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Numeration, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . b Simple Addition, . . . . . . • * * * * * * * * * . . . . . . . . . . . . . . . . . . . . * * * * * * * * 12 º Simple Subtraction, . . . . . . . . . . . . . s = * * * * * * * * * * * * * * * * * * * * * * * * * 15| Simple Multiplication, . * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 18 || Simple Division,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . “ 23 Addition of Federal Money, . . . . . . . . . . . . . . . . . ............. • 30 Subtraction of Federal Money, . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Multiplication of Federal Money, ... . . . . . . . . . . . . . . . . . . . . . . . 34 Division of Federal Money, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • 36 Reduction, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . " - Compound Addition,. . . . . . . . . . . * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 60 Compound Subtraction, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 | Compound Multiplication, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 | Compound Division, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S0. Simple Proportion,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Compound Proportion, . . . . . . . . . . . ................. . . . . . . . . . . 100 Practice, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Tare and Tret, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Interest,... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Compound Interest,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Oil Insurance, Commission and Brokage, . . . . . . . . . . . . . .” -------. . . . 121, Discount, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 º Equation of Payments, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 | Barter, . . . . . . . . . . . . . . . . . . . . . . . * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 126 Loss and Gain, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * * * * * * * * * * * 128 º Fellowship, ... . . . . . . • * * * * * * * * * . . . . . . . . . . . . . . . . . . . . . . . . * * * * * * * * 132. - Vulgar Fractions, * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 135 . Reduction of Vulgar Fractions, * * * * * * * * 4 & 6 tº # * * * * * * * * * * * * * * * 136 - Addition of Vulgar Fractions, • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 Subtraction of Vulgar Fractions, * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Multiplication of Vulgar Fractions, . . . . . . . . . . . . . . . . . . . . . . . . . Division of Vulgar Fractions,'........... . . . . . . . . . . . . . . . . . . . . Decimal Fractions, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Addition of Decimals, • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subtraction of Decimals, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | Multiplication of Decimals,... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - Division of Decimals, # * * & # * . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . Reduction of Decimals, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . & ### Proportion in Decimals, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compound Proportion in Decimals, ... . . . . . . . . . . . . . . . . . . . . . . . . . . gº Mensuration, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i Involution, .... • * . . . . . . . . . . . . . . . . . . * * * * * * * * * * * * * * . . . . . * * * * * Evolution, . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . Square Root, . . * * * * * * * * * * * * * * * * * * * * * * * * * * * * ........... Cube Root, . . • . . . . . . . . . . . • - - - - - - - - - - - - - - - - - - - - - - - - - Roots of All Powers, . . . . . . * * *.* . . . . . . . . . . • * * * * * • . . . . . Ž Arithmetical Progression . . . . . . . . . . . . . . . . . . . . . . . . Geometrical Progression . . . . . . . . . . . . . . . . . . . . . . . . . . Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ll Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | *"omiscuous Exercise . . . . . . . . . . . . . . . . . . . . . . . . . *. *-º: § PREFACE. §§ presenting this work to the public, the author makes no pre- nsions to having discovered any new spring by which to put the uthful mind into action, nor any new method of communicating || a knowledge of Arithmetic. He has founded his work on the belief at labor and labor only, can insure success in any pursuit; and | that labor should always be bestowed upon those objects which pro- uce the greatest useful result. * | In the selection and arrangement of matter, therefore, those rules || t are of the most general use, have been presented first, and their || | exercises made extensive, that the pupil many early become familiar with their principles, and expert in their application. $ º | The explanations accompanying the rules, are designed to facili- || |tate the progress of private students, and to diminish the labor of | eachers, especially in large schools, where they are unable to give | o each pupil the necessary explanations. s || he MENsun ATIon of Carpenters’, Masons', Plasterers’ and || ºrs’ work, &c., will be found an acceptable part of Arithmetic, I every man of business, and a practical knowledge of it will con- tribute much to the security and satisfaction of both workmen and || mployers, in estimating amounts of work. This has been intro- ced in consequence of numerous applications to the author to || easure various kinds of work, and for instruction in particular || es of Mensuration. -* - - | The system of Book Keeping, is thought to be sufficient for all || e purposes of farmers, mechanics and retailers, in that necessary || anch of a business education. . . How far the author has succeeded in his attempts to compile a || ful work, particularly adapted to the circumstances of the Western || ople, remains for them to judge, and for experience to determine.] The favorable reception of this treatise and the increasing demand | , have induced the publishers to revise, enla vuerwise || rove the work. Such alterations and amendments have been| ide as the experience of the author and of other intelligent and | x will be found more useful, and consequently more umerous testimonials to the merits of the work, have been re- ed; but its general adoption without any efforts to force its || . luction, and its intrinsic worth, are our main reliance; we have ARITHMETIC. &º ARITHMETIC is that part of MATHEMATICs which || treats of numbers. It is both a science and an art;- | the science explains the nature of numbers, and the principles upon which the rules are founded, while the art relates merely to the application of the various || rules. - || All the operations of arithmetic are conducted by | means of FIVE fundamental rules, viz., Numeration, | which includes Notation,) Addition, Subtraction, || wltiplication, and Division. § NUMERATION AND NOTATION. Numeration is the art of representing figures or num- || | bers by words; Notation is the art of representing num- || | bers by characters called figures. || | All numbers are represented by the following charac- | ters, which are called figures or digits. | | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. | nought, one, two, three, four, five, six, seven, eight, nine. | The one is often called a unit, it signifies a whole | thing of a kind; two signifies two units or ones; three || | signifies three units or ones, &c. - || | The value which the figures have when standing || | alone is called their simple value; but in order to denote | | numbers higher than 9, it is necessary to give them ano- || || ther value called a local value, which depends entirely || | on the order or place in which they stand. Thus, when || | we wish to write the number ten in figures, we do it by combining the characters already known, placing a || || 1 on the left hand of the 0, thus, 10, which is read ten. | This 10 expresses ten of the units denoted by 1, but || | as it is only a single ten it is called a unit, and the [ 1 being written in the second order or second place || | from the right hand to express it, it is called a unit of || º || the second order, the first place being called the place || º F- 5 ºśº - ;:...º.º.º.º. NUMERATIon AND NoTATIon. of units, and the second, the place of tens ; ten units || the first order making one unit of the second order. || When units simply are named, units of the first order |are always meant, when units of any other order are || | intended, the name of the order is always added. §§ºš | Two tens or twenty, are written 20. |*Three tens or thirty, “ . . 30. | Four tens or forty, “ “ 40. | Five tens or fifty, 4 & & 4 50. | Six tens or sixty, “ “ 60. | Eight tens or eighty, 4 & & 4 80. | Nine tens or ninety, “ 4 & 90. | Ten tens or one hundred, “ { % 100. || | The numbers between 10 and 20, between 20 and || |30, between 30 and 40, &c. may easily be expressed || |by considering the tens and units of which they are || |composed. Thus, eleven being composed of one ten || and one unit, is expressed thus, 11, twenty-three being || || composed of two tens, and three units, is expressed || |thus, 23. &c. w - | Sixteen being 1 ten and 6 units, is written thus, 16. | Thirty-nine being 3 tens and 9 units, is written 39. I Sixty-four being 6 tens and 4 units, is written 64. || Ninety-five being 9 tens and 5 units, is written 95. || | | Ten tens or one hundred forms a unit of the third || order; it is expressed by placing a 1 in the third place, | ind filling the first and second places with cyphers, i. us, 100. Two hundred is expressed thus, 200. | Three hundred thus, 300, &c. * {} With the orders of units, tens, and hundreds, all the | & º, mbers between one and one thousand may be readily || d its, that is, 4 units of the third order, 2 units of the . . . ; ence the number is written thus, 4 27 the number three huudred and five, the pressed. For example, in the number four hundred | twenty-seven, there are 4 hundreds, 2 tens, and 7 || & cº º 3 O 5 Ten units of the order of hundreds, that is ten hun- dreds form a unit of the fourth order, called thousands, | written thus, 1000. . In the same manner ten units of the fourth order form a unit of the fifth order, called tens of thousands. The following may be regarded as the principles of | Notation and Numeration. 1st. Ten units of the first or lowest order, make one | unit of the second order; ten units of the second order, make one unit of the third order, and universally ten | units of any order make a unit of the newt higher || | order. || | 2d. All numbers are expressed by the nine digits, | and the cypher, and this is effected by giving to the | same figure different values according to the place it i occupies. Thus, 4 in the first place is 4 units, in the | second place 4 tens or forty, and so on. This tenfold || | increase of value by changing the place of the same || | figure is usually expressed by saying that figures in- || | crease from right to left in a tenfold proportion. The | names of the orders are to be learned from the : NUMERATION TABLE, # : . | if : f | | The orders are likewise divided into periods of fix | | places each, according to the following table. º º NUMERATION AND Notation. 7. none of the second, and five of the first order, hence the || number is written thus, ă ă # * } ... of Mj llions. 3. * # * f f t ºr: i # # c ...~ ſ | i ſ # tº d º 5- É. . The periods succeeding those in the table, are Tril- | || lions, Quadrillions, Quintillions, Seatillions, Septil- || T F- P T || might be formed for the sueceeding higher periods. | From the preceding remarks the pupil will readily || understand the reason of the following rule for numer- || |ating or expressing figures by words. ; Rule.—Commence at the right hand, and separate || | e given number into periods, then beginning at the left || | land, read the figures of each period as if they stood || º | || alone, and then add the name of the period. || | Thus, the number 8304000508245, when divided || | into periods, becomes 8,304000,508245, and is read, | Eight billion, three hundred and four thousand mil- || | || lion, five hundred and eight thousand two hundred | || and forty-five. The name unit of the right hand pe- || | riod is commonly omitted in reading. % º Exercises IN NUMERATION. || | Ex. 1. 35 | 10. 37.00054 | 19. 20031025 || 2. 204 || 11. 6130425 | 20. 68723145 || 3. 513 | 12. 2701030 21. 901023406 || 4. 2000 N. ~ 3.3% 5, 3054 14. 6803217 |23. 310275603 || 6, 7428 i 15, 2003005 || 24. 600000501 || 7 10345 | 16. 70032004 || 25. 300040023.0024 || 8.400.24 || 17. 62003005 |26. 80000102051003 || 9. 61304 | 18. 91010010 | 27. 50000021375604 || lions, Oetillions, and Nonnillions, and analogical names || | 13. 3705423 22. 820302008 || Tº :RATION AND NoTATION. | 9 | i | From the preceding tables and remarks, the pupil will || likewise readily understand the reason of the following | rule for notation, or expressing numbers by figures. RULE.—Make a sufficient number of cyphers or dots, and divide them into periods, then underneath these || dots write each figure in its proper order and fill the vacant orders with cyphers. 3. NoTE.—The object of the dots or cyphers, being to guide the learner at first, after a little practice he may dispense with them. Ex. 1. Write down in figures the number twenty millions three hundred and four thousand and forty. Here millions being the highest period named, we | write cyphers to correspond with that, and the period | of units, and then underneath these place the significant || | figures in their proper order, and afterwards fill the vacant orders with cyphers. 0 0 0 0 0 0, 0 0 0 0 0 0 2 0 3 0 4 0 4 0 º The pupil must recollect that cyphers being of no | use except to fill vacant orders, are never to be placed || | to the left of whole numbers. - EXERCISES IN NOTATION. Express the following numbers in figures. EXAMPLES, Seventy-five. . Ninety. x . One hundred and five. . Three hundred and twenty. Nine hundred and four. w Eight hundred and ninety. Two thousand three hundred and five. . Six thousand and forty. . Seven thousand and four. . Eight thousand and ninety-five. . Ten thousand five hundred and fifty-six. . Forty thousand and forty. º | . Ninety-five thousand two hundred and sixty-seven. || . Eighty thousand one hundred and nine. A 2" # . º º NUMERAtion AND Notation. | 16. One hundred and thirty-six thousand two hundred || and seventy five. - || 17. Three hundred and seven thousand and sixty-four. || | 18. Five hundred thousand and five. | | 19. One million, two hundred and forty-seven thousand, | four hundred and twenty-three. . | 20. Ten millions, forty thousand and twenty. |21. Sixty millions, seventeen thousand and two. | 22. One hundred and four millions two hundred and ## *ś four thousand and sixty-five. |23. Five hundred and three millions, one hundred and || two thousand and nine. - . 24. Ninety one thousand and two millions, and four. || 25. Sixty billions, three millions and forty-one thousand. || |26. One billion, one hundred million, one thousand and Orle, - | The Roman method of representing numbers, is by | means of certain capital letters of the Roman alphabet. || XVIII eighteen II two XIX nineteen III three XX twenty IV four 9 ºr XXX thirty I O Il • | V five XL forty | WI six L fifty || VII seven LX sixty || VIII eight LXX seventy | IX nine LXXX eighty X ten XC ninety | XI eleven C one hundred || XII twelve - CC two hundred | XIII thirteen - CCC three hundred | XIV fourteen CCCC four hundred || XV fifteen D five hundred | XVI sixteen º M one thousand XVII seventeen MDCCCXXXVIII 1838 Nº. As often as any letter is repeated, so often is its value re- || ated. º, º fore a greater one, diminishes its value. *...º.º.º. . &&. : Ž Explanation of characters. 11 || QUESTIONS. º: £ What is Arithmetic : When is it a science When || | is it an art 2 What are the fundamental rules of arith- || metic : What is numeration ? What is notation ? What does a unit signify What does two signify Three, &c. : What is meant by the simple value of a unit? || What does the local value of a figure depend on ? How do you write the number ten in figures : Why is the one in this case called a unit of the second order? How many units of the first order does it take to make a unit of the second order How many units of the second order does it require to form a unit of the third order? &c. Repeat the principles of notation and numeration. Repeat the names of each of the first nine orders as ex- pressed in the numeration table. Repeat the name of each of the periods. Repeat the Rule for numeration. |Repeat the Rule for notation. EXPLANATION OF CHARACTERS. | Signs. . Significations. iſ — equal ; as 20s. = £ 1. more ; as 6 + 2 = 8. less; as 8 — 2 = 6. 3: into, with, or multiplied by ; as 6 × 2 = 12. by (i. e. divided by ;) as 6–3–2 = 3; or, 2)6(3. :: : proportionality; as 2:4:: 6: 12. | V or, V Square Root; as V 64 = 8. W . Cube Root; as V 64 = 4. . . RULE. - .- º | Place the numbers with units under units, tens under |tens, &c. Begin the addition at the units, or right hand; |column, and add together all the figures in that column;| |then, if the amount be less than ten, set down the whole| sum: but if greater than ten, see how many tens there || |are, and set down the number above the even tens, and |carry one for each ten to the next column, and proceed|| | with it as in the first. || | Proof–Begin the addition at the top of each column, |and proceed as before, and if the result be the same, it |is presumed to be right, || ExAMPLEs. (1) (2) (3) 4 3 2 . . 2 13 1 2 1 2 1 3 i : ; i i 1 2 3 1 j - N. | 97 9 sum 8.9 7 sum 96 8 sum 14 sum| (5) Here 4, 2, 1, 2and 6 make 15. In fifteen there! is one ten and five units. Set down the five units under the units column, and carry one for the ten || to the next or tens column. jº i : 3 9 4 || 3 2. Then 1,4,3,4,9 and 3 make 24; in 24 there are || two tens, and four over: set down the four under || the column of tens, and carry two to the next or H 3. hundreds column &c., to the last, where the whole| : amount may be set down. N. º A 5 7 4. 6 8 2 5 % % 8 4 5 SIMPLE ADDITION. |（57 * © ® <* ?， CN， n^ _ º cº • • • • © ® °C • • • • • cº № QO OYO QO CNR ■< ∞, ∞， CR Cº r-4 •• №cº do № cº qo (^^ <+ № cº cael © ® ： ao do º №- © ®© ?> C， CN， QO co QO CQ <ł tº <+. eo cº cae <， >， cº QO CQ QO CQ QD CQ se cº <º ： º do cº • • • • • <ſ ) **śºw 3 8 6 1 1 9 3 2 3 6 5 3 2 8 8 0 3 4. *W*Wº §4% ºo QO CIÒ CNR tº CNR _ º cº <* ao cº so |-: • c） Go tº so go `（<* ap ， cº do cº QO ， CQ （Q CR （~ cae cº cº cº CR №. _ ºo º do № cº ao 2 so ao № cº do c> >^ cº nº cº • • • • • • ！ cº <* *> <*<* ?> tº < > ≡ ?>， co QO È* CN- 3，5`` № cº qo <* ºº º Stº cº tº | Cº. <º ： •• • • • • • ） © ?> → Cº QO CD 1 & sc.<* Qo • ***** APPLICATION. & º 1. Add 224 dollars, 365 dollars, 427 dollars, and 784 <º ： • <* CN， QO CQ QO ， cº cº 7 || 42 FXAMPLES. 98754 14107—5 first remainder 2351—1 last remainder 7 first divisor §§§º 7 . add: 5 first remainder *š 12 true remainder spirie division. (3|98437 sºs—3 × 8 + 1 =10 Rem. || º Exercises. º º Divide 9756 by 35 Quotient 278 Rem. 26 8491 by 81 104 67 44767 by 18 2487 1 || 92017 by 56 1643 9 | 38751 by 48 807 15 || 734071 by 72 101.95 31 || APPLICATION. & . 1. Seven boys have 161 apples, which they divide qually among them. How many does each have? Answer, 23. || 2. What is the quotient, if 8736 be divided by 8, and || at quotient by 4? Ans. 273. | 3. If 350 dollars be equally divided among 7 me hat will be the share of each? Ans. 50. 4. How many times are 27 contained in 952? || y trees will there be in each row? Ans. 148. || 6. Several boys who went to gather nuts, collected || 741, of which each boy received 431. How ma bys were there? Ans. 11. || 7. If the expense of erecting a bridge, which is 15036 || ollars, be equally defrayed by 179 persons, what must || ach pay? ºn Ans. 84 dollars. || . Suppose a man receive in one year 2920 dollars; e that his expenses for the year amount to 1769 dol- || ... How much will he save in a year? | Ans. His income will be 8 dollars a day; he will save || 1 dollars in a year: ºº ºl ºº: º * r much a day is his income at that rate; and sup- || SIMPLE DIVISION. 29. Questions. What is division? What do you call the number that is to be divided? What do you call the number you divide by? What do you call the number obtained by division? What do you call that which is left when the work is done? | When the divisor does not czceed 12, how do you per-li form the operation? … * { When the divisor exceed 12, how do you proceed? How do you prove division? ºf How may the operation be performed when there are || cyphers at the right hand of the divisor? - How may it be performed when the divisor is the exactl| |product of two numbers in the multiplication table? How do you obtain the true remainder in the last case?|| PROMIScuous ExERCISEs N THE PRECEDING RULEs. | | 1. If the contents of five bags of dollars, containing |$295, $410, $371, $355, and $520, be divided] | equally among 25 persons, how much is the share of | each? Ans. $78.04 | 2. A man possessed of an estate of $30,000, disposed |of it in the following manner: to his brother he gave |$1500, and the balance to his 5 sons, to be equally di- |vided among them. What was each one’s share? * § 3 ; 3. Ans. $5700. | 3. What number is it, which being added to 9709, | will make 1109012 - An ioiº." 4. Add up twice 327, three times 794, four times: | |31196, five times 15880, six times 95280, and once | |33304. Ans. 812,344. # 5. Three merchants have a stock of 14876 dollars, |of which A owns 4963 dollars, B 5188, and C the re-Î |mainder. How much does C own? Ans. 4725 dolls. H= | 30 - FEDERAL money. FEDERAL MONEY, OR MONEY OF THE UNITED STATES. TABLE. 3 & " " .. , 10 mills make 1 cent 10 cents 1 dime 10 dimes . I dollar 10 dollars . 1 eagle These denominations bear the same relation to each || other as those of units, tens, hundreds, &c. Federal | | money is therefore added, subtracted, multiplied, and |divided by the same rules as Simple Addition, Subtrac- tion, Multiplication, and Division. º | ADDITION of FEDERAL MONEY. | Place the numbers one under another, with mills on | |the right, cents, dimes, &c., in succession; observing to |keep mills under mills, cents under cents, &c. Then | |proceed as in simple addition. - | When halves or fourths of a cent occur, find their |amount in fourths, and consider how many cents these |fourths will make, and carry them to the column of cents. | ExAMPLEs. * |Eagles. Dolls. Dimes. Cents Mills. Dolls. Ds. Cts. || 3 T S 9 5 7 8 : : ; i ; Nore.--In common business, transactions, eagles, imes, and mills are not used: dollars, cents, and frac- |tions of a cent, are the only denominations kept in || |accounts. FEDERAL MONEY. EXAMPLEs. Ds. cts. Ds. cts. 34 , 62 427 , 68 56 , 31 342 , 31 27, 82 427 , 26 23, 68 793, 84 27 , 42 273 , 42 169 , 85 2264 , 51 EXERCISES. (1) (2) (3) Ds. cts. Ds. cts. Ds. cts. 468 , 31 927 , 24 273 , 45 723 , 62 768 , 32 846 , 37 845 , 92 427 , 56 283 , 75 736 , 25 792, 34 846 , 91 846 , 31 587 , 62 674 , 75 428 , 62 842 , 27 273 , 25 | Ds. cts. One half is two-fourths; and one half more make || 437 , 624 four fourths, and three fourths more make seven || | 386 , 81 # fourths, and one fourth more make eight fourths, and 5 one half (or two fourths) more make ten fourths. | 243 , 183 Four fourths make one cent, then ten fourths make | 427 , 37% two cents, and leave two fourths, or one half cent. || 428 , 124 Set down the # cent, and carry the two cents to the º next column. . | 1923 , 124 Ds. cfs. Ps. cts. Ds, cts. 274 , 814 27 , 683 56 , 064 362 , 874 36 , 814 32 , 124 421 , 183 28 , 62% 36 , 25 625 , 314 37 , 934 42 , 62% 241 , 563 24, 624 54 , 814 APPLICATION. % | 1. Add 48 dollars 20 cents; 14 dollars 58 cents; 100 || dollars 25 cents; and 84 dollars 36 cents. . . . | Ans. 247 dollars 39 cents. | 2. Add $7,624, $34,314, $72,064, $41,314, $25, |683, and $87,433 together, and tell the amount. | | . . . . Ans. $268,433. || | 3. Bought a hat for $4.25 cents; a pair of shoes for || | $2,25; a pair of stockings for $1.25, and a pair of gloves || for 75 cents. What is the cost of the whole? º { Ans. $8 50 cents. || | 4. If I buy coffee for $1,183, tea for $2,50, cloves | |for 87%, mace for 933, cinnamon for $1,874, raisins for || | $2,683, nutmegs for 37%, candles for 874, and wine for || |$1,933, what must I pay for them? Ans, $13,25. £3. §§ º Questions. . . . . º | What relation do mills, cents, dimes, &c., bear to each || | other? . . & - - ... : : - - - - | | How are the addition, subtraction, multiplication, and || | division of Federal money performed? -- º | How do you place the numbers to be added? § How do you proceed when halves, fourths, &c., occur? | SUBTRACTION OF FEDERAL MONEY. I RULE.—Place the less under the greater, with dollars || under dollars, and cents under cents; then, if there are || no fractions, proceed as in simple subtraction. . . . . ; If there is a fraction in the upper sum and none in || the lower, set it down as a part of the remainder, and || proceed as before. ; : : & 3. ...& ... & & zs: “...: ::. . . . . ; º § If there is a fraction in each sum, and the lower be || less than the upper, subtract the lower from the upper, i. and set down the difference. || If the lower fraction be greater than the upper one, { borrow one cent, and call it four fourths, and add them || to the upper fraction, and subtract the lower one from || e sum. . Pr of-As in simple subtraction. FEDERAL MONEY. 33 EXAMPLES. Ds. cts. Ds, cts. Ds. cts. 32 , 62 43 , 683 75 , 683 21 , 31 21 , 25 24 , 12% $11 , 31 $22, 433 51 , 564 Not E. –Three fourths cannot be taken from two Ds. cl.g. fourths: then borrow one cent from the two cents, 271 , 62% which has four fourths in it: add the four fourths to 332 - 93 # the two fourths, this makes sir fourths; subtract -- 2 three fourths from sia fourths, and three fourths (#) —— remain. Set down the # and add one to the next 138 2 68%. 