000 ; and the 
 difference between the sum he receives and the sum to be invested, 
 or $45, must be his brokerage. 
 
 ■ 
 
 . 
 
MHBBii 
 
 PErwCENTAOE. 
 
 197 
 
 /•9. An ngont roccivos $3838.80 to lay out in purcliaso of ^. 
 
 goods, (Icdiictiiig liis coraniLssiou at 5 % ; what value of goods 
 shall tho owner receive? Ans. $3056. 
 
 ^ 10. An agent receives $581.85 to lay out in coal, deducting 
 
 his commiission at 2J % ; what value of coal can he purchase ? -6\> )-6^* 
 
 ^11. I remit a commission merchant in Chicago $2050, for 
 the purchase of flour; the Hour costs $10 a barrel, and his 
 commission is 2.V % ; after deducting his connnission, how 
 numy barrels will the money enable him to send me ? 
 
 Auii. 200 barrels. 
 
 -•12. A broker has $11000 to invest, after deducting his fees 
 at the rate of ^- % ; what sum is there to be invested?-^ /<:^ $/ <; . jy, /,y. 
 
 -13. Sent an agent in New Orleans $01890 to purchase a 
 quantity of cotton ; if he deduct his commission at 3 %, and 
 with the bahui'^.e purchases 700 bales, how much is the cost per 
 bale % Jus. $90. 
 
 or 
 
 INSURANCE. 
 
 278. Insurance is an obligation by which one party is bound 
 to indemnify another for a loss which he may sustain. 
 
 Property I^,.>URANCE has reference to houses, mills, ships, 
 goods, &c. 
 
 Health and Li. E Insurance have reference to risks on 
 health and life. 
 
 The party taking the risk is the Insurer or Underwriter, and 
 the party protected the Insured. 
 
 279. The Policy is the written contract made by the parties. 
 
 280. The Base is the amount insured; and 
 
 281. The Premium is the percentage charged for the insur- 
 ani^e. 
 
 !ii 
 
 Wha is Insuriince ? Property insurance? Health or Life insurance? 
 Tiie PoUcv ? The Base ? The Premium ? 
 
198 
 
 PRACT'OAL ARITHMETIC. 
 
 2S2. To find the premium of insurance. 
 
 1. What is the premium on a policy of $3840, at 4^ per 
 
 cent. 1 
 
 Since the premium is 4^ %, 
 or .04.}, on $3840, it must he 
 .041 of 83840, or $172.80. 
 
 Rule. Multiply the amount insured hy the rate per cent, of 
 premium expressed decimally. 
 
 operatic:^. 
 $3840 X .04J = $172.80, Ans. 
 
 Examples. 
 
 ' 2. What is the premium for insurance of $3500 on a dwell- 
 ing-house, at 2 % ? Ans. $71.20. 
 . 3. Wiiat is the cost of insuring a cargo of corn,, worth 
 $5000, at 3 %, and $1 for the policy? - / •• J^^r^t.^, 
 ,. 4. If a person 36 years old takes out a life policy for $5000, 
 payable at death, or when he is 45 years old, how much will 
 he ha^'e paid, if he shall live to the latter age, the anrual pre- 
 mium being $541.30 1 Ans. $4871.70. 
 
 - ' 5. What is the cost of insuring $7500 on a store and goods, 
 
 6. A ship and cargo are insured for $08000, at ?t\ % ; if 
 they should be destroyed by fire or storm, what would be the 
 actual loss of the insurers? Ans. $94815. 
 
 PROFIT AND LOSS. 
 
 283. Profit and Loss are commercial terms denoting gain 
 and loss in business transactions. • • 
 
 284. The Base is the cost price, or the quantity on which 
 the gain or loss accrues. 
 
 The Eo'amples are like those of the four cases in Percentage, 
 and may be solved by the application of rules already given, 
 or by analysis. 
 
 Explain the operation. Kepeat the Rule. 'What do the terms Profit and 
 Loss denote ? What is the Base ? How may the examples be solved ' 
 
 
^mm-'^.^'^'' 
 
 PERCENTAGE. 
 
 199 
 
 CASE I. 
 
 285. To find any given per cent, of profit or loss on a 
 quantity. 
 
 1. If a house be bought for $8500, and sold at a gain of 12 
 per cent., "vvhat was the gain 1 
 
 Since tlie lioiiye was boiiglit 
 
 OPERATION. for 88500, and sold at a gain of 
 
 $8500 X .12 = $1020, Jns. 12 %, the gain v/as 12%,or .12, 
 
 of $8500, or $1020. That is, 
 
 The profit or loss is equal to the base multiplied hy the rate 
 per cent., expressed decimally. 
 
 ^2. Mr Edson bought flour on speculation fur $23-10, but he 
 was obliged to sell it at 15 % below cost; how much wao his 
 Joss? yiw5. $351. 
 
 y 3. Sold goods for which were paid $8500, at a gain of 21 A 
 per cent. ; how much was the profit ? -=*/<? '«. / • ^'^ 
 ^ 4. A man sold a farm, for which he gave $5000, at 9 per 
 cent, below cost ; how much was hit loss? Ans. $450. 
 
 ^ 5. If cloth cost $4 a yard, at what price must it be sold to 
 gain 25 % ? 
 
 Solution. 25 %, or |-, of $4, is $1 ; $4 + §1 = C5 per yard, Aks. 
 
 ^6. If sugar cost 12 cents a pound, at what price must it be 
 marked to make 33 J % profit ? Ans. IG cents per pound. 
 
 .7. To make a profit of 12^- % per pound, at what selling 
 price must tea, that cost 80 cents per pound, be marked ? ™ ij o £/' 
 
 8. A trader having some shop-worn articles that cost him 
 $130, has marked them down 10 % below cost; what is his 
 reduced price? Ans.^lll. 
 
 9. A merchant bought GO casks of molasses, each contain- 
 ing 63 gallons, for $1512 ; at what price per gallon must it be 
 sold for him to gain 15 per cent. ? Ans. 4G cents per gallon. 
 
 Explain the operation. To what is the profit or loss cqu.il \ 
 
mism 
 
 200 
 
 PRACTICAL ARITHMETIC. 
 
 CASE II. 
 
 286. To find the rate per cent, of profit or loss. 
 
 1. If a house be bouglit for $8500, and sold for $1020 more 
 than the cost, what was tJie £ix\n. per cent. ? 
 
 Since the gain is $1020 
 
 OPERATION. on $8500, it is -I^gJj = -12, 
 
 1020 -f 8500 = .12, or 12 %, Jns. and since .12 = 12 %, the 
 
 gain was 12 %. That is, 
 
 The rate j^cf cent, of ijrojit or loss is equal to the p'ojit or loss 
 divided hy the base, with the division extended to hundredths. 
 
 — 2. From a cask containing 84 gallons, 14 have leaked out; 
 what per cent, is the loss? Ans. 16r;- %. 
 
 — 3. If a merchant sell cloth, which cost him $4.50, at a profit 
 of $.90, what is the gain per cent. ? ~ f^X' '1 
 
 - 4. When cloth, which cost $4.50 per ^ard, is sold at $6, 
 what is the gain per cent. ? Ans. SSg- %. 
 
 ,5. Bought a horse for $250, paid for heeping him $10, and 
 sold him for $234 ; what was the loss per cent. % Ans. 10 %. 
 
 ^ 6. A merchant bought 50 barrels of flour for $10.20 a bar- 
 rel, and was obliged to lose $85.00 in selling it ; what percent, 
 was his loss? Ans. 1G| %. 
 
 - 7. If I sell -i- of a lot of land at -| of its cost, what is the 
 profit per cent. ? Ans. 60 %. 
 
 "8. If I sell f ot -i cord of wood at the cost of a cord, what 
 is the gain per cent. ? Ans. 33| %. 
 
 - 9. Sold a house for $5000, and gained $500 ; what would 
 have been the loss per cent, if 1 had sold it at $4000 ? 
 
 Ans. Ill %. 
 
 CASE III. 
 
 287. To find the "base of profit or loss. 
 
 - 1. What is the cost of a house, if in selling it at a profit of 
 $1020, the gain is at the fate of 12 per cent. ? 
 
 Explain the operation. To what is the rate per cent, of profit or loss 
 equal ? Explain the opertation. 
 
 I 
 
■ m t amm tmf mm'V^- -' 
 
 PERCENTAGE. 
 
 201 
 
 1 
 
 m 
 
 
 OPERATION. 
 $1020 X 100 
 
 500, Ans. 
 
 Since 12 % of tho cost is $1020, 
 1 % of it is i\r of $1020, or $8;"), 
 and 100 %, or the -whole co^^t, is 
 100 times $85, or $8500. 
 
 Or, since 12 %, or .12, of the 
 
 Or, 
 
 $1020 -r .12 = $8500, ^7is. 
 lost is $1020, tlie cost must be r^.j of $1020, or $1020 -h .12 = $8500. 
 Tliat is, 
 
 2'he base is equal to the profit or loss divided hy its rate per cent., 
 expressed decimalhj. 
 
 ^2. If by selling cloth at $.90 per yard above cost, the gain 
 is 20 %, what was the cost? Ans. $4.50 per yard. 
 
 -3. A house was sold at a loss of $15, which was 1;^ % of the 
 cost; what was the cost? Ans. $1200. 
 
 -4. Sold coal at 125 % of its cost, and thereby gained ^1,75 
 per ton ; what was the cost per ton 1 
 
 Solution. 125 % - 100 % = 25 % ; $1.75 ^ .25 = $7 per ton, Ans. 
 
 -5. Sold cloth at 95 % of its cost, and thereby lost $.30 on a 
 yard, what was the cost per yard? ylns. $G. 
 
 ..6. Sold coal at $8.75 per ton, and thereby gained 25 % ; 
 what was the cost per ton ? •=. "^f * oo 
 
 Solution. Since $8.75 is 25 % more than the cost, it is equal to 
 125 % of the cost. If 125 %, or 1.25, of the cost is eriual to $8.75, the 
 cost must be equal to j.-J--^ of $8.75, or $8.75 -^ 1.25 = $7. That is, 
 
 The cost is equal to the selling price divided hy 1 increased hy 
 the rate per cent, of profit, or hy 1 diminished hy the rate 'per cent, 
 of loss, expressed decimally. 
 
 -7. Sold a horse for $204, and lost 15 % ; what did the horse 
 cost me 1 ' Ans. $240. 
 
 ~8. By selling molasses at $.55 a gallon, I sufl'er a loss of 
 8^ % ; what was the cost price 1 ^J^ ^x, (o ^Lt 
 
 ~9. By selling wheat at $1.70 a bushel, I gain 18 % ; what 
 was the cost price? Ans. $1.44 + . 
 
 _10. If 700 barrels of flour are sold for $4550, and thereby 
 at a gain of 4 % ; what was the cost per barrel? Ans. $6.25. 
 
 To what is the base equal? Explain the solution of example 6. To what 
 is the cost equal? 
 
 [: 
 
202 
 
 PRACTICAL ARITHMETIC. 
 
 li! 
 
 
 EEVIEW EXERCISES. 
 
 -^ 1. If wood, which should be cut 4 feet in length, fall short 
 3 inches, what per cent, should the price be reduced to corre- 
 spond? A71S. GJ %. 
 ^ 2. A merchant paid $25 commission on sales amounting to 
 $1250 ; at what rate per cent. Avas the commission? •= » ^/}^4^ 
 ^ 3. If I pay a premium of $5C0 for insurance at 5 %, what ** 
 is the amount for which the i)roperty is insured? = W I foo^, 
 y 4. Sold corn at $1.00 a bushel, and gained 25 % ; what per 
 cent, should I have gained had it been sold at $.88 a bushel ? 
 
 Solution, ti'mce $1.00 is 25 % more than the cost, it is equal to 
 125 % of the cost. If 125 %, or 1.25, of the cost is equal to $1.00, the 
 cost iniist he equal to jl^ o/$1.00, or $.80. 
 
 Since the cost is $.80, if it should be sola for $.88, the gain, would he 
 $.08 on $.80, and if the gain should be $.08 on $.80, it would he ;§^ = 
 .10, or 10 %. Therefore, dec. 
 
 ' 5. Sold tea at $.88 a pound, and made 10 % ; what percent, 
 should I have made if I had sold it at $1.00 a pound ?t=t 1 1 j, / 
 
 * G. Sold a horse, at a loss of 12 %, for $132 ; what per cent, 
 should I have gained if it had been sold at $159 ? Ans. G %. 
 y 7. Goods marked at 30 per cent, above cost, having d<'pre- 
 ciated in value, were sold at 28 % off the price as marked ; 
 what per cent, did I gain or lose ? 
 
 Solution. Since 30 % aboce cost is 130 % of the cost, 28 % ofj'the 
 marl'ed ijrice must he 28 % of 130 % of the cost, or 36| % of the cost. 
 130 % - 3Gf % = 93§ %, and if sold at 93f % of the cost, they must 
 have been sold at 100 %, or cost, less 93§ %, or at a loss of 6§ %. 
 Therefore, dec. 
 
 y 8. How much is made or lost by selling goods marked 25 % 
 above cost, at 20 % off ? Ans. Nothing. 
 
 _ 9. Sold 2 farms for $G000 each ; on the one I made 25 %, 
 and on the other I lost 25 % ; what did I gain or lose on the 
 whole? Ans. Lost $S0O. 
 
 Kkview Questions. AVImt is Commission? (274) Brokerage? (275) In- 
 surance? (278) Profit and Loss ? (283) "NV hat is the base of Commissioner 
 lirokcragc? (275) Of Insurance? (280) Of Profit and Loss ? (284) 
 
 i } »• 
 

 INTEREST. 
 
 203 
 
 ^ 
 
 ///tf 
 
 INTEREST. 
 
 288. Interest is percentage allowed for the use of money 
 or for value received. 
 
 The PRiN'CirAL is the sum on interest. 
 The Hate of interest is the rate per cent, for one year, or 
 any given time. 
 
 The Amount is the sum of the principal and interest. 
 
 289. Legal Interest is that reckoned by the rate per cent, 
 established by law. 
 
 UsuiiY is illegal interest. 
 
 The rate per cent, of legal interest varies in different states 
 and countries. 
 
 In most of the United States, Canadas, and Nova Scotia, the legal 
 rate, especially where no other is mentioned, is 6 per cent. In some 
 of the States it is 7 per cent., and in others, any rate which may be 
 agreed upon between the pariies. 
 
 The rate for one year, and at 6 per cent., is to be understood in this 
 book when no other is named. 
 
 M. 
 
 SIMPLE INTErtEST. 
 
 290. Simple Interest is that which is reckoned on the given 
 principal alone for the given tine. 
 
 291. Since the interest of $1, at G j^er cent, for 1 year, or 
 12 months, is 6 cents, for I month it will be yV of G cents, 
 which is 5 mills, and for 2 months, 2 times 5 mills, or 1 cent ; 
 also, 
 
 Since the interest for 1 month, or 30 days, is 5 mills, the 
 interest for G days, or \ of 30 days, will be 1 mill, and for 2 
 days, 3 days, &c., as many sixths of a mill as there are days. 
 Hence, the 
 
 What is Interest? The Principal? The Eate? The Amount? Legal 
 Interest? Usury? AVhat is the legal rate in most of the United States? 
 In some of the others ? What rate is understood in this book when no other 
 is named? Wh.at is Simple Interest ? .,;.. k..^' 
 
204 
 
 PRACTICAL ARITHMETIC. 
 
 Table. 
 
 Interest of $1, at G per cent., 
 For 12 months, or 1 year, is $0.06. 
 2 months, „ ^ year, „ 0.01. 
 
 5) 
 
 1 month, ,, --^ year, ,, 0.005. 
 
 G days, ,, i month, „ 0.001. 
 „ 1 day, „ ^V month, „ 0.000|. 
 292. Since the interest of $1 at G per cent, for 2 months is 
 1 cent, or -—-^ of the j^rincipal, for 200 months, or IG years and 
 8 montlis, it must be 100 cents, or eqaal to the principal; and 
 as this holds true of any sum, at that rate, we have the 
 
 Interest for 200 mo. 
 
 IOC mo. 
 
 50 mo. 
 
 25 mo. 
 
 20 mo. 
 
 10 mo. 
 
 » 
 
 » 
 
 if, 
 
 it 
 
 » 
 
 G§ mo, 
 
 5 
 
 2 
 
 1 
 
 1 
 
 2" 
 
 2_ 
 
 B 
 
 1 
 
 ¥ 
 
 1 
 
 6 
 
 1 
 
 TIT 
 
 i_ 
 
 15 
 1 
 
 "SIT 
 
 mo. 
 mo. 
 mo. 
 mo. 
 mo. 
 mo. 
 mo. 
 mo. 
 mo. 
 mo. 
 
 Table. 
 
 or IG y. 8 mo,, equal the whole Principal. 
 ,, 8y. 4mo., 
 
 j> 
 
 )> 
 
 5> 
 
 4 V. 2 mo,, 
 2 y. 1 mo., 
 1 y. 8 mo., 
 
 G mo. 20 d.. 
 
 5J 
 
 j» 12 y*j 
 
 „ GO d.. 
 
 ioi 
 A of 
 iof 
 
 1 
 
 1 
 
 _l_ 
 10 
 
 i_ 
 
 10 
 
 _1_ 
 
 10 
 
 30 d., 
 
 15 d., 
 
 12 d,, 
 
 10 d., 
 
 6d., 
 
 3d., 
 
 2d., 
 
 Id., 
 
 
 4 ^'■ 
 
 It) 
 1 
 
 
 
 100 
 
 
 loi 
 
 T~J^ 
 
 
 J of 
 
 1 
 10(T 
 
 
 iof 
 
 1 
 
 Tiror 
 
 
 iof 
 
 100 
 
 1 
 
 
 
 1000 
 
 
 ^of 
 
 I 
 loao 
 
 
 J of 
 
 1 
 lOoTT 
 
 
 iof 
 
 1 
 
 1000 
 
 293. In computing interest, it is customary, among business 
 men, to regard a month as a twelfth part of a year, and any 
 number of days less than a month, as a part of 30 days. 
 
 Repeat the Table. In what time is the interest, at 6 per cent., equal to 
 the whole principal? 
 
INTEREbT. 
 
 205 
 
 In dealing with the United States government, however, 
 each day's interest is ^Ij of the year's interest. 
 
 CASE I. 
 
 294. To find the interest, the principal, rate per cent., and 
 time, being given. 
 
 1. Let it be required to find the interest of $G40 for 3 years 
 7 months and 3 days, at 7 per cent. 
 
 FIRST OrERATION. 
 
 Principal, 
 Eate, 
 
 Int. for 1 y. = 
 
 $G40 
 .07 
 
 $44.80 
 
 40320 
 22400 
 13440 
 
 74G + 
 
 SiiicG the rate is 7 %, or 
 .07, the interest of $040 for 
 1 year is .07 of $040, or 
 $44.80. 
 
 If the interest for 1 year 
 is $44.80, for 3 years 7 
 months and 3 days, or 3.59J- 
 years, it must be 3.59^ times 
 $44.80, or $100.9006 +. 
 
 Since the interest for 
 1 year is 7 %, or .07 of 
 $640, or $44.80, for 3 
 years it is 3 times 
 $44.80, or $134.40. 
 
 7 months is equal to 
 6 months and 1 month. 
 If the interest for 1 
 year is $44.80, it must 
 be, for 6 months, or one 
 
 T ^ p o ^ ^(^^f of ^ year, one half 
 
 Int. for 3 y. 7 mo. 3 d. = $160,906 + of $44.80, or $22.40 ; 
 for 1 month, or one sixth of six months, one sixth of $22.40, or 
 $3,733 +; and for 3 days, or one tenth of a month, one. tenth of 
 $3,733+, or $.373 +. 
 
 Int.for3.59^y. = $160.9066 + 
 
 kSECOND OPERATiO'.vi. 
 
 Principal, 
 Pate, 
 
 Int. for 1 yr. 
 Int. for 3 y. 
 
 $640 
 .07 
 
 = $44.80 
 3 
 
 J) 
 
 6 mo. 
 1 mo. 
 
 = $134.40 
 = 22.40 
 
 3.733 + 
 3d. = .373 + 
 
 How is it customary to regard each month and each day ? In dealing with 
 the United States government, what is each day's interest? Espluin the 
 first operation. The second. 
 
 :.U1 
 
 ill 
 
206 
 
 PRACTICAL ARITHMETIC. 
 
 If the intorest for 3 years is i|134.40, for 6 months $22.40, for 1 
 month $3,733 +, and for 3 days $.373 +, it must be for 3 years, 7 
 months, 3 days, the sum of those interests, or $100,900 +. 
 
 General Kule. Multiply the principal hy the rate, expressed 
 decim/illy, and the iwodud hy the mnnhcr expressing the time in 
 years. Or, 
 
 Find the interest for one year hy multiphjinrj the principal hy the 
 rate, expressed decimally, and then, for the given time, hy aliquot 
 parts. 
 
 To find the amount, add the principal and interest together. 
 
 Ans. $L53.G8. 
 Ans. $30.08. 
 
 Ans. $1752. 
 
 Ans. $794.50. 
 
 Ans. $5,043 + . 
 
 Ans. $22.23. 
 /. Ans. $47.25. 
 . ^715. $157.16. 
 
 Examples. 
 
 Find tlie interest of 
 
 / • '2. $960.50 for 2 years, at 8 %. 
 
 ^3. $150.40 for 4 years, at 5 %. 
 
 ^4. $1700 for 5 years, at G %. 
 
 • ^5. $8000 for 3 years, at 1-^^ %. 
 
 • ^ 6. $9080 for 2 years G months, at 3;^- %. 
 • ..7. $71.20 for 1 year 8 months, at 41 %. 
 
 ^8. $30.16 for 1 year 10 months, at 7 %. 
 r 9. $56.78 for 3 years 11 months, at 10 %. 
 .^ 10. $300 for 2 years 7 months 15 daj^s, at 6 J 
 ... 11. $444 for 6 years 5 months 7 days, at 5^- % 
 ^ ^12. $19000 for 2 years 2 months 2 days, at 9 %. ---. ^'jJ^^'fSO 
 ^13. $2000 for 5 years 4 months 10 days, at 7/^ %. 
 
 Ans. $782.72 + . 
 
 '14. Find the amount of $575 for 2 years 6 months 15 days, 
 
 at 6 %. * ^725. $662,687 + . 
 
 .'•15. Find the amount of $1234.56 for 8 years 9 months 
 
 10 days, at 7 %. Ans. $1993.128 + . 
 
 295. It has been shown (Art. 292), that interest upon any 
 
 sum at 6 per cent., in 200 months, is equal to the principal ; in 
 
 2 months, to 1 hundredth of the principal ; in 6 days, to 1 
 
 •v thousandth of tlie principal, &c. Hence, to find interest at 
 
 6 per cent., we may 
 
 Repeat the Rule. What has been shown with regard to the relation of 
 interest at G i)er cent, to 200 mouths? To 2 months? To 6 days? 
 
immtu -mf' tmi^- 
 
 INTEREST. 
 
 207 
 
 74- 
 
 
 
 TaJce such aliquot parts of the principil as (he numher express, 
 ing the time is of 200 months. Or, 
 
 Multiply the principal hj the numher expressing one half of the 
 entire months of the time as hundredths and one sixth of the 'days 
 as thousandths. 
 
 10. What is the interest of $::440 for 3 years, 11 months, 
 21 days, at G%? ' 
 
 . FIRST OPERATION. 
 
 Principal, ^2440 
 
 Int. for 2 y. 1 mo. 
 
 ft 
 it 
 
 >> 
 
 1 y. 8 rno. 
 2 mo. 
 
 Int. for 3 y. 11 mo. 21 d 
 
 = J of Princij 
 
 al 
 
 — 
 
 p05 
 
 _ 1 
 
 
 nr 
 
 244 
 
 15 d. =iof,J^„ 
 
 
 
 
 24.40 
 G.IO 
 2.44 
 
 Princij:)al, 
 
 SECOND OPERATION. 
 
 - ^081.94 
 
 $2440 
 .238,1 
 
 19^20 
 7320 
 4880 
 1220 
 
 Int. for 3 y. 11 mo. 21 d., pSlM 
 
 The second oj^eration may be analysed thus :— 
 
 Since the interest of any sum at 6 %, for 2 months, is equal to 01 
 of the prmcipal, for 3 years and 11 montlis, or 47 months it must l)e 
 as many times .01 of the principal as 2 months are contained times 
 m 47 months, which are 23|-, and 23i times .01 are .235 
 
 Since the interest of any sum at 6 % for 6 dayn is equal to 001 of 
 the principal, for 21 days it must be as many times .001 as 6 divs 
 are contamed times in 21 days, which are 3^ times, and 3^- times .001 
 
 4b 
 
 How m.y we find interest at 6 per cent. ? Explain the second operation. 
 
 11 
 
 ► » it 
 
'"-J! .V '1 '^rtr II Mm 
 
 208 
 
 PRACTICAL ARITHMETIC. 
 
 W '• 
 
 ih 
 
 K- ■ 
 
 m 
 
 ft 
 
 i 
 
 
 Since tlio interest for '.) years and 11 months i.s .235 of the princi- 
 pal, and for 21 days .dOo^, it must be for 3 years 11 months 21 days, 
 .235 + .003.1, or .238.], of tho principal, and .23bi- of |2440 is 
 $581.1)4. 
 
 Find tliu interest of 
 / 17. 1^4.24 for 5 mont'hs and G days, at 6 %. Ans. $1.07. 
 
 18. $19.00 for 2 years and 11 months, at G %. /, , .^3' f 3 
 
 19. $75 for 1 year 4 months 20 days, at G %. Ans. $0.25. 
 
 20. $1000 for 2 years 4 months 9 days, at G %. 
 
 yi?is. $141.50. 
 
 (■ \, 21. $2000 for 8 months, at G %. Ans. $80. 
 
 ^ 22. $000.80 for 15 months, at G %. Ans. $45.00. 
 
 ^23. $30400 for 42 days, at %. ^ ^O.h'^'^0 
 
 Solution. $36400 x .007 = $254.80, Ans. 
 
 24. $1200 for 25 days, at G %. 
 
 25. $3540 for 57 days, at %. 
 
 Ans. $5. 
 Ans. $33.G3. 
 
 296. When the time is sliort, we may, as a convenient 
 method of finding interest at 6 per cent., 
 
 Talce one hundredth of the principal, hy removincj the decimal 
 point two 2)laccs toivard the left, for the interest for two months, or 
 sixty days ; or one thousandth of the p)rincipal hy removing the 
 decimal point three places toward the left, for the interest for six 
 days ; and then find the interest for the given time hy aliquot parts. 
 
 2G. Eequired the interest of $4800 for 93 days, at %. 
 
 Principal, 
 
 Int. for GO d. 
 30 d. 
 3 d. 
 
 FIRST OPERATION. 
 
 = yj^ of the principal = 
 
 »> 
 
 ¥ "I Toir 
 
 ^ nf 1 
 2(1 "^ lOJJ 
 
 5> 
 
 J) 
 
 $4800 
 
 48.00 
 
 24.00 
 
 2.40 
 
 Int. for 93 d. 
 
 - $74.40 
 
 What is a convenient method of finding interest at 6 per cent, when the 
 time is short ? 
 
 ^ 
 
5. 
 
 i 
 
 ' 
 
 % 
 
 INTEREST. 
 ,. . . , SECOND OPEKATION. 
 
 1 iincip.il, 
 
 Int. for G (1. = ^^V^ of the princii.al - 
 
 03 a. - G d. 
 
 2u9 
 
 ytns.$2.S7. .*' A*" 
 
 ^i 
 
 2400 
 4 SO 
 240 
 Int. for 93 d. = int. for G d. x L5; = ^tIIo 
 In tlic .second operatio.., .since tlie interest for G days is .001 of tlie 
 imncpal, or $4.80, for 03 days it miust be a.s numv Ji.ue.s $4.80 as (] 
 days are contained tinges in 1)3 days, ^vlnch are 15^ ; and 15i limes 
 Ci-±.bO are o<4.40. 
 
 Required tlie interest of 
 ^27. $140 for 123 days, at G %. 
 ^2S. $44.8v7 for 4 months and 9 days, at G 7 
 .li9. $3000 for G3 days, at G %. ' '°' ^^ ,..^, ^^^^ ^.^ 
 
 -30. $1 120.G0 for G months and 20 days, at G %. An^. .$.37,35 
 .31. $8000 for 15 days, at G %. ^ ^M 
 
 ^3-. i;?1880.8y for 1 month and 3 days, at G %. 
 
 0Q7 T.^ , -//i6. $10.34 + . 
 
 297. lor any other rate tlian G per cent., we may, wlien 
 more convenient than to apply the rule (Art. 294), 
 
 Find the interest at 6 per cent., and then increase or diminish 
 this interest hy such a part of itself as will give the interest at the 
 given rate. 
 
 It will be observed that interest at 3 % is l of the interest at G /, 
 at o/i, equal to the intore.st at G % - x of itself, at 7 % is equal to 
 the interest at 6 % + ^ of itself, &c. 
 
 What is the interest of 
 --33. $1385.50 for 23 days, at 7 %? Ans. eG.19G + . 
 
 -^4. $3G00 for GG days, at 7 % ? ^ns. $4G.20. 
 
 ^30. $1600 for 21 days, at 5 %? ^... $4.GG. 
 
 -36. $15G00 for 13 days, at 5 7 ? - 4 
 
 o 
 
■¥• 
 
 !:io 
 
 PRACTICAL ArwITIIMr.TIC. 
 
 t ! 
 
 I i 
 
 f 
 
 -k ; 
 
 H^ ' 
 
 ; : 
 
 ■ I 
 
 -37. 821.40, f(,r 11 months, nt 7 % ? Ans. 81. "7. 
 
 '.*JS. .^.-iJO for 11 months, tit 5 % < .-/^.y. 8.210. 
 
 -.30. 8-'500 for I year and G months, at 4 % ? y/«.v. 81«. 
 
 ,40. 87r)0.40 for 2 years and 3 months, at 9 % ? 
 
 ////.s'. 8ir)1.9r)0. 
 • 41. 834 t.45 for 2 years, 2 montlis, 3 day.s, at 7 % ? 
 
 ylns. 852.44+ . 
 x42. 8GS.75 for 1 year, 4 montlis, 10 days, at 7 % ? .-s-- ^ (j Wj ij 
 ^43. 83970.18 for 2 years, 4 months, 8 days, at 8 % ? — ^^f^^.Q, 
 
 What is tlie amount of 
 ^ 44. 880 for 1 year, 5 months, 12 days, at % 1 ylns. 8SG.9G. 
 ^ 45. 8241.20 for G months and 20 days, at 7 % ? 
 
 Alls. 82r)0..'S. 
 
 - 4G. 8500 from May IGth to November 3d following, at 
 10 %? Alls. 8523.33 + . 
 
 I lore, hy Art. 2.5.5, we find, by taking the number of entire cak-ndav 
 montlis from the tirst date, and the exact nuiubcr of day.s left, tliat 
 the time is 5 months and IH days. 
 
 - 47. 8345.94 from Decend>er 8th to March 4th following, 
 allowing the intervening February to have 28 days, at 7 % ? 
 
 Ans. 8351.59. 
 -" 48. 8800 from June 20, 18GG, to July 8, 18G7, at 5 % ? 
 
 Ans. 8842. 
 
 -49. 81000 from August 31, 18G5, to September 15, 18G8, at 
 
 6 %? Ans. 81182.50. 
 
 298. When it is required to compute interest very exactly, 
 count the actual number of days in each calendar month in- 
 cluded ill the time, and make each day's interest ^^j of a 
 year's interest. That is, 
 
 MuUiphj the interest of the pindpal for one year at the given 
 rate hy the number of days in the time, and divide by 365, 
 
 How is the time found in example 46? "When it i« lequired to compute 
 interest very exactly, how must the time be counted ? What must each 
 day's interest bo made ? How is the interest found ? 
 
 ' 
 
u 
 
 :4+. 
 
 INTKUKST. 
 
 211 
 
 t 
 
 - .'^O. Wliat is tlio interest on a govenmuont bond of 8000 lur 
 K)() (lays, Jit G % ? 
 
 SoLL-TK.x. Tlu. interest of $500 for | y,,,,,- at - is mi U ilw 
 interest IWr 1 year is $'M), lor 100 days, or ,^i''' ul' 1 year, it must be 
 
 ^ .31. Wiiat is the interest on a note of 81000 from Jinie 1.") 
 to October 31 following, at 7 /I, % ? ,/,,, .c,27.»;0. 
 
 ^ 52. What is tho interest of 8G400 from February 21, Ls(;4 
 to January 30, 18G.7, at 5 % ? ^;/,,,, ;"<2U8.yr,.' 
 
 CASK IT. 
 
 299. To find the rate, the principal, interest, and time 
 being given. 
 
 1. At what rate per cent, must 8i)GO.rjO be on interest to 
 gain 8 li5 3. G8 in 2 years? 
 
 OPERATION. Sincj the interest of 
 
 Int. of 89G0.50 at 1 % = 819.21. •^'•"'J'-'''^ ^<'i' 2 years, at i;/, 
 
 8153.G8 -f- 8I9.21' =. 8. ' '" ^^'^-^^ ^^ "'"'^t re(iiur(j 
 
 as many times 1 % to gain 
 8153.G8 as $19.21 is cmitaincl times in $15:3.(58, which is 8. Tliere. 
 lore, etc. 
 
 KULE. iJivide the given mtered hj the interest of the j)nnciml 
 for the given time at 1 ^x^r cent. 
 
 Examples. 
 
 At wliat rate of interest Avill 
 ^2. 875 gain 8G. 25, in 1 year, 4 months, 20 days? Ans. G 7 
 
 ^3. 83000 gain 852.5, in 2 years and 4 months? - /'- °" 
 - 4. 83G00 gain 8 1G.20, in GG days ? j^g^j y ' ' 
 
 -6. 8150 gain 830, in 4 years? j^'^^ 5 y 
 
 ^ 6. 8444 gain 815G.G95, in 6 years, 5 months ? Ans. oh %. 
 
 CASE III. 
 
 300. To find the time, the principal, interest, and rate 
 being given. 
 
 1. In Avhat time will the interest of 89G0.50 be 8153 G8 0^ 
 8 per cent. 1 , c 
 
 I • V, 
 
 Explain the operation. Repeat the Rule. 
 
212 
 
 PRACTICAL ARITHMETIC. 
 
 V 
 
 OPERATION. Since the interest of 
 
 Int. of $OG0.50 for 1 year = $76.84. $060.50 for 1 year, at 8 per 
 
 8153.68 -r 876.84 = 2 cent., is §76.84, it must 
 
 require as many years to 
 
 gain $153.68, as $76.84 is contained times in $153.68, wliich is 2. 
 Tlierefore, «Stc. 
 
 EuLE. Divide the given interest by the interest of the ^principal 
 jw 1 year^ at the given rate. 
 
 Examples. 
 
 In what time will the interest of 
 2. 83000 be 8525, at 7 % ? 
 • 3. 8700 be 863, at 6 % ? 
 4. 84080 be 8668.10, at 5 %? 
 6. $444 be 8157.16, at 5-^ %? 
 6. 8225 be 877.40, at 6 % ] 
 
 Ans. 2 y, 6 mo. 
 
 X^ '. I' (o 
 Ans. 3 y. 3 mo. 9 d. 
 
 Ans. 6 y. 5 mo. 7 d. 
 
 Ans. 5 y. 8 mo. 24 d. 
 
 CASE IV. 
 
 301. To find the principal, the interest, time, and rate 
 being given. 
 
 1. AVliat i)rincipal will gain 8153.68, in ? years, at 8 per 
 cent. ? 
 
 OPERATION. Since the interest of $1 
 
 Int. of 81 for 2 years = 8.16. for 2 years, at 8 %, is $.16, 
 
 8153.68 -f 8.16 = 8960.50. it must require as many 
 
 dollars of principal to gain 
 
 $153.68 as $.16 is contained times in $153.68, or $960.50. Tlierefore, 
 cV-c. 
 
 EuLE. Divide the given interest by the interest of %\ for the 
 given time, at the given rate. 
 
 Examples. 
 
 What principal will gain 
 / 2. 863, in 1 year and 6 months, at 6 % ? 
 
 Ans. 8700. 
 
 Explain the operation. Repeat the Rule. Explain the operation of Case 
 IV. Roueat the Rule, 
 
 I 
 
 i-: 
 

 V 
 
 the 
 
 INTEREST. 
 
 213 
 ^«.'. 88000. 
 
 /3. $1752, in 3 years, at 7^% % ? 
 
 /■i. .^oSl.94, 111 3 ypars 11 months 21 clays, at G %? 5t<t',UiAi^/ -^ 
 '5. $9.j38 in G months and 20 days, at 7 % ? Jus. 8241.20. 
 ,G. $151,675, in 2 years and 3 months, at 9 % ? Jns. ^jlo, 
 
 APPLICATIONS. 
 
 /I. If a man gives his note July 1.5, 18GG, for 8400, on in- 
 terest, ^yhnt sum will pay it January 21, 18G7 ? Ans. $U2 40 
 
 / 2. If a capitalist has $20000 loaned, one half at G %, an<l 
 the other at 7 %, what interest will it gain in 2 years, 2 months 
 
 ^,l'^'\^:' • , Ans. 82800! 
 
 /.I llow much must be paid for the use of $1200 for 2 
 months and 3 days, at 1 % a month ? 
 
 Solution, ^f^^^e interest at I % for 1 month u m of the principal, 
 for 2 month, and 3 days, or 2^, months, it will he 2,^ times .01, or 
 .0^10 of the principal, and .02 -,\ n/$l200 is $25.20. 
 
 . 4. If you lend $250 at 2 % a month, and are paid at the end 
 of 15 days, what amount must you receive? Ans $'^5'^ '50 
 
 / 5 Borrowed $200 the 20tli July, at 11 % a month ;7iow 
 much was the interest on the 4th of the foflowing October ? 
 
 - ,„, Ans, $7.50. 
 
 / 6. When $4,208 is paid for the use of $194 for 4 months 
 and 12 days, what is the rate of interest ? Ans. 6 %. 
 
