SOtTHERK CONFEDERACY ARITHMETIC, Coiuiuoii Schools and .Academies, WITH A ii II i PKAOTiCAL SYSmi- OK HOOK-KEEPING l>i SINGI^i: ENTRY. Bi Rev. CHARhKS Jv KKVKl.CTT, A.M., AUTHOR OF THK KTHKBN CON^KDHUArV CLASS READBFd. v;tC. . V.1 1". lit ''I'' A IT G I S r A , G A . : POINTED AND I'UrMSinF.P HX. J. ^ rVTBBSON * CO. 1861. COL. GEORGE WASHINGTON FLOWERS MEMORIAL COLLECTION DUKE UNIVERSITY LIBRARY DURHAM. N. C. PRESENTED BY W. W. FLOWERS u t^ THE SOUTHERN CONFEDERACY ARITHMETIC, FOR Common Schools and Academies, WITH A PRACTICAL SYSTEM OF BOOK-KEEPING BY SINGLE ENTRY. By Rev. CHARLES E. LEVERETT, A.M., AUTHOR OF TUB SOUTUKUN COXKEDKRACY CLASS READERS, ETC., ETC. A U G U S TA, G A.: PRINTED AND PUBLISHED BY J. T. PATERSON ach too little, and ngain too nnu-l) Tb^ VI PREFACE. consequence lias been embarrassment to tbe common order of minds, and a tendency to make a most pleasing science an occult study, and a useful and necessary bvauc-b of pcience a distasteful task. In tliis way, that course, vrithout -wliicb' no education is complete, eitl\er in its relation to the sciences connected with the reasoning powers, or in its practical addresses in commercial engagements, and in the daily appli' cations of domestic finance, is made to sustain a most depre- ciated stand. It has been the author's aim to avoid prolixity. To acquire numerical skill, a pupil should not become vreary of his work. Arithmetics, generally, because too long, are tire- some, and the end — which the scholar always has in view — appears — ■ "Tho weary wild, that lengthens an. v>'e go." This book is purposely short, and, as for the most part, Hhe secret of scholarly success is the culture from the review — the nail to fasten science to the mind — it should be repeat- edly used before the higher volume of the series is taken up. It has been an additional . aim to give, at each progressive Step, clear explanations. In these, when it could be done, the rule for operations is introduced, and by this plan has been avoided what to young- pupils is a very mystifying and objectionable paj-t *of Arithmetical study— the dry formula of directions.' In regard to the explanations, much must be left to the teacher, and he is an unintelligent one who does not make plain, by illustrations, certain truths, which no Arith- metic, unless it be expanded to an unreasonable length, could communicate. In this connection, it may be well to say that it would have been perfectly easy to have introduced a very frequent analysis, and with it have blended questions, after rUEFACE. Vll the manner of some Aritlimcticiaus, in order to cDnibme tho mental and written dcpavtnaents; but, after trying all forma, his experience has held it preferable to keep- tbcni separate. When too many things arc presented together there is con- fusion, and the contemplated object is not secured. For tho encouragement of tbe pupil, even if his be not a mind, strictly speaking, mathematically framed, it mny be • said, if he have well studied . the Intellectual volume of this series, ho can hardly fail, by the time he has completed this, with its review, to comprehend to a fair degree what Arithmetic has been defined, the art of numbers, and the science or knowledge of computation. The marginal notes, it is thought, will prove highly useful and acceptable as a ready index of subjects :ind a form, for questions, without the formality of their shape. This new feature has received from intelligent teachers very gratifying approbation. The divisions into Introductory, Higher, Frac-r tional. Proportional, Commercial, lladical and I'.iiscellancous Arithmetic, it is hoped, will be a help both to the scholar and instructor. For the convenience of the latter, a key, containing answers, will accompany the volume. Columbia, S. C, Feb., 18G4. 216341 COISTTEILTTS. INTRODUCTORY ARITHMETIC. Page. Definitions ........ i notatiok ..--... 2 Numeration -■-•-- 3 Addition - - - - - .7 Subtraction - - - - - - - 1 7 AfuLTUM.ICATION - . . . . . 23 DrviSTON - - - - - - - ' - 32 Miscellaneous Examples - - - . . kj Reddction ■'----- 42 Tables and Examples ■---.. 43 Miscei^laneocs Tables - . - - - . - 59 FoREiuN Coins --■-... 59 The Meascri:ment of Corn, Pkas, Potatoes, in Bulk - CO Reddction of Lower to Hu;in:i; N'amkh - - . ^;4 Miscellaneous Examples - ■ - GC American Money ---... c,1 Definitions - - • - - - 07 Table ■ • - •■ - - - . . q-j Examples in Addition ■■-... c,<} Subtraction - - ... 70 Multiplication - ■ Division . . . . ^ . Miscellaneous Examples --•... CONTENTS. HIGHER ARITHMETIC. Page- Defixitioks - - - - - - - - 76 CoMrouxD Addition - - , - - - 76 Compound Subtraction - - - - - - 80 Compound Multiplication ..... 83 Compound Division ... - - - - 85 PiiOPKRTiES OF Numbers ..... 88 •Prime Factors •.....-- 89 Greatest Common Divisor • - - - - 90 Least Common Multiple ...--- 91 Miscellaneous Examples - - - - . 92 FRACTIONAL ARITHMETIC. Definitions ....-.-- 93 Vulgar Fractions ...... 93 Various Kinds and Examples - - - - - 93 Miscellaneous^ ExAMPLlss • - - - - 103 Decimal Fractions ....-••- 105 Definitions - - • - - - - 105 Table - - ■ . - - - - - 105 Explanation op Fractions .----. 106 Notation - - - • - ■ - - 106 Addition ....... 107 Subtraction - - - - ... . - 108 Multiplication - - - - ■ - - 109 Division .....--- 110 Reduction • - - - - - - HI Miscellaneous E.xamples - - - - . • - 114 Circulating Decimals - - - ■ - • 116 Continued Fractions . • - • - .- - 117 Duodecimals • - - - - ■ - 118 Analysis - - - • - - - - 522 CONTENTS. XI PrvOrM^IlTIONAL ARITHMETIC. Page. Ratio Axn I'soi-ortiov, oh Simtlk Rule of Tiiijer - - 12 1 CoJiPouND Pi;oi'oiiTio\, oi{ Douiu.B Rule ov Tureu - - • 127 COMMERCIAL ARITHMETIC. IXTERKST, SlMI'LI'; - - - - • - - 130 i>fTi:REST, CoMIMUXl) - ' - - - - 1?,5 Discount - - - - - - - - ];i8 COM.MIISION - - - ^ - - - l:-?!) FELI.OWSliU' ()!t PAIlTNK.r.SIIIP, SlMPLK .... 141 FELLOWSiKI' t^l i'aU7N-!;i;SHII', DoUDLE - - - I'l?) In'suraNi K - - - - - - -141 Profit and Loss - , - . • . . . i/i", Equation (■'■• INym nts - • - - - - 149 Barter - - - - - - lijl Practu-i: ........ 152 ExCI!AN!.r. - - - - • - - - 134 Guaci.vg - - - - . - - - - 1.09 ToxxAui: -..--.-- 161 Anxuitii> - - • - - - - 163 • AlLIGATIkN, >iK,UAL ...... 1G5 Allioation. .Xi.tiirxate - - - - * - - 10(5 Tark or .\t.' -.v V ,-i K ...... 1G7 Position, h;\) .: - - - • - - - 1G9 Position, Douulk - - - • 170 MiSCELLAV! <•' -■ '- \ , .:.s ...... 172- )rCAL ARITHMETIC. Intolution - - - - - - - 174' EvOLUTI.pN,- : lOxTRA'.TION 01' SqUARE RooT - ■ • 17t) Applicaiidv 1 ^j'^uake Hoot - - • • - 179 CucR R.ni;- - - - - • - 18:J E.VTRArTiov oi" ('liiiE Root - - • - - IHt MiSCELLA - '" \Mi'LK3 ..... Ib'J CONTENTS. MISCELLANEOUS ARITHMETIC. Page. AuITHMFViOAL PllOGKESSION , . . ... 187 GEOMSTiaCAL Progression - - - - - 190 PsKMUTATJOJf AND COMBINATION - , - - - - 191 MuKtiCRATION - - - - - - - 193 MiscKi.i.ANEous Examples ------ 202 APPENDIX, BOOK-KEEPING BY SINGLE ENTKY. ran l-)Av Book - - - - - - 207 Tnio LEn(5E!i - - . - - - - - 207 TiM^ CU-.i! Book ...---- 207 Ti Book ■ - 207 .Tu:, i>ii.!.;- .\>:n Notes- Payable - . . - - 207 The Bills axi> Notes Receivable - - - 207 Rkmauks on Noft;s - - - - - " - 208 C():MMKn(jAL FoKMS - - - - - . - . 209 Ti;;- ' •> ::i of Day Book - , - - • - - 211 Til 10 L'i^a.'i OF Ledger ------ 213 TiiE FoFiM oi-'- Cash Book ------ 215 Tu!-: FoKM OF Baj[;k Book . . . . -^ 216 iiiE Fi>KM oi'^ Bills and Notes Payable - - - - 217 Tii. ''■:ni:i or Bills a.nd Notes RECEivAiLt: - - - 218 COMMON ARITHMETIC. SECTION I. SIMPLE NUMBERS. Article \, Arithmedc — a Greek derivative — signifies Antiiiiutii! d^-- simply the ajt of numbers. It h now understood to com- prehend in its exprosision the science or knowledge of computation. 2, A number is what is used to describe a quantity, and wimt m nmniKr is either a unit, as the number omc ; or a collection of linits, as Uco, ten, one hundred. 3* A number is either simple or compound. It is simple Kumhcrs sim- when it expresses a single collection of things, vis jive trees : |j!^„X "^"'^ compound, a collection of varieties*, as Jive treeft and. six apples. i. Arithmetical science shows the values and connections ^^I'l'^'^'^jJJjj .j,., of numbers ; art, their structure, either in complex form, or explained. common relation. 5< An arithmetical operation is icorlrdonehy i\\Q employ- An oppration: nient of numbers ; the result of the work is the ansicer. 6« A rule is a direction to a result; and a sum, the pro- .\ rule; a sum. posed problem for the exercise ol' a rule. 7, The analysis 6? a sum is its separation info component \ ■*"'" ""'*'■ ^ ' Vzod. parts.. 8. There are six diiferent names in arithmetical u^e, and tI"' manber of , , . '" , . . 1 • , 77 the arithmotiC'il what they express is more or less introauced into alt opera- b.isis, and ex- t ions not si mphi elementary : these are,. '*^"'^' 9, JS'otat ion,' Numeration, Addition, Subtraction, Mvltipli- ;Y,!!!'^me's*'-^"" ration and Division. 10. Notation — the simplest expression of a number — Notation d«- sfeows how to read or write an arithmetical proposition. lit The forn» in which an arithmetical proposition is Tinee forms oi given, is in letters, figures, ox tcords : the first is known as notation. Roman notation ; the second, Arabic ; the third, Verbal, or, as when we \\\\t — . CO o e S .2 ' ^.1 f 3 z5 t— zs «.- rs ^ 2 • 'i^ ■o ~ O "- o -° ,-r •M' tc W «^. tB" « o to C — ^ «i "> — "O tn _o -O tn c ^ * 3 TS ot' trv c = 5 = <— » = C o s= •- S 'Z .E <^ •^ 3 y-five billions, five hundred and sixty-three millions, eight hiwidred and one thousand, and ten. The table ex- 25. By the adoption of a name for further periods, this ^1"^®*^'^"^*'^^' table can be indefinitely extended. For common purposes, ■ it is sufficiently large. Numeration (|.e- 26. Numeration is simply the art of reading numbers that have been expressedby figures. i<'igiues.ot o:i( h 27. .Each period always contjiins three figures, except the ponort. j^^^^ which may have onefigure, or two or three figures. In the table given, it will be seen, that in the fifth period, no hundreds are given. ■I'o I'-ii-i iinni- 28. To read numbers : • I. Separate, by a comma, the number into periods of three figures each, beginning at the right hand. If. Name the order of each figure, beginning at the right hand ; as units, tens, hundreds, and so on, to the extent. required. III. Then commence at the left hand, and read each period as if it stood alone. EXERCISES IN A'UMEKATION. 29* Let the pupil sejjarafe by commas the follovCing num- Ifers, viz "^ 468765201 32575654321 718950 985674 M2 6785403 25fi87J45 112543764 •32789532140 .3642005428 5001400 30. To write ntiinbers : I. Commeuce at the* 'f'ft. hand, and write each ptriod in ordfr. II. When a fiill vticant ptMJod occurs, ciphers must occupy the spasce. -, . ■ 1 : : „ , Note. — With- a little care, the pupil will be able to ■write any number. Let him remember that each period is to be expressed. When no nuiTiber i's given, a cipher must be emplovod. No unit, ten. or himdred must be left Avithour expression. I'.xeri-i.sos in 1. iinmevation. 2. 3. 4. 5. 6. 7. ■ 8. 9. 10. 1 .1 wi'ifp ninn- m 48- i 11. •127 12. .5«54 • 13. 9705 14. 32693 15. 57865 16. 69432 ! 17. 234356 18. 78.5643 1 19. 305400 ! 20. >U?IERATION. EXA?1PLES. 31. 1. ■3. 3. 4. 5. 6. 7. 3. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 30 602 100,000 Write the following in figures: Twenty-five, Thirty, Seventy-eight, Four hundred, Six hundred and two, Nine hundred and sixty-five, One thousand and. one, Fifty-three hundred and fifty-six, One hundred thousand, One hundred thousand and eishty,' Six hundred tliousand and two. One million, three hundred and twenty-one thou- sand. Two hundred and twenty. five billions, four bun- dred and sixty-three millions, seven hundred and ninety-eight thousand, two hundred and thirty-four. Three billions and si\ty-five, 3,000,000,065 Nine billions, two millions, twenty-five thousand, one hundred, Forty billions, one hundred and twenty-seven mill- ions, and ninety. nine. Three trillions, sixty-one thousand, and seven. One hundred and twenty billions, one thousand, and one hundred. Two "billions, twenty, two millions, two hundred and twenty-two thousand, two hundred and twenty- two, 2,022,222,222 One trillion, one hundred thousand. l".x;imples lor practice in tim- iDcration. NOTATION AND NUMERATION, 1. The earliest method of denoting numli^rs was prob- j,,,, ,;,,||, ,( ably that of renresonling each unit by a separate siiin. No ni.tiio.i of .». •,. ,, ' . . °. J / 1 .•! ci, . I- noting Tiuni- perlfctly convenient method was found, until ihc Arabic i)«>ri-. figures, or Digits, and the present Decimal system were employed. These figures, 0, 1, 2, .3, 4, 5, 6, 7, 8, 9,^denote wimi tho un- nothing, one unit,. two units, three units, and so on. lloto." " '^' 2. To express numbers in excess of 9, availment is made to exj>res.« » 'if the law that assigns higher values to figures, according ^,"'^"" '" '"*" " lo their position. According to (his law, any figure at the KOTATION AND NUMERATION. A law in fijrnrei- The result of the position. Different value: expressed by the figure 1. The zero— its nse. Uaity of tlie second order. Third order. lnterniedi;Ui' numbers. A more consis- tent nomencla- ture. , The numeni- tion table. left of another figure expresses ten times the value that it would express if it ©ccupied the place of the figure at it.« right. Consequently, a higher srder of units arises in suc- cession. 3. In illustration of this law, let the different values .expressed by the figure 1 be noticed. Standing by itself, or at the right hand of other figures, 1 represents 1 unit of the first order ; in the second place towards the left, thus 10, it represents 1 ten, which is one unit of the second degree ; when in the third place, thus, 100, it represents 1 hundred; which is one unit of the third order, and so on. The ciphers here employed are. without value in themselves. They are only used to occupy a place. 4. The units of the second order, or the tens, are suc- cessively named 10, ten, 20, twenty, 30, thirty, and so onto 90, ninety. The units of the third order, or tlie hundreds, are named 100, one hundred, 200, two hundred, 300, three hundred, and so 'on to 900, nine hundred. The numbers between 10 apd 20 are named 11, eleven, 12, twelve, 13, thirteen, and so on to 19, nineteen. The intermediate num- bers in other tens are similarly denoted, but their designa- tion is taken from the names of their respective units ; thus. 21,' twenty-one, 22, twenty-two, 23, twenty -three, and so on : 31, thirty-one, 32, thirty-two, &c.; 41, forty-one, &c. Com- pound names applied to the number.s between 10 and 20, might be more consistent, thus, 11, ten-one, 12, ten-two, 13, ten-three, &c.; but those in common use are sufficiently intelligible. . • 5. As the first three places of figures are appropriated to simple units, tens and hundreds, so every succeeding three places are appropriated to the units, tens and hundreds of" higher denominations. Adopting a name for every three degree of units, the table can be indefinitely extended. The tabk ex- terided. 0:2:0 o .2 Ul XJl a -a b^ » ^ c 0i s S S -c 'S 233, 104, 395, 473, 258, 333, 573, 820, 765, 569, 321, 560. Write in figures 1. Three octillions, four quintillions, tvi'o millions and twenty-nine. >2. Twenty-five septillions, three hundred and thirty-five quadrillions, thirty-seven billions, two thousand and five. ADDITION. 3. Ten decillions, eight sextillions, one thousand and one hundred. 4. Three hundred quadrillions, four billions and sixty-five. 5. Sixty-five octillions, thirty-five trillions and ninety-five. 6. Four sextillions, two thousand and three. ADDITION. 32t Addition is the putting together of one or more num- Addition dc- herfi, to find their amount. ^^ ' 33« The sign + is called phis, and signifies more. When t'^- ^>g" v^^»- placed between two numbers, it shows that addition is to be performed ; thus, 3-f-4, is 7. 34« The sign of equality is =, and it signifies that the The sign ..r quantities between which it stands are equal to each other ; ''i""^''*-^ ^ thus, 3+4=7. 35< These signs are not employed when the figures to be wii.n these added are placed vertically : thus, 5 ;;'«!;( ''"•*' "'« 4 . 7 16 36* The latter is the common form, and when the num- wheutiievi-ni- hers are more than simple collections of units or ones, as in '"' ♦'^'■nnsbo-t. the example, it is, if not quite necessary, the most easy and convenient arrangement. 37« Write the numbejs to be added together in order, t„ .„m mnn- placing units under units, tens under tens, hundreds under '"'' Hundreds, and so on. Draw* a line beneath, as in the exam- ple, and if the amount or sum be less than ten, set it under that column ; if it be ten or more, put down the units or ones as before, and add the tens to the next column. The sum of the last column is to be written in full. 38« Example 1. — Add together the numbers 50, 46, 68. When the numbers have been placed as directed, the right hand or unit column is first added ; thus, 8 and 6 and 0, are 14 uiiits, or I ten and 4 units. The 4 is then ]lut tmdor the first or unit column, and the 1 ten is added to the second column, or the column of tens ; thus, 1 and 6 are 7, and 4 arc 11, and 5 are 16 tpn=, or 1 hundred and 6 tons. The 6 tens is placed OPERATION. 50 Rxplauation of tho way to ndd 46 68 164 ADDITIOJS'. under the second column, or the column of tens, and the I hundred, which, if there had been another column, would have been added to that, is set at the left hand of the 6. Sums for.prar- tjcoip addition. (2.) 58 416 (3.) 5 421 (4.) 5936 1862 (5.) 3650 2905 233 3856 21 3217 Ans. 707 Am 9. 4282 Ans, 7819 Ans. 11 1 1 1 1 * (6.). 575 (7.) 3360 (8.) 56780 (9.) 1309652 8026 4527 2300 275943 32 9802 45695 322568 232 3765 37821 534576 7495 4930 26308 323355 Ans. 2769 39376 -» 486789 .(10.) 3567820 (11.) 5 (12.) 39605067 495323 74 4214503 36532 532 7809 4789 4789 890 532 36532 20 74 495323 5060 • 5 ' 3567820 » 25 Ans, PROOF. A shorter * of addine. Reading Uu' 39. To find if the columns have been added correctly, the simplest method is to observe the s'ame process, only commencing at the top, and adding downward. Ifthe figures correspond, it is the f^roof sought of correct addition ; if they do not, the variation shows evidence of error. 40. In adding a column, omit the names of the figures.. Thus, in the 12th sum, instead of saying 5 and 9 are 14 and 3 are 17 and 7 are 24, say briefly, 5, 14, 17, 24, and then setting down the 4 units* or ones, and carrying the 2, say 4, 10, 12, 21, 27, and so pn. This is called reading the col- umns, and is done by good accountants. Frequent practicft will render it easy. * From the Latin unus, which means one. ADDITION. - '.} Example (13.) Add 7959, 6543, and 3487 together. Examples f.r (14.) Add 6954, 78421,5678, 34,5659, 432178, 598765, i>r»21234, 56789015^and 325.to^ether. (15.) Add 32, 361, 4500,51189, 67891, 3606,48572, 72684, 30605.4336, 211667, 5870, 59865, and 999 together. (,16.) Add 71 19, 59435, 625, 32,59601, 43C656, 5987451, and 23205 together. (17.) Ajark. — The sign that denotes a dollar, or dollars is $ ; Ttiodoii.ir.<.^ii:i. thus, $1 represents one dollar ; $5, five dollars; $100, one hundred dollars. ^ , PRACTICAL EXAMPLES. E.VAMPLE (1.^ There are due to my factors in Charles- pnK-tioaJ vx- ton. fur money advanced on cotton, S 150, to uiy grocers, '""'''''■" $275, to the machinist forn steam-engine, S050, to the dry- goods merchants, $159, to the hardware dealers, $87, and to the Rank of the Slate of Sontfi Carolina, for a note dis- counted, S2895: what is the amount of this indebtedness? (2.) The State of A' ir<£inia harl, in the year 1790. a pop- Papulation of ulation of 748,308; of Maryland. 319 '28 ; of North Car- f,ru!r"^'''" lina, 393.751 ; -of Tennessee^ .35,791 ; of Kentucky. 73,077 ; of Georgia, 82,548 ; and of South Carolina, 249,073 : what was the aggregate ? (3i) The same States had, in 18-30, the first named a population of 1,211.405; ih" second, 447,040 ; the third, 737.987; the fourth, 681,901; the filth, 687,917 ;,ihe sixlh. 516 823; and the last. 5^L1''5 : what was the whole, the Northern States. 10 ■ ADDITION. Free jjopuia- .(^O The States forming the Southern Confeder»cy, iu ^'Ste st^aTefin' 1861, had, according to the last census— South Carolina, a hhl. " free population of 301,271 ; Georgia,* 595,079 ; Florida. 78,686 ; Alabama, 529,164 ; Mississippi^ 354,699 ; Lou- isiana, 376,913 ; Arkansas, 324,233; Texas, 420,651; Virginia, 1,105,196: what was the total ? ^i Siave wvpuia- (5.) The same Confederate States had. at that date, a iTerite'state"" ^'^^'^ population— South Carolina, of 402.541 ; Georgia, ,u'i86L ' " 460,232 ; Florida, 61,753 ; Alabama, 435.132 ; Mississippi, 4.36,696; Louisiana, 332,520 ; Arkansas.lir,104 ; Texas. 180.388 ; Virginia, 490,887 : what was the entire number I The United (6.) The United States has 9,334 miles of coast ; the Idlrite'states'" Confederate States, 28,803 : what is the whole ? sea-coast. (7.) The State of Maine has of sea-coast and shores, of 'rhej5ea-coast of bays, sounds, dsc, and of rivers to tide-head, a total of miles, < ' 2,452 The State of"Neurope. 272.000,000 A^sia, 755;000,000 America. 20%000,000 Africa, 59,000,000 Australia, (kc, 2,000,000 What was the entire number : 12 . ADDITION. Population of (13-) According to races, the population of the globe was, the ranes. j^ I860-- Caucasian, 369,000,000 Mongolian, 522,000,000 Ethiopian, 196,000,000 American, 1,000,000 Malay, ^00,0 00.000 What was the total? Population reii- (14.) In religion, the population of the' globe is thus Kiously divided. (]jyj(jgj^ • . Christians— Protestants, 89,000.000 Romish Church, . 170,000,000 Jews, 5,000,000 Mohammedans, ' 160,000.000 Heathen, 788,000,000 What is the sum ? Lo.sse.s of the (^^O ^" ^^^ '^^^^ ^^ ^^^ Revolution, the losses sustained British and by the English and Americans, including in some cases the the Revolution- wounded, and the surrenders at Saratoga and Yorktown, iry Wnr. hsLve been computed as follows : Lexington, April 19, 1775, Bunker Hill, June 17, 1775, <• Fort Mouhrie, June 28, 1776, Flatbush, Augiist 12, 1776, White Plains, August 26, 1776, Trenton, December 25, 1776, Princeton, January's, 1777, Hubbardstown, Aug. 16 and 1 7, 1777, Bennington, August, 16, 1777, Brandy wine, September 11, 1777, Stillwater, September 17, .1777, Germantown, October 4, 1777, . ^ • Saratoga, Oct. 17, 1777, surrendered, Red Hook, October 22, 1777, Monmouth, June 25, 1778, Rhode Island, August 27, 1778, Briar Creek, March 30, 1779, Stoney Point, July 15, 1779, Camden, August 16, 1779, King's Mountain, October 1, 1780, British. American. Loss. Loss. 273 -93 1054 454 205 11 400 200 409 400 1000 9 400 100 800 ■ 800 800 100 500 1100 600 350 600 1200 6752 500 32 400 130 260 211 13 400 600 100 375 610 950 96 . ADDITION. • 1'^ Cowpen^, January 17, 1781, 800 72 Guilford Court Houso. MarrI, 15, 1781, 532. , -IPO ' Hobkirk Hill, April 25, 1781. ' 400 400 Eutaw Springs, September, 1781, 1000 550 Yorktown, October, 1781, surreiKlered, 7072 What was the entire loss on either side? (16.) Gen. Washington was born A. D. 1732, and lived 67 years : in what year did he die ? (17.) How many days are there in the twelve calendar months, January having 31, February 2S. March 31, April 30. May 31, June 30, July 31, August 31, September 30, October 31, November 30," December 31 ? (18.) William the Conqueror began to reifrn in England in the year 1006, and rrigned 21 years; William II, , 13 years ; Heurvl.. 15 ; Stephen, 39; Henry 11., 35; Richard I., iO ; John, 17 ; Henry HI., 56 ; Edward I., 35 ; Edward 11., 20 ; Edward HI., 50 ; Richard H., 2'3i In what year was Richard dethroned ? (19.) From the creation of tlie world to the flood wa.^ 1656 years ; front? that time to the building of SolomonV Temple, 1336 years; thence to the birth of our Saviour. 1003 years : in what year of the world was our' Lord born"? (20.) If you invested in the Hank of Georgia, $5000 ; in the Mississippi State Funds, $2550 ; in the Savannah City Stock, $1550 ; in the Southwestern Railroad Company, 82000 ; and in the Port Royal Railroad, 85500 : what would be the amount thus placed ? (21.) A father in his will leaves to his eldest daughter, ^5000 ; to his youngest, $3550 ; to his four sons, $300 each ; and to them also, severally, the same sum left to his eldest daughter : what was the entire amount ? • (22.) A. planter sent to his factor, at one time, 4 billes of cotton, which weighed 1437 pounds; at anothor 6, which weighed 2100 ; at a later date 10, which weighed 3575 ; and afterwards 25, which weigliod 8000 : what was the whole nu nber of pounds, and how many bales ? (23.) A factor received on consignment 300 busheb of rice; a few days later 2169; then 5560 ; afterwards 4175 ; and last, 2358 : how many bushels in all ? '. (24.) There are 60 seconds in a minute, 3000 In an hour. 86,400 in a day, 604,800 in a week, 2,419,200 in a month, and 31,557,600 in a year: how many are all these combined together? (25.) A merchant in Mobile received frr m New Orleans, ten casks of hardware, weighing each 19604-875+1000+ It • ADDITION. 1268+999+25614-3248+1545+1862+1300: what was fhe .weight altogether ? (26.) The sums on the debit side of a merchant's books for three different periods were as follows : 832.68 15789.51 $2134.c0 $789.61 1643.42 4357.00 5.48 347.31 429.66 14.52 56.25 3906 25 23.07 3.33 78.16 1832 43 741.50 259.30 6789.75 1150.00 745.S3 . 5432.10 212.13 567.49 874.32 56.15 2.12 75.15 83.33 16.79 7128.23 79.20 8901.31 4728.53 56.02 47.14 ' 2.25 9.15 87.42 773.19 940.42 ,940.43 ^544.96 ' 59.75 366.03 337.16 42.58 267.30 22.81 569.87 24.56, 18.76 256.00 842.15 1530.21 .55.02 47.96 55.02 268.34 • 81.15 586.75 367.35 46.53 142.04 2269.54 • •487.20 1678.39 693. 1509.99 Ans. The porioii Rematik. — In the last three sums, the 'period marks ''u?«hin"the between the second and third columns, distinguish the lioiiiir.-^ from dollfw's and cents ; thus, the bottom figures in the last sum, 1509.99,' are read fifteen hundred and nine dollars, and ninety-nine cents. ^ : _, . -r-, ■ :— I'olnmns to be Note. — When the columns of figures are very long, let them be ^.iepar.tted wher. separated by Imes, as in the last example ; and tlie separated amounts \pry long. placed by themselves, added for the required answer. i ilH ADDITION. 15 The census of the United States, in 1850, and that of 1*^60, were as follow? : what were the totals ? CENSUS OF 1850. .) States. Fr«e. Slave, Total. Alabama, 428,779 342.844 771,623 Arkansas, 162.797 47,100 209.897 Kr"*^'" California, 92.597 92,597 Connectic\it, :nO 792 370,792 Delaware, 89,242 2,290 . 91,532* Florida, 4S.135 39,310 87,445 Georgia, 524,503 381,682 906,165 Illinois, 651,470 851,470 Indiana, 983.416 938,416 Iowa, 192.214 192.214 Kansas, Kentucky^ Louisiana, Maine, iMaryland, Massachusetts, Mississippi, .'Missouri, Michigan, Minnesota, New Hampshire, Xew Jersey, New Y6rk, North Carolina, Ohio, Oregon, Pennsylvania, Uhode hlaud,* .South Carolina, Tennessee, Texas, Vermont, Virginia, VV'isconsin. 'rKKRITORIES Colorado, Dakotah, Nebraska, Xevada, Xew Mexico, Ttah, Washington, Dis. of Columbia, Ans, 771.424 210,981 ^982,405 272,953 244,809 517,762 583.169 ,583,169 492.666 90,368 583.034 994,514 994,514 296,648 309,878 606,526 594.622 87,422 682,044 397,654 397,654 6,077 6,077 * 317.976 317,976 489.319 236 489,555 3.097,394 3,097,394 5^0,491 288,549 869,039 1,980,329 1.980,329 13.294 13,294 2.311.786 2.311,786 147.545 147.545 283.523 384.984 668,507 763.258 289,459 1,002,717 154,431 58,161 212,592 314,120 314,120 949,133 472,528 1,421,661 305.,'^91 305.391 t'clisus ivf t U.S.TfiT ries in is:.' 61.547 (51.547 11,354 26 1 1,380 » 48.020 8,687 51.687 ir Census of tlie U^ite.l States ' I'li.-u?' Ill", the .r. S. 'i'crrito- •i<'^ 111 i»;(i. ADDITION. ■ CENSUS OF ISOO. (i'<:) i^TWES. Free. Sl.Wi!. TOIAL. AlAbrima, 529,164 435,132 964,296 Arkansas, 324,323 111,104 435,427 California, 380,015 380.015 Connecticut, 460,151 460,151 Delaware, 110,420 1,798 112,218 •Florida, 78,630 61,753 • 140.439 Georgia, 595,097 462,230 1.057.327 Illinois, 1,711,753 1,711,753 Indiana, 1,350,479 1,350,479 Iowa, 674,948 674.948 Kansas, 107,110 107.110 . Kentucky, 930,223 225,490 1,1.55,713 Louisiana, 376,913 332,520 709,433 Maine, * 628 276 628,276 Maryland, 599,846 87,188 687,034 Massachusett s, 1,231,065 1.231.065 Mississippi, : 354,699 436,696 791,395 Missouri, 1,058,352 111,965 1,173,317 Michigiin, 749,112 749,112' Minnesota, 102.022 162,022 New Hamps hire. 320,072 326,072 New Jersey, 672.03'I 672,031 New Yojk, 3,887,542 3,887,542 North Carol i na, 061,556 331,681 •S82 667 Ohio, 2,339,599 ■2,339,599 Oregon, 52.464 52,464 . Pennsylvania 2,906,370 2,906.370 Rhode Island L . .. 174,621 174,621 South Caroli na, 301.271 402.5M .703,812 Tennessee, 834083 .275,784" 1,109,847 Texas, 420,651 180,388 601,039 Vermont, 315,1 16' 31.5,110 Virginia, 1,105,198 490,887 1,596,083 Wisconsin, 775.873 775.873 Tep.hitories. Colorado, 34,197 34,197 Dakotah, 4.S39 4.839 Nebraska, ■ 28,632 10 28,842 Nevada, 6,857 G,857 Now" Mexico 03,517 24 93,541 Utah, . 40.260 29 40.295 Washington, 11.578 11,578 Dis.o.f" Colli mbia, 71.895 3.181 75.07t) Aihs. . RUBTK ACTION. 17 The primary mode of formino; numbers by joinmg oneThepriucipieoi unit to another, and the sum of those to a third, and so on, * justifiid by the selfevident truth that if two unequal quantities be equally increased, their difference is the .•r sale ? 8. A farmer bought 200 sheep, and gave 300 dollars for them ; a yoke of oxen, which cost 65 dollars ; a horse, 125 dollars ; a cow, 40 dollars ; and he paid toward the purchase 100 bushels corn, valued at 75 dollars ; 200 bushels oats, at 80 dollars ; 800 weight of blades, at 50 dollars, and gives his note for the balance ; what is the amount of his note ? 0. If you purchased 100 oranges for 250 cents, 200 limes tor 538 cents, 5 bunches of bananas for 950 cents, and 25 cocoa-nuts for 150 cents, what would they come to in cents ? what in dollars and cents ? what, if for bad fruit, 338 cents were taken off, would the amount be ? 10. In France, during the Reign of Terror, there were guillotined, hy sentence of the Revolutionary Tribunal, nobles 1278, women of the sajiie clas.s 760, wives of artisans 1467, religieuses 350, prie^its 1135, persons 'not noble 13,623 ; what was the entire number ? Wlnlt is the dill'r. ence between this and the aggregate of other victims ))ui to death in other forms? — women killed in La Vendee ]5.tH)0, women dying fom yiief and f(iar 3.748, chiKIren killed 22.000, men 900,000, victims at Lyons 31,000, at Nantes 32,000. iMULl'lPLICATlON. •■18. Mahivlicaiion is a process by which is found, quickly, MulUpiif'.iii' > .-'-.. ', •' I ^ defined, the amount ot a given number. , 4^, 'I'he given nundier is called iho mulliplicand ; •1"'- \'",'J'^'',j'.'"^' nne which is used to disc-over the required amount is called Multipii'nn i the mullipJicr; and the answer found is called the ■prndvct. J*,' '"^ The multiplicand and multiplier are called factors — that is ) the malitrs of the product. ■2-1 MULTIFLICATIOX. The sign X- 50i The sign X placed between two numbers, denotes their mvltiplicafion togoJher ; the ?e4wZ/ of the niuliipHca- pounVnumbw^' '"^^ '^^ Itnown -is z compound numher : thus. Gx3=18 shows, , . that by the use of the factors, 6 and 3, the result, or the compnund number 18 has been found. duct thoug^*^ 51 • The product or result is invaria})ly the ^cr^ne, whether the figures are fhp factors stated as above, or in the reverse order, 3x6. To facilitate ^^* "^^ facilitate multiplication, it is necessary to keep in nuiltiplication. memory the sum of each of the nine first numbers, or digits, • as they are called,* repeated from one time to nine times ; that is, the products of each of the nine digits by themselves and by each other. The common tables usually extend through 12 ; and the following copies them, as it is conve- "hient to know more than the products of 9x9. MULTIPLICATION TABLE. 1 2 3 4 6- 6 7 8 9 10 11 12 24 The inultipljea- tiun table. 2 4 6 8 10 12 14 16 18 20 22. 3 6 9 12 15 18 21 24 27 1 30 33 1 36 4 8 12 16 20 24 28 32 36 40 44 48 ■! 5 10 15 20 25 30 1 35 40 45 50 55 60 6 i 12 18 24 30 36 42 ' 48 54 60 66 72 84 1 14 21 28 36 42 49 56 63 70 77 8 16 24 32 40 48 56 04 72 80 88 06 9 18 27 36 45 54 63 1 72 81 90 99 108 10 1] 20 22 30 33 40 50 60 70 80 90 1 100 110 120 44 55 66 77 88 99 110 121 13t 12 24 36 48 60 72 84 96 108 120 1 132 144 * So named from an old custom of counting upon the fingers, and derived from the Latin word digitus, a finger. MULTIPLICATION. -0 53. This table was made by writing, as iu the upper The muUipii. n- row, the uumbers 1, 2, 3, 4, otc. The second by adding *g'","be't''^ '"^' these numbers to themselves, and writing them directly under the first; thus ,1 and 1 are 2; 2 and 2 are 4 3 3 and 3 are '6, etc. The third row by adding the second to the first; thus 2 and 1 are 3; 4 and 2 are 6; and 8 are 9, etc.; this contains the first row, it will be observed, three times. The fourth row is formed by adding the third row to the first, and so on with the rest. 5-1. When the formation of the table is comprehended, ^,^^, ^^^^ the mode of use will be apparent. If, for instance, the uso. product of 8 by 5, that is 5 times 8 were required, we look for 8 in the upper row, then directly under it, in the fifth row, is found 40, which is 8 repeated 5 timtjs. lu like manner, we find other numbers. 55. To multiply one number by another, is to rc/;ea< — w"eplnui^^^^^ as is done in addition — ^the first number as many times as a i>rui" ■ - there arc units, or ones, iu the second number. '* ^ '"' Example 1. — In one pasture are 8 sheep: how many,, ,,. ,. . :irc there in 5 pastures, each with the like number t jx.rfoini,-.i i \ i;y addition. i'Y multiplicatiox. ^l]^. ''■• '8 8 S s . 8 8 5 Ans. 40 An^. 40 2. A field is marked out into 12 tasks : how many tasks would there be iu 12 fields similarly divided? OrKRATIO>f. KXI'LANATJON. E.tplar.ation '1 ~r ^ ■ ^^ i 11 lrl.