3, as in simple subtraction. EXERCISES. Ds. cts. Ds. cts. Ds. cts. 65 , 49 520 , 314 436 , 314 35 , 124 210, 12% 243 , 183 Ds. cts. Ds. cfs. Ds. cts. 273 , 624 237 , 564 732 , 314 124 , 373 142 , 874 261 , 684 APPLICATION. 1. Subtract $432,683 from 1000,933. * . . . * * Ans. $568,25. || | 2. Subtraction shows the difference between two || | numbers; what is the difference between $37,624 and || | $93,874. Ans. $56,25. | 3. Bought goods to the amount of $545,95, and paid | at the time of purchase $350. How much remains un- | paid? . Ans. $195,95. | 4. A merchant bought a quantity of coffee, for which | he paid $560. He afterwards sold it for $610,874. | How much did he gain by the transaction? º | Ans. $50,874. FEDERAL Money. - Questions. - How do you place the numbers in subtraction of | Federal Money? | How do you perform the operation? | If a fraction occur in the upper line or minuend, what | do you do with it? * - . w If a fraction occur in each, how do you proceed? | Suppose the lower fraction is greater than the upper || one, how do you proceed? ~ - - - 'How do you prove subtraction of Federal Money? MULTIPLICATION OF FEDERAL MONEY. RULE.—Set the multiplier under the multiplicand, | and if there be no fractions, proceed as in simple multi- || |plication; observing to separate the cents from the dollars || | in the product. º | If there is a fraction in the sum, multiply it, and see | how many cents are in the product; set down the frac- || |tion that is over, and proceed as before. | Or if the multiplier exceeds 12, multiply the sum, | omitting the fractions; then multiply the fraction, and || * || add the number of cents contained in the product, to the | product of the rest of the sum. º ºs - EXAMPLES. - Ds. cts. Ds. cts. Ds. cts. 12, 50 10 , 564 23, 624 -- — $118, 124 $50, 00, $21, 124 Ds. cts. Ds. cts. | 10,874 125 times 4,183 24 times 3 are || 125 one half make 24 72 fourths: four || 5435 into 125 go 62 1672 tained 18 times in 2174 - 1087 one; that is, *624 one half, mak- — —ing 624 cents. $100,50 times, leaving 836 72 fourths, mak- FEDERAL MONEY. 35 * : EXERCISES. 1 Multiply $145,183 by 7 Ans. $1016,314 2 7,874 by 47 370,124 3 28,683 by 68 1950,75 | 4 42,314 by 58 2454,124 | 5 137,624 by 67 9220,874 6 79,004 by 207 16354,034 || APPLICATION. | 1. What will 8 pounds of cheese come to, at 18 cents | a pound? Ans. $1 44 cts. 2. What is the value of 12 yards of linen, at 35 cents | a yard? … Ans. $4 20 cts. 3. What cost 29 yards of cloth at $2 25 cts. a yard? Ans. $65 25 cts. 4. What will 213 barrels of flour cost, at $525 cents a barrel? Ans. $1118 25 cts. | || 5. Bought 321 barrels of cider at $125 cts. a barrel. || | | What did it amount to? Ans. $401 25 cts. || 6. What will 580 bushels of salt cost at $1 124 cts. a bushel. Ans. $652 50 cts. 7. What is the value of 2 pieces of cloth, one contain-H ing 38 yards, and the other 26 yards, at $3.874 cts. a yard? Ans. $248. | 8. What will be the cost of 132 pieces of linen at || |$17,374 cts. each? Ans. $2293 50 cts. || 9. What will 8 cords of wood amount to, at 4 dollars || |50 cents a cord? Ans. 36 dollars. H 10. Sold 213 barrels of flour for 6 dollars 25 cents per | barrel. What is the amount? Ans. 1331 dols. 25 cts. || | 11. Bought 308 pounds of coffee at 21 cents a pound. || |What is the amount? Ans. 64 dols. 68 cts. | | 12. Bought 217 gallons of brandy at $1 183 cts, per || |gallon; and sold it for $1.374 cts. per gallon. What was || the amount paid for the whole; the sum it sold for; and || |the gain? - | | Ans. Prime cost, $257 683: sold for $298373; gain, || | $40,683 / | DIVISION. | RULE.—Divide as in simple division. When a re- ||mainder occurs, multiply it by 4; and add the number| lof fourths that are in the fraction of the sum (if any) to || its product: divide this product by the divisor, and its | ||quotient will be fourths, which annex to the quotient. | Proof–As in simple division. z. Ds. cts. Ds. cts. Ds, cts. | 2,4522 3)63,183 2)25,374 || º | 22.61 21,064 12,0s, || Ds. cts. ; : z- 2. : |25636, sig5.1s, 32)78800(24,62,] 148 128 ºf 18 Here 18 cents remain; multiply 16 4 18 cents by four, brings them to 4 - fourths of a cent; add the #, this ſº makes 75 fourths: divide 75 fourths 25); j(3 place in the quotient. ExERCISEs. Quotients 5.6 75 by 25, and # are obtained, which * or * | 1, FEDERAL MONEY. 37 | APPLICATION. 1. If 7 pounds of butter cost $1,89 cts., what is the value of 1 pound? Ans. 27 cts. || | 2. If S lbs. of coffee cost $2,04 cts., what is the price || | of one pound? Ans. 254 cts. | 3. Bought 29 yds. of fine linen for $65,25 cts., what was the price per yard? Ans. $2,25. 4. Paid $58,75 cts. for 235 yds. of muslin, what was it per yard? Ans. 25 cts. 5. A piece of cloth containing 72 yds. cost $450, |what was it per yard? . Ans. $6,25. Questions. How do you perform division of Federal Money? -- How do you proceed when a remainder occurs? … º PROMISCUOUS EXERCISES IN THE PRECEDING RULES. 1. Bought 18 barrels of potatoes, each containing 3 bushels, at 25 cts. a bushel, what did they cost? ź Ans. $13,50. 2. A farmer sold 30 bushels of rye at 87 cts. a bushel, 30 bushels of corn at 53 cts, a bushel; 8 bushel of beans at $1.25 cts. a bushel; 2 yoke of oxen at || $62 a yoke; 10 calves at $4 a piece; 15 barrels of cider at $2,374 a barrel, what was the amount of the whole? w . Ans. $251,624. 3. What will be the price of four bales of goods, each | bale containing 60 pieces, and each piece 49 yards, at || 374 cents a yard? Ans. $4410. || 4. Add $324,434 cts. $208,094 cts. and $507,904 cts. together, and divide the sum by 2, and what will be the result? Ans. $520,213. 5. Divide 400 dollars, equally, among 20 persons. || What will be the portion of each person? Ans. $20. 6. Divide 1728 dollars, equally among 12 persons. || What does each one of them share? Ans. $144. || 7. If 240 bushels cost 420 dollars; what is the cost| of one bushel at the same rate? Ans. $1.75. º REDUCTION. : | REDUCTION is the changing of a sum, or quantity, |from one denomination to another, without altering th | value. . . § 3. . . ; | CASE 1. o | To reduce a sum, or quantity, to a lower denomination || º than its own. | | RuLE.—Multiply the sum, or quantity, by that num- | | ber of the lower denomination which makes one of its | own. . | | | | If there are one or more denominations between the | denomination of the given sum, and that to which it is | to be changed, first change it to the next lower than its | own; then to the next lower, and so on to the deno-| |mination required. DRY MEASURE. | | | | | w TABLE. . || 2 pints (pts.) make 1 quart, qt. || 8 quarts - 1 peck, pe. || 4 pecks - 1 bushel, bu. || NoTE.—This measure is used for measuring grain, alt, fruit, &c. . | ExAMPLEs. . NoTE.—1. To reduce bushels to pecks, multiply by º, because each bushel has 4 pecks in it. 1. Reduce 23 bushels to pecks. § 3.3 : . . 3. . . . . ; ; 3. 4 3. Reduce ºbushes to recks. Amt. 140 peeks. . To reduce pecks to quarts, multiply by 8 each peck has 8 quarts in it. Jº º REDUCTION. 39 || &º: 3. Reduce 27 pecks to quarts. pe. 27 8 *** Amt. 216 quarts. 4. Reduce 43 pecks to quarts. Amt. 344 quarts. NoTE.—3. To reduce quarts to pints multiply by 2, I because each quart has 2 pints in it. º 5. Reduce 43 quarts to pints. qt. 43 2 Amt. 86 pints. 6. Reduce 32 quarts to pints. Amt. 64 pints. Reduce 34 bushels to pints. bu. 34 4 Multiply the bushels by 4 to bring || *sº them to pecks. - 136 8 Multiply the pecks by 8 to bring || º them to quarts. 1088 2 And multiply the quarts by 2 to bring *ś them to pints. . Amt. 2176 pints. # EXERCISES, 7. Reduce 56 pecks to pints. Amt. 896 pints. || 8. Reduce 47 bushels to quarts. Amt. 1504 qt. 9. Reduce 85 bushels to pints. Amt. 5440 pt. 10. Reduce 63 pecks to quarts. Amt. 504 qt. 11. Reduce 132 bushels to quarts. Amt. 4224 qt. 12. Reduce 234 bushels to pints. Amt. 14976 pt. || NoTE.—4. When several denominations occur, reduce || || the highest denomination to the next lower one, and this | | again to the next lower, and so on; observing to add || the amount of each denomination, the number there is || of that denomination in the given sum. tº &º REDuc º EXAMPLES. * 1. Reduce 23 bushels, 3 pecks, 5 quarts, 1 pint, to pints. - - bu. pe. qt, pt. Tron. * * * * * , Multiply the bushes by 4| ºn to bring them to pecks, and | 92 * 3 - ?: 3. ... 3 which makes 95 pecks. * Multiply the pecks by 8 to || bring them to quarts, and add || the 5 quarts, which makes || 765 quarts. . . . . . [. 3. tº ºt * * Multiply the quarts by 2 to bring them to pints, and || add the 1 pint which makes || § % º * * * * º > gº tº º º, ſº : # .* º º Multiply by º add the 3, and set down the * º s uce 13 bushels, 2 pecks, 7 *...*.*.*. REDUCTION, . 41 || 2. Reduce 24 bushels, 3 pecks, 1 quart to quarts. Amt. 793 qt. 3. Reduce 7 bushels, 3 pecks to quarts. Amt. 248 qt. 4. Reduce 3 pecks, 2 quarts to pints. Amt. 52 pt. 5 6 . Reduce 7 quarts, 1 pint, to pints. Amt. 15 pt. . Reduce 32 bushels, 0 pecks, 1 quart to pints. . Amt. 2050. | 7. Reduce 5 bushels, 1 peck, 0 quarts, 1 pint to pints. Amt. 337 pt. || 8. Reduce 43 bushels, 1 peck to pints. Amt. 2768 pt. Questions. What is reduction? For what is case first used? . . How do you reduce a sum to a lower denomination| than its own? | How do you reduce bushels to pecks? Why do you multiply by 4? How do you reduce pecks to quarts? Why do you multiply by 8? How do you reduce quarts to pints? How do you reduce bushels to pints? AvoikDUPOIs WEIGHT. TABLE. 16 drams (dr.) make 1 ounce, OZ. | 16 ounces & 1 pound, lb. 28 pounds - 1 quarter of a cwt. Qr. 4 quarters, (or 112 lb.)”. 1 hundred weight, cwt 20 hundred weight 1 ton, T. | Note—By this weight are weighed, tea, sugar, coſºl |fee, flour and other things subject to waste, and all the |metals, except silver and gold. º | * The gross hundred weight of 112 pounds is nearly out of use ;| º || the decimal hundred weight of 100 pounds is taking its place. | r . . . . ExAMPLEs. | 1. Reduce 23 tons to hundred weight. 20 2. Reduce 34 hundred weight to quarters. cwt. 34 4 | Amt. 136 quarters. | 3. Reduce 42 quarters to pounds, qis. 28 336 84 §º - . .” Amt. 1176 pounds. 4. Reduce 73 pounds to ounces. lbs. 73 16 438 ºn Amt. 1168 ounces. 5. Reduce 54 ounces to drams. REDUCTION. - 43 º - 6. Reduce 35 tons to drams. tons. 35 20 700 cwt. 4 * 2800 qr. 28 22400 5600 78.400 lb. 16 470.400 78.400 1254400 oz. 16 7526400 1254400 Amount. 20070400 drams. **** ſ EXERCISES. | 7 Reduce 24 pounds to drams. Amt. 6144 dr." 8. Reduce 36 hundred weight to pounds. * * Amt. 4032 lb. I 9. Reduce 73 quarters to ounces. Amt. 32704 oz. I 10. Reduce 2 tons to pounds. Amt. 4480 lb. I 11 Reduce 4 tons to drams. Amt. 2293760 dr. A 3: REDUCTron. % f 12. Reduce 3 tons, 13 cwt., 2 qu., 14 lbs., to pounds. || T. cwt. qr. lb. -><: 3 3 - 13 - 2 - 14 20 60 Or thus: 13 T. cwt. qr. lb. — 3 - 13 - 2 = 14 73 . 20 - 292 . 4 2 - šš §§§ 294 294 28 28 §§§% 3.4% º 2366 2352 588 588 ... z. §§§ºś 3 #ºš 8246 pounds 8232 14 8246 pounds. 13. Reduce 2 tons. 15 cwt. 2 qr. to quarters. | Amt. 222 qr. || 14. Reduce 3 tons. 25 lb. to pounds. Amt. 6745 lb. | 15. Reduce 5 cwt. 3 qr, 14 lb. to ounces. | - . Amt. 10528 oz. || 16. Reduce 2 cwt. 2 qr. 14 ounces to drams. || Amt. 71,904 dr. º º - º º º . º º TROY WEIGHT. TABLE. 24 grains (gr.) make 1 pennyweight, dwt. 20 pennyweights - 1 ounce, oz. 12 ounces . % 1 pound, lb. | Nore-By this weight, jewels, gold, silver, andl REDUCTION. EXAMPLES. 1. Reduce 32 pounds to ounces. lb. 32 12 Amt. 384 ounces. 2. Reduce 23 ounces to pennyweights. OZ. Amt 1032 grains. | 4. Reduce 53 pounds to grains. lbs. #. 53 12 636 20 12720 24 50880 25440 Amt. 305280 grains. | EXERCISEs. º 1. Reduce 24 ounces to grains. Amt. 11520 gr. 2. Reduce 32 pounds to pennyweights. Amt. 7680 dwt. || || 3. Reduce 132 pounds to ounces. Amt. 1584 oz. | 4. Reduce 234 ounces to grains. Amt. 112320 gr. || º * * | 4 REDUCTION. | | 5. Reduce 463 pounds to grains. Amt. 2666880 gr. | | 6. Reduce 47 pounds, 10 ounces, 15 pennyweights to || | pennyweights. . Amt. 11495 dwt. || | 7. Reduce 5 pounds, 6 ounces, 4 pennyweights, 20 || | grains to grains. Amt. 31796 gr. || §§§§§4%º APOTHECARIES WEIGHT. TABLE. . 20 grains (gr.) make 1 scruple, sc. B 3 scruples §§ 1 dram, dr. 5 8 drams # 1 ounce, Oz. 3 12 ounces # 1 pound, lb. º | NoTE.—By this weight apothecaries mix their medi- || |cines, but they buy and sell by Avoirdupois Weight. | EXERCISES. . Reduce 32 pounds to ounces. Amt. 384 oz. . Reduce 43 ounces to drams. Amt. 344 dr. . Reduce 27 drams to scruples. Amt. 81 sc. Reduce 37 scruples to grains. Amt. 740 gr. : Reduce 28 pounds to drams. Amt. 2688 dr. . Reduce 36 ounces to scruples. Amt. 864 sc. || . Reduce 27 drams to grains. Amt. 1620 gr. || . Reduce 23 pounds to grains. Amt. 132480 gr. | . Reduce 3 pounds, 5 ounces, 2 scruples to scru-|| { % z. Amt. 986. sc. || | 10. Reduce 7 ounces, 5 drams, 14 grains to grains. | Amt. 3674 gr. || | 11. Reduce 27 pounds, 7 ounces, 2 drams, 1 scruple, || |2 grains, to grains. Amt. 159022 gr. | §ºšš% CLOTH MEASURE. N. TABLE. 4 nails (na.) make I quarter of a yard, qr, 4 quarters - 1 yard, yd, 3 quarters - 1 Ell Flemish, E. Fl. 5 quarters - 1 Ell English, E. E. 6 quarters - 1 Ell French, E. Fr. i-By this measure cloth, tapes, linen, muslin, | e measured. REDUCTION. 47 EXERCISES. | 1. Reduce 24 yards to quarters. Amt. 96 qr. 2. Reduce 32 quarters to mails. Amt. 128 ma. 3. Reduce 27 yards to nails. Amt. 432 na. 4. Reduce 46 Flemish ells to quarters. - Amt. 138 qr. 5. Reduce 27 English ells to quarters. - Amt. 135 qr. 6. Reduce 34 French ells to quarters. Amt. 204 qr. | 7. Reduce 45 Flemish ells to nails. Amt. 540 na. || 8. Reduce 36 English ells to nails. Amt. 720 na. 9. Reduce 54 French ells to nails. Amt. 1296 ma. 10. Reduce 13 yards, 3 quarters to quarters. Amt. 55 qr. 11. Reduce 3 quarters, 2 nails, to nails. Amt. 14 na. | 12. Reduce 24 yards, 2 nails to nails. Amt. 386 na. | 13. Reduce 13 E. ells, 2 qrs., 3 nails to nails. - Amt. 271 na.| LONG MEASURE. TABLE. 12 inches (in.) make 1 foot, ft. 3 feet gº i yard, yd. 54 yards * 1 Rod, Pole, or Perch, p. 40 poles º I Furlong. - 8 Furlong & 1 Mile. º 3 Miles & 1 League. 60 Geographic, or 60 sºliºs"' degree. | NoTE.—This measure is used for length and dis- | |tances. - . | | A Hand is a measure of four inches, and is used in | | measuring the height of horses. - - A Fathom is 6 feet, and is chiefly used in measuring || the depth of water. - | #3 º 3. º % §§§ { §§ Reduction. - EXERCISEs. { . Reduce 23 leagues to miles. Amt. 69 m. || . Reduce 43 miles to furlongs. Amt. 344 f. . Reduce 27 furlongs to poles. Amt. 1080 p. 1 . Reduce 56 poles to yards. Amt. 308 yd. || . Reduce 132 yards to feet. Amt. 396 ft. || . Reduce 76 feet to inches. Amt. 912 in. || . Reduce 24 miles to poles. Amt. 7680 p. . Reduce 32 furlongs to yards. Amt. 7040 yd. || . Reduce 86 poles to inches. Amt. 17028 in. || . Reduce 26 leagues to yards. Amt. 137280 yd. |} . Reduce 52 miles to feet. Amt. 274560 ft. . Reduce 5 leagues to inches. Amt. 950400 in. I | 13. Reduce 24 degrees to statute miles. Amt. 1668 m. || | 14. Reduce 12 miles, 3 furlongs, 25 poles to poles. º w Amt. 3985 po. 15. Reduce 14 leagues, 2 fur & º ; ; longs to poles. . . 16. Reduce 3 leagues, 2 miles, 6 furlongs, 18 poles to . Amt. 20779 yds. || º LAND, OR squaRE MEASURE. |144 square inches make 1 square foot, ºft. | | 9 square feet - I square yard, yd. |304 square yards - 1 square perch, p. || 40 square perches - I rood, r. 4 roods * 1 acre, * : * a. 3. § 3. . º. . . ... º.º. §§ - * , , , , , , * … .º.º. 3 } #: . Note:-This measure is used to ascertain the quan-li of lands, and of other things having length and || |breadth to be estimated. exercises. 1. Reduce 27 acres to roods. 2. Reduce 53 roods to perches. REDUCTION. . Reduce 28 perches to square yards. | Amt. 847 sq. yds. . Reduce 36 square yards to square feet. 324 ft. || | . Reduce 27 square feet to square inches. | . > Amt. 3888 in. || . Reduce 34 acres to perches. Amt. 5440 p. . Reduce 42 roods to square yards. š. % x Amt. 50820 sq. yds. || . Reduce 24 square perches to square feet. º 3 6534 | . Reduce 32 roods to square feet. Amt. 34S480 ft. || . Reduce 23 acres to square inches. | Amt. 144270720 sq. in. . Reduce 11 acres, 2 roods, 19 perches to perches. || Amt. 1859 p. ; 1 I I 12. Reduce 17 acres, 3 roods to perches. º 3. . Amt. 2840 p. | 13. Reduce 12 acres, 2 roods, 12 perches to square | yards. Amt. 60863 sq. yd.] | || CUBIC, OR SOLID MEASURE. . | TABLE. 1728 cubick inches make 1 cubic foot 27 feet 1 cubic yard 40 feet of round timber, or i Ton or load 50 feet of hewn timber, ; on or loa 128 solid feet 1 Cord of wood | | NoTE.—This measure is employed in measuring |solids, having length, breadth, and thickness to be esti- ºil mated. º | | | ExERCISEs. ºf 1 Reduce 29 cords of wood to cubick feet. | 2 Reduce 32 cubic yds. to feet. Amt. 864 c.f. I || 3 Reduce 23 cubic feet to inches. Amt. 39744 c. | || 4 Reduce 32 cubic yds. to inches. Amt. 1492992 c. in. || || 5 Reduce 2 cords of wood to inches. Amt. 442368 c. in. | 6 Reduce 3 cords, 10 feet to feet. Amt. 394 ft | 7 Reduce 1 cord, 3 feet, 136 inches to inches. Amt. 226504 in ::::::::::::::::: LIQUID MEASURE. TABLE. ~ 4 gills make 1 pint pt. 2 pints (pts) 1 quart qt. | 4 quarts ; : , "ººz. . . . 1 gallon gal. 42 gallons 1 tierce te. | | 63 gallons 1 hogshead hid. 2 hogsheads 1 pipe or butt pi. “ 2 pipes ; 1 tun. T– : | NoTE.—This measure is employed in measuring |cider, oil, beer, &c. º º § EXERCISES. - tº 1 Reduce 23 tuns to pipes. Amt. 46 pi. 2 Reduce 43 pipes to hogsheads. Amt. 86 hbd. || 3 Reduce 34 hogsheads to gallons. Amt. 2142 gal. || 4 Reduce 27 tierces to gallons. Amt. 1134 gal. 5 Reduce 53 gallons to quarts. Amt. 212 qt. || 6. Reduce 724 quarts to pints. Amt. 1448 pt. || 7 Reduce 37 pints to gills. Amt. 148 g. || 8 Reduce 12 pipes to gallons. Amt. 1512 gal. 9 Reduce 4 hogsheads to quarts. Amt. 1008 qt. || 10 Reduce 32 gallons to gills. Amt. 1024 g. || 11 Reduce 2 tuns to gills. Amt. 16128 gills || 12 Reduce 32 gals 3 qts. to pints. Amt . 13 Reduce 2 hogsheads, 27 gals. 3 qts. t º ~ - Amt. 615 qt. | Reduce 3 tons, 1 hogshead, 15 gals. 1 qt. to pints. MOTION, OR CIRCLE MEASURE. TABLE. º 0 seconds ("sec) make 1 minute minutes 1 degree 0 degrees I sine 12 sines (or 360 degrees) 1 revolution | NoTE–This measure is employed by astronomers n navigators, &c. H º s : - o - º - | REDUCTION. EXERCISES. 1 Reduce 5 sines to degrees. Amt. 1509 | | 2 Reduce 8 degrees to minutes. Amt. 480" || 3 Reduce 6 minutes to seconds. Amt. 360 sec. | 4 Reduce 12 sines to seconds. Amt. 1296000 sec. | 5 Reduce 3 sines 15 degrees to minutes. Amt. 6300 min. | TIME. TABLE. 60 seconds (sec) make 1 minute min. 60 minutes 1 hour H. 24 hours 1 day 7 days 1 week 12 months (or 365 days) 1 year. | NoTE.—The true year, according to the latest and | | most accurate observations, consists of 365 d. 5 h. 48 m. and 58 sec: this amounts to nearly 3654 days. The com- | mon year is reckoned 365 days, and every fourth or || | leap year one day more on account of the fraction omit- | ted each year, which being put together, every fourth year is added to it, making leap year 366 days. The year is divided into 12 months as follows. The fourth, eleventh, ninth and sixth, Have thirty days to each affixed, And every other thirty-one, Except the second month alone, Which has but twenty-eight in fine, Till leap year gives it twenty-nine. OR THUS : Thirty days hath September, April, June, and November, February hath twenty-eight alone, And each of the rest has thirty one. . . . ; | When the year can be divided by four, without a re- | mainder, it is bissextile, or leap year. º REDUCTION. EXERCISES. 1 Reduce 42 years to months. Amt. 504 m. 2 Reduce 23 days to hours. Amt. 552 h. || 3 Reduce 36 hours to minutes. Amt. 2160 min. || 4 Reduce 25 minutes to seconds. Amt. 1500 sec. | 5 Reduce 14 days to minutes. Amt. 20160 min. || | 6 Reduce 52 hours to seconds. Amt. 187200 sec. 7 Reduce 13 weeks to hours. Amt. 2184 h. 8 Reduce 12 weeks to minutes. Amt. 120960 min. ||| 9 Reduce 3 years to minutes, allowing 365 days to | each year. Amt. 1576800 min. - . .” -- | | 10 Reduce 15 years and 6 months to months. | | Amt. 186 m. | 11 Reduce 4 weeks, 3 days, 22 hours, to hours. | | 23 Amt. 766 h. || | 12 Reduce 7 years, 24 days, 43 minutes, to seconds. - Amt. 222828.180 sec.| STERLING MONEY. TABLE. 4 farthings (qr) make 1 penny d. 12 pence 1 shilling s. 20 shillings 1 pound - Farthings are usually written as fractions of a penny, # three farthings. * 1 Reduce 14 pounds to shillings. 2 Reduce 23 shillings to pence. 3 Reduce 34 pence to farthings. 4 Reduce 4 pounds to pence. - ------------------- 3::::::: 5 Reduce 13 shillings to farthings. Amt. 624 q 7 Reduce 13 pounds 14 shillings, to pence. ... . tºº - - - - - s sº ---- ºr º % A - it. * 2. r º 8 Reduce 3 pounds 15 shillings 6 pence to farthings. º Amt. 3624 q REDUCTION, FEDERAL MONEY. TABLE. 10 mills make 1 cent 10 cents I dime 10 dimes 1 dollar 10 dollars 1 eagle EXERCISES. 1 Reduce 5 eagles to cents. Amt. 5000 ct.| 2 Reduce 3 dollars to mills. Amt. 3000 m. || | 3 Reduce 15 dimes to cents. Amt. 150 ct. || | 4 Reduce 3 eagles, 5 dollars to cts. Amt.3500 ct. 5 Reduce 7 dollars, 3 dimes, 6 cents, to mills. N. Amt. 7360 m. || As eagles, dimes and mills are not used in accounts, they will generally be omitted in the subsequent exer- cises of this work. # 4 fourths, or 3 thirds, or 2 halves, make 1 cent. 100 cents #3% & § 1 dollar. | 6 Reduce 125 cents to halves of a cent. § Amt. 250 halves. || || 7 Reduce 32 cents to fourths of a cent. - Amt. 128 fourths. | 8 Reduce 23 dollars to cents. Amt. 2300ct. # 9 Reduce 25 dollars 15 cents to cents. Amt. 2515 ct. || 10 Reduce 15 dollars 374 cents to halves of a cent. || Amt. 3075 halves. || 11 Reduce 21 dollars 15 cents to thirds of a cent. º | Amt. 6345 thirds. || |12 Reduce 5 dols. 374 cents to fourths of a cent. || x Amt. 2150 fourths. || | 13 Reduce 15 dollars 334 cts. o thirds of a cent. || Amt. 4600 thirds. || NoTE. To reduce dollars to cents annex two cyphers: || || thus 53 dollars are 5300 cents. * * : # 3:33 To reduce dollars and cents to cents, place them to- *** **, ºgº, REDuction. | gether without any separating point, and the amount ||will be cents. Thus 35 dollars 24 cents are 3524 cents. || § Questions. |For what purpose is Dry measure used? |For what is Avoirdupois weight used? |For what is Troy weight employed? |For what is Apothecaries weight employed? |For what is Cloth measure employed? |For what is Long measure used? |For what is Land or Square measure used? |For what is Cubick measure employed? |For what is Liquid measure employed? |For what is Sterling currency used? |For what is Federal currency used? | . CASE 2. s | To reduce a sum or quantity to a HIGHER denomina-l |tion than its own . || | RULE.—Divide the sum or quantity by that number of |its own denomination which makes one of the denomina- |tion to which it is to be changed. | | When there are one or more denominations between the denomination of the given sum and that to which it is || to be changed; first change it to the next higher than its || own, and then to the next higher, and so on. | Remainders are always of the same denominations as || he sums divided. . <--- * DRY MEASURE. ExAMPLEs. I Reduce 25 pints to quarts. pts. º NoTE-Divide by 2, because every 2 pints 2)25 make one quart. In 25 are 12 two's and — 1 over, that is 12 quarts and 1 pint. qt.12—lpt||| Reduce 43 quarts to pecks, qt. || Divide by 8, because every 8 qts. make 8)43 peck. In 43 are 5 eights and 3 over, — REDUCTION. 55 | | 3 Reduce 26 pecks to bushels. bu. || Divide by four because every 4 pecks 4)26 ll make 1 bushel. In 26 are 6 fours and *éseasºas # * |2 over; that is 6 bushels and 2 pecks, bu.6—2 pecks 4 Reduce 359 pints to bushels. pt. | Divide pints by 2, brings them 2)359 |to quarts; divide quarts by 8, brings — them to pecks, and divide pecks by 8)179—1 pt. 4 brings them to bushels. 4) 22–3 qt. 5 b. 2 p. 3 qt. I pt. | 5 Reduce 81 quarts to bushels. A. 2 bu. 2 pe. 1 qt. | 6 Reduce 134 pints to pecks. 8 pe. 3 qt. 7 Reduce 194 pints to bushels. 3 bu.0 pe. I qt. Questions. What is reduction? For what is case second used? How do you reduce a sum to a higher denomination || |than its own? | When there are one or more denominations between || |the denomination of the given sum and the one to || |which you wish to reduce it, how do you proceed? Of what denomination is the remainder always? How do you bring pints to quarts? How do you bring quarts to pecks? How do you bring pecks to bushels? How do you bring pints to bushels? | : AVOIRDUPOIS WEIGHT. ğ. || 1 Reduce 65 cwt. to tons. , Result 3 tons 5 cwt. ii. | 2 Reduce 27 quarters to cwt. Res. 6 cwt. 3 qr.| | 3 Reduce 109 pounds to qr. Res. 3 qr. 25 lb.] | 4 Reduce 123 ounces to pounds. Res. 7 lb. 11 oz. [] | 5 Reduce 234 drams to ounces. Res. I4 oz. 10 i. | 6 Reduce 4274 drams to pounds. Res. 16 Ib. 11 oz. 2 dr.] || 7 Reduce 175 quarters to tons. Res. 2 tons 3 cwt.3 qr. | 8 Reduce 6745 pounds to tons. Res. 3 tons 25 lb. REduction. TROY WEIGHT. * * * || 1 Reduce 378 ounces to pounds. Result,31 lbs. 6 oz. | 2 Reduce 235 pennyweights to ounces. * || * * * Res. 11 oz. 15 dwt. 3 Reduce 748 grains to pennyweights. . Res. 31 dwt. 4 grains. 4 Reduce 678 pennyweights to pounds. Res. 2 lbs. 9 oz. 18 dwt. 5 Reduce 732 grains to ounces. . . Res. 1 oz. 10 dwt. 12 grains. 6 Reduce 14752 grains to pounds. 3. Res. 2 lbs. 6 oz. 14 dwt. 16 gr. APOTHECARIES WEIGHT. . 1 Reduce 432 ounces to pounds. Res. 36 lbs. 2 Reduce 782 drams to ounces. Res. 97 oz. 6 dr. 3 Reduce 91 scruples to drams. Res. 30 dr. 1 scr. || 4 Reduce 192 grains to scruples. Res. 9 sc. 12 gr. || 5 Reduce 256 scruples to ounces. . º | Res. 10 oz. 5 dr. 1 scr. || 6 Reduce 12660 grains to pounds. . . . Res. 2 lb. 2 oz. 3 drs. . CLOTH MEASURE. 1 Reduce 60 quarters to yards. Res. 15 yds. || 2 Reduce 60 quarters to English ells. || Res. 12 E. ells. |} 3 Reduce 60 quarters to French ells. || . Res. 10 Fr. ells, hº 4 Reduce 60 quarters to Flemish ells. Res. 20 FI. ells. || 5 Reduce 52 nails to quarters. Res. 13 qr. || 6 Reduce 123 nails to yards. Res. 7 yds. 2 qr, 3 na. || 7 Reduce 543 nails to English ells. 8 || LoNG MEASURE. duce 36 miles to leagues. ... Res. 121 -longs to miles. Res. 9 m. 3f. f º arouotion. 