 >7. Lent $114 at 7 % interest; on its return it had gained 
 $13.30 ; how long had it been on interest? Ans. 1 y. 8 m. 
 / 8. In what time will any sum double itself by simple in- 
 terest, at 6 % ? =..4»,., j^ % , , , ,, 
 
 Solution, ^ince the interest in one year is G % of the inirccipal, it 
 must require, for a sum at interest to double, or for the interest to equal 
 100 % of the principal, as many years as 6 % is contained times in 
 100 %, or 16§ years, which is 16 years and 8 months. Therefore, d'c. 
 /9. In what time will any sum double itself by simple in- 
 te rest,at7%? A t Ij^^ yj ^f /.., ;,^,, .^^^i^. 
 
 Give tlie .solution 0^^^:^^^^^ no., ni^^i^per cc^t. must the~iZCt 
 be to e<:ual tliu rnncipal ? Give the solutiou to example 3. 
 
214 
 
 PRACTICAL ARITHMETIC. 
 
 / 10. If I am paid 837.20 as the interest duo on monov lent 
 for 2 years and 17 days, at 7 %, what was the sum lent ? 
 
 I Ans. $200.00. 
 
 ' / 11. What would be the difference between the interest of 
 $10000 from July 1st to January 1st following, computed first 
 by months, and then by days, counting 305 days to the year? 
 
 Ans. $2,405 + more by the latter method. 
 
 fj 
 
 PRESENT WORTH. 
 
 302. The Present Worth of a sum of money, payable at a 
 future time without interest, is such a sum as, Ijeing placed at 
 interest, at the given rate, will amount to the debt when it 
 becomes due. 
 
 The Discount is the interest on the present worth, deducted 
 or abated from the apparent value of the debt, for present pay- 
 ment. It is the difference between the real and apparent value 
 of the debt, and, for distinction, is called true discount. 
 
 303. To find the present worth of any sum. 
 
 1. Find the present worth of $480, due in 4 years, without 
 interest, money being worth 5 %. 
 
 Since $l,at 5 % interest, in 
 4 years, amounts to $1.20, it 
 will require as many dollars 
 to amount to $480, at the 
 same rate, for the same time, 
 as $1.20 is contained times in $480, or 400. 
 
 Rule. Divide the given sum hj the amount of $1 for the given 
 time, at the given rate. 
 
 To find the discount, subtract the present worth from the 
 given sum. 
 
 Examples. 
 
 Find the present worth of 
 ' 2. $250, due in 6 months, at G %. Ans. $242.71 +. 
 
 / 3. $900, due in 72 days, at 7 %. Ans. $887.57 + . 
 
 OPERATION. 
 
 Amount of 81 for 4 y. = $1.20 
 $480 ^ $1.20 = 400 
 
 s 
 
 AMiat is Present "Worth ? Discount ? Explain the opeiatiou. Repeat the 
 Kale. IIow do you find the discount ? 
 

 V 
 
 INTEREST. 215 
 
 • 4. ^C,r^O, due in 1 year and 1- months, at 8 %. fv>v / "^ 6iy3^ 
 / 5. $347.25, due in 2 years, 7 months, 15 days, at G %. 
 
 y/W6\ $300. 
 What is the discount on 
 
 / G. $072, due 2 years lience, at G % ? y//^.^. $72. 
 
 / 7. $350.75, due in 93 days, at 6 % ? ylns. $5.3G + . 
 
 ^ 8. $750, due in 2 years, 3 months, 20 days, at 7 %? 
 
 Ans. $101.23 + . 
 304. Since the p'cscnt worth corresponds to the pnncijMtl, 
 the debt to the amount, and the discount to tlie interest of the 
 principal for the time and at tlie rate given, tlie rule also 
 applies, wlien the time, rate, and arrmint are (jicen, to Jliid the 
 2>ri)icijjal. 
 
 What principal will amount to 
 
 ^d. $1114.18, in 2 years, at 8 % ? Jus. $9G0.5(). ' 
 
 y 10. $3041.20, in GG days, at 7 % ? Ans. $3595.04 + . 
 
 /ll. $145.G7, in 123 days, at G %? J^y..^,^. ,42./ V^i- 
 
 /12. $4748.10, in 3 years, 3 months, 9 days, at 5% ? 
 
 ylns. $4080. 
 
 APPLICATIONS. 
 
 / 1. What is the present value of a note for $385, payable in 
 9 months without interest, money being worth G % ? 
 
 Ans. $3G8.42. 
 / 2. What is the ditTerence between the interest and discount 
 of $1050, due 10 mouths hence, at G % ? Jns. $2.50. 
 
 / 3. Bought goods for $1831.53" cash, and at once sold them 
 for $1986.48, on credit of G months; how much did I make, 
 money being worth 5 % 1 Ans. $100.49'. 
 
 / 4. A man was offered a horse for $225, cash in hand, or for 
 $230, payable in 9 months ; if he accepts tlie latter, wlien 
 money is worth 7 %, how much will he gain by the choice ? 
 
 Ans. $0.48. 
 
 To what does the present worth correspond? The debt? The discount? 
 To what does the rule apply ? 
 
•^r 
 
 11 « 
 
 21G 
 
 PRACTICAL ARITHMETIC. 
 
 lUNK DISCOUNT. 
 
 305. A Bank is a joint stock company, or an incorporation, 
 for the purpose of receiving deposits, loaning money, or issu- 
 ing notes or bills for circulation. 
 
 306. A Promissory Note is a written promise to pay ab- 
 solutely a ceitain sum of money, for value received. 
 
 The Face of a note is the sum made payable. 
 
 Days of Grace are the three days usually allowed for the 
 ]\iyment of a note, after the expiration of the time named on 
 its face. 
 
 A note i?? nominally due at the ex];tiration of the time named in it, 
 and is legally due at matnriiij, or at ihe expiration of the days of ^race. 
 
 AVhen the last day of grace occurs on Sunday, or a holiday, tho 
 note is payable the day bel'ore. 
 
 The time when a note is nominall}^ and when legally due may be 
 indicated l)y writing the number of the days with a line between. 
 them. Thus, May ^^'^^ 
 
 A note is discoiinted when bought for less than its face. 
 
 The time to ru7i, or tcr7n of discount, of a note, is the tune from the 
 day of discounting to the maturity. 
 
 307. Bank Discount is an allowance to a bank for payment 
 of money on a note before it is due. 
 
 It is the interest on the face of the note for the term of 
 discount. 
 
 The Proceeds, or Avails, of a note is the sum paid for it, 
 or the face of the note less the discount. 
 
 ] 
 i 
 
 CASE I. 
 
 308. To find the bank discount or proceeds of a note. 
 
 — 1. What is the bank discount and proceeds of a note for 
 8500, for 00 days, at G % ? 
 
 What is a Bank? A Proniissoiy Note ? Tl)e Face of a Note? i 'iiys of 
 Grace? 'Wlicu is a note legally due ? What is Bank Discount? The Pro- 
 ceeds ? The time to run ? 
 
 1' 
 
.»iwy« j i iwij. w wi i iW tii ilvw«^ '» 
 
 »i 
 
 I 
 
 II. 
 
 INTEREST. 
 OPERATION. 
 
 Interest of $500 for GO d. - 8;I()0 
 
 30 (I. .- 2.50 
 
 217 
 
 5J 
 
 3J >5 
 
 » ;> 
 
 3 (1. 
 
 .25 
 
 Interest of $500 fur 93 d. = $7.75, discount. 
 $500 - $7.75 = $492.25, proceeds. 
 
 PtULE. Find the interest on the face of the vote, for the term of 
 discount, at the given rate, and it will give the discount. 
 
 The discount subtracted from the face of the note will give tin: 
 2^'oceeds. 
 
 Notes discounted usually draw no interest till maturity. If, liow- 
 ever, a note is at interest, the face of the note is the amount al 
 maturity. 
 
 The dilference between hank and true discount is equal to the in- 
 terest on the true discount. 
 
 V 
 
 lil^ Examples. 
 
 - 2. A sixty days' nate'for ^GOO was dated and discounted on 
 the same day, at 6 % /rehired the discount and proceeds. 
 
 Ans. Discount $G.30; proceeds $593.70. 
 ^ 3. A note for $250, in 4 months, was dated and discounted 
 Dec. 31, at 1 % a month ; required when due, and the amount 
 of proceeds. Ans. Due ^1"-"=^^ | ,,,^3; proceeds $239.75. 
 
 Find when due, time to run, discount, and proceeds, of tlie 
 following notes : — 
 , 4. $1650-j^V New York, July 5, 18GG. 
 
 Four months after date, for value received, I promise to pay 
 Horatio Sheridan, or order, one thousand six hundred fifty 
 -/^V dollars at the Manhattan National Bank. 
 
 [Sfanip.] Charles N. Tliayer. 
 
 Discouni^l Sept. 5, at 7 %. 
 
 Ans. Due Nov. ^ | ^ ; to run 2 mo. 3 d. ; discount $20,217; 
 proceeds $1G--0.183. 
 
 Explain the operation. Repeat the Kule. How does bank discount com- 
 pare with true discount ? To what is their difference equal ? 
 
Y 
 
 218 
 
 PRACTICAL ARITHMETIC. 
 
 / 5. ^rmO. St. Louis, June 10, 18GG. 
 
 Ninety days after date, I promise to pay at the order of S. 
 Clark & Co. five thousand dollars, value received. 
 
 [stam,,.] AVilliam Kaspar. 
 
 Discounted July 13, at 6 %. 
 
 Ans. Due Sept. ^ ^ ; to run GO d. ; discount 850 ; proceeds 
 $4050. 
 
 CASE II. 
 
 309. To find the face of a note, its proceeds being given. 
 
 1. What must be the face of a note at 90 days, which, when 
 discounted at G %, will give $492.25 ? 
 
 OPERATION. Since $1 discounted for 
 
 Proceeds of $1 for 93 d. = $.9845. the given time and rate 
 
 $492.25 -f 8.9845 = 500 gives 8.0845 proceeds, there 
 
 must he required to give 
 8402.25 proceeds, as many dollars as 8'0845 is contained times in 
 8402.25, or 500. 
 
 Rule. Divide the i?roceeds of the note hy the ])roceeds of $1, 
 for the given time and rate. 
 
 I 
 
 Examples. 
 
 -' 2. Find the face of a four months' note which, when dis- 
 counted at 1 % a month, yields 8230.75. Ans. 8250. 
 
 ^ 3. The proceeds of a sixty days' note discounted at G %, are 
 8593.70 ; required the face of the note. _Jpv>./ ^ ho (t.oq 
 
 -^ 4. I wish to obtain $3755 from a bank ; what must be the 
 face of the note, payable in 90 dayS; at 7 % ? Ans. $3824.15 + . 
 
 ^ 5. Find the face of a two months' note which, whv :i dis- 
 counted at 2 % a month, yields $57G. J^^yyv^ - :^p (fO l>%^-^ 
 
 /6. A merchant owing $994.50, gave a 30 days' note, which 
 was discounted at 6 % ; required the face of the note to pay 
 the exact debt. Ans. $1000. 
 
 Exi>lain the operation. Repeat the Rule. 
 
 ^ 
 
 It i 
 
^> 
 
 I-NTERUbT. 
 
 219 
 
 •?1, 
 
 V 
 
 . 
 
 - 
 
 ANNUAL INTEKEST. 
 
 310. Annual Interest is sim})le interest on the principal 
 and on each year's interest of the principal duo and nnpaid, 
 when the note is written " with interest annually.'* 
 
 This interest is sanctioned by the courts of so^ne States, in 
 the nature of damages for the detention and u.se of interest 
 after it is due. 
 
 311. To compute annual interest. 
 
 1. What IS the interest due on a note of 8800, interest pay- 
 able yearly, on which no payments have been made, at the 
 end of 3 years and 9 months ? 
 
 OPERATION. 
 
 Int. of ^800 for 3 y. mo. = .^LSO.OO 
 
 Int. of $800 for 1 y. = ;?48 ; and 
 
 Int. of $48 for 2 y. 9 nio. + 1 y. 9 nio. + inc., or lor 5 y. 3 nio. = L'i. 1 2 
 
 Annual interest, 6105.12 
 Here the interest of the principal for the time is §180, and I'or each 
 year is 348. The first year's interest, after Ijeconiing due, remains 
 unpaid 2 y. 9 mo., the second year's, 1 y. 9 mo., and the third year's, 
 9 mo. The interest of $48 for 2 y. 9 mo. + 1 y. 9 mo. -}■ 9 mo., is 
 e<pial to the interest of $48 for 5 y. 3 luo., or $15.12. Hence the 
 aimual interest is $180 + $15.12, or $195.12. 
 
 ItULE. Find the interest of the princ'qnd for the whole time ; 
 find the interest of one year's interest for the sum of the times each 
 ycafs interest remains unpaid, and the sum of these interests icill 
 be the annual interest. 
 
 Examples. 
 
 2. "William Norton has L. Dixon's note, dated June 1, 18G4, 
 for $500, with interest to be paid aniuially, at 6 % ; what was 
 due June 1, 18G7? ylns. $595.40. 
 
 3. What is the interest due on a note of $200, interest pay- 
 Wliat is Annual Interest ? Expluiu the operation. Kepeat the Rule. 
 
^, 
 
 ttma 
 
 9 
 
 220 
 
 PRACTICAL ARlTriMKTIC. 
 
 I 
 
 I '■ 
 
 I 
 
 1 
 
 jihlt' annnall}', on whicli no paj'mcnts have been made, at tlio 
 end of 2 years, G niontlis, 3 days? Ans. $31.05. 
 
 4. AVliat was diK; July 1, 18G7, on a note dated May 1, 
 18G3, for $780, with interest payable annually, at G % ] 
 
 5. What was due January 1, 18G8, on a note dated July 1, 
 18G5, for $1000, with interest payable annually, at 7 %? 
 
 Aus. $1184.80. 
 
 PARTIAL PAYMENTS. 
 
 312. Partial Payments arc payments in part of a not\ 
 l)ond, or like obligation. 
 
 The payments, being receipted for by writing upon the back 
 of the obligation, are called Indorsements. 
 
 313. When settlements are made at the end of sliort periods, 
 or within a year, and on notes for any time, in States where 
 there are no prescribed methods of reckoning interest in case 
 of partial pa3-ments, it is customary to compute by the 
 
 Mercantile Rulk 
 
 Find the amount of the princmd frora the time it began to draio 
 interest, and the amount of each indorsement from the time it icas 
 made until settlewent. 
 
 Subtract the sum of the amounts of the payments from the 
 amount of the p'incijjal, and the remainder will be the balance 
 due. 
 
 In mercantile accounts, the rule may lie applied to each specified 
 period for whicli they are allowed to run, so that a new principal 
 may he formed at the end of every three, six, twelve months, &c., 
 according to the custom of merchants in balancing their ledgers. 
 This method has, even in application to notes, been sanctioned by 
 the courts of some of the States. 
 
 What are Partial Payments ? "What are called Indorsements ? By what 
 Rule is it customary to compute when settlements are made within a year? 
 Pepcat the Mercantile Rule. 
 
INTEREST. 
 
 221 
 
 Examples. 
 
 (^ ) $1728. Chicago, January 1, 18G3. 
 
 For value received, I promise to pay Kayniond, JMurris, and 
 Co., or order, on demand, one thousand seven hundred and 
 twenty-eight dollars, with interest. 
 
 [Stamp] liufuS OgdeH. 
 
 Indorsements. March 1, 18G3, $300; May IG, 18G3, $150; 
 September 1, 1863, $270 ; December 11, 1863, $135. 
 
 What is due Dec. 10, 18G3 ? 
 
 OPERATION. 
 
 Principal, 
 
 Interest for 11 months and 15 days, 
 
 Amount, 
 
 First payment, 
 
 Interest for 9 months and 15 days, 
 
 kSecond payment. 
 
 Interest for 7 months. 
 
 Third payment. 
 
 Interest for 3 months and 15 days, 
 
 Fourth payment. 
 
 Interest for 5 days, 
 
 Balance due December 16, 18G3, 
 
 1728.00 
 91). 30 
 
 
 1827.30 
 
 
 1300.00 
 
 
 
 14.25 
 
 
 
 150.00 
 
 
 
 5.25 
 
 
 
 270.00 
 
 
 
 4.72 
 
 
 
 135.00 
 
 
 
 .11 
 
 879.33 
 
 J 
 
 
 1 
 
 $948.03 
 
 (2) $700. Montville, February 4, 18G4. 
 
 For value received, we jointly and severally promise to pay 
 Joseph Perry or order, on demand, seven hundred dollars, 
 with interest. Jackson Fields. 
 
 [Stamp.] Edwin lieed. 
 
 Review Questions. What is Interest ? (288) The Principal ? (288) The 
 Rate of interest ? (288) The Amount ? (288) Le-al Interest ? (289) Usury"' 
 (2Mt) 
 
{ i r 
 
 
 i 
 
 •1 ' 
 
 1 1 
 
 i 1 
 
 MM 
 
 j 
 , 1 
 
 m. 
 
 
 
 Ki ' 
 
 ' 
 
 
 
 
 
 11 
 
 
 
 III 
 
 f 
 
 
 ' 
 
 
 1 
 
 
 i 
 
 
 
 
 1 
 
 ! , 
 
 
 i 1 
 
 \\l 
 
 ■ 
 
 « . 
 
 222 
 
 PnACTICAL ARITnMF.TIC. 
 
 iKDORSEMENTfl. Dccoinhor 18, 1804, $10-1 ; June 24, 1805, $200 ; 
 Septciiilie-r 11, 1805, $!:!() ; July 5, 18GG, $00. 
 
 What was due on this note November 28, ISGG ? 
 
 Ans. $227.87 + . 
 
 (3.) $500. Milwaukee, January 1, 18GG. 
 
 For value received, three months after date, I promise to })ay 
 to the order of Andrew Benson, five hundred dollars. 
 [.stamp.] James Lockwood. 
 
 Indorsement. January 1, 1807, $200. 
 
 What was duo April 1, 18G7, at 7 % interest 1 
 
 Ans. $331.50. 
 
 314. The United States Courts have adopted the following, 
 called the 
 
 United States Rule. 
 
 Find the amoiint of the principal to the time of the first lyay- 
 ment ; if the imyment cqiials or exceeds the interest, suUract it from 
 the amount, and regard the remainder as a new pincipcd. 
 
 If any imyment he less than the interest due, find the amount 
 of the same principcd iip to the time when the sum of the payments 
 shcdl first equal or exceed the interest due, and subtract the sum of 
 the payments from the amount; the remainder regard as a new 
 principal, icith ivhich proceed as hefore. 
 
 Examples. 
 
 (1.) $1000. Washington, January 1, 18GG. 
 
 For value received, I promise to pay Albert White, or order, 
 oire thousand dollars, on demand, with interest. 
 '^tamp.] George W. Reves, 
 
 Indorsements. April 1, 1866, $24 ; August 1, 1866, $4 ; De- 
 cember 1, 1806, $6 ; Februaiy 1, 18<37, $00; July 1, 1867, $40. 
 
 What will be due June 1, 1870 ? Ans. $1121.90. 
 
 Eepeat the Uuiteil States Kule. 
 
INTEREST. 
 
 223 
 
 SS, $200 ; 
 
 27.87 + . 
 
 18GG. 
 c to i\iy 
 
 wood. 
 
 ^331.50. 
 [lowing, 
 
 rst pcoj- 
 it from 
 
 amount 
 
 :iymcnts 
 
 ' sum of 
 
 a new 
 
 8GG. 
 order, 
 
 ves. 
 
 i; De- 
 
 10. 
 
 21.90. 
 
 OrEU.\.TU)N. 
 
 Principal, 
 
 Interest to April 1, 18GG, 3 mo., 
 
 Amount, 
 
 First payment, April 1, 18GG, 
 
 "r> 
 
 icmainder for a new principal, 
 Interest to Feb. 1, 18G7, 10 mo.. 
 
 $1000.00 
 15.00 
 
 $10 IT). 00 
 21.00 
 
 8'JOl.oo 
 
 40.55 
 
 Amount, 810-10.55 
 
 Second pay't, Aug. 1, 18GG, less than int. due, $4.00 
 
 Third „ Dec. 1, 18GG, 
 Fourtli ,, Feb. 1, 18G7, exceeds 
 
 Remainder for a new principal, 
 Interest to t'"uly 1, 18G7, 5 mo.. 
 
 Amount, 
 
 Fifth payment, July 1, 18G7, 
 
 Ilemainder for a new principal, 
 Interest to June 1, 1870, 2 y. 11 mo., 
 
 Amount due June 1, 1870, 
 
 >> 
 
 )> 
 
 G.OO 
 GO.OO 
 
 '0.00 
 
 $1)70.55 
 24.2G + 
 
 $994.81 + 
 40.00 
 
 $954.81 + 
 1G7.09 + 
 
 $1121.90 + 
 
 (2.) ^(S2b{^. Boston, October 1, ISO t. 
 
 For value received, we promise to pay Madison Wells, or 
 order, on demand, six hundred twenty-five -/y"^^ dollars, with 
 interest. 
 
 [Stamp.] Bancroft, Stetson, & Co. 
 
 Indorsements. January 1, 1865, $200 ; Nov. 1, 18G5, $20 ; 
 January 1, 1866, $300. 
 
 How much was there due May 1, i8G6 ] Ans. $143.79 + 
 
 Review Questions. "What is Simple Interest? (290) In computing 
 interest, how is it customary to regard mouths and days ? (293) "What is 
 each day's interest in dealing with the United States Government ? (293) 
 
Hi il 
 
 i , 
 
 i ; 
 
 ! ( 
 
 i I 
 
 224 
 
 rilAt'TlCAL AIlITlIMETrC. 
 
 i'X) .i?240O. New Voik, May IG, isr.4. 
 
 For value rcci'ivtnl, I jtromise to \niy J. L. Wi'ston and 
 Company, or order, on demand, twenty-four hundred dollars, 
 wiili interest after three montlis. 
 
 istainj,.] J. M. .Meigs. 
 
 Indorskments. Au^'ust Ifi, 1HG5, 8100 ; May 31, 18GG, .^G7.80. 
 
 How much was due November 30, 18GG, interest at 7 % ? 
 
 Ans. 82295.71. 
 
 (4.) $r,GGO. New Orleans, May 1, 18G:J. 
 
 For value received, I i)romise to pay to the order of Louis 
 De Bois five thousand six luuulred and sixty dollars, (jii de- 
 mand, with interest. John Vincent. 
 
 [Stamp.] 
 
 Indorsements. June IG, 18G4, 8578,:^ ; Jan. 31, 18G5, 8lG() ; June 
 IG, 18GG, 8-120. 
 
 How much was due February IC, 18G7, interest at 5 % 1 
 
 Ans. 85538.71. 
 
 315. The preceding rule modified, so as not to allow of com- 
 puting interest between payments for any period less than a 
 year, is the 
 
 Connecticut Rule. 
 
 JVhen at least a year's interest has accrued at the time of a jJatj- 
 mcnt, and also in case of the last jJaijment, proceed accord'uuj to the 
 United t-lates Hide. 
 
 When less than a year's interest has accrued at the time of any 
 payment, except the last, take the difference between the amount of 
 the principal for a ichole year, and the amount of the payment for 
 the remainder of the year after it was made, for a new principal. 
 
 When the interest tchich has accrued at the time of a payment 
 exceeds the payment, find the interest only upon the pnincipal. 
 
 IvEVlEW Questions. What is the General Eule for computing simple in- 
 terest ? (291) The Rule for interest at G per cent. ? (295) 
 
INTEREST. 
 
 no-; 
 > • t' 
 
 (1.) ^1000. New Haven, July 1, 1804. 
 
 For vhIiu! ivccivcd, I promiso to pay to the order of II. IJ. 
 Lacon, one thousaii'l dollars, on demand, with interest. 
 
 Iticluird liusscll. 
 
 (SUIMI).] 
 
 Indorskmknts. Jainiary 1, 1805, $I(>>>; Sc^ttoiiilHr 1, l^(j(i, 
 $-22:j.l)!) ; Deceiiil.iT '2'i, l^sG(i, $12. 
 
 How much v/as due January 1, ISO" ? 
 
 Ans. $804. 
 
 316. Tiu; (Jeneral Assembly of Vermont, in 18GG, by law, 
 established the following' as the 
 
 any 
 hit of 
 lit for 
 
 ml. 
 
 le 111- 
 
 Veu.mont Rule. 
 
 On all notes, bills, or other aimilar obligations, ichefhcr nuide 
 payable on demand or at a sjiecified time, with intekkst, where 
 jKiyiiituts are made, such imyments shall be applied: first, to liqui- 
 date the interest that has accrued at the time of such imyments ; 
 and secondly, to the c.rtin;jiushment of the j^i'iiicijud. 
 
 On all notes, bills, or other similar oblitjations, whether made 
 2Kiy(d)le on demand or at a specified time, with inteiiest annu- 
 ally, the annual interests that remain unpaid shall be sidfect to 
 simple interest, from the time they become due to the time of final 
 settlement; but if in any year reckoning from the time such annual 
 interest began to accrue, payments have been made, the amount of 
 such payments at the end of such year, tcith interest thereon from 
 the time of payment, shall be applied : first, to lirpiidafe the simple 
 interest that has accrued from the unpaid annual interests; 
 secondly, to licpiidate the annual interests that have become due; 
 and thirdly, to the extinguishment of the principal. 
 
 Examples. 
 
 (1.) $5000. ]\rontpelier, \t, Dec. 1, 18G7. 
 
 For value received, we promise to pay to the order of James 
 ]\Iason, five thousand dollars, on demand, with interest. 
 [^stamp.] Richardson, Bentley, & Co. 
 
 Indorsements. June 1, 1869, $400 ; December 1, 1869, $2200. 
 
V 
 
 I 
 
 I ? r 
 
 i| 
 
 ''•■\ 
 
 (I 
 
 if 
 
 il 
 h 
 
 u 
 
 j f 
 
 !♦ 
 
 1^^ 
 
 Jl* 
 
 1 ^ 
 
 22G 
 
 PRACTICAL ARITHMETIC. 
 
 What was due June 1, 1870 ? y/?2.s. 83090. 
 
 (2.) 81000. Pomfret, Yt., Oct. 1, 18G2. 
 
 For value received, I promise to pay Andrew Baldwin, or 
 order, one thousand dollars, three years from date, with in- 
 terest annually. Charles Dayton. 
 
 Indorsements. April 1, 1864, S50 ; June 1, 1805, 8400 ; AuLjust 
 
 1, 1865, $200. 
 
 What was due at maturity ? 
 
 Ans. $520.43, 
 
 COMPOUND INTEREST. 
 
 317. Compound Interest is interest upon principal and in- 
 terest, the two being comliined at regular intervals of time 
 and converted into a new j^rincipal. 
 
 The interest may be made a part of the principal, or com- 
 pounded, annually, semi-annually, quarterly, &c., according to 
 agreement. 
 
 318. To find the compound interest of any sum. 
 
 1. What is the compound interest of 8G00 for 2 ycar« and 
 G months, at 6 % ? 
 
 OPERATION. 
 
 Principal, $(300.00 
 
 Interest for 1st year, ' 3G.00 
 
 Amount, or 2d principal, 
 Interest for 2d year, 
 
 Amount, or 3d principal, 
 Interest for 6 months, 
 
 Amount for 2 y. and 6 mo. 
 Given principal, 
 
 863G.00 
 38.1G 
 
 $G74.1G 
 
 20.224S 
 
 $094.5648 
 GOO 
 
 Compound interest for 2 y. ?nd 6 m., $94.3848 
 
 "What is (:i!omponud lutcrest ? How often may the interest be compouudod? 
 Exi>hvin the operation. 
 
5. 83090. 
 
 18G2. 
 dwiii, or 
 witli iii- 
 ayton. 
 
 ; AuLjust 
 
 §j2G.4;3. 
 
 1 and in- 
 of time 
 
 or com- 
 rdiiig to 
 
 cars and 
 
 npouudod? 
 
 INTEREST. 
 
 007 
 
 WW t 
 
 The intercut for the first year is $36', and the amount .?G3G, wlii.li 
 is made a second principal 
 
 The interest for the second year is $3S.1G, and the amount $G74.1(!, 
 wliich is made a third principal. 
 
 The iiiterest Jur G months, the time at tlie end of the entire year^ 
 IS §20.2248, and th.e amount 8G94.384S. 
 
 Subtracting tlie given principal from the last amount, the com- 
 l^ound interest is $94.3848. 
 
 liULE. Fi7id the amount of the prinnpal for the first interval, 
 and make it the principal of the second interval; then, the amount 
 of the second principal for the second interval, and make it the 
 imncipal of the third, and so on for all the entire intervals. 
 
 If there he a part of the time more than the entire intervals, 
 find the amount for it. 
 
 The last amount ivill he the amount at compound interest, and 
 it, less the given principal, will be the compound interest. 
 
 Examples. 
 
 <^ 2. What is the amount of $100, at G % compound intnrcst 
 ^^^^^y^j^^^[ . Ans.m^.lO,-'. 
 
 " 3. What is the compound interest of $G00.50, at 5 7 for -^ 
 years I jiiTym^.^^tl^ 06^ "^'J^'b' ' O'^^ ' ' >6 
 ^ 4. What is tho compound inte<4st of $300, at 7 %, for 3 
 years, 4 montlis, and 15 days ? ,^ ;■' .. - Ans. 877.15 + . 
 
 ^ 5. AVhat is tlie amount of 88G0, at 4 % half-yearly com- 
 pound interest, for 3 years ? Ans. $1088.17 -f- . 
 ^ G. What is the amount of $500, at 5 % compound interest, 
 for 4 years, 2 mouths, and 15 days ? Ans. $614.08 + .' 
 319. The process of computing compound interest may be 
 abridged by means of the folio wiuir 
 
 I^epeafc the Rule. How may the proccs.s .^f computing compound uiterest 
 be abridged ? 
 
,i 
 
 ,! f 
 
 i I 
 
 1 
 
 h 
 
 ( 
 
 I 
 
 i: 
 
 ii \' 
 
 
 PRACTICAL AKITHMETIC. 
 
 Table, 
 
 SHOWING THE AMOUNT OP ?1 FROM 1 TO 20 YEARS, AT 21,, 3, 5, 6, AND 
 7 I'EK CENT., COMPOUND INTEREST. 
 
 Years. 
 
 24 per cent. 
 
 3 per cent. 
 
 5 per cent. 
 
 6 per cent. 
 
 7 per cent. 
 
 '7 
 Years. 
 
 1 
 
 1 
 
 1.025000 
 
 1.030000 
 
 1.050000 
 
 1.060000 
 
 1.070000 
 
 2 
 
 1.050G25 
 
 1.060900 
 
 1.102500 
 
 1.123600 
 
 1.144900 
 
 
 3 
 
 1.07G890 
 
 1.092727 
 
 1.157625 
 
 1.191016 
 
 1.225043 
 
 3 
 
 4 
 
 1.103812 
 
 1.125508 
 
 1.21550G 
 
 1.262476 
 
 1.31079G 
 
 4 
 
 5 
 
 1.131408 
 
 1.159274 
 
 1.27G281 
 
 1.338225 
 
 1.402552 
 
 5 
 
 G 
 
 1.159G93 
 
 1.194052 
 
 1.340095 
 
 1.418519 
 
 1.500730 
 
 6 
 
 7 
 
 1.188G85 
 
 1.229873 
 
 1.407100 
 
 1.503G30 
 
 1.G057S1 
 
 7 
 
 8 
 
 1.218402 
 
 1.266770 
 
 1.477455 
 
 1.593848 
 
 1.71818G 
 
 8 
 
 9 
 
 1.248862 
 
 1.304773 
 
 1.551328 
 
 1.689478 
 
 1.838459 
 
 9 
 
 10 
 
 1.280084 
 
 1.343916 
 
 1.628894 
 
 1.790847 
 
 1.9G7151 
 
 10 
 
 11 
 
 1.31208G 
 
 1.384233 
 
 1.710339 
 
 1.898298 
 
 2.104852 
 
 11 
 
 12 
 
 1.344888 
 
 1.425760 
 
 1.795856 
 
 2.012196 
 
 2.252191 
 
 12 
 
 13 
 
 1.378511 
 
 1.468533 
 
 1.885649 
 
 2.132928 
 
 2.409845 
 
 13 
 
 U 
 
 1.412973 
 
 1.512589 
 
 1.979931 
 
 2.260903 
 
 2.578534 
 
 14 
 
 15 
 
 1.448298 
 
 1.5579G7 
 
 2.078928 
 
 2.396558 
 
 2.750032 
 
 15 
 
 10 
 
 1.484505 
 
 1.604706 
 
 2.182874 
 
 2.540351 
 
 2.9521G4 
 
 16 
 
 17 
 
 1.521618 
 
 1.652847 
 
 2.292018 
 
 2.692772 
 
 3.158815 
 
 17 
 
 18 
 
 1.559658 
 
 1.702433 
 
 2.406619 
 
 2.854339 
 
 3.379932 
 
 18 
 
 19 
 
 1.598650 
 
 1.753506 
 
 2.526950 
 
 3.025599 
 
 3.616527 
 
 19 
 
 20 
 
 1.638G16 1.806111 
 
 1 
 
 2.653297 
 
 3.207135 
 
 3.^'r.0685 
 
 20 
 
 ^ 7. What is the compound interest of $400, ai '■ /, for 20 
 years and 6 months 1 Jl'- fO^Z-'iS'^^' yvj.^- y^. 
 
 Solution. Amount of $1 for 20 years = $3.207135 ; interest of 
 $1 for 6 months = $.03; $3.207135 x .03 = $.09621405 = interest of 
 amount for 6 months ; $3.207135 - $1 = $2.207135 = compoimd in- 
 terest of $1 for 20 years ; $2.207135 + $.096214 = $2.303349 = 
 compound interest of $1 for 20 years and 6 months ; and $2.303349 
 X 400 = $921.3396, or $921.33 + = compound interest of $400 for 
 20 years and 6 months. 
 
 '^ 8. What is the amount of $100, at a semi-annual compound 
 interest of 2} %, for 10 years ? Ans. $163.86 + . 
 
Years 
 
 ^ I 
 
 INTEREST 
 
 229 
 
 9. ^Vliat is the amount of $50, at 7 % compound iutore^t 
 
 Here, find tlie amount for 20 years, and then the amount of that 
 sum lor 10 years, by aid of the table. 
 
 3 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 12 
 13 
 U 
 15 
 
 IG 
 17 
 18 
 19 
 20 
 
 \ 
 
 REVIEW EXERCISES, 
 s 
 
 1. If on settlement with a merchant I give my note for 
 
 p400 payable in G months, with G % interest, how nnicli nuist 
 
 be paul when the note becomes due I Aiis. $.")5G2. 
 
 v^ 2. A certain sum lent at G % produced $250, between Julv 
 
 5, 18G5, and December G, 18C6 ; what was the sum ? 
 
 Ans. $2935.42 + . 
 3. My money at interest doubles itself, I find, in just Mr 
 years; Avhat is the rate per cent. ? yi^^^^ j y' 
 
 V 4. On the 15th of July, ISGG, I paid $G5, the interest due 
 on a note of $250, at G % ; from what date did the interest 
 commence? yi/i^. March 15, 18G2. 
 
 =^5. How much more is the bank than the true discount on 
 $800, for 3 years, 4 months, and 18 days? Ans. $27.80 + . 
 -h G. Having a gold watch to sell, one man offers $220 payable 
 in two years, and another oft\;rs me $200 cash in hand ; whicii 
 is the better ofter, and how much ? 
 
 Ans. $200 cash in liand, by $3.58 + . 
 / 7. I have received a note dated April 10, 18GG, for $500, 
 payable six months after date ; required when it becomes due! 
 the time to run if discounted Aug. 11, and the proceeds at 6 %. 
 
 Ans. Due Oct. ^^,3 ; time to run, G3 days ; proceeds, $494.75. 
 
 8. How much more is the compound than the annual in- 
 terest of $1300, at G %, for 4 years ? Ans. $1.13 + . 
 
 REViinv Questions. What b the Freseut Worth of any sum ? ( J02) TI.e 
 Discount ? (802) The Rule for finding the present worth ? (.303) What is -i 
 Promissory Notn? (.Wi) The Face of a Note? (300) Davs of Grace? (300) 
 Bunk DLscount? (307) llulo for finding bank discount? (308) 
 
■n 
 
 ■ 
 
 \ 
 
 \. 
 
 
 ; ','' 
 
 'i- 
 
 230 PRACTICAL ARITHMETIC. 
 
 RATIO AND PROPORTION. 
 
 320. Ratio is the measure of the relation of two like quan- 
 tities. 
 
 It is determined by dividing the first quantity by the second. 
 
 Thus, 
 
 The ratio of 6 to 3 is 2, or of 88 to $2 is 4. 
 
 321. The Terms of a ratio are the two quantities compared. 
 The Antecedent is the first term of the ratio. 
 
 The Consequent is the second term of the ratio. 
 
 322. The relation of antecedent to consequent may be indi- 
 cated by writing : , or the sign of division, between two 
 numbers. Thus, 
 
 G : 3, or G -f 3, indicate the ratio of G to 3. 
 
 The sign : is an abbreviated form of -f., and has a like meaning. 
 
 Some few American authors determine ratio bv dividing' the conse- 
 quent by the antecedent, after the old method of La Croix, which has 
 become quite obsolete in the country where it originated. 
 
 323. A Simple Ratio is a single ratio of two terms. Thus, 
 
 8 : 2 expresses a simple ratio. 
 
 324. A Compound Ratio is the product of two or more 
 simple ratios. Thus, 
 
 (G : 5) X (2 : 3), or -5- x |, expresses a compound ratio. 
 
 325. From the definition of ratio, follow the 
 
 PEINCIPLES. 
 
 1. Eatio can only exist between quantities of the same name and 
 hind. 
 
 2. The ratio is equal to the quotient of the antecedent divided by 
 the consequent. 
 
 What is Ratio ? Terms of a ratio ? The Antecetleiit? The Consequent? 
 How may the relation of antecedent to consequent be indicated ? AVhat is 
 u Simple Ratio? A Compound Ratio? Give the first Principle. The second. 
 