-lr.,l > .1 1 t U' WOrK 111 til- :\rulliiplu-and, 12 Multij)ly first by 2, }i,s though given esami«!.- 2 were the only multiplier ; then by 1, and set the first figure of the result in the place of tens, or the second column,* and then the 1 by the 1, and set it in third column, or th:it of hundreds. Then draw a line beneath, add the two results, and the jiroduet will be as shown in the cxaiiiple. 56. The same process is to be observed whether the 'r>'^„'f!!^? '" ■ \ ample have few or many figures. Multiplier, 12 24 12 Product, 144 * Weu'e tlie wuici ( iMLiniij m muid rcpctiti id for Wi,nt of a better term. 2'i MULTIPLICATION. 3. lu a day are 2-i hours : how many hours are there in 28 days ? OPERATION. EXPLANATION. ExiManation of work-. Multiplicand, 24 IMultiplier, 28 In this example, we first say t^ times 4 is 32, and then place the 2 — a line being drawn — under the 192 8, or in the units column. "We 48 then say 8 times 2 is IG, and add to that the 3 of the first multipli- Product, 672 cation, which making 19, is set at the left of the 2, the whole being 192. We then proceed to the second figure 2 in the multiplier, and say 2 times 4 (or preferably twice 4) is '8, which is put down in the • second or tens column; and as nothing remains over to be carried, we proceed to the nest and say, 2 times (or twice) 2 is .4. This is placed at the left of the 8 in the hun- dreds column, a line beneath drawn, and'the results added for the product, as above. A partial pro- g-j^ ']^\^q product obtained by the multiplication of a single figure of the multiplier is called a partial product. In the last example, 192 and 48 are such. 58« From these examples, we draw the following plain directions for multiplying a number : Direction for' I. Place the uiultiplier under the multiplicand, and draw multiplying i- „ Kgneath II. Commence at the right hand, and multiply the mul- tiplicand by each figure in the multiplier, and place the first figure of each partial product directly under its' mul- tiplier, and the others in order at t.hc left. III. When the multiplication is completed, draw a line beneath, and add for an answer the results of the multi- plication. Proof. 59, To prove a sum, simply reverse the position, of mul- tiplicand and multiplier, and proceed by the directions. 4. Multiply 4320 by 345. 5. Multiply 58760 by 4632. 6. Multiply 897432 by 3G91. 7. Multiply 94870 by 3323. 8. Multiply 660761 by 4232. 9. Multiply 3010102 by 6654. 10. Multiply 34208 by 6985. When ,1 cipher ^^' When a cipher occurs between the figures of the is in the muiti- multiplier, it must, as much as anyvOther figure, be set in ^'"■'''' its place in the partial product. MULTIPLICATION. 11. Multiply 9567 by 8005. Xotc. — This simply sho^vs the value of ciphers in certain positions. 0507 8005 47835 '653000 Ana. 765S3835 12. Multiply 90574 by 8903. 1 3. Multiply 79876 by 7005. 14. Multiply 37965 by 2002. 15. Multiply 54032 by 2561. Rkmark., — It is to be' observed, that multiplying-, or multiplied by a number, 5 for instance, is 0, and will, if it be the multiplier, form a partial product, unless the briefer plan, as shown in the 11th example, be adopted. To explain the point in the following example, both methods are given. Tlic briefer i>s the better. 16. Multiply 3665 by 5002. 3665 5002 7330 1832500_ A«s. 18332330 obbo 51102 J330 0000 0000 18325 Ans. 18832330 01. When ciphers are at the right hand, the same can a oi, be extended beyond their usual place, and simply brought |?j',^,^i'.' down, before we commeuce with the multiplication of the nest figure. 4567 17. Multiply 4507 by 300. 300 Ans 1370100 i>hf at ti li.'iii'l. !-. Multiply Multiply Multiply Miiltil.lv 90700006 80507700 10. Multiply 21. Multiply 23. Multiply 25. :\ruitipiy 51000504 6530050 20. 69007 2035000 807065 203040 22. 357555 2300.\iiiHple of foc- roi-jng multipli- <;;ition. 31. Multiply 54 by 21. OPERATION. 21=7X3. Multiplicand, 1st Factor of Mul., 2d Factor of Mul, Product, , 1134 Remark. — The factors of 21 are the numbers 7 and 3 ; and it is plain that 7 times 3 times a number, arc 21 times that number. 32. Multiply 355 by 40, or its factors 8 and 5. 33. Multiply 6789 by 63. 34. 565x72=howmaDy? ' 35. 36789 x42=how many? 36. 96569 x36=i=how many? '37. 23467 X54=how many? How to dispose g4^ When ciphers are at the fight hand both of th- hlnlum'j'iiVand multiplicand and multiplier, it is sufficient to multiply iu jiiid multiplier. ^^^^ ^.^^^g already stated, and then to place the full number of ciphers at the right of the product. ' ^ 395 38. Multiply 39500 by 65000. 65 1975 2370 39. Multiply 4500 by 6500. Am. 2587500000 4500 6500 Ans. 29250000 MULTIPLICATION. -•' 10. Multiply 345600 by 28700. 41. 2G900X.;54300. 42. 0785600x353200. 43. 27000x690. 44. 53600x70.500. 45. 840320x543000. 46. 2680000x135600. 47. 901000x75300. TKACTICAL EX'AJIPLES. I. What is the cost of 10 pounds of coffee at 15 centsa pi'^^'J^i'luK"',: : pound? ' 'i cation. , 2. What is the value of 350 bushels of corn at 75 cents a bushel ? o. There arc 320 rods in a mile : how many rods are til ere in 265 miles ? 4. Suppose the Charleston Mercury to have 32 columns, and 16 paragraphs on an average in each column : how many paragraphs would there be in the paper ? ■ 5. Suppose the same paper to have 9 words in each line Vof 512 paragraphs : what number of words would there be? ,6. If a regiment contain 780 men : what, if they were viof like number, would there be in 120 regiments ? 7. What is the value of 250 aci-es of land, at 636 an acre ? 8. In its annual revolution, the earth moves about 19 miles the second : how far does it move in a weekjsupposing that in a week are 604,800 seconds? 9. How many pounds of cotton in 432 bags, each bag containing 337 pounds? 10. Sixty-five men can build a waJl in 45 days : how long would it take one n»an to build a wall 18 times as long? II. Two men travel in diffeVeut directions; one at the rate of 65 miles a day, the other 45 : how far will they be apart in 15 days ? 12. What must I pay for 12 barrels of flour, at §7 a barrel ; 250 bushels of rice, at 83 a bushel ; 5 barrels of molasses, at SI 8 a barrel j and 3 hogsheads of bacon, at §75 a hogshead ? , / Multiplication is simply an abridged method of finding ?^"'''!p''V'**'^'" the sum ot several equal quantities by tlie repetition otinetiiod to tini ouc of those quantities. Hsum. When the product of factors, consisting of several figures, 1 .' . . u- 1 1. c • .1 The rationale oi is required, it is necessary to multiply each ngure in thcniuitipiicatton. multiplicand by each figure in the multiplier, and denote the several products in such order that they shall represent their respective values. When simple units a,re employed I' the multiplier, the product of each figure in themulti- MULtlPLICATION. . piicaud is of tlie same degree as the figure multiplied; that is, units multiplying units give units, units multiply- ing tens give tens, units multiplying hundreds give hun- dreds, etc. When tens are employed as the multiplier, the product of each figure in the multiplicand is one degree higher than the figure multiplied : that is, tens multiplying units give tens, tens multiplying tens give hundreds, tens multiplying hundreds give thousands, ete. When hundreds are employed as the, multiplier, the product of each figure in .'the multiplicand is Uvo degrees higher than the figure multiplied, and so on. * OPERATION. We commence by placing To multiply • 87468 the multiplier under the mul- By 5847 tiplicand, so that the units of the same order fall in the saoie | 612276 column. This being arranged, | 3498720 v.'e observe that to multiply ' 69974400 87468 by 5847, is to take the 437340000 .multiplicand 7 times, 40 times, 800 times, and 5000 times; . 511425396 then to add together these partial products. We can first find the product of 87468 by 7, which gives 612276. The second operation reduces itself to multiplying the multiplicand by the figure *4, considered as expressing simple units in writing a to the right of the product, and in placing the result, as in the written operation, below the first partial product. In like manner, in order to perform the multiplication of 87468 by 800, it is sufficient to mul- tiply that number by 8, which gives 699744, then .annex two ciphers to the right of -this product ; we thus have a third partial product, which is placed below the two pre- ceding products. So also to perform the multiplication of 87468 by 5000, it suffices to multiply by 5, to annex three ciphers to the product and write the result, 437340000, below the first three products. Adding the four partial products, we have the total product 511425396. In the multiplication of the 4, or the tens figure in the above, we may conceive that we have written one under another, 40 numbers, each one being 87468, and that wjj add them to obtain the required product. It is, however, evident that these 40 numbers form ten divisions, each con- taining 4 times 87468. Adding.and multiplying the result by 10, or what is equivalent, annexing a 0, we obtain 3498720 for the product of 87468 by 40. A similar reasoning applies to the third number 8; for MULXirLICATION. 31 800 numbers equal to 87468, and placed one under another, ' form, evidently, 100 divisions of 8 numbers, each equal to 87408, or 100 numbers, equal to the product of 87468 by 8 ; that is, 00!)7400 : also, to the fourth number 5, that 5000. It is customary to dispense with the ciphers to the riijht '*"''^ 'li-ivn.-a- 01 the partial product; but we write each partial product of oipiitTs: below the preceding one, advancing it one place to the left ; that is, we make the first figure multiplied occupy the same column which the figure by which we multiply, occupies. To determine if an error has happened iu the process, of multiplication, the following method of trial, which er^ror! """" depends on the peculiar property of the number 9, and ,„, . .,. „ which is called casting out the nines, may be practised: «f tho nines. Add together the figures of the product- horizontally, i-),,, , rejecting the number 9 as often as the sum amounts to that number, and proceeding with the excess, and finally denote the last excess. Perform the same operation upon each of the factors; theu multiply together the excesses of the factors, and cast out the nines from their product. If the excess of this smaller product be equal to the excess of the larger product first found, the work may be supposed to be right. Such test is, however, not infallible; for if a » product happen to contain an error of just 9 units of any degree, the excess of its horizontal sum is not thereby altered. To understand why the excess above nines found in the The rea.'^on horizontal sum of a product, must be equal to the excess ^^^'^'" found in the product of the excesses of the factors, it is to be noticed that by the law of notation, a figure is increased nine times its value by its removal one place to the left; and hence, however far a figure is removed from the place of units, when its nines are excluded, its remainder can only be itself Hence, any number, and the horizontal sum of its figures, must have equal remainders when their nines are excluded. This being understood, let it be observed, that since factors composed of entire nines will give a product consisting of entire ninus, it follows that any exce.-;i g5. Division is a process used to find how many times ' '^"^ '• one number is contained in another. X- .V, e ^v.,. 1 66* T^6 ^"'st number used for this purpose is called the -Names employ- ,.. , ,. hiit.,, it ■, • -, ed in division, dwisor ; the sccond IS called the dividend; and the third, or result, is called the g'MO^J'/enf. . The overplus or g7t When any number is over to the performed division, rernaiuder. ., . ii i .1.7- *■ . it IS called the remamder. ^■j,..^^^^,f 6S« Three signs are employed to express division, vi-^ion. namely, -j- j — ') ; and are thus used, as in the case where 15 is to be divided by 5 ; 15-r-5 ; -V-; 5)15. Wj)on tiie divi- 69. When the divisor exceeds 12, and the divisionary ""n^^u^ed'^"^^^ ^"^^^ ) ^^ "'^cd, one is drawn corresponding to it ( on the right of the dividend, and the quotient placed against it; thus, 15)135(9 quoticRt. Division of two 'J^^ Division is distinguished as sJiort and lonff. ^-'hort division. '^i* Short division is the method used when the results simply are written, in consequence of the divisor^,being 12. or less than 12; for its performance observe the following " directions: 72. I- Write the divisor on the left of the dividend. Short division. Begin at the left, and having divided the figure, or the fewest figures in the dividend, that contain the divisor, set each quotient figure under its dividend. II. When there is a remainder after any division, annex to it the next figure of the dividend and proceed as before- Ill. Should any figure of the dividend be less than the divisor, put down for its quotient, and annex the next figure of the dividend for a new dividend. IV. Should there be a remainder after the division of the last figure, set the divisor under it, and annex the result to the quotient. . ' • 73. To discover if the work is ciorrect, multiply the Proof. divisor by the quotient, and if there-js a remainder add it to the product. This will give the dividend if the division has been correctly performed. EXAMPLE 1. Divisor, 3)6396, Dividend. Quotient, 2132 3 Proof, 6396 DIVISION. 2. Divi(Je t340 by 5. OPKHATION. Divisor, 5)7o40, Dividend. Quotient, 14US 5 7HV, KXPLANATION. Explannt.on o In this sum we say 5 into w"'''* ^^ opera 7, 1 time (or, preferably/'**" once) and 2 over; setting down the 1 under the 7, and annexing the 2 that was over to the 3, we say 5 setting down tlie 4 under the Proo into 2o, 4 times and o over i5, and annexing the 8 that was over to tlie 4, we say 5 into J>4. G times and 4 over; setting the under the 4, and annexing the 4 to the 0, we r>ay 5 into 40, 8 times, which placed uiicicr the 0, concludes the sum. 3. Divide 8JU0 by 5. 4. Divide 42:UU by 4. 5. Divide 7G-'l by G. 6 Div^lc N()7.'n)y 7. 7. Divide 9") 1 80 by l>. 8. Divide 67674U by 8. 9. ])ivide 27840 by 5. No/r.—As in tliis example, the divif^or 5 is not contained xvhcm the dnr in the tirsl figuie 2, oi' the dividend; w,e say & into 2"7y^nd po'e-\<^eedstb^ then proceed. ] 0. Divide 56128456788 % 12. n. Divide 4r)6780Di3:.'77 by 11. 12. Divide 768492Ul(/-340 by 10. Divide 507><0-l3i'l by 9. Divide ()470.)4.'i4 by8.— ^«.s. {^045679 and 2 over. Divide 324.o<)'75 by 5. 67^5-4^1=:'' 7S9n4-f-5= ? 18. 8-.'3.")0(i-f- =:' 19. 5i:;70U?-H;'=? 20. ":-'1i;M)nf;-^<'— ' first figure <>< the dividend 13 14. 15. 16. 17. i-ONG DIVISION 74, \< -nil ilio divi.sor exceeds 12, ii is cUv'^tomary to when long div employ the method called Lvtif/ Divmo . In this, the '''^" '^ "^^^'^ entire work is i)Ut down j the mental calculation, as in short division, being dispensed with. I^y, To perform Lt)ng Division — ' I. Write the divisor and dividend as in short division, and draw a curve on the right ot the dividend • 4 ■ ■ ' '4i DIVISION. How to perform II. Divide the smallest number of figutes in the left ol long division, j-j^g- dividend, that will contain the divisor, and set the result, as the first figure of the quotient;, at the right of the divicleud. III. Multiply the divisor by each now quotient figure, and set the product under that part of the dividend taken for division. What is a par- IV. Subtract the product from the figures over it, and tiai dividend? ^^ ^^^ remainder annex the nest figure of the dividend for a new partial dividend. V. Divide the partial dividend, and proceed as before. until the whole dividend is exhausted. 21. Divide 46755 by 15. OPERATION. EXPLANATION. 15)46755(3 117^4ns.{ In this example, we say 15 into Explanation of 45 45 3 fcimey, wliich 3 is put in the t}w> work I . . — I quotient ; then we multiply the 17 divisor by the 3, and set the result. 15 45, under the 46 in the left of the — dividend; we next subtract the 45 25 ?rom the 46, and having drawn 15 a line beneath the 45, put "the — reixainder, 1, under the 5, and 105 draw down the next figure in the 105 dividend, 7; we then say 15 into 17, 1, and putting this 1 in the quotient, multiply the divisor by the 1, and place the 15 under the 17 ; drawing a line beneath, we subtract the 5 from the 7, and to the 2 annex the next figure 5 in the dividend, and so on until the division i? completed. 22. Divide 678954 by 25. 23. Divide 78956 by 35. 24. Divide 954321, by 45. 25. Divide 345687 by- 50. When the divisor contains three or more figures, some- times it is difficult to discover how many times the divisor is contained in the figures separated as a partial dividend. Often the difficulty is obviated or decreased by trying the first figure of the divisor into the first figure or first two figures of the partial dividend. vSuch trial indicates nearh the true figure. DIVISION. ;i'> 26. Divide 436940074 by 64237. OPKRATION. EXPLWATION 64237)436940074(6802 Ans. 385422 515180 513896 128474 128474 111 this example, we Unci how many times 6, the first figure of the divisor, is con- tiiiued in 43, the first two figures at the left of the dividend. It is found to be 7, and 7 is contained 6 times. This is the limit The iiuiit. or «- ,-, . , ,, , tent m hjvisiou- or extent. i>y trial, we nud :.ry trial. it cannot be 7, for 7 -times 6 are 42, which subtracted fr^m 43, leaves 1 to be joined to the next figure 6, for a new partial dividend. But 4, the s'econd figure of the divisor, is not contained 7 times in 16, therefore 6 and not 7, will be the first figure in the quotient. 27. Divide 56679670 by 3456. 28. Divide 89765132 by 94432. 29. Divide 7543215 by 876504. 30. Divide 37651245 by 508912. 31. Divide. 29543278 by 896302. 32. 35678-=-412= r 33. 56945-4-375= ? 34. 88003-4-510= 'f 76, AVhen the divisor is 10, 100, 1000, etc., cut from Whm. ^tho^diw the right hand side as many figures as there are ciphers in etc. ' the divisor. The figures at the left will be the quotient, and thi)se at the risjht the remainder. 35. Divide 846"by 10. Am. S4-6, or 84 quotient and 6 remainder. 36. Divide 95607 by 10. 37. Divide 74.568 by 10. 38. Divide 76543 by 100. Ans. 765-43 39. Divide 234678 by 100. 40. Divide 35987654 by 1000. Aas. 35987-654. 41. Divide 5976U20 by 1000. 42. Divide 9396781 by 1000. -^ 77t VVhen the divisor is a compound number (Art. 50), a compound di a short method to find the quotient is to separate the divisor 'nto^itsl-a^tors^ into factors, and proceed as in the Ibllowing : 43. Divide 626800 by 40, that is, by its factors 5x8=40. OPERATION. ^^^ ...Oj. 1st Factor, 5)626800, Dividend. j. - t-^. v..j. 2d Factor, 8)125360, 1st Quotient. 15670, 2d Quotient, or Result. M DIVISION. Oiviaion factors. by 3 44. Divide 1678 by 24. 45. Divide 1896 by 35. 46. Divide 3564 by 54. 47. Divide 6780 by 56. 48. Divide 32960 by 21. 78« When the divisor can be resolved into 3 factors the' like process is to be performed, thus : OPERATION. Work perforin- 49 pjvide 990576 by 108, or by the factors 9X6X2=108. 9)990576 6)110064 2)18344 9172 Quotient. 50. Divide 187236 by 252. 51. Divide 1255872 by 192. 52. Divide 12393327 by 189. iuv,o„ n,^..» o..^ 79. Should there be remaindei's to the different divis- •Tneu sneie are . 1.1 ii .1111 ^• • i remainders to lous, multiply the hist remainder by the last divisor but i.m. ""^^ '^'^'one, and add in the preceding remainder; then multiply this result by the next preceding divisor, and add in the remainder, and so on until the first remainder is added; the sum obtained in this way will be the true remainder. Nute. — Should any remainder be a cipher, then let a cipher be added. OPERATION. If the remain- der is a cipher. 53. Divide 15956 by 280, that is, by the factors 7X5x8=280. 7)15956 5)2279-3 Remainder, 8)455-4 Rem. Work showing the true remain- lidr. 56-7 Rem. True remainder as seen by the following division : 280)15956(56 1400 1956 1680 276 By application of method (Art. 79), 7x5+4=39; 39x7+o=:276, the true remainder. 54. Divide 3765452 by 126. i>rvisioiV. Oh. 543/650^-4156= ? o/. How many times is qin ^S. What would you J vfde sl;^^^ '" "^^^3 ? quotient 475 ? ^''^'^ ^^7b54 by, to have for its 59; If a dividend is G94505 tlio . .- rerna.nder i 19 : what is the d?;is;%'^"'*^''"^ ^^06, and the 61.^)ivide 'iS^^r ''" ^"^^^-^ b«-ath, t^?|f^r^. •578934128 56 "20676-6 Ans. 189 168 213 196 174 168 62. Divid6 89765643 by 29 63. Divide 53740912 W 356. 64. Divide 9655468 b3 4..78. ticu h.i„i „„eV they've,, pTp?; "'""" '"'"'■ "" 9uo. ,if tunes may be eurric! down toThi I ""V'"'' """"^^^ ind divided therewith, wien th ' Til 7"" '^"'' "^ "'^'*« in exact number of tiles in the Li 'T 'l""' ^^"tained •emainder at the end c' th one, f "'h''^'''' "'" ^^ ^ ■ pai-t of. the dividend and is t?. i /• • i \'' '•«»ia'"der is ient will be smaller thn. un "t "'^"^ ' ^"^ ''« <1^°- !ivii- ,unatt.eief,. ^.^^ ^^^ .J^.^ence at the left, because .^hc dundend being the sum of the partial products of the divisor, by the units, tens, hundreds, etc .of the quotient, all these partial pro- ducts are luin^'le'l ^"^ ^'*'' another, so that it is impossible to eommencoV^sf ^^'•fi"^ out the product by the units, V ,vg ^^,„.^, etosed divisor. . Since in division, the dividend is a product of which thecert»iu truths divisor and quotient are two factors, it follows that to divide •the dividend by a certain entire number, the quotient is by this change divided by the same entire number. . For, as, after this change, the quotient multiplied by "the divisor must produce a dividend a certain nuuiberof times greater or less than the first dividend, it follows necessarily, the divisor remaining the same, that the quotient must be the >' '"le number of times greater or less.- )n the contrary, if, without 'altering the dividend, we under the divisor a certain number of times greater or smaller, the quotient is thereby rendered the same number of times smaller or greater. Then by dividing the divi- , dend and the divisor by the same number, we do not change ;£lie quotieat; since, if, by the change of the dividend, we ^*^ MlSCELLANEOtS EXABIPLKS. divide the quotient by a certain number, the second cbaugt renders it the same number of times smaller or greater. Thus, the compensation leaves it the sami-. MISCELLANEOUS EXAMPLES ON .THE INTRO.DUCTOEY PART. SECTION I. Kxamplos in Votation, Siiditjo)!. ."■'jilitniofion. tfultiplicaiion ind Division. Example 1. — Write in Koman Notation 9, 99, 100, 1000, 2060. 2. Write in figures six, sixty, six hundred, ono thousand, nine liuiidrcd. 3. Write in figures three trillions, t>wo hundred and forty billions and five. 4. Write in words 35,644,760,027,201. • 5. Add 500, 650, 4709, 810, 80, 1, and 29 together. 6. Add 2, 20, 35, 595, 6a6, 890, and 5 together. 7. From one hundred take ten. 8. From two hundred and fi^fty take eighty. 9. From one hundred take one. 10. From four hundred and sixty-two take sixty- five. 11. Add 5000, 659, 7005, 89, and 100 together. , 12. Add 99, 990. 709, 56, and 47 together. 18. How many times are 483 contained in 568943!' 14. How many times are 6340 contained in 2076907 ''. 15. Howmanvtimesai-eSoOOcontainedin 37695843? 16. Add 55+664-706+94+2 together: 17. Subtract 56785 from 1578790. 18. 9340760—65007= ? 19. 725631—483...= ? 20. What is the difference between 1776 and 1861? 21. What is the product of five thousand eight hundred and seventy, by two hundred and sixty-five? 22. What is the product of thirty-nine thousand seven hundred and thirty-six, by eight hundred and twenty-five ? MISCE^iLANEOUS ESAMPLES. -i' 23.' Subtract 4596006 from 58560C7('. Example in 24. How many times arc 64:Z (iontaiiicd in 1000 ? PubtSon. 25. Howmany times arc 71o2 coiitiuncd in 10000? MuitipHc-.toou 26. IIow many times are ;»SU:i contained in 1000000 ? pivisio-. 27. Subtract U)056784 from 875648201. 28. 68541 — 3954 arc how many? 29. Twenty thousand times thirty thousand are how many ? 30. Sixty-five thousand times five thousand and dighty-ibur arc how many ? 31. What is the product of eiglity-onc thousand Iwo hundred and seven, by three thousand one hundred and forty-five ? 32. >yhat is the product of thirty-seven thousand five hundred and sixty -five, by two thousand and fifty-two ? j 83. From four thousand four liundred and tvventy- tiinc, take two thousand and sixty-eight. / 34. Front five thousand seven hundred and s^vehty- Jioven, take six hundred and eighty-lour. 35. Multiply ten millions by ten. 36. Multiply thirty millions bj' one hundred. 37. Multipl}' one thousand by one thousand. . 38. From one hundred thousand take one. " 39. 65-|-405+76y-f8905, are how many? 40. 3695-^340, are how many? 41. 785U6x695, are how nmny? 42. 597092 — 579, are how many ? 43. How many times are 5684 contained in 16784320 ? 44. How many times are 1804 contained in 286784? 45. How miiny times are 695 contained in 457803? 46. From one thousand take nine hundred and ninetj'-nine. 47. Multiply seven millions six thousiad and thirty bv thirty -three. *48. Divide 87658910 by 795. 49. Divide 59374390 by 622. 50. Divide 38427695 by 3652. 51. Multiply 9867846 bv 5890. Ri'DUCTION. • EKDUCTION. COMPOUND NUMBERS. SECTION II. Uoduciuivi de- ^^* i^^duction is tiio change of numbers from one uw.d. name or denomination to that of another, but without change of value. '!y« numi-.erij 82. When numbers are to be changed fiom a higher io''d1fferlnf ^^ ^ lov/er name, mv.ltiplication is employed; but when 'i-tnis. tlie change is from a lower to a higher one, we use division. <>>mt)ounu^ 83. I^umberB subject to such changes are called "ul!}eHiJ^ih^L^^'^^P^''^^^d,m distinction to t\\esi77i2)le numbers already ■<.hc..nges. Considered. A compuiind 84. A compouncl number is made of two or more 11"^^^'"'^*^; unlike denominations; thus dollars, cents, dimes, pounds, shillings and pence, are of this cluss. What are not 85. It should be notcd, that while the sevcr.il parts mimbers"^ °^ '^ compound number, as pounds, shillings-!, pence, are of different name, they are classed together as relatively alike ; but pounds and dollars, and grains and minutes, etc., having no common bond, cannot be reckoned as compound numbers. All of such nature are expressions of unlike values. HoH to ppduoe 86. To reducc a compound number to one of a lower r°»^!r''^ name, observe the following directions : Multiply the highest denomination in the number to be changed, by that figure which indicates, how mapy ones or units of the next lower denomination are con- • tained in one of the highest, and add to the product the parts of the same value with the multiplier, and 80 on. rabie of values 87. The following tables of values should be thor- loi memory, oughly memorized. The tables of refef-ence beneath TaWes for refer- them arc not to bc Studied, but used for comparing work done by the pupil. They simply show, in a con- densed form, the tables of values, thus: 1 £=20.s.=240(?.=960gr. or farthings. REDUCTION^ 4.-i ENGLISH MONEY. S8- 4 Farthings, marked qr.* make I penny, d. 12 Pence make 1 slnliing, ,«. 20 F^hillings make 1 potind, £. 21 Shillin!L!;s make 1 i>innea: Tuble of Km crlish moni^v Reference Tabic. S. ' d. 1 1 = 12 s:0 = 24(1 = 4S = 0(U) Example 1 - operation. 10 20 lis 12 8 21.5.?. 12 2588rf. 4 -E(*diice £10 15s. M. 2qr: to farthing^.; EXPLANATION. Ten poiinds=200s., and the ^:^,?^|-"^'' 155. added make 215s.; thi.s number multiplied by 12=2580, iiiid with the 8^/. added, 2588 ; this multi)died by 4=10:352, and the 2 added, 10854, which is tlie answer. Woi)\qr. Ans. \ 2. Reduce £2518. Gd. 3q}\ to lUrthings. ^otc- — It is convenient to place in small figures, as in the first example, above the numbers of the sum, the several multipliers. o. Reduce £37 16s! f>d. to pence. 4. Reduce £l9 Os. 4^^. to farthings. 5. Reduce £40 -85. 5.Vrf. to farthings. 6. Reduce £32 3s. 9d. to pence. 7. Rc 6s. Qd. to pence. 12. Reduce £48 12s. 6.^Z. to farthings. 13. In 5 guineas, how many pence are there Kxamples in Knglisn mnii(> Qr. is quarter, the quart'.^r or fourth part of a penny. My iMfysure. ■'i-^ REDUCTION. 14. Eod^^ce 50 guineas to shillings. 16. Eeduce 2 guineas to shillings. IS^ote. — Whenever there is omission of any term, as that of shillings in the 4th sum, it does not afteet the multiplication by shillings; thus 19 is to be multiplied by 20, though the order of shillings is not i!i the sun:. DRY MEASURE. 8Si 2 Pints, pt. make 1 quart, qt. 8 Quai-ts make 1 peck, pk. 4 Pecks make 1 bushel, bush. 36 Bushels make 1 chaldron, ch. bush. Note. — By this table, all dry articles, as grain, salt, coal, vegetables, etc., are measured. Example 1. — How many pints are there in 1/ bushels, 3. jiecks, 5 quarts and 1 pint? 4 8 ^2 lo bush. Spk. b qt. \ pt. 4 m pk. 8 509 qt. Reference Table. pk. qt. pt. 1 = 2 1 = 8 = 16 4 = 32 = 64 .4715. 1020 pt. k.j.iuripies m '^- ^^luce 10 bushels, 3 pecks, 5 quarts to pints. ih-v i,ie;.,s-are. 8. Eeduco 25 bushcIs, 2 pecks, 6 quarts to quartp. 4. How many pecks in 35 bushels ? 5. How many pints in 28 quarts ? 6. How many pints in 3 pecks and 6 quarts? 7. How many quarts in 2 bushels and 2 pecks? 8. Eeduce 144 chaldrons to bushels. 9. Eeduce 46 chaldrons to pecks. 10. Eeduce 25 bushels, 3 pecks, 7 quarts, 1 pint, tc' • pints. KEDUCTION. 11. Eeduce 75 bushels and 2 pecks to quarts. 12. Eeducb 250 Sushels to pecks. .LIQUID MEASURE. 90. -i Gills, gi. make 1 pint, pt. ■ -:q«' 2 Pints make 1 quart, qt. 4 (Quarts make 1 gallon, gal. )\\l Gallons make 1 barrel, 6^*/. Go Galloiis (iil.]x2) make 1' hogshead, hhd. 2 llogsheadiS make 1 pipe,/)/. 2 Pipes or 4 hogsheads, make 1 tun, tun. The tierce in tables called 42 gallons, is omitted. 'X» it does not represout the tierce used in trade, which baa sometimes many moi*e gallons. lieferctice Table, hin. pi. hhd. bbl. gal. qt. pt. gi. • 1= 4 1 = 4= 8= ^2 I = SU= 126= 252=U)UB 1 = = (>3 = 25.=^ 504=201(5 1 = 2 = =126 = 5o4=luU8=40;52 1 = 2 = 4 = =2.")2 =lOOc^=::ul(!=8U64 B}' this are measured all liquids, except milk, ale and l)ocr. Tiie Confederate States gallon of liquid n\easure i-i 231 cubic inches. Example 1.— Ill 6 pipes', 3 hogsheads, 15 gallona '. an4 3 quarts, how many quarts ? "^ 2 ' 63 4 pi. 3 hhd. 15 gal. ?> qt. 15 63 iu;o 4 Ans. 3843 qt-^ 46 . KEDUCTION. Eemark. — When a number to be added consists of two figures, as the 15- in this sum, it is more con- venient to place as above, than to use it as we do sin- gle numbers. 2. Eeduce 15 pipes, 1 hogshead, 3 gallons to quarts. 3. Reduce 1 tun, 30 gallons, 2 quarts, 2 pints, •:■■ gills to gills. 4r. Reduce 1 hogshead, 15 gallons, 3 quarts to pint?. 5. Reduce 1 barrel, 2 quarts to quarts. 4 I bbl. = ol^ gal. 2 qt. i explanation. 4 _ j ■ In this example, a!< 124 I the f gallon=2 quarts. 2 { we simply add i of 4 j to the quart, one-half 126 ! gallon beino- 2 quarts. 2 A /IS. 128 qts. 6. Reduce 1 hogshead, 1 barrel, 3 quarts, 2 pints to pints. 7. Reduce 1 barrel, 3 quarts to gills. 8. Reduce 1 hogshead, 15 gallons, 3 quarts, 2 pints. 3 gills to gills. 9. In 3 hogsheads, 12 gallons, 3 quarts, how many quarts ? 10. In one pipe, 1 hogshead, 5 quarts, how many quarts ? 11. In 1 tun, 1 barrel, how many pints ? 12. In 2 hogsheads, 14 gallons, how many gallons'' AVOIRDUPOIS WEIGHT. iipf"ht"^°" SI* 1*> I>i'aehms, dr. make 1 ounce, o~. 16 Ounces make 1 pound, Ih. 25 Pounds make 1 quarter, qr. 4 Quarters make 1 hundred weigiit, not. 20 Hundred weight make 1 ton, t. Reference Table, t. act. qr. lb. oz. dr. 1 = 16 1 = 16 = 256 1 = i>5 = 400 = 6400 1 = 4 = 100 = 1600 = 25600 I = 20 = 80 = 2000 = 32000 = 512000 REDUCTION. Such articles as sugar, coffee^ tea, cotton and metals, with the exception of gold (ind silver, arc weighed by this weight. .1 In the old tables, liS lbs. was called a qr., and 112 lbs. a cwt.; but now the standard qi*. is 25 lbs., and the cwt. 100 lbs. Example 1. — Reduce 17 tons, 8 huiidrcd weight, -^ qiiarters, 13 pounds, to pounds. . 20 4 2.') nt/:..^cwt. s qr la //>. 20 i'AS cw 1 . 4 1395 qr. 25 6975 2790 18 .ins. 34888 lbs. 2. Reduce 6t. 7cwt. 2qr. 2010., to oz. 8. Reduce 25t. 5cwt. Iqr. lOlb. 2oz., to oz 4. Reduco 181b. lloz. 12dr., to dr. 5. Reduce 2t. 15oz. 14dr., to dr. 6. Reduce Oqi-. 171b. looz. 5dr., to dr. 7. Reduce 3t. 22 lb., to oz. 8. Reduce 8qr. 15oz., to dr. 9. In ocwt. 201b., how many pounds ? 10. In 9qr. 121b. 5oz., how many- ounces? apothecaries' weight. 92, 20 Grains, 0 1 = 8 = 24 =s 480 12 =-96 = 288 = 5760 4S REDUCTION. Medicines are compounded by this weight, but bought and sold by Avoirdupois weight There is no difference in the pound, ounce and grain of this and Troy weight, but the ounce is differently sub- divided. ' Example 1. — Eeduce 6 pounds, 7 ounces, 5 drachms, 2 scruples, 12 grains, to grains. '12 8 3 20 6tb 7s 53 29 12gr. 71) 8 637 8 ■:;^^t, 1913 20 Ans. 38272 f/r. 2. Reduce 25rb. 8oz. 2dr. 15se., to scruples. 3. Reduce 801b. 9oz. Idr., to drachms. 4. Reduce 21b. 3oz. 2dr., to scruples. 5. in oOfb. 8oz. 2dr. 12sc., how many scruples? 6. In 9oz. 5dr., how many drachms? 7. In 50tb. 4oz. Idr., how many scruples? 8. In 121b. 2dr., how many scruples ? 9. In 291b. 6oz. Idr. IGsc, how many grains? 10. In 5oz. 2dr. 15sc., how many grains? ' TROY WEIGHT. 93. 24 Grains, gr. make 1 pennyweight, ^j?o^ 20 Pennyweights make 1 ounce, oz. 12 Ounces make 1 pound, lb. Reference Table. lb. oz. pwt. gr. 1 = 24 1 = 20 = 4H0 1 = 12 = 240 = 5760 This is the standard measure for gold, silver, jewels, corn, bread and liquors. REDUCTION. 41) Example 1. — Reduce 15 pounds, 9 ounces, 14 penny- weights, 12 grains, to grains.- 12 20 24 151b. 9oz. 14pwt. 12 gr. 12 189 oz. 20 3794 pwt. 24 . Ans. 51068 gr. 2. Reduce 211b. 9oz. J4pwt,.12gr., to grains. 3. Reduce 5oz. 14pwt., to pennyweights. 4. Reduce 551b. lOoz., to grains. 5. Reduce 9oz. 15pwt. 15gr., to grains. ■C. Reduce IGpwt. 6gr., to grains. 7. Reduce 101b. 6oz., to -pennyweights. 8. In 131b. IGpwt. logr., how many grains? 9. In 351b. 14pwt. lOgr., how many grains? 