㺠57| | 3 Reduce 295 poles to furlongs. Res. 7 f. 15 p. |4 Reduce 286 yards to poles. Res. 52 p. I 5 Reduce 365 feet to yards. Res. 121 yds. 2 ft. | 6 Reduce 759 inches to feet. Res. 63 ft. 3 in. I 7 Reduce 253 inches to yards. Res. 7 yds. 0 ft 1 inch. || 8 Reduce 2792 poles to leagues. Res. 2 1.2m.5f 32 p. SQUARE MEASURE. … 1 Reduce 287 roods to acres. Result 71 a. 3r. |2 Reduce 245 perches to roods. Res. 6 r. 5 p. 3 Reduce 756 square feet to yards, Res. 84 yds. || 4 Reduce 4731 square yards to perches. | --> Yals. Res. 156 p. 12 yds. º 304 4731 Bring the 304 yards and the 4731 yards 4 4. both to fourths, and divide. The remainder - 48, is fourths of a yard; divided by four, T | Tº - - , brings it to yards, the true remainder. 121 | 18924 156 p. rings it to yards, rue remainde 121 682 605 774 726 *ś 4 || 48 Rem. ***** 12 yards. 5 Reduce 3575 square inches to feet. | Res. 24 feet 119 inches. || 6 Reduce 1728 square perches to acres. | - Res. 10 a. 3r. 8 p. CUBIC MEASURE. || || 1 Reduce 789 cubic feet to cords. Result, 6 c. 21 ft. ||2 Reduce 343 cubic feet to yards. Res. 12 yds. 19 fill || 3 Reduce 9386 cubic inches to feet. Res. 5 fl. 746 in.]] 4 Reduce 703539 cubic inches to cords. || Res. 3 c. 23 ft. 243 in. REDUCTION. | LIQUID MEASURE. | |1 Reduce 25 pipes to tuns. Res. 12 T. 1 P. |2 Reduce 34 hogsheads to pipes. Res. 17 P. ||3 Reduce 1575 gallons to hogsheads. Res. 25 hbds.| |4 Reduce 163 quarts to gallons. Res. 40 gal. 3 qt. | ||5 Reduce 6048 pints to tuns. Res. 3 tuns. | º MOTION. ||1 Reduce 1440 seconds to minutes. Result, 24 min. || |2 Reduce 720 minutes to degrees. Res. 12 deg. || |3 Reduce 342 degrees to sines. Res. 11 sines 12 deg. || |4 Reduce 443907 seconds to sines. & || | Res. 4 sines 3 deg. 18 min. 27 sec. I |1 Reduce 1800 seconds to minutes. Result, 30 m. || |2 Reduce 720 minutes to hours. . Res. 12h. |3 Reduce 744 months to years. r Res. 62 yrs. || ||4 Reduce 4649 minutes to days. Res. 3d. 5h. 29 min- ||5 Reduce 48888 minutes to weeks. , º, ... ; ; , | | Res. 4w. 5d. 22hrs. 48 min. | STERLING MONEY. | |1 Reduce 78 shillings to pounds. Res. 3ſ. 18s. || |2 Reduce 93 pence to shillings. Res. 7s. 9d. || |3 Reduce 39 farthings to pence. Res. 9d. 34r || |4 Reduce 656 pence to pounds. Res. 2.É 14s. 8d. || |5 Reduce 781 farthings to shillings. Res. 16s. 3d. Iqr. || |6 Reduce 6529 farthings to pounds. § | ſº º Res.66. 16s. 0d. l qr, FEDERAL MONEY. || Reduce 250 halves to cents. Res. 125 cts.]] Reduce 128 fourths to cents. Res. 32 cts. || educe 2343 cents to dollars. Res. 23 dol.43 cts. || |4 Reduce 15374 cents to dollars. Res. 15 dol. 374 cts. || |5 Reduce 6150 half cents to dollars. Res. 30 dol. 75 cts. || ote.--To reduce cents to dollars, separate two figures || he righthand for cents; those on the left will be dollars.[ REDUCTION. 59 3. PROMISCUOUS EXERCISES. 1 How many bushels in 738 quarts? Ans. 23 bushels 2 quarts. 2 In 7 bushels, how many pints? Ans. 448 pints. || 3 How many cwt. in 5356 ounces? Ans. 2 cwt. 34r. 261b. 12 oz. 4 How many drams in 3 qr. 23 lb. 14 oz.? | Ans. 276.16 drams. 5 How many grains in 9 oz. 14 dwt. 3 gr.? | Ans. 4659 gr. 6 How many pounds in 7432 dwt. Ans. 30lb. 11 oz. 12 dwt. || || 7 How many scruples in 15 lb. 1 oz. 6 drams? § . Ans. 4362 scr. || 8 How many ounces in 216 scruples? Ans. 9 oz. . 9 How many furlongs in 2346 yards? Ans. ; ; ; 10f. 26p. 3yd. || | 10 How many poles in 3 leagues? Ans. 2880 poles.| 11 How many yards in 84 nails? Ans. 5 yd. I qr. | 12 How many perches in 4719 square yards º * Ans. 156 perches.| | 13 How many square yards in one acre? | | Ans. 4840 sq. yds.] |14 How many hogsheads in 9728 gills? | # Ans. 4 hbds. 52 gal.]] | 15 How many pints in 2 pipes? Ans, 2016 pints.]] 16 How many minutes in 3 days 6 hours? Ans. 4680m. || 17 How many hours in 2 weeks and 4 days? | | Ans. 432 hours. || |18 How many shillings in 27 four-pences? Ans. 9s.]] 19 How many cords of wood in 9334 cubic feet? Ans. 73 cords 20 feet. || |20 How many cubic feet in 9 cords? Ans. 1152 feet. |21. How many inches round the globe, which is 360 de-|| |grees of 69% miles each? Ans. 1,585,267,200 inches, | Enumerate the answer. . iº coMPound ADDITION. Compound ADDITION is the art of collecting several || |numbers of different denominations into one sum. RULE. | Place the numbers so that those of the same denom-ji |ination may stand directly under each other, observing || | to set the lowest denomination on the right, the next || | lowest next, &c. : | Then add up the several columns beginning with the | lowest denomination: divide the sum by as many of the | number of that denomination as it takes to make one of H. the next; and so on. - Proof–As in Simple Addition. DRY MEASURE. ExAMPLEs. ## The first column on the right makes || 1 five pints. Five pints make two quarts || I and leaves one pint. Set down the one || pint under the column of pints and carry|| } the two quarts to the column of quarts. I It pt º The column of quarts with the two quart - added makes twenty four quarts. Twen- ty-four quarts make three pecks and leave || 1. no quarts. Set down 0 under the column || of quarts and carry the three pecks to the — column of pecks. ºš º tº umn of pecks with the three pecks added makes fourteen lecks. Fourteen pecks make three bushels and leave two pecks. | et down the two pecks under the column of pecks and carry the aree to the column of bushels. , - - The column of bushels with the three bushels added makes | -four bushels. Here set down the whole amount. - | 3.3%anºrºż%22&%% ºw?&%% ºxº COMPOUND ADDITION. 1 § . .. bu. pe. qt. pt. bu. pe. qt. bu. pe. qt. p 23 3 7 1 4 3 7 3 7 || || 34 2 6 1 5 2 6 2 6 1 || 42 3 5 1 6 || 5 0 5 0 51 1 4 1 7 () 4 | 4 || || 23 2 3 ]. 8 3 3 3 3 1 || || 14 1 2 1 4 I 2 0 2 1 || | 1 || 3 4 | 3 2 4 2 1 1 | 202 3 2 1 40 3 7 3 2 7 0 || APPLICATION. || 1 Add 2 bu. 3 pe.; 7 bu. 3 qt.; 4 bu. I pe. I pt.; 6 bu. || |4 qt. 1 pt.; and 3 pe. I qt, together. ge | Amount 21 bu. 0 pe. I qt. 0 pt. | 2 Add 3 bu. 2 pe.3 qt. 1 pt.; 7 bu. 7 qt. 1 pt.; 3 pe. I pt.; 4 bu. 5 qt.; 4 bu. 3 pe.; 8 bu. 3 pe. 7 qt. 1 pt. together. || %;. Amt. 29 bu. 2 pe. 0 qt. 0 pt. || 3 Add 7 bu. I pt.; 3 pe. 7 qt. I pt.; 6 qt. 1 pt.; 9 bu. 3| | pe. 6 qt. 1 pt.; 3 bu.3 qt.; 4 bu. 1 pe. Amt. 25 bu. 2 pe. 4 In a wagon load of grain contained in seven sacks, | viz: in the first 4 bu. 3 pe. I qt.; in the second 5 bu. 7| | qt. 1 pt.—the third 3 bu. 1 pe. 1 pt.—fourth, 3 bu. 2 pe. |6 qt.—fifth, 5 bu-sixth, 4 bu. I pe. I pt.:—and in the |seventh 6 bu. 1 pe. 1 pt. How many bushels? % § º § ſº Questions. What is Compound Addition? * / How do you place the numbers to be added? ; | Do you place the greater or smaller denominations|| | in the right hand column? i | Where do you begin the addition? : | When the first column is added, how do you proceed|| | with the sum? # | | When you divide the sum by as many of that denom- | ination as make one of the next; which do you set down, || || the remainder or the quotient? . | | What do you do with the quotient? compound ADDITION. | In what particular does compound addition differ from || simple addition? * | Do you carry one for every ten in compound addition? || Since you do not carry one for every ten, how many | do you always carry? A. One for as many of any de-|| nomination as make one of the next. Here the pupil will have something with which to compare simple addition, in which he carries one for every ten. This comparison will improve and correct his understanding of the elementary rules. AVOIRDUPOIS WEIGHT. - T. cwt. qr. lb. T. cwt. qr. lb. oz. dr. (1) 15 3 2 15 (2) 7 11 2 16 4 13| 4 8 3 9 15 7 3 8 16 7 | 82 19 1 10 138 19 1 12 8 13|| 163 8 3 17 42 8 3 19 12 4 || 34 15, 2 24 , 357 6 2 8 3 3 || *†: 300 16 1 19 PoTHECARIES WEIGHT. (2) 84 132 16 1427 14 ; : "...º. -- º # ...:I- - # : . l WEIGHT. lb. oz. dwt. lb. oz.dwt. gr 1) 47 10 12 (2) 185 2 1920 TRoy wei compound Abortion. *ś CLOTH MEASURE. 3/ds. gr. na. E.E. gr. ma. E.Fl. qr. na (1) 75 3 2 (2) 72 3 2 (3) 19 2 3 163 || 3 536 2 1 728 I 2 245 2 () 847 1 3 142 0 1 738 3 1 1453 () 2 816 0 0 1785 2 3 41 2 () 32 || 2 3009 1 1 LONG MEASURE. L. M. fur. P. Ayd...ft. in. (1) 5 2 4 17 (2) 3 2 11 º 16 I 3 10 I 1 9 72 0 5 24 2 0 8 | 526 0 3 12 3 1 10 834 2 6 34 2 0 4 38 0 3 12 6 2 7 1493 2 2 29 * SQUARE MEASURE. A. R. P. A. R. P. (1) 39 2 37 (2) 487 2 17 º 62 1 17 25 3 28 | 68 0 38 67 0 32 º 129 3 12 45 1 16 | 532 1 18 26 0 29 832 2 2 compound Appirion. &:------~~~~~~~~~~ CUBic MEASURE. yd. ft. in. cords. feet. in. (x) 75 22 1412 (2) 37 119 1015 9 26 195 9 110 159 3 19 1091 48 127 1071 28, 15 1110 . 8 111 956 49 24 218 21 9 27 186 is 67 135 122 sº || 3 In four piles of wood; the first containing 32 feet 149|| inches; the second 121 feet 1436 inches; the third 97 || feet 498 inches; the fourth 115 feet 1356 inches; how; much did the whole amount to? sº , , º, Ans. 2 cords, 110 ft. 1711 in. 4 In six boat-loads of wood: the first containing 22|| cords 114 feet, 987 in.; the second 18 cords, 121 feet, 1436 in.; the third 21 cords, 109 feet, 1629 in.; the] fourth 15 cords, 82 feet, 1321 in.; the fifth 16 cords, 98|| feet, 1111 in,; the sixth 24 cords, 89 feet, 987 in. How| much did they contain? ~~~~ X- Ans. 120 c. 105 ft. 559 in. "16 15 ºg áš 740 1 18 COMPOUND ADDITION. 65|| - * MOTION. O 1. f / SIN. O H f f (1) 17 55 48 (2) 1 25 49 51 || I 37 51 2 4 21 36 || 28 19 45 4 19 47 18 19 19 37 1 25 25 39 || 67 13 1 10 15 24 24 || 3. Add 5sin. 10° 46' 38; 11° 37' 18"; Isin. 17912 || |18"; 2sin. 52"; Isin. 15° 12'23"; and 11° 57'29" to-I gether. Ans. 11 sin. 6° 46'58". || 4 Add 45'; 1sin. 9° 18'; 14° 21' 34"; 2sin. 8° 13' || 54"; 4sin. 79 12, 19"; and 47° 32” together. | & Ans. 8sin. 10° 20' 37". TIME. … Y. M. we, d. h. H. min. sec. | (1) 17 11 (2) 3 5 20 (3) 20 52 40 | 172 9 2 3 17, 122 12 35 35 7 3 6 22 68 9 17 4 10 0 4 16 135 17 12 6 0 3 19 24, 35 28 31 7 II 3 22 371 7 12 STERLING MONEY. º s. d. f s. d. f s. d. | (1) 2 3 4 (2) 7 9 44 (3) 4 6 4 || 7 I 2 13 7 63 47 19 7 || 9 7 3 4 5 2 159 5 3 || 5 2 24 10 18 104 78 6 114|| 23 13 114 º | 4 Add g763 7s. 4d. e894, 9d.; Elgº 17s. 2d.;| £459 15s.; £473 12s. 8d together. | Ans. £1898 16s. 11d. *ś * *:::::: º **** * # - |66 compound subtraction. § | 5 Add the following sums: viz. É69 18s. 7d.; £175 | |2s. 6d.; £1582 19s 4d.; £175 13s. 9d.; £143 13s. 8d.;| |and £212 0s. 7d Ans. É2359 8s. 5d. || &::... º. COMPOUND SUBTRACTION. | CoMpound SUBTRACTION is the art of finding the dif-|| |ſerence between two numbers consisting of several de- |nominations. | RULE. | Place the numbers, as in compound, addition with the |less under the greater: then begin at the right hand || |denomination and subtract the lower number from the || |upper, and set down the remainder. | If the upper number of any denomination be less than || |the lower one, add to the upper one as many as it takes || |of that to make one of the next; subtract the lower num-l |ber from the amount and set down the remainder as || |before. § 3. 3 § | Proof–As in simple subtraction. EXAMPLEs. ź. bu. pe. qt. pt. 42 3 6 1 31 2 3 || 11 1 so We cannot take 7 quarts from 5 quarts; then | borrow 1 from the 2 pecks. One peck has 8 || quarts in it: 8 quarts added to the 5 quarts, make 13 quarts. Take 7 qts, from 13 qts, and || 6 qts. remain. Set down the 6 qt. º ºn Because I borrowed I from the 2 quarts, I || st add one to the 3 below it, which makes the lower figure 4. Now || pecks from 2 pecks we cannot take: then borrow one bushel from the I. |9; that bushel has 4 pecks in it; 4 p. and 2 p. make 6 p. Now 4 p. |from 6 p. and 2 pecks remain, which set down. . . . | Because I borrowed 1 from the 9, I must add 1 to the figure below || to 2 make 3. "Take 3 from 9 and 6 remain. Set down the 6, [. compound subtraction. bu, pe. qt. pt. bu. pe. qt. pt. 8 2 7. | 8 ! 3 0 4 : 6 1 4 2 5 3 3 1 0 3 2 5 l bu. pe. qt. bu. pe. qt. pt. 95 3 2 28 2 2 () 22 0 1 14 3 5 1 APPLICATION. 1. From a granary containing 94 bushels, 2 pecks, 7 | quarts, have been taken 43 bush. 3 pe. 5 qr. How much | remains? Ans. 50 bush. 3 pe. 2 qt. | 2. From a wagon load of corn containing 63 bushels, || 3 pecks, 4 qts., have been sold 27 bush. 3 pe.7 qt. 1 pt. |How much remains unsold? Ans. 35 bu. 3 p. 4qt. I pt. Questions. What is compound subtraction? In what particular does it differ from simple subtrac- |tion? tion? Where do you begin the operation? When the upper number of any denomination is less than the lower one, how do you proceed? x & . Do you borrow one from the next? Do you call the number you borrow, one ten, as in simple subtraction? What do you call it? the case may be? What do you do with it then? Ans. I reduce it to quarts, or to feet, or to furlongs, &c. according to the nature of the case; then add these ğ | How do you place the numbers in compound subtrac- Ans. I call it one peck, or one yard, or one mile, as | corrpound subtraction. || to the upper figure on the right, subtract the lower figure || | from the sum, and set down the remainder. . | When you borrow one from the upper figure, why do | | you add one to the figure below it? - | NotE.-Upon a clear conception of the principles involved in these | questions, depends the pupil’s correct knowledge of the science of | | Arithmetic. AVOIRDUPOIS WEIGHT. | tons cut. qr. tons cut. qr. lb. cwt. qr. lb. oz. dr. || | From 45 11 3 52 12 3 15 17 0 || 0 0 0 || | Take 15 10 2 24 10 0 26 6 3 21 15 9 7 |Rem, 30 1 1 28 2 2 17 10 0 6 0 | 1. Subtract 76 tons, 18 hundred weight, 3 quarters, | from 195 tons, 2 hundred weight, 2 quarters. . | Ans. 118 tons, 3 cwt. 3 qr. || | 2. Subtract 14 pounds, 6 ounces, 3 drams from 20 || | pounds, 2 ounces. Ans. 5 lbs. 11 oz. 13 dr. || APOTHECARIES WEIGHT. 3 3 fB 3 3 B gr. 1090 1 6 48 9 6 1 4 106 2 7 1 10 0 2 8 | 983 :0 7 . || 3. From 59th 1323 take 53f6 7 # 53. Ans. 5ft, 53 53. || 4. Subtract 14593 13 from 69ffi. Ans. 54th 23 73. : TROY WEIGHT. | lb. oz. dwt. gr. lb. oz, dwt. gr. lb. oz, dwt. gr. || 10 6 18 () 8 3 0 2 106 0 0 15 || 4 0 220 2 1 18 6 10 6 2 20 || 6 6 154 4. Subtract 14lb. 6oz. 11dwt. from 22lb. 12dwt. 6 gr. || Ans. 7lb. 6oz. Idwt. 69r. From 16lb. take 12lb. 11oz. 10duct. 11gr. || Ans. 31b. 0oz. 9dwt. 13gr. || COMPOUND SUBTRACTION. 69 | |perches. 288 A. 2 R. 29 P. | Take 25 2 39 36 3 || 14 7 10|| CLOTH MEASURE. Syds. Qrs. na. E.E. qrs. na. E. Fl. qrs. na. From 71 3 1 42 () 2 5.1 2 2 Take 14 2 3 19 2 3 42 2 1 Rem. 57 0 2 22 2 3 9 0 1 4. Subtract 95 yards, 3 quarters, 2 nails, from 156 yards, 2 quarters, 3 nails. Ans. 60 yds. 3 qr. 1 nail. 5. Subtract 14 English ells, 1 quarter, 2 nails, from 52 English ells, 3 quarters, 2 nails. Ans. 38 yds. 2 qr. LONG MEASURE. L. M. fur. L. M. fur. P. wds. ft. in. From 24 1 7 56 || 0 19 6 2 10 Take 18 2 4 10 0 7. 20 3 2 7 Rem. 5 2 3 46 () () 39 3 0 3 4. Subtract 45 miles, 5 furlongs, 20 poles, from 320 miles, 3 furlongs, 36 poles. Ans. 274 M. 6 F. 16 P. || 5. Subtract 15 yards, 2 feet, 6 inches, from 36 yards, 1 foot, 11 inches. Ans. 20 yds. 2 ft. 5 inches. LAND, OR SQUARE MEASURE. A. R. P. A. R. P. Yds. ft. in. From 96 3 36 195 2 2 25 2 72 *śº Rem. 71 0 37 158 3 ]. 10 4 62| 4. Subtract 36 acres, 2 roods, from 900 acres, 3|| roods, 16 perches. 864 A. I. R. 16 P. 5. Subtract 72 acres, from 360 acres, 2 roods, 29|| CUBICK MEASURE. yds. ft. 2??. cords. ft. in. 79 11 917 349 97 1250 17 25 1095 192 127 1349 61 12 1550 156 97 1629 ; : º º r- ||70 COMPOUND SUBTRACTION. | 1. From a pile of wood containing 432 cords, 27 feet, | and 1432 inches, have been hauled 156 cords, 92 feet, | 946 inches: how much remains? . . Ans. 275 cords, 63 feet, 486 in. 2. From a bank of earth containing 2984 yards, 18 | feet, have been taken 1436 yards, 21 feet: what re- | || mains? Ans. 1547 yds. 24 feet. LIQUID MEASURE. Thhd.gal. qt. pt. T. h.hd.gal. qt. pt. 2 3 50 1 0 100 1 19 2 1 1 2 16 3 1 99 || 28 3 1 11 33 1 1 || 3. If I purchase 2hhd. of wine, and to oblige a friend | send him 29gal., what quantity have I left? . | . Ans. 1hhd. 34gal. || 4. Bought 1 pipe of wine, 4hhd. of brandy, 2 barrels | | of beer; I have since sold 93 gallons of wine, 29 of ||brandy, I barrel of beer: how much of each have I | remaining? . . . . . Ans. 33gal. of wine, 223gal. of brandy, and 1 || | barrel of beer. . MOTION. . o t tf Sin. o f f; 79 21 31 6 10 12 48 41 41 52 3 8 39 29 . 37 39 39 & 1 33 19 º A circle being 12 sines, how far has the hand of all watch to pass, after having gone through 4 sines, 23°|| 15, 29m 2 * . - . . Sin. o f iſ 12 0 0 0 4 23 15 29 º l - CoMPound SUBTRACTION. 71 2 A person residing in latitude 27° 32' 45" north, wishes to visit a place 52° 24′ 18' north. How many degrees, minutes, and seconds northward must he travel? Ans. 24° 51' 33’. f TIME Y. M. v. d. ho. min. sec. H.min.sec. Y. M. 6 9 3 1 3 40 20 16 29 33 18 11 1 6 2 6 2 57 36 7 36 44 9 10 5 3 2 () 42 44 4. From 900Y. take 111 Y. 6m. Ans. 788 Y. 6m. 5. If I take 1 Y. 1M. from 6Y. what space of time will still remain? . Ans. 4. Y. 11 M. NoTE.—To ascertain the amount of time passed be- tween two events, set down the year, month, and day of || the latter event, and place those of the former below it, and subtract. × | 6. A bond was given 24th July, (7th month) 1809, and paid off 13th August, 1821. § Jrs. mo, ds. 1821 8 13 1809 7 24 % 12 0 20 7. The declaration of independence of the United || States passed Congress, 4th July, (7th month) 1776;| | and the declaration of the late war with Great Britain, 18th June, (6th mo.) 1812. How many years, &c. be- | | tween them? Ans. 35yr. 11mo. 14d. STERLING MONEY. % S. d. £ 8. d. . f S. d. 146 19 104 47 6 7; 419 7 6 7 19 93 28 5 104 227 8 94 139 0 0} |72 compound MULTIPLICATION. || 4. Subtract £2009s. from £1000 11s. 113 d. 3 || - . #. Ans. £800 2s. 113 d. || | 5. I have a purse of money containing £1000 2s. || | 4%d.: if I take out £60 7s. 83d. what sum will be left? | # , . - Ans. £939 14s. 7#d. coMPOUND MULTIPLICATION. CoMpound MULTIPLICATION is the art of multiplying|| | numbers composed of several denominations. #3 CASE 1. When the multiplier does not eacceed 12. % RULE. 3. | Place the number to be multiplied as directed in || |compound addition; and set the multiplier under the | lowest denomination. . | || Multiply as in simple multiplication, and divide the | product of each denomination by as many as it takes of || || that to make one of the next greater; set down the re-|| | mainder (if any) and carry the quotient to the product} | of the next denomination. . | | Proof–Double the multiplicand and multiply by half | || the multiplier. . § 3. . EXAMPLEs. Bu. pe. qt. pt. 7 times 1 pint make 7 pints; 2 pt. || 7 2 5 I make 1 qt.; then 7pt, make 3 qt. and || a 7 leave 1 pt. Set down the 1 pt. and || 2. a carry the 3 qt, to the product of the — - next figure. | 53 2 - 6 1 7 times 5 qt. make 35 qt, to which || dd the 3 qt, which make 38 qt.; 8 qt. make one peck; then 38 qt. || ake 4 pecks and leave 6 qts. Set down the six quarts and carry the I. ecks. * | º times 2 pecks make 14 pecks; add the 4 pecks, makes 18 pe. ;| pecks make one bushel; then 18 pe. make 4 bushels and leave 2 || ecks. Set down the 2 pecks and carry the 4 bushels. | 7 times 7 bushels make 49 bushels; add the 4 bushels, makes 53|| ushels, which set down, and the work is done. . COMPOUND MULTIPLICATION. 73|| Bu. pe. qt. pt. Bu. pe. qt. pt. 9 3 6 I 23 2 5 l 5 . 8 49 3 0 1 189 I 4 () 1. In one vessel are contained 29 bushels 2 pecks and || 5 quarts: how many in 9 such vessels? º ~ Ans. 266 bu. 3 pe. 5 qt. | 2. If one tub will contain 8 bu.3 pe. 5 qt. how much | will 11 such tubs contain? Ans 97 bu. 3 pe. 7 qt. || § CASE 2. | When the multiplier exceeds 12, and is the ea act product || º of two factors in the multiplication table. . RULE. º Multiply the given sum by one of the factors, and the product by the other factor. - H. | Proof–Change the factors. EXAMPLES. .” º 1. Multiply 3 bushels, 2 pecks, 7 qt. by 24. Product. || bu. pe. qt. bu. pe. qt. º 3 2 7 3 2 7 . 6 4 22 1 2 14 3 4 ; : . . 4 - w 6 89 1 0 Proof 89 1 0 OR THUS : . bu, pe. qt. bu, pe. qt. 3 2 7 3 2 7 11 0 5 20 3 0 89 1 0 89 1 0 2. Multiply 7 bushels, 3 pecks, 5 quarts, by 36. Product, 284 bu. 2 pe. 4 qt. 3. Multiply 19 bushels, 2 pecks, 3 quarts, by 42 MULTIPLICATION. CASE 3. | | When the multiplier exceeds 12, and is Not the product | of any two factors in the multiplication table. | º, RULE. - º | || Multiply by the two factors whose product is the least || |short of the given multiplier; then multiply the given || |sum by the number which supplies the deficiency; and || |add its product to the sum produced by the two factors. ExAMPLEs. º 1. Multiply 21 bushels, peck,7auarts, by 28, Prod. | bu. pe. qt. bu. pe. qt. 21 1 7×3 21 1 ºx2 . 5 3 3 & || – OR THUS : - - || 4 7 | 429 4 product of 20 450 & 3 product of 21|| 64 1 5 product of 3 42 3 6 product of 2 || 493 & 1 product of 28 493 & 1 product of 23| 2. Multiply 19 bushels, 3 pecks, 7 quarts, by 34. Product, 678 bu. 3 pe. 6 qt. 3. Multiply 7 bushels, 3 pecks, 4 quarts, by 59. 4. Multiply 9 bushels, 3 pecks, 2 quarts, by 47. 5. Multiply 15 bushels, 1 peck, 7 quarts, by 78. 6. Multiply 12 bushels, 2 pecks, by 92. 7. Multiply 17 bushels, 3 quarts, by 98. | | 8. How many bushels in 104 sacks, each containing| |7 bushels, 2 pecks, 3 quarts? || | 9. How many bushels of wheat on 125 acres, con-| | compound MULTIPLICATION. 75 || t . CASE 4. l º |When the multiplier is greater than the product of any || | two numbers in the multiplication table. . RULE. || Multiply the given number by 10, as many times | |less one as there are figures in the multiplier. | || Multiply that product by the left hand figure of the |multiplier. | || Multiply the given sum by the units figure of the |multiplier; the product of the first 10 by the tens figure || |of the multiplier; the hundreds product by the hundreds | |figure of the multiplier, and so on, till you have multi- |plied by all the figures of the multiplier except the left |hand one. . || | Add all the products together, and you have the pro- |duct required. | EXAMPLES. 1. Multiply 3 bushels, 3 pecks, 1 quart, by 456. | bu. pe. qt. Product 1724 bu. 1 pe. | 3 3 1X6 . i. 10 Because there are 3 figures, multiply | 2 times by 10. Multiply that product by || 2. o J - the keft hand figure (4) of the multiplier. 37 3 2X5 Multiply the given number by the units | 10 figure (6) and set the product beneath.| Multiply the 10's product by the tens H º figure (5) of the multiplier. aſs 0 : *...?. several º § { 1512 2 0 22 2 6 189 0 2 1724 1 0 Product compound M. ULTIPLICATION. º 2. Multiply 53 bushels, 2 pecks, 7 quarts, by 2345. || bu. pe. qt. r: 53 2 7x5 º 10 0 537 6–1–4 10 & *H, product of the 2000 & << 300 “ “ 40 gºš Product of the 2345 1 6 1 1 5 i 125970 1 sº bers in this way. 3. Multiply 72 bushels, 1 peck, 2 quarts, by 4723. Product,341531 bu. 3 pe. 6 qt. | 4. Multiply 13 bushels, 2 pecks, 4 quarts, by 5124. || Questions. What is compound multiplication? In what does it differ from simple multiplication? | When the multiplier does not exceed 12, how do you || proceed? . * ..." || How many do you always carry? º How do you prove compound multiplication? || How do you proceed when the multiplier exceeds 12, nd is the exact product of two numbers in the multi-|| ication table? . "º Note:-Let the pupil try experiments, by multiplying simple num- | When the multiplier exceeds 12, and is not the ex ct| roduct of any two numbers in the table, how do you || roceed? § - º COMPOUND MULTIPLICATION. 77 | How do you proceed when the multiplier is greater || || than the product of any two numbers in the table? : AVOIRDUPOIS WEIGHT. A tons. cwt. qrs. cult. qr. lb. oz. dr. 23 12 3 7 3 14 9 6 % 4 6 94, 11 0 47 I 3 8 4 tons cut. gr. cut. gr. lb. .oz. dr. 7 15 3 7 3 24 12 14 8 9 5. Multiply 7 tons, 16 cwt. 3 qr. by 24. Product, 188 T. 2 cwt. 6. Multiply 3 cwt. 2 qr.21 lb. 14 oz. by 30. Product, 110 cwt. 3 qr. 12 lb. 4 oz. 7. Multiply 3 tons, 7 cwt. 2 qr. by 34. Product, 114 tons 15 cwt. APOTHECARIES WEIGHT. # 3 3 B # 3, 3 B gr. Th. 3 3 B gr. 4 S 2 1 53 10 () 2 12 17 5 6 1 4 5 9 12 23 5 3 2 TROY WEIGHT. lb. oz. dwt. lb. oz. dwt. gr. lb. oz. dwt. gr. . 67 5 16 43 () 8 10 113 6 0 6 | 2 - 4. 6 | 134 11 12 4. Multiply 41 lb. 6 oz. 18 dwt. 2 gr. by 7. Ans. 291 lb. 0 oz. 6 dwt. 14 gr. 5. Multiply 91 lb. 4 oz. 14 dwt. 16 gr. by 8. || || Ans. 731 lb. 1 oz. 17 dwt.8 gr. || || % COMPOUND MULTIPLICATION. |yd. qr, na. E. E. qr, na. E. Fl. qr, na. E. Fr. qr, na.| | 20 2 3 37 4, 2 18 0 3 14 1 3|| | 6 8 12 9 | |Tºa o 2 | **** | 5. If 19 yd, 1 qr. 2 na. be multiplied by 5, what num- || |ber of yards will there be? Ans. 96 yds. 3 qr. 2 na. || | 6. Multiply 56 Ells Eng, 3 qr, by 9. -- | º º | LONG MEASURE. | deg. m. fur. p. l. m. fur, p. m. fur. p. yd. ft. in. || || 8 || 3 36 4 2 2 29 18 3 20 1 2 10|| || 4. Multiply 6 deg. 40 m, 7 fur, by 10. i. Ans, 65 deg. 61 m. 2 fur. Multiply 44 m. 6 fur. 20 p. by 7. | Ans. 313 m. 5 fur. 20 o *::::::: * : LAND, OR SQUARE MEASURE. * a.º. r. p. a. r. p 19 3 20, 10 0 33 * 0. º r. p • • 49; 2 17 t * º 6 9 2 - # - - - - * * * * * 4. How many acres will 10 men reap in one day, i. lowing them 1 acre 3 roods 11 perches each?, 5. Multiply 63 acres 3 roods 18 perches, by 11. | º Ans. 702 A. 1 R. 38 P. H. 6. How many acres in 15 lots, containing 17 acres, 2 || oods, and 20 perches each? Ans. 264A, IR. 20P.]] COMPOUND MULTIPLICATION. 79 | * , CUBIC MEASURE. cords. ft. in. yd. ft. in. 7 28 1327 19 23 1421 - || 6 8, 43 44 1050 159 1 1000 cords. ft. in. Ayd. ft. in. 21 56 1432 27 13 1291 7 . 9 5 In a pile of wood are 14 cords 92 feet; how much in 24 such piles? Ans. 353 c. 32 ft. || 6 In a cellar, are contained 42 yards 25 feet; what || are the contents of 23 such cellars? Ans. 987 yd. 8 ft. || LIQUID MEASURE. hhd.gal. qt. Thhd.gal. qt. pt. pi. hndigal. qt.pt. 8 43 2 1 2 16 3 1 4 I 19 3 1 || 4 I0 5 || 34 48 0 4 Multiply 3 T.2 hhd. 50 gal. 2 qt, by S. d Ans. 29 T. 2 hhd. 26 gal. 0 qt. || 5 Multiply 4 hld. 41 gall pt. by 10, | | Ans. 46 hid. 33 gal. I qt. 0 pt. % MOTION. sin.” C f sin. O / !, 3 27 48 1 24 48 25 7 . 