 \J 
 
 i'>ii 
 
 mmt 
 
I 
 
 KATIO AND rilOPORTION, 
 
 231 
 
 3. The antecedent is equal to the jproduct of the conseqiient and 
 ratio. 
 
 4. The consequent is equal to the quotient of the antecedLid 
 divided bi/ the ratio. 
 
 Also, sinco ratio may be exjiressed by a fraction : 
 
 5. The ratio is not changed, if loth the antecedent and cov^equent 
 are muUiqilied or divided hij the same number. 
 
 Exercises. 
 
 Write tlie ratio of 
 1. 3 to 5. Ans. 3:5. 4. 3 x 2 to -i x 3. Ans. (3x2): (4 x 'i). 
 
 2. 8 to 7. Ans. 8 : 7 
 
 3. to 4. 
 
 5. i^- to \. 
 G. 2 to A. 
 
 Ans. I : \. 
 
 ylns. •§. 
 
 11. ^to^? 
 
 12. 3 vd. to ;^y(l. ? 
 
 Ans. 3. 
 ylns. 71. 
 
 Ans. ]. 
 
 Ans. 2. 
 
 Ans. T,"n. 
 
 '> u 
 
 Ans. 3,1. 
 
 Wliat is tlie ratio of 
 
 7. 12 to G? Ans. 2. , 10. ^, to ^? 
 
 8. 2 to 3 ? 
 
 9. $1G to $4? 
 
 13. Reduce tlie ratio 6 : 30 to its smallest terms 
 
 1 4. Eeduce to a simple ratio 8 x 3 : G x 2. 
 
 15. Reduce to a simple ratio A x ^ -. ^ x JLr, 
 IG. Find the ratio of G h. 20 m. to 2 h. 
 
 17. If the antecedent is 15. G, and the ratio G, what is tlie 
 consequent? j,^s^ o.G. 
 
 18. If the consequent is }/l and the ratio \, what is the an- 
 tecedent ? jins, ^^, 
 
 PROPORTION. 
 
 326. A Proportion is an equality of ratios. Thus, 
 
 8 : 2 = 16 : 4 is a proportion. 
 
 The equality is generally indicated by writing : : between 
 the ratios. Thus, 
 
 8 : 2 : : 1 G : 4 indicates a proportion, 
 
 nive the thinl Principle. The fourth. The fifth. AVhat ia Proportion ? 
 How is the equality generally indicated? 
 
V •' o 
 
 w tJ w 
 
 PRACTICAL ARITHMETIC. 
 
 and may be read, the ratio of 8 : 2 is equal to the ratio of 16 
 to 4, or 8 is to 2 as IG is to 4. 
 
 327. The Terms of a proportion are tliose of its ratios. 
 
 The ExTKEMES arc the first and fourth terms. 
 
 Tlie Means are the second and third terms. 
 
 A Peoportional is any one of the terms. 
 
 A Mean PRoroiiTiONAL is a term repeated between the 
 other two. Thus, 
 
 In 1 2 : G : : G : .3, G is a mean proportional. 
 
 ! i 
 
 i I 
 
 I I i 
 
 PRINCIPLES. 
 
 328. 1. In every 2^yoim'tion the p'odud of the means is equal to 
 the irroduct of the extremes. 
 
 For, in the proportion 6 : 3 : : 4 : 2, since the ratios are etpud (Art. 
 320), we have !} = ^. Now, these eipial fractions, reduced to equiva- 
 lent fractions having a common denominator, give ^^^ = * x 3' 
 and by dropping the common denominator, 0x2 = 4x3. Hence, 
 
 2. Either extreme is equal to the iJrodud of the means divided 
 hi/ the other extreme. 
 
 3. Either mean is equal to the p'oduct of the extremes divided 
 hy the other mean. 
 
 4. The fourth term is equal to the quotient of the third term 
 divided hy the ratio of the first to the second. 
 
 Exercises. 
 
 Find the missing term in 
 
 1. 27 : 3 : : 54 r ( ). Ans. 6. 5. ^^^ : t : : 15 : ( )• Ans. 12. 
 
 2. 12 yd. : 4 yd. : : $9 : ( ). 6. ( ) : f : : |- : f. Ans. ^. 
 
 Ans. $3. 7. 81.50 : $7.50 ::(,): 3 bu. 
 
 3. 20 rd. : 25 rd. : : ( ) : $10. 0'' Uns. | bu. 
 
 Ans. $8. 8. 2 gal. 2 qt. : (oV : : $4 : $80. 
 
 4. 51 : (;j) : : 16 : 32. 
 
 Here, in example 8, the 2 gal. 2 qt. must be reduced to an equivji- 
 lent single denomination l)efore proceeding to find the missing term. 
 
 "What are the Terms of a proportion? Tlic Extremes? The Meaus? A 
 Proportional? A ]\Iean Proportional? The Principles? 
 

 RATIO AND PROPORTION". 
 
 233 
 
 8DIPLE PROPOimOX. 
 
 329. In Simple Proportion the terms of two equal simpKi 
 ratios are compared. 
 
 It ap})lies to the solution of questions, in which three given 
 quantities are so related that a fourth may be determined from 
 them, by equality of ratios. 
 
 Of the three given quantities, two of them must be of the 
 same name, and constitute a ratio, and the tliird nmst ])e of 
 like name with the required quantity, so as to constitute with 
 it a second ratio, 
 
 330. To solve questions by simple proportion. 
 
 1. If 8 yards of cloth cost ^GG, what will 32 yards cost? 
 
 It is evident that 8 yards 
 bear the same relation to 
 
 OPERATION. 
 
 yd. yd. § S 
 
 8 : 32 : : GG ; cost of 32 yd. ^2 yards, that the cost of s 
 
 4 
 
 ^GG X 0'^ 
 
 yards bears to the cost of 32 
 
 yards ; hence, the proi)()r- 
 
 = $264 = cost of 32 yd. tiou 8 yd. : 32 yd. : : $6G : 
 
 ■^ cost of 32 yd. 
 
 Since the product of the means divided by either extreme must 
 
 give the other extreme (Art. 328), the required extreme is equal to 
 
 ($66 X 32) -^ 8, or $2G4. 
 
 Or, if 8 yards of cloth cost $66, 32 yards, which are 4 times 8 yards, 
 must cost 4 times $66, or $264. 
 
 KuLE. Armnge the given terms so that, from the nature of the 
 question, the ratios shall be equal. 
 
 Find, then, the required term by dividing the irroduct of the 
 second and third terms by the first ; or, by dividing the third term 
 by the ratio of the first term to the second. 
 
 All questions in proportion may also be solved by Analysis. It is 
 recommended to the learner to solve the examples that follow liy 
 both methods. 
 
 What are compared in Simple Propoitiou ? To what (juestions does i5iuij)lo 
 Proportion apply ? Explain the operatiou. Repeat the llulo. 
 
ii ' 
 
 i I 
 
 i' 
 
 234 
 
 PRACTICAL ARITHMETIC. 
 
 Examples. 
 
 i 
 
 1 
 
 11 
 
 ^1 i 
 1 ' 
 
 1 
 
 
 1 
 
 M 
 
 l! 
 ''■ 1 
 
 1 
 
 
 ii 1* 
 
 h 
 
 Lli 
 
 ll 
 
 2. If 12 bushels of wheat cost $10, what Avill 30 Ijushols 
 cost 1 Aiu. $40. 
 
 3. If tlic rent of a farm of 183 acres is $273, what will b(^ 
 the rent of a farm of 61 acres '] Ans. $91. 
 
 4. If 08 bushels of potatoes cost $50, how many bushels can 
 be had for $10 ? Ans. 28 bushels. 
 
 5. When 12 yards of cloth ca'i be bought for $10, how 
 many can be bought for $72? yim. 54 yards. 
 
 0. A person completed a journey of 40 miles in 5 hours ; 
 how far, at the same rate, can ho travel in 45 hours ? 
 
 Ans. 300 miles. 
 
 7. If 5 yards of cloth cost $G|, what will 12 j yards cost? 
 
 Ans. $15.81. 
 
 -y 8. If 385 kilos of sugar cost $03, how many may be bought 
 'fbr $18 ? r- IIO/lc^.^2,. fi-l^h^'- 
 
 0. If I can complete a piece of work in 32 days, by working 
 8 hours a day, in what time can I do the same by working- 
 only hours a day % Ans. 42| days. 
 
 10. If I borrowed of a friend $300 for 8 months, for hoAV 
 long a time should I lend him $200 in return 1 
 
 Ans. 12 months. 
 
 11. If 100 workmen can do a piece of work in 12 days, how 
 many can do the same in 8 days ? , -o -... ' > 
 
 12. How many yards of flannel | of a yard wide are required 
 to line a cloak which has in it 12 yards of cloth i of a yard 
 wide 1 Ans. 8 yards. 
 
 13. If I give $2 for .4 of a cord of wood, how much must I 
 give for :^ of a cord ? Ans. $1.25. 
 
 14. If ylj- of a ship cost $9750, what will -j^ of it cost ? 
 
 Ans. $42000. 
 
 Review Questions. What is Ratio? (320) Terms of a ratio? (321) A 
 Simple Ratio ? (323) A Compound Ratio ? (324) Principles of ratio? (325) 
 
 "mv . ii. i jui aaBwwq— i 
 
PvATIO AND rnOPORTION. 
 
 235 
 
 15. Two niimhers arc to oacli otlier as 15 to 31, aiul tlio 
 smaller is 75 ; what is the greater ? ytu.^. 170. 
 
 16. Two numbers are to each other as 3 to 2, ami the greater 
 is 210 ; what is tlie smaller ? =rz-/Jj^(^ 
 
 17. If 3 cords 5 cord feet of wood will purchase 1 T. 5 cwt. 
 3 qr. of hay, what quantity of wood will he re([aired to pur- 
 chase 1 ton of hay? Aii)<. 2 C. G+ c. ft. 
 
 18. If a railway train moves 150 miles in 5^ hours, iu what 
 time will it move 225 miles % Ans. 8 h. 15 lu. 
 
 19. If 8 boarders consume a certain quantity of provisions 
 in 10 days, how many days would it have lasted if 2 more had 
 been in the company ? Am, 8 days. 
 
 20. A besieged fortress has provisions for 3 weeks, at the 
 rate of 14 ounces a day for eacl. man; at what rate per day 
 
 , must the provisions be distributed so that the place may hold 
 out 5 weeks? Am. 8r- oz. 
 
 21. If the fore-wheel of a coach, which is 7 feet G inches in 
 circumference, turns round 70-400 times in iroinu; a hundred 
 miles, how often \vill the hind-wheel, which is 'J feet 2 inches, 
 turn round in going the same distance ? Ans, 5 7 GOO times. 
 
 22. If 125 bushels of wheat grow on 4 acres 84 square rods, 
 how much land will be required to produce G50 bushels % 
 
 Am. 23 A. 84.8 P. 
 
 23. If a stick 7 feet liigh cast a shadow 5 feet in length, 
 what is the height of a spire that casts a shadow 129 feet in 
 length] Am. 180 ft. 7^ in. 
 . 24. A besieged town, containing 22400 inhabitants, has 
 provisions to last 3 weeks ; how many must be sent away that 
 they may be able to hold out 7 weeks ? Am. 12800. 
 
 , 331. When it is required to divide a quantity into parts 
 which are proportional to given numbers, we may 
 
 Add together the projwrtional mimhers; then the sum of these 
 
 Review Questions. What is Proportion? (32G) Terms of a Proportion? 
 (327) Principles? (328) 
 
fi 
 
 230 
 
 PRACTICAL ARITIlvrETIC. 
 
 i I 
 
 ►Sinco till; ,t,dven .sum 
 is to be be(|iu'atlic(l to 
 the three sons, iu tlie 
 pro})()rtiun of 3, 4, ami 
 5, if we divide $18000 
 into li + 4 + 5, or 1:^, 
 
 iiinuJicrs will he to any one of them, as the quantity to be divided is 
 to the 2)art correaiioiidinrj to that miinber. 
 
 25. A gentleman ])equeathe(l $18000 to his tliroe sons, A, 
 1>, und C, in the proportion of 3, 4, and 5 ; how nmcli was tlie 
 part of eacli 1 
 
 OrEUATION. 
 
 3 + 4 + - 12. Tlien, 
 12:3:: .1^18000 : ,^4500, A's part. 
 12 : 4 : : $18000 : $(;000, B's part. 
 12:5:: $18000 : $7500, C's part. 
 
 equal })arts, A M'ill have 3, 15 4, and C 5, of these parts. Ileiice, A's 
 pait is yi- of SISOOO, or $4500, L's ^.^- of $18000, or $0000, and C's 
 j5^7 of $18000, or $7500. 
 
 20. Tw(j men own a field of G40 acres in common, and tlieir 
 
 • lespective shares of it are in tlie ratio of 7 to 9 ; if divided, 
 liow many acres would belong to each?-- y^/ J^/0'X'\. "'.'/,/) 
 
 27. Divide 4720 into 3 parts which shall be in the propor- 
 ' tion of i, §, and 4. Ans. 1200, IGOO, and 1U20. 
 
 28. In a certain town of 4500 inhabitants, the yontli are to 
 
 • the adults in the ratio of 13 to 12 ; required the number of 
 each. Ans. Youth, 2340 ; adults, 21 GO. 
 
 COMPOUND PROPORTION. 
 
 332. Compound Proportion is an equality between a com- 
 pound and a simple ratio. Thus, 
 
 3-4 
 
 ^ ■ o : : 45 : 96 is a compound proportion. 
 
 It applies to the solution of questions which would require 
 several simple proportions. 
 
 333. To solve questions by compound proportion. 
 
 1. If 4 men can earn $96 in 8 days, how much can 10 men 
 'earn in 6 days ? 
 
 How may we pi-oceed when it is required to divide a quantity into parts 
 wliich are proportional to given numbers? Explain the operation. "What ia 
 Compound Proportion ? To what does it apply ? 
 
livi(kd is 
 
 ^'.a 
 
 
 
 RATIO AND morORTION. 
 
 237 
 
 
 
 • 
 
 OPERATION. 
 
 
 
 4 
 
 8 
 
 \ p : : %^d^ : amount roqnircd ; or 
 
 
 4x8 
 
 : 10 
 3 
 
 X G : : |9G : aiuount rcfiuirod. 
 
 
 Then, 
 
 $00 X 10 X G 
 
 = $180 = amount required. 
 
 If 4 men can earn $9G in 8 days, 1 man can earn in the same time 
 \ of $00, or $24, and 10 men can earn 10 times $24, or $240. 
 
 If 10 men cuti earn $240 in 8 days, in 1 day tliey eau earn \ of 
 $240 or $:iO, and in 6 days G times $30, $180. 
 
 Or, if 4 men can earn $9G in 8 days, 10 men in the same time will 
 earn ^^^ of $9G, and in days, % of "^^ of $9G, or $180. 
 
 IIULE. Wrlk the given number of the same kind as the ansivcr 
 far the third term; and arrange the rernalnlmj numhers, each pair 
 if the same land, as m simple iwoiiortion. 
 
 Find the required term hg dividing the product of the third and 
 second terms hg the product of the first terms. 
 
 It is recommended to solve the questions by both cumpoiind pro- 
 portion and analysis. 
 
 Examples. 
 
 2. If a pasttire of IG acres will feed G horses for 4 months, 
 how many acres will feed 12 horses for 9 montlis? 
 
 Ans. 72 acres. 
 
 3. If 3G yards of cloth, 7 quarters wide, be worth $98, what 
 • is the value of 120 yards^of equal quality, but only 5 quarters 
 
 wide 1 r- ^ X--^ J ' JCJ ^ 
 
 4. If 64nen can mow IG acres in 4 days, how long will it 
 take 10 men to mow 40 acres, at the same rate? Ans. G days. 
 
 .5. If a man travel 90 miles in 3 days, by walking 8 hours a 
 day, in what time will he travel 540 miles, by walking G hours 
 a day ? \ Ans. 24 days. 
 
 G. If a family if 9 spend $600 in 8 months, how much will 
 serve a family of 24 for IG months ? Ans. $3200. 
 
 Explain the operation. Kepeat the Rule. 
 
<ll 
 
 If 
 
 i- 
 
 238 
 
 PRACTICAL ARITHMETIC. 
 
 7. If 1l' liorsos ill 5 days plougli 11 acres, liow many luirses 
 would plougli 33 acres in 18 days] Ans. 10 liorst'S. 
 
 8. If .^L'OdO will support a garrison of 150 men 3 months, 
 liow long will 8C000 sui»port 4 times as many men 1^=^jJ^ ^ <'• <'i^, 
 
 9. Tlu! interest on 8200 for 4 montlis being $1, what will1)o 
 the interest on i^n^dO for 1 year and 3 months. Ans. 8-U.2.'). 
 
 10. AMien 12 men can put f of a cargo on board of a sliip 
 in 'l of a day, liow long would it require 5 men to load 3 sucii 
 vessels? yl?<.s. 13^ days. 
 
 11. If 8750 will defray the expenses of 12 men for 14 weeks 
 2 days, how much will 5 men require for 22 weeks G days ? 
 
 Ans. §500. 
 
 12. If 18 men can dig a trencli 30 yards long in 24 days, by 
 working 8 hours a day, how many M'ill dig a trench GO yards 
 long in 04 days, working G hours a day? Ans. IS men. 
 
 13. If 29 men in 5 days of 12 hours each can reap 32 acres, 
 how many acres can 20 men reap 81^ days of 13 hours 
 each] Ans. 40 acres. 
 
 PARTNERSHIP. 
 
 334. Partnership is the association of two or more persons 
 in business, under a certain name, with an agreement to share 
 the profits and losses. 
 
 The Partners are the individuals associated. 
 
 The Company, or Firm, is the business associatiom 
 
 The C.U'ITAL, or Stock, is the money, or property used in 
 business. 
 
 A Dividend is the profits to be divided. 
 
 The profits and losses of a firm are usually distributed 
 among the partners in proportion to their respective shares in 
 the Ijusiness. 
 
 Review Que.stions. A Simple Proportion? (329) The Eule? (330) 
 Wliat is Partnership ? The Partners ? The Company or Firm ? The Capital 
 or Stock? A Dividend ? 
 
 ''I -ii.'r«irii 
 
RATIO AND rilOPOUTION. 
 
 230 
 
 ' liorses 
 
 liorscs. 
 
 iioiitlis, 
 
 AVill 1)0 
 
 CASE I. 
 
 335. When the capital of each partner is employed for 
 equal time. 
 
 1. A, V), aiul C enter into partiicr.sliip ; A fiirnislies 8'')0(H), 
 li ii?-Jr)(lO, and C .J7500 ; tliey gain i?GOO ; what is each partner's 
 sliare ? 
 
 OPERATION. 
 
 Stock - $15000 = -jV'o7^^ = i of entire stock j 
 
 -.JUU -■ -J ,-ou(J f, 
 
 7r,on „ 7 r^{)i) .. j 
 
 A's 
 B's 
 
 C's 
 
 
 
 
 Entire stock = ^ 15000 
 
 Hence A's share - i of 8C00 = $200 
 B's „ - I of^GOO ..: $100 
 C's „ = i of $000 - $300 
 
 >) 
 
 Entire gain = $000 
 
 Since A's stock is equal to ^, B's to ^, aiul C's to \, of the entire 
 stock, A must have f,, B \, and C ][, of the gain ; licnce, A'.s .share is 
 equal to l^ of the $GOi), or $200, B's to ^ of the $000, or $100, and C's 
 to I of tlie $000, or $300. 
 
 EULE. Aj^portiun to each ^;rtr//«f7' such a part of the gain or 
 loss as his stock is of the entire stock. 
 
 Examples. 
 
 ♦ 2. A, B, and C form a joint capital for conducting a busi- 
 ness, of which A contributes $1500, B $1950, and C $2100. 
 At the end of a year the profits are $1GG5 ; wliat sluire sliouhl 
 each receive 1 Ans. A $450, B $585, and C $G30. 
 
 . 3. A, B, and C traded togetlier. A put into the business 
 $240, B $300, and C $120. They pined $350; what was 
 each partner's share of the gain ? - ;';/ .^-*_.^- .. '^^ -"^^ f Y/^^^S'^'"-^ 
 4. A, B, and C had 108 tons of freight" oh "botird a ship, 6i j 
 
 which A had 48 tons, B 30, and C 24 ; but in a storm 45 tons 
 w^ero washed away ; wliat was each man's share of the loss ? 
 
 — ■ " — "» p 
 
 What ia the Rule?/.^ i (^Js f:^ / OL^ / ^ 
 
 -/;- 
 /'.^w*.- 
 
■■i 
 
 ■■ 
 
 7 -F-- 
 
 
 ii 
 
 
 i 
 
 : : 
 
 « 
 
 \ I 
 
 ; } 
 
 ■ J 1' 
 
 ■ ' 'i 
 
 
 » ' '1 
 
 
 
 
 1 
 
 »> 
 
 
 
 It ! 
 
 M 
 
 J ' 
 
 1 ' ] 
 
 ^, 
 
 1 i 
 
 a t 
 
 
 1 
 
 
 .-■\ti.iirim 
 
 240 
 
 PRACTICAL ARITHMETIC. 
 
 • 5. A and B, with equal stocks, clear in trade $2000; A 
 is to have 3 parts of the profits, and B 2 parts, because A 
 managed the business ; what is each one's diare of the profits, 
 and how much did A receive for his services 1 
 
 ylns. A's share $1200 ; B's share $800 ; and A for services 
 $400. 
 
 336. The rule applies to distributing the assets of bank- 
 rupts, and other like apportiofjiments. 
 
 G. A bankru]it, whose property is only worth SGOOO, owes 
 A $8000 and B .^12000; what should each of these creditors 
 receive ? Ans. A $2400, and B $3000. 
 
 » 7. Three persons rent a pasture for the summer; the first 
 i puts in 21 horses, the second 17, and the third 47. The rent 
 is $307 ; what part of it must each pay 1 
 
 Ans. The first $75.84 + , the second $61.40, and the third 
 $169.75 + . 
 
 • 8. A gentleman left by will to his wife $5000, to his elder 
 son $3000, and to his younger son $2500. But it was found 
 that the property, after paying debts, Tvas only $7475 ; how 
 should this be apportioned under the w41l ? 
 
 Ans. To tlie wife $3559.52 + ; to the eider son $2135.71 + ; 
 and to the younger son $1779.76. 
 
 CASE 11. 
 
 337. When the capital of the partners is enployed for 
 unequal times. 
 
 1. A and B trade in company ; A puts in $500 for 8 months, 
 and B $600 for 10 months. They gain $240 ; what is each 
 partner's share of it ? 
 
 OPERATION. 
 
 As $500 for 8 mo. = $4000 for 1 mo. 
 B's $000 „ 10 mo. - 6000 „ 1 mo. 
 
 Entire stock the same as 810000 for 1 mo. 
 How does the Rule apply ? 
 
RATIO AND PROPORTION. 
 
 241 
 
 bank- 
 
 Hence, iVs sliare = j%%% = | of !?240 = $9G 
 
 _; .<!ooo_ _ :i of .v;->4.0 - !^14-4 
 
 ii'6 
 
 )) 
 
 Entire gain - $2-10 
 
 $500 for 8 months is the same as $4000 for 1 month, and $G00 for 
 10 months is the same as $6000 for 1 month ; honee the entire stock 
 is the same as $4000 + $6000, or $10000, for 1 month. 
 
 If $10000 gain in a certain time $240, $4000, or i; of that snm, 
 must gain f of $240, or $90, and $6000, or ^ of the sum, '•} o\: $24<», 
 or $144. 
 
 Rule. Multiply each partners stock hj the time it teas in- 
 vested, and apportion the gain or loss in proportion to the products. 
 
 lonths, 
 is eacii 
 
 Examples. 
 
 » 2. A, B, and C commenced trade together tlie first of June, 
 on $G000 put In by A ; the first of August B put in .$9000, 
 and the first of September C put in $12000. At the end of 
 the year their gains amounted to $4500 ; what was each part- 
 ner's share'? Ans. A's $1400, B's $1500, and C's $1000. 
 
 3. A, B, and C enter into partnership ; A put in $500 for 
 ISmontlis; B $380 for 13 months; C$270 for 9 inof'^l>s. 
 They lost $818.50 ; what was each man's share] 
 
 Ans. A's $450, B's $247, and C's $121.50. 
 
 4. Jones and Smith rent a pasture for $275 ; Jones puts in 
 80 sheep and Smith 100, but at the end of G months they each 
 dispose of half tlieir stock, and allow Hall to i)ut in 50 sheep ; 
 wliat should each pay toward the rent at the end of the 
 year ? 
 
 Ans. Jones $103. 12J, Smith $128.90,:;, and Hall $42.9G'-. 
 
 • 5. A and B entered into partnership for 1 year. A at first 
 put in $500, and at the end of 5 months he put in $150 more ; 
 B at first put in 8G00, and at the end of 9 months took out 
 $200. Their year's profits were $G82.50 ; what was each 
 man's share ? Ans. A's $352,50, and B's $:;30. 
 
 Explain tlie operation, llcpciit the llule. 
 
1 i 
 
 242 
 
 PRACTICAL ARITriMKTIC. 
 
 {' 
 
 C. A builds a mill at a cost ( '' ^35000 ; 2 months after its 
 completion B buys stock in it ol .. o the amount of $11000 ; 
 and in 3 months more C purchases also of A .^-1000 worth of 
 stock. Thej^ run the mill for 7 months, and gain during that 
 time .^9700 ; -what portion of this belongs to each ? 
 
 Ans. A's share $7205.71 + , B's share $2177.55 + , and C's 
 share $310.73 + . 
 
 , 7. S, T, and Y entered into partnership. S kept his stock 
 
 in 1 year ; T put in -}- as much as S, and for 10 months ; Y jmt 
 
 in £- as much as S, and for 4 months. Tliey gained $3400 ; 
 
 what was each one's share of the profit 1 
 
 'Ans. S's share $2400, T's share $400, and Y's share $G00. 
 
 EQUATION OF PAYMENTS. 
 
 338. EcLuation of Payments is the process of finding tlie 
 average or equitable time for paying several sums due at dif- 
 ferent times. 
 
 339. The Equated Time is the date at which the items due 
 at different times may be justly paid together, 
 
 340. The Average Term of Credit is the time that must 
 elapse before the equated time. 
 
 CASE I. 
 
 341. To find the equated time when the terms of credit 
 b3gin at the same date. 
 
 '■ 1. I owe, July 1, to John Wentworth, $G00, of which $200 
 is due in 2 months, $300 due in 4 months, and $100 in 8 
 months ; required the equated time of paying the several items 
 at once. 
 
 Review Questions. What is a Compound Proportion ? (332) TheKulc? 
 (333) Partnership ? (334) The Ruhi when tlie capital of each pax-tuer is em- 
 ployed equal times ? (335) When for unequal times ? (337) 
 
 What is Equation of Pajments? The Equated Time ? The Average Term 
 of Credit ? 
 
RATIO AND PKOrORTION", 
 
 2t3 
 
 ftfT its 4 
 
 11000; 
 ortli of 
 ig that 
 
 and C's 
 
 is stock 
 83400 ; 
 i $G00. 
 
 ling tlie 
 e at dif- 
 
 ems due 
 
 at must 
 
 f credit 
 
 ch $200 
 30 in 8 
 •al items 
 
 :he Rule ? 
 tuer is ciu- 
 
 rage Term 
 
 •J mo. X 
 4 „ X 
 8 „ X 
 
 OPERATION'. 
 
 200 - 400 mo. 
 300 - 1200 
 100 - 800 
 
 M 
 
 GOO ) 2400 mo. 
 
 4 mo. 
 July 1 + 4 mo. = November 1, Ans. 
 
 A credit on $:i(H) t'^r 
 
 2 mo. is e(|ual to ti croilit 
 
 du $1 for 200 times 2 mo., 
 
 or 400 mo. ; a credit ou 
 
 $300 for 4 mo., to a credit 
 
 on $1 for 300 times 4 mi»., 
 
 or 1200 mo. ; and a credit 
 
 01. $100 for S nio., to a 
 
 credit on $1 for 100 times 
 
 8 mo., or 800 mo. 
 
 Hence, tlic entire credit is ef^ual to a credit on $1 for 2400 mo. ; and a 
 
 credit on $1 for 2400 mo. is ef[ual to a credit on $(500 for ^,',(7 of 2400 mo., 
 
 or 4 mo. ; hence, 4 mo. from July 1, or November 1, is the equated time. 
 
 Rule. Multiply each term of credit by the number denoting it.^ 
 debt, and divide the sum of the 2Jroducts by the number denoting the 
 sum of the debts; the quotient icill be the average term of credit. 
 
 The average term of credit, added to the date of the debts, kHI 
 gice the equated time. 
 
 When any of the items have cents, if 50 or more, reckon them as 
 one dollar, but if less than 50 cents, neglect tlieni. Also, wiieu any 
 result has a fraction of a day, if it is h or more, reckon it one day, 
 otlierwise neglect it. 
 
 Examples. 
 
 / 2. Eeqiiired tlie average credit for tlie payment of $500 pay- 
 able in 2 months, $1000 in 5 months, and $1,300 in 8 months. 
 
 Ans. G months. 
 ^ 3. I owe $1G00 payable now, and $800 in 90 days; Mhat is 
 the average term of credit ? Ans. 30 da^s. 
 
 ^ 4. Required the equated time from Marcli 1st, at which to 
 pay $200, of which $40 is due in 3 months, $G0 in 5 months, 
 and the remainder in 10 months. Ans. October 4th. 
 
 / 5. May IGth, 18GG, Albert Day owes $19U.50 payable in 30 
 days, $150.15 in GO days, and $300 in 90 days ; wliat is the 
 equated tune? Ans. July 20, 18GG. 
 
 Explain the operation. Kepeat the llule. How do you proceed when ai;}' 
 of the itemis have cents ? When any result h;\s a fraction of a day ? 
 
fT- 
 
 24 i 
 
 PRACTICAL ARITHMETIC. 
 
 ! 1 
 s ■ 
 
 \\ ' : 
 
 2; ' 
 
 I' ; I 
 
 i 
 
 ^ 
 
 i 
 
 CASE 11. 
 342. When the terms of credit begin at different dates. 
 
 1. 
 
 Alexander English 
 18GG. 
 
 New Haven, January 1, 18G7. 
 
 Bought of James Miles & Co. 
 Oct. 7, Merchandise, on 90 days, $1000. 
 Nov. 15, „ net cash, 800. 
 
 Dec. 20, „ on 60 days, GOO. 
 
 What is the equated time of the payment of this bill ? 
 
 Eeckoned, for 
 
 - convenience, from 
 
 ' "^ ' the earliest date 
 
 OPERATION. 
 
 Due Nov. 15, days x 800 = 
 
 „ Jan. 5, 51 
 
 „ xl000 = 
 
 51000 
 
 „ Feb. 18, 95 
 
 „ X G00 = 
 
 57000 
 
 J> 
 
 that any one of 
 
 5) the debts is due, 
 
 2400 ) 108000 days, the credits of the 
 45 days. ^^^0, $1000, and 
 
 Nov. 15, 18G6 + 45 days = Dec. 30, 18G6, Ans. ^.^^^' ^''^' ^^*'P''; 
 ' ^ ' ' tively, 0, 51, and 
 
 95 days, from Nov. 15. 
 
 The average term of these credits, by Case I., is 45 days ; hence, 
 
 45 days from Nov. 15, 18CG, or Dec. 30, 1866, is the equated time. 
 
 Rule. Sdcd the. earliest date at which any one of the debts 
 became due, and therefrom reckon the terms of credit 
 
 Midtiphj the terms of credit of each item by the number denoting 
 its item, and divide the sum of the products by the number denoting 
 the sum of the items; the quotient will be the average term of 
 credit. 
 
 The average term of credit, added to the selected date, will give 
 the equated time. 
 
 Examples. 
 
 2. Purchased the following bills of goods : July 1, a bill of 
 $200, on 2 months; July 20th, a bill of $G00, on GO days; 
 Aug. 1, a bill of $1000, on 30 days. What is the equated time 
 of payment? Ans. September Gth. 
 
 Explain the oiieratioii. Repeat the llule. 
 
 i^ 
 
RATIO AND PROPORTION. 
 
 245 
 
 lates. 
 
 [8G7. 
 
 3 & Co. 
 
 [1 
 
 ed, for 
 
 ce, from 
 
 2st date 
 
 one of 
 
 is due, 
 
 ;s of tlie 
 
 300, and 
 
 , rc'.spec- 
 
 51, and 
 
 ; hence, 
 time. 
 
 the debts 
 
 denoting 
 
 denoting 
 
 term of 
 
 will give 
 
 a bill of 
 days; 
 ted time 
 .ber Gth. 
 
 3. I owe Oliver Bates as follows : April 1, for cash, $1400 ; 
 May 1, for merchandise, $500; Juuo 1, for flour, $1100. 
 What is the average date of the items ? Ajis. April 28th. 
 
 4. R. Hicks & Co. have sold a merchant the following 
 bills: Jan. 1, merchandise, $735; Jan. 21, corn, on 30 days, 
 $640.50; Feb. 1, lumber, $100; March 12, merchandise, on 
 30 days, $200. If they should receive in settlement for the 
 whole a note, from what date ought it to draw interest 1 
 
 Ans. February 3d. 
 
 343. When the items have the same term of credit, we may 
 First find their average date, and then add the common term of 
 credit for the equated time. 
 
 5. Purchased goods, on 4 months, as follows : April 1, a bill 
 of $1450; May 7, a bill of $1250; June 5, a bill of $850. 
 Required the equated time of payment. Ans. August 2yth. 
 
 G. Sold the following bills of goods, on G months : Jan. 15, 
 a bill of $3750; Feb. 10, a bill of $3000 ; March G, a bill of 
 $2400 ; June 8, a bill of $2250. At what time should a note 
 be made payable, that will settle for the whole 1 
 
 Ans. Sept. 2d. 
 
 AVERAGINCx OF ACCOUNTS. 
 
 344. The Balance of an Account is the difference between 
 its debtor and creditor sides. 
 
 Accounts are subject to interest after the expiration of the 
 term of credit. 
 
 345. Tlie Averaging of an Account is the process of fin<l- 
 ing the equated time of paying the balance, or the date at 
 which the balance of the account becomes due, or subject to 
 interest. 
 
 How may we proceed when the items have a common term of creilit? 
 What is the Bahmco of an Account? "When are accounts subject to interest? 
 What is the Averaging of an Account? 
 
2]o 
 
 PRACTICAL ARITHMETIC. 
 
 If '1 
 
 346. To find the equated time of the balance of an account. 
 
 1. When does the balance of the following account become 
 due? 
 
 Dr. Franklin Fuller. Cr. 
 
 18G6. 
 
 
 
 i 18(10. 
 
 
 
 March 10, 
 
 To Mdse. on 4 mo. , 
 
 $200 00 
 
 : April 11, 
 
 By Cash, 
 
 SCO 00 
 
 „ 28, 
 
 >) 5) ji " ») ; 
 
 1GO,00 
 
 July 10, 
 
 > 5 » » 
 
 140 00 
 
 April 1 
 
 „ „ net, 
 
 140^00 
 
 Aug. 2!), 
 
 1 
 
 100 00 
 
 Due April 1, x 140 
 July 10, 100 X 200 
 Sept. 28, 180 X 160 
 
 >> 
 
 )) 
 
 OPERATION. 
 
 Due April 11, 10 x (50 
 
 20000 „ July 10, 100 x 140 
 
 28800 „ Aug. 29, 150 x 100 
 
 >5 
 
 OoO 
 14(100 
 150(10 
 
 500 48800 
 
 300 29G00 
 
 19200 
 
 300 
 200 = 9G da. 
 
 290OO 
 
 Balances, 
 
 200 19200 April 1 + 96 days = July 6, Ans. 
 
 We select, for conveuience, April 1, the earliest date at which any 
 of the items of accouut is due, as the poiut of reckouiug, and find tlu' 
 aggregate of the terms of credit of the debit items, witli reference to 
 the selected date, to be equal to the credit of $1 for 48800 days ; and 
 the aggregate of the terms of credit of the credit items to be equal to 
 the credit of $1 for 29600 days. 
 
 Striking the balance, it appears that at the selected date, $200 sub- 
 ject to a credit equal to the credit of $1 for 19200 days, is against 
 Franklin Fuller. But the credit of $1 for 19200 days is eipial to that 
 of $200 for ^l^ of 19200 = 96 days ; hence, the |200 dollars is not 
 due in equity till 96 days after April 1, or till July 6. 
 
 If, however, the balance of items and ot terms of credit had been 
 on different sides of the account, the balance of items would have 
 been due before, instead of after, the selected date. 
 
 Rule. Select the earliest date at Schick any of the items oj 
 account become due, and therefrom reclcon the terms of credit. 
 
 Multiply each term of credit hy the number denoting the corre- 
 sponding item, and divide the balance of the sums of the ftroducts 
 hy the balance of the sums of the items of the account. 
 
 •1 
 

 Cr. 
 
 RATIO AND PROrORTlON. 
 
 LMT 
 
 The qnollcnt iclll be the ilmo, tchlch must he added to the, 
 f^'h'cfcd dak, icJien the tiro h<dnnces are on the same side of accnuiif, 
 lad- suBl-li ACTED .//Y^^/i the selected date, icheu the balances are on 
 (liferent sides, to obtain the equated time of the account. 
 
 SGOOO 
 140 00 
 100 00 
 
 (IDO 
 14(100 
 15000 
 
 20COO 
 
 Examples. 
 
 2. AVliat is tlie balance of the following account, and, allow- 
 ing each item to be on 30 days, when does it become due ? 
 
 Dr. 
 
 Robert Smith (^: Co. 
 
 C^ 
 
 r. 
 
 1866. 
 •Tune 30, 
 July 15, 
 
 To IMerchnndiso, ^550 00 
 „ ! 850 00 
 
 1806. ! ' I 
 
 July ], By Mercliaiidise, .' $400 00 
 
 July 5, 
 
 300 00 
 
 ylns. Balance 8700; due Aug. 1;>. 
 
 3. Required the date, at which a note, given for the balance 
 of the following account, should begin to draw interest. 
 
 Dr. 
 