10. In 601b. how many pennyweights? ALE OR BEER MEASURE, ^4, 2 Pints, j)t. make 1 quart, qt. A»e or beer 4 Quarts make 1 gallon, ffai. 36 Gallons make 1 barrel, bar. 54 Gallons make 1 hogshead, hhd. Refo'encc Table. hhiL bar. gal. qt. pt. 1 = 2 I = 4 = 8 1 == 36 = 144 = 288 1 = 1] = 54 = 216 = 432 1 Gallon contains 282 cubic inches. ' 5 50 REDtiCTlON. Example 1.— Reduce 25 hogsheads, 3 quarts, 2 piiite^ to pints. 4 2 25hhd. Sqt 2pL 54 Cloth measure. 100 125 1350 gal. 4 5403 qt. 2 Jfote. — Though no gallons are named in the sum, we still have to multiply by the number of gallons that ' make a hogshead. A71S. 10808 pt- 2. Reduce 5hhd, 2bar. 3qt. Ipt., to pints, 3. Reduce 30hhd. 3bar. 2pt., to pints. 4. Reduce 15hhd. 15gal. 3 qt., to quarts, 5. Reduce 12hhd. 20gal. 2qt., to pints. 6. Reduce 25gal., to pints. 7. Reduce Ibar. 3qt., to pints. 8. In 8hhd., how many quarts ? 95. 2i* 4 4 CLO^fH MEASURE, Inches, m. make 1 nail, na. Nails make 1 quarter, qr. Quarters make 1 yard, i/d. Quarters make 1 Ell Flemish, E. Fl Quarters make 1 Ell English, E. Er E. Eng. Reference Table, yd. E. FL qr. 1 li 1 If na. 1 4 12 16 20 in. 2( 9 27 /]6 45 Cloths, carpets and all articles of the yard ineusurement are sold by this. * The fractional \, and other like expressions in thu Tables^, will be explained under the head of fractions/ REDUCTION. 51 Example 1. — Reduce 36 yards, 3 quarters, 2 nails, to nails. 4 4 36 yd. 3qr. 2na. 4 147 qr. 4 Ans. 590 na. 2. Reduce 2.5yd. 2qr., to nails. 3. Reduce 350yd. 3qr. 3na., to nails. 4. Reduce 3qr. 3na., to nails. 5. Reduce 12yd., to quarters. 6. In 28E. Fl., how many nails? . 7. In 17E. Eug., how many quarters? 8. In 60E. Fl., how many yards? 9. In 37E. Eng., how many yards? 10. In 45yd. 3qr., how many quarters ? LONG MEASURE, 96, 3 Barleycorns, i.e. make 1 inch, //?. j.ons,' mea^wn. 12 Inches make I foot, ft. 3 Feet make 1 yard, i/d. 5 J Yards, or (5:1x3) 16^ feet, make 1 rod,"r(?. 40 Rods make 1 furlong, /ur. 8 Furlongs make 1 mile, mi. 3 jMiles make 1 league, l. '69j^ Statute miles, nearly, or 60 geographical miles, make 1 degree or circumference of the earth, c?e6 ' REDUCTION. A cord meas- ure. 40 Cubic feet make 1 ton of timber, t. 16 Cubic feet make 1 cord foot, eft. 8 Cord feet, or 128 cubic feet make 1 cord, Reference Table. cu. yd. cu. ft. cu, in. 1 = 1728 1 = 27 = ' 46656 This is used to measure what has length, breadth' and thickness. A cord of wood is 4 feet wide, 4 feet thick and 8 feet long. ' " A cube defined. A Cube is a figure of six equal squares, called faces; the sides of the squares are called edges. The face on which a cube stands is called its base. If the edge is one yard, it will contain 3x3=9 square feet; therefore, 9 cubic feet can be placed on the base; and hence if the figure «vere 1 foot thick, it would, contain 9 cubic feet; if • • it were 2, it would contain twice as many; if 3, 27 feet. Tofindttiecon- The Contents of a cube are found by multiplying tents of a cube, together the length, breadth and thickness. Theio.sainhew- Round timber is estimated to lose one-fifth by ,ng timber. gquaring. Example 1. — Reduce 18 cubic yards, 18 cubic feet, ] 5 cubic inches, to inches. Ans. 27 1728 18 cu. yd. IScu.ft. 15 cu. in, 27 126 18 18 324 cu. ft. ' 1728 2592 648 2268 324 659872 cu. in. REDUCTION. 2. Reduce Ic. 5cu. yd. 20cu. ft., to inches. 3. Reduce 16cu. yd. 15cu. ft., to feet. 4. Reduce 12cu. yd. lOcu. ft., to inches. 5. Reduce 5c., to cubic inches. 0. In 25 cords of wood, how m\ny cord feet ? 7. In 30 cords of wood, how many cubic feet ? 8. In 120 feet round timber, how many inches TIME MEASURE. , 101. GO Seconds, .sec. make 1 minute, m. 60 Minutes make 1 hour, h. ' "^'"'^^ "^^•'"'"•^. 24 hours make 1 day, d. 7 Days make. 1 week, wk. 4 Weeks make 1 lunar month, I. m. 12 Months make 1 calendar year, c. yr. 13 Mouths, I day and 6 hours make 1 Julian year, J.i/r. Reference Table. I. 711. wh. d. h. m. sec. \= CO 1= 60= 3600 1 = 24= 1440= 86400 1 = 7 = 168= 10080= 604800 1 = 4 = 28 = 672= 40320= 2419200 13, ^; ,=52J^3±=365i=8766=525960=31557600 J.r/it-. 1 = Note. — As the length of the year is 365 days and 6 hours, the odd hours in 4 years make 1 day, which is added to every fourth year, in the month of February. '^<'^r J*''""- The year, thus increased, is called. Leap Year. Years exactly divisible by 4, as 1860, 1864, 1868, are leap years. The following verse memorized, is of use to recall the number of days in each month : Thirty days hath September, TheriavKof tiie April, June and November; * All the rest have thirty-one Except the second month alone, And that has eight and twenty, clear, But nine and twenty each Leap Year. moath. 5& EEDUCTION. Example 1. — Reduce, 5 years, 9 months, 6 days, 12 hours, 15 minutes, to minutes. 12 7 24 60 5 yr. 9 mo. 6 da. 12 hr. 15 m. 12 In this sum let it 2791440 15 69 mo-. 4 be noted that no weeks are named. 276 wk. 7 1938 da. • 24 7752 3876 12 46524 hr. 60 Ans. 2791455 m. 102. 2. Reduce IJ. yr. to minuted. 3. Reduce 5mo. 5da. 16hr., to minutes. 4. In 25yr. 2wk. 5da., how many hours ? 6. In Ida. 18hr. 20ra., how many seconds ? 6. In 15hr. 35m. 40sec., how many seconds? 7. Reduce Ic. yr. to minutes. 8. Reduce 3mo. 5wk. 5da., to hours. 9. Reduce 2J. yr. to minutes. 10, Reduce Se. yr. to hours. REDUCyON. id MISCELLANEOUS TABLES. 103. 12 Units, or things, make 1 dozen. 12 Dozen make 1 gross. 12 Gross, ox 144r dozen, make 1 great gross. 20 Units, or things, make 1 score. 196 Pounds make 1 barrel of flour. 100 Pounds make 1 quintal of fish. 200 Pounds make 1 barrel of pork. 18 Inches make 1 cubit. 14 Pounds of iron or lead make 1 stone. * 21-5 Stones make 1 pig. 8 Pigs make 1 fother. 24 Sheets of paper make 1 quire. 20 Quires make 1 ream. i04. FOREIGN COINS- COUNTRY. ' • GOLD COINS. SILVER COINS. I>EN0MI.VATl6x. 1 VAlDf,. BENOMINAWON'. 1 v.\h;e. Austria, . Ducat, . •Z 28 Seudo. . 3 Ct«. IB, 1 01 5 Belgium, Bolivia, . 25 Francs, 4 72 5 Francs, . 9t> 8 Doubloon, . 15 .58 Dollar. . 1 05 4 Brazil, . :i(J.ooo Kois, 10 90 .5 2000 Reis. . 1 01 ;j €hi!i, \0 Pesos, 15 .3 New Dollar, . 97 Denmark, . 10 Thaler. . 7 flO 2 Kigsdaler, . Shilluig. new. 1 09 4 England, Sovereigil, now, . 4 t!fl 3 22 7 England, Sovereign, average. 4 84 8 Shilling, average. 22 2 France, . 20 Francs, average, 3 84 o 5 Francs, average, 90 8 .Germany, nortli. 10 ThsJer, . 7 00 Thaler, 71 7 Germaay, south, Dueat, . 2 28 3 Guilder or Florin, 41 2 Mexico. Doubloon, average, 15 53 4 Dollar, avcr.ige, 1 04 9 Netherlands, . 10 Guilders, . 3 99 2i4(jNUders, 1 02 3 New Granuda, 10 Peso.s. new. . 9 0" 5 Dollar, 1857, . 96 8 Peru, Doubloon, old, . 35 56 I>ollar, 1855, . 93 r, Portngal, Crown, 5 81 3 Crown, 1 16 (4 Rome, 2y, Sendi, new, . 2 60.0 .Scudo, . 1 04 7 Russia, 5 Roubles, 3 97 6 Rouble. . 78 4 Spain, 100 Heals, . 4 90 3 Pistareen, new, . 20 1 Sweden, Dueat, 2 20 7 Rix dollar. 1 10 1 Turkey, . 100 Pi.istres, 4 37 -J 20 Piastres,, . 86 5 TuHcnny, Seqnin. 2 .30 n Florin, 27 4 iVo^r. — The above values are computed at the Mint rate of $18.60 per ounce standard (9-10 fine) for gold, and ^1.21 per ounce standard for silver. Note. — The English pound, or pound sterling, is valued at e4.44c. 4m. (Art. 285). The French franc is valued at 18. i cents (Art. 286). BEDUjCTION. » RULES FOR MEASURING CRIBS, HOGSHEADS, ETC. V"^ To find the number of cubic feet iu any square crib or box, multiply the length by the breadth (in I'eet) for the number of square feet on the floor, and this product by the depth, for the required number of cubic feet in the box or room. Thus if a room be 12 feet long by 6 wide, it contains 12x6=72 square feet on the floor, and if 5 feet deep, it contains 72x5=360 cubic feet. To find the number of bushels: A cubic foot contains 1728 cubic inches — and a bushel about 2160 (accurately, 2150.42) inches. A cubic foot is therefore 1728-2160= 4-5 or 8-10 of a bushel. A wine gallon contains 281 cubic inches. A cubic foot therefore contains about 7 1-2 and a bushel about 9 1-3 wine gallons. Corn is usually put up on the cob or in the shuck, while it is sold by the bushel or barrel of shelled corn. The proportion of shelled corn to corn on the cob is nearly uniform, but compared with corn in the shuck' it varies considerably — depending on — 1, the size of the ears — 2, the Avay it is shucked, and — 3, the way it is packed or troddcu in. One bushel of shelled corn is equal to two bushels of corn on the cob, to about three bushels of corn in slip shuck (say 21 to 3i), and to about 4 of corn in full shuck (say 4 to 4^). If a crii) of corn on the cob is 12 feet long, 10 wide, and 8 deep, it will hold as follows: 12 Lensrth in feet. 10 Widlh. 120 Square feet on floor. 8 Depth. 960 Cubic feet, 8 =8-10 Multiplier for bushels. 7680 (The right hand figure cut ofl) number of bushels of corn on the cob — 768. 2—7680 384 Number of bushels of shelled corn. S — 768 Bushuls — if in slip shuck. 256 Bushels of shelled corn. 4 — 768 Bushels — if in full or whole shuck. REDUCTION. 192 Bushels of shelled corn. 5 — 38-4 Bushels of shelled corn. 7G 4-5 Barrels of shelled corn. \Nbte. — If the corn be not level, it must be made so or averaged. A concise rule for finding the" contents, in shelled corn, of a crib of corn put up in the cob. Multiply together the length, breadth and average depth, expressed in feet. Multiply this product by 4, and cut off one figure from the right, for the answer in bushels of shelled corn. EXAMPLE. In a crib 15 feet long, 12 feet wide, filled 9 feet deep with corn on the cob, how many bushels of shelled corn ? 15 Length in feet. 12 Width. 180 Square feet on floor. 9 Depth. 1620 Cubic feet. 4 Multiplied for bushels. 648,0 (One decimal cut off) 684 bushels shelled corn. If the corn be in slip shuck, multiply the cubic feet by o, and if in full shuck, by 2, and cut off one figure as decimal for the answer in bushels of shelled corn. A concise rule for reducing corn on the cob, to barrel?? of shelled corn : Take 8 per cent, of the product of length, width luid depth, expressed in feet. EXAMPLE. In a crib of corn on the cob 20 feet long, 10 wide, and 9 deep, how many barrels of .shelled corn ? 20X10=200—200X9=1800 8 per cent. 14400 cut off 2 decima]s=^144 bbls. i For all grain, wheat, shelled corn, etc., which are sold as they stand, multiply together the length and bre;ulth and depth in feet for the number of cubic feet; multiply REDUCTION. this by 8 and cut off one decimal for the answer in bushels. EXAMPLE. A box of wheat is 12 feet long, 4 wide and 5 deep: how many bushels does it contain ? 12x4=48— 48 x5=:=-240 192,0— 192 bushels. RULE FOR PEAS IN THE SHELL. Multiply together the length and breadth and depth in feet for the number of cubic feet; divide this product by 20 for the number of bushels of shelled peas. EXAMPLE. In a room, of unahelled peas, 20 feet long, 15 wide, and averaging 6 feet deep, how many bushels of shelled peas ? 20X 15=300.— 300X 6=1800.— 1800— 20=90 bushels. To find the number of bushels in a hogshead, barrel, or other vessel of a circular base, and approximating a cylinder in form, measure the inside diameter one-third of the jfi^ay down from the top, and the depth, in inches RULE. Multiply the diameter in inches by itself, and the pro- duct by the deptli. Then multiply by 86 jl, and cut of 5 decimals for the answer in bushels. EXAMPLE. In a hogshead whose depth is 40 inches, and the diameter (one-third from the top) 30 inches, how many bushels ? 30 diameter in inches. 30 900 40 depth. 36000 36} 216000 108000 18000 1314000 (5 decimals)— 13 bushels, 14-000. REDUCTION. • ^^ To find the nutnber of bushels in a potato bank, piled in the form of a cone : RULK. Multiply the diameter at the base by itself, and the product by the height in feet. Then multiply by 21 and cut oif 2 figures for decimals for the answer in bushels. EXAMPLE. In a potato bank, the diameter being 6 feot at the base, and the height 5 feet, how many bushels ? 6x6=36—36x5=180 21 180 860 . ^ 3,780=37 8-10 bushels. If the potatoes do not come to a poiiit at the top, but round considerably; then divide the 180 by 4 for the answer—say 180— -4::i=45 bushels. TO MEASURE BY A MEASURlNCi ROD. Cut a rod exactly 51 1 inches long, and measure it oflF into 4 equal parts. Each part \\^11 be a line * bushel. A box just as long, wide and deep as this would contain exactly one bushel. Subdivide each line bushel into ten equal parts calling them tenths. When the dimensions are found with this rod, the product of length, breadth and depth is the answer in bushels. ♦ EXAMPLE. A crib is 10 line bushels long, 8 wide, and 5 4-10 (or 5-4) deep : how many bushels does it contain ? 10 line bushels long. 8 wide. 80 54 320 400 432,0 number of busbels— 432. * 12 906-1000 inches. 64 REDUPTION. If tlie crib is full of corn on the cob, divide by 2 to reduce it to shelled corn, and so in other cases. Example 1. — How much clear corn in a bin 5 feet high, 8 wide and 16 long, the corn being in full shuck? 2. How much clear corn in a bin 12 feet wide, 10 high and 20 long, the ears being in slip shuck ? o. What is the quantity of corn in a bin 61 feet wide, 8i feet high and 101 feet long, the ears being without shuck? 4. How many bushels of peas in a room 20 feet wide, 30 long and 42 high? 5. How many bushels of potatoes in a bank, the diameter at the base being 8 feet and the height 7 feet? 6. A erjb is 9 line bushels long, 7 wide and 3 3-10 (or 3, 3) deep: how many bushels? 105, To change numbers from a lower to a higher a ' iowtn^ to""ttname, without change of value, we employ division. This Higher uama. jg jj giinpje reverse of the process, used in reducing from a higher to a lower denomination. Example 1. Keducc 10354 farthings to £.* OPERATION. I EXPLANATION. <.f 4)10354 I We first divide by 4, since there can be 12)2588 2qr. ' only one- fourth as • • many pence as far- 20)215 8d.. things. By this divi- sion are found 2588d, Ans. £10 15s. 8d. 2qr. and a remainder of 2qr. To reduce these pence to shillings, we divide by 12, since there can be only' one-twelfth as many shillings as pence, and we get 215s. and also have a remainder of 8d. To reduce these shillings to pounds, we divide by 20, and find for result, £10 and a remainder of 15s. Placing by the side of this figure, £10, the several remainders in their proper order, we find that 10354 farthings are, when reduced, £10 15s. 8d. 2qr. 106. In a similar manner, like examples are to be per- formed according to the following directions: Divide the sum directed to be reduced by the number tiom a lower to of its denomination that makes it higher; divide that » higin-r nai!ie. ^^g^jj, ^^, ^^^ number of its denomination, and so on. The final quotient and the several remainders are what were to be found. otlanatiou -fk. How to i-fdueo * Ex. 1.,. Art. 88, shows the reverse process. pren REDUCTION. b-' 107. It will be seen by the coiiiparison of Ex. 1, Art. The two kin.i* 88, with Ex. 1, Art. 99, that the two kinds of reduction X'*"'"'^ prove each other. Example 1. — Reduce 1365 inches to rods. OPERATION. 12)1865 Note. — In this ex- ample it will be seen 3)113 ft. 9 in. that the division by 51 is easily performed 5])37 yds. 2 ft. I by doubling that — I number, and the 11)74 I dividend, 37, so that — I they stand 11)74. Ans. 6 rods, 4 yd. 2 ft. ill. I Such change never Diyi,iing ^.y r,v<, affects the value of a term, while it relieves from the o^;^^''.'^^''/ •'-''■ embarrassment of fractional division. A similar method is to be observed in dividing by any number connected with \, h, etc., only changing the divisor and dividend into fourths, thirds, etc., as the case requires. 2. Reduce 455 pints, Dry measure, to higher denomi- nations. 3. Ucduce 29795 cubic inches to feet and yards. 4. lleduce 177564 farthings to £. 5. lleduce 65432 shillings to £. 6. lleduce 59678 pence to shillings. 7. lleduce 965480 pence to £. 8. Jlcduce 3764354 pounds to tons. 9. Reduce 545509 grains, Apothecaries' weight, to pounds. 10. In 2500 nails, how many yards? 11. Reduce 5665 rods to miles. 12. Reduce 3567 links to miles. 13. In 9657840 cu. in., how many cords? 14. Reduce 985067 sq. r. to sq. miles. 15. What number of circles in '1296000 seconds? 16. How many gallons in 835 gills? 17. In 896574 seconds, how many calendar months? 18. In 765325 ounces, how many tons? 19. In 57850 links, how many miles? 6 tiG REDUCTION. MISCELLANEOUS examples; 08. Example 1. — Reduce £,75 15s. dd., to pence. . 2. Reduce £5 6s. 7d. 3qr., to farthings. 3. Reduce 6078095 farthings to £. 4. Reduce 695432 pencp to £. .'■. Reduce 6bush. 2p. 6qt. Ipt., to pints. , ^ «■. Reduce 50t. 15cwt. 2qr. 161b. Tioz. 8dr., to drachiv.^. 7. Reduce 3c. yd. 20c. ft. 1435c. in., to cubic ijiche?^. ;>. Reduce 6mi. 5fur. 6ch. 2rd. 161i,, to links. 9. Rcdiiec 141b. 95- 55- 29- 16gr., to grains. 10. Reduce 651b. 9o^. 15pwt. ISgr., to gTaii;8. 11. Reduce 5yd. 2qr. 3na., to nails. 12. Reduce 5^89 65 links to miles. 13. Reduce 694205c. in. to cubic yards, 14. Reduce 787650na. to yards. 15. Reduce 3yd. 2ft. 6in. 2b. c, to barley corn.s. 16. Reduce 4m. 4fur. 25rd., to rods. 17. Reduce 26sq. m. 30a. 2rd. 33sq. rd., to square rods. 18. Reduce 9gal. 3qt. 2pt. 3gi., to gill^. 19. Reduce 6378 gills to gallons. 20. Reduce 12circ. e.s. 15° 45' 10", to seconds. 21. Reduce 5mo. 3wk. 16h. 27sec., to seconds. 22. Reduce 3m. 6ch. ^rd., to links. 23. ReducQ 3t. 181b., to ounces. 24. Li 75a. Gx- 5sq. rd., how piany square inches ? 25. In 30mo. 3w. 6da. i5h., how many minutes? AMERICAN MONEY. 67 * AMERICAN MONEY. SECTION III. W}> Arr:ran3fanr^/h tlv.^ currencv of tlio Soutlieru g^Sg^*;!,"'" iifcclcracy. ■ , Statesi ■ no. It ^:-^ '■-•'■ - ->.o-^^. ■V.IImv-^. K^.ii/isions. lies, cents and mills ni. The .rroM ■"■ .u".....i.v _;lc and dollar. 112i The silver cniii is me (.ioiiar, iiaii-*;(.)i;aV;( f|i:;v. iiav. dime, half-dime, aud threo-cerit piece. il3. The nickel is ' > numeicially 'ciilled tiic,'-'"*!' • '^' ■ , ,. . J; ■ ly ii copper , 'pev. ' - . ^ ■ ^ ■ co:r. AME*IICAN MONET TABLE. I.14, 10 mills luake 1 cent, marked cf. 10 cents make 1 dime, marked d: 10 dimes make 1 dollar, marked >'■< 10 dollars make 1 eagle, marked U. K'j^ih. Jh. 1 = *cents and" cipher; dollars become cents by the annexation of two ■Bills. ciphers, and mills by three; and eagles become dollars by changel'to doi- ^^^ annexation of one cipher ; thus, 60 cenis are 600 mills ; •*r8. 65 dollars are 6500 cents, 65,000 mills ; 75 eagles are 750 Examples of , „ ' ' ) a ohaiige. dollars. Example 1. — Write 45 dollars, 46 cents, 6 mills, in mills. OPERATION. 45 dollars=4500 cents. Add to these 46 cents. 4546 cents. Annex one cipher to change to mills, 45460 mills. Add 6 mills. Anst. 45,466 mills. 2. In 35 dollars, 63 cents, 5 mills, how many mills ? ^. In 865, how many cents ? 4 In $550, how many cents ? how nsany mills ? 5. In S5.60, how many mills ? 6. In 82.50, how many cents ? 7. In $100, how many cents ? 8. In 5 E. and §6.40, how many cents ? Mills changed 120i To change mills to cents, the right hand figure joints; todoi- ^^^^ ^^ ^^^ ^g.. ^^^ ^^jjj^ ^^ (jollars, the three right hand figures. Cents changed 121* To change cents to dollars, the two right hand to dollars. figures must be cut off. Remark. — When dollars are multiplied by dollars, the answer is in dollars ; when by cents, the answer is in cents ; and when cents are multiplied by cents, the answer is in mills. AMERICAN MONEY. 69 9. Change 36445 mills to dollars, cents and mills. Ans. 36,45,0, or e?6.45c. 5m. 10. Change 6954320 mills to dollars, cents and mills. 11. Change 78654461 mills to dollars and cents. 12. Change 5567905 mills to dollars and cents. 13. Change 3890076 mills to dollars, cents and mills. 14. Change 3650123 mills to dollars and cents. 15- Change 984060 mills to dollars, cents and mills. 122« The mill is simply an imaginary coin, and in com- The mill an im- mcrcial transactions hardly known; thus, in the sale of *g'"'"'>' <^<''°- articles, amounting severally to 62 J 431 1.3U Which added, are $2,371 The trader does not express the J, I and J in mills, but adds them as f'ractions, and writes the same as above, not as the amount equally is, $2. 37c. 5m. Remark. — The pupil must be particular in putting thepojnt^nof U)'b« separation point between the dollars and cents; as also, J\<'gi«'^*«d;»'»0' whert adding, to place cents under cents, and dollars under ment of eimiiar dollars. _ ^■*"^«*- 123* When one figure only expresses the cent, a cipher How to write m is to be placed at its left; thus to write four dollars and ^^^^^ ^^*^'* ' six cents, we d^ not write $4.6, but $4.06. 124i Iq t.he following examples, in the Addition o/A«idition of American Monet/, the answers can, be found by the table, ^6^^'.*^*° ^^^' or in the commercial form of adding fractions. Example 1. — John bought 6 pair socks for $1.25, a vest for $2.25, a coat for $9 37^, 6 handkerchiefs for $1.62 J, and a cravat for 75e : what was the cost ? OPERATION. Socks, $1.25 = $1.25 or $1.25 Vest, . • 2.25 = 2.25 2.25 Coat, 9.371 = 9.375 9.371 Handkerchiefs, 1.621 = 1.625 1.621 Cravat, .75 = .75 .75 m Ans. $15,250 $15.25 2. Purchased 1 box of candles, for $7.50; 1 box raisins, $3,371; 1 keg of buckwheat, $2,621; 1 barrel of flour, $9,871 : what was the amount of the purchase ? 3. Bought 1 bag of coffee for $15.621 ; 5 sacks of salt, $4,371; 1 barrel of molasses, $14.45, and 1 box of starch, $3.371 : what was amount of bill ? 70 AMERICAN MONEY. 4. If you owe to A $437.50^ .to B 665 ;^to G.|5.3YJ j t. i) 02 J : what is the whole amount? , 5. If your father's State tax is 819.37^; liis town tax S25.12i ; his poor tax |G.62i, and his hridge tax $5.87 -i : what is the' sum of the whole ? ^ G. Add $125, $t)5.37|, $60.62.1, and ei235.87Hogether; Subtraction of 125, The fSuhtractton of American money is suhstar;- Aineriftau cur- j.- ^^ j.-i i i- • ^ i reroy. tially that or .simple numbers. Example 1. — If my income is $2500 a year, and ni\ pv,,f>n(i;inves are 62437.50: what is the surplus ? : ;:hatioiJ. explanation. 2500.00 I Having for convenience put the two 2487.50 I ciphers in the place of cents in tlu -- — : I minuend, we say from 0, nothing, $62.50 i which set as a cipher in the units- phuc of cents, we then say 5 from \eu (Art.' 40, remark) 5 ; placin<^ this in the tens-place of cents, we carry the 1 ]jorrowed to 7, and then proceed with the subtraction as in the ruje for simple numbers. 2. A man buys a horse for 387.50, what change is h% to • receive from a hundred dollar bill handed the seller ? 3. If yon pay for a cari-iage $450, and for a pair of lioi'.ses ?>337.7''^ ^'""v v-"'--i> m,.,-,. ,1,,,.c: flw. '-arriage cost than the htn-ses? , 4. AVhat is lIh.- u!ii'.n.-in.c ueiwrL-u ?775.37l', and !^5e2.r2i? • 5. Wh;;t would 8595 deducted from $1000.50 leave ? G. How much more is 62000.60 than 8999.99 ? 7. Deduct $735.39 from $862.21, and state the remain- der ? 8. What is the differcnoe between $59.69, and $96.95 ? The miiitipiiciv 126. The Multiplication of American 'money is similar !'«" cfr-"lf V. i'l process to that of -simple numbers. Example!. — What will 28 pieces of cotton bagging cost at $15.50 a piece? operation.' 15.50 I Kemakk. — When the multiplicand, 1^ t?K^ nmuipif- 28 i as in this example, has cents, the two >i. I right hand figures in the result must 12400 i be separated by the point for cents. 3100 ; Ans. $434.00 AMERICAN MONEY. 2. What is the cost of 40 barrels of liouv at $C)i a barrel ? ' -PEKATiON. I Remark. — Here 4CX J=forty halve.s=20 40 i wholes; or it could be said §6 ■ =86.50 G^ ! 40 240 I . $260.00 20 I S260 I 2 What is the cost of a firkin of butter, containing^ libs., at 25 i per* lb? .']. What will 350 bushels rough rice cost at .87 i per bushel '( Eemark. — If this operation is performed by writing When tJio mui- .875, nameil, when so written, decimals, for the multiplier, *^?-,!'^or*'dep;-"^ which is 87 cents 5 mills, three figures on the rigV Uaua J^^''-;- <*f5"«! f" side, in the result, are to be marked off, the first on the ' • ' (Extreme right, by a comma, fur mills ; the next itwo by a 'int for cents. Our preference is for the other form, ais i 'ing in common use, and practically best. ' . , 4. I purchased a flock of sheep, numbering 225,at'^2i t acli : what did the whole cost 'I 5. What will 66 bushels of oats cost at .33 J per bushel ? 6. What must I pay for 52 barrels of potatoes, at §3] ]ier barrel'? 7. What will oSO'acres of land, cost, at f;151 per acre? 8. How much has to be paid for 20 railroad share?, valued at $95,875 each? 127. To find the cost of articles soldV the 100 or 1000, T» smi tho eosi, ■il'tev multiplying th'e quantity by the pricey we cut off two pp^^ioo'^ ^^^ -fi^oi. figures on the right hand of the product, if the price be by the XOO; and three, if by the 1000; the remaining figures r%jresent the answer, in the same denomination, as the price. , ' ' ■ ' 9. What will 5750 bricks cost at $10 .per thousand ? 5750 10 Am. $57,500 or $57.50 c. mills. * Per, the Latin particle, signifying for. 71: AMERICAN MONEY. 10. Bought a raft of boards, containing 3345 feet, at $12 per thousand ; what did the same anaount to ? 11. What is the value of 3475 feet of timber at $2 per hundred ? 12. What must be paid for 450 feet of boards at $8 per thousand ? Articles sold by '^^ ^^^ ^^^ worth of articles sold by the ton : having ♦he toi^. ' multiplied by the given numbers, we strike off three figures from the right of the product, and divide the ^^^'^-.'o!!"''""" '^^ remainder by 2 for the answer. This answer will be in fho 13tll SUnii , , •' . . 1 • n mi the same denomination as the price oi a ton. ihe reason of this division is, because the ton consists of 20001bs., and the example proposes a number less than that. 13. What cost 1637 weight of blades, at $10.50 the ton ? OPERATION. 1637 1050 81850 16370 2)1718,850 A71S. ^8.59 14. What will be the cost of 26761bs. of plaster, at $2.65 per to»? 15. What will 9501bs. of hay cost, at $12.50 per ton? 16. What will be the freight of 56781bs. of iron, at $9 per ton ? 17. What will be ihe cost, by railroad, from Charleston to Memphis, on an invoice. of merchandize, weighing 8560 tons, at S7 per ton ? >)ivi5ion of 128. The Division of American money is to be pcr- Amcricau cur- fo,.„,g^^ ^^ j^ ^^jj^plg Q^nibers. _ i Dow to divide 129. When the sum to be divided consists of'dollars, tioUais. annex two ciphers at the right, in the place of cents, plac- ing, always, the separation point between the dollars and cents. The answer will be in dollars and cents. v,hat is .ione 130. Should there be a remainder, it is to be expressed, ivhen there is a fractionally, as in the following example; or if it be desirable to pursue the i.nquiry further, by annexing a cipher to the dividend, the next division will give mills, and so on. riie foi-mer, or REMARK. — The first way IS the preferable one, for the •iL^'be'p^cfeTrld. reason already given, that in business transactions, we do not write beyond dollars and cents. AMERICAN MONEY. 7o Example 1, Divide $9.67 by 5. OPERATION. 5)9.67 1.93 and 2 over, which fractionally written is f, making the answer $l.93|^. l>ut if performed so as' to have millH in the result, it would be d(inc thus: 5)9.67,0 1.93,4 which is to be read §1.93 cents, 4 mills. Note. — This subject will be treated of in decimal fractions. 2. Divide $535 by 17. OPERATION. 17)535.00(31.47 iS 51 EXPLANATION. It will be noticed in this example, that as no cents w-crc given in the sum, two ciphers have been put in 25 1 the cents place; while in the (juo- 17 tient, two figures at the right hand — have boon marked off, showing the 80 . answer to be $31.47 and the frac- 68 tional expression, fS^. Had three — I ciphers been annexed to the divi- 120 dend instead of. two, the figure on 119 the extreme right would have been mills. 1 I 3. Divide $25.44 by 16. 4. Divide ^536 by 145. 5. Divide $1000 into 250 equal parts. 6. Divide $6532.50 into 105 equal parts. 7. If 1575 be divided equally among 8 persons, what will be the share of each? 8. Bought 56yds. of straw matting, for $20 : what was that per yard, 't 9. Hired a carpenter for a month of 26 working days, at $22 : what was the expense of his services a tlay ? 10. Sold 20 bags of Sea Island cotton for $1975 : what was the worth of a single- bag ? 11. At $6 the barrel, how much flour can be had for $2.58 ? 12. At 75 cents per pound, how much tea can be bought for $9 ? 13. How many barrels of apples can be bought for $45.50, at $3,-50 per barrel? AMERICAN MONEY. 14. Hew long, witli the wages of $1.12 J a day, •■will it take a laborer to earn $40.50 ? A'ote. — In this exr.niplc (Art. 107, Note Kx. 1), beeaus*' C'f tlie fractional >}, double botli divisor and dividend, and theu proceed. 15. For C47.50, bow many yards of broud cloth can be had iit $2.37^- per prd? . 16. At S7.''^0 a ton. how many tons of coal can be bought for >5255 ? * 17. If a bag of coffee, containing IGUlbs., cost $24 : wiiat will be the price of a j,on,iil ? MISCELLANEOUS F.XAMPT.ER. Ici« i^x.vr.iPLE 1. — Which o...;.. ..,-. j,;.,.^,, _..j Lusboli of oiats at 75 ceutte per bushel, or 124 yards of calico, at 7 cents per yard ? What is the difference ? 2. What is the difference in the cost between 5 toils, of coal at ?;i7.60 per ton, and 12yds. of cloth, at $3.50 per yard. 3. How many acres of larid at o3 each, may be bought with the value of 46yds. of cassimdre at $3 per yard ? ' 4. How many acres of land ;Tt $3.50 per acre, may be purchased with the value' of 120hhds. of molasses at 30 cents per gallon. 5. Wlnit is the cost of 21b. Ooz. 5pwt. of silver at' 27 cents per pwt. ? JVot/i. — Ixeduce to pennyweights and then multiply by 27. This gives the answer- in cents. 6. What is the value of 8 tons, 9cwt 2qr. ISlbs. of sugar, at 13 cents per lb. ? 7. What is the price of 2 bushels and 3 pecks of rice at 25 cents a peck ? 8. What will 51b. 7oz. salts come to at 9 cents per lb? AMERICAN MONEY. 9. A mercliant sold a remannt of ciotli for $25.50 ; thora W'^re Gyd-^. : what was that a \ip.i\ '.' J,\//, , — iU'iiuct' Gyds to nails, ace _..".' cents. 10. VVliat will 150 acres, 3rd. 18 perches amount to, at ;1.40 per perch ? 11. Bought 3 tous, 9cwt. of irou for $450 : what was ihat per CAvt. ? • , 12. Tf,l ton of hay cost $12^, what -will 5 cost? 13. If 1 lamb cost §2.25, what, at the .same rate, will 15 cost? 1-4. If 85 bushels of corn cost §63.75, what is that per bushel? 15. If a butcher purchase 17 beeves for §18Q, 25 sheep U)V §(37.50 : what is the value of all '! V'h.:\'i of the beeves ; nd sheep each per head? 10. A farmer buys 3(3 sheep, and p:iys for them with •'> cows, valued at §15 each, ami a wagon worth §79 : what lio the sheep cost each ? » 17. An estate valued at §21,000, is to be divided amonu 1 children, when the widow has received liet portion ul .>ue-third. What are the shares ? 18. Bought a plantation for §4500 ; and paid for it with 10 shares in the South Carolina Railroad, §100 a liare ; 20 in the Savannah Railroad, at §125 a share: what amount will be called fur to meet the '■• '■■'••■ '^ 76 COMPOUND ADDITION, HIGHER ARITHMETIC. PART SECOND. COMPOUND NUMBEES. A fompound 132» A compound number consists of two or more number. denominations of like character, as expressed in the tables of currency, weights and measures. What are com- 133« Pounds, shillings and pence are of this class, tcTi^.'^ "*"'" ^^ before stated ; also, dollars, cents, dimes, bushels, quarts, pecks, etc. ; but these or any other denomina- tional values cannot be united to form a common wnatarenot ^'"^ • thus, it is impossible to say, £5 and $5 — , although both money representatives — are £10 or §10, and so of other like numerical classes. 134. Ill adding comijound numbers, place always addition is per- the ditterent values or measures in columns oi a formed. Himilar class, and in the order of the tables, com- mencing at the right hand with the lowest value named. When set in order, as in English money, pounds under pounds, shillings under shillings, pence under pence, farthings under farthings, add the right hand column, and divide it by the number of this denomination, to make one of the next higher : the quotient is to be carried to the next column, but the remainder placed beneath the added column. COMPOUND ADDITION. . 135. The proof is the same as in simple addition (Art. 39). COMPOUND ADDITION. ENGLISH MONF.Y. OPERATION. £ s. d. qr. 20 12 4 Example 1. — 15 10 6 3 14 15 10 2 10 5 9 3 25 19 8 3 85 4 7 3 Ans. 101 IC, 7 2 2 3 3 EXPLANATION. Explanation of The amount of the sum. first column is 14qr. =3d. 2qr. The 2qr. are placed in the column of farthings, and the 3d. (the small figure below) carried to the ponce column, malcing 43d.=3s. 7d. The 7d. is now placed in the pence column, and the 3s. added to the shillings column, making 56s.=£2 IGs. The IGs. is placed in the shillings column, and the c£2 added to the £, making the answer as above. JSfote. — It is convenient to put the division numbers above the columns, and the carrying ones below in small figures, as in above example. £ 5. d. qr. 20 12 4 30 15 9 2 16 14 8 3 17 12 6 1 19 5 4 3 3 3 10 2 Ans. £87 12s. 3d. 3qr, 2 3 2 £ s. d. qr. 4. 250 17 9 2 500 16 8 3 1250 9 3 25 7 5 2 35 6 4 1 £ s. d. qr. 6. 55 17 10 2 16 15 9 1 14 13 1^ 2 5 12 5 1 29 16 3 2 d. qr. 3. 12 5 4 2 5 19 11 3 16 15 4 *? w 18 6 9 3 7 12 8 1 £ s. d. qr 36 9 3 19 5 10 2 45 15 11 3 120 18 9 3 16 12 8 1 £ 5» d. qr 20 19 10 1 85 17 9 18 12 8 3 9 14 5 o 10 15 1 3 COMPOUND ADDITION. DRY MEASURE. hn. vh. qt. />t bu. pk. qt. pt O I 1 9. 1 6 15 2 1 15 .) ^55 o 9 1 bl) tr;* 1 7 25 3 1 AVOIRDUPOIS WEIGHT. tons.ctct. qr. lb. oz. dr. } 12. 14 14 2 15 18 5 i 15 3 24 14 3 1 12 2 22 12 15 j 9 2 15 10 10 3 14 9 8 to9is.ciDt.qr.ru. oz. dr. 13. 22 17 2 20 15 14 ' 15 15 3 10 13 5 12 2 8 9 12 4 9 1 15 7 8 16 10 2 22 5 7 ArOTHECAEIES WEIGHT. 14. lb. 5. 24 7 17 11 3G 6 15 9 9. 3 5- B- gr. 2 1 16 7 2 19 7 13 1 9 5 7 1 15. lb. 25 15 14 11 4 2 10 y. (/r, 16 15 19 11 16 TEOY WEIGHT. .56. lb. 45 9 11 8 10 17 3 14 , 3 9 15 10 8 20 17 3 15 16 17 lb. 35 40 12 6 2 5 4 oz. pjivt. qr. 10 12 14 9 15 13 3 14 20 7 8 10 3 14 15 9 17 IS 10 18 20 llhd. IS !:) 14 JS i9 i'O.Mroij.Mj ADDITION. * ALE AND BEER MEASURE. qal. qt. pt. "31 8 1 1 50 47 15 12 13 3 19. hhd. hd. gal. at. p1. 75 25 3 '^ 3 15 1 40 50 9 1A 1 2 i'J.Fl. or. JUL ]2-^ '■ :■ '&.E, qr. »< ■111 !.S) :; 4S ;•] LONG MEASURK. ■ ; • rur.po. ft. in 210 15 5-15 10 2 41. 