9 27 14 36 § 16 13 1545 3 If a planet move through 2sin. 15° 23' of its orbitl in one day; how far will it advance in 8 days. || . 3. Ans. 20sin.3°4 || ºn compound DIVISION. ºr, º, . TIME. w § . º • .. weeks. d. h. d. h. min. sec.| 3 5 23 3, 14 25 36 8. 9 | | 53 8 30 5 16 32 9 50 24 | || 4 If a man can perform a piece of work in 2 yr. 3 mo., | |how long would it take him to perform 5 such?' º Ans. 11 yr. 3 mo.] | 5 If a laborer dig a drain in 2 weeks, 3 days, how long| |a time would he require to dig 9 such drains? º Ans. 21 weeks 6 days. || STERLING MONEY. - fº s. d. f; s. d. f. s. d. 246 13 3: 14 6 04 111 11 104 || . 11 9 10 d. . f. . s &. d. by 5 Prod. 186 13 114 by 9 — 507 17 93. | CoMPOUND DIVISION. | Compound Division is the art of dividing a sumſ |which consists of several denominations. " * † Case 1. * When the divisor does not exceed 18. . . RULE. . . . . . fº | Divide the several denominations of the given sum, one after another, beginning with the highest, and set} respective quotients underneath. . . . ] When a remainder occurs, reduce it to the next lower| denomination, and add it to the number of the next de-l omination, and divide the sum as before. . tºº *…*** COMPOUND DIVISION. 81 | EXAMPLES. bu. pe. qt. pt. in to 9 # | 7) 25 2 6 l Here 7 into 25 bu. 3 times and 4 remain. Set down the 3. -w Reduce the 4 bushes to pecks, 3 2 5 1 which makes 16 pecks: add 16|| | pecks to 2 pecks, which make 18 pecks. Now 7 into 18 pe. 2 times, and leave 4. Set down the 2. … Reduce the 4 pecks to quarts, which makes 32 qts. Add 32 qt. to 6 qt.—makes 38 qt. º 7 into 38 qt. 5 times, and 3 remain. Set down the 5. Reduce the 3 qt. to pt.—makes 6 pt., add 6 pt. to 1 pt. makes 7 || pints. . 7 into 7, 1 time. Set down the 1, and the work is completed. bu. pe. qt. . bu. pe. qt. h 2) 8 2 6 3) 9 3 6 | 4 l 3 4 Divide 34 bu. 3 pe. 6 qt. between 9 persons. | Ans. 3 bu. 3 pe. 4 qt. || 5 92 bu. 3 pe. belong equally to 7 persons; what is the | share of each? w Ans. 13 bu. I pe. CASE 2. When the divisor eacceeds 12, and is the eacact product of two numbers in the multiplication table. º RULE. || Divide the sum by one of the factors, and the quotient | by the other. | || Multiply the last remainder by the first divisor, and || |add the first remainder for the true remainder, as in || |simple division, note 2. . º EXAMPLES. I Divide 89 bu. 3 pe. 7 qt. by 28. || Quotient 3 bu. 6 qt. I pt.—18 lem, D 5– º compound Division. bu. pe. qt. | 4 || 89 3 7 | (7 22 1 7–3 Rem. gº 3 o 8–5 Rem. 4 First divisor 20 3 First Tenn. True rem. 23 Quarts 2 . 28)46 pints Pt. 1 and a rem. of 18 pints undivided. | 3. Divide 78 bushels, 3 pecks, 4 quarts, among 32|| persons; what will be the share of each? || | Ans. 2 bu. I pe. 6 qt. 1 pt. and a remainder of 24|| | pints undivided. x--> § | When the divisor is more than 12, and is Not the eract || | product of any two numbers in the multiplication table. || Divide the highest denomination of the given sum, as || case 2, simple division; and reduce the remainder, if any, to the next lower denomination; add the number| of that denomination to the result, and divide as before. 1. Divide 77 bushels, 1 peck, 7 quarts, by 23, Quotient, 3 bu. 1. º COMPOUND DIVISION. 83| EXAMIPLES. bu. pe. qt, bu. pe. qt. pt. 23)79 1 7(3 1 6 1–H3 pint remaining. || 69 - - 10 4 23)41(1 peck 23 18 8 23)151(6 quarts 138 ***** 13 2 23)26(1 pint 23 ** Rem. 3 pints 2. A boat load of corn, containing 4927 bushels, 3|| |pecks, is owned equally by 29 persons: what is the |share of each? … Ans. 169 bu. 3 pe. 5 qt. I pt., and a rem. of 1 pint. | Questions. What is compound division? When the divisor does not exceed 12, how do you || |proceed?' * | When a remainder occurs, what do you do with it? || Where the divisor exceeds 12, but is the product of two || |numbers in the table, how do you perform the operation?|| | How do you find the true remainder in the latter case?|| | When the divisor is more than 12, and is not the pro-|| |duct of any two numbers in the table, how do you per-|| |form the operation? # ..." * ound Drvision. AVOIRDUPOIS, WEIGHT. #. | tons cut. qr. lb. lb. oz. dr. |6)37 17 3 27 7)46 12 14 6 6 1 9–1 rem. 6 10 15–5 R. | tons cut. qr. cult. gr. lb. oz. dr|| |8)92 3 3 9)75 3 23 14 12|| f | 5. A quantity of iron weighing 473 tons, 19 cwt., |3 guarters, is owned equally by 22 persons; what is the |share of each? . | | Ans. 21 T. 10 cwt. 3 qr. 16 lb. 8 oz. 11 dr. Rem. 14 dr. APOTHECARIES WEIGHT. 3 B fb 3 3 B gr. IB 3 4)23 7 5 1 5)41 6 7. 2 14 5 10 7 1 8 & 6 2–4 rem. | . . ; 3 B 3 & B gr. | 6)46 9. 1 2 7)03 7 5 2 § { | 5. Divide 127th 33 63 into 17 equal parcels: howl |much in each parcel? Ans. 7th 53 63 29 16 gr. 8 rem. | | 8)34 10 15 7)45 11 16, 22 | | i. or dwt. i. or diet, gr. | - compound orvision. - CLOTH MEASURE. yd. qr. ma. Ells E. qr. na. 5)27 3 l 6)37 3 2 *śº 5 2 l 6 1 1–H4 rem. yd. qr. ma. Ells E. qr. ma. 7)45 3 8)37 3 1 2 LONG MEASURE. l. m. f. 771. f. p. ºd. 6)37 2 2 7)46 7 17 3 6 0 7 6 5 25 2 gd. ft. in. m. f. p. 9d. ft.) 6)53 2 9 7)87 6 23 4 2. ****** 5. A traveller has a journey of 946 miles, 6 furlongs, to perform in 26 days; how far must he travel each | |* Ans. 36 m. 3 ſ. 12 p. 8 rem. LAND, OR SQUARE MEASURE. A. R. P. - A. R. P. yds. 7)37 3 27 9)423 3 28 2 5 I 26–H5 Rem. 47 0 16 13+5 Rem. | 3. A farm containing 746 acres, 3 roods, 29 poles, is || to be divided equally between 9 heirs; what is the share || |of each? Ans. 82 A. 3 R. 38 P. and 7 rem. || | CUBIC MEASURE. | cords f. !yd. ft. in. | 8)97 48 9)148 16 493 | 12 22 16 13 13984-7R.| compound prvision. || 3. A boat load of wood, containing 92 cords 87 feet, |is to be divided between 3 persons; what is the share || |of each? %: ; Ans. 30 c. 114 ft. 1 rem. || || 4. A quantity of earth, containing 6987 yards, 25 | |feet, is to be removed by 29 carters; how much must |each remove? - Ans. 240 yd. 26 ft. *::::: LIQUID MEASURE. tuns.hhd.gal. º hhd. gal. qt. pt. 5)37 3 45 6)57 36 3 1 7 2 21+3 Rem. 9 37 2 1+1 Rem. tuns hid. gal. ... . hhd.gal. qt. pt. 7)84 2 32 8)93 43 3 I | 5. A quantity of liquor owned equally by 27 persons, he whole quantity being 431 ha. 47 gals; what is | |the share of each? Ans. 15 hlid. 62 gal, 1 qt.; 17 rem. | MOTION. || , , Sin. O ' " 16 45 36 9)11 28 48 54 TIME. ºf we da, ho, min. sec. -2 0 13 42 42 da, ho. min. sec. 7)37 16, 28 32 º . º compound Division. 87 || STERLING MONEY. f s. d. f s. d. 6)82 14 6 - 8)143 7 10. k3 15 9 ~ 17 18 53 s. d. ~. f s. d. 7)78 10 11 9)98 17 1 tº §º r f s. d. É s. d. 19)36 16 3(1 18 9 6 Divide 113 f 13s. 4d. by 31. What is the quotient? : 2' Ans. 3ſ., 13s. 4d. 7 Divide 189ſ. 14s, by 95. Quotient, 14, 19s. 11d.-H PROMISCUOUS EXERCISES : | 1 In 35 dollars how many cents? Ans. 3500. 2 How many miles are there in 98 furlongs? | . Ans. 12.M. 2fur. | 3 How many weeks are there in 365 days? | w Ans. 52we. 1 da. 4 In 84 half cents how many cents? Ans. 42 cts. || 5 In 8 tons 15 cwt., how many hundred weight? . Ans. 175 cwt. 6 How many perches are there in 63 roods? | . Ans. 2520 square per.| 7 How many pounds in 157s.” Ans. £7 17s. | 8 In 175 pecks how many bushels? Ans. 43bu.3pe. 9 In 7642 cents how many dollars? Ans. $7642cts || | 10 In 103 pints how many quarts? Ans. 514t. Ipt. || 11 How many minutes are there in 720 seconds? | Ans. 12min. || 12 In 7 hogsheads, 33 gallons, how many gallons? * : . . . . . . Ans. 474 gal. … x. single Rule of THREE. PROPORTION; OR, º THE SINGLE RULE OF THREE. PRopoRTIon is an Equality of RATIos;” | That is, four numbers are proportional, when the first has the | same ratio to the second as the Thind has to the Founth. Thus, | as 12 : 4 : : 24 : 8; or as 4 : 12 : : 8 : 24. #. | The ratio of 12 to 4 is 3 and the ratio of 24 to 8 is 3. Or, the | ratio of 4 to 12 is , and the ratio of 8 to 24 is . Then | Four numbers are proportional, when the first is as many times the second or the same part of the second, as the third is of the fourth. || Or, when the ratio of the first to the second equals the ratio of the third to the fourth. The two quantities compared are called the TERMs f the ratio: the first is called the ANTEcºDENT, and he second the CoNSEQUENT. In any series of four proportionals, the first and fourth terms are called the ExtREMEs, and the second and third the MEANs. The product of the Means, equals the product of the Ex- || tremes. Thus in either series above, 12×8=96, and || Now suppose we have the three first terms of a series in propor- cond and third terms by the first, and the quotient will be the # urth term. In this manner let the fourth term be found in each || : 9 is to what? - ?ns. 33.1, 6 is to what? Ans. 9. 8 is to what? ...Ans. 36. : .: SINGLE RULE OF THREE, 89 And, if the answer must be greater than the third || term, set the greater of the remaining two terms in the second, and the less in the first place; but, if the an- swer must be less than the third term, set the less in the second, and the greater in the first place. ; : When the first and second terms are not of the same denomination, reduce one or both of them till they are; and, if the third consist of several denominations, reduce || it to the lowest, then | | Multiply the second and third terms together and || | divide the product by the FIRST, and the quotient will ! be the fourth term or answer. | Note.—The answer will be of the same denomination as the || third term; and, in many instances, must be reduced to a greater || | denomination. - EXAMPLE8. | 1. If four pounds of sugar cost 50 cents, what will || | 24 pounds cost at the same rate Ans. $3,00. | 1st Term. 2d Term. 3d Term. # # : ' ...? § 3 \-y-' Q- } In this question the answer is º required to be money: therefore . lbs. lbs. # cł8. money (the 50 cts.) must be in || ! As 4 : 24 : : 50 the third place. Because 24 || 50 pounds will cost more than 4 || * pounds—the #.". (24 lbs.) || . 4)1200 must occupy the second place; H w and the remaining term (4 lbs.) || *ś the first || 3,00 . ... º.% | 2. If 24 pounds cost 300 cents (or 3 dollars ;) how || | many pounds may be purchased for 50 cents at the ſi | same rate 2 Ans. 4 lbs. || --~~ . cfs. cts. lbs. In this question the answer is re- || | | As 300 : 50 : : 24 quired to be in pounds; therefore || & 50 pounds (the 24) must be in the third || lace. º Because 50 cts, will purchase less || 300)1200 than 300 cis--the }. (50 cts.) || must occupy the second place; and || 4 the remaining term (300 cts.) the ſº first. †. single Rule of three. | 3. Bought a load of corn containing 27 bushels, 3 || pecks, at 50 cts. per bushel, what did it cost? || : Ans. $13,874. || bu. pe. cts. * : *. . . . % ſ : 27 3 : : 50 Because pecks occur in the second || term, the first and second are reduced || º, § 3 to pecks. , it 5 . $13,87; . | | 4. What are 42 gallons worth, if 3 gallons 2 quarts || | cost $1,20? Ans. $14,40. | gals. 4ts. gals. D. cts. . . . . ; || | 3 2 : 42 : : 1,20 ; : ºf || 4 || 4 . As quarts occur in the first term, | — — the first and second are reduced to 14 168 . quarts. sº 120 . : 14)20160 5. If 8 bushels 2 pecks cost $4,25, how many bush- els can I purchase with $38,25? Ans. 76 bu.2 pe. D. cts. D. cis. bu, pe. 2::--------> ---------- 4,25 : 38,25 : : 8 2 º tº 34 4 As two denominations occur § 3; 3: ... - _ in the third term, it is reduced * * * * . . . . . to the less; hence the resu 11475 to bushels. ...; ju.2 pe. | singº Rule or Thºr. 91 || º 6. What will 5 lb. 6 oz. 5 dwt. of silver-ware cost at $1,50 per ounce? Ans. $99,37%. 02. lb. oz, dwt. D. cfs. 1 : 5 6 5 : : 1,50 gº & 20 12 As dwts. are in the second §§§ gº term, the first and second 2 () 66 must be reduced to dwts. 20 1325 150 66250 1325 20) 198750 $99,37; N. | 7. When 3 yards and 8 feet of plastering cost $1,40, | . what will be the cost of 16 yards' Ans. $5,76. !yds. ft. yds. D. cts. 3 8 : 16 : : 1,40 9 9 35 144 140 5760 144 35)20160(5,76 cts. . | 8. How many yards of cloth can be purchased for 95 | j Čollars, if 4 yd. 3 qr. cost $9,50? × - || | Ans. 47 yd. 2 qr.; or 47; yd. || # $ct. $ ct. 3/d. Qr. qr. || | As 9,50 : 95,00 : ; 4 3 4)190(47 yd. 2 qr. || sing LE RULE or Three. Note.—The operation may, in many instances, be contracted by || | dividing the second or third term by the first; or the first by either || | of the others, or by any number that will divide the first and either || | of the others without a remainder; and, using the quotients instead || of the original numbers. & 8 : ||| 9. If 24 yards cost $96, what will 8 yards cost? | Ans. $32. | yds. Wyds. D. yds. Wyds. D. 24a : 8a : : 96c or 24a 8 :: 96a §§§ 4 3c Ans. 32. — 4 10 If 36 bushels cost $72; what will 12 bu. cost? || . . Ans. $24. || bu. bu. D. bu. bu. D. 36a : 12a : : 72c, 12)36 : 12 : : 72a 3c 24 3a l 24 APPLICATION. || 1 When 4 bushels of apples cost $2,25, what must || le paid for 20 bushels? Ans. $11,25. bu. bu. D. cfs. | || 4 : 20 :: 2,25 | 2 How many yards of cloth can I buy for $60, when || yards cost $12? Ans. 25 yds. || 12 : 60 : : 5 tº . 3 If 6 horses eat 21 bushels of oats in a given time; } low much will 20 horses eat in the same time ! || . Ans. 70 bu. | 4 If 20 horses eat 70 bushels of oats in a certain time now much will 4 horses eat in the same time? || . Ans. 14 bu. || 5 If a family of ten persons use 7 bushels 3 pecks o wheat in a month; how much will serve them when here are 30 in the family? Ans. 23 bu. 1 pe 6 If 1.4 lbs. of sugar cost 75 cents, how many pound be bought for three dollars? An 7 If 4 hats cost 12 dollars, what will 2 SINGLE RULE OF THREE. 93 || 8 If 20 yards of cloth cost $85, what will 324 yards || | cost at the same rate : | I Ans. $1377. || | 9 If 2 gallons of molasses cost 70 cents, what will 2 hogsheads cost Ans. $44,10.| | 10 If 1 yard of cloth cost $3,25 cts., what will be the |cost of 6 pieces, each containing 12 yds. 2 qrs. ' | … Ans. $243,75 cts. || 11 If 3 paces or common steps of a person be equal to || |2 yards, how many yards will 160 paces make : || Ans. 106 yds. 2 ft. || 12 If a person can count 300 in 2 minutes, how many || can he count in a day ! Ans. 216000.| 13 What quantity of wine at 60 cts. per gallon can be || | bought for $37,80 cts. Ans. 63 gal. 14 If 8 persons drink a barrel of cider in 10 days, how || |many persons would it require to drink a barrel in 4 days?|| º Ans. 20. || | 15 If 8 yards of cloth cost $12, what will 32 yards || | cost 7 Ans. $48. || | 16. If 3 bushels of corn cost $1,20, what will 13 bush-|| |els cost 2 Ans. $5,20, | 17 If 9 dollars will buy 6 yards of cloth, how many || | yards will 30 dollars buy . . Ans. 20. || | 18 If a man drink 3 gills of spirits in a day, how much || ||will he drink in a year ! Ans. 34 gal. I pt. 3 gi. | 19 If 12 horses eat 30 bushels of oats in a week, how |many bushels will serve 44 horses the same time ! || | || . Ans. 110.| | 20 If a perpendicular staff 6 feet long, cast a shadow 5|| | feet 4 inches, how high is that tree whose shadow is | | 104 feet long at the same time ! Ans. , 17 feet. EXERCISES, 1 If 12 acres, 2 roods, produce 525 bushels of corn, | |\how many bushels will 62 acres, 2 roods produce 2 sº | w Ans. 2625 bu. || | 2 If 7 men plough 6 acres, 3 roods in a certain time, |how many acres will 96 men plough in the same time º Ans. 92 A. 2 R. 11 Per. 12 yd.- 3 Suppose 3 men lay 9 squares” of flooring in 2 days; how many men must be employed to lay 45 squares in || 4 If 7 pavers lay 210 yards of pavement in one day; |how many pavers would be required to lay 120 yards in || |the same time? | Ans. 4 payers. || | 5 If 2 hands saw 360 square feet of oak timber in 2| |days; how many feet will 8 hands saw in the same time?|| Ans. 1440 feet. 6 An engineer having raised a certain work one hun- dred yards in 24 days, with 5 men; how many men must employed to perform a like quantity in 15 days? Ans, 8 men. If 3 paces or common steps be equal to 2 yards; how any yards will 160 such paces make? . . . . . . . . . Ans. 106 yd. 2 ft. If a carriage wheel in turning twice round, advance 33 feet 10 inches; how far would it go in turning round| 63360 times? Ans. 203 miles.| Sound flies at the rate of 1142 feet in 1 second of ne; how far off may the report of a gun be heard in 1 ute and 3 seconds? ... . . . " " ' ' ' Ans. 13 miles, 5 fur., 0 poles, 2 yd. 10 If a carter haul 100 bushels of coal at every 31oads; many days will it require for him to load a boat with || 500 bushels, suppose he haul 9 loads a day? . bu bu da da As 300 : 8600 : : 1 : 12. 11. If 8 men can reap a field of wheat in 4 days; how ny days will it require for 16 men to do it? Ans. 2 days. ollars 50 cents; what *...*.*.*. SINGLE RULE OF THREE. 95 || | 13 If 7 pounds of cheese cost 873 cents; what must I | | pay for 122 pounds? Ans. 15 dol. 25 ct. || | 14 If 1 ounce of silver cost 72 cents; what will 3 pounds || | 5 ounces come to? Ans. 29 dol. 52 ct, | Why do you multiply the second and third terms to-l gether and divide by the first? - - What will 24 pounds If 4 pounds cost 50 cents, divide the *::: ; , 50 cents by 4, gives the price of ºf bacon º at jº. Ct. 1 pound: thus '4' into 50 124. If ior every 4 pounds 1 pound cost 124 cents, 2 pounds | # . lb. lb. ct, ct. will cost twice that; three pounds | As 4 : 24 :: 50 : 300 three times, and 24 pounds will cost || - 24 times 124 ct.; that is 300 cts, or 3 || dol. - º Cts. * If the second and third terms be x 4) 50 multiplied together, and their product || - divided by the first, the result will be || # the same as it is when the third is di- || 124 the price of 1 lb. vided by the first, and the quotient 24 multiplied by the second. tº 50 - 25 - * || 300ct, the price of 24 lb. º | 15 If 15 yards of broad cloth cost so dollars; what will 75 cost. Ans. 400 dollars. ... 16. A man bought 14 yards linen for $250 cts.; what . 2 * … – sº . - Ans. 624 ct. | 17 If 321 bushels of salt cost $240 75 cents; what|| | was it per bushel? Ans. 75 cents. is the worth of 1 qr. 2 na. at the same rate. º 18 If the moon move 13 deg. 10 min. 35 sec. in one, - "| day; in what time does it perform one revolution? | deg. min. sec. deg. da. || As iſ ſo 35.36% ... i & 4 * * * * * * . º º sº Ž żº º: º.º. º º 3. º º §§ 5 sINGLE RULE or three. | 19. If a staff 4 feet long, cast a shade 7 feet on level |ground; how high is a steeple whose shade is at the same time 198 feet. Ans. 1.13% | 20 If a man's annual income be 1333 dollars, and he |spend 2 dollars 14 cents a day; what will he save at the | end of one year? Ans, $551 90 ct.| 21 Suppose A. owes B. 791 dols. 60 ct, and can pay only 374 cts. on the dollar; how much must B. receive? | Ans. $296 85 ct.| 22 Bought 3 casks of raisins, each containing 3 cwt. | 1 qr, 14 lb.; how much did they cost at $6 21 ct. perſ PROPORTION_DIRECT AND INVERSE. º & | Hitherto, proportion has been treated in general terms;| w remains to consider the two kinds, DIRECT and || | º, ºr Jºiº ºn º ** * * * * º # * # | | DIRECT PRopoRTION is that in which more requires| | more, or LEss requires LEss. Thus: º yd. yd. dol. If 2 yd. cost 4 dol., 124 yd. being As 2 : 124 : : 4 more than 2 yd., will cost more than d. yſl. dol. And, if 124 yd. cost 248 dol.; 2y . . . . . That is, more yards require more º money, and less yards cost less money º - º º 12 4 days # It is supposed that ss; and less requires mo If 12 men built a wall 4 days; how many in do it in 8 days • * *Here it is supposed that 12 tº m. m. performed a piece of work in 8 da y: 12:6 ºbe.”" ...ºy. ** ****, this will req That i ãº. iversel | As4:8i , ag. qa. . m. m. day is8: 4 directly;: 12:6 * * * * SINGLE RULE OF THREE. 97 || f s º | All the past exercises in proportion are Direct—the | |following will be º INVERSE PROPORTION. Questions in Inverse Proportion, may be stated and | |solved by the same rule that is given for Direct Propor-|| |tion. || EXERCISES. || 1 If 6 mowers mow a meadow in 12 days; in what || |time will 24 mowers do it? Ans. 3 da. || | 2 If a man perform a journey in 6 days, when the |days are 8 hours long; in what time will he do it when || || they are 12 hours long? Ans. 4 da. || || 3 If, when wheat is 83 cents a bushel, the cent loaf || |weighs 9 ounces; what ought it to weigh when wheat is || : $1 244 cts. a bushel? - Ans. 6 oz. || | 4 If 100 dollars principal in 12 months gain 6 dollars || |interest; what principal will gain the same interest in || |8 months? . . . º Ans. $150. | | 5. If 12 inches long and 12 inches wide, make 1 || |square foot; how long must a board be that is 9 inches|| ide, to make 12 square feet? Ans. 16 ft.ll. | 6 A. lent B, 500 dollars for 6 months; how long must || |B. lend A. 220 dollars to be equivalent? . . || Ans. 13 months, 19 days.-H| 7 There is a cistern having a pipe that will empty itſ in 12 hours; how many pipes of the same capacity will || mpty it in 15 minutes? - Ans. 48 pipes. | | 8 A certain building was raised in 8 months by 120 || |workmen, but the same being demolished, it is required|| |to be rebuilt in 2 months; how many workmen must be || |employed? . Ans. 480 men, || | 9 If for 48 dollars 225 cwt. be carried 512 miles; how || |many hundred weight may be carried 64 miles for the same money? Ans. 1800 cwt.]] "A month is estimated at 30 days, unless a particular month bell eferred to. 10 If 48 men can build a wall in 24 days; how many || men can do it in 192 days? Ans. 6 men. 11 How many yards of carpeting 2 ft. 6 in breadth, will cover a floor that is 27 feet long and 20 feet wide? || | 12 What quantity of shalloon that is 3 quarters wide, || ||will line 73 yards of cloth that is 14 yd. wide? || | 3. . Ans. 15 yd. || | 13 How many yards of matting that is 3 quarters wide, || will cover a floor that is 18 feet wide and 60 feet long?|| º Ans. 160.| 14 In what time will $600 gain the same interest that $80 will gain in 15 years? Ans. 2 years. § 3. §§ º | When is the proportion direct? . . hen is it inverse? & g . . 3. & w hy is the proportion inverse in the last question? | º A. because it is more money requiring less time.] Why is the proportion inverse in the 11th question? || A. because the shalloon is narrower than the cloth at is less width requiring more length. . Why is the 10th question inverse? Miscuous ExERCISEs. ertain steeple standing upon level ground, cast * to the distance of 633 feet 4 inches, wh perpendicularly erected, casts asha at is the height of the steeple al s & ::3% each person now? º A ns - 3 Bought 215 yards of broad cloth at 6 dollars a ya, | SINGLE RULE of THREE. 99 || i what was the prime cost, and how must I sell it per yard |to gain $135 on the whole. | | Ans. prime cost $1290,00; to be sold for $6,623 per |yard. 4 If 100 men can complete a piece of work in 12 days;| |how many can do it in 3 days? Ans. 400 men. | 5 If a board be 43 inches wide; how long a piece will | |it take to make 1 square foot? Ans. 32 in. ii 6 A pole, whose height is 25 feet, at noon casts a |shadow to the distance of 33 feet 10 inches; what is the |breadth of a river which runs due East at the bottom of |a tower 250 feet high, whose shadow extends just to the |opposite edge of the water? Ans. 338 ft. 4 in. | 7 A plain of a certain extent having supplied a body of |3000 horses with forage for 18 days; how long would it] |have supplied 2000 horses? Ans. 27 da. || | 8 A piece of ground 1 rod wide and 160 rods long,| |makes 1 acre; how wide a piece must I have across the | end of a farm 32 rods wide to make an acre? | | 3 : Ans. 5 rods. || ||| 9 I have a floor 24 feet long, and 15 feet wide, which III ||wish to cover with carpeting that is 3 quarters of a yard|| ||wide; how many yards must I buy. | Ans. 53 yards, 1 foot. I | 10 How much land at $2,50 an acre must be given in || |exchange for 360 acres worth $3.75 an acre? || | . . Ans. 540 acres.| | 11 What is the weight of a pea to a steel-yard, which || |is 39 inches from the centre of motion, will balance a || |weight of 208 lbs., suspended at the draught end 3 quar- |ters of an inch? Ans. 4 lb.]] | 12 If $28 will pay for the carriage of 6 cwt. 150|| Amiles; how far should 24 cwt. be carried for the same || |money? Ans. 37.4 miles.| DOUBLE RULE OF THREE. compound PROPORTION | THE DOUBLE RULE OF THREE. | DIRECT AND INVERSE. | |portionals combined. - - - - - - - - * * ~ sº | Five, seven, nine, or other odd number of terms, |is always given to find a sixth, eighth, or tenth, &c., or Compound PropoRTION is two or more series of pro-|| |answer. x -- " - "…& Rule for the Statement. . . . . . . . . . Place the numbers that is of the denomination in which |the answer is required to be, in the third place. Then:| | Consider separately each pair of similar terms and || |place them agreeably to the rule for SIMPLE PROPoRTION. | on, Wºrk by two separate statements in simple propor| | Reduce the several pairs of terms to similar denom |inations as in single proportion, and the last to the lowes | denomination given: Then nº ºn || | Multiply the two initials, or left hand terms together |for a DIvisor, and the other three for a DIvi DEND. | Divide the latter by the former, and the quotient v #1, wer. in th :--------- o which the • It would be well for the pupil to work each sum both ways. 㺠&iºn, DOUBLE RULE OF THREE. 101 || . In . The answer is required to bel | On 106 # , .3% Q As 6 : 18 rods given in rods: then rods must be || d § d : : 40 the third term. If 6 men build || 3. Cla. . . ** 40 rods, 18 men will build more; I As 8 : 20 then more (18 men) must occupy || §§§ & the second, and less (6 men) the H à first place. . 48 *. If 8 days produce 40 rods, 20 || . days will produce more; then || —. more (20 days) must occupy the 48)14400(300 second, and less (8 days) the first 144 place. §§ $º 00 the lower pair must be alike.— |days, how many bushels will serve 60 horses 36 days, - what must be paid for the carriage of 4cwt. 32 miles? |how many lb. will suffice 38 men 16 days? N. B. The first pair, or two || upper terms must be alike. Also || |That is, both must be men or both days, both hours or both bushels, &c.|| 2. If 6 men in eight days eat 10lb. of bread, how | |much will 12 men eat in 24 days? Ans 60. H men 6 : 12) . . 10 lb º days 8 : 24; & Contracted. §§§ 6 : 12 2 . 288 . ; ; ; ; ; ; 10b. . 