 Strickland S: Hooper. 
 
 Cr. 
 
 ^. 
 
 By Cash, 1 1 .^700 00 
 
 I 
 
 4ns. Dec. 31, 18GG. 
 
 4. Required the fjice of a note which must be given for tlie 
 balance of the following account, and the date at which it 
 should begin to draw interest. 
 
 Dr. 
 
 J. F. Gould. 
 
 Cr. 
 
 1866. 
 May 16, 
 June 3, 
 July 1, 
 
 ToMclse. on 60 a.,: 
 
 55 55 55 00 ,, 
 55 55 55 30 ,, 
 
 $300 
 
 49 
 150 
 
 00 
 60 
 00 
 
 i 1866. 
 1 Miiy 20, 
 July 19, 
 
 i 
 
 By ]\I(lse. on 3') d., 
 
 55 55 55 00 ,, 
 
 8200 00 
 , 200 00 
 
 1 
 
 1 
 
 Ans. Face of note 809. GO ; on interest from June 2. 
 
 What is the Kule ? 
 
243 
 
 PRACTICAL ARITHMETIC. 
 
 SETTLEMENT OF ACCOUNTS. 
 
 347. Merchandise Balance is the balance of the items witli- 
 out interest. 
 
 348. Interest Balance is tlie balance of the interest of the 
 items of tlie two sides of an account. 
 
 349. Cash or Net Balance is the balance when the mer- 
 cliandise and iiitei-est balances have been added to the proper 
 sides of the account. 
 
 350. Tlie Settlement of an account is ascertaining the 
 balance at any specified time. 
 
 351. Since the merchandise balance is understood to be 
 subject to interest from the date of its being due (Art. 344), it 
 follows, that 
 
 The cash or net balance, at any date subsequent to the equated 
 time, may be found by adding to the merchandise balance its 
 interest up to date ; or at any date in-evious to the equated time, 
 by SUBTRACTING fwm the merchandise balance its interest for the 
 time intervening. 
 
 INTEREST METHOD. 
 
 352. The cash balance of an account drawing interest may 
 be determined by means of the interest on the items of the 
 two sides. 
 
 1. Let it be required, on January 1, 1867, to find the cash 
 balance of the following account ; and also the equated time, 
 allowing each item to draw interest from date, at G per cent. 
 
 Dr. P. T. Montgomery & Co. Cr. 
 
 1806. 
 
 
 
 1806. 
 
 
 1 
 
 N(.v. 2, 
 
 To Merchandise, 
 
 §60000 
 
 Nov. 17, 
 
 By Cash, 
 
 $800 00 
 
 Dec. 2, 
 
 » >, 
 
 700 00 
 
 Dec. 2, 
 
 1807. 
 
 >» >» 
 
 200 00 
 
 1 
 
 „ 17, 
 
 >, »» 
 
 1000 00 
 
 1 
 
 ' Jan, 1, 
 
 i» »» 
 
 100 00 
 
 1 
 
 What is Merchandise Bah\iice ? Interest Bahinoe ? Cash or Net Balance ? 
 The Settlement of an Account? How may the cash or net balance be found? 
 
Cr. 
 
 10000 
 
 >■' 
 
 R\TIO AND PROPOIITION. 
 
 2:9 
 
 
 OPERATION'. 
 
 Int. on Sf)0O for 00 dii. 
 
 = $0.00 
 
 Int. on $H00 for 45 da. ^ .SO.oo 
 
 „ 700 „ 30 „ 
 
 = 3.50 
 
 „ 200 „ 30 „ = 1.00 
 
 „ 1000 „ 15 „ 
 
 = 2.50 
 
 100 „ „ = 0.00 
 
 $2300 
 
 $12.00 
 
 $1100 $7.00 
 
 1100 
 
 7.00 
 
 
 ^UUv. hill $1200 Int. bal. $5.00 
 
 Cash bal. = $1200 + $5 = $1205, Ajx. 
 
 CompiitinLj the interest on each of these items, from the time of 
 becoming clue to the given date, January 1, we find th(i bahince of 
 the interest to be $5, and of the merchandise $1200, and both on the 
 del lit side ; hence, the cash balance, January 1, ltt(J7, is $1200 + $5, 
 or $1205. 
 
 KuLE. Comimte the interest of each item for the time inter- 
 vening between its being clue and the time of settlement. 
 
 Find the sum of the interest on each side, and also the sum 
 of the items ; and if the interest balance and the merchandise 
 balance fall on the same side of the account, the cash balance is 
 their SUM ; but if on different sides, it is their diffehence. 
 
 When an item is not due till after settlement, its interest must be 
 subtracted from the interest of its side, or added to that on the other 
 sid 
 
 e. 
 
 >c 
 
 Examples. 
 
 /v 2. E. Holmes owes C. Simmons $G00 due in GO days from 
 September 1, and ^200 due in 30 days from December 2 ; and 
 Simmons owes Holmes $700 due in 30 days from August 3 ; 
 how much balance ought Holmes in equity to pay, should tln^ 
 account be closed on January 1, the rate of interest being 
 7%? Ans. $00.77. 
 
 3. I owe James Conant the following bills : May 10, for 
 merchandise, on 60 days, $300 ; June 3, for flour, on GO days, 
 $50 ; July 1, for labour, on 30 days, $150 ; and he owes me a 
 bill dated May 30, on 30 days, of $300, and another dated 
 July 19, on GO days, of $200 ; required the cash balance I owe 
 him on September 1, the rate of interest being G %. Ans. $.78. 
 
 Exi)laui the oiieratiou. Repeat the llulu. 
 
 o.y^^ 
 
A . 
 
 2.j0 
 
 niACTlCAL AianiMETIC. 
 
 n 
 
 TAXES. 
 
 353. A Tax is a sum of moiu'y a.sso.ss('<l upon a pcrisuu, or 
 upon property, for governinont or public use. 
 
 A Poll Tax is a sum assessed upon each male citizen of a 
 (M'rtain ago, without regard to property. 
 
 354. Real Estate is immovable property, such as laiid.^, 
 houses, t^c. 
 
 355. Personal Property is movable property, such as money, 
 stocks, cattle, &c. 
 
 356. Assessors are officers appointed to take an inventory 
 of taxable property, and a list of taxable polls, and to appor- 
 tion to each person the tax to be raised. 
 
 357. To assess, or apportion, a town or other tax. 
 
 1. The tovrn and county tax of ^liddlebury, for 1800, was 
 $12200. The real estate of the town is valued at $1040000, 
 and the personal property at $560000. There are 500 polls, 
 each taxed $2.00. How many mills is the tax on a dollar ; 
 and Avhat is A's tax, who has $5000 of real estate and $1500 
 of personal property, and pays 1 poll tax ? 
 
 OPERATION. 
 
 $2 X 500 =^ $1000, amount of poll taxes. 
 
 $12200 - $1000 = $11200, amount of property tax. 
 
 $1040000 + $500000 = $1000000, amt. of taxable property. 
 
 $11200 -f $1000000 = 7 mills, tax on $1 of property. 
 
 $5000 + $1500 = $0500, A's taxable property. 
 
 $0500 X .007 = $45.50, A's property tax. 
 
 $45.50 + $2.00 = $47.50, A's entire tax. Hence, 
 
 The tax on one poll, multiplied by the number of polls, will 
 give the amount of poll taxes. 
 
 What i3 a Tax? A Poll Tjix? Real Estate? Personal Property? 
 Assessors? 
 
 i 
 
TAXKS. 
 
 2.-) 
 
 The amount of poll taxes sul/racffd from the whole amount of 
 tn,c to he apportioned, wilt give the amount (f propertij tax. 
 
 The amount of propcrtij tax divided Inj the u-hole taxable pivpcrti/, 
 xvill denote the tax on one dollar of proper t>/. 
 
 Each individaaVs taxable ptropertii mulliplieil b>/ the number 
 denoting the tax on one dollar , with hid poll tax added, if ani/, 
 ivill give his tax. 
 
 Having (leterniinetl tlie number of mills the tax of property is on 
 $1, assessors usually faeilitate computation, Ly fiudiii,^' the tax on 
 §2, §3, &o., and arranging the same for use in a table. Suppose the 
 tax on $1 is 16 nulls, then we could have the following 
 
 Table, 
 
 SHOWING TAXES AT THE RATE OP 16 MILL? ON §1. 
 
 61 
 
 pays 
 
 $.01G 
 
 611 
 
 pays 
 
 6.17G 
 
 62;") 
 
 ])ay.< 
 
 i 6. 10 
 
 
 j> 
 
 .032 
 
 12 
 
 
 .192 
 
 35 
 
 >> 
 
 .56 
 
 3 
 
 j> 
 
 .048 
 
 13 
 
 
 .208 
 
 45 
 
 »j 
 
 .72 
 
 4 
 
 >> 
 
 .(»G4 
 
 14 
 
 
 .224 
 
 55 
 
 ?5 
 
 .88 
 
 5 
 
 j> 
 
 .080 
 
 15 
 
 
 .240 
 
 G5 
 
 J? 
 
 1.04 
 
 G 
 
 5> 
 
 .09G 
 
 IG 
 
 
 .256 
 
 75 
 
 5> 
 
 1.20 
 
 7 
 
 >J 
 
 .112 
 
 17 
 
 
 .272 
 
 85 
 
 »J 
 
 1.3G 
 
 8 
 
 )> 
 
 .128 
 
 IS 
 
 
 .288 
 
 95 
 
 • 1 
 
 1.52 
 
 9 
 
 >> 
 
 .144 
 
 10 
 
 
 .304 
 
 100 
 
 >5 
 
 l.GO 
 
 10 
 
 M 
 
 • IGO 
 
 20 
 
 
 .320 
 
 1000 
 
 .. 
 
 IG.OO 
 
 2. Find by the Table A's tax, his valuation being 61950, 
 and he paying for 2 polls, at $1,50 each. 
 
 orERATiON. "^^^^ ^^^ ^^^ ^^^ ^^ '-•'^^^"^' '^"^^ removing 
 
 .^1900 pays *^30 40 ^^^^ ^lecimal point two phices to the right, 
 
 '" 50 '' HO ^^^ ^^'^^'^ 630.40, the tax on .^lOOU. 
 
 " '- — The tax on §5 is $.08, and removing the 
 
 Jj«1950 „ 631.20 decimal point one place to the right, we 
 
 2 poll taxes, 3.00 have $.80, the tax on .s5(). 
 
 A's tax $34.20 '^^^^ ^^^ °^^ ^ 1^'^^^''^' ^^ ''^^•^*^ ^''^^^^' '"^ 83.00. 
 
 Adding the results, we have A's tax on 
 61900 + $50, or $1950, and for 2 polls, $34.20. 
 
 How do we find the amount of poll taxes? The tax on one dollar? Each 
 individual's whole tax? Explain the operation. 
 
 ii 
 
252 
 
 niACTICAL ARITHMETIC. 
 
 DUTIES. 
 
 358. Duties arc taxes or imposts levied by government npon 
 Commodities. 
 
 359. Excise Duties are imposts ui)on articles produced, and 
 used or consumod in tiio country, and also on licences. 
 
 A Special Tax is a duty which is payable without regard 
 to th(! amount of an individual's income. 
 
 Stamps are marks to bo attached to articles, as evidence 
 that the duty upon them is paid. 
 
 Internal Eevenue is the income of the government derived 
 from excise duties, special taxes, stamps, &c. 
 
 CUSTOMS. 
 
 360. Customs are duties upon imports, and upon the ton- 
 nage of vessels. 
 
 361. An Invoice is a bill of merchandise, from the seller to 
 the importer, iiiving the price and quantity of the goods. 
 
 362. An Ad Valorem Duty is a certain per cent, upon tho 
 value of an .'Licle estimated fmrn the invoice. 
 
 363. .f\ Specific Duty is a certain sum upon an article, with- 
 out regard to its value. 
 
 364. Tare is an allowance made for the weight of the cask, 
 ])ox, bag, Szc, containing the goods. 
 
 "When the tare cannot be satisfactorily deterniined from the invoice, 
 it may be estimated by actual weighing, or, on some articles, by u 
 schedule furnished by the treasr.ry department. 
 
 Leakage is an allov/ance for waste of liquors in casks, and 
 Breakage is an allowance or iiquci's in bottles. 
 
 Gross AVeighi" is the weight before any allowances are 
 made, and Net ^y eight is the weight of the goods alone. 
 
 Wliat are Duties? Excise Duties? Stamps? Internal Revenue? 
 Customs? An Invoice? An Ad Valorem Duty? A Specific Duty? Tare? 
 
 Leakage ? Gross Weight ? 
 
STOCKS. 
 
 253 
 
 lent upon 
 need, and 
 it regard 
 evidence 
 t derived 
 
 the ton- 
 seller to 
 ids. 
 
 upon the 
 
 le, witli- 
 
 ;lie cask, 
 
 2 invoice, 
 iles, by a 
 
 sks, and 
 
 ices are 
 ine. 
 
 Revenue ? 
 r? Tare? 
 
 Exercises. 
 
 1. Ilolderman Si Co. have imported 5000 weight of raisins 
 in (jaarter boxes ; what is the duty, allowing for tare 29 %, 
 and the rate of duty to be 5 cents a pound ? 
 
 Solution. 5000 x .2!) = 1450 ; 5000 - )^i50 -r. 3550 ; 3550 x 
 .05 = $177.50 diitf/, Ans. 
 
 ^ 2. What is the amount of ad valorem duty at 30 % on goods 
 invoiced at $5G00 ? y^r>>-^ J (fiU^ * 00 ^ 
 /" 3. I have imported 200 bags of liio coffee, at 25 kilograms 
 each; if the tare is 2 %, and the duty 5 cents a pound, how 
 much will be the duty] Ans. 8540.13. 
 
 s^ 4. A merchant has imported GOOO pounds of cassia in mats , 
 {dlowing 9 % tare, and the rate of duty to be 20 cents a pound, 
 how much duty will he be required to pay? Ans. 81092. 
 
 "^ 5. What is the duty, at 20 % ad valorem, upon 5 tons (of 
 2240 pounds) of steel invoiced at 22 cents a pound ? =. ^J-j^'fOjf 
 
 STOCKS. 
 
 365. A Corporation is an incorporated body, or company, 
 authorised by law to act and to be considered as an individual. 
 
 366. A Share is one of the equal parts into which the pro- 
 perty of a corporation is divided. 
 
 367. Bonds are obligations securing the payment of a cer- 
 tain sum of money at a specified time. 
 
 The principal bonds of the United Status are, — 
 (5's of '81, payable in 1881, bearing interest at 6 %, in gold. 
 5's of '81, payable in 1881, bearing interest at 5 %, in gold. 
 5-20's, redeemable after 5 years, payable after 20 years, bearing 
 interest at 6 %, in gold. 
 
 What is a Corporation ? A Share ? Bonds ? 
 
 M 
 
254 
 
 PRACTICAL ARITIIMKTIC. 
 
 \i 
 
 ]0-4(V.s, redeciiialilo ai'ier 10 yi-'ar.s, payable after 20 years, bcariii^^ 
 interest at 5 %, in j^oM. 
 
 4V.S of 1S8G, payable after 1886, bearing interest at U %, in ,l:uM. 
 4's of 1901, payable after U)01, bearing interest at 4 %, in gol<L 
 
 368. Stocks is a general term api)lical)lc to the bonds of 
 governments, and Uie bonds and shares of incorporated com- 
 panies. 
 
 Stoi^ks are at imr^ or at tlieir face value, when quoted at in(»; 
 ahove par when quoted at more than 100 ; and helow par when 
 ({uoted at less than loo. Thus, 
 
 A stock quoted at 00, is at dii % (jf its face value. 
 
 A Coupon is an interest certificate attached to a bond. 
 
 BroIv'!ia(;e for buying or selling stock is reckoned ou the par 
 value of the stock, and usual ly at the rate of ^ to ^ %. 
 
 11 
 
 \ 
 
 ' t 
 
 Exercises. 
 
 1. How much must be paid for 85000, U. S. 10-40's, at 110, 
 and brokerage at I yj A ns. ^55 12.50.- 
 
 2. "What amount of U. S. 10-40's, at 110 and brokerage, can 
 be bought for 85512.50? 
 
 3. How many 8100 U. S. 4 per cent, bonds, at 91, can be 
 
 bought with 854G0 ? 
 
 Ans. GO. 
 
 4. HoAv much is tlic yearly income, in currency, from 820000 
 in U. 8. 5's, when gold is at 105| ? Ans. $1057.50. 
 
 5. A\'liat rate of interest will an investment in a G % manu- 
 facturing stock, at 80, yield? Ans. 7.^ %. 
 
 G. At what rat(^ must a 10 % bank stock be bought, to pay 
 8 % on the investment? Ans. 125. 
 
 7. At what rate must a 7 % stock be bought, to pay G % on 
 the investment? 
 
 8. "What sum must be invested in an 8 % railroad stock, at 
 112, to yield a senu-annual income of 8G00 I Ans. $1GSOO. 
 
 Whr.t arc Stocks? "When iiro Stocks at par? AVhen below par? When 
 above par? What is a Cnupoii? On wliat is the Brokenigo of stocks 
 reckoned ? 
 
i, in 1:1 'M. 
 in gold. 
 
 bonds of 
 ited com- 
 
 .1 at 10!' ; 
 par wlicu 
 
 n the WW 
 
 s, at 110, 
 ^5512.50.- 
 
 }Vi\<xo, can 
 
 , can be 
 Am. GO. 
 
 82000(» 
 1057.50. 
 
 nianu- 
 
 ^^^- T^i %. 
 ;, to pay 
 Ins. 125. 
 G % on 
 
 stock, at 
 $1GSOO. 
 
 ? When 
 
 of stocks 
 
 1;XC1IANGE. 
 
 255 
 
 EXCHANGE. 
 
 369. Exchange is the method of remitting money from ono 
 place to another, by drafts, or written orders. 
 
 370. A Bill of Exchange, or Draft, is a written order 
 directing one person to pay money to anotlicr, or to Ids order. 
 
 The Drawfu is the perj^eii who sii^nis the hill, ov dr;it't : the 
 Drawee is tlie person directed to j>,iy tin; suine ; tin; I'avki: i> the 
 perstiH to wlioni the money is directed to l)e ]k\u\ ; the Indouser is 
 the person who transfers his rii^'ht to a l)ill, or draft, hy indorsing it ; 
 and the Holder is the person who has legal })ossession of it. 
 
 371. Tl'.e Indorsement of a bill, or draft, is done by the 
 payee writing his name upon its ))ack. 
 
 372. The Acceptance of a bill, or draft, i.- done by the 
 drawee writing bis name after the word '"Accepted" acro.-^s 
 its face. 
 
 The drawer, accept'^r, and each indorser of a liill, ov diaft, nv--. 
 liable ibr its jiaynient. If the drawee decliius to pay or aeetpt a 
 hill, or an uceeptor to make payment, it is customary for tin- Judder 
 to employ a Notary Public to give a notice called a Protect to ilu- 
 drawer and each of the indorsers, so as to hold them legdly for the 
 l^aynieiit. 
 
 llirec da)js of r/race are usually allowed on hills and drafts, after 
 the time specilied has expired. 
 
 373. The Par of Exchange is the comparative value of the 
 money of two jdaces. Hence, 
 
 Exchange is said to be at ^w?", when a bill sells ibr its face; 
 Sit a. 2>nmium, ov ((hove jmr, Vi'hcn i'ov more than its iace ; and 
 at a discount, or hclow jmt, when for less. 
 
 374. The Rate of Exchange is a rate per cent, of the face 
 of a bill. 
 
 375. The Course of Exchange is the par of exchange as 
 aflected l)y the rate of exchange. 
 
 What is Exchange? A Bill of Exchange? ] Tow is the Imlorscnu-'iit of a 
 bill (lone? The Acceptance? Who are liable fen- the iiaynieiit of a liijl or 
 draft? How many tliiys of grace are aUoweil? Tlie l*ur of Exchange ? The 
 Hate of Exchange? The Course of Exchange? 
 
 i' 
 
256 
 
 PRACTICAL ARITHMETIC. 
 
 DOMESTIC OR INLAND EXCHANGE. 
 
 376. Domestic or Inland Exchange is between drawer and 
 drawee in tlie same country. 
 
 I 'J 
 
 CASE I. 
 
 377. To find the cost of an inland bill or draft. 
 
 1. What is the cost of tlie following draft, at l\ % dis- 
 count ? 
 
 'Sv/ ^fa/i/j /ta?^ /o fj/ Ao-nioj <5^. C//t(f '000(1 , oi ou/ti, (!/ /te 
 kJ^ /io<«in>u/ ,^Lc^aiCat<S, aru/ ytcace /o /Ae accotoi/ oJ: 
 
 c/uiian lyc^cfocctoc/i. 
 
 ^y/ jz-ou-t^j. 
 
 At 1^ % discount, 
 
 OPERATION. the cost of $1 of ex- 
 
 $1 X .98.5 = $.985, cost of $1. clumge will he $.985, 
 
 $.98;5 X 1000 ^-.-- $985, „ draft, and Cf $100o, looo 
 
 times $.985, or $985. 
 
 2. AVliat must lie paid in Cleveland for a draft of $2000 on 
 Philadelphia, at 30 days, when exchange is at 2 % premium ? 
 
 OPERATION. ^^ 2 % prc- 
 
 $1 X 1.02- $1.02, cost of $1 at sight. miuni,theco.stof 
 
 $1.02 X 2000 =-- $2040, „ $2000 „ ^] "^ ^"^f 
 $2000 X .005.5 = $11, bank dis. for 33 da. ^102 
 
 ""TTT"" , P 1 p.. If the cost of 
 
 $2029, cost of draft. ^, ^ •,. • 
 
 ' $1 at sight i.s 
 
 $1.02, the cost of $2000 will be 2000 times $1.02, or $2040 ; and 
 
 since the bank discoimt on the face of the draft, for the given rate 
 
 and time, is $11, the cost of the draft umst be $2040 -$11, or $2029. 
 
 liULE. 7/ ike hill is at sight, mnliijihj the cost of $1 of exchaiuje 
 hi/ the numher demtiiuj the face of the bill. 
 
 ■\Vhat is Domestic or Inland Exchange? Repeat the Rule. 
 
:tmi «?^ij«wiomfc-/w?">iw 
 
 EXCHANGE. 
 
 257 
 
 rawer and 
 
 t. 
 
 U % dis- 
 
 \ discount, 
 $1 of ex- 
 l be $.985, 
 00(», 10(10 
 , or $985. 
 
 $2000 on 
 remium \ 
 
 2 % pru- 
 I, the cost of 
 f exchanLTe 
 jlit will be 
 
 the cost of 
 t sight is 
 2040 ; and 
 given rate 
 , or $2029. 
 
 f exchaiuje 
 
 lie. 
 
 // the hill is payahk after si^jkt, reckon the hank dhrount of the 
 /■■we of the draft for the time and i\ree dai/s' grace, and deduct it 
 Jrom the irroduct. 
 
 Exam] les. 
 
 o. What must be paid in Detroit, where the rate of interest 
 IS 7 %, lor a draft of $500, at GO days, on Nashville, at ^ 7 
 ^^^^^"^"^^^ Ans.^m.'ilt 
 
 ^ 4. Wiiat must be paid at Mobile for a draft of $1940, on 
 Truvidence, exchange being at 1] % premium ? An.-^. .^19G-l'.L>5. 
 5. What is the cost of a draft of 1^020, for 90 days at 8 7, 
 exchange being at a discount of { % ? 
 
 G. A merchant in Portland wishes to discharge a de])t of 
 eoOOO in Xew Orleans : required the cost of a draft for that 
 sum, for 2 months, at G %, (exchange being at 1 % premium. 
 
 Ans, $299«.50. 
 CASE II. 
 378. To find the face of an inlmd bill or draft. 
 \. What is the face of a draft on Cincinnati for GO days, at 
 1 % premium, which can be bought for $1000 ? 
 
 OPERATION. At 1 :: preniiiini, 
 
 $1 + aOl = $1.01, cost of $1 at sight. 81 of exchange for 
 $1 X .0105=: .0105, bank dis. for G3 da. <'*^ '^'ivs can be 
 
 bouglit for $.9995. 
 
 8.9995, cost of 81 at GO da. If 81 of excliange 
 
 can Ix; Ijoiight for 
 ^.9995, as many dollars of excl:ange can be bought for §1000, us 
 ^.9995 are contained thnes in §1000, or $1000.50. 
 
 liULE. Divide the given sum hy the cost r/ 81 of t.rchange. 
 
 Examples. 
 
 2. Required the face of a draft that may be purchased for 
 i075, excliange being at 11 % premium. Jns. 8G()00. 
 
 §t 
 
 i 
 
 ft . 
 
 li'l 
 
 'Ml 
 
 m\ 
 
 Kxplain the operation, lloix.'iit the Kule of Case II. 
 
 r. 
 
' ! 
 
 258 
 
 PRACTICAL ARITHMETIC. 
 
 1 1 1 1<» ■■> 
 
 3. I wish to romit to a mercliaiit in Cluirlostoii §19t90; 
 wlint is tlie face of a draft for that amount for 30 clays, at 2 % 
 discount ? Aiis. $20000. 
 
 FOREIGN EXCHANGE. 
 
 379. Foreign Exchange is between drawer and drawee in 
 difT«'rcnt countries. 
 
 Foreign ])ills are usually drawn in sets, and to prevent loss or 
 delay each bill of the set is remitted in a different manner, and 
 when one of the set has been paid, the others become worthless. 
 
 380. The Intrinsic Par of Exchange between countries is 
 the mint value of tlieir coins. 
 
 381. The Legal Value of some foreign coins, as fixed by 
 present laws of the United States, is shown in the following 
 
 Table. 
 
 Places. 
 
 Oreiit Britain, 
 
 France, 
 
 Be1;,nuTn, 
 
 Turin, Genoa, 
 
 Anisterda-ni, Hague, 
 
 llussia, 
 
 Havana, 
 
 Spain, 
 
 Oon.'tnntinople, 
 
 Portugal, 
 
 O.intoTi, 
 
 Donomiuations of Monoy. 
 
 Valtie. 
 
 1 pound (£) = 20 shillings, or 10 florins = 
 
 1 franc = 100 centimes = 
 
 1 franc = 100 centimes = 
 
 1 lira = 100 centesimi = 
 
 1 tiorin or guilder = 100 centa = 
 
 1 silver ruble = 100 coi)ecks = 
 
 1 dollar — 8 real plate = 110 real \ellon = 
 
 1 real plate = 2 real vellon = 
 
 1 piaster = 100 aspers = 
 
 1 millrea - lt'()0 reas = 
 
 1 tael = 10 inace = 100 candarins — 1000 c;\sli — 
 
 $4.84 
 
 .LSi; 
 
 .18C> 
 
 .18(J 
 
 .40 
 
 .75 
 
 LOi> 
 
 .10 
 
 .0.'> 
 
 LTJ 
 
 1.48 
 
 382. British money is usually expressed in pounds, shillings, 
 pence, and farthings ; but, by means of the new coin, tlie 
 florin, the decimal system of money-reckoning has been com- 
 menced, and is made use of by the Bank of England, and by 
 
 other large business establishments. 
 
 What is Foreign Exchange? The Intrinsic Par of Exchange? The Legal 
 Valui? What is the value of a pound? A franc? A lira of Turin? A 
 florin of Amsterdam? A silver nihle? A dollar of Havana? A real plate 
 of Spain? A piaster of Constantinople? A millrea? A tael of Canton? 
 How is Britisli money usually expressed? 
 
n $;i0190; 
 ays, at 2 % 
 ^5. $20000. 
 
 drawee in 
 
 vent loss or 
 lanner, and 
 ) worthless. 
 
 lountries is 
 
 as fixed l)y 
 i'ollowing 
 
 I Value. 
 
 isli - 
 
 S4.84 
 .18(5 
 
 .18C> 
 
 .40 
 
 l.Oii 
 
 .10 
 
 .05 
 
 1.12 
 
 1.4S 
 
 s, sliilliniTs, 
 
 i coin, the 
 
 been coni- 
 
 nd, and by 
 
 ? The Legal 
 i)f Turin? A 
 
 A rciil jilata 
 ;l of Cantou? 
 
 EXCHANGE. 
 
 2o,) 
 
 Of British or Eii,i,'li.sli currency, 4 farthings (fur.) make 1 pcnnv" 
 (d.), 12 pence 1 shilling (s.), and 20 shillings Ipound ; also, 2 shil- 
 ling.s 1 llorin (11.), 10 florins 1 pound, or sovereign. 
 
 A shilling may he written 5 tenths of a llorin, or as 5 hundredths 
 of a pound, and 6 pence as 25 hundredths of a florin, or 25 thou- 
 sandths of a pound. Thus, 
 
 £3, 7 fl. Is. 6d. may be expressed £3.775. 
 
 383. Exchange in this country on England is reckoned at a 
 certain per cent, on the former value of a £ sterling, whieli 
 was at the rate of £9 = $40, or £1 = $4.44-^-, instead of the 
 present value of $4.84. Hence, 
 
 The present commercial par of exchange is at about 9 per 
 cent, above the old, or nominal par. Thus, 
 The nominal par of £1. = |4.444 + 
 
 Add 9 per cent. = ..399 + 
 
 The nominal par at 9 per cent, premium ^ $ 4.843 + , or $4.84. 
 
 The intrinsic value of a new English sovereign of recent coina-'c is 
 $4.86, so that the intrinsic par of e.xchange on England is about 9? 
 per cent, premium on the old or nominal par. 
 
 384. To compute foreign exchange. 
 
 1. Required the cost of the following bill, at 9.^ per cent. 
 premium : — 
 
 M'^scco. 
 
 •//te accctni/ o/^ 
 
 
 How is cxcluuige in this country on England usually reckoned? What is 
 the present commercial par? Intrinsic value? 
 
2C0 
 
 PRACTICAL ARTTnMETIC. 
 
 i. .1 
 
 h.*'- 
 
 ' 
 
 ! 
 
 OPERATION. 
 
 £9 = $40 X 1.095, 
 140 X 1.095 
 
 £3000 = 
 
 9 
 
 $4.8G|. 
 
 At 9.} % prominm 
 
 the co.st of £0 will 
 
 be $40 X 1.09'), and 
 
 the cost of £1 will 
 
 , $40 X 1.005 
 
 be ^- y , or 
 
 i.8G^ X 3000 - $14000. 
 
 $4.80§. 
 
 If £1 costs $4.86.1, £3000 will cost $4.80.1^ x 3000, or $14600. 
 
 > 2. If J. P. Clay, of Baltimore, should purchase a bill of 
 £2200, at 8 % premium, what would it cost in United States 
 money? Ans. $10500. 
 
 '/ 3. How much must be paid in New York city for a bill on 
 Liverpool, for £1173, 5s., exchange being at 8i % premium ? 
 
 Ans. $5057.67. 
 
 y 4. What will be the cost, in Washington, of a bill on Havre 
 for 2570 francs, exchange being at 5.14 francs to a dollar] 
 Solution. 2570 -t- 5.14 = $500, Ans. 
 
 ! 5. What must be paid for a bill on Amsterdam for 2626 
 
 liorins, at 1 % premium? Ans. $1000.90. 
 
 ,* 0. If I pay $14000 for a bill on London, when exchange is 
 
 at 9-J- % premium, what is the face of the bill in English 
 
 money ? 
 
 At 9 J % premium, £1 can be 
 
 OPERATION. remitted for $4. 865. 
 
 £9 =» $40 X 1.095 If £1 can be remitted for 
 
 *4Q ^ $4.86fT, as many pounds can be 
 
 -^ X 1.095 = $4.86.^. remitted for $14001), as $4.80?^ is 
 
 £1 
 
 $14000 ~ $4,805- = 3000. contained tunes in $14000, or 
 
 3000. 
 
 7, At 5.H francs to a dollar, how many francs in Paris will 
 be required to remit $500 ] 
 
 8. When exchange is at 8^ % premium, what is the face, in 
 English money, of a bill on London, which can be bought for 
 ^505 7. 07? ^ws. £1173, 5s. 
 
 Keview Questions. What is Exchange? (309) A T.ill of Exchange? 
 (:rO) Tar of Excbfiniic ? (373) Course of Exchange? (375) . 
 
m*/!' ■ - - ■tutmt'mmmmms^ja^ 
 
 KtVIliW KXERCISF.S. 
 
 2G1 
 
 , premium 
 f £9 will 
 1.09"), and 
 if £1 will 
 
 1.005 
 
 4600. 
 
 or 
 
 a bill of 
 lid States 
 . $10500. 
 
 a Vjill on 
 mium ? 
 I5G57.67. 
 
 on Havre 
 ollar 1 
 
 for 2626 
 11060.90. 
 
 change is 
 I English 
 
 £1 can be 
 
 litted f(tr 
 
 Is can be 
 
 $4.8G;-i is 
 
 14600, or 
 
 Paris will 
 
 e face, in 
 )ught for 
 1173, 5s. 
 
 Exchange ? 
 
 
 REVIEW EXERCISES. 
 
 ' 1. What is the ratio of 6 weeks to three days ? Arts. 11. 
 • 2. Reduce the ratio 8 : 72 to its smalle.-st terms. Ajis. },. 
 
 K 3. If the antecetent of the ratio 2.11 .s 31.65, what is the 
 consequent 1 Ans. 15. 
 
 > 4. If the consequent of the ratio 2.11 is 15, what is th.^ 
 autecedent ? s=:^;^y * L A' 
 
 * 5. If the product of the means of a proportion is 2^, and one 
 
 of the extremes is I, what is the other? Am. \\. 
 
 6. Wliich is the greater ratio, 9 : 10 or 3 : 4 ? Am. 9 : 10. 
 
 \ 7. Required the .simple ratio equivalent to (2 : 3) x (5 : 7) 
 
 ^ (1 '■ '■)• Am. 10: 147. 
 
 Y^' II' a ship sails 155 miles in 12 hours, how far will it sail 
 
 in GO hours? Am. 775 miles. 
 
 v: 9. If 2 men can -build 803 rods of fence in %'l days, hou- 
 
 long will it take them to build 73 rods \ — ^^ ^r^y/ '^^An \^ 
 
 ^' 10. If a hound makes 27 leaps while a hare makes 25, and 
 
 their leaps are of equal length, how many lea^.s must the hound 
 
 make to overtake the hare, if the latter has 50 leaps the start < 
 
 Am. 675. 
 >^11. A, B, and C trade in company; A \)\\l in .$3000, II 
 $2000, and C $1000 ; they gained Ul % on the whole capital; 
 what was each partner's share of the gain \ 
 
 Am. A's $375, R's $250, and C's $125. 
 ^i2. If Hendricks, William, Arthur, and Frank should shart; 
 an estate of $54000, in the proportion of the numbers of tho 
 lett-^rs in their first names, how nmch will be the share of each / 
 Ai.H. Hendricks' $18000, William's $14000, Arthur's $12000, 
 Frank's $10000. 
 
 ^ . 13. If 3i yards of carpeting f of a yard wide cost $7|, what 
 should Of yards |- of a yard wide cost? Ans. $24.78 + . 
 
 llEViEW QuEsiTluNS. What U Eauatiou of rayuieuts? (3^8) Etiuated 
 Time? (339; 
 
 ll 
 
'W 
 
 2G2 
 
 PRACTICAL ARITHMETIC. 
 
 J*^ 14. A Commenced business January 1, with a capital uf 
 
 SOOOO ; April 1, he is joined by B with a capital of $10000 ; at 
 
 the expiration of the year they had gained $1320 ; what was 
 
 each partner's share of the gain 1 Am. A's $720, B's $G00. 
 
 jy 15. What is the equated time of paying 83000, if 8500 of 
 
 it is payable in 4 months, $1000 in eight months, and the 
 
 remainder in IG months? Ans. 11 mo. 10 da. 
 
 IG. In 18GG, Samuel Ashton charged me for cash May 15, 
 
 ^ ^$400, and Nov. 2, $1000 ; and I charged liim for cash, April 3, 
 
 $800, and August 31, $900 ; allowing interest at G %, what 
 
 will be the balance due me January 1, 18G7 ? An;^. $329.45. 
 
 . 17. A certain corporation has laid a tax of $1500 upon its 
 
 fj capital stock of $75000 ; how much will A, who owns $1200 
 
 of stock, be required to pay ? -J^{'>^^^- •'^.^*'^- DC 
 
 \ 18. What is the amount of duty, at 30 % ad valorem, on an 
 
 ^ importation of woollens, invoiced 500 yards at 15s. per yard, 
 
 allowing $4.84 as the value of a pound sterling ? Ans. $544.50. 
 
 '\ 19. When exchange is at 9^ % premium, what is the face 
 
 of a bill on London that can be bought in Cincinnati for 
 
 $2182.94. Ans. £448, lis. 
 
 Exercises in Analysis. 
 
 <^ 1. If $120 will purchase 4^ acres of land, how many acres 
 will $480 purchase ? ^'7f^. /f^dMi 
 
 Solution. $480 art 4 times $120 ; hence, i7$120 will 'purchase 4.\ 
 acres, $480 will purchase 4 times 4^ acrss, or 18 acres. Therefore, <i'C. 
 
 '^ 2. If the income from 49 shares in a certain mill is $735, 
 
 how much will it be from 7 si>ares ? = f]j / ^b' ^0 
 
 ^ 3. A and B trade in company. A paid in G times as much 
 
 as B, and they gain $1974 ; what was each one's share of the 
 
 gain ? Ans. B's $282, A's $1G92. 
 
 - 4. A, B, and C trade in company ; A put in ^ of the stock. 
 
 Review Questions. What is the Rule for finding the equated time when 
 the terms of credit begin at the same date? (341) When the terms of credit 
 begin at different dates ? (342) 
 
ICEVIEW J-XiillCIiSES. 
 
 L>G3 
 
 pital of 
 (000 ; at 
 hat was 
 I's .fGOO. 
 8500 of 
 and the 
 3. 10 da. 
 May 15, 
 , April 3, 
 %, what 
 $329.45. 
 upon its 
 IS 81200 
 
 !m, on an 
 per yard, 
 $544.50. 
 the face 
 inati for 
 448, lis. 
 
 my acres 
 
 iirchase 4J 
 refore, d'C. 
 