14 3 .16 9 9 9 25 3 20 14 2 36 12 2 19 13 4 16 7 1 7 12 2 12 5 2 G 8 4 .7 ^ ' 'J- iili. I'"- i 1 . 12 9 5 9 25 31 2 5 GO 50 3 3 19 39 2 5 14 25 3 6 G 37 '> 8 LAND OR SQUARE MEASURE. sq. yd. sq. ft. sq. in. 97 . 4 104 22 3 27 105 • S 2 37 7 IL'7 sq.mi. a. r. rd, sq.iid. 2fi. 2 60 3 37 2'5 375 2 25 21 7 450 31 20 11 30 1 25 19 CIRCULAR AND ASTRONOMICAL MEASURE. ") 17 36 29128. 7 25 41 21 1 8 15 10 09 6 29 27 49! 29 8 18 29 IG 7 09 04 58 5 25 25 G 14 26 7 08 0^ 80 •COMPOUND SUBTRACTION. 30. SURVEYORS MEASURE. mi. fur. ch. rd •25 8 3 10 24 5 6 09 23 4 5 8 23 3 4 7 31. mi. fur. ch. 35 ' 5 2 25 6 4 23 7 5 20 8 6 rd. 9 SOLID OR CUBIC MEASURE. cu. yd. cu. ft. cu. in. 32. 65 25 1129 37 20 132 50 1 1064 22 10 17 33. c. eft. 87 9 26 16 19 c. 65 35 25 15 eft. TIME MEASURE. 34. mo. 9 <) o 7 6 4 3 2 1 5 , 2 1 2 3 yr. da. h. "min. sec. 35. 89 59 20 13 12 25 40 10 12 37 5 90 19 19 25 6 5 4 15 20 5 6 14 5 6 COMPOUND SUBTEACTIOK. poirrid^Eubtr.ic- ^S®' '^^ ^^^ *^® difference between compound num- Don is to be bers, jjlace the less number under the greater of sim- per o.me . 1^^^ denominations, and beginning at the right hand, subtract as in simple numbers. Should tbe figure ia the subtrahend, or lower line, exceed the one above, add to the number in the minuend as many as it takes of that denomination to make one of the next higher, and then take the subtrahend from the upper figure or minuend so increased. Set down the remainder, ,!{\nd carry one to the next denominator in the lower line, and bo on. COMPOUND SUBTRACTION. SI ENGLISH MONET. Example 1. — From 45 pounds, 10 shillings, 6 pence, o fai'things, take 25 pounds, 9 shillings, 5 pence, 2 farthings. £ 5. d. qr. 45 10 6 3 . 25 9 5 2 Ans. £20 Id. Iqr. 2. Pi^om 35 pounds, 13 shillings, 7 pence, 2 farthings, take 25 pounds, 17 shillings, 10 pence, 3 farthings. OPBRATION. £ s. d. qr. 2(J lli i 35 13 7 2 29 17 10 3 EXPLANATION. In this example, the upper less tJian the lower, p'" 4 farthings=l pen- ny, are borrowed and fi. „ " i^„- Explanation o^ gure being ^0*1^ in «xi.m-, .471.'?. £5 15s. 8d. 3qr added to the 2=6; and the 3 of the subtrahend being subtracted, the remainder, 3^ is set down in the far- things place, and Id. =4 farthings that was borrowed, is carried to the next figure, 104-1=11. As that also ('xceeds the number of the minuend, we borrow and add 12 pence to the 7=19, and say 11 from 19 is 8*; iind setting it in the pence place, carry the. l8.=12 pence that was borroAved to the next figure, 17+1=18. As tlvvt exceeds the ujiper figure, we borrow and add 20 shillfngs to that, and say 18 from 33=15. That being set in the shillings place, and the borrowed. 1 = 2U shillings carried to the pounds, we have, after the next subtraction, the answer. In a similar way perform all examples, taking care to operate with the numbers expressed in the tables that apply to the proposed sum. £ 5. d qr. £ s.. d. qr From 50 15 4 3 4. From 37 18 9 U Take 37 14 5 2 Take 18 19 5 »; £ s. d. qr. £ ■S. J. qr From 25 19 9 2 G. From 15 6 6 •J Take 5 16 8 3 Take 14 7 6 82. COMPOUND SUBTRACTION. DRY MEASURE, hu. 7. From 38 pk. 6 qt. 3 bu. 8. From 12 f t Take 25 4 1 Take 9 7 Mi' LIQUID MKASCRK. hhd. 9. From 13 'f qt. pt. •0 10. From' 165 gal 15 2 1 Take 2 39 2 1 Take 59 36 3 1 AVOIRDUPOIS WEIGHT. ion. cwt. qr. lb. oz. dr 11. From 45 17 3 24 15 9 Take 7 18 2 14 13 7 ton. cwt. qr. lb. o»» 12. From 200 5 2 20 13 Take 33 12 1 23 15 APOTHECARIES WEIGHT. lb. o 3 B gr,\ lb- 3 o 9 gr. 13. From 3 15 14114. From 4 4 Take 5 2 7 171 Take 7 5 2 14 TIME MEASURE yr. da. h. mm. sec 15. From 95 89 16 15 15 Take 75 57 23 30 17. yr. da. h. min. ace. 16. From 59 Take 13 6 15.20 45 Note. — 'In subtracting dates, 30 days is counted a month. •Fo &fld the dif- 17. What is the difference in time betw'oen March Ja^ "* 15^ 1863, and January 13, 1869 ? yr. mo d. From 1869 1 13 Take 1863 3 15 Rem. 9 2^ Note. — In this sum, Januarj' is called the let month, and put down 1 ; March the 3d, and put down 3. 18. Calculate the time from July 6, 1857, to Decem- ber 9, 1860. . COMPOUND MULTIPLICATION. *3 19. Calculate the time from May 5, 1859, to August 4, 1869. 20. Calculate the time from March 15, 1862, to May 19, 1865. 21. IIow long is it from September 9, 1861, to November 4, 1870 ? COMPOUND MULTIPLICATION. 137. In Compound Multiplication, the multiplier iS) The moH.piie? without exception, an abstract number, that is, a^mbcJ"'* number separate from any particular object; as 4, 9, 15, 25, etc.; hence the product is simply the multipli- cand repeated as many times as the multiplier ex- presses. 138. To perform compound multiplication : having to pcrfonn placed the multiplier under the lowest denomination oompoundmnt- of the sum, multiply this and divide the product by the number it takes of that denomination to make one of the higher; having set down the remaindei', carry the quotient to the product of the next de- nomination, and so Ofi. ENGLISH MONKY. Example 1 OPEKATIOHr. £ S. 20 —Multiply 5 . 7 By explanation. d. qr. Here we E»pia»fciion ot 12 4 say, first, 2*""*' 9 2 |x6=12=3d. ^ jAs there is : — no remainder Product, £32 68 9d. Oqr.jwe write in - 4 8 'the farthings place and carry the ']d. to the next number, after its multiplication; thus 9x0=54 and4-3=57; this divided byl2 gives 4H.and 9d. over. We then put 9d. in the pence place, and carry the 4s. to the shillings after its multiplication ; thus, 7 X 6= VI and+4= 46 ; this divided by the 20 gives £2 Os over. The (Js then is put in the shillings place, and after the £5 has been multi- plied by G=«30 the 2 is added, which finishes the sum. 84 COMPOUND MULTIPLICATION. The proof. Multiplication is proved by division: thus. 6)£32 68. 9d. Oqr., i# £b Ts. 9d. 2qr , the operation will be seen in Gom- pound division (Art. 140). LONG MEASURE. rd. yd. ft. in. b.c. 40 5J 3 12 3 2. Multiplicand, 15 4 2 6 1 Multiplier, 4 Product, 1 fur. 23 2 1 1 1 16 1 23 2 2 7 1 In this example, when we reach the denomination of yards, we say 4x4=16 and 3 added 19; by the multiplication of this by 2=38, and by the multipli- cation of the numbers to divide it, 5 J by 2=11, 11 into 38=3 and 5 over; 5-^2, to bring it back to the right denomination, gives 2iyd. : one-half yard=lft. 6in. must be added to the feet and inches. LIQUID gal. qt. pt. gl 3. Multiply 5 3 12 By 6 MEASURE. gal. qt 4. Multiply 8 3 By pt. gi. 1 3 7 1 AVOIRDUPOIS WEIGHT tons. cwt. qr. lbs. oz. 5. Multiply 14 17 3 20 10 By dr. 15 4 TROY V lbs. OZ. pwt. gr. 6. Multiply 50 7 14 19 By 9 HEIGHT. lbs. OZ. 7. Multiply 25 7 By pwt. gr 11 8 DRY ME bu. pk. qt. pt. 8. Multiply 30 3 6 1 By 10 ASURS. bu, pk. qt. 9. Multiply 25 2 5 By 3 COMPOUND DIVISION. 85 ALR AND BEER MEASURE. hhd. gal. qt.pt.\ . Mu Itiply 75 37 8 1 By 25 Tihd. gal. qt. pt. 11. Multiply 37 25 2 1 By 15 CIRCULAR AND ASTRONOMICAL MEASURE. 12. Multiply By 7 28 26 48 20 13. Multiply By 6 17 25 14 6 TIME MEASURE. yr. da. h. min. sec. 14. Multiply 20 65 20 50 30 By 12 COMPOUND DIVISION. 139. Compound Division is the process by 'whicli weco,„po„m) ^jj. find how often a given number is contained in a dividend vfeion defined, of compound values of similar nature. 140. To perform the work, divide the highest denomi- nation by the given number and set down the quotient; if How it Ik per- there is a remainder, reduce it to the next lower denomi- *"^™* ' nation ; to the result add the given number of that de- nomination and divide as before; and so on until the whole, has been divided. Example 1.— Divide £32 68. 9d. Oqr., by 6 (Art. 138). EXPLANATION. • £32-^6 gives a quotient of ^ , , £o, and a remainder ot JtJ. sum iu com- These £2 reduced to shillings p«"»'» "i» two or more numbers: arrange these in a horizontal line, ^jyitij^g, and divide by anj prime number that will go into two or more of them exactly, and place in a lower line the quo- tients and undivided numbers. Proceed in the same man- ner until there is no prime number greater than 1 that will divide without remainder any two of the numbers. The figures beneath the lower line and the divisors multi- plied together will give the answer. Example 1. — Find the least common multiple of 3, 4 and 8. Ans. 24. OPERATION. 2)3 4 8 2)3. 3 1 2 3x1x2x2x2=24. 2. What is the least common multiple of 6, 8, 12, 18 and 24 ? 3. What is the least common multiple of 33, 44 and' 55? 4. What is the least common multiple of 9, 11, 17, 19 and 21 ? A71S. 223839. 5. What is the least common multiple of 4, 14, 28 and 98? 6. What is the least common multiple of 2, 7, 5, G and 8? 7. What is the least common multiple of 4, 12, 20 and 24? 8. What is the least common multiple of 2, 7, 14 and 49? 92 MISCELLANEOUS EXAMPLES. MISCELLANEOUS EXAMPLES. » 168. Example 1.— From £2 10s. take 17s. 9d. 3qr. • 2. From an ingot of silver, weighing 51b. 3oz. 15dwt., six silver spoons, each in weight,. 2oz. lOdwt. 12gr., were made : what remained ? 3. From some merchandize, weighing, 18t. lOcwt. 3qr., were sold, 15t. r4cwt. 2qr. 141b. : what was left? 4. A merchant had' 455t. of sugar, but sold 225t. 16cwt. 3qr. 201b. : what was over ? 5. A merchant sold from a piece of broad cloth, con- taining 30yd. 3qr., 16yd. 2qr. 3na. : whal quantity had he left? 6. An importer of wine sold from an invoice of 38 tun, Ihhd. ISgal. 2qt. Ipt.,'l7 tun, 3hhd. 42gal. : how much had he left ? 7. From 50hhd. 34gal. 2qt. of ale, were sold 29hhd. 36"gal. 3qt. : what remained ? 8. A .planter divided a plantation, containing 869 acres, 3rd., into 5 parts : what was in each ? 9. What is the 10th part of lyr. 3mo. 2w? 10. What is the 5th part of 20h. 49m. 50sec? 11. If a bale of English cloth, containing 356yd., cost £3425 2s. 4d. : what is the pi-ice per yd. ? 12. Bought 40 loads of wood, each measuring Ic. 3c. ft. 7cu. ft., at $4.50 per cord : what was the entire quantity and cost? 13. If a ship sail 2° 15' 29" a day: what will she have sailed in 25 days ? 14. If five laborers dig a ditch, 6rd. 5ft. deep, in 3 days, what will they have done in 15 days ? 15. What will 61b. 75 53 29 lOgr. come to, at 5 cents per grain ? 16. What is the cost of 2t. 12cwt. 3qr. 201b. of sugar, at 15 cents per lb.? VULGAR FRACTIONS. 93 FRACTIONAL ARITHMETIC. PART THIRD. VULGAR FRACTIONS. A fraction dfl- 167. A fraction is one of the equal parts of a unit, or a collection of units. w^y Tulgar •ir»o rni c • i ■ . , . tractions, and loo. ine traction, under present consideration, called how expressed; vulgar, from being in most common use, is expressed by two numbers, vertiqally placed, with a separating line ; thus, i- one-half; ii two-thirds ; | three-fourths. 169. The number that denotes into how many parts the ^r'^d^K"'"* unit is divided is called the denominator. Its place is ' always below the line. Thus, in the fraction i, 2 is the denominator, and it denotes that 1 has been separated into two parts.* • 170. The figure above the line, as it names or numerates (that is numhers) how many parts of a Unit, when divided Thn numerator are used, is called the numerator • thus, in the expression '^'^^^^'^' ^, one part of the two parts into which the unit was sepa- rated is denoted to be used. 17L Taken together, the numerator and denominator are called the terms of the frac'ion. Terms of a frao- Uon. Note. — For convenience of expression, we always say numerator and denominator. 172. A fraction is simply a peculiar form of writing a Fractions aro divisor and dividend, the denominator being the divisor, Kk^™^ and the numerator the dividend ; thus -} is the same as 0-^3=3 ; or J is the same as 1-4-2= i. Here the value of the fraction is seen to be the quotient of the numerator divided by the denominator. ^4 VULGAR TRACTIOKS. A iro er fiae- ^'^^' Fractions are variously known. A proper fraction tion" ^ has a less number for its • numerator than for its denomi* nator, as -f , f. , i. • An improper 174. An improper fraction has either equal numbers in fraction. j|^g terms, thus, f ; or, is greater in its denominator ; thus -^ •^^' A simple frac- Yib!' A simpU fraction \^ that which has whole numbers tion. in }3oth numerator and denominator ; thus, i, -f-. A componnd 176. A compound fraction is a fraction of a fraction, or ira.;tion. ^^^ fractions connected by the word .of; thus, ^ of -f. 177 A mixed number is a whole number and a fraction Admixed num-^^.^^^ .^ ^^^ ^^^.^ . ^^^^^^ 5^^ jgj^ 178. A Complex fraction is one whole numerator or fra^tiolu'^'' denominator, or both, and has a fraction or- mixed number in 51 ^ 4 9t one or each of its terras ; thus, -_ 4. — - — 6 oj o4 179. Kemark. — As the addition and subtraction of frac- tions involves some preliminary work, we commence with . examples in v ultiplica/ion. ■10 muKipiy a Example 1.— Multiply f by 3. traction by o OPEUATION. ^liole number. f X3=-f- Ajl^. It will be seen that the work is done by the multipliea- • tion of the numerator by the whole number. 2. Multiply i by 3. 3. Multiply i by 2. I 4. Multiply i^s by 4- 5. Multiply h by *. Ans. ■fj=6. 6. Multiply iS by 8. 7. Multiply 2^5 by 7. 8. Multiply i-o by 6. 9. Multiply s by 5. 10. Multiply f by 7. t 1 • Sometimes, when the numbers are larger, the work ib Biraplified by dividing the multiplier by the denominator, and multiplying by the numerator; but, on the whole, the former method is preferable. .11. Multiply i by 25. Ans. \\ By the division of the denominator -J- of 2o=^^y.?'=^ =8i. 12. Multiply t by Sn. 13. Multiply ^ by 53. 14. Multiply ft by (35. ^Vo 2 (» • -T-«*. Here, we multiply the numerator of each fraction by all the denominators, except its own, for the new numerators ; and all the denominators together, for a common denominator. 2. Reduce |, f- and -I- to a common denominator. o. Reduce 'i, f and f to a common denominator. 4. Reduce f , ^} and 4 to a common denominator. 5. Reduce -I, -^, -^f and f to a common denomi- nator. 6. Reduce j-, yV, -f and if to a common denomi- nator. 7. Reduce J, f , -f and ^ to a common denominator. 8. Reduce -fV, f, f and f to a common denomiaator. 9. Reduce 7^, | and 5 to a common denominator. JVote. — When there is necessity, as in this example, reduce to simple fractions. 10. Reduce 42, 5-f and 1 J to a common denominator. To reduce frac- 188. EXAMPLE 1. — Reduce i, t^, -j^ and i-}- to their ttons to the least oommon denominator. least common »iu!tiple. OPER.vnOX. iij-f -N -f-i ii I ■^fi^x3=27, 1st numerator. ■ffx5=30, 2d numerator. x|x7=28, 3d numerator. 4fX 11=33. 4th numerator. 2x2x2x3x 1x1=72, least cora- 1 ^5 1 uioii multiple of denominators. lIen<-^>, f, i\, -?aMd^i=|-z-,f8-, 4A;uidf|. Ans. Here, as the i'ractio.is are in the lowest terms, we find the least common multiple of the denominators for a <-ommon denominator. Then. dividini<; this by each denominutoj", we 'niidtiply each (luoiient by its owtt numerator. 2)4 6 9 12 2)2 3 9 6 3)1 't !» 3 \ ll(;ak tbactioxs. 90 1'. Reduco f, -i\, f and \ to n coimnon deuomi'i:)- lor -fr- of a >hi!HiiM- to il\ ToZrcditi'o ;i fraction of » hichor to on* of a terror denom- ination. :{. Reduce -i^j-, tV, -/g- and VV- 4. Reduce i, -,V, -A- and -/--. 5. Reduce f, -gV. H and -iV- 0. Reduce -,V, ii, -A- and -3^ y4«c- .14 7 X?,2J>- -'*■''*• 3 15 0' 31 5 U ; 7. Eoduce 3, 4, -f nnd-iV- S. Reduce -,V> i< i and ;. 0. Reduce ■?, -iV- -/o- and -jV- 10. Reduce MJ, i and -§-. 189. ExAMPLK 1.— Re.iuci fraction ol'a i'ai'lliing. orEKATION. AYe multiply the fraction to be reduced by S'tch numbers as are neceH>. licduce 4- of a circle to signs, degrees, etc. 7. Reduce -^ of a .ton to cubic feet and inches. 8. Reduce -^ of a hhd. to quarts, pints, etc. 9.#lcducc -^f- of a cord to'cvibic feet and inches. 10. Reduce -i\- of a eii"cumferenc longs and rods. ., 192. Ki':.\^JpL-E ].— Redu< . fraetionof a shilliiiij;. . Ol'EkA'il' .. 'V/.-f 2^r.=M>4g/'.' and Ts.' (tjQ which tht;y arc to be reduced") =48^?''. ;' hence; 4f is fouiid =^s.. Ans. Here, the .given quantity is reduced to the lowest denominaiioa possible, foi* a numerator; and a unit of the hi::-)icr denomination is reduced to a like denominat!"n with the ndtiierator, for a denominator. 2. Reduc-e 9.5 to the fraction of a pound. - 3. Reduce 2pk. Iqt. 1-iVpt- to the fraction of a bushel. 4. Reduce 2oz. 42dr. to the fraction of a pound. 5. Reduce 5 cord feet, 8eu. ft. and 1036-|-in: to the fraction of a cord. 6. Reduce 8ft. 576in. to t?he fraction of a ton. 7. Reduce 2fu]-. 25rd. to the fraction of a mile. 8. Reduce 15° 40' to the fraction of a sign. / 9. Reduce oOgal. oqt. Ipt. to the fraction of a hogshead. 10. Reduce 5ch. 8rd. 20H. to the fraction of a furlong. •e to miles, fur* :.^qr. to the VULGAR FRACrJONS. ' 101 ADDITION OF FRACTIONS. 193. Kx AMPLE 1.— Add -,V iU'd -iV to,^■ethol^ Tbo additicm »f ■f^. Ajis. OPEHATIOJf.: The denominators boinjr ofitho 8a#io order, the numerators are added and written over the common denominator. j!^ote. — When the fractions ai>e compound, complex, or mixed numbers, reduce to simple fraictioiis. 2. Add too;ether -,V, n^s-, -rV i^» nV and -jV- ^«-5.^ -H-=1tV- 5. Add together -^, if, if and ff. ' 6. Add together f and i. OPERATIONS , ,.; f=-|4; |=if-i-2 for the least cpmuionj^ileiiomi^fition, 2 4 ailU 2 4)- 2 4- *^"'>- , • 7. Add together |, -,V and -j^g-. 8. Add together «, f, -,V and -^V- ? 0. Add together 3;] and 51. Ans. 9|i. Note. — In adding mixed nun;jjbei*8, ifeis oftQp'more convenient to add the whole numb^rfjflby thoEpiaelves, and then the fractions. 10. Add together 6s, 7} and 4^. 11. Add together 41, 5s and 6|. 12. Add together 2^, J, f and i. f 13. Add together 1 of i, I of f of 1. 14. Add together ?i I and -^ off.- .4n*-. 2if H- ^*' 15. Add together !_ f of i and f . "*5, 16. Add together is. and id. , OPERATION. ■.2^d. Afis. 17. Add together fib. is ?o and B - 18. Add together Jgal. iqt.. and fpt. ilns. S-iftV-pt- lOL' , VV^iG-^U FRACTIONS SL'KTKACTION OF FRAOTIOXS. The subtrao- li}t. ExImple 1. — From :i take ',. tion of frsc- » tiona. OPERATION. :|— ,l=f. A71S. ^ ^ Here, the denominators being of tlie same order, a common dT- the ditfercnce of thfe numerators is found, and placed nom-nator. Over the cottimon denominator. 2. Fromi.4f take -aV. 3. From -ii take ii. N'ote. — When tlie denominators are not alike, reduce them to a common denom'inator, and then, subtract as above. 4. From-S take -V. i ' OPERATION. |=|- and 2=f=o- A)is. 5. .From f take -f . 6.. From, I- take -f. ' 7i From^'?^- take i. 8, From ^ take -H. 9. From i- off take i off. A71S. -^. 10. From 9i take If. Wheu Uiere .we iVb^6\— Whenever there are compound, complex or comppund or mixed numhers, brini? them to their simplest forms, complex trac- ■, y i tioBSror mixed and then subtract, nwnbars. 11. From 23-iV take 15-,V- 12. From 120 take 40f . N^ote. — Change the whole number to 5ths. 13. From 265 tak« 52f . VULGAR FRACTIONS. 103 Ml SCELL ANKOUSEX AM P LKS. t IftS. Example 1. — If lib. of beef in worth S.J eentR, what is the value of 9. libs. ? ■ ;« '1. Bought a firkin of butter eontiihiin j 40] U»., and was charged '111 per lb. : what did it ainotmt to ? 3. When barle}"^ is worth $1| ft biishol, what will 24>V bushels cost? ^ ' 4. If 25bbl. of fldnr coist $12'^]. what will O^bbl. cost ? 5. When corn is soiling for ',' dollar a bushel, what must be paid for 250 J bushels y 6. If you Avish to divide $130 among soiue laborers, so that 7 of ther.i should have each f as rhuch as each of the other 3, what would you give? ' 7. lf2Jyd. of cloth will pay for 33;}lb of caudloS, what quantity of cloth Avill pay for 7] times S3 Jib. of candles ? '* 8. If for -iV of a bushel of pO»tatoes you pay 42 cents, how many could be had for SfS-j*^? Am. lOibush. 0. How niau}- cubic feet are there in ~^x~ of a. cord of wood, and what is it worth at 2">2 ))cr cord? Am. 104Aft.; U\. 10. 32 is -§■ of how many times \ of 12 ? Ans. 9 times. U. Sold 73^bush. corn for §04|-.J : what Avill be the amount, at the same rate of G4bufih. ? Ans. $50. 12. 28 is i{ of how many times 8 ? 13. How many cubic feet in a box that is Onft. long, r>f ft. wide, and 3-J^ft. deep ? OPKRATION. t) a X 5 4 X 3i= V X V X -V»= ^^3*6^= ^i^= 11 7ii. .l«s. 14. In a box that is 8ift. long, 4fft. wide, and 3*ft. dc(^p, how -many cubic feet are there ? 15. Of the inhabitants of a towii in Alabama, I arc planters, f merchants, | students and professional men, -J- mechanics, and 142 others variously engaged : what iR the number? Ana^. 5040. 104 VULGAR FRACTIONS. 16. If the cargo of a ship be worth $72,000, and if f of J of ■?■ of the cargo ^ be worth f of i of -f^ of the ship : what is the value of the ship ? 17. In a school in Georgia, 2 the schoLars study arithmetic, ;} algebra, -^ geometrj-, and the others, in number 10, study engineering: how many scholars are there ? Ans. 200. 18. The factors of,;a certain number are 32i, 15-f and 19f : wh^'^t is f of f of |- of the number ? >^ Ans. 82231-^. 19. How much cloth that is -f of a yd wide, will it take tp line a cloak, cpntaining 84 yd. whjch is -H of a yd. wide ? Ans. 12 f | yd. 20. What is f of a barrel of flour worth at $6f per barrel ? 21. A man,' can- build 33 J rods of wall in 24 J days, by laboring 123 hours per day : in how many days of 9f-- hours, will he bui]d 1^ times as many I'ods? 22. A gardwi whose breadth is 10 rods, and whose length is If times its breadth, has a wall 3^ feet thick around it : what was the cost of digging a trench 2f feet deep;in which to lay this wall, at f cent per cubic foot ? Ans. $62.941. , 23.YThe distance f^om the earth to the sun is about 95,000,000 miles : in.what time would a railway car run that distance, at the speed of 372 miles an hour, allowing 3651 days in a year ? Ans. 288yr. 363d. 13h. 20min. 24. What is the value of 1^3 of a day ? Ans. 16h. 36min. 55-iV- DECIMAL FRACTIONS. 'lOA DECIMAL FEACTIONS. 190t A Decimal Fraction is one that has its denorai- ^'.^^ ^^*"°;r,'''' nator written in tQnths, or similar classes of num- fined."" ' bers. 197. This denominator is not usually expressed, How the de- bufit'is known from the annexation to its figure 1 o/knwn^*'"^ '" as many ciphers as the decimal demands. 198. A decimal fraction is distinguished from a How a dooimai whole number by a dot, known as the decimal i>o<'^^, [[n°ui>ho.'i. ''"''' placed at the left of the decimal ; the first figure at the right of the point being tenths, the second hun- dredths, the third" thousandths, etc. ; thus .b=-^tr', NUMERATION TABLE OF DECIMALS. 199. In the decimal table, w^hich is read from left How the depi- to right, it will be seen— as tenth denotes— that the ;-;\^|^'^u|^" «:■ 2 " K T3 q S'^^S'^ — CO o 'V( o) =*-( S o • © tj o o "tf r::: o -1.^ c= i- c ^ o S 1^ » 5;-.5^j-Cr;rtfl Billioi Hund Tens Hund Tens ( Thouf Hund Tons. 9 8 7,6 5 4,8 2 1 0234 5 6789 AVhole numbers. Decimals. 205. When a mixed numb^ has the decimal pointy it must be as a fraction of a unit of the order repre- sented by the preceding decimal figure ; thus, in th« mixed number .2^, the \ is half of a tenth ; in .22^, it is J of a hundredth. "Notation C'f de- f^imal frartions. NOTATION OF DECIMAL FRACTIONS. Write in figures : Example 1. — Thirty-five hundredths. . Ans. .35. 2. Fifteen thousandth. 3. Fifty -five tenths of milliontbs. NoU. — It saves confusion in writing whole numbers and decimals together, to place the word decimal before the fraction ; also, when a decimal only is ex- pressed. DECIMAL FRACTIONS. 107 4. Five hundred and decimal two thoii:3andLhs. 5. Foiu'- tlioiit^and and decimal six thousandths. 6. Six hundred and fifty and' decimal throe thou- sandths. 7. Eighty-five and decimal seventy-six thou- sandths. 8. One hundred and twenty and decimal fifty thousandths. 9. Two hundred and sixty and decimal fifty-six tenths. 10. Three hundred and decimal six hundredths. 11. l'>c(-imal three thousandths. 12. Dccinial t^Vo hundredths. 206. Addition, subtraction, multiplication and di- How aiiditk-u. vision of decimals ate performed precisely as whole formed/"", numbers. Example 1. 2. 0.56721 3.42501 Add toge ther 4.56 86.902 Sum, 41.462 Proof, 41.462 8. 805.78402 270.23521" 6543.12678 5005.67812 2.16105 13182.98548 '2-22 12211 12.90222 12.09222 534.21005 60.51054 6.05678 .00525 325.60789 . Remariv. — Bern embev always to place tenths under tenths, hundredths under hundredths, thousandths under thousandths, etc., and the point in the sum total directly uiider the numbers of the work given, to be done. 5. Add 2.40, 545.2, 676.'0006 and 3.57580. 6. Add 1765.5, 52.4301, 6.0065 and .9536. 7. Add 5544, 39.7, 678.0212, .00007 and 31.5.' 8. Add .70l», .560, .4809, .395, .56789 and .3366. 9. Add 29.59, 5000.004^ 200.03 and 56.547. 10. Add decimal three hundred and three thou- sandths, thirteen, decimal forty thousandths, two DECIMAL FRACTIONS. hundred and decimal two thousandths, thirty -five milhons and decimal fourmillionth8,decim^al thirteen thousandths. .Ans. 35000213.358004. SUBTPtACTION. OPERATION. 207. Example 1. From 9.6542 Take 8.5431 Rem. Proof, 1.1111 9.6542 36.05946 28.75437 7.30509 From 54.367890 Take 39.43^07565 69.057546 33.556327 Note. — In examples like the 3d, place ciphers over the figures in the svibtrahend that have none, and then say 5 from 10, 5, and carrying 1 for what was borrowed, say 7 from 10, 3. 4. From 965.43445 take 45.395426. 5. From 874.32154 take 3.4506077. 6. From .895456 take" .000543670. 7. From 1.50050000 take .60051789. 8. From 23.0567 take 22.14675. 9. From 67469. take .56543. 10. From fifty -four millions take decimal fifty-four millionths. Note. — Let the pupil be careful to jilace the separat- ing points one under the other. DECIMAL FRACTIONS. 109 MULTIPLICATION. 208. Example 1.— Multiply .30 by .30. OPERATION. .30 .30 210 108 .1290 Eemauk. — In the mul- „.^ , , ,. ,. ,. pj . , , ^\h.'^t numbers* tipUCatlOll OI,aeCUnalS Ob- in result to Vo the 1st and P'^'"'^^^ ''"'• serve, as in 2d examples, to point oif as many Iri^ures tor deci- mals in the product as in both multiplicand and multiplier are of this kind. Multiply 50.425 By , 2.5 28212500 112850 141.0025 Note. — The ciphers annexed arc merely foV conve- nience. The value of the sum would be the same if the multiplier had been placed at the extreme right. Such is the preferable form ; thus, 50.425 2.5 8. Multiply By .7284 .00023 21852- 14508 282125 112850 4. 141.0025 .5082 .30 34092 17040 .204552 .000107532 5.M\iltiply 507 by 3.24. 6. Multiply .435 by 345. 7. Multiply 5.95 by 03.32. 8. Multiply .2350 by .3453. 9. Multiply 05.30 by 234.5. 10. Multiply .0000 by .X)005. Note. — When there are not as maDy decimal figures 110 DECIMAL FRACTIONS. wherf-a.defi-_ in tho procliict US there are decimal places in mul- inafllA-ai-es* ex-^^P^^*^^' ''"^^ mulliplickncl, pliiCG ag many ciphers at the ists ufpioduct. left as the deficiency requires. 11. Multiply .15 By. ' .05 .0075 To multiply by 10. 100, etc. ' 12. Multiply .6250 by .08. 13. Multiply .5945 by 009. To miiltiply by 10, 100, etc., move the decimal point ag many places toward the right as there are ciphers in the multiplier. This makes each figure in the '.multiplicand 10 times what it previously wa?*. Hence, the result is ten times as great as the multi- plicand. • , 14. Multiply .0467 by 100. Ans. 4.67. 15. Multiply .00454 bv 1000. 16. Multiply 8.764 by ^300. ' Ans. 2G29.2.' 17. Multiply .4)004 by 5600. ' 18. Multiply 3.253 by 2.900. . DIVISION. The number of decimal places so bo in quo- 209. Example 1.— Divide .845 by .18. OPERATION. .13). 845(6.5 Am. 78 05 65 Note. — There must be a* many decimal places in the quotient as the decimal places in the dividend ex- ceed those in the divisor. The excess in this example is 1. . 2.4).1248(.052. 120 48 48 The excess here is 3. Divide 2.3462 by 2.11. 4. Divide 94.0056 by .0.8. . 5. Divide .17638 by .369. 6. Divide 61064.14 by .4506. DECIMAL FRACTIONS. Ill 210. When there arc moredecimal places in the divisor Wiien u.,; ■ .« than in the dividend, annex, ciphers to make the work clmainn^aiTi- practicable. ''"r 7. Dividc*1941.855 by .7846. Ans. 2.475. 8. Divide 26043.16 by .3527. 211. Should the number of -figures iii the quotient be When the qm- less than the excess of decimal places in the dividend over [g!;"' f^»mheru those of the divisor, prefix to the quotient tlie number of < iphers which the deficiency requires. 9. Divide .08750 by 1.1. Ans. .0700. 10. Divide .000702 by .12, .l»,.9. .0006. _ 212. When it is desirable to extend the division beyond ^-:. „ ,;_j;i.,. the given numbers, ciphers can be annexed. When it is not *'■'- auuexeii. wished to go beyond a certain figure, or when it can be continued beyond the given niimber, we annex the sign4- ; thus, 4 divided by .7=''.571+. • . 11. Divide .306656 by .5 to thd extent of Ihousandths. • OPERATION. 5\306656 .611+ 12. Divide .5876i\ by .6 to hundredths. inaifr.«lr..u»l Decimal fractfons arc divided by 10, 100, 1000, byio, etc. moving the point as many "places toward the left as there arc ciphers in the divisor. By this, each figure in the dividend becomes only ^g as much as it previously was, and «n the result is correspondent; thus, 567.5^t'0=56.75.- 13. Divide 7846.987 by 1000. Am. 7.846987.- 14. Divide .9;)67 by 300. . 15. Divide .7986 by 500. 15. Divide .54789 by 6000. KBDUCTION. 213. Example 1.— Reduce i to a decimal iVaotion. I^Tt^n'^^vr: Remark. — The value of a fraction is the quotient iin'-U^" '(rising from the division of its numerator by the denom- inator. This value remains unchanged, though a cipher or ciphers be annexed. Thus, to reduce if wo arin<'x, say. three ciphers, and divide by its denominator 4. OPERATION. I Here, aa mimy decimals .4). 3000 are pointed off ill the result — i as thQ decimal places in the .750 j dividend exceed those inthe divisor. 112 DECIMAL FRACTIONS 2. Keduce -i*,- to a decimal. 3. Reduce -3^ to a decimal. 4. Reduce -^q- to a decimal. 5. Reduce -^ to a decimal. 6. Reduce ^ to a decimal. 7. Reduce to a decimal -of three figures f, -f-, i, -^. 8. Reduce to a decimal of five figures -^, -^^, -^f. changed *to a 214. A decimal fraction is changed to a vulgar fraction, vulgar fraction. by vFriting beneath it its proper denominator; thus, .5= .05= 0,^5=— B^i- .VOO 10 0- 9. Reduce .15. 10. Reduce .350. 11. Reduce 3.5. 12. Reduce 5.65. . 13. Reduce 6d. and 3qr. to the decimal of a shilling. . 3qr.= td.==.75; thus, 6d. 3qr.=6.75. Am. 14. Reduce 10s. 5d. Iqr, to the decimal of a pound. OPERATION. 4)1.00qr. 12)5.2500d. 20)10.437503. lqr.= .25d.; 5.25d.=.43758. 10.43'75s,=£521875. £521875. Ans. Explanatory re- REMARK.: — Ciphers *are annexed to the lowest denomi- ti(5*n*^ *"* opera- nation, and its division performed by the number of that denomitfation which it takes to make one of the next higher. The quotient is annexed as a decimal to the next higher number to be reduced, and so on to the result. 15. Reduce 6oz. 18dwt. 15gr. to the decimal of a pound • Troy weight. Ans. .5776041661b. 16. Reduce 65 23 29 2gr, to the decimal of.a pound. 17. Reduce 5yd. 2ft. 6in. to the decimal of a rod. OPERATION. 12 e.Oin. ■ 11 2.500ft. 5.8333+yd. 9 11.6666+half yd. 1.0606+rod. Ans. DECIMAL FRACTIONS. 11;; 18. Reduce 43a. or. 25rd. 20yd. Gft. SOin. to the deci- mal of a sq. mile. 19. Jleduce 25ft. lG75iu. to the decimal of a cord. • 20. lleduce 3qr. ona. to the decimal of a yard. 21. Keduee 2pk. 6(]t. Ipt. to the decimal of a bushel. 22. Reduce £421875 to shillings, pence and farthiugs. Ans. 8s. 5d. Iqr. OPERATiaX. EXl'LAN'ATIOX. £421875 ^ The first multiplication Expianatfon 20s. , changes the decimal of the''"" "'"'■ — ' pound to the shillings and 8.437500 its decimal of a shilling; 12d. ; the second multiplication changes the decimal of the r^.J.VJO shillings to pence, and so on. 4qr. As many places are pointed off in each product for deci- 1.00 mals as there are decimals 1 in the example. The answer I.-: the several deuominations On the left of the^ecimal point. N^ote. — The ciphers omitted in the multiplication are to l)c counted with the other figures, to determine the posi- tion of the point. They were omitted for convenience. 23. It^duce .9375 of a gallon to qt. pt. and gi. Ans. 8qt. Ipt. 2gi. 24. Rediice .7694 of an acre to rd., etc. 25. Reduce .84 of a lunar month to wk., etc. Ans. 3w. 2da. 12h. 28rain. 48.'5ec, 114 MISCELLANEOUS EXAMPLES. MISCELLANEOUS EXAMPLES. 215. p]xAMPLK 1. — A grocer sold 16 bushels of potatoer. at Q.iS) per bushel ; o baga- of coffee at 315.G:i5 per bag; and 14it barrels of flour at $G.50 per barrel : what was the. amount 'r"" ^ 2, A liousekeeper purchased 561bs. of sugar at 12.5 per lb.; 151b. of currants at 18.75 per lb.; 51b. of almonds at 16.33 per lb., and 61b. of starch at 6 25 per lb,': what was amoitnt of purchase? 3.- Bought a keg of butter, containing 961b., at 21.5 ■per lb., but sold one half pf it for 25 ceats per lb. ; what was paid for the whole and received for the half? 4. A mcrcliant sold -V'y.'lyd. of cloth at W-Od per yd. ; what was his gain, the cost to him being 5.555 per yd. ? 5. An advc^iture has gained for 6 pei'sous ^175.035 : what is the share of each ? G. A merchant bought 14Jhhd. of wine at $75,333 pe* hhd , and sold them at public vendue, when they brou^tt only .^00.50 a hhd., exclusive of commissions and c^her expenses -t-SS : what was bis loss ? 7. Bought 19 barrels of flour for 095.125: w^at was the cost of a single barrel? 8. Imported 65 bags of coffee at a cost of i'996.555: what was the cost of a bag ? 9. A trader bought iJ, barrels of pugar a/ $18. 7 5 per bbl : wiiat was the cost of each barrel ? / 10. A bag cf cotton, weighing ooOIb., ^as shipped to Liverpool and sold at '6'6M per lb., Atnerican money. Deducting various expenses, $0,125, wb^t did it net the shipper? / 11. A factor sold 26cwt. 3qr. cf rirfc, in Liverpool, for £2 IGs. per cwt. : what was the amoutit ? 12. Bought 30bush. opk of wheat at $1,375 per bushel : what .was amount of purchase? IB. Bought 2 bancls of flour at §6} jer bbl ; 4 bushels cf corn at"62l cents per bushel; 6ib. of coffee at 16j cents per lb.; 101b. of sugar- at 61 conta per lb., and 6Ib. or butter at 18 J cents per lb. : what did they avcuanb to ? A'Oi'i'. — Change fractious to decimals. MISCELLANEOUS KXAMPLE8. liO 14. WIi:it would be the cost, of building 37ni. Clur. 22rd. of railroad at §8355.