10 — 48)2880(60 Ans. 10 £3%zºś 60 Ans. 0 || 3. Suppose 4 men in 12 days mow 48 acres, how |many acres can 8 men mow in 16 days? Ans. 128A. || | 4. If 10 bushels of oats be sufficient for 18 horses 20 ! |at that rate? Ans. 60bu. || | 5. Suppose the wages of six persons for 21 weeks be || |288 dollars, what must 14 persons receive for 46 weeks?|| . Ans. $1472. || 6. If the carriage of Scwt. 128 miles cost $12.80, | Ans. $1.60.| | | 7. If 371b. of beef be sufficient for 12 persons 4 days,|| An 16Sb. 10; oz.] gº 8. If a man can travel 305 miles in 30 days, when || |the days are 14 hours long, in how many days can he |travel 1056 miles, when the days are 12; hours long? || | Ans. 116 days.--2540. || 9. If the carriage of 24cwt. for 45 miles be 18 dol-|| ars, how much will it cost to convey 76cwt. 121 miles?|| . Ans. $153 26 cts.--720. || 10. A person having engaged to remove 8000cwt. in || |9 days; removed 4500cwt. in 6 days, with 18 horses: || |how many horses will be required to remove the balance || n the remaining 3 days? Ans. 28 horses.| | 11. If 3 men reap 12 acres 3 roods in 4 days 3 hours, |how many acres can 9 men reap in 17 days? | | Ans. 153 acres.| 3 : 9 a. r. Analysis. - d. h. d. : : 12 3 If 3 m. reap 12 a. 3r. 4 3 : 17 4 1 m. rea 4 a. 1 r. and 12% 12 — 9 m. reap 38 a. 1 r. 51 204 ... If 4 d. 3h. reap 38 a.| 3. 9 1 r. 1 d, reap 9 a. and || gº 17 d. reap 153 a. Ans. || • * * * º º Is he day here estimated at twelve hours. PRACTICE. 103|| f 12. If 40 men build 32 rods of wall insday |ing 10 hours each day; in how many |build 48 rods, working 12 hours a day? Ans. 6 days, 8 hours.l. Men men Multiply all the initial]. 60 : 40 terms (or 60, 32, and 12)| rods rods days together for a divisor; and 32 : 48 ; : 8 the other four for a divi- hours hours dend. . N. 12 : 10 § | 13. If 36 men dig a cellar 60 feet long, 24 feet wide, |and 8 feet deep, in 16 days, working 16 hours per day, how many men can dig a cellar 80 feet long, 40 feet|" | wide, and 12 feet deep, in 20 days, working 12 hours perſ| |day? Ans. 128.]] Questions. What is compound proportion? i. How do you state questions in compound proportion?|| Which terms do you multiply together for a divisor? || Which for a dividend? ; : What other method is there? PRACTICE. | PRACTICE is a short and expeditious method of per- |forming various calculations in business. . 2. CASE 1. When the given price is LEss than one dollar. ---...- | RULE.—Set down the given number as one dollar, |and take such aliquot part” or parts of that number, asſ|| the price is of one dollar, for the answer. ſº | *An aliquot part of a number is any number that will divide ti | without a remainder; thus 4 is an aliquot part of 20; and 8 of 40;| |and 25 cents is an aliquot part of 100 cents, (or $1.) because 25 cts.} | ºuined in 100 cts, an even number of times, without a remain-l. ºr. x. ź. * Roods. roods 2 1 : | + .| º ; perches 20 10. , , 2 i i 2, I' # < | 10. * . ; : . . i ; i º _l "I T 8 5 * 4. 2 T 5. EXAMPLES. come to at 25 cts. | | Ans. $206 50 cts. * | 826 825 bushes, atone dollar a bushel, will * | – cost 826 dollars: at 25 cents, or 4 of a || $ 206,50 dollar, it will cost one fourth as much. H | 2. What will 934 gallons of molasses cost, at 50cts. | a gallon? * * * Ans. $467.| What will 1832 bushels of salt cost, at 75 cents a | Ans. $1374. || At 50 cents the cost will be # as much as-at one dollar. " F | - # as much as-at 50 cts. - | 916 cost at 50 c. . . . -- ; ::: ;-3 º *::: . - --- º ºg practice. 105 || cts. $ As before; the cost at 50 cts, will | 50 | }. 1832 be 3 as much as-at 1 doliar. | | | } e At 25 cts, it will be 4 as much || | 25 | } → as—at 1 dollar. & g ! 916 ct, at 50. . -> | 458 ct. at 25. º $1374 ct, at 75. || | 4. What will 680 pounds of sugar cost, at 10 cents a || | pound? $68. | 5. What will 742 pounds of pork cost, at 6: cts. a | pound? Ans. $46,374 cts. | 6. What must I pay for 371 pounds of bacon, at 12; cts. a pound? Ans. $46,374 cts. || | 7. How much will 8750 bushels of rye cost, at 62; || cts. a bushel? Ans. $5468 75 cts. || 8. How much must be paid for 4360 square feet of || | marble, at 874 cents a foot? Ans. $3815. || ||| 9. What will 468 square yards of plastering cost, at || | 183 cents a yard? Ans, $87 75 cts. || | 10. How much will be the cost of laying 856 perches.| | of stone, at 933 cents a perch? Ans. $802 50 cts, | 11. What will the digging of a cellar, containing 180 | cubic yards, cost, at 20 cents a yard? Ans. $36. || | 12. What will be the cost of hauling 248 cords of || wood, at 314 cents a cord? Ans. $77 50 cts. || *13. What must be paid for 432 perches of stone, at || || 374 cts. a perch? - Ans. $162. º 14. How much must be given for 724 days labor, at || 564 cents a day? Ans. $407 25. | 15. What will 742 bushels cost at 10 cts. Ans. $74.20 || || 16. 732 15 109 80 | 17. 732 20 146 40 | 18. 475 25 118.75 | | 19. 684 30 205.20 | | 20. 756 . 35 264 60 || 21. 927 40 . 370 80| |22. 824 . 50 412 00| |23. 682 55 375 10, | 24. 341 60 20460 | || 25. 784 70 548 S0| —F. PRACTICE. . What will 352 bushels cost at 64 cts.? Ans.#2200.| 436 124 54 50 || 724 - 183 135 75 | 956 314 298 75 | 742 374 278 25|| 274 433 119 874|| 732 564 411 75|| 845 624 528 124 || 684 683 470 25 || 274 814 222 624 | 386 933 361 874 CASE 2. || When the given price is MoRE than one dollar. || | RULE.—Multiply the given sum by the number of || | dollars, and take the aliquot part or parts for the cents, |as in Case 1. s š º EXAMPLEs. || 1. What will 342 cords of wood cost, at 3 dollars 75|| ents a cord? Ans. $128250 342 cords at $1, will cost $342; at $3|| it will cost 3 times $342; at 50 cts, itſ will cost # as much as it will at $1.; and at || TJ 25 cents, # as much as it will at 50 cts. ; i. 1026 which added together, will be the cost at # | 1282 50 What will 250 acr. costat $4.624 Ans. $1156. 435 5 874 2555 624 || 6 12s ºn 1672 124 || 737s 3.683. 5 75 6.933 3 564 * 26s; tº 25897 00 PRACTICE. 107 || CASE 3. | When the given quantity consists of several denom-|| : inations. d RULE.—Multiply the given price by the number of || | hundred weight, acres, yards, or pounds, &c. and take the | aliquot parts for the quarters, roods, feet, or ounces, &c. || EXAMPLES. | 1. What will 240 acres, 2 roods, 10 perches, cost at || | $15 25 cents an acre? Ans. $3668 574 cts. || 2 r. | | | 1525 240 61000 3050 | 10p. ; T624 º 954-H2 rem. 366857; | 2. What will 29 yards, 4 feet, of stone pavement cost, | at $225 cents a yard? Ans. $66.25 cts. N. 3 square feet | | | 225 : 29 2025 450 1 # 75 25 6625 PRActror. || 3. What will 32 pounds 8 ounces of silver cost, [. || at $15,624 a pound? . Ans. $510 414. || 16 3124 4686 2 oz. J. 7814 % -::. . . . ; 2604–1–2Rem. | || 510 413 - || || 4. What will 27 cwt. 3 qrs. cost, at $23.50 cts, a || | cwt.% - - - "... " - * Ans. $652.124. || | 5. What will be the cost of 47 lb. 10 oz. (Troy) at || |$1.25 cts. Ans. $5979. || | 6. What will 64 yds. 3 qrs, cost, at $225 a yard? || | Ans. $145 684. || || 7 Sold 83 yards 2 qrs, of cloth at $10 50 a yard; | what does it amount to? Ans. $876 75. § | 8. What will the laying of 28 squares, 75 feet of floor- |ing cost, at $2.25 cts. a square? Ans. $64,683. | 9. What is the cost of 27 cords, 96 feet of fire wood, at | |º|º to what is the value of 42s gals, 3 at at #10 ºl |a gallon? . . . Ans. $600 25 cts. || | 11. What is the value of 765gals. 3 qt. 1 pt. at $2183 || cents a gallon? . . . . . . Ans. $1675 344 cts. || | 12. What is the value of 5 hlids. 313 gals. at $47 12 || |cts. a hogshead? º Ans. $259 16 cts. 13. What is the value of 17 hbds. 15 gals. 3 qts, at $64.75 cts. per hogshead? Ans. $1116 93 cts. 7m. || 14. What is the value of 120 bu. 2 pecks, at 35 cents || bushel? Ans. $4217 cts. 5 m. 15. What is the value of 780 bu. 2 pecks, 2 qts. at $1 || 7 cts, a bushel? Ans. $913.25 cts.--|| 16. What is the value of 1354 bu. I peck, 5 qts. 1 pt. ſt 25 cts, a bushel? - Ans. $33860 cts, 5m.-H. 17. What is the value of 35 acres 2 roods 18 perches, t 54 dollars 35 cts. an acre? Ans. $1935 53 cts. 9. N. ſ TARE AND TRET. 109 | . Questions What is practice? . What is the rule for the solution of questions in prac- | | tice? . | What is an aliquot part? Are 50 cts, an aliquot part of 100 cents? What part of $1 is fifty cents? What part of $1 is 334 cents? What part of $1 is 25 cents? What part of $1 is 124 cents? What part of $1 is 10 cents? | What part of $1 is 20 cents? | What part of $1 is 5 cents? | What part of $1 is 4 cents? | What part of $1 is 64 cents? TARE AND THET. | TARE AND TRET are allowances made on the weight || |of some particular commodities. - | | Gross weight is the weight of the goods, together with || ll the vessel that contains them. - †† | Tare is an allowance for the weight of the vessel. | Tret” is an allowance of 4 lb. for every 104, for || | Neat weight is the weight of the goods, after all allow- | | ances are made. .” . | 3% RULE. º Subtract the tare from the gross, and the remainder | | is the neat weight. g - | | ExAMPLEs. || || 1 Bought a chest of tea, weighing gross 63 lb., tare 8 || |lb.—what are the neat weight and value, at 85 cents || |per lb" . . . . . . . . . . . º || | "Astret is never regularly allowed in this country; no account of it il | is taken in this work. º 63 gross—or, weight of the chest and tea 55 lb. 8 tare—or, weight of the chest — ~. * 425 55 neat—or, weight of the tea itself 425 || $46,75 value. | | 2 Bought 5 bags of coffee, weighing each 97 lb. gross, |tare of the whole 7 lb.--what are the neat weight and |value, at 25 cents per lb." Ans. 478 lb. neat—$119,50.] || 3 The gross weight of a hogshead of sulphur is 1344 |lb.; the tare 138 lb.--what are the neat weight and its |value, at $4.75 per 100 lb. ? 1. | Ans, 1206 lb. neat—$57,284.j || 4 Bought 3 barrels of sugar, weighing as follows, viz: |236 lb. gross, 23 lb. tare—217 lb. gross, 22 lb. tare—|| |225 lb. gross, 23 lb. tare—what are the neat weight and |value, at $8 per 100 lb.? Ans. 610 lb. neat—$48,80. || 5 Sold 3 hogsheads of sugar, weighing each 12 cwt. 2" |qrs. 14 lb. gross; tare 2 cwt. 1 qr, 27 lb.--what are the |neat weight and value, at $11,50 per cwt.% | i. Ans. 35 cwt. 1 qr, 15 lb. neat—$406 918 *| 6 What is the neat weight of 15 tierces of rice, weigh- ing 48 cwt. 3 qrs. 12 lb. gross; tare 6 cwt. 12 lb., "...] what is the value, at $5,25 per cwt.% *ś. . . º. º.º. *** Ans. 42 cwt. 3 qrs, meat- | 7 What is the neat weight of 28 hogsheads of tobac ||weighing 201 cwt. 3 qrs. 12 lb. gross; tare 3140. " " " Ans. Tºgewºº, sî-ssó9 10, ets.[ | 8 Bought 17 bags of grain, weighing 3561 lb. gross;| TARE AND TRET. III || | 10. In 14 hogsheads of sugar, weighing 89 cwt. 3 qrs.) ||17 lb. gross; tare 100 lb. per hbd.—how much meat | weight, and what is its value, at $9 per cwt.” º Ans. 77 cwt. 1 qr. 17 lb. neat—$696,614. | 11 What are the neat weight and value of 16 hbds. of |tobacco, each weighing 5 cwt. I qr. 19 lb. gross; tareſ | 101 lb. per hbd., at £2 6s. 10d. per cwt.” | ; : Ans. 72 cwt. 1 qr. 4 lb. neat—£169.5s. 43d.| | 12 Bought 6 hds. of sugar, each 1126 lb. gross; tare|| | 117 lb. per hlid.—what are the neat weight and value at $8,75 per cwt.” Ans. 6054 lbs. $529,724. | 13 What are the neat weight and cost of a hogshead |of sugar weighing gross 986 lb.; tare 12 per cent, (or |12 lb. for every 100 lb.) at $8 per meat hundred pounds? º lb. lb. lb. lb. lb. lb. lb. lb. || |As 100:986: 12:118, Oras 100: 88::986: 868, lb. lb. 986 gross. 868 118, tare. - 8 dol. 868, neat weight. $69,44 the value. 14 What are the neat weight and value of 4 hbds. of sugar weighing gross 4500lb. tare 12 lb. per cent. at i. ||75 percent.? Ans. 3960 lbs. neat —$346,50.| | 15 Bought 10 hlids. of sugar, each 920 lb. gross; tare| | 10 lb. per cent.—what are the neat weight and value || |at $9.25 per cwt.” Ans. S280 lb. neat— $765,90.| | 16 S. k 3 casks of alum, each 675 lb. gross; tare 13|| |lb. percent.—what are the neat weight and value at $4, ||25 percent. . Ans. I762 lb. neat—$74,874375.j | - Or, 1762 lb. neat—$74,88.5nearly. I | 17 What is the neat weight of 4800lb gross: tare 12 lb.[ |per cent.”, º Ans. 4224 lb.i. | 18 What are the neat weight and value of 4.hhds. of |sugar, each 12 cwt. 1 qr. 14 lb. gross; tare 12 lb. perſ | cwt. at $12,20 per cwt.” | Ans. 44 cwt. 22 lb neat—$539 194 cts.]] --~! 12 INTEREst. || 19 Bought 17 hds. of sugar, weighing 201 cwt. 2 qrs. || | 13 lb. gross; tare 10 lb. per cwt.—what are the neat || |weight and value at $14 per cwt.” Ans. 183 cwt. 2 qrs. 13 lb. neat— $2570 624 cts. INTEREST. INTEREST is an allowance made for the use of money. | Principal is the sum for which interest is to be com-l | Rate per cent. per annum is the interest of 100 dol- | lars for one year. Amount is the principal and interest added together. | When the time is one year and the rate per cent. is any | | \, \ number of dollars. | RULE.—Multiply the principal by the rate per cent, |and divide by 100; the quotient will be the interest for || º --- . . . . . . . º | 1. What is the interest of 500 dollars for 1 year, at | per cent. per annum? §: :::::. } K. & * * * * * $500 º, º * * § º § 3. $º ºf . º: º * … § 3. 100+-$3000 Ans. . . . 'hat is the interest of 225 dollars for er cent. per annum? A hat is the interest of 384 dollars 50 5 dollars per cent, per annum? Ans. hat is the amount of $275 for 1 ye INTEREST. § | 5. What is the interest of 1654 dollars 81 cents for 1 || | year, at 5 dollars per cent. per annum? Ans. $82 74-H. | 6. What is the interest of 1500 dollars, for 1 year, at || | | dollar per cent. per annum? Ans, $750. | 7. What is the amount of $736, at 6 per cent. per || | annum, for 1 year. . 780 dols. 16. || | 8. What is the interest of 524 dollars, for 1 year, at || 54 dollars per cent. per annum? Ans. $2751. ||| 9. What would be the interest of 842 dollars, for 1. i year, at 5% dollars per cent. per annum? Ans. $4631. CASE 2. When the interest is required for several years. RULE.—Find the interest for one year, and multiply | || the interest for one year by the number of years. º º X- .. EXAMPLEs. .” . | | 1. What is the interest of 500 dollars, for 4 years, at | 6 dollars per cent. per annum? º $500 6 3000 4 . $120 00. Ans. || | 2. What will be the interest of 540 dollars, for 2 years, | at 5 dollars per cent. per annum? Ans. $54 00, || 3. What would be the interest of 482 dollars, for 7| | years, at 6 dollars per cent. per annum? Ans. $20244.]] 4 What is the amount of $736 814 with 7 years, nine || | months interest due on it, at 6 per cent. per annum? || jº Ans. $1079 434. | | Note.—If the interest is required for years and months, || multiply the interest for 1 year by the number of years, and take the aliquot parts of the interest for I year, for || the months. INTEREst. * 442087, interest 1 year 7 # 30946,124 interest 7 years 433 interest 6 months º #+ 3 34261,78 interest for 7 yr. 9 mo. 73681,25 principal - —$107943 03 amount | | 5. What is the amount of $362.25 for 4 years 6 mo. I | at 6 per cent. per annum? Ans. $460 053.]] º : * : . When the interest is required for any number of months, weeks or days, less or more than one year. || RULE.—Find the interest of the given sum for one || | year Then, by proportion, § As 1 year ...” Is to the given time, - So is the interest of the given sum (for 1 year) To the interest for the time required. Or take the aliquot parts of the interest for one year for the given time, as in note, Case 2. * 1. What is the interest of $560 for 2. at 5per ct, per annum? º º º 5600 - $7000 interest for 2 years 6 months. INTEREST. 115|| 2. What is the interest of 325 dollars, for 4 years and || |2 months, at 4 dollars per cent. per annum ? | Ans. $54 16 cts. 6m. || 3. What is the interest of 840 dollars for 5 years and || || 3 months, at 4 dollars per cent. per annum? - Ans. $17640. || || 4. What is the interest of 840 dollars, for 5 years and |4 months, at 7 dollars per cent. per annum? - . > Ans. $31360. || 5. What is the interest of 560 dollars, for 4 months, at |6 dollars per cent. per annum? 56() 6 ** m. m. § cts. $ cts. 100)33 60 As 12 : 4 : : 33 60 : 11 20 Ans. - 6. What is the interest of 1200 dollars, for 15 weeks, 1. | at 5 dollars per cent. per annum? Ans. $1730. | 7. What will be the interest of 240 dollars, for 61 || days, at 43 dollars per cent. per annum? Ans. $1 90.--|| | 8. What is the interest of $1000, for 14 months, at 7 || | per cent. per annum? Ans. $81,664. || {| 9. What is the interest of 450 dollars, for 6 months || || and 20 days, at 53 dollars per cent. per annum? *sº Ans. $1375. | d 10. What is the interest of 375 dollars 25 cents, for 3 || | years 2 months 3 weeks and 5 days, at 6 dollars per ct. | per annum? . Ans, $72.92.-- || | II. What is the amount of $736 for 28 weeks, at 10 | | per cent. per annum? ~. Ans. $775 63. | H! CASE 4. | | To find the interest of any sum for any number of days,|| º as computed at banks. - * | RULE.—Multiply the dollars by the number of days, | and divide by 6; the quotient will be the answer in mills. | The interest of any number of dollars for 60 days, at || | 6 per cent. will be exactly the number of cents; and if || any other rate per cent. is required, take aliquot parts, | and add or subtract according as the rate per cent. is | li more or less than 6. - 3 : . . . . . . º 3–2. ----> ~! INTEREST. | ExAMPLEs. N. | 1. What is the interest of 563 dollars, for 60 days, at || |6 per cent. per annum—and likewise at 7 per ct, per an.? || | $563 Ans. $5,63 at 6 per cent. || º 60 $6,56.8 at 7 per cent. || 6)33780 § 3 ; ; | in. at --~~ | 6per; 5630 mills, 1 || || $56.30 | cent.) º 938 # Interest at 7 per cent. 6568 mills. | 2. What is the interest of 854 dollars, for 30 days, at || | 6 per cent. per annum? * Ans. $4.27. || || 3. What is the interest of 1100 dollars, for 48 days, at || | 6 per cent. per annum? Ans. $880. || | 4. What is the interest of 3459 dollars, for 75 days, at || | 6 per cent. per annum? Ans. $43 23 cts. 7 m.—H || | 5. What is the interest of 1500 dollars, for 60 days, at || | 5 per cent. per annum? Ans. §1250. || | . . . . CASE 5. . º | The amount, time, and rate per cent, given, to find the # | : rincipal. º | RULE.—Find the amount of 100 dollars for the time || |required, at the given rate per cent. || | Then, by proportion, as the amount of 100 dollars for || he time required, (at the given rate per cent.) is to the mount given, so is 100 dollars to the principal required. |} 3. > > ExAMPLEs. º 1. What principal, at interest for 8 years, at 5 per ct. || |per annum, will amount to 840 dollars? Ans. $600. ars ~ || — 140 : 840 :: 100 : 600 3.3% INTEREST. 117 || | 2. What principal, at interest, for 6 years, at 4 per || | cent. per annum, wiil amount to $1240. Ans. $1000 | 3. What principal, at interest for 5 years, at 6 per ct. || per annum, will amount to 2470 dollars? Ans. $1900. CASE 6. The principal, amount, and time given, to find the rate | per cent. - RULE.—Find the interest for the whole time given, by || subtracting the principal from the amount. || Then, as the principal is to 100 dollars, so is the in- || terest of the principal for the given time, to the interest || of 100 dollars for the same time. - Divide the interest last found by the time, and the quotient will be the rate per cent. per annum, º: $; Or by compound proportion. EXAMPLES. º 1. At what rate per cent. per annum, will 600 dollars | amount to 744 dollars, in 4 years? Ans. 6 per cent. || $ $ $ $ || , $744 amount As 600 : 100 :: 144 : 24. | 600 principal yr. § † **ś 4) 24 (6 rate per cent. 144 interest Or by compound preportion: As 600 : 100 $ 8 yr. yr. : : 144 : 6 rate per cent. : 1 | 2. At what rate per cent. per annum, will $1200 || | amount to $1476, in 5 years and 9 months? †† - Ans. 4 per cent. || | 3. If 834 dollars, at interest 2 years and 6 months, | amount to 927 dollars 82% cents, what was the rate per || | cent. per annum? Ans. 44 per cent. || compound INTEREST. | . CASE 7. § | To find the time, when the principal, amount, and rate | | per cent. are given. | | RULE.—Divide the whole interest by the interest of || | the principal for one year, and the quotient will be the | time required, or by proportion. 㺠! . . . . . EPAMPLEs. & || | 1. In what time will 400 dollars amount to 520 dol-|| | lars, at 5 per cent. per annum? Ans, 6 years. || || 400 520 | 5 400 - | — 3& $ $ Y. Y. | 2000, 20)120(6 20 : 120 : : 1 : 6 Ans. || |per cent, per annum? º -- Ans. 7 years. || |...ºpºlºdºlº, tº per ºpe, ºff | amount to 1281 dollars 25 cents, how long WaS it at in- | | * COMPOUND INTEREST, | Compound interest is that in which the interest for || |one year is added to the principal, and that amount is || the principal for the second year; and so on for any || |nº of ºf tº tº **__ | RULE.—Find the amount of the given sum for the first | yearby simple interest, which will be the principal for the | second year; then find the amount of the principal for || || the second year for the principal for the third year; and || o on for any number of years. || mainder will be the compound interest required. || What is the compound interest of 150 dollars for 5|| 4 per cent. per annum? || %3 ºxº COMPOUND INTEREST. 119 | $150 $150 4 6 inst 1st year ,600 int. 1 yr. 156 amount 1st year . 6,24 int. 2d year $156 162,24 amount 2d year º 4 6,48.9 int. 3d year 6|24 168,72.9 amount 3d year 6,749 int. 4th year. | 162,24 175,47.8 amount 4th year | ` 4 3. 7,01.9 int. 5th year |6|48.96 182,49.7 amount 5th year 150,000 principal * 32,49.7 compound int. for 5 years. 2. What is the compound interest of 760 dollars, for || 3 years, at 6 dollars per cent. per annum? Ans. $145 17 cts. 2 m.-H. 3. What is the compound interest of $242 50 cts., | for 4 years, at 6 per cent. per annum? º Ans. $63 65 cents. 4. What is the amount of 1300 dollars, for 3 years, | at 5 dollars per cent. per annum, compound interest? & Ans, $1504 91 cts, 2 m.—H || 5. How much is the amount of 3127 dollars, for 4| years, at 44 dollars per cent. per annum, compound in-l |terest? Ans. $3729 00cts. 5m. || * . . Questions. || What is interest? | What is the principal? | What is the rate per cent. | | What is the amount? , | | How do you proceed when the interest for several || | years is required? w H. 20 compound INTEREST. 233 | What is to be noted if the interest is required for || | years and months? . . || | When the interest is required for any number of || | weeks or days, less or more than one year, how do you || | perform the operation? | | How do you proceed to find the interest, at 6 per cent. | | for any number of days, as computed at banks? | | What is to be observed when the interest is at any || | other rate than 6 per cent.? | | How do you proceed, when the principal, amount, and || | time are given, to find the rate per cent.?. . | | How do you find the time, when the principal, amount, | and rate per cent. are given? . . | | What is compound interest? How is compound interest computed? º PROMISCUous ExERCISEs. | 1. What is the interest of 620 dollars 25 cents for 5 || | years, at 5% per cent. per annum? | Ans. $170 56 8 m + || | 2 What is the interest of $420, for i year, at 7| | per cent. per annum? Ans. $29,40.| 3 What is the interest of 1450 dollars, for 60 days,|| t 6 per cent, per annum? Ans. $14.50 cts.] What is the compound interest of $626 25, for 3| ears, at 54 per cent. per annum? tº Ans. $103.91.--|| 5. What is the interest of $1659 for 3 weeks, at 4 perſ| per annum? N. : Ans. $3.824-H || 6 In what time will 500 dollars amount to 1000 d lars at 8 per cent. per annum, INSURANCE, comMISSION, AND BROKAGE. 121 || INSURANCE, COMMISSION, ANO BROKAGE. | INSURANCE, Commission and Brokage, are allowances | | made to insurers, factors, and brokers, at such rate per || |cent. as may be agreed on between the parties. § RULE. | Proceed in the same manner as though you were re-|| |quired to find the interest of the given sum for one year. EXAMPLES. 1 What is the commission on 625 dollars, at 4 dollars || |per cent? | - $625 4. Ans. $25,00 2 What is the commission on $1320, at 5 per cent.?| Ans. $66. || 3 What is the commission on 3450 dollars, at 44 dol- |lars per cent.? * Ans. $155,25. 4 The sales of certain goods amount to 1680 dollars: ||what sum is to be received for them, allowing 23 dollars |per cent. for commission? Ans. $1633,80.| || 5 What is the insurance of $760, at 64 per cent.? || | º Ans. $49,40. || | 6 What is the insurance of 5630 dollars, at 73 dollars| |per cent.? . . . . Ans. $436 32 cts. 5 m. || 7 A merchant sent a ship and cargo to sea, valued at] ||17654 dollars: what would be the amount of insurance, |at 183 dollars per cent.? Ans. $3310 124 cts.] 8 What is the brokage on 2150 dollars at 2 per cent.?|| - Ans. $43. 9 When a broker sells goods to the amount of 984| | dollars 50 cents, what is his commission, at 14 dollar per || |cent.? Ans. $12.304 cts.-H..] *10 If a broker buys goods for me, amounting to 1650, | %. º º - sº | dollars 75 cents, what sum must I pay him, allowing him. # : Ans. $24.76 cts. 1 m.-H. i | |li per cent.” urance, Commission, and Brokage? º || How do you proceed to find the Insurance, Commis-I |sion, or Brokage? : ... º.º.º. - . | | In what does this rule differ from interest? It takes|| ||no account of time. || Questions. **** | % DISCOUNT. | | Discount is an abatement of so much money from any | |sum to be received before it is due, as the remainder || would gain, put to interest for the given time and rate | per cent. * ... g || Find the interest of 100 dollars for the given time at the given rate per cent. , , , , , , , , || Add the interest so found to 100 dollars, then by pre-I on, , , il As the amount of 100 dollars for the given ti Is to the given sum, . So is 100 dollars, To the present worth ‘om the given sum, and the remainder will be the int. . . . . . E–When discount is time, it is found precisely like t MPLEs. ºn ExAMPLEs. , the present worth of 420 tº at 6 per cent, per a : £ 3 is DISCOUNT. $ $ $ $ $ 6 112 : 420 :: 1U0 : 375 2 12 100 *ś 112 | 2 What is the present worth of 850 dollars, due in 3| | months, at 6 per cent. per annum? . w # , , . Ans. $837 434 cts.