 I is $735, 
 
 as much 
 ,re of the 
 .'s $1692. 
 \\Q stock, 
 
 time when 
 ns of credit 
 
 B \ of it, and C the rest ; on divi<liiii(, C's share of the gain 
 was $495 ; what was tlie gain of each of the other partners I 
 
 Solution. If A put in } of the dock, B ^, and (J the rat, C mud 
 have put 171 1 — {} f 1), or ,'{j, of the dock. 
 
 If {^Q of the dock (jaiii $4!)'), ^ of the dock must gain ] of^^Uo, or 
 
 If ^^f of the stock f/ain $10"), since A put in J, or f^, of the stock, liis 
 share of the gain must he 2 times $l(io, or 8;j:i(); and xince D put in .\, 
 or ,'g, of the stock, his share of the yain must be 5 times §105, or ^^-IC), 
 
 I'henfore, d'c. 
 
 ^ 5. A ship worth $G0200 was entirely lost, | of her belonging 
 to A, \ to Jj, and the rest to C; what loss will each sustain, if 
 $35000 of her was insured ? 
 
 Aa^. A $3150, 13 $G300, and C $15750. 
 
 -^ G. A merchant owes a certain sum, \ of which is due iti 2 
 months, ^ in 4 months, \ in 5 months, and the balance in G 
 months ; what is the average term of credit \ 
 
 Ans. 4 months 5 days. 
 
 7. A borrowed $2000 for 6 months ; 4 months before it was 
 due he paid $1000, and 2 months before it was due he jiaid 
 $G00 ; how long after the expiration of the G months may the 
 remaining $400 remain unpaid ? 
 
 Solution. x\ credit on $1000 /ar 4 months is cqiuvtdcnt to a credit 
 on $1 for 4000 montlts, and a credit on $GOO/c»r 2 months is c(pcit;alent 
 to a credit on ^\ for 1200 months. 
 
 J/ence the ivholc credit is equivalent to the credit of $1 for 400(> + 
 1200, or 5200, months; and tlie credit of $1 for 0200 mouths is 
 equivalent to the credit of ^-iVO for ^Jg of 5200, or 13 months. 
 
 Therefore, dc. 
 
 8. Sold, March 5, 18GG, p-oods to H. Mitchell amounting to 
 $1G00 on 6 months ; April 5, he paid $200, and on August 5, 
 $800. When in equity should the balance be paid ? 
 
 Ans. December 5, 18GG. 
 
 9. If I owe $2500 payable in 4 months, but to ac^ ^mmodate, 
 
 Review Questions. What is the liahiuce of an Account ? (344) Averag- 
 ing; of an Account ? (345) 
 
 
 
 I 
 
T 
 
 204 
 
 PRACTICAL AftlTDMliTlO. 
 
 !» 
 
 i» 
 
 pay $1500 down, how loii;:^' in ((jiiity sliould f 1)0 pf-rmitted to 
 keep the remainder ufU-r the expiration of the 4 months? 
 
 Av.'i. G montlis. 
 
 10. A iiarc starts 2.^ of its leaps in advance of a hound, and 
 
 takes 4 h}ai)S to tljo liound's 3 ; but 2 of tiie liound's leaps an; 
 
 rqual to 3 of the hare's; how many leaps must the liound take 
 
 to overtake the liarc ? 
 
 SoiiUTioN. 1/ 2 of the hound's leaps equal 3 of (he haro\t, 1 of (lie 
 honv'Ps is equal to \\ of the hare's, and If the hound takes 3 leai)s to 
 the Itare's 4, he takes 1 leap to the hare's \\. 
 
 If 1 of tlie hound's leaps is equal to ]}, of the hares, and he tai-*.^ 1 
 h'aj) to the hare's 1 J, he gains in takiwj 1 leap, 1 .^ - 1 \, or ,V of a harc.i 
 Imp. 
 
 If he gains ^ of a hare's leap in takiyuj 1 leap, he will gain Sf) h-api 
 of the hare, or ocertake the hare, in taking 'is many leaps as ^ is eon- 
 tained times in 25, u'hich are 15(.>. 
 
 Tlierefore, djc. 
 
 ^ 11. A tiiief 'laving gone 51 miles, an officer set out to over- 
 take hii7., and for IG miles travelled by the thief, the otlieer 
 travels 19 miles, llow far will the officer have travelled before 
 
 A.i^. 323 mil 
 
 es. 
 
 ^ the thief is overtaken \ 
 
 /^ 12, A starts from IJoston toward a town 10 miles distant, 
 walking at tlie rate of 2] miles an hour; an<l 2 hours after, W 
 starts from Boston upon the same route, by coach driven at the 
 rate of 9 miles an hour; in what time, and how far from Ltts- 
 ton, will B overtake A i 
 
 Ans. In 40 minutes, and G miles from Boston. 
 
 ■/^ 13. If 12 barrels of corn will pay for 10 cords of wood, and 
 48 cords of wood will pay for 8 tons of hay, how many barrels 
 of corn will pay for 15 tons of hay ? 
 
 Solution. If 8 tons of hai/ can he paid for h}j 48 cords of wood, 15 
 tuns can be paid for bij ^^ of 4S cords, or 9'> cords. 
 
 If 10 cords oj wood can he paid for by 12 barrels of corn, 90 cor Is 
 
 Keview Questions. What is the Rule for tinding the equated bnlnnce of 
 an account? (340) What is I\Iorchaudiao Baltiuoe? (347) liitcn.st Ual.UKX- .' 
 (;HS) Cash IJahmce ? (34U) 
 
UKVIKW KXHRCl.sns. 
 
 205 
 
 ) 
 
 
 nf u'ooJ, or lo tons of hay, can he j-iaid forhy tum'ii 12 f/anels,ov K'Ji 
 b'trnis. 
 
 IVicrejore, <N. 
 
 '^^ 14. If 10 calves am wortli as inucli as colts, and 00 colts 
 Am worth as much as 112 shoep, how many shcop an^ worth 
 lus much as 50 calves ? y//<.s-. .OO. 
 
 . 15. If 8 men can do the work of '.\'2 women, and 2 women 
 can do the work of o l)oys, how many men can do the work of 
 24 boys? y//w. 4 men. 
 
 ->. 10. If the r( lativc value of oak wood to spruce is as 2 to 1, 
 and that of spruce to pine as 7 to S, how many cords, com- 
 posed of spruce and pine in equal parts, will t'(pial 00 cords of 
 oak? y///.s'. 112 cords. 
 
 y 17. A, B, and C together can do a piece of work in 20 days, 
 A alone can do it in GO days, and I] alone can do it in 80 days, 
 in what time could C working alone do it? 
 
 Solution. If A, B, and C tojetlitr can do the work in 20 dajs, in 
 1 day the J/ can do i}^ of it. 
 
 If A ivorkiiKj alone can do the icork in GO days, in 1 daj/ lie can do 
 ^^^ of if, and if B working alone can do it in 80 days, in 1 day he ctn 
 do -Jfj ; hence A and B working together can do ^^, + ,,'„, or .J[-q, of ih-' 
 work in 1 da>/. 
 
 If A and B ivorkinrj together can do t,^,^ ^f ^^"^ work in 1 day, siucn 
 A, B, and C VJorkivg toijelhcr can do :^^^ of it in 1 daij, C must do the 
 difference between .}q and ^^^ or tt^q = -^V <fihe work in 1 da>/. 
 
 If C can do ^ (/ the work in I day, working alone, he can do (lie 
 vc/iole of it in an matii/ days as -^^ is contained times in 1, or in 48 dugs. 
 
 Therefore, dec. 
 
 X 18. A piece of work can be done in a day of 1 1^ hours by 2 
 
 or 12 boys ; in what time could it be done 
 
 men, or D 
 
 by 1 
 
 man, 
 
 women 
 2 woni 
 
 en, and 3 boys, working together? 
 
 Ans. In 10 hour: 
 
 IIEVIEW Questions. What i\re Taxes? (;r;3) A Poll Tax? (.35.3) How !>* 
 a town tax assessed or aitportiontil? (."-''r) What are Duties? (358) Interniil 
 Jleveuue? (301) Customs? (303) Exchange? (3GU) Doniostic Exchan^'- ? 
 (o76) Forei-n Exchange? (379) 
 
 tttjjttamt^ 
 
2G(j 
 
 PRACTICAL ARITilMKTIC. 
 
 ^ 19. A carpontor is ofrt-rod $325 to do a job of work, wliich 
 lio can do in \'2\ days, his journeyman in l^*! da3'.s, and liis 
 aj>[)n'ntice in 20 days. If tlicy slionM do it to,L<('tli<'r, in wliat 
 time conld it he completed, and how much would each earn I 
 
 ytns. In 5\?^ days; the carpenter 8150, the journey man 
 $100, and the apprentice 875. 
 
 I'i' 
 
 1 ' I 
 
 INVOLUTION. 
 
 385. A Power of a numher is either the number itself, (jr 
 tlie product obtained by taking the number >• /eral times as a 
 factor. Thus, 
 
 2* = 2, is the first power of 2, 
 2^ = 2 X 2 = 4, is the st cond power, or square of 2, 
 2^ = 2 X 2 X 2 = 8, is the third power, or cube of 2, 
 2* = 2x2x2x2==! G, is i\\v. fourth poAver of 2, 
 and so on, the exponent (Art. 103) of the power denoting the 
 number of times the number 2 is taken as a factor. 
 
 386. Involution is the process of raising a given number to 
 a required power. 
 
 This may be effected, as is evident from the definition of a 
 power, by tiie simple process of multiplying. That is, to 
 raise a number to any power, 
 
 Find the product of the numher taken as a factor as many times 
 as is denoted by the exponent of the required jpoiver. 
 
 Exercises. 
 Raise to tlie powers denoted by their respective exponents : 
 
 ,1. 
 
 252. 
 
 Ans. 625. 
 
 5. 
 
 (f)^. 
 
 Ans. -5 J. 
 
 
 
 222. 
 
 Ans. 484. 
 
 ^6. 
 
 (^D*. 
 
 Ans. 50>^. 
 
 ,3. 
 
 813. 
 
 Ans. 531441. 
 
 7. 
 
 1.23. 
 
 Ans. 1.728. 
 
 '4. 
 
 55. 
 
 Ans. 3125. 
 
 ■8. 
 
 .133. 
 
 Ans. .002197. 
 
 What is a Power ? What does the exponent of the power denote ? What 
 is Involution? How is a number raised to any power? 
 

 INVOLUTION. 
 
 
 
 2G7 
 
 9. Wliati.^ 
 10. Wluit b: 
 
 tlio sfjuarc of 78 ? 
 the cube of .00 ? 
 
 
 -•/ 
 
 .-tns. r.osi. 
 
 us. .()007'J1». 
 
 387. From 
 
 the definition of a power may 
 
 be 
 
 derived the 
 
 
 rULNX'irLES. 
 
 
 
 
 1. The product of two or more, powers of a 
 denoted hij the i>iniL of their e.rpoiwnts. 
 
 num 
 
 her 
 
 is tlifif power 
 
 For, since 2' 
 = 25. 
 
 X 22 = (2 X 2 X 2) X (2 X 2) 
 
 = 28 
 
 , 28 
 
 X 22 =: 2» + « 
 
 What 
 
 2. Any power of a nvmher raised to a power is that pau-,;- of iJf 
 number denoted by the product of the exponents. 
 
 For, since (2^)2 = (2 x 2 x 2) x (2 x 2 + 2) = 2", (2'';= = 2^'<2 = 2". 
 
 3, Any power of a number divided by a power of the same num- 
 ber is that power denoted by the dijierenee of their exponeuts. 
 
 For, since 2' -i- 2= = (2 x 2 x 2 x 2 x 2) -;- (2 x 2) = 2'', 2" ^ 2'' 
 = 2»-2 = 2^. 
 
 Ans. 21 
 ins. 7i'9. 
 
 
 Exercises. 
 
 \. What power of 2 is 2^ x 2- ? 
 
 - 2. What is the product of S*^ x 3«? 
 3. \ 'hat is the 7th power of 2 ? 
 
 r;oLi riON. 2' = 2* X 23 = 16 x 8 = 128, Ans. 
 
 A. ' Hiat is the 8th power of 5 ? An<i. 300025. 
 
 I. What power of 15 is 15- raised to tlie 3d power / 
 
 .Ins. l.V'. 
 
 6. Find the 2d power of 5^. Am. 15025. 
 
 7. What power of 17 is 17^ ^ \1^] Ans. \1\ 
 . 8. Find the quotient of 10^ - 10^ Ans. 100. 
 
 - 9. What is the 8th power of \ ] Ans. 15^-]^^. 
 10. Kequired the value of {i^f. Ans. 202144. 
 
 ,_11. Required the 10th poAver of 3. Ans. 59040. 
 
 What is the first I linciple? The second ? The thinl? 
 
 I 
 
la 
 
 wmemn 
 
 268 
 
 PRACTICAL AUITIIMliTIC. 
 
 I ) 
 
 EVOLUTION. 
 
 388. A Root of u nuiulxT is ouu uf the ticjual factors taken 
 to ibriii tlic iiuitiIht. 
 
 389. Tlw Second, (»r Square Root of a number is one of its 
 two equal factors. 'J'luis, 
 
 Tlu; square root of 25 ^- 5 ; since 5 x 5 -- 25. 
 
 The Third, or Cube Root of a number is one of its tlnvo 
 equal factors. Thus, 
 
 The culje root of 125 = 5 ; since 5 x 5 x 5 ^^ 125. 
 
 390. Tlu^ Radical Sign, v', or Fradhnial Ej'iionciii!>, are used 
 to denote roots. Thus, 
 
 V4, or 4-', denotes tlic second or Hfiuarc root of 4 ; 
 \/8, or 8''^, denotes the third or ciihc root of 8 ; 
 
 4 _ 1 
 
 \^l(j, or IC, denotes i\\Q fourth root of IG ; 
 and so on, the figure or figurtjs, called the Index, written o^•e^ 
 the radical sign, or the denominator of the fractional exponent, 
 denoting the degree or name of the roots. 
 
 The index is usually omitted in denoting the square root. 
 
 391. Evolution is the process of finding or extracting the 
 roots of numbers. It is the reverse of Involution. 
 
 A Perfect Power is a number whose root can be exactly 
 obtained ; aiid an Imperfect Power, or Surd, a number whoso 
 root cannot be exactly obtained. 
 
 392. The Root correspondir g to any perfect power mny bo 
 found by factoring the power. Thus, 
 
 r)y resolving 19G into its prime factors, 2, 2, 7, and 7, we 
 find its square root is 14, since one of its two equal factors is 
 7 X 2 - 14. 
 
 v'27000 = 30 ; since the factors of 27000 are 2, 2, 3, 3, 5, 
 and 5 ; and one of the three equal factors is 2 x 3 x 5 = 3<.>. 
 
 Wliat is a Root? The Square Hoot of a number? The Cul.e Rout? Ry 
 what aro roots denoted? The degree or name of the roots? What is Evolu- 
 tiou? A Fcrfcc . Power? Au Inipcrt'ect Tower? 
 
EVOLUTION. 
 
 2G9 
 
 ors taken 
 )ne of its 
 
 its tliivu 
 
 12,-). 
 
 are usvil 
 
 4; 
 
 ten over 
 q)oiU'iit, 
 
 root, 
 ting the 
 
 exactly 
 r w]ios(! 
 
 may ha 
 
 1 7, we 
 ctors u 
 
 "^ '\ ^\ 
 ' - 30. 
 
 lot? Hy 
 s Evolu- 
 
 In finding tlie approximate root of an imperfect power, mu], 
 for convenience, in finding the root of large numbers, other 
 methods are used. 
 
 S QUAKE KOOT. 
 
 393. A method of extracting tlie square root of numl)ers is 
 derived from the foHowing 
 
 PKINCirLES. 
 
 1. 2'hc square of any integral nuiulnr is c.q^rcsscd iy tirice (W 
 f/iun// j'ldccs of figures as the number^ or bij twice as mamj less oiu; 
 
 For tho first ten nunil)ers are 
 
 1, 2, 3, 4, r,, 0, 7, 8,- 5), 10, 
 
 and their s(|uares arc, 
 
 1, 4, 9, IG, 25, 3(5, 49, 04, M, 100; 
 
 also, the scpiare of 99 is 9801, of KM) is 10000, oi 999 is 998001, of 
 1000 is 100(1000, and so on. Hence, 
 
 2. If a point be placed over every second figure in any integral 
 number, beginning tcith units, the groups (rr periods of figures thus 
 formed will correspond respectively to the units, ti:\s, iiUNDllKits, 
 tf'c'., in its square root. 
 
 3. Every integral 'number expressed by mo're than two places of 
 figures is equal to the square of the tens in its ROOT, plus the pro- 
 duet of twice the tens p)lus the miits, multiplied by the units. 
 
 For, take any number composed of tens and \Hiit8, as 30, separate 
 it into its tens and units, s(piare it, keeping the [iroducts distinet, 
 and we have 
 
 :m; = 30 + 0, and 30* -- (30 + 0) x (30 -t- U) 
 
 30 X 30 = 
 
 3U» 
 
 
 = 900 
 
 X 30 ) _ 
 
 30 X {>\ ~ 
 
 
 (30 X 0) X 2 
 
 = 30O 
 
 X :- 
 
 
 &' 
 
 = 36 
 
 30- 
 
 30- 
 
 + (30 X 0} X 2 + = 
 
 = 30- + (00 + 0) X = 1290 
 
 * 
 
 What is the first riincii-le? The second? The third ? 
 
■■H— i^H 
 
 ■r 
 
 270 
 
 PRACTICAL ARITHMETIC. 
 
 
 4 
 
 \ 
 
 1 
 
 r 
 
 r 
 
 : i 
 
 
 ' 
 
 ir 
 
 1 
 
 
 li 
 
 % ' 
 
 n, ' 
 
 If, now, we denote the tons by t, and tlie units by u, we have the 
 fonuulii, 
 
 (t + ?<)2 = t"' + 2 (t X u) + M- = «- + (2< -t w) X u, 
 
 or that wliicli was to be proved. 
 
 This formula is geiu'ral, since all numbers expressed by more thnn 
 two places of li<,'ures may be considered as composed of tens and units. 
 
 394. To extract the square root of a number. 
 
 1. Let it be required to find the square root of 129G. 
 
 OPKUATION. 
 
 1296 ' 
 
 30-' 
 30 
 
 = 900 
 
 30 
 
 X 2 
 
 GO 
 6 
 
 6 = 
 
 396 
 
 396 
 
 ■1 
 I 
 
 liOfi j3G 
 9 I 
 
 or 
 
 GO 
 
 39G 
 
 G 
 GG X G = 
 
 396 
 
 Since the S({uare of any integral number is expressed by twice as 
 many places of figures as the numl)er, or by twice as inaiiy less one, 
 the root of 12f)G must be expressed by two places of figures. 
 
 If the root is ex])ressed by two places of figures, it must be com- 
 posed of tens and units ; hence, 125)6 must be equal to the square of 
 the tens in its root, ]ilus the product (jf twice the tens plus the units, 
 multiplied l>y the units. 
 
 Since the square of tens is always hundreds, we seek the greatest 
 number of tens whose square is contained in the 12 hundreds, which 
 is 3 tens. 
 
 AV'e write the 3 as the tens' figure in the required root. 
 
 3 tens s([uared, or 5) hundreds, sul)tracted from 1296, leaves 306, 
 which nuist contain the product of twice the tens i»lus the units, 
 multiplied by the units. 
 
 If the 396 coiitaint'd only the product of twice the tens multiidied 
 by the units, dividing by twice the tens must giv(! the units ; and 
 making a trial divisor of twice the tens, or 60, we find the probable 
 units of the root to be 6. 
 
 The 6 units added to twice the throe tens, and multiplied l)y 6 
 units, or (60 + 6) x 6, equals 396, and completes the square. We 
 write 6 as the true units' figure of the root. 
 
 How many places of figures exjiress tlic square of any intcgi"al number? 
 What is always the sciuare of teu»? Exi'laiu tlie operation. 
 
•^^ 
 
 EVOLUTION. 
 
 271 
 
 ; and 
 
 i. ^Vii 
 
 T]icrer)re the square root of 120f) is 3 tor^ + fi units, or 3(!. 
 Had tlie root been expressed by more than two jilaces of fii,Min's, 
 tlje principle ajjpliud in the operation would liavi; l)i'i'n tlu; sanir, 
 since we may liavi; tens and units of units, tmis and units of teas, tens 
 and units yf hundreds, kc. 
 
 The ojieration may 1)e illustrated by dia;^'rams. Thus, 
 Let there l>e a st^uare of 12i)6 square inclu's, wliose sidi' is requiriMl. 
 
 The grL-alc'st square (»f the tens iu 1200 
 square inches is 1)(»() square inciit-s, npin- 
 senting a sij^uare whose side is 3 tens, or 30, 
 inches. 
 
 DOO square inches taken out of \2'i)(\ s<iuare 
 
 m in. 
 
 •a 
 
 + i 
 
 ii 
 
 30 
 
 30 - 
 
 •s> 
 
 <> "^ inched leaves 3!)()S(|uare im lies, wliicli l>elon;^' 
 900 180 ' ^" ^^^^ *^'^ ^^^^ sides, so as to preserve the form 
 
 of a s(|uare. 
 
 " 30 i'lT + 6^in! If 300 scjuare inches belon;.,' to two sides, the 
 
 width would be as numy inches as the num- 
 ber MeiiotiuL; the len;^th of two e(jual sides, or GO, is contained times in 
 31Ki, which is(J. V>\\, ihe whole len;.;th ol'llie additituial portion is that 
 
 of twice »)ne of the e(|ual sides 
 plus the side of a small square, 
 whose side is the wiiitli of the 
 additional portion, or (J inches ; 
 (GO + G) 6, or GG X G is 3:JG ; and 
 
 
 G6in. 
 
 
 •^i 
 
 
 
 -1 
 
 
 
 30 in. + 30 + 
 
 G HI. 
 
 3f)G s([uare inches subtracted, leaves no remainder. 
 Therefore, 12!)G is a square whose root is 3G. 
 
 llULE. Separate the f/lven nvruhcr into periods, by j^ointimj 
 fjVcnj second Jlgiu'e, bcijinning ivith the units' Ji'jnre, prvceedinij to 
 the left if ifhole numbers, and to the rigid if dcciinals. 
 
 Find the (jreatest number ichuse square is contained in the left- 
 ha. id period, and icrite it in the root ; subtrart the square of this 
 from the lifi-hand period, and to the remainder annex the next 
 period for a dividend. 
 
 Take twice the root found, regarded as tens, for a trial divisor; 
 divide the dividend by it, and write the result as a second part of 
 the root. 
 
 Had the root been expressed by more than two figuies, would the principlo 
 applied in the optiiition still apply? Why? E.xplaiu the opcratiou aa 
 illustrated by diagrams. Recite the Rule. 
 
 Is 
 
l\ I 
 
 H* 
 
 . 
 
 — I ad 
 
 PllAfTlCAL A lUTUMETIC. 
 
 To fhe Iriiil (liv(.<or add /Iir last part of t/ir root; inultliihj the 
 ri'suU hjj the last part of the ruot, and subtract the product from the 
 dirideral. 
 
 If there are more periods, continnc the operation as before. 
 
 Tlie trial divisc^r ])c;ing an incomplete divisor, the (iguiv of the Mot 
 jiiay Polnc'tillu•^^ Ik; too large ; in such a cii.se, substitute for it a root 
 1. less. 
 
 Examples. 
 
 2. Mxtract tlio pquaro root 
 of 1 SCO. 24. 
 
 3. Extract the square luut 
 of 102'J12G1. 
 
 OI'KRATION. 
 18G(J.24 ! 
 
 43 
 
 2 
 
 • 
 
 01 
 
 ';j{ATI()N. 
 
 1O2012G4 
 
 3208 
 
 80 
 
 10 
 
 2G6 
 
 
 
 CO 
 
 
 
 '^ 1 
 
 129 
 
 
 3 
 
 
 
 
 2 
 
 
 
 
 
 s;^ X 3 - 
 
 240' 
 
 
 
 (;2 X 
 
 o 
 
 ~ 
 
 124 
 
 
 {>G.O 
 
 17.24 
 
 
 
 G400 
 
 
 
 512G4 
 
 
 8G.2 X .2 - 
 
 ' 17.24 
 
 
 
 8 
 
 04'08 
 
 X 
 
 8 : 
 
 _ 
 
 r)12G4 
 
 
 In tlie operation of the .'M exauqile, a oceurrin;j[ in the root, we 
 annex a cipher to the trial divisor, ami to the dividend auutlu r 
 period, and ]troceed as before. 
 
 Find tlio square root 
 
 .4. Of 77841. Jns. 27'J. -7. OniGOt. ^/k 10s. 
 
 .5. Of201G. .8. ()f.4r)9G84. -//^^•. .078. 
 
 . G. Of 10.497G. v///.s\ 3.24. 1.0. Of 31040025. .Ins. 5025. 
 
 .10. What is tho square root of .0003272481 ? Ans. .0180i). 
 
 . 11. What is the square root of .00001849 1 Ans. .0043. 
 
 395. When there is a remainder after using all ti.e periods, 
 it indicates that the given inunber has not an exact square 
 rttot; but the approximation maybe continued ]jy annexing 
 jieriods of decimal ciphers. 
 
 If a figure of tho root jn'ove to be too Ijhitc, what is to be dono? How do 
 we itroceod wlion a occurs in the '"'ot ' ^V li ii, itoes a remuiiuler, after using 
 all the iicriods, imlicatc ? How ni..y *.£i '.iT>}uoAivii,uion be coutiiiued? 
 
^£a 
 
 -r— r- 
 
 iphj the 
 roia the 
 
 the I'M.t 
 it a root 
 
 ro iiiot 
 
 3203 
 
 I'ltot. we 
 aiiutluT 
 
 n.-<. lOS. 
 
 //,>•. .G7S. 
 
 (6-. 5025. 
 
 .0180U. 
 
 ^ .0043. 
 
 periods, 
 
 ■t square 
 
 luu'xin.!.,' 
 
 How do 
 after using 
 ucil ? 
 
 EVOLUTION. 
 
 273 
 
 • 12. Extract tlie square root of 12 to tliroe docimal i)la(^^s. 
 
 yiu.s. 3.1«i t + . 
 
 , 13. Extract tlie square root of l.G to two dor-iinal places, 
 
 . 14. Extract the square root of .002 to four decimal places. 
 
 yins. Mil f . 
 
 , 15. Extract the square root of 5. yim^. 2.23() + . 
 
 • IG. What is the square root of .5 to four decimal places I 
 
 y//;.s\ .7071 f-. 
 
 396. When the given miiulier is a rnmmon fraction, or a 
 mixed numher, reduce it to its sim})lest form, and if the nume- 
 rator and denominator are both peifect squares, extract the 
 square root of each separately; hut if not, reduce the fraction 
 to an ecjuivalent decimal, and extract its root. 
 
 What is the square root 
 . 17. Of li:^? Jus. \l ,20. Of 37v:: ? .-//'■<'. Gl. 
 
 A US. I. 
 
 ■i 11 
 
 ,21. Of;? 
 
 4ii.^\ .!i3o4 h. 
 
 IS. Of "l!"^? 
 .VJ. Ofv^^? Ans.l^. .22. Ofl7,V;.] J //.v. 4.1501) i . 
 
 • 23. What is the value of n^42025 ] An.'i. 205. 
 
 • 24. What is the value of VX;-;; ] Ans. .0330'J +• 
 
 APPLICATIOXS. 
 
 " 1, A certain numher of boys spent iS^'-Gl, c uh impending as 
 many cents as there were boys; what was the number of boys? 
 
 ytns. 19. 
 ,^ 2. A square pavement contains 2073G square stones, i\\\ of 
 the same size ; what number compose one of its sides ? ; '■ ' 
 '^ 3. If 39G0 hills of corn, each hill an equal distance from 
 another, are planted in a square, how many hills are there in 
 each row ? Ans. 03 hills. 
 
 How do we inoceed when the given number is a common fraction or a 
 mixed number? 
 
 RiVlEW Ql-i:.sttons. Wlmt is a Power? (.38.5) Invohitinn ? CJSO) A 
 Koot" (oS8) Evolution? (oO\) The S<iuair Root of a number? (.'?S'J) 
 
 ) I 
 
 > J 
 
^ 
 
 271 
 
 PRACTICAL ARITIIMKTIC. 
 
 n 
 
 ^ 4. A gPTioral lias an army of 1 1 1")7() int'ii ; how many must 
 lif placi! ill rank and lih; to foi i tliciu into a S(juaro I .///.<. ."i 7 <). 
 ^ 5. What is til.- It'iigtli of o t' of the* (([ual sides of a s(|uare 
 aero ? yliu. !-.<; 1 i i(>d>. 
 
 >— G. How niiicli ".viil it cost to enclose 10 acn-s of hind in th(^ 
 form of a .s(|iiar(', at ^.i)0 a rod? ylns. ijUG. 
 
 *^ 7. How iiuich will it cost to enclose a hectare of land in tlu! 
 form of a s(|uari', at 8.lM a inoter? ^///.s\ sl<"). 
 
 — 8. A general, trviiiL' to mass his armv of l-'iHO men into a 
 square, found he had 'M men over ; rtMjuiied the number in 
 rank and lile. Ail^. 124. 
 
 CUBE liOOT. 
 
 397. A method of extractiii'' the cube root of numbers is 
 derived from the foilowinLT 
 
 :( 'I 
 
 rrjNcirLES. 
 
 1. T/ii' oihe of anij laU'ijml number coudlsls of fhrrr. timc^ afs 
 man;/ jihur.i of fujures as Ihc numher, or of ihrC'^ iimes as rnamj 
 less one or tvw. 
 
 Fur ihc fiisl ten umuber.s aru 
 
 1, 2, 3, 4, 5, G, 7, 8, !), 10, 
 
 and tlu'ir cubes are 
 
 1, 8, 27, G4, 125, 21G, 3-i:5, 512, 72!), lOOO ; 
 
 also llie cube off)!) is 0702:)!), (jf 100 is IdOOOOO, of !)!)9 is 9970021)09, 
 of 1000 is 1000000000, and so on. Hence, 
 
 2. If a point be placed over every third fifjure in any integral 
 number, heyinning with units, the groups or j^criods of figures thus 
 formed will correspond respectively to the units, tens, hundukds, 
 dc, in its cube root. 
 
 3. Every integral number expressed by more than three places of 
 figures is ecjual to the cube of the tens in its root, plus the sum of 
 
 "What 13 tho Cuho or Third Root? What is the first Principle? The 
 ficCDud? The third? 
 
ly nmsfc 
 IIS. ;57<). 
 L s<]iiare 
 f rod-i. 
 1 in tlio 
 
 ] ill tliti 
 
 1 into a 
 uber in 
 //.y. 124. 
 
 iilxTs is 
 
 (/.5 Ilia II !j 
 
 0. 
 
 10; 
 )70l)2!)00, 
 
 integral 
 'ures thus 
 
 NDllKDS, 
 
 places of 
 ic sitm of 
 
 ii)le? Tho 
 
 EVOLUTION-. 07. -t 
 
 three iime^ the square of thr fnr<, three llnm the rmhdof the tens 
 (nid NNi/s, ami the ,>jnare of the unit,, mnltipUed hj the units. 
 
 Fur, let ;{(! 1)0 any numluT tonipo.sr.l of u-ns aii.l units, 
 TliL'ii, -Mj = :>,n + (;, ;i,„i^ 1,^. \,.^ ;,j,^,^ ^^.^ ,^.^^'.^ 
 
 3G'=3()- + (;io X -2) x (\ + (r\ 
 
 .>f-.Itii.lyii.g tl.i.s sr^uare by 36 = 30 + 0, ati.l keepin-,' tlie nroducS 
 ui.-itiiict, wl' oljtain ' 
 
 3G''' = 3()'' + (30'- X 3) X n + (30 x 3) x Vr + C- 
 = 30=' + (3(r-: X 3 + 30 x 3 x (J ■{- {;•-) x (j 
 An<l 30 f 
 
 uO- X 3 =2700 
 
 30 X 3 X f; ^ r.io 
 
 (30^ X 3 + 30 X 3 x fi + (;■) X (; .. .^ X f) .- locnd 
 
 .3G^ = 303 ,. (3,,^ X 3 + 30 x 3 x (! , (;"-) ^ (; ^ "g"^ 
 
 U\ iioNv, AVf .leiiote the tens by ^ and the units by u, we have tlio 
 lornmhi, 
 
 (t + ny =. i-i + (f- X 3) X /< + (i X 3,) X «•• + m' 
 == i'' + (i- X 3 + ^ X ii X 3 + M-) X ?^, 
 or that whioli was to be jnovcd. 
 
 This I'unuuhi is general, since all intc-ral numbers exprossod by 
 tun or more places of figures, may be considered as composed of tens 
 and units. 
 
 398. To extract the cube root of a number. 
 
 1. Let it bu required to lind tlu> cube root of 405224. 
 
 Oi'JLU.VnoN. 
 
 70^ 
 
 70-' X 3 
 
 70 X 3 X 4 
 42 
 
 = 14700 
 
 - 840 
 
 16 
 
 40.5224 I 74 
 ^ 343000j' 
 "02224 
 
 (70'^ X 3 -f- 70 X 3 X 4 + 4^) X 4 =^ loooG x 4 = C2224 
 
 Since the cube of any integral num])er is expressed by three times 
 as^ianyjlgui-8s as tlie number, or three times as many less one or 
 
 What is the formula ? "Why is it general ? 
 
270 
 
 rPvACTrCAL ARTrnMETTC. 
 
 » 
 
 two, tlu; root of ■i()')224 must Ito oxprospod by two places of fi^Mircn, 
 juid, tlHM'i'forc, must consist of ti!iis and units. 
 
 Since tlio cube of tens is always tliousiinds, wo sock the fjreatcst 
 mind)cr of tens whose cube is contained in 405 ihousands, which is 
 7 tens. We write the 7 as the tens' fii^ure in the rcfjuircd root. 
 
 7 tens cubed, or 343 thousands, subtracted from 4()."):i:24, loaves 
 (■(2224, which must contain the jtnubuit of throe times the s((uare of 
 th(! tens, tiiree times tlie tons by the luiits, and the s(|uart' of the 
 iinils, multiplied by the unit.s. 
 
 If the 02221 contained only three times llu; square of the tens l)y 
 the units, dividinj^ by three times the s(piaro of tho t< lis iiiu>t j^dve 
 lh(! units ; and making,' a trial divisor of three times the S(piare of the 
 tens, or 14700, we lind the [irobable units of the rcjot t(t be 4. 
 
 Addiiii; to threi! limes tlu; s(piare of the tons, or 14700, the product 
 of thrive times the tens by tho unit.s, or SJO, and tin- sipi iri- of the 
 units, or l(j, we have the sum Ifjo.^G, ;ind tlii^, multiplied ]>y the 
 units, equals 02224, which completes the cube. We wr te 4, as the 
 true units' (ij^ure of the root 
 
 Therefore the cul)e root of 40r)224 is 74. 
 
 Hail the root been expressed bj-^ more than two places of figures, 
 t.lie princi[)le appliod in the operation would have been the sanu', 
 i^ince we may have tens and units of units, tens and units of terts, tens 
 and units oi' hnndreds, &c. 
 
 The operation may be illustrated by diagrams. Thus, 
 Let there be a cube cjf 405224 solid inches, whose edge is required. 
 
 The greatest cubi' of the tens in 405224 
 solid inches is 343000 solid inches, represent- 
 ing a cube whose edge is 
 7 lens, or 70 inches. 
 
 343000 solid inches 
 taken out of 405224 solid 
 inches, leaves 02224 solid 
 
 inches, which belong to three sides of the solid, 
 i?o as to preserve the form of a cube. 
 
 The addition upon one side of the cube must 
 be. 70 inches long by 70 inches wide, and if 1 inch thick, will Ix? 
 4000 solid inches, and upon the three sides will be 3 times 4*J00 or 
 \4700 solid inches. 
 
 How many places of figures express the cube of any integral number? If 
 the root is expressed by two places of figures, of what must it be composed? 
 AVhat is always the cube of tens? Explain the operation. How may the 
 operation bo illustrated? 
 
KVOLUTION. 
 
 ivliich is 
 
 )0t. 
 
 i, loaves 
 ijUiiro of 
 {' of tlie 
 
 1^77 
 
 • Ifiis hy 
 u>l ^'ive 
 ire of the 
 
 i-ii (jf the 
 I l.>y the 
 4, ius the 
 
 f figures, 
 ht* sjuiie, 
 tens, tens 
 
 required. 
 I 400224 
 
 present- 
 
 :, will 1)0 
 41)00 or 
 
 inber? If 
 oin posed ? 
 .V may the 
 
 Jf 14700 solid iiiolu'H will miikc 1 inch of thic' ess, (;2224 solid 
 imdies will make us many inciies C)f thickness, a> i470U inulid inches 
 are eontained times in ()2224 solid inches, or 4 inches. 
 
 L> 
 
 
 
 lint. Ill-sides the three additions to the sides, there are recjuinid, upon 
 three edu'es, tlirt*e oilier additi(jns of 70 inches long, 4 inches wide, and 
 
 if 1 inch thick, there will be for the three 
 840 solid inches. 
 
 1^ ^-.-^ \^ To complete the cuhe an addition is 
 
 |- 1 — I " rt'([uiivd, u])'>n oni; eoi-iier, 4 indu's long, 
 
 4 inches wide, and if 1 inch thick, will 
 1)e IG solid inches. 
 
 liut all the additions are 4 inches thiek, 
 and (14700 + 840 + 10) x 4 - (;l'22 I 
 .solid inches, which completes the cube. 
 Therefore 405224 is a cube whose root is 74. 
 Rui.K. Si'parate the given iiumher into ])e)io(l^, Inj j)(>intiriij 
 crtTi/ iJtird /((jure, leqinning icith the units' fujure, proceediuij to 
 the left if ichule nuwhcrs, and to the right if decimals. 
 
 Find the greatest number vdiose cube is contained in the left-hand 
 period, and write it in the root ; subtract the cube of this root front 
 the left-hand period, and to the remainder annex the next jieriod 
 for a dividend. 
 