(521 per mile ? 15. A merchant bought 20hhd, of tobricco at $57J per hhd., and sold them for S15o5i : what was his proUt? 16. Wliat is the value of .575cwt. of coal at £'d Gs. Gd. 2qr. per ton ? 17. What is the value of 10 bales of Sea Island cotton, the average of the bnles being 3201b., at 4;j| cents per lb. 18. What must be paid for 52 weeks* board, at the rate •^Hch week of !^4.o7J ? - 19. Ao 2ri cents a bushel, J^hat number of bushels can be had for $200^ ? 20. A piece of land is 39.5 rods long and 25. 5G roda wide : what will, it cost to wall it at G2t} ccnta per rod ? 11(3 CIRCULATING DECIMALS. OIRCULATI A (.i lJi.CiMA L>. •inr/de'fined^' . ^^^^' ^'^ Circnlnlifuf Derm , I is simply a decinnil frac- tion, repeating invariably the same figure once or many times. When a single 217. When confitied to a sirisrle fijrure it h cnllcd a '.poUnd: ., ,, „.,_ ,^ ii-* i ,.„ jj ^.fJJy^, Single repotend, a.s .60S, etc.} beyond this, a compound : imd: . one, as 010101, etc.; but when combined with other . deci- whcm niixed: mals preceding its cxprcyKion, a mixed repetend, as .8333. hen ft finite ^^^ figure ov figures preceding the repetend is- r;ill"d the "•I- finite p:irt of the cx])ressioH, as S. ii.ivv the repe- !21S. A Tepetoyd is obtained by reducing a vulj^ai iiac- imdishaiL fJQj^ jq ,, decimal M'hen there is any. prime factor, except 2 and 5, in the denomiBator, and not numerator, as ^=^.HoS, etc. ; 4=''"^5''i> etc. The way trf !il». To avoid repetition in a simple number; the same "^"- is written once, with a point above it j thu:^, 4=.,'). ; but in a compound, the first and last figufcs are thus desig- nated . -^3^^= 1 08. wiiat is a per- 220. When the number of figures in the n!p9tend is lect repotend. ^^g less than the number of units in the denominator of the equivalent vulgar fraction, it is called a perfect repV tend; thus, -f gives the perfect repetend, .14'J857. ^■'»fVinixud ^*^^' 'I'o fi?id the value of a mixed repetend, iii?certain ietend. the valuc of the finite part and cf the repetend separately, and add the re?uUs. When one- 222. KxAMPLj! 1. — What js the value of .270? ninth IS re- ■ . duc-ed to n dcci- .275=27+-4-5=-Hi+-r& f=iU=/aV- ^ns. Note. — When a decimal begins to repeat at the third place, the two' first figures will be so many hundredths, ■ and the repeating figure so many ninths of another hun- dred: 2. What is the value of .345 ? 3. What islhe value of .67123 t 4. Change .4444, etc., to a vulgar fraction. Ans. \. 5. Change .9999, etc., to a vulgar fraction. Xote. — When \ is reduced to a decimal it produces a CONTINUED FilACTIONii. 117 quotient of\llll, etc., that is the repetend 1 ; 4 is the^^''V'" ». "^eci- 1 ' ' . ' _ ' * niiil begins to value of the repetend 1 ; the value of 833, etc., or the''<"peat ut tho '■ ' ■ } 7 third place. repetend 3, must ho three times as much, or -f; 4, f; 5, ■^; andi), A=il. 6. Chance .5833, etc., to a vuli;ar fraction. Here, the figure 5=-i\ and the remaining part of the fraction is -f of -jV) that is -v,^=-/u- J add these, A==io"~l"'/(r =iu-"=T^- ^"^- ft" changed back will give .5833. 7. Change .3744, etc., to a vulgar-fraction. 8. Change .40355, etc., to a vulgar fraction. 9. Change .863630, etc., to a vulgar fraction. When -gV is changed to a decimal it produces .010101, ^^^'":" (^"r"^"®- etc. The decimal .030303, etc., is three times as much, cLianL'ed"to a and is -,3^j.— -gijj The decimal 363636, etc.,- is thirty-six^^^^'"''''- times as much, and is -f 9=-i*r. ' 10. Change .47647 to a vul.Q;ar fraction. 11. Change .•24' to a vulgar fraction. 12. Change .42 to a vulgar fraction. 13. Change .72 to a vulgar fraction. Ans. -{-5= iV 14. Cli;^!'>.<' On:^ to a vul^-ar -i'mction. .!;/«. ~-^»-=-T-i-T. COXTIXUKO rKACTIONS. S'J3. A Continual Fracfian is one which has for its ^ contiiniod numerator a unit, and for its denominator a whole number fined, plus a fraction, and si) on ; thus, ' ' >'+ete., is a continued frartion. S24. The investiiration of the nature and properties of its investi^n^ this fractional formula belongs to the study of Algebra, ^XiRsTAige^ and, hence, wo give simply a passing reference to the sul> bra. ;^ ject. « Example. — Reduce -J^ to a continued fraction. Ol'EKATIO.V. i^= '-+,%- and A=i+| . ■ hence, il= ^^^^_ ■ Again, 1=++^. j^^^^^^ ii=i.x_^. 118 DUODECIMALS. EXPLANATION, We here dlvkle botli terms of -ff by 17^ the numerator of the fraction, and get i_,.^4.. then dividing both terms of A by 9, Ihe numerator, we get i.|_A . substituting then this value of -^ in ,the expression l\ ^L wehavei^=J_}_j. ' ' t-f, etc. 225. In any continued fraction, ^-.^1 '' >4-i, the several aimple fractions arc known as integral, fractiovis, because ' the denominators are integers j thus, ^ in the above is the first integral fraction, \. is the second, -f is the third, and ^,is the fourth. Sometimes we call ^ the first approxi- mating or converging fraction; ^^_j. ^^^ second, and so on. DUODECIMALS. Duod-'.tiimtis 226. DuodecHiials are concrete* fractions, and are chiefly ^^' ' used in the measurement of surfaces and solids. How changed. ^ 22^. They 3X0 added, subtracted and divided, as other compound numbers ; but, in certain cases, they are multi- plied differently. How t)iey do- 228. Duodecimals decrease uniformly from the highest crease. ^^ ^^^ luwest denomination, by the constant divisor, 12. The measures. 220. The measures used lor their change are the inch or prime, the second and the thirds The measures 2 SO. A foot divided into 12 equal parts, is, ia each into twoluhH. division, called an inch or prime; an inch, also s'.milarly divided, is called a second ; and a second, a third ; thus, 1 inch or prime, marked 1'=-jV of a foot. 1 second, " l"=-iV of-iV oi' tIt '^f a foot. 1 third " r"=nVof-,Vof-,Vorl788 " of a! foot, and so on, for minuter divisions, when required. What are jndi- 231. The distinguishing points are known as indices. ft. y y\ ■!. Add Oft. 0' :V' and 7ft. 2' l'\ 5. Add 15rt. C/ V .-vnd ()ft.-. 8^H'^ (). Add 20ft. 9' 8'^ ^V and Oft. 6' 4" ivr V duodecimal ad the dition. SUBTRACTION. 23'J. Example 1. — What is , the difference between '.)K. 3' 5^' Q"' and 7ft. 3' C' 1'''. OPKRATION. I EXl'LAN'A HON'. As the G of the minuend is le.ss than the 7 to be sub- tracted from it, we add 13, and say 7 I'rom ! >^== 1 1. That put down and carry- ing 1, for the borrowed number, to the 6, we say 7 From -0 Take 7 12 3^ 3 12 5' G 12 7 6 Rem. lft.4r 10^ Proof, 9 3 5 from 17, which is the 5 added to 12, and put down the remainder, 10, and so on. 2. What is the difference between 40ft. G'' 6" and 29ft. 3. What is the difference between 12ft 7' 9^' CV' and 4ft. 9' V 0'''? Ans. .7ft. 10' T' 9^'^ 4. What is the difference between 'ISft. 8' T' G""'. and 12ft 7' 8" b'''? 5. What is the difference between 19ft. 9' S'' 1''', and 14ft. 5' 9'' 8'^'? 6. What is the difference between 30ft. 8' d'' 10''^ and ?9ft.. 9' 8'' 11''^? 120 * ■ DUODECIMALS. MULTIPLICATION. how muitipii- 233. The multipiioatioa of decimals consists in niulti- Jbrined!^ ^^^' V^P'^S ^^^^^ ^^^'^ ^^' the multiplieaud by each term of the multiplier, couimeucing with the highest unit of the mul- tiplier and the lowest of the multiplicand, and making the indices of each product equal to the sura of the indices of the factors. Example l.—A board is '5ft. b' 4^^ in length cVid 3ft. 5^ 4^' in breadth : nvhat are the contents ? Ans. 18ft. 9' 5''' \f''\ Kxplanation of - ^'t ^'t. ! EXPLANATION. work of rnuiti- o D 4 i " In this work, as the 4 'X plication. , 3 5/ 4// I g£^__22^'---l' (12-r-r2— 1), we set down under the seconds, and add the 1' to the next product of 5'X3ft. =16', which, reduced by dividing by 12'F=lft. 4', we then write down the 4' and carry the 1 to the next prod uct=16fb. In the same way, we multiply the multiplicand by the next figure, 5', and this line being written down, we mul- tiply by the 4 in like manner. The products are then added for the answer. 2. How many square feet in aboard 17ft. 6in. long and 1ft. 7 inches wide ^ Ans, 27ft. 8' 6". o. What quantity of boards will it take for a floor 14ft. 8' 3" long and 13ft. 6' 9" wide? yi».s-. 199ft. 2* 4" 8'" 3'";. 4. How many feet in a plank l2ft. 4' Ions', 2ft. 3' wide, ,ind4'thickr' • ^Ans. 111ft. OrEHATlON. 12 12 5^ 4// 3 6' 4// - 10 4' Q'' 2 .'V ?/'' g/// V iV 9/// 4//// 18ft. 9' 0" 5'"; 4"" 12 2 QPKnATIOX. 4' length. 3' width. 24 3 - 8' 1' 0" 27 9' 4' 0" thickness. 11 1ft. Q' 0" DUODECIMALS. 121 \ 5. What are the couteuts of a block of marble that is ft. 9' 3" loHg, 3ft. 2' 4" wide, and 2ft. 5' 7" thick ? Am. 60ft. 0' 10" 4'" 5"" 1'"". 0.' What number of cubic feet in a granite block 3ft. ;/^i. wide, 2ft. Sin. thi6k, and 12ft. Gin. loui?? Ans. 105ft. 6' 7" 6'". How many square j'ards in the walls of a room 14ft. lonsTj'llft. Gin. wide, and 7ft. llin. high ? Alls. 46jd. 3' 8". What will it cost to plaster a room 20ft. G' long, wide, and 9ft. 6' high, at 24 cents per scfuurc yard ? Ans. $25.18f. How many square yards of oil cloth will it take to an entry that is IGft. Bin. long and 8ft. 7in. wide ? 1^. If a load of wood be 8ft. long, 3ft. 9in. wide, and It. pin. hiiih, how much does it contain ? Ans. Ic. 4c. ft. 3cu. ft. Ill How many feet of boards will it take to make 12 uuxeA whose interior dimensions are 4ft. 5', 3ft. G', and / 2ft. 7ui., the boards being 1' thick ? 12.uIow many solid feet in a stick of timber 25ft. Gin, t. 7in. broad, and 3ft. 3in. thick ? 13. \Iow many cords in a pile that is 25ft. 7' long, 5ft. 4in. hiji, and 2ft. 9' wide ? . 14, \\hat will a marble slab cost that is 7ft. 4' long and 1ft. Sin, \vide, at §1 per foot? ,A?iS. ^9. IG 3. in VISION. 2JJ4» E>qfLMPLE 1. — In 1G5' how many feet? ' lG5-^12==13ft. 9in. Ans. 2. In 2G0\ how many feet and inches ? 3. In 43G^'" how many feet ? Ans. 2ft. 6' 3" 11'"- OI'BUATIDN. 12)43G7"" ■ ■ 12)363. 11'" 12)30. 3" 4. In 5280'" h 5. In 28800 G. In 5G450 2. 6' many feet ? tw many feet ? \w many feet '/ 122 . ANALYSIS ANALYSIS. Analysis de- 235. Anali/ah, in arltlimetic, is the separation of a svai into its component parts, with the relative bcariags of he numbers in that question to each other. It dispenses 230. It (^ffers u .simple and practical method to pcrf'rm nOes'^"'^"^*' an operation without the formality of rules j and it is • ery useful in aiding ^lutions when rules, as in practical qac8- tions, occurring daily, are not remembered, and cannoG be conveniently referred to. 237. 1q analysis we reason gtcp by step from the in- quiry to the result. Example 1. — If 6cwt, of hay cost $13, what will Qcwt. cost? ^ ^ Ans. m. Expianalion of The auijysis is that one hundred weight costs one sixth *"'" as much as six hundred weight. Since 6cwt. cos^ 12, 1 costs i 12=2 ; 9 costs 9 times as much as Icwt., a'ld that is 9 times i.l2=:lh.. 2. If 9 men can dig a trench in 25 days, how mmy will it take to do the work in 5 days ? 3. A ship's company have .provisions to last J2 men 8 months : how long would these last 18 men? 4. A hare has 39 rods the start of a houni, but the «N l^outid runs 27 rods while the hare runs 24 : how many rods must the hound run to overtake the haie ? Ans. Bfil . 5. The United States commander in Fort Sumter bad 21b. of bread per day for' each soldifer, for 10 days; but, by private dispatches, learning that his goveinmeut would relieve him soon, he wi.shes to stave off surrender 15 da3's : to do that, what must be the daily allowf'i^^e, say to 80 men 6. 24 is I of what number ? ' 7. 70 is -i%- of what number ? 8. If ^ of a cask of wine cost $54, what will 4 casks cost ? " 9. A man sold ,a watch fpr $56, which was |- of its coat : S<>0 ; C S-4. ANaItSIS OF Tim ABOVK. The pasturage- of 2 horses' for v} wceka is the ■same as Explanation by that of I horse 2 times o weeks, or (\ weeks; that of (> '^"■'^y-^'s- horses 2^ weeks, the same as that of 1 horse (i times 2} weeks, or lo weeks ; and tliat of 9 horses H weeks, the Bamc iis that of 1 horse !) times lis, or I'J. weeks. The ad- dition of the weeks together. ()4-l;'i4-l-=o3 weeks ; hence, A'.s share of payment is -jVof S'>1> ('56^o:J=:iXU^Sl2) ; "s i^ of $6(3— f.:5Jj and O's i| of S()t)=:§24. Remark. — The pupil should begin an analysis from the term which is of the ftame name or kind as the required answer. lo. The inheritor of an estate spent i of it in 9 months, and -f of the remainder in 12 mouths more, when he had only iJt'WO left : what was the estate when he received \t ? \. Divide $l7i).40 among .3- persons, so that A shall twice as much as B,' and three times as much as what is the am^junt when so divided ? ' ■ > Two men had the Siinie income. The first saved i of his each year ; but the second, by spending §20(t a year more than the first, w«s, at the end of 5 years, $Wb in debt; what was the income? Aiif^. Slo-it. K). If it take 44 yards of cai^peting, liyd. wide, to dover a floor, how manv yards of the kind, i wide, will it take ? ' Ans. (i.f . 17. If an acre of land cost ^ off of -f- of 350, what will 8 J acres co.st? Ann §10. 18. If :5U gallons of ale are worth 89f, what, at that rate, will 5i cost? . . Ans. $!l.G(). PROPORTIONAL ARITHMETIC. PART FOURTH. RATIO AND PIIOPOKTION, OR SIMPLE RULE OP THREE. Ratio defined. M umbers rela- tivuly viewed. ]{iiw ratio is in- diciued. Tlie terms. Th<» Qnt<5ce- dents. The cocsc- 'luents. First and see- I'md couplets. Tlie work of a l^roi)ortion. X ratio of equal- ity. t >f 'sreater ine- iinafity ; of le.ss iuo'iuality. 238. Ratio is that r'elation of one quantity or number to anotlier of the same kind, by which is found their' equality or inequality. ■ 239. When numbers are relatively viewed, the one which measures the other is considered as the standard, arid the quotient resulting from the division by the stand- ard is the ratio or relative value. 240. Ratio is indicated by a colon (:) in the first term ; by a double colon (::) in the second; and by a colon in the third (:); thus, ' 4:8::G:12 ; that is, 4 is to 8, as 6 is to 12. 24L The two quantities compared are the terms of the ratio ; the first terms, or 4 and 6, 'in the above expression, being the antecedents, and the second, 6 and 12, the con- sequents. Here, the antecedents are the standard. The 4 and 8 are known as the first couplet, and 'G and 12 the second. 242. When two couplets have, as in the example, the same ratio, their terms are proportional ; hence, 243. A. proportion compares the terms of ts5;o equal ratios. 244. When the antecedent equals tlrtj consequent, the ratio being one, it is called a ratio of equality; thus, 4:4 = 1. ; when it is greater, or more than One, it is known as a ratio of greater inequality; thus, 6:2=3; when it is less than one, a ratio of less inequality ; thus, 2:lC=i. RATIO AND PROl'ORTION. 125 245. When there is but one antecedent and one conse- a simple ratio, quent, the ratio is said to be simple; thus, 12:4=3. 2146. When tlie corresponding terms of two or more a compomui simple ratios arc multiplied together, the ratio arising from '^''^'' it is called compound ; thus, 4: 2= 2 4:2^2 6: 3= 2 G:2=3 12: 4= 3 24:4=6 and 288:24=12 are compound ratios ; and, .as the illustration shows, are equal to the product of the sim- ple ratios of whicli tliey are composed. 24'/. The 1st and 4th terms of a proportion are called t^^p extremes; •the extremes; the 2d and 3d the means; and the product ^^ '"eans. of the extremes is equivalent to ihat of the means ; thus, in the proportion, 3:9::12:36; 3X36=-- 108; and 0x12=108. .\ 248. The 4th term of a proportion is found by multi-To find (he plying the second and third terms together, and dividiug "" '' '^'^"'' by the first; thus, 3:9::12: ?; 9x12=108-4-3=36, the term sought. • Example 1. — What is the 4th term in the proportioii, 3: 9:: 4 ? 5:15:: 3 ? 6: 8::12 ? 8: 4:-: 4? Remark. — Any term of a proportion can be found To find other when the three others are given ; for the product of the '^'""'*- extremes, divided by either mean, gives the other; and the product of the means, divided by either extreme, gives the other. In the solution of problems, two of the three given num-T^.f,r,ft,,<,Qu„,. bers must be of the same kind ; the third like the one re- bers to bo of , , ' correspondent quired ; thus, n.sturc. 2. If 4 men build 8 rods of wall in a day, how many will B |)uild ? Ans. 12. 4 m^n : 6 men :: 8 rods : 12 rods, the number sought. 3. If a staff 6ft. long cast a shadow 12 feet, what shadnvv will be cast at the same time by a steeple 60 feet high ? 4. If a staff 6 feet long cast a shadow 9 feet, what is the hei^-ht of a tower w;hose shadow, at the same hour, extenus 198 feet ? and what is the ratio ? Ans. 132ft. ; Ilatio, 22. 126 RATIO AND PROPORTION. 5. If board for 52 w'eeks amounts to $182, what is it for 39 weeks? Ans. ^iHQ/oO. G. If I pay for 48 yards of cloth S87.7->, what, at the same rate, will 144 cost? Aiis. S-Ol.T;^. 7. If S'O soldiers require 11,250 rations of bread for a month, how many will be necessary for a garrison of 600? Ans. 18,0G0. 8. If 12 men can build a house in 20 days, how many can do the work in 5 days ? and what is the ratio ? Ans. 48 men ; Ratio, 4. 9. If 80 gallons, in an hour, run into a reservoir tbat will contain 1400, and 30 run out, in what time will it be filled? ' -^"s. 28 hours. Soiuiions by Eemark. — A uscful Avay ,to solvc qucstions like these analysis prefer- is by analysis. As in busiue^ss operations such method is ^' usuully adopted, we would adKse it as practically the bet- ter plan. 10. 3 bricklayers build 6 rods of a inundation in a day.: how many rods would 5 build ? If 8 men build 6, 1 will build J of 6=2 j if 1 build 2, 5 will build 5 times 2=10. 11. If 25 men perform a certain work in 35 days,- how long will it take 9 to do the same ? 12. If 271b. of butter will buy 451b. of sugar, bow much can be had for SOibs. of butter? 13. If an engineer's salary for throe years amount to $3600, what will it be in 9 years ? 14. Iff of a yard ol cloth cost f of a dollar, what will 21 yards cost? ^«s. S4.8ij. Here we say, if f cost -f-, ^ costs A of i=-(Vx5 for the whole =ffX2J, that is -I =-Yb^) which, by the annexing of two ciphers ibr cents, =4.bG. Note. — Let it be remembered that a mixed number, as in this sum, is to be reduced to a fraction. 15. What is the price of Gfyd. of cloth, if '}yd. cost 16. What is the 'cost of 3Sdoz. of wine, if jdoz. coat I7H 17. What ifi the value of SSJ barrels of ale, if 4i bar- rels cost $15? 18. A merchant owning f of a ship, sella | of his s-haro for S 15,000 : what is the value of the ship ? .. COMPOUND PROrOKTION. 127 19 llow many yards of silk, f yd.' wide, will it l^kc to line ITiyd. of camlet iyd. wide ? 'f^ 20. II', when flour is worth ^tl per barrel, a 5 cent' Idaf weigli 4oz., what ought it ^ weigh when flour is Worth " $8 per barrel ? , " i' 21. If the earth revolve 366 times in 365 days, in what time does it revolve ojx^e? Aus. 23h. SG-jVm. 22. If ,62a. 3r. 2pi: of land cost £615 93. 3d., what will 1 5a 2r. 3rd. cost 7 - ' 23. If-,V ofaship cosfctl350, what isf of her worth? 24. IJorroved $25l) 'for 6 months ; for how long must $35J be le«t to repay the favor ? • • 25. Ii"l2iyd. of silk, Jyd. wide, will make a dress, how many yards of cambric, that is 1| wide, will Hue the same ? Aus. 6-(*r- COMPOUND PROPORTIOX, OR DOUBLE RULE OF THREE. 249. Compny.nd Proportion is an equality which em- compounu pr»- braces in its consideration simple and compound ratios ; portion defmod thus, ^ 3-12) 1^' ., \ ::18:9 is a compound proportion, and 4S:24::18:9 is the same in a simple form. 250. Que.stidns can be reduced to a simple form, and aome of complicated expression are easily solved, when the terms have been carefully arranged. ExA.Mi'LE ].— If 3 men in 4 hours can thresh 15 bushels of ric6, in how ;uany hours can 2 thresh .') '( Aiiti. 2. ^1/ Simple Proj)ortion. OPERATION. KXrLAXATION It will be Peen that inj;,p,^„^^^„ ^ the first proportion the qucetion 2:3::4:6, and 15:rK:():2, An?. amount of labor i.^ not made the subject of incjuiry, but the time that 2 men will take to do the work of '6 men. Finding that to be 6 hours, the second iiujuiry is,' in what time 5 bu.ihels can be threshed, if l.> bu.^!»cls are .in G hours, and the time i.s found to be 2. 128 COMPOUND PROPORTION. Why called compound. How to solve questions. 2. If poo gain $6 in 12 mobths, what will be the gain of $3400 in 18 months ? Ans. $306. OPERATION. I EXPLANATION. 100:3400 :: 6:204 and Here are stated three of ,12 : 18::204:306 T the given things in a' sim- ple proportion ; that $100 are to |-3400 as $6, the interest, are to the answer: or, if $100 gain $6, what will $8400 gain ? 100:8400::6:204. This being done, there remain the terms 12 months and 18 months to be disposed of, and forming a second proporti'on under tue first, we say, 12 months are to 18 months::204 (the interest on 3400 for 6m.) :306 the answer. . The reason why questions conducted thtis aio called compound proportion is evident, for the answer sought is not only in proportion to the principal, but also in propor- tion to the time; and, therefore, in the compound pro- portion of the interest multiplied by the time. » To solve questions in compound proportion, make for the 3d term that which is of the same kind or denomina- tiqn with the answer. Then, take any two of the remain- ing terms that are alike, and arrange them as in simple proportion. In a similar way, arrange any other two terms of the same kind, and multiply the continued product of the 2d terms by the 3d term, and divide this result by the' continued product of the 1st terms ; the quotient will be the term required. 3. Ten men can build 25 rods of fence in 6 days : how many men will it take to build 30 rods in 3 days ? Ans. 24. OPERATION. 25:30::10:12 and 3: 6::12:24 Aiis. EXPLANATION. With the statement of the first three terms, which. are in direct proportion, we have the two terms, 6 and 3, placed in what is called an inverse proportion to the an- swer, and in this way, because the more men are engaged, the less time is needed. These terms are therefore to be put inversely from the order necessaryjif the proportion had been direct. Instead of 6 to 3, say 3 to 6. wiien the con- Note. — This same rule applies when there are three or three or 'more, niore conditions to the question. 4. If a family of 6 persons spend $300 in 8 months, what amount is necessary to a family of 15 persons for 20 months? ' Ans. $1875. COMPOUND PROPORTION. 5. If a family of 6 persons spend §600 in 8 months, how many dollars will be required for a family of 10 per- sons in 14 months ? Ajis. §1750. 6. A trader, with a capital of $300, gained $75 in 3 months : how much would he gain at the same rate with a capital of $1000, in 1 year (12 months)? 7. If 10 acres feed 15 head of cattle for 20 days, how many acres would feed 400 head for 90 days ? 8. If 50 men can dig a ditch 100 yards long and 4 wide in 30 days, how many can dig one 400 yards long and 5 wide in 5 days ? ft. A ship's company of 16 use Ihhd. of water in 1 month (30 days), how long, if the number be increased to 24, will 40hhd. last? 10. If 100 men can build a wall 300 feet long, 2 feet deep, and 6 feet high, in 10 days, in how many days can 50 men build 50 feet of wall, 3 feet deep and 4 feet high? 129 stated thus, 300 2 6 50 Remark. — This somewhat complicated question may be " 50::10: 3 4 100. The terms are in direct proportion to the answer, because more requires more ; that is, the longer the wall, the longer the time for its completion j the deeper the wall, the longer time, and the higher, the longer time. The last proportion is inverse, because more men need .less time, and it is the only one to be stated inversely. 11/ If a twopenny loaf weighs 8oz. when wheat is 68. '9d. per bushel, how much bread may be bought for 3s. 4d. when wheat is selling 13s. 6d. per bushel ? Ans. 51b. 12. If 40 men can dig a trench 40 yards long, 8 deep, and 8 yards wide, in 8 days, how many men would be em- ployed to finish a trench 100 yards long, 12 wide, and 3 deep, in 24 hours (1 day) ? 13. If 25 persons consume 300 bushels of corn in 1 year, how many bushels will 139 consume in 8 mouths, at the same rate ? Ans. 1112. 14. If the cost per railroad of 12cwt. 3qr., for 400 miles, is $57.12, what will be the cost of 10 tons for 75 miles?- , Ans. $168. 10 COMMERCIAL ARITHMETIC. PART FIFTH. INTEREST. Intereat de- fined. Names con- nected with it, 251. Interest is a premium for the use of borrowed money. 252. The money on which interest accumulates is called the Principal; and the principal, with its interest, is called the Amount. Premium dif- 253. The premium is a legalized value, but is not uni- lot^tates!^'^^'^ ^'^^^ ^^ ^^"^ Confederacy. In South Carolina, the rate or annual per centage is 7 per cent., that is, 7 cents on the 100 cents, or 7 dollars on the 100 dollars, for a year. In Georgia, it is 8 per cent. ; in Texas, 12 per cent. ; in some of the States of the old Union, 6 per cent. ; in France aad England, 5 per cent. 254. The term per cent, (per centum) is Latin, and "^^^ ^^^"8 P^'' signifies for the hundred j and per an. (per annum) for annuia. the year. Note. — A sum at simple interest becomes double in 16 years, 8 months. How long it *59« Example 1. — The interest on SI, for a year, is 7 takes to double gg^^g [^ South Carolina, what is the amount of interest due on $100 for the same period r Ans. §7. an asBount. OPERATION. Multiply 100 By 7 700 Eemark.— The $100x7 =700; and as the multipli- cation of dollars by cents (Art. 121, Remark) gives cents, the answer is 700 cts. INTEREST. 181 In multiplying dollars by dollars, let the pupil remem- When doiiflrs ber that the result will be dollars; but when dollars arc'^,^,''J^Jf|,'r'£''B^'i multiplied by cents, as in the above example, the answer doUnrs by is in cents. Two right hand figures pointed off for cents, '^ show the answer to be 87. 2. What is the inte-rest of $50, for 1 year, at 7 per oent.? Am. §3.50. OPERATION. 50 7 350 cents. 3. "What is the interest of §90, for 1 yeaj:, at 7 per cent/!* Ans. S6.30. 4. What is the interest of §99, for 1 year, at 7 per cent. ? 5. What is the interest of 6250, for lyr. 6m., at 7 per cent.? ^ Note. — The interest for Gm. (iyr.) is to be added to that of the year. 6. What is the interest of S300, for lyr. 4m., at 7 per cent.? (4m.= J of a year.) A7is.'$2^. When the number of months is an equal part of 12, as How in ?o,w in the last example (12-f-o=4). it facilitates calculation f.'**'^^ c^icu;:,- to take an equal part of the year's interest, according toted. the question, and to add it to the amount for the year. Note. — The same applies to parts of a month and to days. 7. What is the interest of §700, for 2 years and 2 months, at 7 per cent. ? (The 2m.= h 12.) 8. What is the interest of S900, for 1 year, 3 mouths and 15 days, at 7 per cent. ? (The 15d.= J mouth.) A71S. e80.37J. 9. What is the interest of 81000, for 2 years, 1 month and 20 days,*at 7 per cent. ? (The 20 days is fm.) 10. What is the interest of 375, for lyr. 5jn. and 18d., at 7 percent.? (5m.=T^ofl2; ]8d.=-J^ of 30.) 11. What is the interest of ^15G, for 2 years, 8m. and lOd., at 7 per cent. ? (8m.=5 12 ; lOd. ^ 30, or jm.) 12. What is the* interest of 817.49, for lyr., 7m. and Sd., at 7 per cent. ? Ans. U.QSb. INTEREST. When there are Note. — Here, dollars multiplied by cents give cents,^uf cents in a quee- ggjj|.g jj^jtipiieij j^y cents givc or reduce to mills (Art. 121, Renjark). Thus, the answer is $1. 68c. 5m.= J cent.) 13. What is the interest of $459, for 5yr., at 8 per cent.? ' Ans. 183.60. 14. What is the interest of $600.50, for 3yr., at 8 per cent.? ^ns. $144.13, 4 mills. 15. What is the interest of $62, for 3yr.,.at 8 per cent. ? 16. What is the interest on a Confederate State bond, for $1000, for 2yr. 6m., at 8 per cent. ? 17. What is the interest of $156,for ]yr. 9m. and 15d.y at 12 per cent. ? (9m.= f 12m.) 18. What is the interest of $256, for 6m. 20*, at 8 per cent. ? • 19. What is the intei:est of $444, for lyr. 10m., at 6- percent.? (10m.=il of 12.) • 20. What is the interest of $85.30, for 1 yr. 16d., at 6 per cent. ? 21. What is the interest of $550, for 90 days, at 7 per cent. ? (90d.=3m.) ' 22. What is the interest of $150, for 60 days, at t per cent.? (60d.=2m.) 23. What is the interest of $125, for 30 days, at 7 per cent.? (30d.= lm.) 24. What is the interest of $38.55, for 2yr., at 7 per cent. ? What to kc^'p Remark. — Let the pupil remember that when cents are '"^oi^^nc off i^ *^6 question, 4 figures, counting left to right, are to be Dumbefs. pointed off; and that the two, on the right of the point, are cents, and the others tenths ; thus, as in the 14th ex- ample, the numbers in the result are 1441344, which, pointed off as directed, read $144.13 cents, \^, or 4 ;(uills, etc. 25. What is the interest of, $95.50, for lyr. 6m., at 7 per cent. ? 26. What is the interest of $350, for 3yr. 8m. 18d., at 6 per cent;, ? Ans. $79,388. JIemArk. — ^In mercantile transactions it is customary to take for the answer only two figures, cents, at the right hapd of the separating point. This answer would usually b?i:ead $79.38. INTEREST. 27. What is the interest of $326, for 3yi-. 2m., at 7 per «ent. ? 28. What is the interest on $56.52, from March 19, 1859, to January 25, 1862, at 7 per cent. ? 29. What is 'the interest on §598, from July 15, 1860, to Oct. 20, 1864, at 7 per cent. ? ■ 30. What is the interest on $135, from Dec. 10, 1861, to May 17, 1868, at 7 per cent. ? 31. What is the interest on $65.^, from Jan. 1, 1855, to Feb. 1, 1863. at 7 per cent. ? 32. What is the interest on 8100, from June 10, 1860, to June 10, 1865, at 7 per cent. ? 33. What i.s the interest on £27 15s. 9d., for 1 year, at 5 per cent. ? Ans. £1 7s. 9d. Iqr, OFERATIOX. £27 15s. 9d.=27.7875 5 1.389375=£1.389375 20 133 d. 9.4500 4 EXPLAXATION. As the principal is in pounds, shillings,, pence, we reduce the lower denominations to the decimal of a pound (Art.* 214, 14 Ex.), and multiply that, with the pounds prefixed, by the per centage.^ In this ex- ample, the left hand figure is the interest of the pounds denomination. The decimal interest is then reduced back to shillings, qr. 1.80 pence and farthings. It will be noticed that only 3 deci- mal places in the multiplicand ^re used. 34. What is the interest of .£16 9s. 5d. 2qr., for lyr. 6in. 15d., at 5 per cent. ? 35. What is the interest of £75 123. 9d. 3qr., for 2yr. 9m. 20d., at 5 per cent. ? 36. What is the interest of £100 3s. 3d., from March 9, 1862, to June 24, 1864, at 5 per cent. ? , 37. What is the interest of £90 6s. 3d., from Dec. 1, 1862, to Oct. 11, 1863, at 5 per cent. ? 38. What amount of principal and interest, at 7 per ceftt., is due Jan. 1, 1861, on a note of S400, dated Jam 1, 1859; there having been paid, July 1, 1859, 8100; Jan. 1, 1860, $150 ; and July 1, 1860, $50 ? Remark. — In like examples cast the interest from the wh^H naf- dt^e of the note to the specified time, and add to the prin- b"^u'm3.iJo« & flipal ; then, on the several payments, from their dates to *»ote. 9 lo4 INTEREST. the specified time, and add to their collected amount this credit interest ; subtract the same from sum of principal and interest, for answer. 39. What was due on a note of $448.50, at 7 per cent. interest, dated June 15j 1854, when finally settled July 3, 1856, which had these endorsements : Dec. 6, 1854, $75 : April 19, 1855, $125; Dec. 15, 1855, $10; Jan. 25, • 1856, $100? . .' " • Ans. $183.60. 40. What was due, March 26, 1855, on a note for $1000, at 6 per cent, interest, on which had been paid, Sept. 6, 1850, $50 ; July 14, 1851, $150 ; Aug. 9, 1852', $25 ; May 14, 1853, $28 ; Oct. 15, 1853, $125 ; Nov. 11, 1853, «75 ; and Not. 13, 1854, $500 ? Ans. $282.58. To find the jf it j^g desirable to ascertain the principaL when the iinncspal when , . i.i i "^ ■, • , x t)i9 interest, interest, time and rate alone are known, cast the interest on ^'i"\nown.'''^ one dollar for the given time, and divide the interest by, it. 41. The interest of a certain sum, for 2 years, at 7 per cent., is $70 : what is the principal ? Ans. $?00. OPERATION. Int. on $1=7X2(2 years)=14; and 70.00-f-14=500. 42. The interest of a" certain sum, for lyr., at 7 per cent., is ^63 : what is the principal ? Ans. $900. 43. The interest of a certain sum is $350 a year, at 7 per cent. : what is the principal ? 44. The interest of a certain sum, for lyr., at 7 per cent., is $1500 : what is the principal ? 'I'o find the rate When the interest, time and principal are known, we tlineandpritic'i- ascertain the rate, by casting the interest on the principal 7Xii are known, f^y ^\^q given time, at 1 per cent.^ and then dividing the known interest by it. The quotient is the rate. 45. The interest of $500, for lyr., is 835 : what is the rate? Ans. 7 per, cent. OPERATION. ] Here, to accomplish the division 500)35.00(7 of 35 by 500, we annex (Art. 129) 35.00 1 two ciphers to the S35. 46. The interest of $1200, for 2 years,' is ^144 : what is the rate ? Ans. 6 per cent. , 47. The interest of 81500, for 1 year, is. $120 : whut is the rate ? • * • 48. The interest of 82000, for lyr., is. $100: what is the rate ? To find tiie When interest, principal and rate are known, we ascer- jldnoipai! hiter^- tain the time by casting the interest on the given pri^- .-st, and rate are cipal, at the kuown rate, for 1 year, and dividing the * COMPOUND INTEREST. 135 interest by it. The quotient will be the time in years and decimals of a year. 49 What is the time when the principal is ^900, the interest is $63, and the rate is 7 per cent. ? Arts. 1 year. OPERATION. Int. 900=63, and 63-=-63=l. 7 63.00 50. What is the time when the principal is $750, the interest S90, and the rate 6 per cent. ? Ahs. 2 years. 51. For what time must $200 be at interest, at 6 per ^ent., to gain $36 ? COMPOUND INTEREST. 256. Compound Interest is interest on principal and compoun.i in- ... .f. J J 1 . .'^ . T^ rr,, terest defined. interest iinited and making a new principal. The latter being, as an unliquidated debt, added to the former. . 