-- 3 What is the discount of 645 dollars, for 9 months, at 6 per cent. per annum? : Ans. $27 774 cts. | || 4 What is the present worth of 775 dollars 50 cents, I | due in 4 years, at 5 per cent. per annum? ... --> Ans. $646,25.] | 5 What is the present worth of 580 dollars, due in 8| months, at 6 per cent. per annum? Ans. $557,09.-- | 6 What is the present worth of 954 dollars, due in 3| | years, at 43 per cent. per annum? | | Ans. $840 52 cts, 8 m.—H·] || 7 What is the discount of 205 dollars, due in 15| |months, at 7 per cent per annum? | -2. Ans. $16 49 cts. 5 m.--| 8 Bought goods amounting to 775 dollars, at 9 months' |credit: how much ready money must be paid, allowing || |a discount of 5 per cent. per annum? | | 3 Ans. $746 98 cts, 7 m. || ||| 9 I owe A. to the value of 1005 dollars, to pay as ſol-l |lows: viz. 475 dollars in 10 months, and the remainder || | in 15 months; what is the present worth, allowing dis- |count at 6 per cent. per annum? || # 3 Ans. $945 40 cts. 4 m. || | 10 What is the difference between the interest of || |2260 dollars, at 6 per cent. per annum, for 5 years, and || |the discount of the same sum for the same time and rate || |percent.? Ans, $156 46 cts. 2m- Equation of PAYMENTs. | cent.” | What is the discount of 520 dollars, at 5 per * $520 :* --> 5 | $26,00 Ans. | 12 How much is the discount of $782, at 4 per cent.?|| → Ans. $31, 28|| 6 dollars, at 3 per cent.? | nº Ans. $14,28.]] |, 14 Bought goods on credit, amounting to 1385 dol-| |lars: how much ready money must be paid for them, if |a discount of 6 per cent, be allowed? Ans. $1301,90. 15 I hold A.’s note for 650 dollars; but I agree to al- low him a discount of 44 per cent. for present payment: | what sum must I receive?” - Ans. $620,75. * . . . . . . . . . . *::: * * * . | 13 What is the discount of 47 º What is discount? . . . . º 3: ... ; What is first to be done? . . . . . . . £3. 3 interest of 100 dollars, at the | After having found the inter # * : *... º.º.º.º.º. º. º.º. # ºf . * *š º . . |given time and rate per cent, what is next to be done? || After having added the interest so found to 100 dol- ars or pounds, by w at rule do you work to find the dis- endiscount is made without regard to time, how is le debt, the EQUATION OF PAYMENTS. 125 || Proof–The interest of the sum payable at the equa- |ted time, at any given rate, will equal the interest of the || |several payments for their respective times. N. EXAMPLES. | 1 C. owes D. 100 dollars, of which the sum of 50 | | dollars is to be paid at 2 months, and 50 at 4 months; but they agree to reduce them to one payment; when || must the whole be paid? Ans. 3 months. § 50×2=100 50×4–200 100)300(3 months | 2 A merchant hath owing to him 300 dollars, to be || paid as follows: 50 dollars at 2 months, 100 dollars at 5 || | months, and the rest at 8 months; and it is agreed to || make one payment of the whole; when must that time || | be? Ans. 6 months. || || 3 F. owes H. 2400 dollars of which 480 dollars are || | to be paid present, 960 dollars at 5 months, and the rest || | at 10 months; but they agree to make one payment of || |the whole, and wish to know the time? Ans. 6 months. || 4 K. is indebted to L. 460 dollars which is to be dis- || charged at 4 several payments, that is 4 at 2 months, 4 || |at 4 months, 4 at 6 months, and 4 at 8 months; but they || |agreeing to make one payment of the whole, the equa- || |ted time is therefore demanded? Ans. 5 months. || | 5 P. owes Q. 420 dollars, which will be due 6 months || |hence, but P. is willing to pay him 60 dollars now, pro- || |vided he can have the rest forborn a longer time: it is || |agreed on; the time of forbearance therefore is required? || tº . ~ : Ans. 7 months. || | 6 A merchant bought goods to the amount of 2000 || | dollars and agreed to pay 400 dollars at the time of pur- || |chase,800 dollars at 5 months, and the rest at 10 months; |but it is agreed to make one payment of the whole; what |is the mean or equated time? Ans. 6 months. 'I * T T º BARTER." BARTER, Barren is the exchanging of one kind of goods for || |another, duly proportioning their values, &c. : RULE. % | The questions that come under this head, may be || | done by the compound rules, the Rule of Three, or | Practice, as may be most convenient. s -- º § sº : - : ºf . , , ; ; ; ; ; ; ; ; ºf . , , , , , , ---| - ExAMPLEs. - || | | 1 A country storekeeper bought 150 bushels of salt, | at 56 cents per bushel; and is to pay for it in corn, at || | |334 cents per bushel; how much corn will pay for the | salt? ct. ct, bu, bu. As 334 : 56 : : 150 : 252 - OR. 56 - 334)8400 150 3 3 tºº * * * * 56 . . . . . . . . : - stofthe salt, 8400 cts. . . . . “. . . . . . .3, 2 How much wheat, at 1 dollar 25 cents per bushel, ll pay for 35 sheep, at 2 dollars 25 cents a piece? . Ans. 63 bush, 3 How much sugar, at 9 cents per lb. will pay for i || dozen pair of shoes, at 1 dollar 75 cents per pair? 252 bushes of corn. s º 4. How much tea, at 80 cents per lb. will pay for 560 lbs. of pork, at 5 cents per lb.? Ans. 35 lbs. 50 cents; and 5 dollars—how much corn is: its will pay for them? Ans. 53 bush. 4 qts. º: º \. () bush barters with | º f corn, which he . narrºn. 2. : . 2:3... .33.3%:... ? Tººl | 7 A boy bartered 735 pears for marbles, giving 5 | pears for 2 marbles—how many marbles ought he to | have received? Ans. 294 marbles. || 8 A boy exchanges marbles for pears, and gives 2 || marbles for 5 pears—how many pears should he receive || | for 294 marbles? Ans. 735 pears. || 9 A farmer bartered 3 barrels of flour, at 5 dollars 25 || cents per barrel, for sugar and coffee, to receive an equal || | quantity of each—how much of each must he receive, || | admitting the sugar to be &alued at 9 cents per lb. and || || the coffee at 14 cents? " Ans. 68% Ib. nearly. || 10 A bartered 42 hats, at I dollar 25 cents per hat, | with B. for 50 pair of shoes, at 1 dollar 124 cents per | pair—who must receive money, and how much? º . Ans, B. $3,75. 11 Sold 75 barrels of herrings, at 2 dollars 75 cents || per barrel, for which I am to receive 75 bushels of wheat, at 1 dollar 8 cents per bushel, and the residue in money— || | how much money must I receive? º Ans. 125 dolls. 25 cts. 12 Sold 35 yards of domestic, at 20 cents per yard, |and am to receive the amount in apples, at 25 cents per bushel—how many bushels must I have? || Ans. 28 bush. I 13 Gave 35 yards of domestic for 28 bushels of ap-|| |ples, at 25 cents per bushel—what was the domestic | rated at per yard? Ans. 20cts. || 14 What is rice per lb. when 340 lb. are given for 4 | | yards of cloth, at 4 dollars 25 cents per yard? . - . Ans. 5 cts. 15 Gave in barter 65 lbs. of tea for 156 gallons of | rum, at 334 per gallon—what was the tea rated at? || | Ans. 80 cts. per Ib. 16 Q. has coffee worth 16 cents per pound, but in || |barter raised it to 18 cts.; B. has broad cloth worth 4|| | dollars 64 cents per yard—what must B. raise his cloth || |to, so as to make a fair barter with Q? Ans. $5.22.] --- * --- Loss AND GAIN. | 17 B. had 45 hats, at 4 dollars per hat, for which A. || |gives him 81 dollars 25 cents in cash, and the rest in || |pork, at 5 cents per Ib; how much pork will be required?|| | ~ - Ans. 1975 lb. || | 18 Two merchants barter; A. receives 20 cwt. of |cheese, at 2 dollars 87 cents per cwt.; B. 8 pieces of || |linen, at 9 dollars 78 cents per piece; which of them || must receive money, and how much? Ans. A. $20,84. || | 19 If 24 yards of cloth be given for 5 cwt. I qr. of || tºbacco, at 5 dollars 7 cents ºper hundred; what is the cloth rated at per yard? Ans. $1.109. || | 20 A. barters 40 yards of cloth, at 98 cents per yard, |with B. for 284 lbs. of tea, at I dollar 53 cents per lb.;| |which must pay balance, and how much? || | * Ans. A. $4,405. || | 21. A has 73 cwt. of sugar, at 8 cents per lb., for || which B. gave him 124 cwt. of cheese; what was the ||cheese rated at per lb. ? Ans. $.048. | 22 What quantity of sugar, at 8 cts. per lb. must be || |given in barter for 20 cwt. of tobacco, at 8 dollars per || |cwt.% : Ans. 17 cwt. 3 qrs. 12 lb. 23 T. has coffee, which he barters with Q. at 11 cts.l. er 1 b. more than it cost him, against tea, which stands || 1 dollar 33 cents the lb., but he puts it at 1 dollar || sents; query, the prime cost of the coffee? ..., || Amºs. 443+| º, , º, % & . . . ; # - By Loss and GAIN, merchants and dealers compute|| ir gains or losses. ~, : - - I.OSS AND GAIN. 129 | EXAMPLES. || 1 Bought 1234 lbs. of coffee, at 124 cts. per lb., and || |sold the whole for 160 dollars; did I lose or gain by it, |and now much? Ans, gained $5,75. || | 2 Bought 120 dozen knives, at 2 dollars 50 cents per dozen, and sold them at 183 cents a piece; did I gain || |or lose, and how much? Ans, lost $30.| || 3 Bought 1234 yards of muslin, for 174 cents, and || |sold it at 20 cents per yard; what was the gain? 3 N. . . Ans. $30,85. | || 4 Bought 10 chests of tea, each 63 lbs. neat, for 600|| | dollars, and retailed it at 874 cents per lb.; did I gain || |or lose, and how much? Ans. lost 48 dol. 75 ct.| || 5 Gave 285 dollars 25 cents for 4564 lbs. of bacon, |and sold it for 365 dollars 12 cents; what was the gain || |per Ibº Ans. $. 13 cts. | 6 Bought 1234 yards of muslin, for 246 dollars 80 | |cents, and sold it for 215 dollars 95 cents; what did I | |lose per yard? Ans. $. 24 cts. || || 7 Gave 25 cts. per bushel for corn, and sold it at 28 | | cents; what is the gain per cent.” | : Ans. 12 dolls. per 100 dolls. | 8 Sold corn at 25 cts. per bushel, and 4 cts, loss; | ||what was the loss per cent.? Ans. $18,79. 9 Bought 13 cwt. 25 lbs. of sugar, for 106 dollars, |and sold it at 94 cts. per lb.; what did I gain per cent.?| i. . Ans. 32 dolls. 73 cts. || | 10 Bought 126 gallons of wine for 150 dollars, and || |retailed it at 20 cts. per pint; what was the gain per || licent.? Ans. 34 dolls. 40 cts. || | 11 Sold a quantity of goods, for 748 dollars 66 cents, |and gained 10 per cent; what did I give for them.? || * . . . . . ." Aas. 680 dols. 60 cts. || Loss AND GAIN. dols. dols. dols. 110 : 100: ; 748,66 100 **** 110)74866,000$680,60 | 12 Sold goods to the amount of $1234, and gained at I |the rate of 20 per cent.; what was the prime cost? | | & , , ; ; ; gº & Ans. $1028,334| || 13 Sold a quantity of goods, for $475, and at a loss || juſ 12 per cent.; what did I give for them? || dols. dols. dols. 100 88 : 100 : : 475 12 100 88 … SS)17500(539,77+Ans. 14 sold hats to the amount of - $136, at 20 per cent. | |loss; what was the first cost? - Ans. $170.| |, 15 Laid out $755 in salt; how much must I sell itſ for, so as to gain 12 per cent.” 12 I00 16 Bought 32 yards of mole skin for 128 dollars; what must I sell it for per yard, so as to gain 20 per nt.? Ans. 4 dols. 80 cts.--|| 17 Bought 17 yards of silk for 21 dollars; how much || yard rust Isetail it for, and gain 25 per cent.? || T. "Tº Kns, idol. 54 cts.--| Bought 64 yards ºf muslin for 13 dollars 50 gents, º ving a bad bargain, I am willing to lose 8 perſ hat must I sell it at per yard? Ans,19ets,4 m.-H..] - º º * . . . . . . . . . . 1 hats are bought at 48 cents, and sold at 5. the gain per cent.? Ans. 124 OSS AND GAIN. ? 131 || 20 If, when cloth is sold for 84 cents per yard, there is gained 10 per cent.; what will be the gain per cent. || when it is sold for 1 dollar 2 cents per yard? … . Ans. 33 dols. 68 cts.--|| | 21 Bought a chest of tea, weighing 490 lbs. for $122 || |50ct. and sold it for $137 20 cents; what was the profit on each lb.? Ans. 3 cts. || 22 Bought 12 pieces of white cloth, for 16 dollars 50|| |cents per piece; paid 2 dollars 87 cents a piece for || dying; for how much must I sell them each, to gain 20 | per cent.? Ans. 23 dols. 244. || || 23 If 28 pieces of stuff be purchased at 9 dollars 60| cents per piece, and 10 of them sold at 14 dollars 40 |cents, and 8 at 12 dollars per piece; at what rate must || the rest be disposed of, to gain 10 per cent. by the whole? Ans. 5 dols. 568. 24 Sold a yard of cloth for 1 dollar 55 cents, by || | which was gained at the rate of 15 per cent.; but if itſ had been sold for 1 dollar 72 cents; what would have been || the gain per cent.? Ans. 27 dols. 69-H || 25 If, when cloth is sold at $. 935 a yard, the gain || is 10 dollars per cent.; what is the gain or loss per cent., || | when it is sold at 80 cents per yard? . Ans. 5 dollars 88-Hloss. || 26 A draper bought 100 yards of broad cloth, for || which he gave $56—I desire to know how he must sell || |it per yard, to gain $19 in the whole? x #: ; x . ; : Ans. 75 ct. per yard. || | 27 Adraper bought 100 yards of broad cloth for $56;| I demand how he must sell it per yard, to gain $15 in || | laying out $100? 2 : Ans, 64 ct.4 m. || 28 Bought knives at 11 cents, and sold them at 12 || | cents; what will I gain by laying out 100 dollars in || |knives? … Ans. 9 dols. 09-H || | 29 Bought knives at 11 cents, and sold them at 12|| | cents; what did I gain by selling to the amount of 100|| | dollars? Ans. 8 dols. 333-H || % || 30 If by selling 1 lb. of pepper for 104 cents, there | | are 2 cents lost; how much is the loss per cent.? | Ans. 16 dols. || 31 Amerchant receives from Lisbon, 180 casks of | raisins, which stands him in here 2 dollars 13 cents each, nd by selling them at 3 dollars 68 cents per cwt., he ains 25 per cent.; required the weight of each cask, |one with another? Ans. 81 lb. || – - : - FELLOWSHIP. % | Fellowship is a method by which merchants and || |others adjust the division of property, loss, or gain, &c., | in proportion to their several claims. CASE 1. SIMPLE FELLowship. 2. . . ; ºft When the claims are in proportion to the amount of || k, labor, &c., without regard to time. || As the whole amount of stock or labor, Is to each man's portion, a So is the whole property, loss, or gain, To each man's share of it. - º Proof–The sum of all the shares must equal the whole gain, &c. tº || llars by the transaction; what was the share of each? Ans. A. received 85 dols. 334 cts. and B. 42 dollars ... . . $, $, $ $ ºt. Prº 8:42,663 j hole stºk 480 As480:160::12 FELLOWSHIP. 133 | 2 Three workmen having undertaken to do a piece | of work for 275 dollars, agreed to divide their profits in | proportion to the amount of labor each one performed. |M. labored 50 days, N. 65 days, and O. 85 days: what || | was the share of each? . | |- Ans. M. received 68 dols. 75 cts.; N. 89 dols. 37: cts.; |and O. 116 dols. 874 cts. | 3. A merchant being deceased, worth 1800 dollars, is | found to owe the following sums: to A. 1200 dollars, to | B. 500 dollars, to C. 700 dollars: how much is each to |have in proportion to the debt? . Ans. A. 900 dols., B. 375 dols, and C. 525 dols. || 4 Three drovers pay among them 60 dollars for pas- | |ture, into which they put 200 cattle, of which A. had 50, B. 80, and C. 70: I would know how much each had to | pay? Ans. A. 15 doh., B. 24 dols., C. 21 dols. 5 A man failing, owes the following sums: to A. 120 | dollars, to B. 250 dollars 75 cents, to C. 300 dollars, to | D.208 dollars 25 cents; and his whole effects were found || | to amount to but 650 dollars: what will each one receive in proportion to his demand? º Ans. A. $ 88.73.-- C. $221.84.-H. B. $185.42.-- D, $153.99.--| | 6 Abankrupt is indebted to A. 500 dollars 374 cents— |—to B. 228 dollars—to C. 1291 dollars 23 cents—to D. | 709 dollars 40 cents; and his estate is worth 2046 dol-[ | lars 75 cents: how much does he pay per cent, and || | what does each creditor receive? | Ans. He pays 75 per cent, and A. receives 375 | dollars 273 cts.; B. 171 dols.; C. 968 dols. 424 cts.;| | and D. 532 dols. 5 cts. 2. « . . ; … .. 8 | | 7 If a man is indebted to A. 250 dollars 50 cents, to || | B. 500 dollars, to C. 349 dollars 50 cents, but when he | comes to make a settlement, it is found he is worth but || | | 960 dollars, how much will each one receive, if it be in | proportion to their respective claims? || . A. $218 61 cts. 8 m.—H || B. $436 36 cts. 3 m.—H || C. $305 01 ct. 8 if r Ans. 12 CASE 2, Compound FELLowsHIP. | | When the respective stocks are considered with rela- |tion to time. . . . . | Multiply each man's stock by its time; add the several | products together; then: , tº As the sum of the products . 3. || Is to each particular product, ź | So is the whole gain or loss * * * . x-X. ExAMPLEs. . || 1 Three merchants traded together; A put in 120 || ||dollars for 9 months, B. 100 dollars for 16 months, and || C. 100 dollars for 14 months, and they gained 100 dol- lars; what is each man's share? A's. stock 120 × 9 = 1080 B's. stock 100 × 16 = 1600 Sum 4080 | As 4080: 1080::100 : 26,47+ A’s. share. || As 4080: 1600::100 : 39,21++ B's. share. I As 4080: 1400::100 : 34,31+ C's, share. 8. s, M. 120 dollars for 4 months, and N. 300 d ach man receive of the gain? Ans. {M. $34, 71 cts. 6 " (N. $130 18 cis. 8 WU LGAR FRACTIONS. 135|| | 3 Two merchants entered into partnership for 16|| |months: A. put in at first $600, and at the end of 9|| |months put in $100 more; B. put in at first $750, and at || | the end of 6 months took out $250, at the close of the time their gain was $386, what was the share of each? Ans. A's. share was $200,793; B's. share was] $185,20. w - 4 A., B., and C., made a stock for 12 months; A, putſ |in at first $873,60, and 4 months after he put in $96,00| more; B, put in at first $979,20, and at the end of 7 months he took out $206,40; C. put in at first $355,20, and 3 months after he put in $206,40, and 5 months after| that he put in $240,00 more. At the end of 12 months, their gain is found to be $3446,40; what is each man's |share of the gain? A’s. share is $1334,824 Ans. {B's. - - $1271,613–H C’s. - - $839,96 - Questions. - What is Fellowship? By what rule are its operations performed? When is Fellowship simple? When is it compound? In what respect is Fellowship compound? Ans. The proportion is compound: that is, the divi- |sion of property, gain, &c., is founded on the compound| |proportion of the stock and time. vulgar FRACTIONs. | A vulgar FRAction is a part, or parts of a unit ex-| |pressed by two numbers placed one above the other with || |a line between them. As #, ;, &c. ~ The number below the line is the denominator, the] |number above the line is the numerator. llwhich the unit is divided. The denominator denotes the number of parts into The numerator shows how many of those parts are to e taken. || º Fractions are either proper, improper, or compound A proper fraction is one whose numerator is less than || denominator, as # or 3. . . . . " || An improper fraction is one whose numerator is || reater than its denominator, as # or #. - || A compound fraction is a fraction of a fraction, as of || # of 3. . . . . . . . . . . . . . . . . . . . . º . º, º 4, º * ; : 3. - - ; : . * - - A mixed number is a whole number and a fraction. || To reduce a fraction to its lowest terms. * * #: ; º * * RULE. | || Divide the terms by any number that will divide both || ||without a remainder, and divide the quotient in the same || manner, and so on till no number greater than one will divide them: the fraction is then at its lowest terms. || 1. Reduce # to its º * 4)#=#=# result. Reduce ºr to its lowest terms. & 3. Reduce # to its lowest terms. vv. 4. Reduce # to its lowest terms. º Res. # || NoTE.—When a divisor cannot readily be found, divide || he denominator by the numerator, and that divisor by he remainder, and so on, till nothing remain: the last || - sommon measure of the two numbers; with || % º º VULGAR FRACTIONS. 137 %2. 5 Reduce ºn to its lowest terms. Res. #. 85 85)136(1 Here 17 being the last divisor, I. 85 is the common measure of 85|| f 51)85(1 º 51 l 34)51(1 - 85 5 34 17 | —-) = Q– $º 136 8 17)34(2 w 34 | 6. Reduce fºr to its lowest terms. Res. #. 7. Reduce ### to its lowest terms. Res. #. 8. Reduce #}} to its lowest terms. Res. #. CASE 2. To reduce a mired number to an improper fraction. | RULE. | Multiply the whole number by the denominator, and | |add the numerator to the product for the numerator of || the improper fraction, and place the denominator under || it. *: 1. Reduce 12 # to an improper (raction. | 4 - 1 1 2 || 112 Nine 12's are 108; add — a 4 makes 112 ninths. | 2. Reduce 17 jº an improper fraction. Ros. .#. || 3. Reduce 45 # to an improper fraction. Res. 31. | 4. Reduce 24 # to an improper fraction. Res. ***. vulgar FRActions. CASE 3. . . . . . . . To reduce an improper fraction to its proper value. || Divide the numerator by the denominator, and the | quotient will be the whole number; the remainder, if || |any, will be the numerator of the fraction. -- º º EXAMPLES. w º 1 Reduce "# to its proper value. Res. 3 #. || 17 T5)17 2 Reduce "#" to its proper terms. Res. 12 #. || 3 Reduce "#" to its proper terms. Res. 17. ; . || 4 Reduce # to its proper terms. Res. 24 ##. | CASE 4. º | To reduce several fractions to other fractions having a || common denominator, and retaining their value. || RULE. sº i || Multiply each numerator into all the denominators || |but its own, for the respective numerators; and all the denominators together, for a common denominator. || || 1 Reduce #3 and # to a common denominator. || . Res. ####, and #: º 3×3×6=54} Numerators Then we have # for #; # for 3, and # for ; , , || Reduce each new fraction to its lowest terms, and the [ result will prove the work to be right. º || : 2 x ~ 3 Res. #4, #4% ** 358, agº, "" as a common denominat 16 2.88 3 na º & 3. Reduce }, +, #, and § # to º gº sº % UICe #, § 3 ;5 and #, ºś &:3&º $33 V ULGAR FRACTIONS. 139|| | NotE-It is often convenient to use the least possible| common denominator; to find which, divide the denomi-|| |nators by any number that will divide two or more of || || them without a remainder, setting down those that would || |have remainders; then multiply all the divisors and all || | the quotients together. . 412 3 4 5 6 7 8 9 a || Tº T. To T. T., | 2 || 2 | ----, | 1 → | 4X3X2X5X7 ×3–2520 common denom. |Which may be divided separately by 2, 3, 4, 5, 6, 7, 8, | and 9, without a remainder. EXAMPLEs. º 5 Find the least common denominator for 3, #, , , || | #, and #7, and compute their equivalent fractions. . * . R 18.0 2 1 0. 220. 7 5 - 103. ©S. 3 #52 gig, 31 gº 31 J, #4 o 240 com. denom. #| 60× 3=180 # 30 × 7=210 *}| 20×11=220 12× 9=108 6 Reduce {, }, +, +, and #}, to their least common t 2 0 H 1». :- - - , 9 0 1 0 0 1 0 5 1 0 8 1 1 0 || || 7 Reduce {, , , , , to their least common denom- ! ~~ 3% 1 5 0 1 6 & 1 35 | 1 || 0 || | | . 8. Reduce the above fractions to a common denomina-|| |tor, by the general rule, Case 4. . § 5× 10×16X24=19200 7X 8×16×24=21504 9× 8×10×24==17280 11× 8×10×16=14080 1728.0 , 14030. Hiſ 50730, 30735. H. vulgar FRACTIONs. CASE 5. To reduce a compound fraction to a simple one. || --> ‘. . . . . RULE. z ź . | | Multiply the numerators together for a new numera-H |tor; and the denominators together for a new denomina- || |tor. . º | EXAMPLES, . is 1 Reduce # of ; of ºr to a single fraction. | . Res. #. º 3×5×9 135 9 * : * * 4×6X 10 •240 16 H 2 Reduce # of 4 of ## to a single fraction. Res. #. || 3 Reduce # of # of 4 to a single fraction. Res. ...| 4 Reduce # of ; of # to a single fraction. Res. #, | § 3 ; ;& CASE 6. º . | | To reduce a fraction of one denomination to the fraction || | of another denomination, but greater, retaining the same value. . . . . " tº . RULE. º | Multiply the denominator of the fraction by the num- || | ber of that denomination which it takes to make one of || he next, and so on to the denomination required, and || lace the numerator of the given fraction over it. §. ExAMPLEs. . . . . . . . tº 1 Reduce # of a quart to the fraction of a bushel.” || qt. º º 3. * * º º — — = — Result, ºr of a bushel. || 2 Reduce # of an ounce, Troy, to the fraction of a || 3 Reduce # of a nail to the fraction of a yard. Res. #, or ºr of a yard. || 3 Reduce # of a perch to the fraction of an acre. . Res. Hºw: vulgar FRACTIONs. 141|| º * 4 Reduce ºr of a pint to the fraction of a hogshead : Res. ### of a hºld. || CASE 7. To reduce the fraction of one denomination to the frac- tion of another, but less, retaining the same value. RULE. Multiply the given numerator by the parts of that be- |tween it and that to which it is to be reduced, and place || the product over the given denominator for the fraction || required. EXAMPLES. 1 Reduce ºr of a bushel to the fraction of a quart. Res. ; of a quart. 2 Reduce ºr of a yard to the fraction of a nail. - Res. 4 of a nail. 3 Reduce rºw of an acre to the fraction of a perch. Res. # of a perch.| 4 Reduce ºr of a hogshead to the fraction of a pint. || º Res. # of a pint. 5 Reduce riºr of a day to the fraction of a minute. Res. }} of a minute. CASE 8. To reduce a fraction to its proper value or quantity, in whole numbers. || RULE. - Multiply the numerator by the parts of the integer, and divide by the denominator. ſ EXAMPLES. 1 Reduce # of a yard to its proper quantity. - - Res. 3 qr. 2 na. 7 eighths of a yd. 4 eighths of a qr. 4. 4 - 8)28 8)16 3; quarters 2 nails š º º | 2 Reduce # of a pound, avoirdupois, to its proper || || quantity. - - - & Res. 8 oz. 14; dr. | 3 Reduce # of a pound, Troy, to its proper quantity. || Res. 9 oz. . 4 Reduce # of a mile to its proper quantity. || - Res.4 fur. 125 yd. 2 ft.1#inch. H 5 Reduce ºr of an acre to its proper quantity. Res. 1 rood, 30 perch. || 6 Reduce # of a dollar to its proper quantity. | : - Res. 60 cents. || 7 Reduce of a pound to its proper value. || - Res. 6s. 8d. || | 8 Reduce # of a year (365 days) to its proper quan-l |tity. & . . . . . . . Res. 225 days. || ||| 9 Reduce # of a tun to its proper quantity. | | * Res. 3 hlid. 7 gal. | | 10 Reduce # of a ton to its proper quantity. | : Res. 15 cwt. 2 qr. 6 lb. 3 oz.8% dr. | & | To reduce a given quantity to a fraction of any greater || denomination of the same kind, | . - RULE. | Reduce the given quantity to the lowest denomination || entioned for a numerator; and the integer to the same || #º | denomination, for a denominator. º - EXAMPLES; - 1 Reduce 3 qr. 2 na, to the fraction of a yard. | º, Res. # of a yard. || qr, na. -- º, 3 4 * 2 Reduce 2 roods 20 perches to the fraction of a # of an WUL.GAR FRACTIONS. 