 Take three times the square of the root found, regarded as tens, 
 for a trial ditisor ; divide the dividend la/ it, and write the result 
 as a second part of the root. 
 
 To the trial dirisor add three times the first part if the root, 
 regarded as tens, multiplied by the last part, also the square of the 
 la<t part ; inuUiphj the result hij the last part, and subtract the 
 product from, the dividend. 
 
 Give the illustration. Rejieat the Rule. 
 
278 
 
 rUACTICAL ARITIIMF.TIC. 
 
 //* there arc. more. prrio(J.<t, conthinc the oprrnllon as he/ore. 
 ir lIic nunilter fxijres.scd by luiy fi^aire of tlie ruot bhuiiIJ \>io\Ki to 
 be too large, a smullur number must be tukcn. 
 
 Examples. 
 
 2. Find the cnbo root of 70:)ii PJ.iJ 1720 t. 
 
 orr:iiATi()N', 
 
 7()r')91{i.'Jl7JGi 89.0 1 
 
 10-JOO 
 
 2100 
 
 81 
 
 21411 X 9 = 
 
 2370:10000 
 
 100800 
 
 IG 
 
 512 
 
 ToaoiQ 
 
 192000 
 
 237730810 x 4 
 
 950917204 
 
 950947204 
 
 In Iho oporntion, ii dccurrin^' in the root, we annex two ciiihcrs 
 to tlic trial ilivisoi', and to the dividend annex anothi;r period, and 
 ]troceed as before. 
 
 Find tlie cube root 
 . 3. Of 54872. Ans. 38. * 7. Of .000001728. yim^. .012. 
 
 » 8. Of .001000024. yin^..V2\. 
 ^ 0. On070800025. .^??5.102;». 
 
 , 4. Of 030050. 
 
 . 5. Of 04004 S08. An^. 402. 
 
 . 0. Of 444104.947. .//^s. 70.3. [ 10. Of 80.077508101, 
 
 y//^s•. 4.321. 
 
 399. AVhcn the given nur'ber lias not <iii exact cube root, 
 
 tlic operation may be extended hy annexing periods of decimal 
 
 ciphers. 
 
 ^ 11. AVliat is tlie cube root of 20.2 to two places of decimals '? 
 
 Anx. 2.07 + . 
 
 If the nunihor expressed by iiny figure of tlio roct i)rovc too lurge, wliat is 
 ■to lie done? How do we proceed Avhen a occurs in tlic root? How may 
 the operation be extended when the given number has not an exact cube 
 root ? 
 
'"T"^. «|S ''P ■ 
 
 F.VdLirTfOX. 
 
 
 . 12. What is tlic ciilif rool of 1' to {\n\'v plucf"" of decimal.- ,' 
 
 .his. l.l.'r.U + . 
 » l."». What is the ciibo roof of 517 to time [ilaci's of cK'ti- 
 
 400. When the given numbor i- a mmmon fi-<irfloii, or a 
 Ti)ij-ed uniiibn\ icdiici^ it to it.s sii;ii)I('st form, ami if tlm minic- 
 rator and di'iiominator arc both perfect cubes, extract the cube 
 rnot of (-ach separately; but if not, reduce the fraction to au 
 (■(luivaleiit decimal, and extract the root of it. 
 
 What is the cube root 
 . lb Of i^;^? ..///.. 0. I'lT. Of 1!? .//,.. .910 + . 
 
 , ir>. Ofj? Ans. .i\^:\-^. 'U> 0{'M)'i\'\] ylus.:]^. 
 
 , 10. Of ,i::;;,? A„s. ;;. I'll). Ofr;?'/ " Jn.^. 1.1)00 + . 
 
 . 20. What is the value of xM()r/;-„^,, i 
 p 21. ^^'hat is the value of Vrr).;;2.i 
 
 An?. 2.48.") +. 
 
 AI'PLICATIONS. 
 
 1. A block (tf granite in the f(>iiu(»f a cube contains 10.^82.'^ 
 cubic inches ; what is the measure of one of its e((ual i'*\'j,{'A 1 
 
 yliis. 17 inches. 
 ^ 2. Required the depth of a cubic box that shall exactly hold 
 •'I bu.shel. ' y///.s'. 12.9+ inches. 
 
 ■* 3. There is a, range of wood 21.', feet long, feet high, aiul 
 4 feet wide ; how long a cubic pile will it make ? Ans. 8 feet. 
 - 4. Find the area of a side of a cube contaitiing \7[r>.rj; 
 liters. ylns. 00.84 square meteis. 
 
 • 5. There is a cistern of a cubical lorni, which contains 13.'U 
 cubic feet ; what are the length, breadth, and depth of ii ? - // 
 
 , 0. What must be the depth of a cubical cistern that shall 
 contain 570 gallons? Ans.4:.25+ feet. 
 
 How do we piocet' J wliuii the given uuiiiucr is a coiumou fraction or a mixed 
 number? 
 
IMAGE EVALUATION 
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 1.6 
 
 Vi 
 
 <^ 
 
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 7M 
 W 
 
 ^ ' ^J^ 
 
 
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 Photographic 
 
 Sciences 
 Corporation 
 
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 S. 
 
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 280 
 
 niACTICAL ARITHMETIC. 
 
 /• 
 
 MENSURATION. #^ ^^Oy. bx.z^.wvvk 
 
 401. A Point is that wliicli has only position. 
 
 402. A Line is that which has only length. 
 
 A Straight Line is one that has 
 
 Jill its parts in the same direction. A 
 
 Thus, 
 
 The line A B is a straidit line. 
 
 B 
 
 A Curved Line is one which continually changes its direc- 
 
 _,.^-— ---^ tion. Thus, 
 
 ^ -^ The line C D is a curved line. 
 
 403. A Plane is a surface (Art. 199) in which any two 
 points being taken, the straight line that joins tliem will lie 
 wholly in the surface. 
 
 A Curved Surface is one of which no part is plane, 
 
 404. Parallel Lines are such as, A B 
 
 being in the same plane, have the 
 
 same direction with each other. C D 
 
 Thus, 
 
 A B and C D are parallel lines. 
 
 B 405. Two straight lines are said to 
 
 be perpendicular to each othei-, when 
 their meeting forms equal adjacent 
 angles (Art. 208). Thus, 
 Q D The lines A B and C D are perpeii- 
 
 A dicular to each other. 
 
 406. A Eight Angle is one 
 
 forr.ied by a straight line and a 
 perpendicular to ir-. Thus, 
 
 The angle A C E is a right angle. 
 
 An Acute Angle is one which is 
 less than right angle; as the angle -^ 
 BCD. 
 
 "What is a Poiut ? A Line? A KStraight Line ? A Curved Line ? A Plane? 
 A Curved Surface? Parallel Lines? A Plight Aui,4e? Au Acute Anglo? 
 
 6' 
 
MENSURATION. 
 
 281 
 
 6 
 
 B 
 
 ts direc- 
 
 line. 
 
 my two 
 will lie 
 
 B 
 D 
 
 said to 
 !r, when 
 xdjaceiit 
 
 perpeii- 
 
 :d 
 
 -E 
 
 A Plane ? 
 Anurle ? 
 
 c 
 
 An Obtuse Angle is one wliicli is greater than a ridit anglt;; 
 as the angle A C D. 
 
 The sides of an angle are the lines forming it, and the verd'x, 
 of an angle the point of their meeting. Thns, 
 
 In the angle ACE, AC and C E are the sides, and C the 
 vertex. 
 
 407. A Plane Figure is a plane bounded by a line or 
 lines. 
 
 The Peiumeter of a plane figure is its 
 boundary. 
 
 The Base of a figure is the line upon 
 which it is sui)posed to stand. 
 
 The Altitude of a figure is its perpen- 
 dicular height. Thus, 
 
 In the plane figure A C B, the line A B is the ijase, and the 
 line C D the altitude. 
 
 The Diagonal of a figure is a straiglit 
 line joining any two of its angles, which are 
 not adjacent to each other. Thus, 
 
 In the figure A B C D, the line D B is a 
 diagonal. 
 
 The Sides of a figure bounded by straight 
 lines are the bounding lines. Thus, 
 
 A B, B C, C D, D A, are sides of the figure A B C D. 
 
 408. A Polygon is a plane figure bounded by straight hues. 
 A Begular Polygon has equal sides and equal angles. 
 
 A polygon of three sides is called a Triangle, of four sides a 
 Quadrilateral, of five sides a Pentagon, of six sides a Hexagon, 
 of seven sides a Heptagon, of eight sides an Octagon, &c. 
 
 409. Mensuration treats of the measurement of lines, plan<'>, 
 and solids or volumes (Art. 199). 
 
 Wliat is an Obtuse An ^le ? A Plane Figure? The Perimeter of a plane 
 figure? The Base? TheAltituae? The Diagonal? What is u Polygun ? 
 A Regular Polygon ? Mensuration ? 
 
 D 
 
 C 
 
282 
 
 I RACriCAL ARITIIMIITIC. 
 
 |i 
 
 I 
 
 TRIANGLES. 
 
 410. A Triangle is a polygon li.aving 
 tliroe .si(I(;.s, and therefore three angles. 
 Thus, 
 
 The fiLCurc A B C is a trianu;le. 
 
 An Acute-Angled Triangle has three ^ i- ^ jj 
 
 acute angles. Thus, 
 
 The figure A V> C is an acuie-anded trianirle. 
 H I An Obtuse-Angled TijlvNgle has 
 
 one obtuse ande. Thus, 
 
 The figure G H I is an obtuse-angled 
 triangle. 
 
 A Pt [GUT- Angled /^L 
 
 Tpjangle has one right angle. Thus, 
 
 The figure J K L is a TJght-angled tri- 
 angle. 
 
 The side opposite the right angle is 
 called the itYJ'OTlTENUSE, and the side per- 
 pendicular to the base, the pehpendicular. Thus, 
 
 In the figure J K L, the side J L is the iiYroTiiENUSE, and 
 
 K L the PERPENDICULAR. 
 
 411. By Geometry there may be readily demonstrated the 
 
 following 
 
 PKIXCIPLES. 
 
 J 
 
 1. The square of the hjpothenuse of a 
 riglit-angled triangle is equal to the sum of 
 the squares of the other two sides. Thus, 
 
 Let h denote the h}'pothenuse, h the base, 
 and }) the perpendicular of a right-angled 
 triangle, and ^ve have the fonnula, 
 /t2 =.. U' + p\ 
 
 which is illustrated by the diagram. 
 
 y^^ 
 
 ^^\ 
 
 c 
 
 y 
 
 — 
 
 _!_.- 
 
 
 1 
 
 
 
 
 
 IJ 
 
 — 
 
 — 
 
 
 
 
 
 
 
 
 
 "What is a Triangle? An Acnte- Angled Triangle? An Obtuse-Angled 
 Triangle ? A Kiglit-Anglcd Triangle ? AVliat are the sides called ? "What ia 
 the first Principle ? 
 
N'GLE lias 
 
 ise-angled 
 
 L 
 
 K 
 
 :USE, and 
 :ated the 
 
 V/ 
 
 L_ 
 
 
 1 
 
 
 B 
 
 ise-Aiigled 
 ? What is 
 
 I 
 
 MRNSUR.VTIOX. 
 
 283 
 
 2. The h/polhenuse of a rhjld-nnghd triaiiqJe i.^ cqvnJ fn tho 
 Siimre root of the sum of the squares of the othrr two sides; and 
 
 3. Either of the tiro shorter sides of a riohf-eimiled tmwqle i^ 
 equal to the square rout of thr. difcrcnee of the Squares of the 
 h/q/othenusc and the other side. 
 
 Exercises. 
 
 ^ 1. If tlio base of a riglit-aii-led triaii:,do is 00 feet and the 
 perpendicular 45 feet, what is the liypothenuse I 
 
 OPERATION. 
 
 CO^ + 45^ . 3G00 + 2025 = 5G25 ; v/5025 . 75, f;..t. ./... 
 / 2. If the hypotlienuse of a right-angled triangl.. is 75 fwt 
 and one of the other sides GO feet, what is the thir.l .side ? 
 
 OPERATION. 
 
 75'-' 
 
 GO^ = 5G25 - 3G00 .. 2025; s^20J5 = 45, feet, Jus. 
 y 3. A fort which is 15 feet high is surrounded by a moat '^0 
 
 feet wide ■ wliat must be the length of a la.l.ler that will Just 
 
 reach from the outer edge of the moat to the top of the fort? 
 
 Ans. 25 feet. 
 
 / 4. Two men travel from the same place, one due east and 
 the other due north. One travels the first day GO mile, and 
 the other 80 miles. How far apart are th(>y at th.- en.l of the 
 *^'^^'- Ans. 100 miles. 
 
 / 5. A line 3G meters long will exactly reach from the top cf 
 a perpendicular tower standing on the brink of a river, known 
 to be 24 meters broad, to the opposite bank ; what is the hei'dit 
 
 of the tower? j,,,. .)r oo , ,* 
 
 Atis. JO.h.j + meters. 
 
 yC. A tree broken off 30 feet from the ground and resting on 
 
 the stump, touches the ground 40 feet from the stump • what 
 
 was the height of the tree ? j^^,,^ Jq ^^^^^ 
 
 / 7. The rafters of a liouse, each 25 feet long, meet at the ed-e 
 of the roof 15 feet above the attic floor ; required the width of 
 
 the house, ^ ac\ v ^ 
 
 Ans. 40 feet. 
 
 Whafc is the second Trinciple ? The tliinl ? 
 
 i i 
 
284 
 
 rilACTlCAL ARITIIMKTIU. 
 
 (I 
 
 11 
 
 h ' ' 
 
 I! 
 
 I > 
 
 QUADKILATEKALS. 
 
 412. A Quadrilateral is a ])o]ygoii liaving four sides, and 
 therefore four angles. 
 
 413. A Parallelogram i^ a quadrilate- 
 ral liaving its o})po.site sides parallel. 
 
 A PiECTANOLE is a right-angled paral- 
 lelogram ; as the figure A B C D. 
 
 D 
 
 A 
 
 C 
 
 ■J r> 
 
 H , 
 
 E 
 
 G 
 
 F 
 
 A Square is a rectangle having equal 
 sides ; as the figure E F (r H. 
 
 A IviiOMBOiD is a J J J- 
 
 parallelogram having / / 
 
 no riii;ht an^^les : as t / A 
 
 the figure I J K L. 
 
 A lillOMBUS is a rhonihoid havinir 
 
 equal sides ; as the 
 figure M N P. 
 
 i: 
 
 T 
 
 S 
 
 414. A Trape- 
 zoid is a quadrilateral having oidy two of 
 its sides parallel ; as the figure K S T U. 
 
 -^ 415. A Trapezium is a quadrilateral 
 
 having no two of its sides parallel ; as 
 
 W 
 
 V the figure V W X Y. 
 
 AREAS OF TRIANGLES AND QUADRILATERALS. 
 
 416. By (xeometry may he proved, in ]-elation to areas, the 
 followinu' 
 
 PRINCIPLES. 
 
 1. The area of a parallelogram is ajnal to the p'odud of the 
 "base by the altitude. 
 
 What is a Quadrilateral ? A Parallelogram ? A Rectangle? A Rhonibind? 
 A Rhombus ? A Trapezoid ? A Trapezium ? To what is tho area of a 
 parallelogram equal ? 
 

 285 
 
 des, and 
 C 
 
 -^ B 
 
 ig equal 
 
 liaviiig 
 .^ 
 
 rilateral 
 illel ; as 
 
 iALS. 
 
 eas, the 
 
 d of the 
 
 lomboid ? 
 irea of a 
 
 T^ 
 
 !■: 
 
 (' 
 
 MEXSLTrwVTIOjr. 
 
 This has hcon shown to 1»(' tlio case with F 
 a rcctancrlc! (Art, 210), and tliat it, applies- r 
 cfiually to a rhomboid or rliorabus;, appears • 
 from tlie diagram, in which the rlioniboid 
 A BCD is equivalent to tlie rectangle A B K F, 
 of the same base and altitiuk'. 
 
 2. The area of a TRAPKZOID is equal b lite j'loJud <f half the 
 Slim of the parallel sides hij the altitude. 
 
 For any trapezoid A B C D is eqnivahnit 
 to a paralhdogram A B K D of tlie ■^aine 
 altitude, and whos^e base A L is e<pial to 
 11 1, which is half of A B, f C 1). 
 
 3. The area of a tiuanole is equal to 
 the prodiid of half the base hi/ the alti- 
 tude, or of half the altitude hij the base. 
 
 For any triangle A B C is equivalent to one A 
 
 half of the parallelogram B>CEA, of tlie same 
 b;Lso and altitude. 
 
 4. The area of a trapezium, or of any 
 pohjgon, is equal to the sum of the areas of the 
 tri angles irdo which it may be resolved. 
 
 Thus, the trapezium WXYZ is equal to tlie 
 triangle W X Z plus the triangle X Y Z, made by 
 the diagonal X Z. 
 
 L B. 
 
 5. 
 
 The area of a reul^lar polygon is equal to lite p-odud of m 
 
 the perimeter by half the perpendicular 
 drawn from the centre to any one of tlie 
 sides. 
 
 For any regular polygon A B C D E F F 
 may be resolved into as many equal tri- 
 angles as it has sides, by drawing from 
 the centre, 0, the lines O A, B, C, 
 &c. 
 
 To what is the area of a trapezoid equal ? Of a triangle ? Of a trapezium v 
 Of a regular polygon ? 
 
^9 
 
 280 ruACTiCAL auitiimi:t[c. 
 
 Exercises. 
 
 /' 1. AVluit is the area of a board 18.8 foet long and 2.7 fo(^t 
 wide? A US. 50.70 s([. ft. 
 
 / 2. W'liat is tlie area of a board 28 feet long and 15 inclies 
 broad ? Ans. 35 sq. ft. 
 
 . 3. If the base of a iiabLi of a lionsc be 40 feet lonijr and its 
 perpendicular height 20 feet, how many square f(;et of boards 
 
 will be re(]uired to cover it ] Ans. 400 sq. ft. 
 
 / 4. How many acres in a triangular lot, one side m(\asuring 
 32 rods, and the shortest distance from this side to the o})po- 
 site angle being 14 rods? Ans. 1 A. 04 sq. rd. 
 
 / 5. If the parallel sides of a lot be 75 and 33 yards, and its 
 breadth 20 yards, wdiat is the area in square rods 1 
 
 Ans. 35.7 + sq. rd. 
 
 / 0. How many hectares in a rectangular meadow G40 meters 
 
 long and 240 meters wide 1 Ans. 15 hectares and 30 ares. 
 
 X 7. One of the diagonals of a field in the form of a trapezium 
 is 100 rods long, and the perpendiculars from the opposite 
 angles to that diagonal are 70 and 50 rods ; what is the area ? 
 
 Ans. GO acres. 
 417. AVhen the three sides of a triangle are given, Ave may, 
 to find the area. 
 
 Take half the sum of the three sides, suUract therefrom each side 
 sejKiratcli/, midtiphj together the four results, and extract the square 
 root of the jjroduct. 
 
 ^ 8. The sides of a triangle are 13. 84, and 85 rods, respec- 
 tively; -what is its area? Ans. 3 A. GO sq. rd. 
 
 0. The sides of a certain field in the form of a trapezium 
 
 ' measure 30, 35. 40, and 25 rods, respectively, and the diagonal 
 
 "which forms a triangle with the first two sides, 45 rods ; what 
 
 is the area ? Ans. A. 01.8 sq. rd. 
 
 \ 10. What is the area of a regular hexagon, whose sides 
 
 When the t aree sides of a triangle are g'ven, how may the area be found ? 
 
MENSURATIOX. 
 
 
 01)pO- 
 
 iiro each 1 l.G feet, and tlie ixTpeiidifular iVom the r«>iitr<' to a 
 si.lol2.G-tfeet7 ./.,v. i:.:].G:3 ,- .s^. It. 
 
 C I II C L E S. 
 
 413. A Circle is a plane figure bouiidcd by a curved line, 
 all the points of uhich are equally distant from a point Avithin, 
 culled the centre. 
 
 The CiRCUMKEUF.NCH is the bound- 
 ing line ; as the line A E B D. D ^ 
 
 The DiAr^iKTKii is any straight line / 
 
 drawn tln-ough the centre and terrni- / 
 
 nating in tlr- circumference; as the Al- 
 
 liucAJJ. y 
 
 The liADius is any straight lino 
 drawn from the centre to the circum- ]? 
 
 ference ; as the lines C A, C B, or C D. 
 
 419. A Square is said to bo inscribed 
 in a circle when the vertices of its au'des 
 are m the circumference. Thus, 
 
 The square A B C D is inscribed in a 
 circle. 
 
 420. By Geometry there may be proved the following 
 
 IlilNCTPLES. 
 
 1. The CIRCUMFERENCE of every circU is nearhj XUIQ times 
 its diameter. Plence, 
 
 2. The CIRCUMFERENCE is equal to the product of the dUimeter 
 hu'S.lilG; and 
 
 3. The DIAMETER is equal to the quotlmt of the circumference 
 divided hy 3.1416. 
 
 What is a Circle? The Circumference? The Diameter? The Radius? 
 How many times the diameter is the circumference ? To what is the circum- 
 ference equal ? The diameter ? 
 
 il\ 
 
 It • ■ 
 
288 
 
 PRACTICAL ARITHMETIC. 
 
 fi 
 
 
 i 
 
 .:/ 
 
 4. The ARF.A is rqual to the p'oduct of the drcumference hj one 
 half of the radius, or bij one fourth <f the diameter. Iloncc, 
 
 5. The AiiKA is equal to the product of the square of the diameter 
 
 hy .78.'>4 ; and 
 
 0. Tlie DiAAiETER is equal to the square root of the quotient of 
 the area divided hi/ .7854. 
 
 7. The SIDE of ever (I square inscribed in a circle is nenrly .7071 
 times the diameter, or ,2251 times the circumference; also, 
 
 8. The SIDE of ever}) square inscribed in a circle is equal to the 
 square root of half the square of the diameter. 
 
 sN 9. The SIDE of a square equal in area to a given circle is equal 
 to the product of the diameter by .8802. 
 
 Exercises. 
 
 . 1. What is the circumference of a circle whoso diameter is 
 20 feet? Ans. G2.83+ feet. 
 
 ^ 2. What is the diameter of a circle who.5e circnmference is 
 142 yards] Ans. 45.19+ yards. 
 
 / 3. What is the area of a circle whose diameter is 100 yards? 
 
 Ans. 7854 sq. yd. 
 
 / 4. What must be the side of a square stick of timber that 
 can be hewn from round stick 24 inches in diameter ? 
 
 A71S. 10.97 inches. 
 
 •' 5. A wheel is 5 feet in diameter ; what is the length of its 
 tire? Jh^,//:y,y^lrd/^ ' 
 
 •• 0. The area of a circle is 5 acres 14G square rods; what is 
 the diameter? Ans. 34.7 rods. 
 
 7. What is the surface in ares of a circular fish-pond whicli 
 is 50 meters in diameter? Ans. 19 ares 03.5 centares. 
 
 To whfi.t is the area eqiuil ? The side of every inscribed square ? The side 
 of a square equal iu area to a given circle ? 
 
 - \ 
 
s: 
 
 MKNSURATION. 
 
 •J80 
 
 ; hij one 
 cc, 
 
 Jiamder 
 otlcnt oj 
 
 id to the 
 3 is Clonal 
 
 imctcr is 
 1 3 + feet. 
 
 eroncG is 
 + yards. 
 
 yards'? 
 sq. yd. 
 
 )cr that 
 
 7 inches. 
 ;h of its 
 
 ■what is 
 4.7 rods. 
 
 id Avhicli 
 centares. 
 
 ^rhe side 
 
 8. Wha^ is the side of a square eciual in area to a circular 
 plat r)0 fi'et in diameter? Jna. 44.31 feet. 
 
 0. What iTiust bo tlie Icni'th of a tetlier fastened to a horse's 
 neck, that it may s\veep over just one acre ? .liis. 7. l.')G t rods. 
 
 10. How hirgo a square can be cut out of a circular piece of . 
 jdank 300 inclies in circumference 1 ~ j / • ^ l^ ■^ V ^ /^ 6^ -^ */"/ 
 
 11. How many rods in length must be a rope, sucli as, -with ' 
 one end fastened to a stake in a meadow, and the other to the 
 nose of a cow, will allow her to graze over just 2 acres i 
 
 PRISMS AND CVidNDEllS. 
 
 421. A Prism is a volume having two faces equal and 
 I)arallel polygons, and the other faces parallelograms. 
 
 The Bases of a prism are its equal and 
 parallel polygons. 
 
 The Convex Suhface of a prism is 
 formed of its lateral faces, or parallelo- 
 grams. 
 
 The prism is triangular, quadrangular, 
 SzQ,., according as its base is a triangle, 
 quadrilateral, &c. Thus, .. ., 
 
 The diagram represents a pentangular prism, whose bases 
 are A B C D E, and F G IT T K, and whoso convex surface is 
 formed by the faces A B G F, B C H G, kc. 
 
 422. A Cylinder is a round body of uniform 
 diameter, whose bases are equal and parallel 
 circles. 
 
 The Altitude of a cylinder, or of a prism, 
 is the straight line joining the centres of tlie 
 two bases. Thus, 
 
 The diagram represents a cylinder, of which 
 A B is the altitude. 
 
 What ia a Prism .' The Bases of a priaiu? The Convex Surface? A 
 CyUiider? The Altitude ? 
 

 1 
 
 I ! 1 
 
 I ■ i 
 
 f 
 
 X t 
 
 1 
 
 11 
 
 i 
 
 ■i 
 
 i 
 
 
 r' 
 
 
 ■1 '"^ 
 
 j' 
 
 1 i 
 
 f 
 
 \ 
 
 
 i . 
 
 ,i: 
 
 200 
 
 PRACTICAL AUITIIMI:TI0. 
 
 -/ 
 
 423. P>y Ooomctry then! may be estublislicd Llio following 
 
 naxoiiM.Es. 
 
 1. The CONVEX SURFACE of a prism is cijual to the product of 
 the perimeter of the base by the altitude. 
 
 '1. Till' <;()NVKX suiiFACE of a cylinder is equal to the product 
 of the. circumference of the base hij the altitude. 
 
 15. The v.Kluw. SURFACE of a pris7n or ci/lindcr is equal to the 
 co7ivex surface plus the area of the bases. 
 
 4. 2'he CONTENTS of a prism or cylinder are cij[ual to the pro- 
 duct of the area of the base by the altitude. 
 
 Exercises. 
 
 1. What is the entire surface of a square jirisni wiiose side 
 is 4 feet wide and len''th 30 feet 1 
 
 Solution. 30 x 4 x 4 = 480 ; 4 x 4 x 2 = 32 ; 480 +32-512 
 ai{. ft., Ans. 
 
 -f- 2. Kequirod the convex surface of a rolltu* 4 feet in diameter 
 and 10 feet long? y}nf<, 125. GG+ sq. ft. 
 
 3. Kequired the contents of a cylinder DO centimeters in 
 diameter and 10 meters in length. Ans. C.3G + cubic meters. 
 
 4. If each side of the base of a triangular prism be 2 inches 
 and its length 14 inches, what are its contents 1 
 
 5. What are the contents of a stick of timber 22 feet 7 
 
 inches long, 1 foot 5 inches broad, and 6 J inches thick? 
 
 Ans. 17.329+ cu. ft. 
 
 PYRAMIDS AND COXES. 
 
 424. A Pyramid is a body whose base is any polygon, and 
 whose other faces are triangles meeting at a common point. 
 
 To what is the convex K^virface of a prism equal? The convex sui'face of a 
 cyliiulor? The entire surface? Tlie contents of a prism or cylinder? AVliat 
 is ;i Pyramid? 
 
T^^ 
 
 wing 
 
 odud of 
 2)roduct 
 at to the 
 the pro- 
 
 lose side 
 32 - 512 
 
 liamcter 
 + sq. ft. 
 
 LC'tcrs iti 
 meters. 
 
 2 inches 
 2 feet 7 
 {■ cu. ft. 
 
 jon, and 
 )oint. 
 
 irfuce of a 
 !!•? VThat 
 
 291 
 
 S 
 
 MENSirUATlON. 
 
 Tlu! Vkiiti;x is tiie common point at 
 which the trianguhir faces meet. 
 
 The CoNVKX SuriFACH is formed of 
 the triani^ular faces. 
 
 The diiiL^ram represents a pentangular 
 pyramid, whose vertex is S, and wlio.se 
 convex surface is formed by tlic faces 
 A S W, r> S C, C S I), iVc. 
 
 425. A Cone is a 1)ody whose base is 
 
 A a circle, and whose convex surface tapers 
 
 unif(jrmly to a poiiit at the top or vertex. 
 
 The Altitude of a pyramid or cone is 
 a straight line drawn from the vertex per- 
 pendicular to the base. Thus, 
 
 In the diagram the line A B represents 
 the altitude of a cone. 
 
 The Slant IIkight of a pyramid or cone 
 is the shortest straight line that can be 
 drawn fr(jm the vertex to the perimeter 
 or circumference of the base. Thus, 
 
 In the diagram the line A C represents the slant height of 
 the cone. 
 
 426. The Frustum of ^ pyramid or 
 cone is the i)art which reu... ns after cut- 
 ting off the top by a plane parallel to the 
 base. Thus, 
 
 The diagram C D E F represents the 
 frustum of a cone. 
 
 427. J3y Geometry there may bo established the follow- 
 
 ing 
 
 What is the \'crtcx of a pyramid? How is the Convex Surface formed? 
 Whu is a Cone? The Altitude of a pyramid or cone? The Slant Height? 
 The Frustum of a [ yrainid or cone? 
 
292 
 
 PRACTICAL ARTTEMETIC. 
 
 ' i 
 
 PRINCIPLES. 
 
 1. The CONVEX SURFACE of a pyramid or cone is equal to 
 the product of the pjerimeter or circumference of the base hy half 
 the slant height. 
 
 2. The ENTIRE SURFACE of a X)yramid or cone is equal to the 
 convex surface plus the area of the base. 
 
 3. The CONVEX SURFACE of the frustum of a pyramid or cone 
 is equal to half the product of the sum of the perimeters or circum- 
 ferences of the two bases by the slant height. 
 
 4. The ENTIRE SURFACE of a frustum of a j^yramid or cone is 
 equal to the convex surface plus the areas of the two bases. 
 
 5. The CONTENTS of a pyra77iid or cone are equal to the pro- 
 duct of the area of the base by one third of the altitude. 
 
 G, The CONTENTS of a frustum of a pyramid or cone are eqiial 
 to the sum of the areas of the tiro bases phis the square root of 
 their product, multiplied by one third of the altitude. 
 
 Exercises. 
 
 "/I. What is the surface of a square pyramid, each side of 
 whose base is 3 feet, and the slant he'ght 24.05 feet? 
 
 A71S. 153.3 sq. ft. 
 -/' 2. Required the number of yards of canvas that will cover 
 a conical tent the slant height of which is 20 feet and circum- 
 ference of the base 60 feet. Ans. 66|- sq. yd. 
 3. If the slant height of a frustum of a triangular pyramid 
 is 12 decimeters, each side of the one base 15 decimeters, and 
 of the other base 9 decimeters, how many square m.eters is its 
 entire surface? Ans. 5.6449 sq. meters. 
 / 4. If one of the largest of the Egyptian pyramids is 477 feet 
 in slant height, and each side of its base, which is square, is 720 
 feet, what are the contents in solid yards ? Ans. 2003200 cu. yd. 
 
 To what is the convex surface of a pyramid or cone equal ? The entire 
 surface ? The convex surface of the frustum of a pyramid or cone ? The 
 entire surface? The contents of a pyramid or cone? Of a frustum of a 
 pyramid or cone ? 
 
^ 
 
 equal to 
 hy half 
 
 I to the 
 
 I or cone 
 circum- 
 
 " cone is 
 the pro- 
 ire equal 
 root of 
 
 side of 
 
 3 sq. ft. 
 .11 cover 
 
 circum- 
 ; sq. yd. 
 pyramid 
 ers, and 
 rs is its 
 
 meters. 
 477 feet 
 e, is 720 
 ) cu. yd. 
 
 'he entire 
 aie? The 
 jtum of a 
 
 MENSURATION. 
 
 29.3 
 
 ' t 
 
 5. Required the number of cubic feet in a conical stack of 
 hay whose lieight is 21 feeo and the diameter of whose base is 
 ^••5 ^^^^- Ans. 496. 17G cu. ft. 
 
 0. Tlie diameter of the larger end of a round spar is 30 
 inches, that of the smaller end 18 inches, and the length 45 
 feet; required its contents. Ans. 144.31 +°cu. ft. 
 
 7. If the length of a stick oi timber, in the form of the frus- 
 tum of a pyi-amid, be 18 feet 8 inches, tlie side of its larger 
 end 27 inches, and that of its smaller IG inches, how many 
 cubic feet are there in it ? Ans. G1.228 + cu. ft. 
 
 S P H E K E S. 
 
 428. A Sphere is a volume bounded by a curved surface, 
 all points of whicl. are equally 
 distant from a point within 
 called the centre. 
 
 429. The Radius of a sphere 
 is a straight line drawn from the 
 centre to any point in the sur- 
 face. 
 
 430. The Diameter of a 
 
 sphere is a straight line drawn 
 through its centre, and termi- 
 nated both ways by the surface. Thus, 
 
 ^ In the diagram the line C B denotes the radius and D E the 
 diameter of a sphere. 
 
 By Geometry there may be proved the following 
 
 PRINCIPLES. 
 
 1. The SURFACE of a sphere is equal to the product of the clr- 
 
 Review Questions. What is Mensuration? (400) A Triangle ? (410) A 
 Quadrilateral? (412) A Circle ' (418) A Prism? (421) A Pyramid '(424) 
 A Cone? (425) 
 
 What is a Sphere ? The Radius of a sphere ? The Diameter ? 
 
 !!* 
 
 m 
 
^■^^ 
 
 II I 
 
 '■ 'i' ' '■ 
 
 ■: 
 
 
 
 'M 
 
 
 I 
 
 294 
 
 PRACTICAL AUITIIMETIC. 
 
 cum.ference hj the diameter, or to the p'oduct of 3.1416 hy the 
 square of the diameter. 
 
 2. The CONTENTS of a sphere arc equal to the product of the 
 FAirface hy one third of its radius, or to the product of one sixth 
 0/3.1416 hy the cuhe of the diameteu 
 
 Exercises. 
 
 I 
 
 1. What is the surface of a cannon ball whose diameter is 
 9 inches? Ans. 254.46+ sq. in. 
 
 ^ 2. How many cubic meters in a sphere wliose diameter is 
 12 centimeters ? Ans. .000904 + cu. me. 
 
 3. Kequired the contents of a globe 15 inches in diameter. 
 
 Ans. 1707.15 cu. in. 
 
 "' ' 4. What is the surface of the earth, allowing it to be a 
 sphere 7912 miles in diameter i Ans. 19GGG3355.75 + sq. m. 
 
 SIMILAR FIGURES AND VOLUMES. 
 
 431. Two Figures, or two Volumes, are similar, when they 
 exactly correspond in form, without regard to size. 
 
 From the relations of similar figures, or of similar volumes, 
 to each other, which may be proved, by Geometry, we have the 
 
 following 
 
 PRINCIPLES. 
 
 1. The areas of similar figures and volumes are to each other as 
 the squares of their corresponding dimensions. Hence, 
 
 2. The corresponding dimensions of similar figures and volumes 
 are to each other as the square roots of their areas. 
 
 3. The contents of similar volumes are to each other as the cubes 
 of their corresponding dimensions. Hence, 
 
 To what is tlie surface of a sphere equal ? The contents of a sphere ? 
 When are two Figures or two Volumes similar? What is the first Principle? 
 The second? The third? 
 
m 
 
 HMi 
 
 MENSURATION'. 
 
 295 
 
 4. The corretpondiiuf dimenaions of similar volumes are to each 
 other as the cube roots of their contents. 
 
 
 Exercises. 
 
 1. If a triangle wliose base is 20 feet has an area of 200 feet, 
 wliat i^^ tlie area of a similar trianirle whose base is 10 feet % 
 
 o 
 
 Solution. 20- : 10- :: 200 : 50, Ans. 
 
 2. If a circle whose diameter is 1 2 feet has an area of 1 1 .3.00 
 square feet, what is the area of a circh^ whose diameter is 1/5 
 feet? ytns. 170.70 sq. ft. 
 
 3. If it costs $125 to pave a rectangular court wlidse width 
 is to fe(!t, what will it cost to pave a similar court whose 
 width is 30 feet ] Jns. .$70.31 ,',. 
 
 4. If a triangle whose altitude is 40 feel has an area of 1000 
 square feet, what is the altitude of a similar triangh; whose 
 area is 900 feet? ylns. 37.047 ft. 
 
 5. If the weight of a cannon ball 8 inches in diameter is 3G 
 kilos, what is the weight of a similar ball 9 inches in diameter ? 
 
 Alls. 51.25 I kilos. 
 
 6. If a s[)here of silver 1 inch in diameter be worth ^O, Avhat 
 must be the diameter of another sphere to be worth $103081 
 
 Ans. 12 inches. 
 
 7. A bushel measure is 18.^ inches in diameter; what must 
 be the diameter of a half bushel measure of similar form 1 
 
 ylns. 14.08 + inches. 
 
 8. If the side of a cubical box is 2 feet, what must be the 
 side of a similar box which shall contain 3 times as much] 
 
 Ans. 2.88+ feet. 
 
 9. If a cylindrical pipe 20 centimeters in diameter will fill a 
 cistern in Hi; minutes, how long will it take a similar pipe 30 
 centimeters in diameter to fill it ? Ans. 5 minutes. 
 
 10. If two men own together a conical stack of hay, which 
 
 sphere ? 
 ■inciple ? 
 
 Review QuESTlOxNS. AVliat is a Splierc? (42?>) The Radius of a sphere? 
 he Diameter of a sphere ? (430) 
 
 (429) 
 
296 
 
 PRACTICAL ARITHMETIC. 
 