257. No law recognizes such computation, but it is ite computa- equitable and becomes legal when payment of due*'°° equitable, interest is not made on demand. In such case the unsettled interest becomes a principal, and interest, as on any debt similarly placed, can be charged. -.r t , . 1 1 1 I" whai time a Jyote. — An amount at compound interest doubles Bum at com- itself in 11 years, 8 months and 22 days. fs'douwJd"" 25S. The method of computing it consists in find- Method of com ing, as in simple interest, what is due on a specified ^" * '^"' amount, added to that amount, and then annually increased by the interest of the preceding year. loS COMPOUND INTEREST. Example 1. — What will be the compound interest of $500, for 2 years, at 7 per cent. ? Aiis. $72.45 OPERATION. $500.00 principal 1st yr. 35.00 int. for Ist yr., 500x7=35. 535.00 principal 2d year. 37.45 int. for 2d yr., 535x7=37.45. 572.45 amount in 2yr. 500.00 deduct first principal. $72.45 amount of interest. Hote. — It will bo seen that the whole amount of compound interest is found by taking the first prin- cipal from the last sum of ijrineipal and interest. 2. What is the compound interest on $1000, for B years, at 7 per cent. ? Ans. $225,043. 3. What will be the amount of $5000, at compound interest, 7 per ceift., for 4yr. 10m.? Ans. $6911.26. 4. What is the compound interest of $250, for 2yr., at 8 per cent. ? Ans. $41.60 5. What is the compound interest of $939.64, for 3yr., at 7 per cent.? Ans. $211.45. 6. What is the compound interest of £50 98. 6d., for 2yr., at 5 per cent. ? Note. — Keduce shillings and pence to decimals of a pound (Art. 214, 14 Ex). COMPOUND INTEREST. 137 An expeditious way to calculate compound interest is afforded by this TABLE, fihowing the amount of $1, £1, etc., interest compounded annually at 4, 5, C, 7 ajid Sper cent., from 1 to 20 years. 4 per cent. | 5 per cent, i 6 per cent. 1.040000 1.081600 1.1248G4 1.169859— 1.216653— 1.26i)319+ 1.315932— 1.368569+ 1.423312— L480244+ 1.539454+ 1.601032+ 1.665074— 1.731676+ 1.800944— 1.872981+ 1.947900+ 2.025817— 2.106849+ 2.19112.3+ 1.050000 1.102500 1.157625 1.215506+ 1.276282— 1.340096— 1.407100+ 1.477455+ 1.551328+ 1.628895— 1.710339+ 1.795856+ 1.885649+ 1.979932— 2.078928+ 2.18287,5— 2.292018+12, 2.406619+12 2.o26950+i3 2.653298—1.3 060000 123600 191016 ,2T]2477— ,338226— ,418519+ ,503630+ ,593848+ ,689479— ,790848— ,898299— ,012196+ ,132928+ ,260904— ,396558+ 540352— ,692773— ,854339+ ,025600— ,207135+ 7 per cent. 8 per cent. > 070000 144900 225043 310796+ 402552—; 500730+j 605781+i 718186+1 838459+1 967151+ ! 104852—' 252192— i 409845+ 578534+ 759032—' 952164—' 158815+' 379932+ j 616528— i 869684+ ! 080000 166400 259712 360489— 469328+ 586874+ 713824+ 850930+ 099005— 158925— 331639— 518170+ 719624— 937194— 172169+ 425943— 700018+ 996019+ 315701+ 660957+ Table for find- - ing oompound 1 interest. 2 3 4 7. AVhat is the compound interest of $17.25, for 'Jyr. and 7m., at 5 per cent. ? Note. — From the table take the amount of $1, for 2 years, at 5 per cent., and compute the interest on it, for 7 months, us in simple interest. Add this to the amount for 2 years, and we have the interest of $1 for 2yr. 7m. Multiply this by $17.25 for "answer. To find the interest, subtract the principal from amount. 8. "What is the compound interest of $300, for 5yr. 8m. 15d., at 6 per cent. ? Ans. $171.59. 9. What is the compound interest of $600, at 6 per cent., per annum, for 20yr. ? OPERATION. $3.207l35=int. of $1 for 20yr. 600 principal. $1924.281000 int. of $600 for 20yr. 138 DISCOUNT. Here, counting off 4 figures on the left for dollars^: to the unit-dollars-figures in the multiplicand, and the three of the multiplier, 600, we have the answer, $1924.28. 10. What is the compound interest of $950, for 3yr. ■ 6m ■ lOd., at 7 per cent. ? DISCOUNT. Discount dO' fined. Tke per eent- To find the 259. Discount is an amount allowed for the cash payment of a bill, or for the settlement of a debt, before its unexpired term of credit, or the sum charged by a bank for money loaned on a note. 260. The allowance is called per centage, and its rate is according to agreement. When no rate has been named, it is customary to deduct or discount the interest. 261. Discount is found by subtracting the present amount of dis-^^^^^ f^^^^ ^^le amoiint due. Example 1. — What is the present value of $100, payable at the expiration of a year, without interest ? Ans. $93.45 EXPLANATIOK. We divide the given sum by $1, with its in- terest for the time ; jn this example, 7 per cent. OPERATION. 1.07)100.00(93.45+ 963 370 321 490 . 428 620 535 85 2. What is the present value of payable in lyr. 4m. ? , at 6 per cent., COMMISSION. 1'^*^ 3. What is the value of $500, due lyr. bonce, at 8 per cent. ? . 4. What is the discount to be made on a note for $437, which has an unexpired term of 4ni. 15d. ? 5. What is the present value of a note for $350, payable in 1)0 days, which has been discounted for me at the Bank of the State ? Ans. $343.68. Remark. — Discount is always required by a bank in advance ; thus, in having a note for $350 dis- counted as above, the sum received would be the answer to tliat example. Banks aUow a time of notification for payment of Day* of grac«. a discounted note, of 3 days, called daj'S of grace; in the discount, these are reckoned. 6. What is the bank discount of a note of $1000, payable in GO days, at 6 per cent, interest ? Ans. $10.50. 7. My factor having sold for me 30 bags Sea Island cotton, received 'a note from the buj-er for $3456, at 90d. ; this note bQing discounted by the Charleston Bank, at 7 per cent., what amount was credited to my account? ^?is. $3393.50. 8. What is the present value of a note of $450, payable in 90d. at the bank, at 7 per cent. ? COMMISSION. 262. Commission ifi a, certain per centage charged commis!>ioB by a factor, broker, or general agent for the transac- lie is to have -^The method to of the gain, which is $480=$200; and, similarly, thegaTorYoIr"' other is to have -iV=$280. We, simply to find loss or gain, multiply the w'hole gain or loiAs by each of the person's or company's fractional part of the stock. 3. Three merchants form a partnership ; the first furnishes $3000, the second $5000, and the third $7000 ; their gain was $3000 : w^hat were the divi- , donds? ^ns. The Ist., $G00; 2d., $1000 : 3d., $1400. 3. Three merchants, who had severallv furnished $3000, $5000, and §7000, lost by a failure $G00 : what part of the lors belongs to each, and what amount? -'1"«>^- hi,-^-; $120, $200, $280. 4. A, B and C freiglit 1 a ship Avith 270 tons. A's shipment was IXi tons, B's 72, C's 102. In a storm, 90 tons were thrown overboard: what was the loss on 1 ton, and how lu .ny tons did each lose ? Ans, i loss on each ton, and A's loss 32, B's 24, C's 34. 142 SIMPLE PELLOWSBIP. 5. A ship, Valued at $25,200, was lost at sea ; there Was an insurance on her of $18,000 : what was the loss to the owner A, whose investment was to the value of i ; to B, whose was i^ ; to C, whose was i ? ^«s. A, $2400; B, $3600^ C, $1200. 6. A person, failing in business, owes A, $350 ; B^ $1000 ; C, $1200; D, $420 ; E, $85 ; F, $40 ; G, $20; he has, to nieet these amounts, $1557.50 : what will be each creditor's share ? 7. If a town raise a tax of $1920, and all the property be valued at $64,000, what will it be on $1 ? what will a person's tax be, whose j)roperty is ap- praised at $1200 ? Ans. .03 on a dollar; $38. Assessment of When taxcs are assessed, an inventory of all real *"*'■ and personal property of the whole town is first Capitation tax.taken; /and when there is the capitation or poll tax, that of' each one subject to it is ]Dut down. As the polls are specifically rated, that tax is to be first sub- tracted from the whole tax, and the remainder to be assessed on the property. To find the amount, each individual is to be taxed for his property, We find , how much the remaiilder of the whole tax is on $1, and multiply his inventory by it. General tax. In some States there is no capitation tax, and the sum to be raised for the expenses of the Government is collected from each individual, in proportion to his property. In South Carolina, this is on "land and negroes, and is called the general tax. Apart from spe(SRi tax this, there are, in incorporated towns, special taxes on houses, serx^ants, carriages, horses, etc. 8 In a certain town where there is the capitation tax, the amount to be raised is $5999. The real estate is valued at $500,000 and the personal at $300,000. There are 666 taxable polls, each of which • is assessed $1.50 : what is the tax of A, wjiose real estate is valued at $4000 and his personal property at $8000, and who j)ays one capitation tax ? Ans.. $76.50. DOtJBLE FELLOWSHir'. 143 DOUBLE FELLOWSHIP. 269. Double Fellowship varies from Simple Fellow- DouMe felio-*- ship in the iuvestment of shares in a company for^^'P defined, unequal terms of capital and time. Example 1. — A and B form a partnership for the transaction of certain business. A puts in 8300 for 8, and B $400 for 7m.; their gain is $156 : what is the shafe of each ? A's ^300 for 8m.=2400 for Im. , B's 8400 for 7m.=2800 for Im. 5200 for Im. Hero We have the joint stock of $5200 for 1 month,' of which A j)uls in §2400 and B $2800 ; thus, A is to have -iti^=-i\ of the gain, and B ^1=1^; "r, A fj of $156=^72, aud B -^ of $156=$84. li^otr. — Each partner's stock is mfiltiplied by the time of its engagement. 2. Three merchants, A, B and C, enter into part- nership ', A puts in $60 for 4m., B $50 for 10m., anJ $80 for 12m. ; they lose $50 : what is the share of loss to each ? Ans. A, 87.05; B, $14.70 ; C, $28.23. 3. Four traders form a connection ; A puts in §400 for 5m. ; B, $600 for 7m. ; C, §900 for 8m. ; D, §1200 for 9m ; they lost §750 : what loss did each sustain ? Ans. A, §60.77 ; B, $127.63 ; C, §233.38 ; D, 328.20. 4. A commenced business November' 1, with a capital of $3400. On the Ist February he associated with him B, who invested his capital of $2600. In a twelvemonths' time, they had gained §750: what i» the share of each ? A7is. A's, $476.63 ; B's, §273.36. 5. Two merchants having entered into partnership for 16m., invested as follows : A, at tirst, to the amount of $1600, and, at the end of 9m , $900 more; B, at first, $650, and, at the end of 6m., withdrew 1350; they gained §500: what was each one's part? 6. On the 1st January, A commenced business with $940 ; on the Ist February after, he associated with him B, who put in §660 ; on the Isfc June, C was ad- mitted to the firm, with a capital of $1800; at the end of the year, they had gained §992 : what was the share of each ? 144 INSURANCE. INSUKANCE. » Insurance de- 270. Insurance is the engagement of a company to fined. protect, for a specified time, a certain property from loss by fire or other casualty. The policy. 2T1. The Written contract assuring such protection is called the policy. The underwri- 272. The persons pledged to its performance are ters. known as underwriters. The premium. 273. The sum paid for such risk or service is termed the premium. Fire, marine, 274. Fire, marine and life insurance embrace the and hie insii- . , „ , .'■,,. . . ' ranca cover all riSKS lor which policics are givcn, us^per centage 275. The per centage for which insurance against varioua. fire 18 effected, varies according to the nature of the property or its locality ; against marine or sea disas- ters, according to the strength of the vessel, the voyage, or other circumstances in such connection j against the loss of life, according to the age, health, or exposure to sickness or danger of the individual. Example 1. — What would be the premium for the insurance of a house valued at $5000, against loss by fire, for lyr., at | per cent. ? (5000-r-2=25.) Ans. $25. Note. — We simply divide by the denominator of the fractional per centage, ^, and as it is a per cent, diviisor, pont off two figures in the result for cents. ' 2. What would be the premium for insuring a ship and cargo valued at $37,500, from Charleston to Liverpool, at 3^ per cent. ? Ans. $1312.50. OPKRATION. 37500 3^ 112500 18750 1^2.50 KXPLANATIOK. We multiply by 3, and to its result add the re- sult of the multiplication by ^ (37500-^2=18750). 3. What is the insurance on a store and goods valued at $15,000, at 2i per cent.? Ans. $337.50. rROFIT AND LOSS. •4. A merchant owns 5 of a ship, valued at $24,000, 5ind insures his part at 2V percent.: v,]int does he pay for the policy ? ' . A/^s. $450. 5. A shipment of goods from Liverpool, valued at -C.334 10s. Gd., has been insured at IJ per cent. : in- cluding cost of policy tiaper, $1.25, what must be paid in American money, ealcnlatinj^ the pound at ^4.87 ? [ " Ans. 646.80. . 6. I have effected an ihsurauce on 'a friend's house in Augusta, Ga., for $8000, at I percent.; furniture $1500, at 1\V per cent. ;1 stable, hor.scs and cai-i-iage $2000, at 3 per cent. : what is the insurance i* Yl , , _ / . Ans. ii;108.75. 7. A yessel and cargo! worth $65,000, are damaged to the amount of 20 per cent., and there is an insui->- nnce of 50 per cent, on' the lo/>s : how mucR will the owner receive ? Ans. $6500. 8. -What will l>e the annual premium for insuring '$5000, for 12 years, the iifv- of a man 40 years old, at the premium of fl ('(' p;':- c-nt. ? ■PilOllT AND. LO^-6. STG. / V-;//i and J.oss is the name applied to the procor;? pioflt j tised in mercant^e transactions to ascertain the favorable *^'''^'"''' -or unfavorable result of a financial operation. It is a eoai- bination of rules already known, and very important in this connection as showing a method practically valuable to assure a certain per cciitage of profit or loss. Example 1. — Bought 800 lbs. of sugar a* 9 cents per lb., and sold the sarac for 12* cents per lb. : what was the Am. $10.50. EXPLANATION. Here, after the multipli- cations of purchase and sale, the difference is found ho- twccn t'- - '"- » for answer. profit ?, OPE RATIO.?;. I^urchase. ^Sa^e. 800 300 9 m 27.00 :)600 150 :i7.50 27.00 10.50 H 146 PROFIT AND LOSS. 2. Bought 250 yds. of cloth atl;225. per yd.!: what musi it be sold for to gain 7 per ceni. ? ' Am. $601.87, EXPLANA'ilON. . HaviDg found the cost, it is neccF^ary to Multiply that amount by th<; gain pro- posed, and we find the an- swer $39.37; acidiug this ti> the cost, we obtain for an- swer, as above. ' OPIORATIOK. 250 225 1250 500 . fOO '" S5G2.5G=tlie cost. 7 gain proposed. ~W9.37V>0 . I $562^ •, , -SG01.S7 Kepetitior.. of Xote. — In the multiplication of dollars and ceats by poinf off cemin cents, let it be remembered that 4 figures, uiust be pointed, figures. .oft". The two at the extreme right are not in these calcu--' lations usually considered. The next two arc cents ; those at the left of the point are dollars. 3. Bought 14 bbl. flour for $100, and sold the same at $8 per bbl. : what was the gain ? I Ans. $12. 4. Bought 100 bushels corn for 75 cents per bushel/and sold it for'$69 : what was the loss ? . 5. Bought 58 bushels sweet potatoes at 62^- per bushel,/ and sold them at 75 cents per bushel : what was the gain " 6. Bought 60 hhds. of wine at $39 Jhhd., and sold tlj same at the rate of $50 a hhd., but disc-OfUnted 3 per ceul for cash : what was the profit ? ^ 7. Bought 325 bbls. of flour, at ^5.35 per bbl., and sold them at $6.25 per bbl., with a discount of 2 per cent, for cash : what was the profit ? \ 8. Bought 7 hhds. of bacon for §4ii0, and sold them ih[. $500 : what was the gain per cent., ok per 100 ? Ans. S8-H-. Note. — This can be solved by simple proportion, and is the same as if the question had been thus put : If $460.00 gain $40, what will $100 gain"? 460:40:100: 40=the diffocence- between 460 M [and 500. 460)4000(Ahs- $8if 3680 Eeducedby 10)~820 460 or ff, <)T ii PROFIT AND LOSS. 147 9. Bought 4hhcls. of sugar, each weighing 2501bs., for $70.00, and sold them again at 8 cents per lb. : how much was the profit per cent. ? Note. — Find the net profit on the whote, then the per cent, by proportion, as in the 8th example. ' 10. What is the profit, per cent., on a yard of cloth, bought at $4 and sold at $5 ? Cost : Gain :: Par value : Gain per cent.- $4 : $1 :: lOOpcrct. : 25 per cent. Am. Note. — The par value of an article is its first cost. Par vaiu.> a-v fiuc'd. Remark. — Here, we ascertain the differepce between a propoit.om^i the purchase, price and sain, and say, as the purchase statement. price, or cost, is to the total gain, or losa^^so is 100 per cent., or par value, to the gain or loss per cent. 11. Bought 500bbl. pork at §12 per bbl., and gave note at 6m., in settlement : v/hat was the profit, the same having been sold at $15 per bbl., cash ? 12. A merchant purchased 20 chests of green tea at $40 per chest, and settled for them with a note payable in Cm. He afterward sold them for $46 per chest,- at 30 days : what did he gain, if the bank discount were reckoned at 7 per , cent. ? Remark.— Find the cash cost of the 20 chests, by deducting 3|- per cent. (7-4-2=32-) from their cost, at $40. Find, too, the cas'li price of sale by deducting ^ per cent. (12-r-7=-i^ or Im. int.) from the price of 20 chestsX$40. The differance between the cash cost and the cash price of sale is the profit. 13. A merchant bought 25hhd. of molasses at $30 per hhd., and gave his note at 90 days. A mouth later, he sold them at ^o5 per hhd., and received a note' payable in Cm. : what was tlie profit, the bank discount being at the rate of 6 per cent. ? Remark. — Here, 2 per cent, is to be deducted from the price of the sale, as the good;^ Were 1 mouth on hand, and were not paid for until 6m. had expired; making in all 7m. interest, at G per ctat., which, for7m.= 3i percent.- 148 PROFIT AND LOSS. The purchase Erice given to nd selling price. OPERATION. 700 25 3500 1400 14. A grocer bought 75bbl. of cider at S2.25 per barrel; and gave his note, due in 6m. Two months later, he sold them at $3 a -barrel, on a credit of 4m. : what was his profit, the bank discount being at the rate of 6 per cent. ? 15. Bought, an invoice of books for $700 : how m\ich must the same be sold for, in order to gain 25 per cent. ? Ans. $875. EXPLANATION. We here multiply the purchase price by the per cent., and add .the result to the cost. The amount would have been subtracted had the per centage been 175.00 loss. 700. ^875. 16. What must 15 pipes of wine be sold for, which cost $55 a pipe, to gain 12 per cent, on the cost ? 17. What must 35 bags of cotton bring, that cost $75 a bag, to gain 15 per cent. ? * 18. Bought an invoice of figs at 12|- cents per lb. : proving injured, what must they be sold for to lose 10 per cent. ? Ans. 11 cents and 2i mills. A proposition in profit and loss solved. Note.— This is easily solved by proportion ; thus, OPERATION. 100 : 90 :: 12^ 90 1080 45 100)1125 (11} or lie. 2J mills, 100 125 100 Reduce by 25) YW=i EXPLANATION. Here, we say, as 100 is to 90 (=100—10) so is the purchase price to the sell- ing price. 19. A trader sold apples at $1.50 per bbl., and lost 10 per cent, by the sale : what was the cost ? Ans. $1.66-J-- EQUATION Oi- PAYMENTS. 1-19 OPKRATION. 90 : 100 :: 150 : 100 90)15000(lG6f 90 600 540 GOO 510 60 20. A merchant bought a piece of velvet at 85 per yard, but it being daxiiaged, he proposed to sell it, so as to lose 20 per ceat. : how must it be marjced, so that. 10 per cent, be deducted from tlie price at wliich he hud designed to sell it? 90 :80 :: 5 : Ars. Remark. — Here, it is said, as 100, diminished by the per cent, to be deducted, is to 100, increased by the per cent, to be gained, or diminished by the per cent, to be lost, so is the cost to the proposed price of sale.' '21. Bought a mule for $175 : what, so as to lose 5 per cent, on the cost, .shall I ask for it, so that I may diminish the price 20 per cent..? EQUATION OF PAYMENTS. 217. E(j:Hatwn of PaipnenU is an operation to discover E.iuation of the mean time for settlement of debts incurred at pre- payments d<>. vious dates, so that neither party interested shall suffer loss. ExAMPLK 1. — Supposing a person owes me $100, due in 30 days; 8200, .due in 25 days; and $150, due in 10 days, but wishes to settle all at once : what is the mean time of payment ; or, enuivalentl}'. how long is he to keep the money if he wishes to pay all at once ? 150 ECJUATION OF PAYMENTS. OPERATION. aiOO $200 $150 30 25 10 3000 1000 1500 400 5000 These added =9500, which, divided by 450, the amount of the different" debits, gives 21 nearly, and is the time sought. To^ equate pay- Remark. — From this, we lenrn to multiply each debt by the time at ^vhicu it falls due, and divide the product by the indebtednesses, for the mean titne. 2. I owe $500, one half payable to-day and the remainder in 8m. : when is the equated day for settlement ? A7i$. 4m. 3. A owes B three different sums of money — $150 due in 3m., S240 in 60 days, and |300 in 4m. : what is the mean time of payment ? 4. A owes B $500 due in 3m., $350 in 2m., and $200 in Im. : what is the mean time of payment ? 5. A owes B four different sums of money, as follows: $1000 due in 3m., $500 in 6m., $400 in 5m., and $300 in 2m. : as he wishes to settle these at one time, what is the equated date .? 6. A owes B $400, which is, by agreement, to be paid in 4 equal instalments ; the first in 30 days ; the second in 60 days; the third in 90 days, and the fourth in 4m. He wishes to settle all at one date : what is the equated time ? 7. A merchant purchased a certain commodity for which he gftve his notes ; one of $300, payable in 3m., and another of $350, pa3'able in 6ni. One month after the purchase, he proposes to give a single note for the whole amount : for what time must the note be written ? Remark. — The first note having 2m. of unexpired time, and the second 5m., multiply $300 by 60d., and $350 by 150d. (5m.) 8. When is the mean time for payment of $400 due in 6m., $500 in 8m., and $1000 in 12m. ? Ans. 9-^. I I • ■ • 9. Two mcrchanrs had the following bus;nc?f? tiansac-- tious : A [lurehased of ]5 a bill of goods, I^Iay In, 1860, on Sra. crodjf, for 6200 ; May 1, 1860, on 4ni. credit, for $600 ; and B purchased of A a bill of goods, May 15, 1860, on 3m. credit, for ^oOO; June M, l^<60, on 4m. credit, for $900 : when does the balance owned by B fall due? \ '^ 1 As A's dept is found to be due August 20, and TVs 8ept(imber 20, 40 days after A's is due, the question i^ when is B to pa}' the balance of $400 ? \ Remark. — Had A and B each paid their 'debts when the time was up, the question would have been plain ; but as the account is to be settled by B'« paying the balance of S400, he can keep that sum long enough, after Septem- ber 29, to equal A's holding 8800 40 days after it was due; as 8800 for 40 days equals S400 for 80 days=32,000, and this, divided by 400=80. Hence, 80 days from September 29, rs the mean time. 10. Two houses. in Savannah had the fullowing transac- tions : A purchased a bill of B, January 7, 1861, Im, credit, for §800; February 7, 1861, 2m. credit], for S566.66^; February 7, 1861, 3m. credit, for ^433.33J. B purchased of A a bill. January 18, 1861. '2m. credit, for S200 ; February 26, 1861, 4m. credit, for §1200 ; March 1, 1861, 3m. credit, for $800 ; March 26, 1861, 3m. credit, for $800 : when shall B pay to A the balance ? Ans. January 27, 1862. CABTT^R. '278. Barfcrrin the excliange of commercial values. Bartov define ^79. Though\ separately placed, it rightfully belongs to Its qucsiion=» thV solution of (\ucstions by analysis. ' are tlioso oi J'jXAMPLE 1. — How many pounds of butter, at 22 cents per lb., must be given for a chest of tea, containing 751bs., ;>t 80 cputs per lb. ? PRACTICE. OPERATION. 75 80 ' 22)v^0.OO(272if=-,V 4-1 IGO 154 EXTLANATIOS'. The value of the tea being ascertained, the quan- tity of ];»utter to pay for it is found by dividing the tea's value by the price per lb. of the butter. GO .44 lo 2. A has 300yd. of cotton bagging worth 30 cents per yard, which he wishes to exchange for corn at 75 ceat« per bushel : how many, bushels can he have ? Ans. 120. 3. A flour merchant had 200bl)l. of fiour, valued at is; 10.50 per barrel, for which a grocer gave him, in money, $1090, and the balance in Cuba molasses, at 20 cents pcv gallon : how many hhd. of molasses did he receive ? -i. What number of barrels of apples, at $1.20 per barrel, will purchase 20 cords of wood, at §3.50 per cord f 5. A merchant exchanged 1 case, 50pcs.= 1500)al. calico, worth 9 cents per yd., for G0pcs.= 1800yd. long cloth, at 10 J cents per yd.: what was the difierence that he had to pay ? 6. A trader received in exchange for 300pr. brogans, valued at 95 cents per pair, CO hides, at $l.G2i- each :' how much was the balance in his favor? pbactiok! 280. /.'.(c/<. i,- a process for solving questions, by sub • stituting for large multiplications and divisions aliquot parts, such as l, i, 1, etc, - • .i.trrtotjono'" 281. It is mainly a contracted form of rules alreadr '■'■ ''^'"'- considered, but especially of proportion. A.8 its namfe indicates, the method of solving questions proper to i\ can only be acquired by constant use. PRACTICE. 1 :>:; OrERATION. \ clollar)48 i dollar i8=i of k -^4 .12 Example 1. — Wliat is the cost of 48yd. of satinet, at 75 ceuts per yd. ? EXl'LAXATION. We here first find, by dividing the yds. by i dol- lar, the value at J dollar a yard ; and then, taking I of that answer, and adding S36^1?w. it to the sum of the yd. at 4-Hlollar, find the true result. (J+.l = t and \ dollav=.75. ) 2. ^Vhat is the value of 80yd, of furniture chintz, at 20 eonts a yard ? {^ dn}hir=20 cents.) 3. What is the price of 36 barrels of ale, at $3.75 per barrel ? EXPLANATION. We here first multiy»ly by o for the value, at $3 ; then, we take J of 36, for the 50 cents' value; and. kst, \ of 18, whicrh' is ^ the value of 50 cents. $135 4. V/hat is the value of 42yd. of woolion, at 25 cent'* pci- yd. 'i 5. What will 5cwt. 8qr. 161b, cost., at $4.20 per cwt. ? An^. S24.82; 6. What is the price of 60yd. of ladies' cloth, at 62^ ceuts per yd. '( 7. Wh:it will 250yd. of muslin cost, at 372 cents per yPiTd ? 8. What vrill 130vd. cassimere cost, at 87 J cents per yjlrd ? 9. What l>s the value of 160yd, of cotton flannel, at 12^ ceuts per yd. ? 10. What i& the value of 140yd. of broadcloth, at £1 12s. 6d. per yd. ? Ol'EU ATIOX. 36 3 11)8 of 3f 5= 18 of 1 ^= OPEIIATION, £, S. Fov £1 =140 00 For 10s. (•! £) 70 00 For 2s. (i 10s.) 14 00 For 6d. (1 2.S.) 3 10 ,\m, £227 10 EXTLANATION, At £1 per yd., 140yd.= £140. As 10 shilliugs= £i, V\c take for the 10s. ^ £140=£70. As 2 shillings =\ 10 shillings, we take \ £70=£14, which is the value ibr 10 shillings. As. the 6d.=^ of 1, or I of 2 shillings, we fake i £U—£3 10s. These values added give the result sought. 11. What will 36 bushels of wheat cost, at 7s. 6d. per bushel? ■ yl«s. £13 10s. 12. What is the cost of 3cwt., 3qrs. 211bs., at £3 15s. Gd. per cwt. ? OPKRATION. £3 15s. Gd. . 3 11 G G cost of 3 cwt. ^ OCWt. = 1 17 cost of 2qr3. i 2qrs. = 18 10^ cost of Iqr. h Iqr. = 9 b\ cost of I2|lbs.* i 12ilbs.= 4 H cost of Gi lbs. An^, £U 17s. 3^. 13. What will 18yds. cloth cost, at 14s. Gd. per yd. ? U. What will 24y(ls. cloth cost, at 12s. per yd. ? 15. What will 2cwt., 2qrs. 18lbs. sugar cost, at 87.50 per cwt. i* 16. What will Ggajs. 3qts. oil cost, at $2.25 per gal. ? y% «frecto<1. EXCHANGK. Rx«iiijp^'<' (io- 2§2. Excliange is a convenient process for the tran?;- ""*■''' mission of funds to persons residing or travelling abroad; or to parties living in other places of the same country. n«»»ex.!ian;.'r 283. Sucli transmission is effected through the medium of a draft or bill, which is commonly an order from .1 bank to an agent commercially in its connection; or from a banker or broker, whose business is the transfer of such paper to some house with which he has financial under- standing. ;^'!.J*!fV'in'''"'^ 284. When the parties are residents in the same coue- f ry, such draft is called an inland bill ; otherwise, a foreien bill. i»rfig» bill. * 1 quarter— .251 bs. EXCHANGE. 155 285, The value of the pound sterling (Art. 104), is Tim vaiu« of 1*4.44,4 {=S^), at par. But in settling accounts between {,';,y'°"°'' ''^'" this country and Great Britain, or in the purcha.«e of hillf» of exchanire, it is seldom that Britisli currency is to be had at that rate. Usually it is at an advance of 5 to 2*0 ^^^e!'""^'" """" per cent. Sometimes it is at a discount, or below par, or" )>elow the intrinsic value of $4.44,4 mills. a§6. A]d. to a decimal fraction of a pound. Ans. 0.89= 83.95,5m. OPERATION. 16s. 2J,d. 0.8 9 0.S9 Xotc. — In this example, the shillings being even and the farthings less than 12, the addition of the 5 and tht 1 are not required. 10. Change 10s. to a decimal fraction of a pound. Ans. 0.5= 82.22,2 m. EXCHANGE. 157 11. Change 16s. Td. to a decimal fraction of a pound. 12. I wish to remit to London a bill of exchange for £90 : how much must be paid for the bill, when exchange is at 9 1 per cent, premium ? • Ol'KRATIOX. ^90 Add 9^ 8.55 £98.55= Am. money, $438. 13. Imported from l'vnp:;land a bill of goods amounting to £75 6s. 3d. : what, iu American nioney, will that be, when exchange is 5 per cent. ? 14. My factors in Mobile shipped to Liverpool 55 bale.s of cotton, weighing 22,500 pounds; it was sold r.t Lid. ( I?. 3d.) per lb. The freight and other charges amounlrd to £150 15s. Having sold the bill of exchange, which was received in payment, at 9.i per cent, premium, how much would the amount be, in American niouey? Jns. $6724.44. 15. I have requested my factors to transmit to my banker, in Paris, a bill of exchange for' 4644 francs : 'at the par of exchange, what will be the amount in Amcricaa money ? Rkmark. — French currency is computed in francs andi'renc}: funm- centimes; the franc being equal to 18 J cents, American *^^' money, and each centime ^-J-y part of a franc. To change proncb nx francs to American money, at the par of exchange, wcd divide by 100 ; rouoyr"' '^'"" or, what is equivalcntly such division, we point ofl' 2 decimals and have the answer in dollars and cents. 16. Change 25 francs to American money, at the par of ♦exchange. OPERATION-. 25 181- 200 25 12J ^1 [■ the multiplication by J. Ans. $4.68 1. 15^ EXCIIANCr;. 17. Change 325 francs to Americim money, at the par of exchange. 18. ('hange 560 francs to American iuono3;, at the ]>ur of exchange. 19. What will be the value of a bill of exchange un Paris, in American money, for 8550 iVancs, the rate of premium being 5.12 francs per dollar? yl»s. $1(585.20. JV(/fij. — Hero, the exchangc=5fr. 12 centimes, ha.'^ to be addeil. 20. What will be the valu?' of a bill vi exchange on I'aris, in Aniorieao money, for ,1500 francs, there being a discount on French exchange of 5.13 per dollar? A'utr. — Here, the exchange has to be subtracted. To uhange Kkmark. — When it is required to change American ""ench"nu^ey. i"'^°^'y to French, the process is reversed. 21. Change S4.G82 to French money, ut the par of exchange. 22. C'haDge $870.75 to francs. Ans. 4644. OPEKAilON. 18i 870.75 4 4 75)3483.00(4644fr. Ans. oOO 4S3 440 330 300 . 300 300 Xotc. — For the convenience of the division, both divitor and dividend are changed to 4thii. This enables us to avoid the fractional f , but does not change the value of " the result. 23. Change S75.0 to francs, adding premium 5.15. GUAGINii. loll 24, Wliat will be the value of a bill of exchange on Paris, in American money, for 8550 francs, the premium being 5.08 ? GUAGTNG. 287. Gv.''(?i""""(i.li- cubic inches in a gallon liquid measure), carried to f(mr_ "■'' decimal places, is .0034 ; and this decimal ^multiplied by the square of the mean diameter, and by the length of cask, gives the contents in gallons. 2. What are the contents, in wine gallons, of a cask whose length is .36 inches, and whose head and bung diameters are each IG and 19 inches '{ i\» ti' ni'.-n--- o ;i iMlI\<'i Note. — To measure curved vessels, simply multiply the length by the square of the mean diameter, then by 34, and mark off 4 decimal placfs. 3. What arc the contents, in beer measure, of a cask whose length is 4G inches, and whose head and bung diameters are each 24 and 32 inches!'' KoU'. — Tn bocr measure, the decimal .7854 is divided by 282 cubic iucht.s=:the beer gallon. (Art. 94.) 4. What are the contents, in bushels, of a hhd. whose length is 48 inches, and wlin-c bead nnil bmi"- diniiftprs are 34 aud 42 inches '{ .Voii. — In dry me;isurc, the bushel equals 2150 cubM inches. 5. What arc the contents, in bushels, of a cask whoe* length is 44 inches, and whose head and bung diameters are 32 and 40 inches? (). What are the contents, in wine gallons, of a cask whose length is 38 inches, and whose head and bung diameters are each 15 and IS inches ? 7. In a barrel 30 inches deep, and its diameter (od« third from the top) 20 inches, how many wine gallons? 20x20=400. 400x30=12000 34 48000 36000 408000=408 lOgal. J\fote. — Multiply the diitmeter (one third from the top) TONNAGE. 161 by itself, and this by the depth. Then multiply by 34 and cast oflF 4 figures for decimals, 8. In a barrel 34 inches deep and its diameter 18 inches, how many wine gallons ? 9. In a barrel 28 inches deep and its diameter 16 inches, how many wine gallons ? Note. — For the guaging of corn, peas and potatoes, see 60-63 pages. " TONNAGE. 292. The Tonnage of a vessel is her measured capacity Tonnage dc- of freight. fined. The quantity that can be carried is estimated by two Two rules t<> rules ; one known as the carpenter's, the other the gov- '"*''* ^^^ *°°" ernment's measures. Example 1. — What \» the tonnage of a single decked vessel whose length is 80 feet, breadth 21 feet, and depth 18 feet? Aus. 318-,V tons. OPERATION. nage. 21 80 1680 18 13440 1680 95)30240(318-iV. 285 174 95 790 760 30 12 EXPLANATION. We here multiply the length of keel, breadth at main beam, and depth of hold, in feet, together, and divide by 95; the quotient is the number of tons. 162 TONNAG i To measure a Note. — ^^For a double decker take h of the breadth at double decked ^j^g main beaip, for the depth of the hold, and proceed as above. 2. What is the carpenter's tonnage of a single decked vessel whose length is 90 feet, breadth 25 feet, and depth 19 feet ? 