143 | ſ 3 Reduce 6 furlongs 16 poles to the fraction of a mile. Res. of a mile. 4 Reduce 9 ounces, Troy, to the fraction of a pound. Res. of a pound. 5 Reduce 7 hours 12 minutes to the fraction of a day. % Res. #'ſ of a day. ADDITION OF WUILGAR FRACTIONS. RULE.—Reduce the given fractions, if necessary, to single ones, or to a common denominator; add all the | numerators together, and place the sum over the com- | mon denominator. EXAMPLES. 5 3. 2 g 4 iſ 6 # * , ; : § 3 * 8 . - 8 g 7 ºf 2 : 2 - - * — 4 8 19 or 2 # 26 ſº-26; 28 ; * 8 NotE 1.--When the fractions are of different denom- || inators, reduce them to a common denominator, and || - ; proceed as above. (See Note, page 139.) || | 4 Add 3. W ##, # and ºr together. Result 3:... || # 60× 3=180 . # 30× 7=210 *; 20× 11=220 # 15× 5= 75 | | | 12× 9=108 | 5 Add {, }, +, +, and #!, together. Res.4}}. || 6 Add #, †. #, and }} together. Res. 2.É. | 7 Add +, +, and ſº together. Res. #######. 8 Add 3 and # together. Res. 14%r. | 9 Add ºr, H, and # together. Res. 2 #4. | * VUL ADDITION OF WULGAR FRACTIONS. - RULE. | Reduce the given fractions, if necessary, to single || |ones, or to a common denominator; add all the numera-ſº |tors together, and place the sum over the common de-i |nominator. -----> --> ExAMPLEs. 3. &&. I 0. is : i 0. i i 19 or 2; 26 A-26; 28 * *ś # | NoTE 1–When the fractions are of different denom-|| |inators, reduce them to a common denominator, and || |proceed as above. . . (See Note, page 139) || || 4. Add +, +, +, +, and # together... . Result, 3 #| ; º | 1302 ſesſo > 3172 º —} =3;# 960 º #, #, 4, # and # together. Add #, Fr. Pº, and # together. Add +, +, and # together. A. #, #º vulgar FRACTIONs. 145|| 20 Add # of a pound, to ºr of a shilling. Result 15s. 10; d. || | 15 s. d. # of a £ 15 º 5X2=10 *; of 8. S. 0 3; 3X3= 9 15 101*; 19 —} =1#. 15 21 Add { of a pound, to ; of a shilling. . … Result 18s. 3d. 22 Add + of a penny, to # of a pound. || Res. 2s. 8d. Iqr. #. || 23 Add + lb. troy, to #5 of an ounce. . | Res. 60Z. 11dwt. 16grs. || 24 Add + of a mile, to ºr of a furlong. Res.6fu.28p. | 25 Add + of a yard, to ; of a foot. Res. 2ft. 2 in. || 26 Add + of a day, to ; of an hour. Res. 8h. 30min. | | 27 Add + of a week, of a day, and ; of an hour || |together. . Res. 2 days, 14 hours, 30min. || SUBTRACTION OF VULGAR FRACTIONs. . . . . . . RULE. | Prepare the given fractions as in Addition; then sub- |tract the less from the greater, and place the difference |over the common denominator. 1 Take # from #. **::: * ::... 3 Rem. #. x. 2 Take #4 from #. . . . . Rem, #. 3 Take ºr from #. Rem. #. | | 4 Take # from 3. Rem. ...| || || 35 || 7X2=14 º x | º º i 3. ‘. ; 12 com. denom 60 com. denom. # 1×11=11 is 4x6=28 . . * Ans. —- †† Ans. -- 1# 4; 4; 54 Tws From # of a pound take # of a shilling. || s. d. -- § { ºf a pºnd =15 #| 38%-1. & of a shilling = 3; 3x3= 9 º, sº s.15 3. Ans. # * . . . . . § 3. Fron, # of a £ take 3 of a shilling. || | | From 3 of a lb. troy, take # of an ounce. ach. Res. 5in. #. lin. "ºlo.º. 14:...] §§º • TIPLICATION OF VULGAR FRACT Rule. ** º Prepare the given fractions, if necessary; then m ply the numerators together for a new ominators toget her for a new den #&#& 17| vur, GAR FRACTIONs. 2 Multiply ', by . 3 Multiply , by 'º. 4 Multiply 124 by 7%. 123 – 6.3 § 5 Multiply 7; by 8%. 6 Multiply 4, by #. 7 Multiply 4 by 13+. 8 Multiply of 4 by ºr of +}. 9 Multiply 43 by ; of 4. 10 Multiply , of 7 by . * 3. RULE. Prepare the given fractions, if necessary ExAMPLEs. 1 Divide # by . . 8X 4=32 7X 9=63 2 Divide 4 by 3. 3 Divide # by 4. | 4 Divide 1; by 41°r. 5 Divide 34 by 9%. | 6 Divide ; by 4. 7 Divide 4 by 4. - 8 Divide ; of 3 by 3 of 3. | 9 Divide ; of 19 by 3 of 4. 10 Divide 4} by # of 4. 11 Divide # of , by # of 73. | 12 Divide 5205; by # of 91. | 11 Multiply 24 by 1}, and multiply the product b Res. #'s. Res. 243, or 2;. I Res. 964. 2=23: then 68 × 23 – 1449 –963 es. 61}. 1 || y || es. #. DIVISION OF WULGAR FRACTIONS. 3. , then invert| the divisor, and proceed as in Multiplication. iº ** * 6 3 Res. . Res. 4. Res. } 3 Res. * * * : . +. 33 Rac 4.4 fº. Res. 44. || Res. 4. Res. 7i. | scs 7 Res. Hr. Res. 7i. || 148 VULGAR FRACTIONs. PROPORTION IN vulgar FRACTIONs. || tº RULE. | proceed as in Multiplication of Vulgar Fractions. 1 If 3 yd. cost $4; what will # yd, cost? Ans. 50c. || |...} : ; ; ; ; }. Then #x}x}=%=$#-50 cts. º || 2 If bu. cost $4; what will ; bu. cost? Ans. $2,80. |bu, bu. D. - . . . . * : ; ; ; ;. Then #x 4×4=},\;=$2}=$2,80. || 3 If A owned ; of a toll-bridge, and sold 3 of his || share for $684; what is the whole value Ans. $1520.| | of ; ; } :: ***: that is, ºr ; ; ; : ***. Then || || 2 9& ! ׺- 3 #80 =$1520 29 l * * * * * * * * * * . || 4 If I barter 5; cwt. of sugar at 6: cts. per lb., for | indigo at $4+ per lb.; how much indigo must I re-Hi | ceive? ... Ans. 10 lb. 5 oz. 2; dr.-- **ś. . #" ''', ; ár & 1 tº 9 1 # ºr , x. - oz. 2 r. ####. 27 53 . ić - º,4 {, , . § . . . ºf . sh --- 5 If the cent roll weighs 6; oz., when wheat is 68% cents, per bu.; what is the cost of wheat per bu, when it weighs 4} oz? º, . Ans. $1,03}. 4; ; ; ; ; 68; ; then #xºx***="###"=103 N. How many men will reap 4173 acres eap 52% in 64 days? # , º > * * . . . . . 2.5 º ºg º da 5. x 2 x a sis × ºx#. ote-In multiplying, omit the numb * , * DECIMAL FRACTIONs. : 149 | DECIMAL FRACTIONS. | A decimal fraction is a fraction whose denominator is || | 1, with as many cyphers annexed as there are figures in || the numerator, and is usually expressed by writing the numerator only with a point prefixed to it: thus #, ſº, |###, are decimal fractions, and are expressed by .5, |.75, .625. : ~ A mixed number, consisting of a whole number and || |a decimal, as 25+, is written thus, 25.5. :- - | As in numeration of whole numbers the values of the |figures increase in a tenfold proportion, from the right| |hand to the left; so in decimals, their values decrease in || |the same proportion, from the left hand to the right, which is exemplified in the following TABLE. # . A ... rº ă ă ă ſº 5 : § C o .3 tº § -E : : .S. E = 2, 3 E , E = ##3 Fă ă ă is . s: #F #: Tº = 2 + 3 + 3 ######## .# = 5 # 5 # , , ; ; ####### S. c → d c > 3 c.t. c 2: C c > → c c <5 E,& E = ? -5 – , º, E do tº -c. ° E & Cº. 3 #5 E E = tº F : E E - E + EE E - E E E 1 I I I I I I I 1 - 1 1 1 1 I I I II Whole numbers. .” Decimals. | NoTE.—Cyphers annexed to Decimals, neither in-ſ |crease nor decrease their value; thus, .5, 50, 500, be- |ing ſº, Hºw, ##, are of the same value: but cyphers|| |prefixed to decimals, decrease them in a tenfold propor- ferent values. .” . . . . . viz. units under units, tenths under tenths, &c., and add |as in addition of whole numbers; observing to set the point in the sum exactly under those of the given num- .134° 3.45 .24 4.12 .75 21 40.02 .122 15.4 .92 743 ° 35.4 36 76.36 63.25 002 12532 567 425.04 4. ºf sº 1.554 242.45 6 Add .5, 75, 125, 496, and .750 together. ; : 7 Add 15, 126.5, 650.17, 940.113, and 722:2560 * ... : :" *: . . . * * * šº sº * . Place the numbers as in addition, with the less unde the greater, and subtract as in whole numbers; settin f oint in the remainder under those in the give ExAMPLEs. * s > 54.87s * 14:3965 take 45.3 DECIMAL FRACTIONs. 151|| MULTIPLICATION OF DECIMALS. RULE. || Multiply as in whole numbers, and from the right || |hand of the product, separate as many figures for deci-|| |mals, as there are decimal figures in both the factors. || --> . EXAMPLES. . 1 Multiply,612 by 4.12 2 Multiply 1.007 by 041. .612 1.007 º 4.12 . . .041 §ºś * 1224 1007 612 4028 2448 .041287 2.52144 3 Multiply 37.9 by 46.5 Product 1762.35|| 36.5 by 7.27 265.355 || 29.831 by 952 28.399112|| 3.92 by 196. 768.32 || .285 by .003 .000855 | 4.001 by .004 .016004 || .00071 by .121 .00008591 || DIVISION OF DECIMALS. RULE. * . . . .”. sº Divide as in whole numbers, and from the right hand || |of the quotient, separate as many figures for decimals|| |as the decimal figures of the dividend exceed those of || || the divisor. If there are not so many figures as the |rule requires, supply the defect by prefixing cyphers. M. | 1 Divide 86.3972 by 92, 2 Divide 4.13 by 5724. | 92).563972(939í 5724)4.130000(0öigi+ 828 40068 Šºš 850 - 12320 lº 276 11448 || 2996 | 3 Divide 19.25 by 38.5 Quotient .5 || 4 || 234.70525 by 64.25 3.653 1.0012 by .075 . .1606 by 44 .1606 by 44 186.9 by 7.476 ºn wº ºš E.1. When a whole number is to be divided by a || i. whole number, cyphers must be affixed to the ividend, as decimal figures. 3 * . || 12 Divide 3 by 4 ºr Quotient .75 || 13 275 by 3842 071577-1-[. NoTE 2. When any whole number is divided by ano- her, if there be a remainder, cyphers may be affixed to || 4 5 6 9 he divide quotient continued. 25 ºn Quotient a DECIMAL FRACTIONS. **** REDUCTION OF DECIMALS. Case 1. To reduce a vulgar fraction to a decimal. RULE. | Place cyphers to the right of the numerator, until you || can divide it by the denominator, and continue to divide | | until there is no remainder left; or if it be a number || | which will never come out without a remainder, until it |is carried out to a convenient number of decimal places.| EXAMPLES, 1 Reduce # to a decimal. 5)40 - .8 Ans. - 2 Reduce 4 to a decimal. Ans. .875.] 3 Reduce #4 to a decimal. Ans. .70833.--| 4 Reduce #4 to a decimal. Ans. .1762.--|| | 5 Reduce # to a decimal. Ams. 4566.--|| * CASE 2. - tº |To reduce any given sum or quantity to the decimal of || º any higher given denomination. || º RULE. || | Reduce the given sum or quantity to the lowest de-|| | nomination mentioned in it. - - *} | Reduce one of that denomination of which you wish || |to make it a decimal, to the same denomination with the |given sum. º | | Divide the given quantity so reduced by one of the | denomination of which you wish to make it a decimal;| |the quotient will be the decimal required. | . ExAMPLEs. . 1 Reduce 3s.6d. to the decimal of a pound. 3s.6d.— 42 240)42,000(.175 decimals. An 1680 #º 1200 1200 Reduce 2R. 4P. to the decimal of an acre. , , , , , , Answer, .52: 3 Reduce 2 qr. 2 nails to the decimal of a yard. Ans. .62 4 Reduce 5 minutes to the decimal of an hour. . Ans. .08333 5 Reduce 10 grains to the decimal of an ounc apothecaries' weight. Ans. .02083.- 6 Reduce 2 quarts 1 pint to the decimal of a hog |head. . Ans. .00992. Case 3. fraction to its proper value. RULE. * . . . . . . . . . . . . Multiply the given fraction continually by the denom nation next lower than that of which it is a decimal, fo educe a decimal the proper value. What is the value of .375 of a dollar? Ans. 37}ct: &%. 3% DECIMAL FRACTIONs. 155|| 3 What is the value of .235 of a day? || . Ans. 5 hours, 38 min. 24 sec. || 4 What is the value of .42 of a gallon? | . ź Ans. 1 quart, 1.36 pt. || 5 What is the value of .253 of a shilling? Ans. 3.036d. || 6 What is the value of 436 of a yard? . º . . . Ans. 1 qr, 2.976 na. 7 What is the value of 9 of an acre? . . . . . º. . ‘. . Ans, 3R. 24.P.] PROPORTION IN DECIMALS. RULE. State the question as the rule of three, in whole num-II |bers, only observe, when you multiply and divide, to || |place the decimal points according to the rules of multi-l |plication and division of decimals. | š ExAMPLEs. 2 : || 1 If 4.21b of coffee cost 8s. 2.3d., what cost 639.25lb.?|| || 4.2 : 639.25 : : 8 2.3 : 62 6 9.49 Ans. || | 2 When 14 yard cost 13s. what will 15 yards come || |to at the same price? Ans. £6 19s. 3d. 1.71 qr. | 3 If I sell 1 qr. of cloth for 2 dollars 34.5 cents, what || |is it per yard? Ans. $938 cts. || || 4 A merchant sold 10.5 cwt. of sugar, for 108.30 dol-|| |lars, for which he paid 84 dollars 39.12 cents; what did.] |he gain per cwt. by the sale? Ans. $2.27 cts. 7m-H || | 5 How many pieces of cloth, at 20.8 dollars perſ| |piece, are equal in value to 240 pieces, at 12.6 dollars || |per piece? % & 3. ... 3 Ans. 145.38–H pieces.| | , 6. If, when the price of wheat is 74.6 cents per bush-l |el, the penny roll weighs 5.2 oz., what should it be per || |bushel when the penny roll weighs 3.5 oz.? | | Ans. $1 10cts. 8m.—H || How do wºnº 2.--> 1. ow do you perform operations in the rule of three |in decimals? > - bers, placing the points agreeably to former directions. || | 1 If 3 men receive 8.9% for 19.5 days labor, how || much must 20 men have for 100.25 days? || | men ) ºn 3 : 20 . . § 3. * * *ś # r iſ º days | 19.5: 100.25 days | .: 894. : 305E. 0s. 8.2d. | | 2 If 2 persons receive 4.625s, for 1 day's labor, how || || much should.4 persons have for 10.5 days? | | 3 If the interest of 76.5% for 9.5 months, be 15.24f. || what sum will gain 636 in 12.75 months? … A f5 men reap 52.2 acres in 6 days? Ans. 20 men. || , 5 If a cellar 22.5 feet long, 17.3 feet wide, and 10.25 || hours a day, how many days of 8.2 hours, should 9 men || ake to dig another, measuring 45 feet long, 34.6 wide, | and 12.3 deep? 4 ins. 12 days. || MENSURATION. sur-ATION is employed in measuring masons' and || ing timber in all its forms, and for estimating || nº length, superfices, and solids, when MENSURATION. 157| ADDITION. RULE. Proceed as in Compound Addition. EXAMPLES. | pf Ft. I. tº in ºr | | F. I. " F. º 17 9 2 3 11 || 25 6 3. 72 14 2 9 54 35 11 10 14 45 10 11 26 6 0 0 19 4 9 0 14 18 11 10 8 9 || 22 11 5 4 9 | 14 10 11 10 8 12 0 0 4 10|| 10 2 8 4 0. i i 132 4 9 || 4 Four floors in a certain building contain each 1084 || | feet, 9in. 8"; how many feet are there in all? . | -- - - Ans. 4339ft. 2 in. 8". I || 5 There are six mahogany boards, the first measures || 27 ft. 3in., the second 25 ft. 11 in., the third, 23ft. 10in.., || the fourth 20ft. 9in., the fifth 20ft. 6in., and the sixth 18|| feet 5 in.; how many feet do they contain? ~ : | . . . . . Ans. 136ft. Sin.| SUBTRACTION. RULE. Proceed as in Compound Subtraction. % EXAMPLES. 3. | || F. I. " Fº. I. tr Ft. I tº m 'm || ||75 9 9 84 6 4 100 10 8 10 11 || || 14 6 11 72 9 8 97 2 4 6 8|| |612 to || 4 If 19ſ. 10n be cut from a board which contains ||41ſt. 7in. how much will be left? Ans. 21ſt. 9in. || 5 Bought a raft of boards containing 59621ſt. Sin, of |which are since sold 3 parcels, each 14905ft, 5in; ho |many feet remain?” Ans. 14905ft. 5in. | RULE. | Set the feet of the multiplier under the lowest denom-l ination of the multiplicand, as in the following example;| | then multiply as in Compound Multiplication, by each || | denomination of the multiplier separately, observing to || ace the right hand figure, or number, of each product, nder that denomination of the multiplier by which it is | roduced. - 1 Multiply 10 feet 6 inches by 4 feet 6 inches. || Product 47 feet, 3 in. || 10 6. 4 . | A table 10 feet 6 inches long,| 10 feet || meas- 6 and 1 foot wide, will make 42 0 . And 4 feet 6 inches, or 4 feet || — wide, will make 43 times 10%, or || 47 3 0. 474 feet, or 47 feet 3inches. |o %3% 10ft. 6in. or 104 feet long. 4 times 104 make 42ft. | and Atime 10% make 54 ft. || Added, make 473. i. Pf by y . 9 by 6 by F. I. " …, Res. 33 6 6 36 10 7 | 8 62 6 7 9 | 4 92 2 10 6 || # i Multiply > I . 2 8 0 | 3 4 5 : : 10 CASE 2. When the feet of the multiplier exceed 12. RULE. || |__Multiply by the feet of the multiplier as in Compound| |Multiplication, and take parts for the inches, &c. | ExAMPLEs. 1 Multiply 112ſt. 3in. 5" by 42ſt.4in. 6." Fº. I. ºr 113 3 5 6x7=42 673 8 6 . : 7 4715. IT 6 m ºrm 4 S 1 8 9 8. 4758 0 9 4. by 184 8. 23545 0 0 || |4 6 by 81 1 8 5777 9 2 2 * . 8 6 6 I. 7 6 2 . Application. . . . . . . . . . . . 1 A certain board is 28ft. 10in.6" long, and 3ſt. 2 in. " wide; how many square feet does it contain? || Ans. 92ft. 2in. 10' 6". || 2 If a board be 23ft. 3in. long, and 3ft. 6in. wide, | how many square feet does it contain? || . Ans. 81 ft. 4in. 6”. || 3. A certain partition is 82ft. 6in. by 13ft. 3im.; how || many square feet does it contain? Ans. 1093ft. lin.6". || 4 If a floor be 79ft. Sin. by 38ſt. 11 in., how many | | square feet are therein? Ans. 3100ft. 4in. 4”. || | NoTE-Divide the square feet by 9, and the quotient| | will be square yards. ºf | "5 If a ceiling be 59ſt. 9in, long, and 24ft. 6in. broad, | how many square yards does it contain? | 55 6 in. I 29 1 9 8 0 0 6. * …, * 3: . 6 How many yards are containe in a pavement 56 º º ; | º feet 9 inches long, and 18 feet 4 inches wide? || ... ...a...º. ººl 7. How many yards in a ceiling 92ft. 4in, lº 8 How many squares in a floor 37ft. 6in. long, an }lft. 9in. wide? 8 squares, 15 feet.-H. A square is 10 feet long an et wide, or 100 ſuare feet. It is used in estimating flooring, roofing, x ding, &c. . ſº MENSURATION. , 161|| * | || 10 How many squares in a roof 36ft. 4in. long, 15ft. || ll 9in. wide? Ans, 5 sq.72ſt.--| | NoTE 2-To measure a triangle. Multiply the base || | by one half the perpendicular height, and the product || | will be its superficial content. . 11 Let C, H, and G, represent a triangle, whose || | base is 40 feet, and perpendicular height 28 feet; how | many feet does it contain? Ans. 560 feet. || G - # G 5 #. 33 | *..] base C H | C 40 feet H 40 feet | feet. | 40 || 14 half the perpendicular | 160 º 40 560 - § - 12. How many square feet in a triangle 80 feet long || |and 36 feet high? Ans. 1440 ft.) 13 In a triangular pavement 46 feet long, and 24 feet | at the place of its greatest width, how many yards; and || |how many bricks, allowing 41 to every square yard? | - … " - Ans. 61yd. 3ft., and 2514 bricks. | | 14. In the gable ends of a house, which is 63 feet long | |and 22 feet high, from the “square of the building” to || || the top, how many squares? - Ans. 6sq. 93in.] |.. NotE 8–To find the circumference of a circle, when | || the diameter is given: Say, - || | As 7 are to 22, so is the diameter to the circumfer-l |ence; or the contrary, - w | | As 22 are to 7, so is the circumference to the % . |eter. H diam-l MENSURATION. The diameter of a circle is 14 feet; what is the circumference? Ans. 44. As 7: 22 : : 14 : 44 . | The circumference of a circle |is 44ft.; what is the diameter? Ans. 14. cººge diameter % | | NoTE 4.—To find the superficial contents of a circle.] Multiply half the circumference by half the diameter 15 How many square feet in a circle whose diameter || is 14 feet, and circumference 44? - Ans 154 ft. || - half circumference, 22 | half diameter, ; : . . 7 g . . . . . . . 154 feetli 16 How many square feet in a circle whose circum ference is 16 feet? Ans. 20 sq. ft As 22 : 7 : : 16 : 5 half diam.2% 17 How many square feet in a circle whose diamete is 21 feet? . . Ans. 3464 ft NoTE 5.—To find the superficial contents of a globe ſultiply the circumference by the diameter. o cover a globe, whose diameter is MENSURATION 163|| 21 What are the solid contents of a cube whose di- || | lameter is 4 feet? . Ans. 64 feet. š. 4 feet º 4. 16 4 º 64 - | 22 What is the solid contents of a stick of timber 2 || | feet thick, 3 feet wide, and 36 feet long? | | Ans. 216 solid feet. 36 feet º 3 108 2 *ś 216 | 23 How many solid feet in a block of marble 3 feet || |thick, 7 feet wide, and 13 feet long? Ans. 273 Sol. ft. || | 24 In a cube whose diameter is 7 feet, how many solid || | feet? Ans. 343 feet. || 25 How many solid feet in a pile of wood 28 feet || | long, 8 feet wide, and 10 feet high; and how many || cords does it contain? Ans. 2240 feet; 17 cords 64 ft. i. ~ . or, 17% cords. || | 26 In a cellar 36 feet long, 27 feet wide, and 44 feet || | deep, how many solid yards? Ans. 162 yards, i. | 27 How many perches” of stone in a wall 42 feet || | long, 84 feet high, and 2 feet thick? Ans. 28,8 per.| º feet § | . 24,75)714.00(28,8 || | 28 In a 12 inch brick wall, 52 feet long and 36 feel || | high, how many bricks, allowing 21 to every square || | foot of wall? . Ans. 39312. || | * A perch is 16 feet long, 13 ſ. wide, and 1 foot high, or 243 || | solid feet. * . . . . . . % ºl MENSURATION. |that product by the length. half circumference 22 half diameter 7 * 154 16 924 154 * e same. : … "... . . . . . º > -º *º ºx....w º à | 32 What are the solid contents of a cylinder whose | |diameter is 14 feet, and length 16 feet? Ans. 2464ft. 29 In an 8 inch brick wall, 82 feet long and 16 feet || high; how many bricks, allowing 14 bricks for every || |square foot of wall? & Ans. 18368. || 30 In a 16 inch brick wall, 148 feet long and 42 feet || |high, how many bricks, allowing 28 bricks to the square || | foot? Ans. 174048. || 31 How many bricks in 3 walls, the first 68 feet long, |18 feet 6 inches high, 16 inches thick; the second 72 ft. |6in. long, 19ſt. 4in. high, 12in. thick; the third 43ft. 4 in. |long, 12ft. Sin. high, 8in. thick? Ans. 72343.-- || NotE 7–To find the solid contents of a cylinder.”—|| |Find the contents of one end by Note 4, and multiply º 2464 feet || 33 What are the contents of a circular well, 7 feet in |diameter, and 62 feet deep? Ans. 2387 ft. || || 34 What are the solid contents of a tub whose diam- || |eter is 6 feet and height 7 feet? Ans. 198 ft. || 35 How many yards in a circular well, 10 feet diam- || er and 20 feet deep? Ans. 1571; ft. || NoTE 8.—To find the contents of a cistern whose top || and bottom are of different diameters. To three times || the product of the two diameters, in inches, add the square of their difference; multiply the sum by one || third of the depth, and divide the product by 359,05 for || tle; 294,12 for wine measure, and 2788 for malt bush- || els, and the quotient will be the content accordingly. . Acylinder is a long round body, whose diameter is every where || MENSURATION. 165 || EXAMPLES. | || 1 Suppose the greater diameter 80 inches, and || | the less diameter 71 inches, and the depth 34 inches, | what is the content in ale—and also in wine gallons? Ans. $540.42 A. G. “*” )659.72 W. G. | 2 The greater diameter of a vessel is 38 inches, the less || | 20.2 inches, and the depth 21 inches; what is the con- || | tent in ale gallons? Ans. 51.07 gal. || || 3 The top diameter of a vessel is 22in., bottom 40in.., || |and the depth 60in.; what is the content in ale gallons, | and in wine gallons; also in malt bushels. . | 165.1 A. G. 201.55 W. G. 21.26 Malt bu. || 4 How many barrels of 32 gallons each, are contained | in a cistern whose bottom diameter is 8ft. 6in., top Sft., | | and the depth 7ft. 9in. Ans. 96 bbls. 28 gal. :- Ans. | Greater diam. 102 inches 102 | Less diam. 96 96 6 612 6 918 |Sqr. of dif. 36. 9792 29376 gal. 36 sqr. of dif. 32)3.100(96 bbls. #& . 288 ~ 29412 º §§§§ 31 one third of depth. 192 29412 — . SS236 28 gallons. 3. 294,12)011772(3100+gallons. || 66 MENsurATION. | NoTE 9–To find the solid ccntents of a round stick|| |of timber of a taper from one end to the other. Find || || the circumference a little nearer the larger than the li | smaller end; from this, by Note 3, find the diameter: || | multiply half the diameter by half the circumference, i. |and the product by the length.* º - ExAMPLEs. | || 1 What are the solid contents of a round stick of tim- |ber 10 feet long, and 2.61 feet circumference? º | - - Ans, 5.4 feet.-Hi As 22 : 7 : : 2,61 : .83 diameter || 1,305 half circumference .415 half diameter 6525 # 10 length || 5.415750 | | 2 How many solid feet in a log 40 feet long, which girts| |66 inches? Ans. 96.25 ft. || As 22 : 7 :: 66 : 21 in diameter || 33×104×40=13860. 144)13860(96.25|| NotE 10.—To find the solid contents of a globe—i. Multiply the cube of the diameter by .5236. | ExAMPLEs. | 1 What are the solid contents of a globe whose diame- |ter is 14 inches? Ans. 1436.75in.-H. I. | 14×14x14=2744, 2744×.5236=1436,7584. || 2. What are the contents of a balloon of a globular || form, 42 feet in diameter? Ans. 38792.4 ft.--| 3 How many solid miles are contained in the earth, or || globe, which we inhabit? * This method, though not quite accurate, is sufficiently nearthell truth ſor the purpose of measuring timber. INVOLUTION. 167 | | Suppose the diameter to be 7954 miles; then, 7954× | 7954×7954–503218686664 the cube of the earth’s | | axis, or diameter; then, … . 503218686664×.5236=263485304337 cubick miles. Ans. | NoTE.—The solidity of a globe may be found by the | circumference, thus—Multiply the cube of the circum- |ference by .016887—the product will be the contents. INVOLUTION, OR THE RAISING OF POWERS. | The product arising from any number multiplied by || | º any number of times, is called its power, as fol- | | HOWS : w †. 2×2= 4 the square, or 2d power of 2. 2×2×2= 8 3d power or cube of 2. 2×2×2×2=16 4th power of 2. # The number which denotes a power is called its index. || | NoTE.—When any power of a vulgar fraction is re-|| | quired, first raise the numerator to the required power, |and then the denominator to the required power, and | | place the numerator over the denominator as before: || thus, the 4th power of # §–4 Questions. | | What is the product, arising from the multiplication of |any figure by itself a given number of times, called? | | What is the number which denotes a power, called? || | How do you proceed to find any required power of a || | vulgar fraction? - º *~…~~~ºrºrº | [ . 7Tºle of The first nine Powers.T INvoluTroN. | ! ! ! #! I 1| ! 128 256 512| 3. º I 2 3 49|343 64|512 81\729 i 2 What is the cube of 14? || 3125] 15625 6561 2187 16384 78.125 279936 823543 7776|| 46656 16807||117649 32768|262144|2097.152|16777216 590.49531441|4782969 |43046721 EXAMPLES. 1 What is the square of 32? 32 96 . 㺠6561 65536 39.0625 1679616 5764801 1024 Ans. . i4 T. 4. * 3874.20489 19683 262144 1953125 10077696 40353607 134217728 ~~~ º- THE SQUARE ROOT. 169 | EVOLUTION, OR THE EXTRACTING OF | - ** 8 ROOTS. º The root of a number, or power, is such a number, as || being multiplied into itself a certain number of times, will produce that power, Thus 2 is the square root of | 4, because 2X2=4; and 4 is the cube root of 64, be- || cause 4×4×4=64, and so on. THE SQUARE ROOT. The square of a number is the product arising from that number multiplied into itself. | Extraction of the square root is the finding of such a || | number, as being multiplied by itself, will produce the | number proposed. Or, it is finding the length of one || side of a square. º RULE. 1 Separate the given number into periods of two fig- ures, each, beginning at the units place. - 2 Find the greatest square contained in the left hand period, and set its root on the right of the given number: subtract said square from the left hand period, and to the remainder bring down the next period for a dividual. ' 3 Double the root for a divisor, and try how often this divisor (with the figure used in the trial thereto annexed) | is contained in the dividual: set the number of times in the root; then, multiply and subtract as in division, and bring down the next period to the remainder for a new dividual. | 4 Double the ascertained root for a new divisor, and || proceed as before, till all the periods are brought down. | NotE.