 I ! 
 
 is IG foet ill height, liow far (hnvii from the top must one of them 
 take off for his part, if it is y of tiie whole] Ans. 8 feet. 
 
 BOARD MEASURE. 
 
 432. Lumber, or sawed timber, as boards, pLanks, joists, and 
 beams, are usually measured by board measure. 
 
 In Board Measure 1 foot is reckoned 1 foot long, 1 foot 
 broai], and 1 inch thick. Hence, 
 
 433. To find the contents of boards, planks, joists, &c., 
 
 Mnltiply the product of the length and hreadtli, each taken in 
 feet, h)j the numher denoting the thickness in inches. 
 
 When the boards, planks, &c., arc tapering, take half tlio sum of 
 the hreadth of the two ends for the l)readth. 
 
 Since 1 foot board measure is 1 foot or 12 inches long, 1 foot or 12 
 inclies broad, and 1 inch thick, it must be equal 12 x 12 x 1 = 1 44 
 cul)ic inches. 144 cubic inches are containud in 1728 cubic inches, 
 or in 1 cubic foot, 12 times. Hence, 
 
 12 hoard feet = 1 cubic foot. 
 
 \ 
 
 Exercises. 
 
 1. What are the contents of a board 20 feet long and IG 
 inches broad? Ans. 26| bd. ft. 
 
 N 2. How many square feet in 2 planks, each 16 feet long, 18 
 inches wide, and 3 inches thick? Ans. 144 bd. ft. 
 
 • 3. W'hat are the contents of G joists, 14 feet long, and 4 
 inches square? Ans. 112 bd. ft. 
 
 ]__ 4. What is the cost of a stick of timber 24 feet long, 10 
 inches wide, and G inches thick, at 3 cents a square foot 1 
 
 Ans. $3.60. 
 
 ' 5. What are the contents of a plank 22 feet long, and Sc- 
 inches thick, the width of the ends being 16 and 20 inches re- 
 spectively ? Ans. 115^ bd. ft 
 
 How is Lumber usually measured ? How do we find tho coutents of boards, 
 pliviiks, &c.? 
 
/.■ 
 
 MENSURATION. 
 
 GAUGING. 
 
 297 
 
 434. Gauging is the process of finding the capacity of casks. 
 Tlie Mean Diameter of a cask is very nearly equal to the 
 
 head diameter increased by two thirds of the ditference between 
 the bung and head diameters, or by three fifths when the staves 
 are but slightly curved. 
 
 The capacity of a cask is that of a cylinchir of the same 
 length and mean diameter. Hence, 
 
 435. To find the capacity of casks, 
 
 Multiply the product of the square of the mean diameter aihl 
 the length, expressed in inches, hy .0034 for lifpiAd gallons, or by 
 .01 29 /or liters. 
 
 ]\rultip]yiii^' by .00:34 is the same as nlulti])lyill,^' by .7854 and 
 dividing by 231, and by .0129 the same as by .78r,4 and dividing by 
 61.022. 
 
 Since a liter is equal to one cubic decimeter, there will be in a cask 
 1000 times as many liters as cubic nxeters. 
 
 oh 
 
 Exercises. 
 
 'Tl. How many gallons in a cask whoso mean diameter is 18 
 inches, and whose length is 30 inches? Am. 33 + gallons. 
 
 /. 2. How many gallons in a cask 3G inches long, 22 inches 
 bung diameter, and 1 G inches head diameter ? . . 
 
 Am, 48. 9 G gallons. 
 I 3. How many gallons in a cask whose length is GO inches, --'-^r' 
 bung diameter 3G inches, and head diameter 32 inches ?-=s ^JJ-io'' ^''*" 
 
 4. How many liters in a cask 1 meter long, and whose mean 
 diameter is GO centimeters/ Ans. 282.744 liters. 
 
 What is Gauging? The Mean Diameter of a cask ? What is the capacity 
 of a cask? How can we find the capacity of casks? How may the capacity 
 of a cask in liters be found, when its conlents in cubic meters are known ? 
 
r% 
 
 :;i5r- 
 
 \ 
 
 m I 
 
 208 PRACTICAL ARITHMETIC. 
 
 MEDIAL PROPORTION. 
 
 436. Medial Proportion, or Average., treats of mixing dif- 
 ferent articles. 
 
 This suLject has Pomotimea hoen calhul Alligation, from thn 
 mechanical method formerly adopted of liiikin.L;' or tying together 
 ligures by a line, in the process of solving its questions. » 
 
 CASE I. 
 
 437. To find the average value of given quantities of dif- 
 ferent values. 
 
 1. Let it be required to find the average value of a mixture 
 of 8 lb. of sugar, worth 10 cents a pound, ^vith 12 lb., worth 
 15 cents a pound. 
 
 OPERATION. ^ pounds of sugar, at 10 cents a pound, are 
 
 . ,,> r. A OA v.'orth $.80, and 12 pounds of sugar, at 15 
 
 ^. lU X o = S.o'' 
 
 cent.', a pound, are worth $1.80; hence, the 
 
 .Ij X 12 = 1.80 ^^i^^i^ 20 pounds are worth $.80 + $1.80, or 
 ~~ $2.60. 
 
 If 20 pounds of the mi.Kture are worth $2.(50, 
 one pound of it is worth r}^J of $2.00, which is 
 $.13. 
 
 Rule. Divide the entire value of the mixture by the entire 
 quantity, and the result will he the average value. 
 
 Examples. 
 
 y 2. A farmer mixed 8 bushels of oats, worth 50 cents a bu.shcl, 
 12 bushels of corn, worth 65 cents a bushel, and 10 bushels of 
 barley, worth CO cents a bushel ; what was the average value 
 of the mixture a bushel? Ans. $.59^j. 
 
 I ^ 3. A grocer mixed together 18 lb. of oolong tea, at $1 a 
 
 » pound, 6 lb. of souchong, at $.G0 a pound, and G lb. of hyson, 
 
 at $1.20 a pound. How much a pound is the mixture worth] 
 
 Ans. $.9G. 
 
 What is Medial Proportion ? Explain the operation. What is the Rule ? 
 
 
 20 )2.G0 
 
 $.13 
 
^lEDlAL PROPORTION. 
 
 299 
 
 mixing (lif- 
 
 h from tlin 
 111,' to^ret'ner 
 
 ies of dif- 
 
 ;i mixture 
 lb., worth 
 
 pound, are 
 ,yar, at 15 
 lience, the 
 ■ $1.80, or 
 
 Tth $2.60, 
 , wliich is 
 
 '/^e entire 
 
 I bushel, 
 isiicls of 
 ^e value 
 ■• $.591. 
 
 ifc $1 a 
 hyson, 
 worth ? 
 s. $.90. 
 
 e Rule ? 
 
 18 C. 
 
 r 1 lb. 
 I 2 1b. 
 ■| -I lb. 
 ' lib. 
 
 casp: it. 
 
 438. To find what quantities of different kinds must he 
 taken to form a mixture of a given value. 
 
 1. Let it be required to find what quantities of cofTee at 14 
 cents, 10 cents, II) cents, and 22 cents a pound, must be' taken 
 to form a mixture worth 18 cents a pound. 
 
 OI'ERATION. 
 
 ' 14 c, to gain 1 e. take ] lb. ] 
 ^C c., „ „ „ 1 lb. j 
 19 c, to lose 1 c. „ i lb. ^ X 4 - 
 
 22 «•' . n „ -} lb. J 
 
 If we take 1 11>. at 14 cents to form a mixture wortli 18 cents, we 
 gam 4 cents, and to gain 1 cent we must take { of all. 
 
 If we take 1 11,. at 22 cents, we shall lose 4 cent., and' to lose 1 cent, 
 so a.s to balance exactly what we have just gained, w. must take ' of 
 a Jb. * 
 
 In like maimer, we find we must take }. of a ]1>. of the ] G cent kind 
 t« gain 1 cent and to lose 1 cent, t.. exactly halance that gain, we 
 must take of the 19 cent kind 1 11). 
 
 Therefore, we may take ^ \h of the coffee at 14 cents -^ ]1, at Hi 
 cents, 1 11). at 19 cents, and ] 11,. at 22 cents, or hy multiplvit.; these 
 proportionals by 4, the least common mHti|>le of ihe denou'iinalors of 
 the fractions, which will make all the quantities integral, we may 
 take 1 11). at 14 cents, 2 lb. at IG cents, 4 II, at 19 cents, an.l 1 11> at 
 22 cents, to form a mixture ^^■orth 18 cents a pound. 
 
 Also since proportional quantities will remain proportional, when 
 multiplied or divided by any cpiantity, an indefinite nuinb.u- of re- 
 sults, all answering the conditions of the question, may l)e obtained. 
 
 Rule. Find how much must he taken of a hind tchose value is 
 k^s tha.n the given average, to gain 1 of that averarje ; also how 
 mmh must he taken of a kind whose value is greater than the given 
 average, to lose 1. 
 
 In like manner, compare the value of each of the other kinds 
 uith the average value. 
 
 _ If there are fractions in the results, multijHj/ each of the numbers 
 
 Explain the operation. What is tlie HuIq. ' 
 
300 
 
 PRACTICAL ARITHMETIC. 
 
 hii the lead common mvlt'iple of their ilevomliiators, and the several 
 jjroducts wilt he the required imrtSf in integers. 
 
 Wg may also clear olF fractions simply Ly multiplyinj,' each pair of 
 results correspond in;^' to conplets of quantities compared, by the least 
 common mulli[»le of tlie denominators of their own fractions. 
 
 ' ! , ( 
 
 Examples. 
 
 2. How mucli sugar, at 10, 14, 17, and 18 cents a pound, 
 may be mixed together, so that the mixture shall be wortli IG 
 cents a pound ? .^ns. 1, 3, G, and 3. 
 
 3. How much rice, at 4, G, and 11 cents a pound, may be 
 mixed, so that the compound shall be worth 7 cents a pound ] 
 
 4. A grocer wishes to mix brandy at 3, 5, and 7 dollars a 
 gallon, with water, so that the mixture may be worth $4 a 
 gallon ; how many gallons of each kind may be taken 1 
 
 5. A farmer ])ought pigs at $G each, sheep at $9 each, and 
 colts at $10 each; how many may he have bought if he paid 
 for all on an average of $8 apiece ? Ans. 2, 2, and 1. 
 
 CASE III. . 
 
 439. To find the quantities of other kinds, when the 
 quantity of one kind in the mixture is limited. 
 
 1. Let it be required to find how much coffee, at IG, 20, and 
 24 cents a pound, must be mixed with 10 lb., at 18 cents, so 
 that the mixture may be worth 22 cents. 
 
 OrERATION. 
 
 r IG c, to gahi 1 c. take 
 
 >> 
 
 20 c., 
 
 5> 
 
 Ic. 
 
 )> 
 
 41b. 
 ilb. 
 
 ilb. 
 
 Gilb. 
 
 ^^^^=120 lb. 
 
 [24c.,tolosel + l + lc. „^+| + llb.^ 
 10 lb. -f \ lb. = 40 times. 
 
 [go lb. 
 
 By comparing the price of the articles with the average price (Art. 
 438), we tind, by taking \ lb. of the 16 cent article, ^ lb. of the 18 cent, 
 and ^ lb. of the 20 cent, there is a total gain of 3 cents, which may be 
 
 Explain the 'operation. 
 
the several 
 
 ach [)aii' of 
 )y the least 
 
 )I1S. 
 
 a pound, 
 wortli Ki 
 G, and 3. 
 
 1, may bo 
 a pound I 
 dollars a 
 
 ^rth ^\ a 
 1? 
 
 each, and 
 P he paid 
 
 2, and 1. 
 
 hen the 
 
 20, and 
 cents, so 
 
 ■{ 
 
 ^\h. 
 
 10 lb. 
 20 lb. 
 60 lb. 
 
 ice (Art. 
 18 cent, 
 I may be 
 
 MEDIAL PROPORTION. 
 
 301 
 
 balanced Ity taking i + ^ + -^ = 1.^ lb. i»f the 24 cent, at a corre- 
 Bpoiuling loss of 3 cents. 
 
 Jiiit of the 18 cent kind it is rcquiivd to take 10 lb., or 40 times 
 I lb. ; and taking the other proportional parts also 40 times as large, 
 we have, as the required mixture, ()f; lb. of the IG cent article, 10 lb. 
 of the 18 cent, 20 lb. of the iJO cent, and GO lb. of the 24 cent. 
 
 KULE. Find tJie pro])ortioJial quantities d.s in the jwcceili));/ 
 case, and taJce these proportional qxantities as mant/ times as lar</e 
 a.v the proportional quantitij found of the pro2:)osed rate is con- 
 tained times in the given quantity. 
 
 Examples. 
 
 2. A merchant lias sugar at 9, 10, 13, and 14 cents a pound; 
 how much may be mixed with 30 pounds of the 14 cent kind, 
 to make a mixture worth 12 cents per pouiul ? 
 
 3. How much water of no value may be mixed with 50 liters 
 of wine, at $.90 per liter, to reduce the price to ^^75 per liter \ 
 
 Jns. 10 liters. 
 
 4. Sold some cows at 80, 60, and 40 dollars each ; also, 30 
 at 100 dollars each, and found that the whole averaced 70 
 dollars each ; how many may there have been sold of the first 
 three kinds? And. 1, 1, and 30. 
 
 CASE IV. 
 
 440. To find the quantity of each kind, when the entire 
 quantity is limited. 
 
 1. Haw much sugar, of kinds at 10, 12, 18, and 20 cents a 
 pound, must be taken to fill a box with a mixture of 200 
 pounds at 15 cents a pound? 
 
 OPERATION. 
 
 ' 10 c, to gain 1 c. take i lb. "] 
 
 15 
 
 c. i 
 
 12 c., „ Ic. ., ilb 
 
 )> 
 
 >> 
 
 18 c, to lose 1 c. 
 . 20 c, ,, 1 c. 
 
 200 lb. -f -[f lb. = ^-^ times. 
 
 
 3 "■"• L V 37 5 
 
 i lb. i ^ 
 i lb. J 
 
 = -< 
 
 r 371 lb. 
 
 02^ lb. 
 
 62^ lb. 
 I 37i lb. 
 
 llepeat the Rule. Explain the operation. 
 
I 
 
 1: 
 
 ii 
 
 302 
 
 PHACTICAL ARITIIMKTIO. 
 
 We fiiul ihii jjroportionul qiiaiititics (Art. 4138) to ho • lb. of thn 10 
 cent kind, \ Uj. of the Vl cent, \ lb. of the 18 cent, iind \ Ih. of tin) 
 20 cen!. 
 
 'JMk! sum of these quantities is only \\ Ih. ; but the entire (juantity 
 of tilt; jiroposcd mixture must be 2i)() lb. ; hence, cuch of the propor- 
 tional parts must be taken as many times as large, as |JJ lb. is con- 
 tained times in 200 lb., or •'J*'"' times. 
 
 "^ ,. « - 
 
 Taking J- lb., ;\ lb., ;", lb., and \ lb., respectively, -„ times, we 
 
 have Wll \\\ of the 10 cent kind, G2.V lb. of the 12 cent, G2.\ lb. of the 
 18 cent, and L'-T^ lb. of the 20 cent. 
 
 KULI']. Find the pnqoortional quantities as in precedinf/ cases^ 
 and take each of these quantities as many times as large as their 
 sum is contained times in the given entire quantity. 
 
 Examples. 
 
 2. What quantities of tea, worth $.9G, $.90, and .$.78 per 
 pound, respectively, may be taken to form a mixture of 112 
 pounds, at $.88 per pound? Ans. 1^1% 07^^, 27-3^. 
 
 3. A farmer has oats worth 40, GO, and 70 cents a bushel ; 
 what quantity of each kind may he take to make 40 bushels 
 worth i50 cents a bushel ? 
 
 4. A jeweller has gold 18, 19, and 24 carats fine ; what 
 quantity of each may he take to make 1 pound 20 carats fine ? 
 
 Ans. ;f, 2^, -^. 
 
 5. A grocer has sugar worth 7 cents a pound, which he 
 would mix with some at 8 cents a pound, some at 10 cents, 
 and some at 11 cents a pound. How much of each kind may 
 he take to make a mixture of 90 pounds worth 9 cents a 
 pound? 
 
 C. IIow much wine, at $2.40, $2.60, $2.80, and $2.90 per 
 gallon, may be taken to make a hogshead worth $2.70 per 
 gallon? Ans. 18, 9, 9, 27. 
 
 What is the Rule? What is Medial Proportion? (436) What is it Bome- 
 times called? (436) What is the Rule in Case I. ? (437) In Case II. ? (436) 
 In Case III. ? (439) 
 
'. of thft 10 
 lb. of thu 
 
 o (jiuintity 
 Ik; propor- 
 11). is cou- 
 
 tiiiu'S, we 
 lb. of the 
 
 'ing cafte.t, 
 ? as their 
 
 $.78 per 
 c of 112 
 
 29 97 6 
 
 bushel ; 
 
 bushels 
 
 5 ; what 
 
 ats fine ? 
 Ill 
 
 ¥> 2) 4- 
 
 hich he 
 
 cents, 
 
 ind may 
 
 cents a 
 
 90 per 
 70 per 
 I, 9, 27. 
 
 it Boine- 
 :i. ? (io6) 
 
 aKUIKK. 
 
 303 
 
 SERIES. 
 
 441. A Series is a sucoession of numbers that dopend on 
 one anotlifT according to some fixed law. 
 
 The Tkiims of a Series are the numbers composing it ; 
 
 The ExTUKMKS are the first and last terms ; and 
 
 The ^Ikans are all the terms between the first and last. 
 
 442. A series is incrcas'uKj when the terms incrcasf; from 
 k4rt to riglit, and decrcas'uKj when tlnjy decrease f'rcin left to 
 right. 
 
 AKITIIMETICAL SERIES. 
 
 443. An Arithmetical Series, or Progression, is a series in 
 which the terms vary by a common diirerence. Thus, 
 
 2, 4, 6, 8, 10, 12, 
 
 is an increasing arithmetical series, in which 2 is the common 
 di fie re nee. 
 
 444. The first term, the last term, the number of terms, the 
 common diflerence, and the sum of the terms, in an arith- 
 metical series, are so related that, any three of them being 
 given, the other two may be found. 
 
 CASE I. 
 
 445. To find any term in an arithmetical series. 
 
 Let S = the firdt term, 2 = the conmion Jiirerence, and 5 = tlie., 
 number of terms. Then, 
 
 2d term = 3 + 2 ; 3d term = 3 + 2x2; 4th term = 3 + 2x3; 
 5th term = 3 + 2x4. 
 
 That is, when the first term, the common diffsrence, and the num- 
 ber of terms are given, to find any term of the series, 
 
 To the first term add the jjvoduct of the common difference hy 
 the number of terms which precede the required to'm, if the series 
 
 
 What is a Series ? The Terms of a series ? The Extremes ? The Means ? 
 Aa Arithmetical Series ? 
 
304 
 
 ntACTICAL ^ .'.ITHMCTIC. 
 
 I 
 
 
 in increaslrtff ; or from the first ^erm subtract the same if the series 
 is dccreasimj. 
 
 Exercises. 
 
 \ 1. If tlio figos of fivo persons arc in ■•iritlunctical progn^.ssion, 
 tlic youngest being IT) years old, and the common ditlerence 
 is 2, wliat is the ago of the oldest ? Ans. 23 years. 
 
 ^ /-- 2. A merchant l;oiight 34 yards of cloth, and agiTcd to give 
 12 cents for tlie first yard, 12.'- cents for the second, 12r| cents 
 for the third, and so on ; what <lid tho thirty-third yard cost 
 him? Ans. 22!j cents. 
 
 K 3. I iiavo 40 books; the first is worth $1.80, the second $1.77, 
 tlic third $1.74, and so on, each being wortli 3 cents less than 
 the preceding; what is the value of the last book? Ans. $.03. 
 
 
 CASE 11. 
 
 446. To find the common difference and the number of terms. 
 
 Let 3, r>, 7, 1), 11, be an arithmetical series. 
 
 Then, by the preceding case, the last term 11 = 3 + 2x4, and 
 subtracting the lir.st tevni 3, we have 2x4, or the product of the 
 coiimiuu diilerence by the number of terms less one. 
 
 Jlence, to find the connnon difference, 
 
 Divide the dijjh'ence of the extremes by the number of terms less 
 one. 
 
 Also, to find the number of terms, 
 
 Divide the difference of the extremes by the common difference^ 
 and add one to the quotient. 
 
 Exercises. 
 
 1. The extremes of an arithmetical series are 5 and 27.|, and 
 the number of terms 1 1 ; what is the common difference 1 
 
 Ans. 2-}. 
 
 2. The extremes of an arithmetical series are 5 and 27-~, and 
 tlie common difference 2] ; what is the number of terms ? 
 
 How can we fimd the commun difference? The number of terms? 
 
if the series 
 
 rogni.ssion, 
 
 (lilU'icnco 
 
 >'. 2') years. 
 
 iH'(l to give 
 
 , 1 ^2f^ cents 
 
 I yard cost 
 
 22- cents. 
 
 3ond$1.77, 
 
 s less tliaii 
 
 And. |.G3. 
 
 jr of terms. 
 
 [2x4, ami 
 iict of the 
 
 f terms less 
 
 difference, 
 
 2 7 1, and 
 ;nce? 
 Ans. 2|. 
 
 271, and 
 rms? 
 
 ierms ? 
 
 8kkii;m. 
 
 nor. 
 
 3. A [terson travcUbiL; went the first day 3 miles, antl iii- 
 cn-ased luis spewed every day hy 5 miles, till at la.st he weii' ^^'li^ 
 miles in one day ; how many ilays did he travel ? .Ins. 12 days. 
 
 CASK 111. 
 
 447. To find the sum of all the terms of an arithmetical 
 fceries. 
 
 Let 2, 4, fi, 8, lo, IJ, hi; an arithnictical scries, 
 and 1:.', 10, 8, (1, 4, 2, hi- the sumo inverted. 
 
 Tliiii li t- 14 + 14 I- 14 + 14 V 14 = 84, thesuniefhoth scrii'H. 
 
 But 84 is t'(jual to 11, tlic sum oi" tlu; cxtreiiies, niultiiilieil by t!, the 
 number of tunus ; and hall" of 81, or ol" 14 x G, is thu sum of one of 
 the series, llonce, to Ihid the sum of all the tcnus, 
 
 Muldplt/ the sura of the extremes by the nuinO.r oj terms, and 
 take half the product. 
 
 Exercises. 
 
 1. The clocks of Venice strike IVoni 1 to 21 ; how many 
 strokes do one of these clocks make in one day? An.^. 300. 
 
 2. If a person on a journey travel the first day 30 miles, and 
 each succeeding day a quarter of a mile less than he did tlie 
 day before, how far will he travel in 30 days ? Ans. 7'J 1 [ miles. 
 
 3. If 100 eggs bo laid one yard distant from one another in 
 a straight line, and a ])asket be placed one yard from the first 
 one, what distance must a person travel to gather them singly 
 and place them in the basket? Ans. 5 m. 1300 yd. 
 
 GEOMETRICAL SERIES. 
 
 448. A Geometrical Series, or Progression, is a scries in 
 which the terms vary by a common multiplier. Thus, 
 
 3, 9, 27, 81, 243, 
 is an increasing geometrical series, in which 3 is the common 
 multiplier. 
 
 How do we fiiid the sum of all the terms ? Whut is a Geometrical Series? 
 
 U 
 
 i -^ ■-":¥> ■.iSia:*'--^'. . . .-. -■ 4-"i .^ji ■ 
 
30G 
 
 PPwACTICAL AIUTIIxMETlC. 
 
 The Eate, or Ratio, of a geometrical series is the common 
 multiplier. 
 
 CASE I. 
 
 449. To find any term in a geometrical series. 
 
 Let 4 = the first term, 2 — tlie rate, and 5 = the number of terms. 
 Tiieii, 
 
 2(1 term =.4x2; 3(1 term ==4x2'; 4th term = 4x2^; 5th term 
 = 4 X 2^ 
 
 That is, when the first term, tlie rate, and tlie numher of terms aio 
 given, to find any term of the series, 
 
 Multiphj the first term hy the rate raiseil to a 2'>ower tvlun^e 
 exponent is equal to the numher of terms which iweciide the 
 required term. , 
 
 Exercises. 
 
 1. Find the 8th term of a geometrical series whose first term 
 is 6 and rate 2. Ans. 768. 
 
 2. Find the Gth term of a geomtlrical scries whose first tenn 
 is 409 G and rate \. Am. 4. 
 
 3. A gentleman dying left 11 sons, to whom he bequeathed 
 his property, as follows : to the youngest he gave $1024 ; to the 
 next, as much and a half; to the next, 1^ of the preceding sou's 
 share, and so on. What was the eldest sou's portion \ 
 
 Ans. $59049. 
 
 CASE II. 
 
 450. To find the rate of a series. 
 
 Let 4, 8, 16, 32, 64, be a geometrical fcries. 
 Then, by the preceding case, the fifth term 04 = 4 x 2*, and divid- 
 ing by the first term, we have 2^, or the fourth power of the rate. 
 Hence, to find the rate of a geometrical series. 
 
 Divide tlie last term by the first, and extract that root of the 
 quotient ichose index is denoted hy the number of te^ms less one. 
 
 Wliat is the Kate or Ratio of a geometrical series? How do you find any 
 tonu ? The rate ? 
 
^^-Offf 
 
 3 common 
 
 ' of terms. 
 
 ; 5tli'teriu 
 
 terms aie 
 
 ')er ivhose 
 ecede (he 
 
 irst term 
 
 tns. 768. 
 
 rst tenn 
 
 ylns. 4. 
 
 [uoatlied 
 : ; to tlie 
 ng son's 
 
 S59049. 
 
 d di^ 
 
 'id- 
 
 ratii. 
 
 
 
 ^/<e 
 
 ' one. 
 
 
 tiiui any 
 
 
 ■iPi 
 
 SERIES. 
 
 Exercises. ' 
 
 307 
 
 1 Find the rate of a geometrical series whose first term is 6, 
 last term 768, and number of terms 8 ./.,.. •)' 
 
 2. Find tlie rate of a geometrical series whose first term is 
 4()d6, last term 4, and number of terms 6. y^ns, ' 
 
 3. A man paid a debt by making 12 payments in geometri- 
 cal progression, the tirst payment being 83, and the last86141 • 
 
 what was the rate ? . ' 
 
 Ans. J. 
 
 CASE III. 
 
 451. To find the sum of all the terms of a geometrical series. 
 Let 4, 12, an, 108, be u geometrical series; and niultiplyincr it 
 
 by tlie rate 3, we have 
 a vsecond series 
 the first series, 
 
 12, .30, 108, 324, whose sum is 3 times 
 4^2,J6, 108. Subtracting, we have 
 
 twicethesumof tlie firsts - 4, o", 0, 0, 324, or 324 - 4 ; that i. 
 the sum of the firsts ^^;;^^^ which is the dim.rence between the first 
 term and the product of the last term by the rate, divided by the 
 rate less one. Hen^e, to find the sum of all the terms, 
 
 Multi2>Ii/ the last term hj the rate, subtract the first term fram 
 thep'oduct, and divide the result hij the rate less one. 
 
 Exercises. 
 
 1. Find the sum of a geometrical series whose extremes are 
 2 and ILVS, and rate 4. Ans. \1^). 
 
 2. If the descendants of tho 101 persons who landed at 
 Plymouth, in the year 1G20, had increased so as to (h.uble their 
 number in eyery 20 years, how great would have been the 
 aggregate to the year 1860 ? _^^^,^._ 41359-, 
 
 3. A jockey offered to sell some fine horses to a young man 
 not well yersed in numbers, and rec.'ive in payment $1 fbr the 
 tirst, $3 for the second, $9 for the third, and so on. The 
 young man, thinking it a great bargain, agree.l accordingly; 
 what d.d the horses cost him, proyided there were 12 of them \ 
 Ans. $265720. 
 
 How do we find the sum of ull the tu 
 
 rms 
 
I • 
 
 I 
 
 1 
 
 i 
 
 in 
 
 
 m 
 
 308 PRACTICAL ARITHMETIC. 
 
 ANNUITIES. 
 
 452. An Annuity is a fixed sum of money payable at the 
 end of equal periods of time. 
 
 An annuity is said to be forhorne, or in arrears, when pay- 
 ments are not paid when due. 
 
 453. The Amount, or Final Value, of an annuity is the 
 sum of the amounts of all its payments at interest from the 
 time each becomes due. 
 
 454. The Present Value of an annuity is such a cum as, 
 put at interest, will, for the given time and rate, exactly 
 amount to the annuity. 
 
 Pensions^ rents, reversions, life insurance, &c., involve the principle 
 of annuities. 
 
 CASE I. 
 
 455. To find the amount of an annuity at simple interest. 
 
 1. liequired the amount of an annuity of $100, forborne five 
 years, at G % simple interest. 
 
 At the end of the 5tli year there will be due : the fA[\ years pay- 
 ment, or $100 ; the 4th year's payment, $100, plus 1 year's interest, or 
 $106 ; the 3d year's payment, $100, plus 2 years' interest, or $112 ; 
 the 2d year's payment, $100, plus 3 years' interest, or $118 ; and the 
 1st year's payment, $100, plus 4 years' interest, or $124. 
 
 Hence, the sums due are $100 + $106 + $112 + $118 + $124, or 
 $560. 
 
 But the sums due at the end of the 5th year form an arithmetical 
 series, of which the annuity, or $100, is the j^r^^ tcrjn., its interest for 
 1 year, or $6, is the common difference, and the nmnber of years, or 5, 
 is the number of terms. Hence, 
 
 Find the amount of the first ixiyment for the last term of an 
 arithmetical series, and then the sum of the series for the amount of 
 the anmdty. 
 
 "What is an Annuity ? When is an annuity said to be forborne, or in 
 arrears? "What is the Amount of an annuity? Tlie Present Value of an 
 annuity ? How do we find the amount of an annuity at simple interest? 
 
\,J^^ 
 
 SERIES. 
 
 309 
 
 Exercises. 
 
 2. An animity of .$200 lias boon in arrears 8 years ; what is 
 the amount duo, at G % simple interest? Am^. $19')G. 
 
 3. To what will a rent of $150 per annum, payable quar- 
 terly, amount, if forborne for 11 years, at G % simple interest \ 
 
 Ans. 8G51G.37i.. 
 
 4. If a salary of $loO a year be in arrears 10 years, to how 
 much will it amount at 7 % simple interest? Ans, $5917.50. 
 
 CASE II. 
 
 456. To find the amount of an annuity at compound 
 interest. 
 
 1. liequired the amount of an annuity of $100, forborne five 
 years, at C % compound interest. 
 
 At the end of the 5th year there will he cue : the fifth year's i>.'iy- 
 iiient, or $100; the 4tli year's payment, 8100, plu.s 1 year's interest, 
 or $106 ; the .3(1 year's payment, plus 2 years' compound interest, 
 or $112..'jG ; the 2d year's payment, plus 3 years' eompound interest, 
 or $119.1010, and the 1st year's payment, plus 4 years' compound 
 interest, or $12G.247G + . 
 
 Hence, the sums due are $100 + $100 + 8112.30 + $119.1010 + 
 $120.2470, or $503,709 + . 
 
 But the sums due at the end of the 5th year form a geometrical 
 series, of which the anmiity, $100, is the first term, the amount of $1 
 for 1 year, or $1.00, is the rate, and the number of years the number 
 of terms. Hence, 
 
 Find the amount of the first payment at annpotind interest for 
 the last term of a geometrical series, and then tJie sum of the series 
 fm- tlie amount of the annuity. 
 
 Exercises. 
 
 2. What is the amount of an annuity of $200 a year, for- 
 borne 5 years, at 7 % compound interest ? Ans. $1150.140 + . 
 
 3. If a person expends for 30 years $40 per annum for cigars, 
 how much will they cost him at 7 % compound interest? 
 
 Ans. $3778.39 + . 
 
 How do we find the amount of an annuity at compound interest '. 
 
310 
 
 PRACTICAL ARITHMETIC. 
 
 4. If you should deposit $50 every G months in a ftavin;is 
 l)aiik, to how much would it amount in 25 years, at 3 % semi- 
 annual compound interest? Ans. $5639.84 + . 
 
 CASE III. 
 
 457. To find the present value of an annuity at compound 
 interest. 
 
 1. What is the present value of an annuity of $100, to con- 
 tinue for 5 years, at 6 % compound interest 1 
 
 The amount of the given annuity for 5 years, hy the preceding case, 
 is $563,709 + , and the present value of tlie annuity muit be the pre- 
 sent value of the amount (Art. 302), The amount of $1 at compound 
 interest for the given time and rate, from the table, Art. 319, is 
 $1.338225. 
 
 Hence, the present value is $563.709 -f- $1.338225, or $421,236 + . 
 Hence, to find the present worth of an annuity at compound interest, 
 
 Find the amount of the ammity, and divide it hy the amount of 
 %\ at comjwund interest for the given time and rate. 
 
 Exercises. 
 
 2. What is the present value of a pension of $1000 for 4 
 years, at 7 % ? Ans. $3387.207 + . 
 
 3. What is the present value of an annual rent of $154 for 
 10 years, at 5 %? ^?i6-. $1801.13. 
 
 4. Bought an estate for $30000, payable in equal yearly 
 instalments of $5000 ; how much ready money, at 6 %, should 
 discharge the debt at the time of purchase? Ans. $24580.62. 
 
 EEVIEW EXERCISES. 
 
 • 1. What is the third power of 11|? Ans. 1481-^?^%. 
 
 ^ 2. Find the second power of the third power of 5. 
 
 Arts. 15025. 
 
 y. 3. Sold a field for $484, receiving as many dollars per acre 
 as there were acres ; how many acres were there, and what was 
 the price per acre ? Ans. 22 acres, and $22 per acre. 
 
 How is found the presA»>t worth of an annuity at compound interest ? 
 
^ 
 
 savin;is 
 ^ sonii- 
 9.84 + . 
 
 npound 
 to con- 
 
 ing case, 
 the pre- 
 inpoinul 
 . 319, is 
 
 1.236 + . 
 interest, 
 
 wunt of 
 
 for i 
 .207 + . 
 154 for 
 5G1.13. 
 yearly 
 should 
 i8G.G2. 
 
 ^ ' 1 ii5« 
 
 15625. 
 
 ;r acre 
 
 at was 
 
 r acre. 
 
 (St 
 
 Rr-VIEW EXERCI3ES. 
 
 311 
 
 ^ 4. There is a certain room, of a cubical form, which con- 
 tains 1953.125 cubic feet; what is the len<^th of each, of its 
 equal sides ? Aiis. \2.5. 
 
 ^•^5. I have 811 trees, which I wish to set out in a square -^^-^ 
 grove ; hovr many of the trees must be planted in each row 'I -z. OJj 
 
 0, What is the differenco between }j of a solid foot and a 
 solid \ foot, or a cube whose sides arc each ^ of a foot square; ? 
 
 ylns. 3 solid ^ feet. 
 
 ' 7. If a lead pipe 1 inch in diameter will fill a cistern in 4 
 hours, in what time will 2 pipes, each ^ of au inch in diameter, 
 fill the same] yins. 8 hours. 
 
 8. A grocer has two kinds of tea, one at 75 cents a pound, 
 and the other at $1.10; how must he mix them in order to 
 afford the mixture at $1 a pound? 
 
 9. A man had 10 children whose several ages ditfered alike, 
 the youngest being G years old, and the oldest 51 ; what Avas 
 the difference between the aires of the ninth and tenth ? 
 
 - 10. What will be the cost of painting a conical spire at 1 of 
 a dollar a square yard, if the slant height of the spire be 50 feet, 
 and the circumference at the base 2G.7 feet ? Ans. $14.83 -h . 
 
 ■^' 11. If a horse be tethered equidistant from the four corners 
 of a square lot containing exactly 10 acres, what must be the 
 length of the rope to allow him to graze over every part of the 
 lot 1 Ans. 28.27 + rd. 
 
 12. Construct a geometrical series, of which 12 is the first 
 term, and 3072 the 5th term. Ans. 12, 48, 192, 7G8, 3072. 
 
 13. I can purchase a farm for $700 cash down, or for $924 
 to be paid in the course of 7 years, | part of the price at the 
 end of each year. Allowing compound interest at G %, which 
 terms will be the most advantageous to me 1 
 
 Ans. Cash down, by $3G.86. 
 
 Review Questions. AVhat is a Series? (441) Terms of a Series? (441) 
 Extremes? (441) The M. ins? (441) Arithmetical Seric.-s ? (443) Geomotrical 
 Series? (44S) Kate or ItaMo of a Geometrical Series ? (448) 
 
>"- ■ rm. 
 
 
 He 
 
 
 «t 
 
 
 ~<^ li' 
 
 
 '-;.:'»■ 
 
 
 tM^WjM a 
 
 
 Ml 1 
 
 ' 
 
 L'W 
 
 ij 
 
 312 
 
 PRACTICAL ARiTIIMKTIG. 
 
 Exercises in Analysis. 
 
 ^ 
 
 I 1. Roquirod tlio greatest common divisor of C, ;^, and -^^^. 
 
 Sorx'TioN. y!> \i ^'"^ 1^0' ^^f^o.^^'J'^'-^ ^o equivalent fractions haviiKj tlie 
 Ifi'ixt common denominator^ become 5*, -^,';, and If^. 
 
 The greatest common divisor of 24, 15, and 18 twentieth, is 3 t>r^,n- 
 tifiths, or .j',y. 
 
 Therefore, <£•(;. 
 
 / 2. Wliat is the greatest common divisor of ;^, oi, and Gg? 
 
 / 3. What is the greatest number tliat is contained an exact 
 whole number of times in f , -~, y, and 21 lis: .^_. 
 
 4. What is the least common multiple of 2|, mT, and 3|| ? 
 Solution. 24-, 4|, and 3§, changed to equivalent fractions having 
 
 th.e least conwion denominator, become ^y*? "y-, and "J-. 
 
 77/-e ^casi common mulliyle <>f 18, 30, and 27 eighths, is 108 eitjhths, 
 
 Therefore, rtc. 
 