3. What is the carpenter's tonnage for a double decked vessel whose length is 200 feet and breadth 38 feet? Am. ] 520 tons, 4. What is the gOTernmcnt's tonnage for a vessel of single deck v.hose length of keel is 80 feet, breadth at main beam 21 feet, and depth 18 feet? EXPLANATIOK. From the length we tak< f of tlu^ breadth (SO— 12|), and having multiplied the remainder by breadth and depth, divide by 95 for the result required. Government measure exam- ple. OPERATION. 80 12i 67f 21 67 134 H 14151 18 ! 11320 1415 7i 95)254 7 7i(268il^ tons. 190 • "' 047 570 < < / 7<.0 17i flow to meais- . Kemark. — By government rule, for a' single decker, take length, in feet, above the deck, from the fore part of the main stem to the after part of the stern post; the ANNUITIES. 16;) breadth, at the widest part above the main wales, on the outside ; and the depth, from the under side of the deck ^lank to the ceiling in the hold. "). What is the government tonnage for a vessel of the 3 "capacity as the one given in 3d example ? Remark, — To measure a double decked vessel, take the to medsure a length above the upper deck ; for the depth, take \ the do"W« d^ci^^'- width, and proceed as before directed. 6. What is the carpenter's tonnage for a single decker whose length is 100 feet, breadth 25 feet, and depth 20 feet ? 7. What is the government's tonnage for the same r:i;.;icitv? Ans. 447-iV tons. ANNUITIJ]S. r 293, An Amiuity is a sum of money payable to a person Annuity «^- for a certain term of years ; usually, for the life time. ^^- \ 294, An annuity not paid at the stipulated date is said Aif annuity in to be iu arrears ; when it is not to commence until some 'i"p'»''s; future time, it is called a reversionary annuity; but when j^^^ reversion; its payments have commenced, it is said to be in posses- j^^ g^g^j^^jj sion. 295, Annuities are often bought and sold, as other Bought arid • 1 1 gold. commercial values. ■ j 296, The sum of annuities, such as rents, pensions, tj,^, amount of salaries, remaining unpaid, with the interest on each, is "■'^ annuity, called the amount of the annuity. \% 297, To find the value of an annuity iu arrears observe ,p^ ^^^^ j,,^ simply the method to ascertain an amount at interest ; but, talue. for an expeditious way to discover the same, we give the followiusc : i«;4 ANNUITIES. Tal'lo shovriufi ciertain TABLE, 29$. Showing the amount of the annuity of $1, £1, etc., at 4, 5, and 7 per cent, compound interest, for any number of years not exceeding 20. Years 4 per cent. 5 per cent. 6 per cent. 7 per cent. 1 1.0000 1.0000 1.0000 1.0000 2 2.0400 2.0500 2.0600 2.0700 3 3.121G 3.1525 3.1836 3.2149 4 4.2464 4.3101 4.3746 4.4399 5 5.4163 5.5256 5.6370 5.7507 6 6.6329 6.8019 6.9753 7.1532 7 7.8982 8.1420 8.3938 8.6540 8 9.2142 9.5491 9.8974 10.2598 9 10.5S27 11.0265 11.4913 11.9779 10 12.0061 12.5778 13.1807 13.8164 11 13.4863 14.2067 14.9716 15.783S 12 15.0258 15.9171 16.8699 17.8884 13 16.6268 17.7129 18.8821 20.1406 14 18.2919 19.5986 21.0150 22.5504 15 20.0235 21.5785 23.2759 25.129Q 27.888(P 16 21.8245 23.6574 25.6725 17 23.6975 25.8403 28.2128 30.8402 18 25.6454 28.1323 30.9056 33.9990 19 27.6712 30.5390 33.7599 37.3789 20 29.7780 33.0659 36.7855 40.9954 E.iplonation of method. Example 1. — "What is the amount of an annual pension of S500 in arrears for 16 years, at 7 per cent, com- pound interest ? Am. 813,940 Note. — The table shows the 7 per centage of 16 yeari to be 27.8880 ; this multipled by the amount, 500, gives 139440000=813,940. 2. What is the amount of an annual salary of 81500, which has been in arrears for 10 years, at 7 per cent. ? 3. What is the present worth of an annual pension of $120, at 6 per cent., to continue 3 years ? Am. $320.76. Explanation.— We find by the table that 8120= 8382.03; this divided by the amount of ?1, compound interest, gives the quotient of present worth. This in evident, since the quotient,* multiplied by the amount ot ALLiaATTOX. ' ll^f ^1 for 3 years, at compound interest, is $382.03. By reference to the tabic under the article, Interest, the amount of ^1 at compound interest, is seen to be $1.1910; then 3382.03 divided by it gives S320.76. 4. "What is the present value of an annuity of $300, at 7 per cent., to continue 10 years? 5. What is the value of a pension of 890 per annum, la arrears, for 12 years, at 7 per cent. ? 6.. What is the value of a ground rent of S200 per :innum, for 8 years, at 7 per cent. ? » 7. For what can I buy with ca.sh, an annuity of $300, to continue 5 years ? 8. If you lay up $50 a year from the age of 21 to that of 60, what will be the amount at compound interest ? ALLIGATION. i99. AlU 14, IG, 22 and 28 shillings a gallon, in such form that the mixture shall be worth 20 shillin.2;s a gallon. OPERATION. How tofiod t)ie worth of mix- nirog. 8 2 4 128 6 8 gals, at 148 2 " ♦' 16s 4 " " 22s 6 " " 28s Ans. EXPLANATION. Here we write the prices of the different wines in order, in a vertical form; and each price that is lees tbau that of the mixture we connect by a line with one or more of the prices that are greater than it. We then write the difference between the required price TARE OR ALLOWANCE. 1 67 and each of those that arc less, by the side of the larg'er one, with which it is connected; also, the difference between each larger price and the price of the mixture by the side of the less, in its connection. The difference standing by the side of each price is the quantitj' requi^od to form the mixture. * Note. — By calculating the mean prices as given above= p,.ooi 400, and dividing by the g4llons=20, we find by the quo- tient 20 tlio proof of the correctness of the calculation. 2. A grocer wishes to mix 4 different kinds of wine, to obtain a mixture that shall be worth 81 per gallon. The first kind is worth 75 cents per gal. ; the second, 50 cents ; the third, 81.25; and the fourth §2: how many ;r:illnns must he take of each kind ? •J. A grocer wishes to mix four different sorts of tea, worth 5s., 6s., 8s. and 9s. per lb., so as to have a mixture of 871bs. worth 7s. per lb.: how much must he take of each kind ? Am. 21!,' lb. 4. A goldsmith has 4 different kinds of gold, 14, 16. 20 and 22 carats fine ; he wishes to obtain from them a mixture 1 8 carats fine : how many ounces must he take of each 't .5. A drover has sheep worth $2, S4, 86, 88 and $10 each : how m my from these must he take to form a flpck of SO wortli $11 each? 6. A grocer wishes to mix 3 different kinds of su;.;ar, to obtain a mixture worth 8 cents per lb. The first kind is worth 7 cents per lb.; the second, 6 cents ; and the third, 10 cents: how many of each must he take to form the mixture ? 7. A grocei* wishes to mix 4 different kinds of coffee, so as to have a mixture worth 12 cents per lb. The first is worth 8 cents per lb.; the second, 10 cents ; the third and fourth 14 cents : what quantity must he take of eacli ? TARE OR ALLOWANCE. JJOiS. fare is a mercantile term to express an allowance ,^^^^ ,i*h„.,,<. made in the transfer of goods for known or presumed •deficiencies. 168 TARE OR axlowancj: An allowance .it 303. An allowance is made at tlie port of entry on ports of entry, (juti^ljle goods, because boxes, etc., containing imported articles, are not considered any part of the commodities subject to tariff charge. Allowance by 304. Ecsidcs this allowance or tare, Reductions arc an e. ^^de by wholesale merchants on boxes, etc., in the inter- changes of trade. The allowance 305. When goods are not actually entitled to theb( of draft. allowances, a deduction, comnjercially known as draft, h made for waste. This is 9 lbs. on the ton, and proportion- ally for sniaaller weight. j Allowance for 30(». On liquor in casks there is usually the allowance ikjuor*. pf 5 pgj. ^■Qnt. for leakage. On liquor in bottles (particu- larly . porter and other fermenting liquors), for breakage", there is an allowance of 10 per cent. fcrofK weight; 307. (jross weight is the whole weight of the good5 Ni-t woicht before allowance is made ; and net weight is what remain.* after allowances are deducted. Example 1. — What is the net weight of 10 boxes of candles, each weighing 501b8., there being a tare of 41bh. on each box ? ' OPERATION. 50 10 10 4 . Note. — Examples in Tart- are performed by multipli- cation, subtraction, and in some cases, by proportion. r)001bp. 401bs. tare. 40 460 net. 2. At §156 per hhd., what will 4hhds. of sugar amount to, allowing lOlbs. per cwt. tare on the gross weight of 19cwt., 3qrs. 151bs. ? 3. At 16 cents per lb., what is the worth of 4 bags of coflFee, the gross Veigbt of which is 6501b8., allowing 2 lbs. tare on lOOlbs. ? 4. At 8 cents per lb., what is the cost of Shhds. of sugar, weighing gross, Icwt., 3qrs. 14lbs.; 2ewt., 2qr8. 131bs. ; 3cwt,, Iqr. 151bs., allowing tare of 91bs. in each cwt.? 5. .At §15 per cwt., what will lOcwt., 3qrs. 121bs. sugar cost, allowing tare lOlbs. per cwt. ? ATo/e. — Deduct tare from gross weight, and find answer by proportion : if lOOlbs. cost $15, what will the number of lbs. net weight cost ? SINGLE rOSlTION. 169 SINGLE rpSTTION. 308. Single Position is a jurocess to ascertain the true Single Position answer to a question, by assinnuiL^a certain number as the * °® rightful one, or as Icadin-i tn i:. ExA.\iiB.E 1. — I liave a c<'ifain number of sheep in my pasture, and if the number weio increased by J and J, the whole would be 06 : what i.s the number '( Ans. 36. OI'KI.'A I |(l.\. Suppose 12 is the number, then by adding \ of 12=6, and i ot 1.2=4 to 12=22, wo find the supposed number was the true one by the proportion, 22 : 66 :: 12 : 36. Remark. — The operation with 1 2, which gives as ab6te, 22, enablps us to obtain the evidently true answer in the fourth term of the proportion. 309. The examples in siii;^le position can be analyzed without trouble. Thus, in the above c.\;imple, the number to be found is fractionally ^=1 : i of f=^ and i=f ; which added=-^^'-=UG. Hence, ii -';.'•-:=()(), -,\=6, or -^ of the number; then, asf=the whole, iix6=o6. Ans. 2. The master of a school being a.sked how many pupils he had charge of, said tliiit if he had as many more as his present nuaiyer, \ as many more, J and \ as many more, he should have 296: what wis the nuuiber '( An^. i>6. 3. A man. who was asked his ago, said\ that if -f of it were muitiplied by 7, and -g- subtracted from the product, the remainder would be 66 : what was his age ? 4. What is' the number, which multiplied by 7, and the product divided by 6, will have a (juotient of 14 't Ans. \ 2. 5. Two men have the same income. One saves i of his, but the other, by spending twice as much, finds himself, at the end of 4 years, $560 in debt: what is the annual income ? Ans. $420. 6. Throe speculators gained $2400, of which B took 3 times as much as A, and C 4 times as much as B: what was the (share of eachf 13 170 DOUBLE POSITION. DOUBLE POSITION. €> Double Posi- 310. Dov.bJe Position is a method to determine an tion defined, a^ng^yej. \)y the use of two munbers, supposed to .bo the ones sought. • Results differ 311. In this, the results vary fj'om Single Position sim"ie^PMi-^^^y ^^'^^ beilig proportional to the assumed numbers. won. Example 1. — A person having a certain sum, spent $100 more than -J- of ityand had reiiiaining $40 more than ^ of it : wliat had he at first ? OPEUAflON. Suppose, first, he had §150.0, then §100 more than •^= •400= sum spent, and 1100=remainder.; hut $40 more than ^= 795 ' ■ hence, |305=lst error. Suppose, second, he h^d $2000, then $100 more than ^= 500=sum spent, and 1500=remainder ; but $40 more than 1=- 1040 hence, $4,60= 2d error. $1500x4GO:=S690000=rlst assumed No.x2d error. $2000x306— $(310GuO=:2d assumed Ko.Xlst error. 155 ) $80000(|^51G-i^V- ^n^- lib \ 250 . ^ 155 - 950 ', 930 • 20 that is, the difl'crence of the products divided by the difference of the errors, ^ives the result of $51()-iVs-- DOUBLE POSITION 171 KXPLANATION. By the cxiimplo it will be noticed, that 1st, having Process oi yiipposed two nnnibers, wc proceed with them accord-^'''"""'*' jng to the question ; 2d,' that having compared each result with that in- the question, and calling each dif- ference an error, we nuilli])lied the 1st :)ssunied "^i^n- jlj^exarnpu-" ber by the 2d error, and the 2d assumed number by the first error; andJkl, that wc divided the diiference of the products by the dino'-"'> ■■■ of ■(li'^ r>i-ror.. fny t],.-. true answer. Remark — Had one assumed numbq^ been too great whon iho w and the otficr too small, we should have divided the be?^*^are' t™ Piim of the products by the sum of the errors. |_«^ff^i ^^ '«'* ■J. Three perilous speaking of their ages, B said.that' was 10 years older than A, and 0, that his doubled both of theirs: what were their respective ages* the united sum being lUO ? Afi^. A's 20, B's 30, C's 50. o. What number is that, which being divided by 7, .'I'ld the quotient (liininislu .1 by 10, throe times tlic iuainder is 2-4i' ' J.ns. 126. i. Two clerks haw .;.. .j^mo income; A saves } of Ills yearly, but J5, by improvidently spending 5>150 per annum more than A, at the end of 8 years finds himself 8-400 in d^bt : what are their incomes, and hut the annual expenditures of each ? Note. — First, assume that each had S200; second,'. -iiO ; then the ei-n>rs will bo 400 and 200. • J/N. e; . sexp-.^00;..B's§450. 5. A ])lanter purchased a number of hcraes, mules and cows for $ii340. lie paid for each horse $50; tor > ch mule f hh much as for a horse; and for each ^o\v ^ of tlie price of a horse. There wore o times as many mules as, horsps, and twice as many cows as mules : Vs-hut was t'; '•• ••^- 'I'^r ot* each ? 172 MISCELLANEOUS EXAMPLES, MISCELLANEOUS EXAMPLES. 312. Example 1. — A owes B $iOO, tlue in 3mo., $250 in Inio.: what is the mean time of paj'ment? 2. What is the bank discount of $455.GLt, payable in 6mo., at the nato of 7 per cent. ? 3. What commission is a factor to receive on the sales of 35ijhh(l. of sugar, at ^62 per hhd.,at the rate of 2^ per cent., and 2|- per cent, for the guarantee of sale ? • 4. A, B and C jointly purchased a piece of land. A paid I of the price, B i, and C the remaining i. They subsequently determined to dispose of it, and gained $7riO : what was the gain of each ? 5. A grocer wishes to mix three grades of sugar to obtain a n\ixture wo»th 8 cents per lb. The first is worth 6, the second 7, and the third 10 cents: what must he take of each to obtain the desired kind ? 6. What is the amount of $375, at 7 per cent, interest, for lyr. Gm.?" 7. Wlut't is the amount of |!400, at 7 per cent, com- pound interest, for 4 years ? 8. A trader purchased 400 barrels flour, at $4.50 per bbl. payable in Gm. In order to gain 10 per cent, and give a credit of 8m., reckoning bank dis- count at 7 per cent. j)er an., what must he ask per bbl.? 9. There is a fish whose head weighs 141bs., his tail weighs as much as his head a«d -i\ as much as his body, and his body weighs as much as his head and tail : what is its weight? Ans. 801bs. 10. A, B, and C form a partnership. A puts in $500 for' 3m., B S600 for 2|-m., an.s. S1263.23. 13. At 12 per cent, advance, what will S750 be in £ s. d.? 14. A watch sold for $60 lost 12 per cent, by the ' sale : what was the cost? 15. At £'.)o per cwt., what is the value of 611bs. ? 16. Wiiat is the interest of $2650, at 7 per cent., for 90 days ? • 17. What is the government tonnage of a single decker whose length is 50 feet, breadth 12} feet and depth 10 feet? 18. What are the contents, in gallons, of a cask ( whose length is 44 inches, head diameter 28 inches, and bung diameter 86 inches? 19. A owes B $800, payable July 4, 1863, and B ■ owes A $60(>, payable October 6, 1863 : when is the equated time for settlemQut, and what should A pay at that date ? RADICAL ARITHMETIC, PART SIXTH. INVOLUTION inToitttion de- gl3, Tnvolution is the process iised to raise a. iiura- *"° ber to some required power; tlui's, 2x2=4, the 2d power or square of 2 5 2x2x2=8, the 3d power or cube of 2. What the pow- 314. The power is the product of a number multi- '^^^ plied by itself; Tke intiex or 315. Sometimes the power is indieutcd by means *sponeat. ^^ ^ sraall figure placed at the right hand of the root, slightly elevated, called the index or exponent; thus, 3x3 is written 3°, and is read the. square oi-2d power of 3. ' The root, or bar BIS. Tlio tsumber at the basis of the operation *is isl powerT' known'as the root or the 1st power; when once mul- equarc; ' tiplied, it is known as the 2d pow^cr Or . Square ; when cnbe. twice multipfiod, as the 3d power or cube, and so oh. What the ^ 3li7. The squarp of a number consLsts- of twice as cube^coasist of. JT^s'^y figui'GS as the root, or of one less than twice as many; the cube of a number of three times as many as the root, or of one or two less ihan three times as many. ... , _ S18. A short horizontal line over two or more VmCUlUin; ^ . ^^ , • 1 IT- ,1 • figures IS called a vinculum; and, like a parenthesis, Paren.aesis. jiKjieates that the numbers so connected are subject to a similar operation; thus, 8+2x8. is the same as (8+2^)x3=30. , . ' INVOLUTION. 175 8i9t A numbt^r already raised. to a power is involved The index mui- by multiplyine; its index by the index of the po.wcrJiP^.'^'^^y*''^"- to wliich i-t is to be raised.- 320.- A vnl<>:ar'fraetion is involved by involving»the How a vulgar nlmierator and denominator separately; thus, the SdyoWed. power of i i8 = xxxV-=-^i-L=30l. 12. What is the value of. lU* ? 18. \Vhat is tlje cube of 3 ? Ans. 27. ' • ' ■ 3x3x3=27. 14. What is. the cube of 4? « l.i. What is the cube of 25 ? ♦ 16. What is the S(piare of IG-J ? 17. Wiiat is the square of .25 ? Ans. .0625. 18. How many figures are in the cube of 99 ? Aiis. 6. 19. How many figures arc in the cube of 243 ? 20. How many figures arc in the 5th power of 99 ? 176 EVOLUTION. 323. A TABLE OF POWEKS. 1st, 1 2, 3 4 5 6 7i 8 » 2d, 4 9 16 25 36 49 64 81 3d, 8 27 64 125 216 343 512 , 729 4th, IG 81 256 626 1296 2401 , 4096 6561 5th, 32 243 1024 3125 7776 16807 32768 59049 6th, 64 729 4096 ' 15625 46656 117649 262144 . 531441 7 th, J 128 2187 16384 78125 279936 823543 2097152 4782969 8th, 266 6561 65536 390625 1679616 5764S01 16777216 43046721 8th, .')12 19683 262144 1953125 10077696 40353607 1342177*^8 38742048* 10th, 1024 69049 1048676 9765625 '60466176 282475249 1073741824,3486784401 EVOLUTION. Evolution de- fined. iadicated. The index of the root. 324. Evolution. — a process the inverse of Involu- tion—is used to find the'root from t]ie giveo power, and may be defined the Extraction of^oots. Howtherootis 325. The-root is indicated by the employment of what is called the fractional index, or by this symbol, which is known as the radical sign. 326. A figure placed above this sign is the index of the root, and is the same as the denominator of the fractional index." When a number is not given in connection with the radical sign, 2 is to be under- stood. A power and a 3'^'S'- A power and a root can be indicated at the root indicated gj^j-^e time by the index and radical sign ; thus, together. o^g5^32^ and is the cube root of the 5th power of 8. 4 a u im effect '^'^^' Some numbers cannot be extracted, feuch pTvversf are called imperfect powers, and their roots irrational, surd numbers, radical or sur i numbers. For convenience,,the term radical is employed, as before stated. Perfect powers; 329. Those are known' as perfect powers that can tutionai roots. %q extracted, and their roots are called rational. The first ten 33©. The first ten numbers and their squares are, numbers and j 2, 3, 4, 5, 6, 7, 8, 9, 10, the. squares. ^, ^, ^. ^^>^ ^J^ ^^^ ^^^ ^^^ g^^ ^^^ explanation of 331. The numbers in the first line are the square aboyenurai.ers. j,QQt8 of those in the second; the numbers m the second line are called perfect squares. EVOLUTION. 177 u:'!uRAT10N. 10 2i(32 9 62)1 24 12i Example 1 . — What is the Bouarcroot of 1024? Ans.B2. Process ox- ^ plained. EX I" L.\ NATION. ^ Wc first point off tho - number into periods of two figures each, as We wish to find the squares of the tens andhundreds. We then find the i^rcatest square in the 1 , ■\vliicli=3 tens or 30. Squiiring tho 3, which gives *.i .hundred, wo place the 9 beneath the hundreds aijd, subtract; this takes away the square of the tens and leaves 124. , Next, wo double tho divisor, which js the root already found, and divide this remainder — without tho right hand figure — by it, and have in the quotient, the uniks-figure of the root. Annexing this figure to the increased divisor, we multiply again, and find thudesiretl result. Note.— A similar course is to be pursued should there be more iii-ures. ex- prri.ssiou of evo- To extract the square root of a number is simply a simple < to resulve it into two equal factors ; that is, to find aiution""" number Aviiich, multiplied into itself, will produce the "iven number. 2. How large a square floor can be laid with 576 square fdet of boards? OPKKATION. EXPLANATION 576(^24 Here, the area beiu 4 given, we are to find the length of one side. 41)176 176 Remark . Fie. 1 To make the operation intelligible, we form a square whose sides shall be 2 ten8=20 feet in length. T4ie area of this square, 20x20 =400 square feet, and this number taken from 576 leaves 176 to be employed in the re- quired enlargement. This is made on the 2 sides, as in fig- ure 2d (though it could be made on the four), and their breadth is alike, oeing twice 178 ^'\ \^^ EVOLUTION. When the pro- duet of divisor is in excess. 4 ft. the tens^f the/oot=4 or 40 feet. Now, 176 divided by 4i.i^'ites 4 feet, Avhich,' added to the trial diyisoE, 4(J==4f, and it? the entire length of the two 'sides, and 44x4=176; that, is, the length of the addi- tion multiplied by ite breadth gives its area. Squaring the sides, eacl. being ^4, shows 576, and proves the sum. Tfhen tlie product of the divisor by a number is in excess of the divi- dend, mak© the quotient figure smaller. When tbe num. . When the given num- ber is not a per- ber is not a perfect square annex ciphers for new ^ periods. When the trial The quotient figure will be a cipher when the trial divXnd''"^'^'^^^''*^^'^^'' '^ ^i'^-'ater than its dividend. 3. AVIiat is the square root of 3600^ 4. .What is the square root of 3^^0625 ? 5. What is the square root of 5764801 ? 6. What is the length of one side of a lawn which cont.'jns»Oi acres, or 400 rods, if made into a square? 7. What is the square root of 15025 ? 8. If a square field contains ^400 sl][uare rods,, what is the measurement on each side? Ans.SO. 9. What is the square root of 15G.7325? A number both REMARK. — A ntlniber, partly integral and partly mtegrfiiandde-jecinKvl, is extracted in the same way as a whole number; the first point, in such case, must be placed over the units and extend both right and left; thus, 156.7325. The num>,er of The number of integral figures in the root is as Inu^'roof^^^^^^^y ^^ there are periods of integral figui'es in the power; and for each period of decimals in the jiow.er, there Will be a decimal figure in the root. 10. What is the square root of 3371.4207 ? 4.ns. 57.19. • 11. What is the square root of .25? 12. What is the square root of 1.44? Aiu. 1.2. 13. What is the square root of 1 ? Am. i, APPLICATIONS 'IN SQUARE ROOT. 170 Eemarkv — We reducG a Vulgar fraction to its sim- To find the root plest form, and then take the root of tl\e numerator ;',|"„j\''"'s'*'^^'"^° and denominator gopai'atoly. 14. W hat is tttq square root of 30} ? . * ' ' ' Eemark. — When either term, after being reduced, .when either is an imperfect square, -vvo change the fraction to a Jforfect, s^qiwre" decimal, and proceed by directions already given. 15. What is tlic square root of i? Afis. .8Gi). 16. What is the square 'root of i-J^? Ans. f. 17. W!i;;f is nio (square root of i + f+l—- iV? v/l+l+i-T^6=n=f=ll. Ans. 18. W!i:ii i.. ..lo ditYcreuce between v/9 and 9" ? 19. What is the difference between v^lG and •v/9 ? 20. Wiuit is the ditferchce hetweern -v/g^- and P ? 21. What is the dilfereuce between ^/4 and s/Q ? , ' ' ' ■ . Ans. 5. ■ 22. ^bat is the sum of >/30i and 272i ? 28. Wlia.t is I he square root of n/'980} ? , 24. What is the dift'erence between v/81 aud 8P '^ Ar'PLTCATlONS IN SQUARE KOOT. ui;f[nition.^. I i 332. A square is a figure witli Definition of I ij four equal siJcs. and four equal •'^'^'"^''°- H , li iuiirlcs, the angles being where the B R bidea jjieet. 180 APPLICATIONS IN SQUARE ROOT. The vertex. ^ right angle. A parallelo- gram. 333. The point of meeting is the vertex. 334. The sides or lines of a square being perpen- dicular to each other, make each angle a right angle. Example 1. — If an acre of land be laid out in a square form, what will be the length of each side in rods? 1x4=4x40=160. ^ns. A parallelogram is a figure which has its opposite ., sides of equtil length, and its opposite angles equal. In the figure, two para- lellograms are described. A triangle. To find area of an irregular field. 2. 1)1 a room 16 feet long and 11 feet wide, how many square feet are there? Ans. 176 feet. A triangle is a plain figure of three sides and three angles. The area of an irregular field is found by its being marked into tri- angles. 3. What is the area of a tria^ngular piece of land, one side of which is 40 rods, and the distance from the corner opposite that side to the other, 20 rods ? A71S. -2/x 40=400 rods. Definitions of a A right angled triangle has one of its angles a Mgie^aud' its' right angle. The side opposite the right angle is the hypothenuse, the lower line the base, the other the per- pendicular. lu a right angled triangle, the square of the hypothenuse is equal to the sum of the squares of the other two sides. This is practically apparent by the following diagram, in which the small squares on the hy- pothenuse, 25= those of the base, 16+ those of the perpen- dicular, 9=25. parts. What the square of the hypothenuse equals. Base. APPLICATIONS IN SQUARE ROOT. 181 / 4. In a vigllt Diacram and angled triangle, ^^^iCJI^^^' Avhosc base is 4 sq"='re of iiy- yjirus and per- pcnclieiilai'8, \;;hat is tlio hypotlicn- usc i' * foot long, striding in a sLroot SJ feet wide ? APPLICATIONS IN -SQUARE EOOT. i('S eiieumfer- cnce. TJ)e divisions of a cirjuirlfor- «nce. •1 (-•liorrt. A diamrt<>r. A rndius. 4 tangent. 8. If a line 125 feet long will reach from the top of a tower, 100 feet high, to the opposite side of the street, what is the width of the street ? 9. What is the height of a pine, tlie line of the hjpothenuse being 100 feet, and from the foot of hypothenase to the base of the tree f)7 teet !* 33-:?, A circle is a plain figure whoso c'.en trc is every- :"'•-.■.-. V-.. ._„^„^,- ^^'■'^^^''^ equally distant y'""^^ i "'■•< ' ' from tlie bounding curve, called the cir- cumference. S36. The circumfer- ence is cliYidod into 36ii equal parts, called degrees,,, and desig- nated by a small -ci- pher slightly' elevated at the right of the number; thus, 360°. The semi-circumference is this equally divided, or 180°; the quadrant one fourth, or 9. The line which passes through the centre of a circle is :i diameter, as B E. ' 34€;. A straight line from the centre to the circum- ferenvv .s a rridius, as G A, G C, G E . 341. A straii!;ht line, as A E, which touches the cifciiiiifci'cnce m one point,-A, and cannot touch it elsewhei'e, is a tangent. ■ ExA^ViPLK I. — What is the area of a circle whose diameter is 6 feet and circumference 1'.) feet ? ' ■ ■ , Ans. 2Si feet. To find area of a circle. uPKRATION. i Diameter •^ Circumference Or r„ Ji592 9 ' t8":i7432§=328i. =3 27 EXPLANATION. We here, to find the area, multiply-^- the di- ameter by ^ the. circum- ference, or we square i tlie diameter and multi- ply it by 3.141592dec. CUBE ROOT. 18.^ 3. What is the area of a circle whose diameter is ^Ofeet? , ^«.s. 814.15. 3. What is the area of a circle whose dianiete];' is •J8 feet -( 4. WJiat is the area of a circle whose circiinifer- -nce is 314.1592? B14.1592-=-3.1 11 5923=100, tbo- diamcL . 7 lis (which is \ 3.U1592)+100'^=7853.98, the area. Aas. 5. The dfiameter of a circle is 25 : what i^ the side >f the inscribed' sqiuu'O ? x/^l' V31.2.5=^i7.G77. Ans. 0. The diameter of a circle is 36 : what fs the side >f the inscribed square? , .. . . 7. The circumference of a circle. is 314.1592 : whnt K the side of the inscribed square? An^-. CUBE EOOT. 342, A Cube is a solidity of six equal sidea^ aud each a cui- )f these is an exact square, i • . 343'. The cube root of a number is that one 'v.iiich, -^cuber.->ot multiplied into itself three times, produces tlu- i ^ ,■,• \'hose cube is to be evolved. >'■ ' 344, The numbers in the first line of .1... )rder are the cube roots of the corresponding on ■i^ocond. The last are perl'ect cubes. 1, 2, .3, 4, 5, 6, 7, 8, !). 1, 8, 27, 64, 125, 216, 343, 512, 72'J .. EiKAMPLB 1. — What is the length of . each cubical bli.cl inir 1'1\) oi'i >, !:,^- Cube roois anif , 1 " perfect cube:^. ni the Their Humeri- t- ^ ; ; ; CT 1 . j I '•« S u UJ X UCMCTH .q FT !iXi>x9- ■ X81=7J9; is the numbi X,,lr. Tr cube is to lin bor which, n. into it.sclf ll. or njultipliti its 'Squ-iro, the given n i • I To find a oabo. :i..!ied 184 CUBE ROOT. « 2. What is tlio length of each side of a cubical block, containiiii;- lUiJU culiic inches? Ans. 10. 8. Wlutt i.s the cube root of 21024576 ? Ans.'ZlQ. , ♦ . OPERATION. To extract the 12102-1576 cuberoot. 1st liial 4ivisor=20'^ x3=1200^ 8 20x7x>]= 420 ' 7^= 49 1069 1.S024 1st dividend. 2d trial d" .. '''Xo=2187U0 11683 1^70X6X3= 4860 e»= 36""^ LM tuw uivis()r=223596 1341 576 2d dividend. 1341576 , KXPLANATION. First, We separate into periods of 3 figures each, the number, placiuL; a point over units, thousands, etc. JSecohd, VYe find by trial the greatest cube in the left hand period, and jilacing its root as in square root, sub- tract the culic v8«> from fhc left hand period, and to the . remainder annex the next periocl for a dividend. 2Viir(l, AVe s(|u:ire the root figure, and, anoesing two ciphers, multiply this result by 3 for a trial divisor; then we divide ihe dividend by the trial divi.-or, and set the quolicnt as (he f-ecoi.d figure ^.2=58. Ans. 13. What is the cube root of f^ '( 14. What is the cube root of f-J-jr ? 15. What is the cube root of -^g't ■ IG. What is the cube root of of ? 17. What is the cube root of -j-^^s '( 315. Example 1. — AVhat must be the length, depth Aud breadth of a box when these dimeuEions arc alike, and the box contains 491;] cubic feet? Ans. 17. 2. A jeweller has two small golden balls; one is 1 inch in diameter and the other two inches : how many of the smaller will it take to make the larger one ? • Ans, 8. 3. If a, globe of gold, ] inch in diameter, is worth §100, what is the diameter of a globe worth §7200 ? Ans. 8. 4. If the diameter of the sun is 886.144 miles and that of the earth 7912 miles, how many bodies like the earth ^will make one as large as the sun? Am. 1,404,928. 5. If 1000 bodies like the earth arc required to make I like the planet Saturn, and if the diameter of that planet is 79,000 miles, what is the diameter of the earth ? , Am. 7900 miles. <5. What is the length of one side of a cubical corn bin that contains 2500 buf^heb? ^«.s. 14.58 busliels. 7. What will be the length of one side of a cubical block which contains 1725 solid or cubic inches? Ans. 12. ^. What will be -the length of one side of a cubical block of granite, whose contents shall be equal to another S2 feet long. 16 feet wide, and 8 feet thick? vV32x 16x8=16 feet. Aiis. U 166 MISCELLANEOUS EXAMPLES. MISCELLANEOUS EXAMPLES. 346. Example 1. — What is the-square root of 580644 ? Ans. 762. 2. x/ 1679 316= how many? Ans lii96. " 3. How many figures are there in the cube of 99 ? Ans. 6. 4. How many figures in the cube of 40 ? 5. What is the diflference between \K8 and •v/4 ? 6. What is the difference between -^ I. and 1' ? 7. A square table of mosaic contains 80,625 square stones of erjual size : what number is in one of its sides ? 8. A person wishes to form a tract of land, containing 50 acres, 2 roods and 20 rods, into a perfect square : what will be the length of each side ? 9. A room i^ 25 feet long, 20 feet wide and 12 feet high : what number of square feet does it contain ? 10. What is the cube root of ^^ ? 11. -^84.60 45 19= how many? Ans. 4.S9. ■ 12. A pipe, f of an inch in diameter, will fill a cibteru in 5 hours : in what time will a pipe 'A\ inches in diameter fill it? . . Ans. Saimin. 13.; What is the cube «root of 28 ? 14. A general with a force of 3136 men wishes to form them into a square : how many must he place in rank and file? MISCELLANEOUS ARITHMETIC. TAIir SIXTH. ARITriMKTK.'AIi PROGRESSION. 347. A'-ithni-'M'ral R-)/r.wifn is tho torra that describes Arithmeticnl an incraa-iing or d3erja i ii „ uBce are given, the dinereiicc of the extremes /4n— v.=i«tj) bv tlie com- mon difference (6)+ 1=7. ARITHMETICAL PROGRESSION. 189 10. A man being asked the number of his children, eaid thr; the youngest was 8 and the eldest oG, and that the increase had been one in every 3 years : how many had he ? 11. A stone falling descends ] 6 J feet in the first second, and 209 1^ in the last second ; the increase of its velocity a second being 32 i feot : in how many seconds docs it fall ? Ann. 7. 12. How many times does a clock strike in 12 hoi^rs? Ans. 78. OPERA^TIOX. 1 • EXPLANATION'.. 1+12=18x0=78. I We multiply the sum of when the ex- the extremes (L+12) by one half (=6) the number of *''''"^r *"^ ^ ' / J V y number of terms. terms are 13. If a piece of land, 60 rods in length, be 20 rods ^'^on. wide at one end, and at the other terminates angularly, what is its number of square rods? Aiii^. 600. 14. The first term is 5, the common difference 8, and the number of terms 21 : what is the sum of the series ? Am. 1685. OPERATION. EXPLANATION. 8x21=168 We find, in this case, the when the first Add 5 last term by multiplyins^'^U"' <^"'"'"on the common difference (8) number of 1685 by the number of terms *^'^'""*'^ given. (21) and adding first term to the product. 15. The first term is 7, the common difference 9, and the number of terms 25 : what is the sum of the series ? 16. The first term is f, the common difference 3 J, and the number of terms 65 : what is the sum of the series ? 17. A falling body descends 16-i\- feet in the first second of time, and the increase of velocity is 32^ feet each suc- ceeding second : how far will it fall in 8 seconds ? 18. The sum of an arithmetical series is 7, the number of terms 8, and the least term 4 : what is the greatest term? Ans. 14. OPERATION. I EXPLANATION. > 8-h2=4 ; 72-4=18, and 18-4=14.| We divide the when the sum sum of the series (72) by half the numbtir of terms (4), ofaseriea.num- and subtract the gi.ven extreme (4) for answer. and one ™^ 19. The sum of a series is 204, the number of terms ^^^^^j® *^® 12, and the greatest term is 39 : what is the least term ? 20. A falling body descends 1029 J feet in 8 seconds; in the 8th second it falls 24H : how far does it fall in the first second ? Am. 16-iV- 190 GEOMETRICAL rROGRESSION. GEOMETRICAL PROGllESSION. Goomctrical 352, Coometriral Progression is the t-onu applied to that nnd. part or arithmetic which shows the increase, by niultipli- cation, or decrease by division, of a uuiuerical series through a common number. Tiie ttrms, 353 i\yQ Duuibers of these sefics are terms ; the first and last are the extreuie ; the otliers uican. The ratio; 354. The coUiUiou number employed is the ratio. It is when an into- g^ intotrer when the series increases, but a fraction when it ger ; wlien a , '^ ' Traction. dccrca.se.s. An increasing 365. An increasing scries, with the ratio 2 is, 1, 2, 4, cS;,/«en>8 ^» l'^, 32, G4, 1:^.8; a decreasing .series, with the ratio |, ^ is, 128, 64, 32, 10, 8, 4, 2, 1. Five t«rmfl and 366. There are five terms in Geometrical Progression, thoirnamoB. ^^^ three of which known, easily determines the remain- ing two : these are, 1st. The first term. 2d. The last term. 3d. The number of terms. 4th. The sum of all the terms. 5th. The ratio. Note. — By the ratio, we multiply or divide to form the series. Remark. — A proper understanding of this branch of numbers, requires the knowledge of Algebraic equations and logarithms. In this work, a few illustrative examples only are given. Example 1. — The first term is 3 and the ratio 2 : what is the 6th term ? OPERATION. EXPLANATION. wiien the first 2x2x2x 2x 2=2*=32 We multiply the term, ratio and , ^t i number of dl.stterm. first term by that terms arc given. rwiwpr ni' fbp rntift to find the lart — power 01 ine raiio or any t«rm. 96 Ans. whose index is equal to the number of terms preceding the required term for answer. Remark. — It will be seen that the last term is equal to PERMUTATION AND COMBINATION. l'9'l the first term multiplied by the ratio, raised to a power less 1 than the number of terms. 2. The first term of a decreasing progression is 12S, the ratio ^, and the number of terms 7 : what is ihe last term ? (i/=A 128xA=V4«=ti. An,. 3. A sum of money is to be divided ami.ng !(• persons; A is to have $10, B ?oO, and so on: what will the tenth receive? . ^h.s. 8196,830. 4. A lad offered to purchase 17 oranges, and to pay for them at the rate of one cent for the first, 2 cents lor the second, and soon in duplicate order: what was the cost of the 17t]i orange ? 5. The cxtiemes arc -2 and 20,000, and the ratio 10 : what is the sum of the series ? OPEU.'VTION. 20000—2=19998-; 10-1=9; 10998-^9=2222; and ,^;';;^,^^/^^ 2222+20000=22222. Aiis. tio have bsoa civen. EXPLANATION. In this example we divide the difference of the ex- tremes (101)98) by the ratio less 1 (10— 1=9), and add to the quotient the greater extreme. 6. A man trading for a horse offered to pay for him at .the rate of a cent for the 1st nail in his shoes, 8 for the 2d, and so on; there were 32 nafls : what ditl the horse cost? ^H.s. £9,205,100,944, ;'.9.20. 7. A father, at the celebration of his daughter's birth- day in January, gave her §5, and said he would double it on tho first day of each successive week in the fifth year: what was th(; nina that he, through ignorance of Geometrical Progre.*sion, pledged himself to pay? PERMUTATION AND COMBINATION. 357. Permutation is a process to find the different ways pcrmutaUfin in which numbers or things can be placed; Comhmation,^'^^ combin*- their various arrangements in sets or series. Example 1. — In how many different ways can we arrange the first five letters of. the alphabet? Ans. 120. PERMUTATION AND COMBINATION. To find perm ..- ^ ut.ions. 1. a b c d e 2. b c d e a o. d e a b c OPERATION, I EXPLANATION. We multiply together all tlie terms of the natural series from 1 up to the 4. c a b c d, etc. j given number, and find, in 1X1'X3X4X5=120. the last product, the uum- a b c d e= 5 letters, i ber of changes sou2,ht. 2. How many different integral numbers may be ex- pressed by writing once in each number, the 9 digits in succession? Ans. 1x2x3x4x5x6x7x8x9=362880. 3. The solar spectrum consists of 7 colors — red, orange, yellow, green, blue, indigo and violet: in what varieties can these be placed ? Ans. 5040 4. How many changes can be rung ou St. Michael's bells', supposing them to be 8, and allowing 3 seconds to each round ? . . 5. How many different companies, each of 7 men, may be selected from 21 men ? ' ' OPERATION. 21x20x19x18x17x16x11? ,,^,^^ , .^ ^---, — ~, — ,,—-„ =116280. Am. Ix 2x oX 4x 5x 6x 7 EXPLANATION. To find the • Here we form a 1st series of numbers, commencing bination.s'of'" with that (21) to be selected from, and decreasing as ^^y J^"""''^'' o^many times as the other number expresses; and a 2d series, commencing with 1, increasing to the number to be combined for a divisor, and find in the quotient the combinatious sought. 6. The graduating class of the University of the South consists of 80 members, 12 of whom are to have honors and appointments : how many different selections could be made ? , 7. There aje said to be 56 different elements in nature : if one particle in each element will combine with one par* tide- in c:'.ch of the other elements, hovr many combina- tions niiVj be formed ? • Ans. 1540. MENSURATION. MENSURATION. 19i 858. Me.nmradon is that part of arithmetical science S^ed*""" which describes capacities of various kinds. Remark. — The points, lines and surfaces named in Po'^'^s'' i'a<^?' . ^ . ' , . , • 1 • . etc-, not roia. mensuration arc imaginary, and are, simply, aids to a mathematical in([uiry. DEriNITIONS. ^59. Example 1. — A point has neither length, breadth A point, nor thickness, but only position. 2. A line has length, but no breadth or thickness. a line. •- ,. 3- A right line, or straight a ^H,ht^.o^ line, extends only in one direction. \ 4. A' broken line is formed'^ ^.roken line \ _ of two or f»ore right lines. ^ 5. A curved line is one tlmt constantly^ '^""'^ changes its direction. j G. Two lines' are perpendicular to each P^^P^^'^'i'^'"''''' I other, when they touch so as to form right .1 angles. • " 7. Two parallel lines are everywhere Parallel lines, equally distant from each other.- 8. Two lines are oblique to each other when Obiiquo linog. their point of union makes acute or obtu.se auirles. 0. A surface, superficies, or area, has perfi \ 0.2886751 Square 4 0.5000000 Pentatron 5 1 0.G881910 Hexairou 6 | 0.8660254 Heptagon 7 j 1.0882607 Octagon 8 j 1.2071068 Konagon..; 9 i 1.37^7387 Decagon .1 10 { 1.5388418 Tlndeeagon...'-.....! 11 | 1.7028436 Opdecagon 12 I 1.8660254 Area. 0.43y0127 1.0000000 1.7204774 2.5980762 3.6339124 4.8284271 6.1818242 7.6042088 9.365G899 11.1961524 A table for facil- itnnngarealCBl- cul.itions of polygon.s. 16. The side of an octagon is 10 feet and its apotliem 12.07106 : what is its area ? Am. 482.8427 sq. in. Ol'EKATION. I EXPLANATION. 10x10=100x4.8284271=482.842-1-'. I Wc square to find ftr** of one side of the polygon whose area is required, and mul- » r'"'.vgf>n by tiply the square by the tabular area of the polygon iiamed, '', for area. 17. What is the area of a regular triangle, one side '>eing 8 inches ? . 18. What is. the area of a square, one side being 12 inches? 19. What is the area of a heptagon, one side being 3 ■feet? , • 20. What is the area of an octagon, one of the sides measuring 8 rods ? 21. What is the circumference of a circle whose diame-Tofipd the cir- . r •\ 'i I 1 - -/ioe whose diame- ter is 30 feet ? Ans. 94.2480 feet. iVo^c— The diameter of a circle is found by the inverse J/;, Jt°?orac?^" operation : 94.2480-t-3.1416=30. cie. ■ 198 MENSURATION. 23. What is the circumference of a circle whose diame- ter is 20 miles? 24. What is the area of a circle ivhose diameter is 7 ? Ans. 38.4846. OPERATION. I EXPLANATION. Toftnd the area 7x7=40x7854=38.4846. | Multiply the scjuare of the ?h'e^tt^r"« diameter by the decimal .7854. given. 25. What is the area of a circle whose diameter is 9/ Ans. (58.6174. 26. What is the area of a circle whose diameter is 20 miles ? 27. The diameter of a circle is 20 feet : what is the side of the inscribed square? Ans. 14.142ft Ol'EllATIOX. KXI'LANATION. ■ 20'=40U ; 400-4-2=200 ; We extract the square 14.142. root of i of the square of the diameter. 28. What is the side of the greatest square stick of timber that can be hewn from a cylindrical log 36 inches in diameter ? 29. What is the convex surface of a prism whose base is bounded by 7 equal sides, each being 33 feet, and the altitude 22 ? ^Hs. 6082 sq. ft. To find the side of a square in- i ,r.r. Bcnbecf 10 a cir- <*"'J v'-'^'-' cle. A piiim de- fined. Remaek. — A prifm is a solid* with two pimilar equal parallel ftices, called bases, and its other laces parallelograms. OPKRATION. I EXPLANATION. Tofindthecon- 33x7=231x22=5082. 1 Here multiply the pen- rnghrpnsm.°^nietcr of base by the altitude. Whenitisright; Notc.—K prism is Called right when its edges are per- when oblique; pg^tiieular to its bases J when not. perpendicular, oblique. terfotc."*"*^''' It is also triangular, quadrangular, etc., according to its bases. * A solid is a figure which has length, breadth and thickneee. MKNSURATION. 199 30. What is the convex surface of a prism when there are 6 equal sides, each IH inches ia length, and in altitude 14 inches ? Kofe. — When a prism is bounded ^r**"*"®'"?'?^ G parallelograms, it is called a p;iralIelopipedon. Whon these 6 parallelograms are When it n-ctaiigles, the parallolt>|)i|>e(l()n jj- '""°"'*'^'' rrr'tangnlar ; when equal rcctaiigles, wheu a cube. it is a cube. •U. What is the convex .surface of !i prism with 6 e(]UHl sides, ouch 14 inches long, and the altitude '•> inches ? 32. What is the cnnvex surface of a cylinder whose altitude is 50 foet and the diameter of whose base is 20 feet ? Ans: 8 U I .o ^q. ft. J^ote. — A cylinder is a round body, whose diameter is ^ (cylinder d«- \ k ^ 1--- ■-■ 4 unvariable, and whcse ends are equal and parallel circ!,S-t>> 4. OI'UKAilnM. IkXI'LAN AIION 14xl4=196x.7S5t=15:}9384x25=3848 4l In this to find the con- case, having found the arci of the base by nmitiplying t,he ^^"/f '^ * '^y'"*' square of 14 by decimal .7S54, we find the coiiicnts by multiplying that result by the altitude. 35. What are the ontents of a cylinder wlhc^o base diaraet«^r is 1^ and wKdsc u'titutle is 'i'' 36. How miny incbc'* in a cylinder whose b.isc di.iuiL^tcr is 15 and whose altituJe is 6 f 300 MENSURATION. 37. What are the contents of a pyramid whose area base is 60 and altitude 18 ? ' Ans^ 360. A pyramid de- JVote. — A pyramid is a 'solid ^'*®^' ''^v having a polygonal face, called the base. Its other faces are called triangles, and meet at a common point, the vertex. OPER\TION. 60xl8=1080-T-3=860. EXPLANATION. tenKf'a^^'ra" """"^ °~" ^ '^^^ ^^'^^ ^^"^ ^^ multiplied by ml±° a yra ^^^ altitude, and i of that taken for the contents. 38. What are the contents of a pyramid, the area of whose base is 350 and the altitude 25 '( ,39. What is the convex surface of a right pyramid whose *laut height is 28 feet, and the cireuml'erence of whose base is 42 feet ? Ans. 882 sq. ft. OPERATION. I EXPLANATION. Tofind,thecon- 42x21=882. | The circumference of the righrpyramfo! ^'"^6 is multiplied by ^ of the slant height. 40. What is the convex surface of a right pyramid whose slant height is 56 feet, and the circumference of whdse base, is 60ft. ? 41. What are the contents of a cone whose altitude is 30 feet and the diameter of whose base is 10ft. ? Ans. 785.40 sq. ft. A „,,Do. • Note. — A cone is a pyramid with circular base. OrERATION. 10 *X. 7854=785400x30= 23562000-^8=785.40. EXPLA^ATION^ Here, the square of the base multiplied by decimal .7854, and that result by the altitude, with a division by 3, give the contents. 42. What are the contents of a cone whose altitude' ia 25 feet, and the diameter of whose base is 8 feet? ^ns. 418.88 sq. ft. 43. What is the convex surface of a right cone whase slant height is 90 feet and the circumference of whose base is 60 feet? Atis. 27 sq! ft. f-;i',N>iliIlAlW>-N '■■ OPERATION. '' ] EXPLANATION. B0x45=":J700. i We multiply the circum- Tofindthocon- reuce of the baso by ^r thc^lant height. • right conf^" u4. What is the couvox suri'ace of a riijjht cone, whose 4ant height is oO l^et aud bjise circumference 18 feet? Xote. — A cone ii called right, when its edges are per- whmacono : /•■ndicular to its bases ; otherwise, it is oblique. oifliquel'^ i5. What is tlie surface «'>f a *ephere whoae diameter is ■Am. 201. OG. j'Vw.' V o|'herc is asolid body a Nunded by a curved surface, all ;i:irfcs bfting equi-distant from a A poiut within, called the ceutre. "' -i ' diameter, is a straight line hn dnmct^r. •ssing through its centre. Ail diameters 6f the same sphere are equal'. ' " ^' Rkmark. — The surfrice of :i sphere equals 4 great •circles of the same sphere j and, as we find the. area of a circle by multiplying (Ex. 1, App.' Sq. 11.) the circum- ference by \ dianietei', so we find a spliQrical surface, as shown in the explication. OPKRATTOX. 1 EXPLANATION. ^ 8'*:=^64x3.14U)^-:^:i01.UG. j in this case we square the to fi,jd the^nr diameter, and, multiplying it by 3.]41Gdec., find the *""•= jf a^P^^en- surface. ■" 46. What is the Rrirface of a sphere whose diameter is Oft. ? ' Am. 254.46 sq. ft. 47. If the sun's diameter is 896,000 miles, what is its surface ? Ans. 2,522,120,323,072 sq. m. 4S. Whot arc the- contents of a sphere whose diameter is T)!'; y Aiis. 65.450 sq. ft. OPERATION. 1 BXI'LANATION. 5 "" X 3. 1416=78.540 X 5=] . AVe multiply the surfiice to n^rf the c(m- .'10:^.700-=-6=65.450. by the diameter, and divi.do t-mts or sphere ! that by 6. 40. What are the contents of a sphere whose diametcc is 20 feety ■ '■ "■ < ' ^ 50. What arc the coatents of a sphere who?36 diameter i^ 28 rods'!' 51. What are the contents of a sphere whose diameter is 8000 miles? 15 ■ 202 ♦ MISCELLANEOUS EXAMPLES. MISCELLANEOUS EXAMPLES. 361. Example 1. — A merchant bought 12 cases of merchandise for $669 : what would 25 have cost at the same rate ? 2. If 8yds. of cloth cost £20 18s. 5d., what is that per yard ? 3. If 9f barrels of flour cost £21 3s. 8d., what would 17-f cost ? • ■ 4. If f f f a ship is worth £865 5s. 9d., what is the whole worth ? 5. What number, multiplied by J of itself, will pro- duce 7^ ? 6. What number, multiplied by i of itself, will pro- duce 5 J ? 7. A man had 4cwt. oqrs. ISlbs. of tobacco, which, equally divided, he put into two parcels : what was each division ? 51 9. If, when wheat is 6s. 4d. per. bushel, the penny loaf weighs 8oz., what ought it to weigh when wheat is 5s. per bushel ? 10. How many gallons will a cisteru contain, the diame- ter of whose base is 10 feet and the altitude 30 ? 11. If a staff 4 feet long cast a ahadow of 6ft. 8in., what is the height of a column which casts a shadow of 155 feet at the same hour ? 12. If 46 gallons of water run in an hour into a cistern containing 21 h gallons, and 38 are drawn ofi in an hour, in wliat time will it be filled ? 13. What is the square root of 9 times the square of 16 ? 14. A and B go from the same place and travel the same rosd ; but A commences , his journey 5 days before B, and travels at the rate of 29 miles a day ; B follows at the rate of 45 miles a day : in how many days will B over- take A ? • 15. IIow many cubic feet in a cistern 4ft. 2in. long, 3ft. Sin. wide, ard 5i't. 7in. high ? 16. V/baf is the square root of the square root of ^g of the square of -,^a"s ? ^'^^- A- 17. A father, in his will, left 815,000; of this, ^ was for his eldest son, \ to the second son, and i to his 8. Multiply -h of 7-,^ by ^. Aiis. 9. MISCELLANEOUS EXAMPLES. -O"! daughter, and the rest in charities : how was it appor- tioned ? 18. A man owes B $500, to be paid as follows : SI 50 in 8 months, §225 in 6 mouths, and $125 in 4 mouths : if paicf at one time, what would the term for payiucnt be ? 19. A house is 50ft. from the ground to the eaves : ivhat length of ladder would it take to reach the eaves, if the foot of the ladder cannot be brought nearer Ihe house than 30 feet ? 20. If a pipe 6in. in diameter will discharge a certain quantity (f water in 4 hours, in what time will 3 four inch pipes discharge the double quantity ? Ana. 6 hours. 21. A gentk^man left an estate of $17,500 between his widow and son ; the son's share was ■§■ of the widow's : what was the share of each ? 22. What are the prime factors of 1400 ? 23. What is the hour when the time past from mid- night is equal to -^ of the time to noon ? 24. AVhat arc all the integral flictors of 1100 ? 25. How many square feet in a floor 12ft. Gin. wide and 14ft. 8in. long? 26. What is the greatest common measure of 102-7 aud 1781 ? . . • . 27. IIow much wood is there in a pile 4ft. wide, 2ft. 9in. high, and IGft. Gin. long? 28. What is the duty on 300 bags of coffee, each weigh- ing, gross, IGOlbs., valued at 7 cents per lb., 2 per cent, being the tare allowed and 20 per cent, the duty ? 29. What is the square of the cube root of -J- of 30. How many bricks 9in. long, 42in. wide and 2in. thick, will build a wall 6ft. high and 13Jin. thick, round a yard, each side of the same being 280 feet on the out- side of the wall ? 31. What is the interest of §276 for 3 years, at 7 per cent. ? 32. Two men speaking of their ages, one said that f of his age equalled f of the other's, and that the sum of both was 90 : v/hat were their ages ? 33. An invoice of goods was sent from England with instructions to sell them and invest the proceeds in cotton, after deducting a commission of li per cent, on the sales, and 1 on the purchase of cotton. The goods were sold at an advance of 5 per cent, on the invoice price, and the iH ' MISCELLANEOUS EXAMPLES. amount of §12,600 was received : what was the invoice price, and what sum was invested in cotton '/ Am. Invoice, $12,600; luvestmeat, $12,288.1 l-iVr- 34. Divide .$150 among^4 persons, so that when A has i dollar IJ has 4, C i and D i- 35. Said A to B, my horse and saddle together arc worth 5>I50; but my horse is worth 9 times as much as the saddle : what was the 'value of each V 36. A drover, with beeves and sheep, was asked how many he had of each kind. He said there v. ere 174, but that the beeves were -j^ of the whole : how many of each was in the drove ? 37. W hat is the cube of the square root of -f- of •Lf_jof-y? 4§ ^450 88. A urocer mixed !)51bs. of sugar that was worth 7 cents per lb., 65 that was worth D cents per lb., and 25 worth 12 cents per lb. : what was the mixture worth per lb. i* 89. A m:in driving some sheep, cows and oxen, being afiked the number of each kind, said that he had twice as many sheep as cows and three times as many cows as oxen, and that the whole was 80: what number of each kind did be have ? 40. A colonel forming his regiment into ahullow square, found that, when he arranged the men 8 deep on each side of the square, lie had 114 men left; but when he arranged them 4 deep, he wanted 114 men to complete the arrangement : what number did his regiment consist ofy " Ans. 750. 41. A merchant has due a certain sum of money, of which i has to be paid in 2 months, i in 3 months, and the rest in 6 months : in what time ought the whole to be paid ? 42. A man built a house of 4 stories ; in the lower story there were 16 windows, each containing 1:^ panes of glass, each pane 16in. long, 12in. wide; in the second and third stories there were 18 windows, each of the same size; in the fourth story there were 18 windows, each containing (i panes, 18 by 12 : how many square feet of glass were thete in the house ? 43. \Vhat is the difterence between 3 times 6 and 18, and 3 times 18 and 6? • 44. What number added to the 25th part of 2600 will make the sum 145? 45. A merchant lias spices at 9d. per lb., at Is., at 2s., MISCELLANEOUS .EXAMPLES. 20r' at 38. per lb. : how much of each kind must he mix to sell the same at Is. fid per lb. ? ' 46. What is the difference between G dozen dozens aul 5 dozen dffzcns ? ^ 47. Write two millions, two hundred and two thousand, two hnn.ived and twenty-two. 48. What is the value of f of a cwt. ? 49. What is the value of .325 of £1 ? 50. Reduce 12s. 3d. 2fj[rs. to the decimal of a pornid. 2. 2« 51. What is the difference between \/— and -5 ? lo lo 52. Whnt is the value of v/3G0.O0O ? 53. Three merchants, A, 13 and C, freight a ship v/ith tobacco. A put on board 300 tons, B 150 and C 85. In ' ' a storm frherc was thrown overboard 1 25 tons: what was the loss to them several Iy1 54. How many boards 20 feet long and 15 inches v/ide will it take f:r a floor 40 feet long and 30 wide ? 55. A grocer bought a hogf^heud of brandy for §87 on 6m. credit, uiid sold it for cash, "with an advance on cost of $18 : hov/ much 'v^as his gain, allowing money to be worth 7 per cent, per an. ? 5S. What is the difference of time between 3Iay 10, 1860, and August 15, 18G5? 57. Honv many hours from October 10, 1860, at 4 P. M., to January 1, 1833, at 7 A. M.? 58. Two men hired a pastiure for $35 ; A put in 3 horses for 4m., and B 5 horses for 3m. : what ought each to pay ? 59. A groc3» bought a hogshead of molasses for $30, but 8 galloiis having leaked out, he wished to sell the remainder, so as to gain 4 per cent, on the whole cost : at what price per gaUon must it be sold ? 60. Divide ^750 between 3 persons, so that the second shall have t^ as much as the first, aud the third 2 as much as the other two. 61. A merchant sold a piece of cloth fn- §45, and lost *by the sale 8 per cent. : what did the cloth cost 1 • 62. A •eiit^eman being asked the time, said that the. time past noon was equal to -^ of the time past midnight : what was the hour ? 63. What is. the discount of v?260.85 for 1 yedr at^d 3 months, when interest is 8 per cent. ? . ., 64. A persou was born on the Lst day December, 1814, at 4 o'clock ill tjie uiorning : what v/as his age 1st day May, 1864, at 10 o'clock in the morning? -'"<> MISCELLANEOUS EXAMPLES. 65. What is the area of a square piece of laud, the sides of which are 27 chains? Ans. 72a. 3r. 24p. 66. A merchant shipped 14 bales of cotton at £-^0 lOs sterling, 4Hihd. of tobacco at £14 8s. 3A per hhd.,. 210bbl. of rice at £3 9s. 6d. per b^l. : what was the « amount in American money, with exchange 5 percent.? 67. A'n inclined plane is 40 feet long and 5 feet high : what weight will be balanced by a power of 200 feet? Am, 1600. 6S. A lot of land, measuring 39 feet by 16 J, cost $2500 : what was that a square fobt ? 6'J. What is the area of a parallelogram, the height being 360 feet and width 271 yards? . .' Ans. 32,520 sq. yd. 70. A merchant owes in England £350 15s. 3d., but has shipped 15 bags of cotton at the valuation of $90 each : what is the remaining indebtedness? 71. What is the area of a circle whose diameter is 10 ?• 72. AVhat is the commission on the sales of 250hhd. of sugar at $55 per hhd., at 3^ per cent., and 2^ per cent, for guarantee of sales ? , 73. Th'ere is due in England, for a bill of merchandise, £250 sterling: how many dollars must be remitted to pay for the bill, reckoning a £ at $4.87 ? 74. What is the area of a circle whose diameter js 14 ? Am. J6,96S, 75. There are due in Paris, for several pictures, 4565 francs : what amount in dollars and cents must be remitted, reckoning 19 J cents to the franc? 76. A note of $350, dated Octobef 7th, and pay- able 6 months from date, i.s to be discounted on the 22d day October, at the bank discount of 7 per cent. : what must be considered the worth of the note on the day of discount ? 77. What is the value of $1000 of bank stock 'at 105 per cent., or 5 per cent, advance ? 78. What is the least common multiple of 2, 7, 14 hnd 49 ? 79. A merchant shipped to Havana llObbl. flour, at $5.75 per barrel, and received in return 325 boxes of sugar, at $340 per box : what amount does he still ow'e ? 80. The double and the half of a certain number, increased bv 2 J, make 100 : what is the number? APPENDIX. BOOK-KEEPING BY SINGLE ENTRY. Book-keeping is a plan adopted by business men to Rook-iseepinc facilitate the interchanges of trade, and is simply an explained, arithmetical record of inorcantile transactions. It is of two kinds, single and double entry. The latter is used Avhen commercial engagements are very ex- ' tensive and great accuracy is desired ; but the former is most simple, and is sufficiently methodical for ordinary trade. Single Entry is the kind treated of in this work. The Day Book is for all mercantile charges «ind rpf^^, ^^^^ i^j^p,. credits. At short intervals these should be posted, that is, transferred in date and amount to a book called li\\Q Ledger. This contains the summary account ,j,|^^j^^^^^ of each individual whose name is in "the Day Book, with numbers referring to the specific page of trans- actions. The Cash Book shows receipts and expenditures. The cash book. The Bank Book shows the deposits and with- The bank book, drawals of sums lodged in a bank for safety and convenience in transacting business. The Bills and Notes Pai/ablc shows to whom and The biiis paya- when an amount is due ; and *''*'■ The Bills and Notes Receivable, by whom a stated The bills recciv- sum is to be paid, and the time of expiration of"*''^- credit. -08 «0UK-KEK1'1NG. REMARKS ON NOTES. .4; joint and sev A j'oint and Several note can bo eoliccted by either oiraipote. of the signers. ^ 'ihe endorser.. The endorser of note is. liable foi- tLc anion lU. ■Anotcnotnago- A note is uot negotiable ^.vhon the words payable T.iabie. I ^ a ^q order" are omitted. • Notes without All notcs, without the oxprcfisiou " for value re- value. ■ ccived," are valueless.- when'payabie When a uotc is paj'oble to A B or bearer, the to bcaror. t;i;oii.ivio;ii- the bank has advanced money, called accommoda- iiou paper: ^j^^^ paper. Notes are often 'lodged in banks by oiiection holders simply for collection. "When a note remains notes: , unpaid it is protested; or a notification made to par- xotes unpaid, ^:.q^ interested of its non-payment. BOOK-KEEPING. 209 r( . . :;r 1 A ]. FO u \l s. ;;i:GoriAiiLE notes. Charleston, S. C, Feb. 4, 1863. For value i\ . ..ived, I promise to pay Messrs. McCartor and Dawson, or order, fi^^e huiKlreJ dol- lars, on demaiul, with interest. WASHINGTON IRVING. 5500. WANNAH, Ga., June 19, 1865. Four iiioiii; s n' i];i date, I promise to pay Ogle- thorpe Hall, or order, three hundred dollars, for value received. STEPHEN DRAYTON. $300. New Orleans, April 5, 1863. Sixty days from date, we jointly and severally promise to pay Mr. F. Vf. Pickens seven hundred dollars,, for value received. JAMES J. McOAETEB, EDMUND ifAWSON. 5700. FOR.AI OF ORDER. , Mobile, July 4, I860. Messrs IIayne, Aiken & Co. Gentlemen: Please pay to the order of Tlon. C G- Memminger two thousand dollars, and charge to hcdient servant, JEFFERSON DAVIS. Note. — The order, before it can be paid, must be endorsed. 210 BOOK-KEEPING. RECEIPTS. Columbia, S. C, April 19, 1864. Received of Ethelwald Preston one hundred dol- lars, on account. WILLIAM MANNING. Knoxville, Tenn., Aug. 9, 1865. Received of James Montgomerj^ three hundred dollars, in full of all demands. JOSEPH ALDINGTON. $300. FOKM OF. A BILL. Richmond, Va., Jan. 1, 1864. Mr. Henry Addikgton, Bought of Palmerston, Russell & Co. 1 bale Plains, 500 yards, @ SOcts $150 2 bales Honie?pun, 20 pieces, 600 yards, @ 7cts. 42 4 dozer; Blankets, @ $3 per pair 72 4 dozen Scotch caps, @ $4 16 §280 Received Pavmor.t, PALMERSTON, RUSSELL & CO. BOOK-KEEPINO. 211 DAY BOOK. Remarks. 6m. cr. E. January 1, 1862. William "VVoRpswoRxn, Dr. To 5 pes. Calico, 150 yds., @ 10c. 20 pes. Shirting, 600 yds., @, 12'c. 5 pes, Osiiabiuf^s, 100 yds., @ 10c. 10 pes. .Sheeting^ 200 yds., @ '26c $15,00 75,00 lOiOO, 50 i 001 150 00 4m. E. 5. James Argtle, Dr. To 10 pes. Scotch Plaid, 400 yds., @''75c. 300 00 Paid to BAG, Factors. E. 20. William Wordsworth, Dr. To Cash paid his order, 5545 4m. E. 30. William Wordsworth, Dr. To 100 pes. Kersej-, 300 yds., @ 60c. 50 pes. Plains, 100 yds.', (oj, r.Oc. 50 pes. Bro. Homespun, 2000 yds, @5c. 20 pes. Chintz, 600 yds., @ 6c. 1 doz. Ildkfs. 150 00 30 00 100,00 36 00 1,00 817 00 E. 19. William Wordsworth, Dr. To pd. his order to Walter Scott, 1 *19 35 30 daye. E. March 9. James Argylk, Dr. To 2 pes. Kersey, 100 yds., @, 60c. 20 pes. Bro. HoniespuD, 600yds..@5c. 60 pes. Long Cloth, 1200 yds., @10c. 50 SO 120 00 00 00 200 1 'bo E. April 6. James Aegyle, Or. By Cash. 350 00 9. James Akgtle, Dr. To 20 pes. Calico, 600 yds., @,8je. 50 00 ■ E. July 1. Wn.i.iiVM Wordsworth, Cr. By Cash, 600 00 ■ E. 4. Casfi Account, Cr. Rv Cash pd. travelling expen. 360 00 3m. E. 21. Willlam Wordsworth, Dr. To 2 cases Shirtings, 3000 yds., @ 10c. 300 00 ^J2 BOOK-KKEPiKG. 1 Remaeks. E. July 2t, 1862. ' James Argyle, Cr. P>y order on Str.ai-t & Co. )275'< Aug. 5. WirXIAM WOKDSWOUTH, Dr. To 1 case plains, 40 p<>v. To 10 pes, Pliiins, 8uu vtls., (^t, 3*0. 2 doz. lidkfu., @ 2| 90'00 1 ,5!60| 96J50 Wirxi.VJI WORDSWOETU, 1 bale Blankets, 2<(i ; r , ^ 1];^ on 00 00 00 400 600 00 C'.n. . 12. J.'.MEs Argyle, Di-. To 1 cusc: Ga. Plains, lOCO yds., (^ilOi: 1 case Bro. llbnitsp., 1000 yds. @ 6c. 1 case Blankets, 100 pairs, @.$2. 20 pe.^ Loii-- Clolh, 400 yds., (il 10c. 300 60 200 40 Ol' ~1 y note iit 3m. ' 600 00 E. 6. Jas, Argyie, ( 1 . By Cash, >i?A, 41 Dec. 1. Wjt. WoRiiswon'rn, By bal. to new aecoxmt, ^- Ja8. Akgtle, By Cash to bal. Jan. I,1l863. Wm. Worhsworth, To bal. from old acct, ^ ^ ^^ Wm. WoRi'SwoKT.ir, Cr. Cr. Dr. 41 0(1 '■32'? 00 41 10(1 Dr. To 2 bales Broadcloth, 300 yds., @ §2. I '600 ; 00 fi Wm. Wordsworth, By Cash, Cr. 350 0(1 Noti — The letter E shows that the charge hnabcen entered into the Ledgei- J^otc. — It \rill be noticed that a few names and entries only are give;. These are sufficient to show the method. nOOK-KEEPINO. 213 Dr. LEDGER. WILLIAJI WORDS WORTU. i.;.j. Prtge ! ■ 1 ^] 1 1861. Page 1 1 1 .fftn. 1 1 'To am tfr.D.ft] ?l,-0 00 July 1 I Byani'i A-.D.hJ'SdoO 00 '-I'l • §ept. 5 11 '' ,'.0O 00 : ) f, ■ " : 317,00 1 1 Nov. 5 ir. ! coo 00 l'(r>. lii .. 19 3r. 1 Dec. 1 17 Bal.toaewac't' !] 00 rui.v2i 300 00 1 '1 \ut;. f) :n 1 ;>00 00} 1863. an. 1 10 1741100 20 jTo bal. I 4) ooi 21 " am t fc D.B.Ij GUOW .Jan. 10 21 iByam-tfr.D.B.'l 350^00 214 BOOK-KEEPING. Dr. JAMES ARGYLEL Cr. 1862. Jan. 5 Mar. 9 April 9 July 29 Aug. 20 Oct. VZ Pagej I 8 JToam'tfr.D.B 10 I •' '• 10 \ ot. i $300 200,00 i862. Page April 6. 15 I July 21 20 50 00, Aug. 10 17550 95 60' 600,00 UVl 1421 00 Nov. Dec. 6 25 35 62 Byam'tfr.D.B. $3.iO " « 275 " " 137 .. 331 BOOK-KEEPING. 2\b Dr. CASH BOOK. CASH. Cr. 1862. Jan. on 24 1 To Cash I liand I April 6, To Cash fr.J.i I Ar2vle July 1 ToCushfr.W.j Wordsworth To Casli Stu- art A Co. To CiV'ih W. Wordsworth Nov. 6! To (;ash J. Argvle Dec. 6 To Ca.9h J. I Argylc iiooo'oo 350 00 Sept. 5 1S63. Ian. 1 Tobrtl. 600 275 500 331141 327'oO 38 41 33S3 43 Feb. 1 July 4 By pd. not« to Uk. S. C. By pd. rent By Sund. for house By Trav.Exv Cash Sales Bal. to new a'?c"t $550,00 200 00 250 OOJ 350 oo; 2000 j 00 38 '41 3383 43 Note. — Though not inserted in our form, yet all receipts aud expendi- tures are to be entered iu Day Book. 516 BOOK-KEEPING. Dr. BANK BOOK. BANK OF THE STATE. Cr. To deposit Jan. 1 5 7 9 10 14 1.5 17 ■M) Feb. 1 To bal. COO j 00 300 00 450 100 700 155>)l00 149 o: 2C7J33 I •2717 00 1862. Jan. 3 5 8 16 16 CO 451T 00; By check to "Pnnglo & Co, Hy check to Akstuu & Co. By ehgck to llnyne & Co. By uheekto Colcock & Co. By check to Cofitin & Co. By bill. $300 00 200 00 400 00 600 00 300 OOi 2717 iOO' 4517 00 4517 00 BOOK KEEPING. 21^ o CO C5 30 CI o o I- CO 1 • . CO (M 01 o UO CI o o U3 to 00 o I- i-i o o o -^ . o 00 iC 1--5 t— 1—1 CO CO o o r-t I'. CO h- . 1 Ci o 0 CI >; CI d ! § '~ ' 00 1 1— • lO CI , . O I g 1-H S io « CI • ^ o • o • , ~- ■ s 1— e^ 00 f-l 1 l~ o o 1 1 1 kO s o\ "» lO eo cc c< V . \z ' -^ is o o 6 c O . . . t) . . . . ^ d C" a: i=j n 1 J ^/L ^ >^ d <' a 1 Jan. 1. 9. Feb. 5. April G. 10. May 1. 9. 16. l-H to t^ rH Ci eo lO 1 1, .1 ERRATA. 19th page, in the 1st line, hcforc the word "figure," insert "exception of the." 52d page, oth line from top, after "40," insert "-f"' for "X ;" same page, 8th Ex., take out " 3qr." .0.3d page, KUli line from top, after "1200" insert "+" for " X." 65th page, lor Art. " 9'J" read "105;" same page, 17th Ex., for "months" read ''davs." 87th page, in 27th and 28th Examples take out " 25t." and " ,39t." 92d page, r2ih Ex., for " 7cu. ft." read "7cu. in." 114th page, 9th Ex., take out "$18.75 per bbl" and read "$180.75." 176th page, in 325th Article, 2d line, after "symbol" insert "v/,". ^ The attention ol* Booksellers and Teachers is invited to ' Jlritljmctical Series. ^ — ^ Prepared lor ihe use of th«* Schools and Academies — Male and Feuialf —of tlu- Conleder^te States : to be I is^ed as soon as j /acticable. I 1. Lev(*i;4ts Piiiiiarv Aiitlmu'ric. I 2. Lcveivtt.N Mv]]\\\] Aiirlmi^'tir. i •5. L('\erott.s ("oiiiiikhi School Ariilinietir. I i. Lcvcrdts Acadciiiiciil An'tliiiictic ^IMILAU TO TUK PRECKDJNG. HUT ENLAUGED. 5. LcvcrettN Academical Algol)ra. 4 a. Leveretts Plane (jooinctrv. J. T. PATES80N. I. C. TUCKO. J. T. PATBRSdN k i:0., ;itj)0(jraj)l]crs ;ini) ^>ttfr-^r55 '0'inttrs, H HANK NOTES, iHCl'KS. CHANGE BILLS, DONDS, ETC., % Book. Pamphle'^ and Job Work of all descriptions KXKCIJTKD PROMVTI^Y