 5. What is the least common multi})le of 2, 7, and ^ ? 
 
 !^W5. 30. 
 
 6. What is the sum of money with which can be purchased 
 a number of hens at $.75 each, a number of ducks at $.37.} 
 each, and a number of turkeys at $2.0G]- each ? Ans. $8.25. 
 
 7. Find the square root of 225 by factoring. 
 Solution. 225 factored is equal to 5 x 5 x 3 x .3. 
 
 Since the square root of a number is the factor which must he taken 
 twice to form the number (Art. 391), 5 and 3, or one of every two equal 
 prime factors of the numhcr, must he taken to compose its square root. 
 
 Hence, 5x3, or 15, is the square root of 225. 
 
 8. Find the sixth root of 40656 by factoring. Ans. 6. 
 
 9. Find the cube root of f f g- by factoring. Ans. J. 
 
 10. A person being asked the hour of the day, said that the 
 time past noon was equal to i of the time till midnight. What 
 was the time ? 
 
 RuviEW Questions. What is a Unit ? (1) A Quantity? (2) A Number? 
 (;)) Figures? (18) Notution ? (10) Numorution? (17) Addition? (37) 
 Subtraction? (42) Multiplication? (47) Division? (57) 
 
-7^ 
 
 EXERCISES IN ANALYSIS. 
 
 313 
 
 
 1 (V 
 
 .25. 
 
 Solution'. The fimo to midnight is f of itself ; then 5 + 1[, or I, of 
 tlui time to midniijld, is e'lual to the time from noon to midniyht, ichieh 
 i-i 12 /(Onri^. 
 
 If 9 of lh>' time to vii(hnijht is equal to 12 /io)irs, I is equal to }^ of \1 
 /ioi/rs^ or 1 /'onr 20 minutes^ and ;^ is tqual to 4 times 1 hour -1^) 
 minutes, or 5 hours 20 minutes. 
 
 Jlence^ the time was 20 miniUes past 5 d'cloch p.m. 
 
 11. If liie timo of day is such that y- of tlu^ time past noon 
 is g- of tho time past midnii^^ht, what is the hour ? 
 
 Ans. 4 o'cloclc P.M. 
 
 12. "What is tho hour, if !! of tho time past 10 o'clock a.m. 
 is tliG lime till 10 o'clock p.m. ? Jus. G o'clock P.M. 
 
 13. A, B, and C start at the same time from a given point, 
 to travel in the same direction round an island 73 miles in cir- 
 cumference, A at the rate of G, B of 10, and C of IG miles [hi- 
 day ; in Avliat time will they be next together *? 
 
 Solution. Since B travels 4 miles a dav faster than A, he will gain 
 an entire round of the island, or 73 miles, in -]- of 73 daijr, or 18|- da>/s. 
 
 Since C travels 10 miles a day faster than A, he loill gain an entire 
 round in ^^^ of 73 days, or 7/jj days. 
 
 Hence, B cannot he with A except at the end of 18J_- da]/s, or of somo 
 multiple of 18|- days ; and C cawiot he with xi except at the end of 1 1\ 
 days, or of some midtiple of 7i'jy days. 
 
 llierefure, (J and B can hotk he with A for the first time, only aft-^.r 
 the lapse of a numhcr of days cxp>rcssed hy the least common multiiiU 
 of \'6\ and 7^'|j ; and the least common midtiple of 18} and 7,'|j is 3G-L 
 
 Therefore, Ac. 
 
 14. There is an island 73 miles in circumference, and 3 men 
 all start together to travel round it in the same direction ; A 
 goes 5 miles a day, B 8, and CIO; when will they all come 
 together again 1 Aiv-i. In 73 days. 
 
 15. A and B, at the opposite extremities of a wood, 135 
 rods in a circuit, begin to go round it in the same direction, at 
 
 Review Questions. Wliut is a Rule? (11) A Formula ? (00) An Opera- 
 tion ? (8) "Which are the Fundamental Operations ? (0(3) "What ia- a Sign t 
 (38) Symbolsof Operation? (07) A Solution ? (10) Analysis ? (7.5) 
 
It ' 
 
 li 
 
 * 1 
 
 ! 
 
 314 
 
 PRACTICAL ARITHMETIC. 
 
 . the same time; A at the rate of 11 rods in 2 minutes, and B 
 of 17 rods in 3 minutes. How many rounds will each makt; 
 before the ono will overtake the other ? 
 
 A7ts. A IG.l rounds, and B 17 rounds. 
 IC. At what time after 12 o'clock are the hour and minute 
 hands of a watch next exactly together 1 
 
 Solution. The minute hand of a watch passes over 60 minute 
 spaces in an hour, and tlie hour hand over 5 such spaces; hence, the 
 minute hand gains 55 minute spaces in 60 minutes, or 1 minute space 
 in >^r7 of 60 minutes. 
 
 If the minute hand gains 1 minute space in ^^ of 60 tninutes, it will 
 gain 5 minute spaces, or the distance the hands are apart at 1 o^ clock, 
 in 5 times ,j\ of 60 minutes, or 5 minutes 27i\ seconds. 
 
 Ther(for€, the hour and mimde ha7ids ivill be next together after 12 
 o'clock at 5 minutes 27^^^ seconds aftev 1 o'clock, 
 
 17. A person looking at his watch, was asked the time of 
 day. He replied that it was between 4 and 5, and the hour 
 and minute hands were together ; M'hat was the exact time ? 
 
 Ans. 21 minutes 49jj seconds past 4 o'clock. 
 ^-f- 18. If a watch which is 10 minutes too fast on Tuesday 
 'noon, gains 3 minutes 10 seconds a day, what time will it indi- 
 cate at 10^ o'clock A.M. of true time on the following Sunday ? 
 Ans. 40 minutes 36/y- seconds past 10 o'clock a.m. 
 y- 19. A labourer agreed to work 20 days upon the condition 
 that for every day lie worked he should receive $2, but for 
 every day he was idle he should forfeit 50 cents ; he received 
 $25. How many days did he work ] 
 
 Solution. Had he worked all the time, he would have received 20 
 times $2, or $40 ; he therefore lost, hy being idle, $40 - $25, or $15. 
 
 Since for each day that he zvas idle he lost $2 + $.50, or $2.50, he 
 7nust have leen idle as many dags as $2.50 are contained times in $15, 
 which are 6. 
 
 Hence ^ he worked 20 days - 6 days, or 14 days. 
 
 Review Questions. What is an Integer? (96) A Fraction? (131) A 
 Common Fraction? (134) A Decimal Fraction? (J 70) A Denominate 
 Number? (243) A Compound Denominate Number? (243) Keduction ? (80) 
 
 T 
 
TI75r^ 
 
 r.XERnSES IN ANALYSIS. 
 
 
 s, and B 
 ill make 
 
 rounds, 
 minute 
 
 minute 
 hence, the 
 liite sjjiice 
 
 tes, it ivill 
 1 dclocL 
 
 after 12 
 
 time of 
 the hour 
 time? 
 o'clock. 
 Tuesday 
 it indi- 
 unday ? 
 ck A.M. 
 Dudition 
 but for 
 eceived 
 
 eived 20 
 r $15. 
 
 ^2.50, he 
 sin^iru 
 
 (131) A 
 loininate 
 ion? (80) 
 
 20. A boy ap;rocd to work 80 days on condition th.it ho 
 should receive 72 cents for every day he -worked, and forfeit 
 48 cents for every day he was idle; at the exj^iration of the 
 time he was in debt $12 ; how many days had he been idle? 
 
 Ans. 58 days. 
 
 21. A labourer was liired 25 days ; for every day he workecl 
 he was to receive $1.25 and board, and for every day he was 
 idle he was to pay boanl. At the end of the time he received 
 Si23.75. The price of the board was $.25. How many <lays 
 did he work ? y'/w.«r. 20 days. 
 
 22. Two brothers, A and B, have the same income. A saves 
 I of his, but B, by spending $150 yearly more than A, at the 
 end of 8 years finds himself $400 in debt. What was the 
 income of each ? 
 
 Solution. If B, hy ^'pending $150 yearJij more than A, is in debt at 
 the end of 8 9/cars $400, he sjiends each year ^ (f $400, or $50, more 
 than his income. 
 
 Therefore, $150 — $50, or $100, 77Uist be what A saves, Schick is | of 
 the income of each. If j of the income of each is $100, the whole income 
 of each mnst be 4 times $100, or $400. 
 
 Therefore, £c. 
 
 23. A and B have the same income. A saves | of his, but 
 B, by spending $30 each year more than A, at the end of 8 
 years finds himself $40 in debt. How much does each spend 
 yearly ? Ans. A $175, and B $205. 
 
 24. A and B have each an income of $400 a year ; A spends 
 each year $40 more tlian B ; at the end of 4 years they both 
 together save a sum equal to the income of either. What do 
 they spend annually ? Ans. A $370, B $330. 
 
 25. A, B, C, and D bought a grindstone, the diameter of 
 which was 4 feet, each contributing an equal amount, and they 
 wish to grind off their several shares successively. How much 
 may each grind off, no allowance being made for the axle ? 
 
 Review Questions. What is Per Cent. ? (202) Percentaj,'e? (20.3) Tom- 
 niission? (274) Brokerage? (275) Insurance ? (278) Profit and Loss ? (2S3) 
 
! 
 
 W' 
 
 ^ ii 
 
 31G 
 
 PRACTICAL ARITHMETIC. 
 
 Solution. Sinrc each ?'x cvidenllfj entitled to | of thcjlat surface^ wJien 
 thefivd haa <jround off his portion, there will remain ^ of that surface. 
 
 Then, since the ai'eaa of circles are to each othei' as the squares rf their 
 diameters (Art. 431), 
 
 Tlic n'holti stone : />ar^ remaining : ; square of diameter if the whole 
 $tone : square of diameter of part remainivg ; 
 
 Whence, denoting the whole stone by unity, v;e liavo 
 
 1 : 2 : : 4" : square of diameter of part rernainimj; hence, the diameter' 
 of part remaining = J^ x 4- feet = 3.404 /fc^. 
 
 Therefore, A grinds off 4 ft. - 3.464 /i. = .530 /cd = 0.432 inches. 
 
 After the second has ground off his portion, there ivill remain \ of the 
 itone. Whence, 
 
 1 : .1 : : 4^ : square of diameter of imrt remaining; hence, the diameter 
 of part remaining = vA- x 4^ feet = 2.82Sfect. 
 
 Therefore, B grinds off 3.404 feet - 2.828 feet = .6'3G feet = 7.032 
 inches. 
 
 After the third has ground off' his portion, there will remain ^ of the 
 stone. Whence, 
 
 1 : 1 : : 4- : square of diameter of part remaining ; hence, the diameter 
 
 of part remaining ~ vj + 4'- feet = 2 feet = 24 inches. 
 
 Thenfore, C grinds off 2M8 fret — '2 feet ^ .828 feet = 0.030 inches, 
 and there remains as Us share a part 24 incites in diameter. 
 
 2G. A, B, and C bouglit a grindstone GO inches in diameter, 
 whicli cost them $12, 13 and C contributing $3 each and A 
 the remainder. In order for eacli to get Ins share according to 
 the sum contributed, how much must each grind off from the 
 diameter, no allowance being made for the axle, A taking his 
 portion first, and then B his ] 
 
 Ans. A 17.573 + inches; B 12.42G + inches; and C 30 inches. 
 
 27. Four ladies bouglit a ball of silk, 5 inches in diameter ; 
 how much of the diameter must each wind oft' so as to share 
 the silk equally 1 
 
 Ans. 1st, .45+ inches; 2d, .57+ inches; 3d, .82 + ; and 
 4th, 3.14 + inches. 
 
 ]lE%'iEW QursTiONS. What is Interest? (288) Simple Interest? (291) 
 Annual Interest ? (;VLO) Compound Interest? (ol7) Present Worth ? (o02) 
 Discount? (302) Bank Discount? (307) Partial Payments ? (312) 
 
 
MISCELLANEOUS EXERCISES. 
 
 317 
 
 'facp,^ when 
 '.t surface. 
 res cf tht }'r 
 
 the whole 
 
 e diametei' 
 
 2 inc/it'g. 
 'n J of the 
 
 i diameter 
 
 t = 7.032 
 
 n ^ of the 
 
 diameter 
 
 jG incJies, 
 
 iamc'tcr, 
 and A 
 rding to 
 rom the 
 ving liis 
 
 ) inches, 
 
 ;imeter ; 
 io share 
 
 f- ; and 
 
 St? (liOl) 
 
 MiscELLAXKors KXKi:risp:s. 
 
 ' 1. How many times may 1)5 bo subtracted from 22515 ? -= >j / 
 
 - . 2. Wliat part of 3 cents is J of 2 cents ? .- ^ 
 
 ■ ■ *3. On counting my sheep I found that \ +''2 + ^ of tliem 
 
 numbered 80 ; how many had I? Ans. SI. 
 
 — "7^4. Wl'at number, when divided by 5 of 1 of 1^, will pro- 
 duce 1 ? y/zis. \. 
 
 -^5. WJiat is tlio square of 1 ? 
 
 .- — *-V*6. A farmer wishes to sejxarate his farm of 113 A. 115 P. 
 
 into lots of 12 A. 10 P. each; how many such lots can lie 
 have? y//<.v. '.r'j^^V 
 
 7, What is the difference in time of two places 7' of longi- 
 tude apart? 
 
 '^ 8. William Hardy having engaged to travel 5 miles, became 
 
 disabled, and was obliged to stoj) at the end of 3 m. 5 fur. 18.], 
 rd. ; what part of liis engagement had he completed ? —SL 7- d '^'' 7 
 9. How many hills of corn may be planted on an acrc§)f2. 
 ground, provided they are planted, in the square order, 4 feet apart >k 
 each wa}', and none nearer the edge than 2 feet? Ans. 2722. 
 
 — — ■•10. A thief having 20 miles the start of an officer in pur- 
 suit, goes miles an hour, and the officer follows at the rate of 
 8 nules an hour; how long before he will overialvo the thief? 
 
 Alls. 10 hours. 
 
 • 11. A straight plank is 3^- inches thick and G^ inches l)road ; 
 
 what length mu^t be cut off so as to contain G^ cubic feet of 
 
 timber? y/w.9. 41} feet. 
 
 .,..,^^* • 12. A man, who laboured under an impediment in his speech, 
 
 in calling for an article, said, *' Give me a half of a — half of a — 
 
 » half of a — half of a gallon of vinegar." Taken at his word, 
 
 how much did he ask for? 
 
 Ans. 2 gills. 
 
 Review Questions. AVhat is Ratio? (.320) Proportion ? (;>2r>) Com- 
 pound Proportion ? (332) Partnership ? (334) Equation of Payments ? (338) 
 Averaging of an Account ? (345) Settlement of an Account ? (.'j.jO) 
 
318 
 
 PRACTICAL ARITHMKTIC. 
 
 13. Wlicn <^()1(1 sells at a prcmiuin of 45 % in currency, how 
 much can he bought for $1 of currency ] Ans. $.G8;jJJ. 
 
 ■-^ 14. If 30 % is lost hy selling shoes at 84 cents a pair, at 
 what price should they bo sold to gain 20 %? Ans. $1.44. 
 
 15. How much stock at 93J- %, can be purchased for $540, 
 a connnission of | % being charged on the stock purchased 1 
 
 Ans. $578; :l. 
 
 10. A gentleman promised his son a new arithnu.'tic, on con- 
 dition that he would go to a certain orchard through three 
 gates, and get a sufiicient number of apples, that on his return 
 he could leave half what he had and half an api)le more at the 
 first gate, and half the renuiinder and half an apple more at the 
 second gate, and half of what he had left and half an apple 
 more at the third gate, without cutting any, and then have one 
 renuiining. How many must he get / Ans. li). 
 
 ^ 17. A can do a piece of work in 3 days, B can do three times 
 as much in 8 days, and C five times as much in 12 days; in 
 what time would they all do it together? Ans. 21^ hours. 
 
 18. A merchant sold a lot of coffee at 15 cents a pound, and 
 lost 10 % ; soon after he sold another lot, to the amount of $525, 
 and gained 40 %. How many pounds were there in the last 
 lot, and what was the price per pound at which it was sold? 
 
 Ans. 2250 lb., at 23^^ cents. 
 
 19. A man bought a house for $1575, and repaired it for a 
 tenant, who agreed to pay him a rent of $220 per annum, which 
 was 12 % of the money paid for the house and its repairs ; what 
 w^as the cost of repairing it? Ans. $258.33^. 
 
 20. What is the difTcrence between the annual and com- 
 pound interest of $300 for 4 years at G % ? Ans. $.26 + . 
 
 21. A note, on 3 months, dated January G, was discounted 
 March 4 ; for what time was the discount? ylns. 3G days. 
 
 Kkview Questions. "What are Tuxes? (,"353) Duties? (358) Internal 
 Revenue ? (301) Customs ? (303) Exchange ? (309) Inland Exchange? (370) 
 Foreign Excliange? (379) 
 
 ,1 
 
 1 
 I 
 
4'fttfW 
 
 icy, how 
 
 pair, at 
 s. $1.44. 
 
 )r $540, 
 used 1 
 
 , on con- 
 ;h three 
 !s return 
 •e at tho 
 re at the 
 Lin apple 
 lave one 
 /ins. 15. 
 
 ee times 
 lays ; in 
 \ hours. 
 
 ind, and 
 
 of $525, 
 the last 
 sold? 
 /j cents. 
 
 it for a 
 , which 
 ; what 
 58.33^. 
 
 d com- 
 
 fe.26-f . 
 
 ounted 
 6 days. 
 
 Internal 
 {37U) 
 
 V 
 
 MISCELLANEOUS liXERCISKS. 
 
 319 
 
 22. If you should, on June 2()t]i, buy a note dated Jantiary 
 2()th, drawn for $40, on 8 months, what should you pay for it, 
 moiH^y being worth 2 % a month? An^. $37.52. 
 
 23. A, n, and (J join tlu'ir capitals, whiih arc in tin' propor- 
 tion of .', ', and ] ; at tiio end of 4 montiis A witlidraws I <»f 
 his capital, and at the end of D months more tiicy divide their 
 [irolit.s, $2840; what'shuuhl eacli receive] 
 
 Ans. $1020, $1010, and $780. 
 
 24. If 100 Iit(!rs of wino cost 250 francs, what must be tho 
 price per gallon, in Uniled States money, to gain 20 per cent? 
 
 25. If 3 lb. of tea be worth 4 lb. of cofTtie, and lb. of colIVe 
 be worth 20 lb. of sugar, how many pounds of sugar can be 
 had for lb. of tea ? An.i. 40 lb. 
 
 20. An express train leaves Cliicago for a point 120 mih'S 
 west, at 2 o'clock, and goes at the rate of 25 miles ])('r hour ; 
 at what time must a slow tiain, which goi-.s 15 miles in 50 
 minutes, have left, so as not to be overtakfii by the express 
 train .^ Aiiu 12 hours 8 minutes. 
 
 27. The merchandise l)alance of an account is $l'00 due by 
 averaging the account from June 11; what will Ijc its cash 
 vahu^ if payment be delayed till September 11, or for 3 
 months, allowing interest at G % ? An>i. $203. 
 
 28. A guardian paid his ward $G4G0 for $5000 which he 
 ha<l in his hands 4 years ; what rate of intcn.'.st did he allow 
 him? Ann. 7 -'^ %. 
 
 20. IIow many square feet of 1 inch boards may be sawed 
 from a beam 24 feet long, and whose rectangular ends ar(3 10 
 inches by 12 inches, if no allowance is made for the thickness 
 of the saw? ylns. 3S4 sq. ft. 
 
 30. If the weight of a hectoliter of wheat is 75 kilograms, 
 what weight of wheat will fill a bin 2 meters long, 1.4 meters 
 broad, and 1 meter deep? Ans. 2100 kilos. 
 
 REvn.:w Questions. AVhat is a Power ? (.385) A Root? (388) Involution? 
 (386) Evolution ? (390) S(iuare Hoot ? (391) Cubo Hoot ? (391)) 
 
^' 
 
 'n^ 
 
 320 
 
 PRACTICAL ARITHMETIC. 
 
 i ^ 
 
 31. In a mixture called coffee, -^ of the whole plus 2.> poutuls 
 is coff(;e, and -J part less 5 pounds is cliiccory ; what per cent. 
 <'f the whole is chiccor}''? Jns. 20} per cent. 
 
 32. A has due him $144, payable in 7 m'^nths, but the 
 debtor agrees to pay ^V down, and ^^ in 4 months ; in what time 
 should he pay the balance ? 
 
 33. Wliat must be the face of a GO days' note, when money 
 is worth G %, to allow of taking $3958 from a bank ? 
 
 34. If the specific gravity of iron in bars is 7.8 times that of 
 water, and a liter of w\ater weighs a kiloi^ram, what is the 
 weight of a bar of iron 4 meters long, 1 decimeter broad, and 
 3 centhneters thick? Ans. 93. G kilos. 
 
 35. If a pipe G inches bore will discharge a certain quantity 
 of water in 3 hours, in what time will 3 pipes, each 3 inches 
 bore, discharge 3 times that quantity? Jns. 12 hours. 
 
 3G. If the merchandise balance of an account is $G00, due 
 April 21, what was its cash value, interest at 7 %, on the pre- 
 ceding January 1st? Ans. $587.17. 
 » 37. A cat watches a mouse which is distant 24 feet, and 
 stealing towards it, advances 5 feet every quarter of an hour, 
 while the mouse goes away 3 feet in the same time ; now, allow- 
 ing the cat the last quarter of an hour to advance 7 feet, how 
 many hours will she be in catching the mouse 1 Ans. I'l h. 
 -■ 38. A park in the form of a rectangle is 40 rods long and 
 3G rods wide ; what is the length of a straight walk between 
 its opposite corners? Awi. 53.81 + rods. 
 
 39. A circular garden, containing one acre and forty- one 
 square rode, has a gravelled walk of uniform widtli just within 
 the circle, that tp.kes up 12 square rods of the ground; what 
 is the diameter of the garden, and the width of the walk ? 
 
 Aufi. Diameter of garden, IG rd. nearly; width of walk, 4 + ft. 
 
 Review Questions. What is Mensuration? (409) A Point? (401) A 
 Line? (402) A Plane Figure ? (407) When are figures or volumes similar? 
 (431) What is a Series? (441) An Arithmetical Series'.' (441) A Geometrical 
 Scries? (448) An Annuity ? (452) 
 
APPKNDIX. 
 
 02 1 
 
 -■"> ponruls 
 i per cent. 
 J- per cent. 
 
 >, but tlie 
 what time 
 
 len money 
 
 les tliat of 
 lat is the 
 )road, and 
 33. G kilos, 
 
 I quantity 
 
 I 3 inclies 
 
 12 hours. 
 
 SGOO, due 
 1 the pre- 
 $587.17, 
 feet, and 
 ' an hour, 
 Dw, allow- 
 feet, how 
 Ins. 2f h. 
 long and 
 : between 
 >1 + rods. 
 
 forty- one 
 ;st within 
 id ; what 
 
 ,'alk ? 
 Ik, 4 + ft. 
 
 ? (401) A 
 iGS similar? 
 looinetrical 
 
 APPENDIX. 
 
 ROMAN NOTATION. 
 
 458. The Roman Notation, so calK-d because it was u^od hy 
 the ancient Romans, employs, in expressing nuniheis, seven cai.ital 
 letters :— ^ 
 
 I' ^'' X, I^, C, D, M, 
 
 one, five, ten, fifty, one huiulred, five hundrcl, one thousand. 
 
 459. All other numbers may be expressed by combining these 
 letters in accordance with the folio win" 
 
 rPJXCIPLES. 
 
 1. Repeating a letter repeat.^ the number it denotes. 
 
 2. By wnting a letter denotinxj a less number bkforr a letter dennt- 
 imj a greater, the number expressed is the diffi-jrion-ck of the nuuthers. 
 
 3. % writing a letter denoting a less number aftkr a letter denoting 
 a greater, the number expressed is the SUM of the numbers. 
 
 4. A dash (— ), makes the mimber expressed b,j it a thousandfold. 
 
 Roman Table. 
 
 T. denotes One, 
 
 11, 
 III. 
 IV. 
 
 V. 
 YL 
 
 vn. 
 
 Vill. 
 
 IX, 
 
 X. 
 
 XI. 
 
 XII. 
 
 XIII. 
 
 XIV. 
 
 XV. 
 
 XVI. 
 
 Two, 
 
 Three, 
 
 Four. 
 
 Five. 
 
 .Six. 
 
 Scn-en. 
 
 Fig lit. 
 
 Nine. 
 
 Ten. 
 
 Eleven. 
 
 Twelve. 
 
 Thirteen. 
 
 Fourteen. 
 
 Fifteen, 
 
 Si.xteen. 
 
 XVII. denotes Seventeen. 
 
 XVI I r. Eighteen. 
 
 XIX. Nineteen. 
 
 XX. Twenty, 
 
 XXX. Thirty! 
 
 X L. Forty. 
 
 1^. Fifty. 
 
 l-iX. Sixty. 
 
 LXX. Seventy. 
 
 LXXX. Eighty, 
 
 XO. Ninety. 
 
 ^' One hundred, 
 
 ^' Five hundred. 
 
 M. One tliousand. 
 
 X. Ten thousand. 
 
 <**• One itiilliuu. 
 
^4^-^-" 
 
 31' 2 
 
 TKACTICAL ARlTlhMKTlC 
 
 J 
 
 t \ 
 
 
 METPJC SYSTE]\r. 
 
 460. This system, first adopted by the 
 French in 1795, and by lar the most 
 siniphi and coniprehensive that has yet 
 been devised, has now been legalised or 
 adopted by almost all civilised countries. 
 
 The metric system, by legislative action, 
 in 1864, was recommended to be taught 
 in the schools of Connecticut. 
 
 461. Its principal denominations are 
 pronounced thus : — 
 
 Meter, Are, Li'ter, Sture, Gram, Mil- 
 li-me-ter, Cen'ti-me-ter, Dec'i-me-ter, 
 Dek'a-me-ter, Hek'to-nie-ter, Kil'o-me- 
 ter, Myr'i-a-me-ter, Centi-are, Hek'tare, 
 Mil'li-li-ter, &c. 
 
 The scales in the margin exhibit a 
 decimeter, or a tenth of a meter, divided 
 into centimeters and millimeters, and four 
 inches divided into eighths of an inch. 
 
 462. The equivalents expressed in 
 the tables (Art. 232-238) are recog- 
 nised, in the United States, in the con- 
 struction of contracts, and in all legal 
 proceedings. 
 
 
 
 - 
 
 — 
 
 1 
 
 M 
 
 ^ 
 
 — • 
 
 u 
 
 a 
 
 o 
 — ce 
 
 
 = 
 
 2 
 
 L. 
 
 a 
 
 -- 
 
 -^ 
 
 *4 
 
 —^ 
 
 CJ 
 
 CB 
 
 
 4> 
 
 10 
 
 O 
 
 ^ 
 
 
 
 iu 
 
 
 I 
 
 LIFE INSURANCE. 
 
 463. The Expectation of Life is the average number . ' y-jars 
 that persons at different ages live, as shown by life statistics. 
 
 The rates of life insurance are based upon the expectation of life 
 of persons of the age of the applicant for a policy. 
 
 464. The Carlisle Table and the Wigglesworth Table, showing 
 the expectation of life, as presented on the following page, are 
 chiefly used in the United States. 
 
 The former is based on life statistics in Great Britain, and the 
 latter on life statistics in this countrv. 
 

 APPENDIX. 
 
 :vj3 
 
 TABLES. 
 
 o 
 
 o 
 
 ^ 
 
 ^•2 i i .-2 
 5a s: w 
 
 1) 
 
 < 
 
 Carlisle Table. 
 Expectation. 
 
 ? o 1 
 
 ^ 2 1 
 
 < 
 48 
 
 Carlisle Table. 
 Expectation. 
 
 Wigple.swnrth 
 
 Table. 
 Expectatior. 
 
 < 
 
 Carlisle Table. 
 Expectation. 
 
 m 
 
 i 31^.72 28.15 
 
 24 
 
 38.59 
 
 32.70 
 
 22.80 22.27 
 
 72 
 
 8.16 
 
 9.14 
 
 1 i 44.<i8 3(5.78 
 
 25 I 37.86 
 
 32.33 
 
 49 
 
 21.81 21.72 
 
 
 7.72 
 
 8.69 
 
 2 ; 47.55 38.74 ' 
 
 26 
 
 37.14 
 
 31.93 
 
 50 
 
 21.11 21.17 
 
 74 
 
 7.33 
 
 8 25 
 
 3 
 
 49.82 40.01 
 
 27 
 
 36.41 
 
 31.50 
 
 51 
 
 20.39 20.61 
 
 , 75 
 
 7.01 
 
 7.83 
 
 4 
 
 50.7(> 40.73 
 
 28 
 
 35.69 
 
 31.08 
 
 52 
 
 19.68 20.05 
 
 i 76 
 
 6.69 
 
 7.40 
 
 5 
 
 51.25 40.88^ 
 
 29 
 
 35.00 
 
 30.66 
 
 53 
 
 18.97 19.49 
 
 ! 77 
 
 6.40 
 
 6.99 
 
 6 
 
 51.17 40.69 1 
 
 30 
 
 34.34 
 
 30.25 
 
 54 
 
 18.28 , 18.92 
 
 78 
 
 6.12 
 
 6.59 
 
 7 
 
 60.80 40.47 1 
 
 31 
 
 33.68 
 
 29.83 
 
 55 
 
 17.58 
 
 18.35 
 
 79 
 
 5.80 
 
 6.21 
 
 8 
 
 50.24 40.14 ! 
 
 32 
 
 33.03 
 
 29.43 
 
 56 
 
 16.89 
 
 17.78 
 
 80 
 
 5.51 
 
 5.85 
 
 9 49.57 3!). 72 \ 
 
 33 
 
 32.36 
 
 29.02 
 
 57 
 
 16.21 
 
 17.20 
 
 81 
 
 5.21 
 
 5.50 
 
 10 1 48.82 39.23 : 
 
 34 
 
 31.68 
 
 28.62 ' 
 
 58 
 
 15. .55 
 
 16.63 
 
 I 82 
 
 4.93 
 
 5. 1 6 
 
 11 1 48.04 38.64 
 
 35 
 
 31.00 
 
 28.22 ; 
 
 59 
 
 14.92 
 
 16.04 
 
 83 
 
 4.65 
 
 4.87 
 
 12 
 
 47.27 38.02 
 
 36 
 
 30.32 
 
 27.78 ; 
 
 60 
 
 14.34 
 
 15.45 
 
 84 
 
 4.39 
 
 4 66 
 
 13 
 
 46.51 37.41 
 
 37 
 
 29.64 
 
 27.34 : 
 
 61 
 
 13.82 
 
 14.86 
 
 85 
 
 4.12 
 
 4 57 
 
 14 
 
 45.75 36.79] 
 
 38 
 
 28.96 
 
 26.91 \ 
 
 62 
 
 13.31 
 
 14.26 
 
 86 
 
 3.90 
 
 4.21 
 
 15 
 
 45.00 36.17 i 
 
 39 
 
 28.28 
 
 26.47 \ 
 
 63 
 
 12.81 
 
 1306 
 
 87 
 
 3.71 
 
 3.90 
 
 IG 
 
 44.27 35.76; 
 
 40 
 
 27.61 
 
 26.04 1 
 
 64 
 
 12.30 
 
 13.05 
 
 88 
 
 3.59 
 
 3.67 
 
 17 
 
 43.57 : 35.37 i 
 
 41 
 
 26.97 
 
 25.61 1 
 
 65 
 
 11.79 
 
 12.43 
 
 89 
 
 3.47 
 
 3.56 
 
 18 
 
 42.87 • 34.98 ' 
 
 42 
 
 26.34 
 
 25.19 1 
 
 66 
 
 11.27 
 
 11.96 
 
 90 
 
 3.28 
 
 3.73 
 
 19 
 
 42.17 i 34.59' 
 
 43 
 
 25.71 
 
 24.77 
 
 67 
 
 10.75 
 
 11.48 
 
 ! 91 
 
 3.26 
 
 3.32 
 
 20 
 
 41.46 1 34.22 \ 
 
 44 
 
 25.09 
 
 24.35 
 
 63 
 
 10.23 
 
 11.01 
 
 92 
 
 3.37 
 
 3.12 
 
 21 
 
 40.75 i 33.84 1 
 
 45 
 
 24 46 
 
 23.92 
 
 69 
 
 9.70 
 
 10. .50 
 
 93 
 
 3.48 
 
 2.40 
 
 22 
 
 40.04 i 33.46 
 
 46 
 
 23.82 
 
 23.37 
 
 ; 70 
 
 9.18 
 
 10.06 
 
 i 94 
 
 3.53 
 
 1.98 
 
 23 
 
 39.31 , 33.08 
 
 47 
 
 23.17 
 
 22.83 
 
 1 71 
 
 8.65 
 
 9.60 
 
 95 
 
 3.53 
 
 1.62 
 
 -J 
 
 The premiums of life insurance are generally reckoned at a certain 
 snm per $1GG0 of insurance, payable annually, semi-annually, or 
 quarterly. 
 
 The payments may he limited to one or more, by the policy ; and 
 the insurance may be for life or for a certain number of years. 
 
 Jage, are 
 
 Exercises. 
 
 1. A man 52 years of age gets his life insured for $4000, at the rate 
 of |31.89, to be paid semi-annually. To how much will the ])re- 
 miums amount should he pay them for ten years ? Ans, $2551.20. 
 
 2. If a man at the age of 35 insures his life for $2000, on the plan 
 of 10 annual payments of $52.40 on $1000, and should die at the age 
 of 49, how much more will his family receive than the premiums he 
 paid ? Ans, $952. 
 
3l!4 
 
 PRACTICAL ARITHMETIC. 
 
 Table showing the Number of Days. 
 
 ■ 
 
 TO THE SAME UAY OK NKXT 
 
 
 
 KIM)AT \\^' 1 » \ V OK 
 
 
 
 
 Jim. Full. 
 
 Mur. 
 
 Avr. May 
 
 June July 
 
 Aiif:. Snpt. Oit. 
 
 212 243 273 
 
 Xov. ' Dim;. 
 
 
 January . 
 
 3G5 31 
 
 59 
 
 90 
 
 120 
 
 151 181 
 
 30-1 334 
 
 February 
 
 334 3()5 
 
 28 
 
 59 
 
 89 
 
 120 160 
 
 181 
 
 212 242 273 3";5 
 
 
 ]\rarch . 
 
 30(j 337 
 
 305 
 
 31 
 
 61 
 
 92 122 
 
 153 
 
 184 214 245 275 
 
 
 April . . 
 
 275 , 30(3 
 
 334 
 
 3H5 
 
 30 
 
 61 91 
 
 122 
 
 153 
 
 183 
 
 214 244 
 
 
 May 
 
 245 270 
 
 304 
 
 335 
 
 365 
 
 311 61 
 
 92 
 
 123 
 
 153 
 
 184 
 
 214 
 
 
 June 
 
 214 245 
 
 273 
 
 304 
 
 334 
 
 365; 30 
 
 61 
 
 92 
 
 122 
 
 153 1^3 
 
 
 July 
 
 184 215 
 
 243 
 
 274 
 
 304 
 
 335 3t)5 
 
 31 
 
 62 
 
 92 
 
 123 153 
 
 
 Au<i;usfc . 
 
 153 i 184 
 
 212 
 
 243 
 
 273 
 
 304 ' 334 
 
 365 
 
 31 
 
 61 
 
 92 
 
 122 
 
 
 September 
 
 122 1 153 
 
 181 
 
 212 ■ 242 
 
 273 303 
 
 334 
 
 365 
 
 30 
 
 61 
 
 91 
 
 
 October . 
 
 02 ' 123 
 
 151 
 
 182 212 
 
 243 273 
 
 ;{04 
 
 335 
 
 365 
 
 31 
 
 61 
 
 
 November 
 
 01' 92 
 
 120 
 
 152 
 
 181 i 212 242 
 
 273 
 
 304 
 
 334 365 
 
 30 
 
 
 Decem))er 
 
 31 62 
 
 90 
 
 121 
 
 151 182 212 
 
 243 
 
 274 
 
 304 ' 335 
 
 365 
 
 J 
 
 
 When Fchnuuy is incliuletl between the points of time, a day 
 ninst be added in leap year. 
 
 465. The following exercise.^ will furnish additional exanii>leis of 
 the application of the Vermont rule for partial payments (Art. 310). 
 . 1. A note for $1000 was given July 1, 1864, 
 
 Indorsements. January 1, 1865, $100 ; September 1, 186(), 
 $223.99 ; December 25, 1866, $12. 
 
 How much was due, by simple interest, January 1, 1867 ? 
 
 Ans. $803.12. 
 
 2. A note for $700 was given b\'bruary 4, 1 864. 
 Indorsements. December 18, 1864, $164 ; June 24, 1865, $200 ; 
 
 Septeml)er 11, 1865, $120 ; July 5, 1866, $60. 
 
 AVhat was due on this note, by annual interest, Xov. 28, 1866 ? 
 
 A71S. $233.50. 
 
 3. A note for $625,50 was given October 1, 1864. 
 Indorsements. January 1, 1865, $200 ; November 1, 1865, $20; 
 
 January 1, 1866, $300. 
 
 How much was due, by simple interest, May 1, 1866 ? 
 
 A)is. $143.79. 
 
 4. A note for $1000 was given January 1, 1866. 
 Indorsements. April 1, 1866, $24 ; August 1, 1866, $4 ; De- 
 cember 1, 1866, $6 ; February 1, 1867, $60 ; July 1, 1867, $40. 
 
 What will be due, by annual interest, June 1, 1870 ? 
 
 A71S. $1130.62. 
 
 THE END. 
 
mm 
 
 time, ii d 
 
 'y 
 
 xamples of 
 (Art. 310). 
 
 r 1, ism, 
 
 37? 
 
 ns. $803.12. 
 
 865, $200 
 
 S, 1866 ? 
 IS. $233.50. 
 
 « 
 
 1865, $20: 
 
 s. $143.79. 
 
 h $4 ; De- 
 
 •, $40. 
 
 . $1130.62.