-If, when all the periods are brought down, there be a re-] mainder, annex cyphers to the given number, for decimals, and pro- ceed till the root is obtained with a sufficient degree of exactness. || Observe that the decimal periods are to be pointed off from the de- || cimal point toward the right hand: and that there must be as many || whole number figures in the root, as there are periods of whole num- || | bers, and as many decimal figures as there are periods of decimals. sº §: % º º ºº:: THE square Root. | Square the root, adding in the remainder, (if any,) || |and the result will equal the given number. in | ExAMPLEs. . | 1. What is the square root of 5499025? . º 5,49,90,25(2345 Ans. 4 . 2345 Žºš . 2345 43)149 . #&º 129 ~ 11725 — . 3. 9380 464)2090 T035 1856, 4690 23.425 : ... . . .'; ...” | 2 What is the square root of 106929? Ans. 327. 3 What is the square root of 451584? Ans. 672. 4 What is the square root of 36372961? Ans. 6031. | 5 What is the square root of 7596796? | . . . . . . . Ans. 2756.228–H || 6. What is the square root of 3271.4007? | . . . . . . . Ans. 57.19-H || 7 What is the square root of 4.372594? What is the square root of 10.4976? Ans. 3.24. || 9 What is the square root of .00032754? || Ans. 01809-1-| Ans. 3.1622–H || 90025 Proof 10 what is the square root of lot To extract the Squa Reduce the fraction to its lowest terms, then extract || square root of the numerator for a new numerator, he square root of the denominator for a new deno-|| TE–If the fraction be a surd, that is, one w ot can never be exactly found, reduce it to a dec ict the root therefrom. 3. ...: and extr THE SQUARE ROOT. 171 | EXAMPLES. : 1 What is the square root of 4}}}} Ans. #. 2 What is the square root of #### Ans. #. 3 What is the square root of ###! Ans. .93809–H To catract the Square Root of a Mired Number. RULE. Reduce the mixed number to an improper fraction, |and proceed as in the foregoing examples: or, | Reduce the fractional part to a decimal, annex it to || the whole number, and extract the square root there- il from. . | EXAMPLES. § 1 What is the square root of 37}}} Ans. 64. | 2 What is the square root of 27#7 Ans. 5}, i 3 What is the square root of 85.4% Ans. 9.27+|| APPLICATION. | 1 The square of a certain number is 105625: what| is that number? Ans. 325. 2 A certain square pavement contains 20736 square || stones, all of the same size; what number is contained in one of its sides? . Ans. 144. || … 3 If 484 trees be planted at an equal distance from || | each other, so as to form a square orchard, how many || | will be in a row each way? Ans. 22. || 4 A certain number of men gave 30s. 1d. for a chari- | table purpose; each man gave as many pence as there || | were men: how many men were there? Ans. 19. | 5 The wall of a certain fortress is 17 feet high, | which is surrounded by a ditch 20 feet in breadth; how || | long must a ladder be to reach from the outside of the | ditch to the top of the wall? Ans. 26.24+feet. | NoTE.—The square of the : º | longest side of a right angled | triangle is equal to the sum of | the squares of the other two | sides; and consequently, the | difference of the square of the # w | longest, and either of the other, Ditch. | is the square of the remaining one. § The square Root. | 6 A certain castle which is 45 yards high, is surroun-ji | ded by a ditch 60 yards broad; what length must a ladder || | be to reach from the outside of the ditch to the top of the | castle? *. Ans. 75 yards. || | 7 A line 27 yards long, will exactly reach from the | top of a fort to the opposite bank of a river, which is | | known to be 23 yards broad; what is the height of the | fort? x. . . . . . . Ans. 14.142+-yards. | 8 Suppose a ladder 40 feet long be so planted as to || | reach a window 33 feet from the ground, on one side of || || the street, and without moving it at the foot, will reach a | | window on the other side 21 feet high; what is the breadth| | 9 Two ships depart from the same port; one of them || |sails due west 50 leagues, the other due south 84 leagues;| |how far are they asunder?? .# ls. 97.75+ Or, 973+leagues.| š “ . . . gº. . . . . . . ; square? A square is a surface whose || length and breadth are equal, and whose angles (or cor-|| ners) are right angles, (or square.) What is its square root? The square root is the ngth of the side of a square. . . º If the square be sixteen, what is the root? :- º is the root four? What is the square root of twenty-five? he root be thrº e, what is the square? What is the square of five? What is the square root of thirty-six?" How do you point off a number whose square root is | o be extracted? ... . . ; period? What do you annex to the remainder? | º tººk THE squarE Root. 173| Illustration of the Rule for extracting the Square Root. The reason for pointing off the given number into periods of two figures each, is, that the product of any | whole number contains just as many figures as are in || both the multiplier and the multiplicand, or but one less; consequently, the square contains just double as many figures as the root, or one less. . +: T F A E B 120 9 G H 120 1600 D I 18,49(43 16 cº 249 |GHID = 1600 |AEHG = 120 | HFCI = 120 |EBFH = 9 |ABCD = 1849. Suppose the figure ABCD contains 1849 square feet, and that the number consists || of two periods; then there || must be two figures in the r00t. # The largest root whose || square can be taken out of the left hand period, is 4, (or as it will stand in ten’s place in the root, it is 40,) and the square of this is 16 (or 1600.) || This taken from the whole | square ABCD, or 1849, leaves 249. | Now double GH or HI, which is 40, for a divisor, omitting the cypher to leave || place for the next quotient figure, to complete the divi- | SOI. & º 80 into 249 are contained 3 times; this 3 is the width || of the oblong ALHG, or || HFCI. But the square is || imperfect without EBFH; | then annex the three to the divisor. Now multiply this perfect divisor by the last; figure of the root, to get the || quantity in the two oblong | figures, and the small square which comprises the great || square ABCD. Hº * The cube Root. How do you find the divisor? Why do you place the new quotient figure in the units| ace of the divisor? How do you prove the square root? | - w THE CUBE ROOT. | The cube of a number is the product of that number| |multiplied into its square? - | Ertraction of the cube root is finding such a number| |as, being multiplied into its square, will produce the number whose cube root is extracted. " . . ; ź:: . RULE. - - | Separate the given number into periods of three fig-l |ures each, beginning at the units place. Find the great- |est cube in the left hand period, and set its root in the |quotient; subtract said cube from the period, and to the |remainder bring down the next period for a dividual. | Square the root, and multiply the square by three || indred for a divisor. g | See how often the divisor is contained in the dividual, || |and place the result in the quotient. || Multiply the divisor by the last found quotient figure;| square the last found figure—multiply the square by the preceding figure or figures of the quotient, and this pro-|| duct by thirty; and cube the last figure. Add these || three products together, and subtract their amount from | To the remainder add the next period, and proceed as || before, until the periods are all brought down. | When a remainder occurs, annex periods of cyphers || obtain decimals, which may be carried to any conve-|| he cube root of a vulgar fraction is found É to its lowest terms, and extracting the root || and of the denomina- by r 3ducing it he numerator for a numerator, ***** THE CUBE ROUT. 175 tor for a denominator. If it be a surd,” extract the root of its equivalent decimal. EXAMPLES. 1 What is the cube root of 99.252847? 99,252,847(463 Ans. 463. 4×4×4=64 - 4×4×300=4800 35252 463 * 463 Div. 4800×6= | 28800 — | 6×6X4×30= 4320 1389 6×6×6= 216 2778 1852 Subtrahend 33336 - - 214369 | 46X46X300=634800 1916847 463 Div. 634800×3=1904400 643107 - 3×3×46X30= 12420 1286214 | 3×3×3= 27 857476 Subtrahend 1916847 Proof 99.252847 2 What is the cube root of 846045.19% Ans. 439. || || 3 What is the cube root of 259694072? Ans. 638. || 4 What is the cube root of 32461759? Ans. 319. || 5 What is the cube root of 5735339? Ans. 179. || 6 What is the cube root of 48228544? Ans. 364. || 7 What is the cube root of 673373097.125? Ans. S765. | 8 What is the cube root of 7532641? Ans. 193.02+|| | 9 What is the cube root of 5382674. Ans. 1752–H || | 10, What is the cube root of 15926.972504? * * * * Ans. 25.16+|| When decimals occur, point the periods both ways, beginning at the decimal point, and if the last period of the decimal be not complete, add one or more cyphers. | A mixed number may be reduced to an improper || |fraction, or a decimal, and the root thereoſ extracted. |} * A surd is a quantity whose root cannot exactly be formed. A || |quantity whose root can be found, is called a rational quantity. º ||176 THE cube Root. 1 What is the cube root of ºf 2 What is the cube root of # 3 What is the cube root of # 4 What is the cube root of 12 5. What is the cube root of 31*; 3 13ſ § &º º 9 * § 7 º | i SURDS. … º What is the cube root of 74% Ans. 1.93--|| What is the cube root of 947 Ans. 2.092–H || --- APPLICATION. The cube of a certain number is 103823; what is || || that number? Ans. 47 || || 2 The cube of a certain number is 1728; what num- | ber is it? . -- Ans. 12. || || 4 There is a cistern or vat of a cubical form, which || || contains 1331 cubical feet: what are the length, breadth| |and depth of it? . Ans, each 11 feet. || || 4 A certain stone of a cubical form contains 474552 |solid inches; what is the superficial content of one of its || | |sides? - Ans. 6084 inches. || º I Questions. || What is a cube? A cube is a solid body contained by || || What is the cube root? It is the length bf one side | of a cube. | What is the square of the cube root? It is the su- || rficial contents of one side of a cube. | How do you point off a number whose cube root is to º What is the first figure of the root? It is the root of || greatest cube in the first period. || When you subtract the cube from the first peri what do you do? a - How do you find the divisor? º hat is the first step towards finding the subtrahend?|| º second? What is the third? ou proceed? hen a remainder occurs, how do you How do you prove the cube root? º º - º 3. * º šº & THE CUBE ROOT. 177 | | Illustration of the Rule for extracting the Cube Root. | The reason for pointing off the number into periods || |of three figures each, is similar to the one given in the . |Square Root; for the number of figures in any cube will || |never exceed three times the figures in the root, and || ||will never be more than two figures less. º OPERATION. 15,625 25 In º: º: 2× 2 × 2– 8 two periods: of course there X2X & will be two figures in the - … r00t. . 2×2×300=1200 7625 “The greatest cube in the left hand period (15) is 8, 6000 the root of which is 2;” || tºº r: therefore, 2 is the first figure || 5×5×3×30= 1300 of the root, and as we shall || 5X 5X5= 125 have another figure in the . root, the 2 stands for 2 tens, 7625 or 20. But the cube root is the length of one of the sides of the cube, whose length, breadth and thickness are l equal: then the cube whose || root is 20, contains 20X20 || - X20=8000. ~ “Subtract the cube thus || found (8) from sand period, and to the remainder bring || down the next period,” or, subtract the 8000 from the whole given number (15625) and 7.625 will remain. Thus || 8000 feet are disposed of in the cube, Fig. 1. 20ft long, | 80 ft wide, and 20 ft. nigh. . The cube is to be enlarged by the addition of 7.625 feet i which remain. In doing | || this, the figure must be enlarged on three sides, to make it longer, |and wider, and higher, to maintain the complete cubic form. | The next step is, to find a divisor; and this must be the number of |square feet contained in the three sides to which the addition must be t! |imade. . . - . 2 º' }: | Hence we “multiply the square of the quotient figure by 300.” | |That is, 2X2X300=1200: or 20X20 × 3–1200 feet, which is teſ | superficial content of the three sides, A, B, and C. . ſubrº ROOT. This “divisor (1200) contained in the dividual” || (7.625) 5 times: then 5 is the second quotient figure; that || is, the addition to each of the three sides is 5 feet thick; if| 1200 feet cover the three sides one foot thick, 5 feet || thick will require 5 times as | many; that is 1200×5–|| 6000. | But when the additions || are made to the three squares || there will be a deficiency along the whole length of || the sides of the squares be-|| tween the additions, which || must be supplied before the cube will be complete. These || deficiencies will be three, as || I may be seen at NNN in Fig. 2; therefore it is that || we “multiply the square of the last figure by the prece- ding figure, and by 30," | (that is, 5×5×2X30,) or || 5X5×20×3=1500which || is the quantity required to || supply the three deficiencies. || Figure 3, represents the solid with these deficiencies || supplied, and discovers an- other deficiency, where they | approach each other at ooo. Lastly, “cube the last fig-H wre;” this is done to fill the deficiency left at the corner, in filling up the other defi-|| ciencies. This corner is limited by the three portions|| applied to fill the former va-i. cancies, which 5 ... º. breadth; consequent cube of 5 will be the so contents of the corner. F ºxºxº- | ROOTS OF ALL POWERS. 179 The illustration is much better made by means of 8 blocks of the following description: One cube of about 3 inches diameter; three pieces each 3 inches square, * inch thick; three pieces each # inch square, 3 inches long; and one cube inch. A set of these should || belong to the apparatus of every Professional Teacher. |A GENERAL RULE FOR EXTRACTING THE --- ROOTS OF AI, L POWERS. 1 Point the given number into periods, agreeably to | the required root. - - 2 Find the first figure of the root by the table of pow-i ers, or by trial; subtract its power from the left hand || period, and to the remainder bring down the first figure in the next period for a dividend. 3 Involve the root to the next inferior power to that which is given, and multiply it by the number denoting|| the given power, for a divisor; by which find a secondſ figure of the root. || | 4 Involve the whole ascertained root to the given || power, and subtract it from the first and second periods. || Bring down the first figure of the next period to the re- mainder, for a new dividend; to which, find a new divi- sor, as before; and so proceed. . Note—The roots of the 4th, 6th, 8th, 9th, and 12th powers, may be obtained more readily thus: º For the 4th root take the square root of the square'. root. For the 6th, take the square root of the cube root. - For the 8th, take the square root of the 4th root. | * For the 9th, take the cube root of the cube root. | For the 12th, take the cube root of the 4th root. - EXAMPLES. 1 What is the 5th root of 916132832? 9161,32832(62 Ans. . 7776 6×6×6×6×6=7776 º 6×6×6×6×5=6480 div.[ 6480)13853 - | 916132832 62×62×62×62×62=916132832|| 916132832 - . . . . . . . . . . " ... * * º ARITHMETICAL Progression. 2 What is the fourth root of 140283207936? Ans. 612.É. 3 What is the sixth root of 7827577.89696 Ans. 96. 4 What is the seventh root of 1947,54273881? Ans. 41. || 5. What is the ninth root of 13526054605946882 || i. , ; ; ; ; ; ; ; ; ; , , º Ans. 48. - . . . . * #. ARITHMETICAL PROGRESSION. | Any series of numbers increasing by the constant addi-|| |tion, or decreasing by the constant subtraction of the same || |number is called an Arithmetical progression. When || |the series is formed by the constant addition of the same| |number, it is called an ascending series, but when the |series is formed by the constant subtraction of the same|| number, it is called a descending series, thus: "| | 1, 3, 5, 7, 9, 11, &c. is an ascending series. || ||20, 17, 14, 11, 8, 5, &c. is a descending series. || | The number which is added or subtracted continually || |is called the common difference; in the ascending series|| |above, this is 2, and in the descending series it is 3. |The numbers which form the series are called the terms || |of the series or progression. The first and last terms|| |are called the extremes, and the other terms are called |the means. - | || | In any arithmetical progression there are five things|| "|to be considered, any three of which being given the ||other two may be found, viz. || || 1. The first term. 2. The last term. 3. The common difference. 4. The number of terms. 5. The sum of all the terms. CASE 1. number of terms|| erence. º ARITHMETICAL PROGREssion. 181 | vide it by the number of terms less I, the quotient will || |be the common difference. | 1. A man had five sons whose several ages formed an || |arithmetical progression, the youngest was 13 years old, and the eldest 25 years; what was the common differ-|| ence of their ages : - | 25 – 13 = 12 and 5 –1 = 4 and 12 + 4 = 3 Ans. || The reason of the rule will be evident by considering || this example. The 12 years is the number that it is | necessary to add to the youngest son's age to produce || the age of the eldest ; now in order to obtain the age of the eldest by adding the common difference to the age | of the youngest, it would be necessary to add the com- mon difference once less than the number of sons, hence || | we divide the sum of the additions by their numbers, | that is the difference of the extremes, by the number of | terms less one, in order to get the common difference. 2. If the extremes be 5 and 50 and the number of terms || 10 what is the common difference 2 Ans. 5. || 3. A man is to travel from Cincinnati to a certain place | in 12 days, and to travel only 12 miles the first day, increasing the distance traveled each day by an equal quantity, so that the last day's journey may be 78 miles; required the daily increase ? Ans. 6 miles. | 4. If the amount of $4 for 10 years at simple interest || be $6,40 what is the annual interest ? Ans. 24 cts. || CASE 2. The common difference, the first term, and the num- |ber of terms being given, to find the last term. | | Rule.—Multiply the common difference by the num-l |ber of terms less 1, and to the product add the first term, the sum will be the last term. | The reason of this rule is obvious, it being just the re-|| verse of the last. . - | | 1. If the first term be 5, the common difference 2 and || and the number of terms 40, what is the last term 2 . Ans. 83. 40 – 1 = 39 × 2 = 78 + 5 = 83. # zºº º Rithmetical progression. lan bought 50 yards of calico, giving 6 cents for the first yard, 9 for the second, 12 for the third, and so |on; what did the last cost Ans. $1,53. || 3. Required the fortieth term of an increasing arith- metical progression, whose first term is 10 and common difference 7. Ans. 283. 4. If a man put out $400 at simple interest at 6 per cent, there will be due at the end of the first year $424, at the end of the second year $448, what would be due |at the end of 10 years. . . . . . Ans. $640.] CASE 3. º The first and last terms and the common difference being given to find the number of terms. RULE.—Divide the difference of the extremes by the |common difference, the quotient increased by 1, will be |the number of terms. | The reason of this rule is plain by considering the |preceding case. | 1. If the extremes be 3 and 51 and the common diſ. ce 6, what is the number of terms? Ans. 9. | 2. If the extremes be 4 and 1000 and the common dif- |ference 12, what is the number of terms ? Ans. 84. § 3. ...}} * Case 4, - -- .* |find the sum of all the terms. | RULE.—Multiply half the sum of the two extremes || |by the number of terms, the product will be the sum of 1. A man bought ten yards of calico at 2 cents for the |first yard, 6 cents for the second, 10 cents for the third, and so on, what was the cost of the whole 2 s: | In this example by case 2 we find the last yard co |38 cents, we then add the cost of the first yard and th - e last together, and take half of their sum, whi es 20 cents for the average price of each yard, an is multiplied by 10, the number of yards, gives #2,0 f the whole. The solution of this examp º sº GeoMETRICAL PROGREssion. 2. If the first term be 4 and the last 40 and the number| of terms 13, what is the sum of all the terms ? Ans. 286. 3. What is the sum of the first 100 numbers in their | natural order, that is, 1, 2, 3, 4, 5, 6, &c.; Ans. 5050. 4. How many strokes do the clocks of Venice, that go | on to 24 o’clock, strike in a day ! º Ans. 300. | 5. If 100 eggs were laid 2 yards distant from each |other, and a basket placed 2 yards distant from the first, what distance must a person travel to gather them singly into a basket ! Ans. 11 miles, 3 fur. 180 yds. 6. Required the sum of the first ten thousand numbers| that cre divisible by 3. (3, 6, 9, 12, &c.) 3, … . º. Ans. 150015000. GEOMETRICAL PROGRESSION. A series of numbers increasing by a constant multipli-|| |cation, or decreasing by a constant division, is called a Geometrical progression. w Thus, 1, 2, 4, 8, 16, &c. is an increasing geometric | | series. And 162, 54, 18, 6, &c. is a decreasing geo- |metric series. The constant multiplier or divisor, is called the ratio ; in the first of the above series the ratio | is 2, and in the second 3. - # | . There are five things to be considered in any geomet- rical progression, viz. * 1. The first term. # 2. The last term. … . º 3. The number of terms. 4. The ratio. 5. The sum of all the terms . . ; CASE 1. | The first term, the ratio, and the number | being given, to find the last term. £ºžsº *…*.*.*.*.*.*.*. º RULE.—Raise the ratio to a power, whose index is || |one less than the number of terms; then if it be an in- |creasing series, multiply the first term by the power, |otherwise, divide by it, the result will be the last term. || | 1. What is the 8th term of the geometrical progression | |whose first term is 4 and ratio 2. | Here the 7th power of 2 is found to be 128, which || | being multiplied by the first term 4, gives 512 for the h term. is . . . * || The reason of the rule as applied in this example will || terms up to the 8th, the first term must be multiplied by || |2, that product by 2, that product again by 2, and so on || | till the 8th term would be found by 7 such multiplica-) |tions, and it is plain that the same result will be found| |by first using 2 as a factor 7 times, and then multiply- |ing by 4 the first term, thus: | | 2×2×2×2×2×2×2=128×4=512 the 8th term, | or 4×2×2×2X2×2×2×2=512 the 8th term. | | 2. Required the tenth term of an increasing series|| |whose first term is 15 and ratio 22 Ans. 7680.| | 3. The first term of an increasing series is 12, and the |ratio 3, what is the 18th term Ans. 1549681956. 4. A boy purchased 18 oranges, agreeing to pay only| price of the last, at 1 cent for the first, 4 cents for | second, 16 cents for the third, and so on, what was || |the price of the last tº Ans. $171798691,84.]] - The extremes and ratio being given, to find the sumſ |of the series. | | RuLE.—Multiply the last term by the ratio, from the product subtract the first term, and divide the remainder| by the ratio less 1, the quotient will be the sum of the The reason of the rule may be shown in the followingſ inner, take any series as, 1, 3, 9, 27, 81, now mult § he ratio 3, and - , 9 º º jº & º it produces the series 3, GEOMETRICAL PROGRESSION. 185 | ||what it will, it is evident that the sum of the second se-|| ||ries will be as many times the first as is expressed by the | ratio ; subtract the first series from the second and it will | give 243–1, which is evidently as many times the sum | of the first series, as is expressed by the ratio less 1, || therefore 243—1 divided by 3–1 will be the sum of the |first series, that is if the last term be multiplied by the ||ratio, and the first term subtracted from it, and the re- | mainder be divided by the ratio less 1, the quotient will | be the sum of the series. . zº. NotE.-When the last term is not given, as in the] |following questions, it must be found by case 1. 1. If the first term be 1, the ratio 3, and the last term | | 177147, what is the sum of the series : 177147 3 3 531441 1 | 2) 531440 265720 Ans. 2. A rich miser thinking 1000 dollars too great a price # for 12 fine horses, offered to give 4 cents for the first, | 16 cents for the second, 64 cents for the third, and so | | on in fourfold proportion to the last; what would the 12|| horses come to at that rate 2 Ans. 223696,20.| || 3. A goldsmith sold 1 pound of gold at 1 cent for the | first ounce, 4 for the second, 16 for the third, &c.; what || | did it come to ? t Ans. $55924,05. || | 4. What sum would purchase a horse with 4 shoes, |and 8 nails in each shoe, at 1 mill for the first nail, 2 || |mills for the second, 4 mills for the third, &c.? º . Ans. $4294967,29 cts. 5 mi. | 5. A merchant sold 30 yards of silk velvet, at 2 pins | |for the first yard, 6 for the second, 18 for the third, &c. | |and sold them at 1000 for a farthing; what did the vel- -T- |vet amount to, and what was gained by the sale, sup-H |posing the first cost to have been 100 £ per yard 2 || | “” & Gained 2144669294 fis. 3}d. || | 6. Suppose a body put in motion should move the |length of one barley corn the first second of time, one || | inch the second, and three inches the third, and so on, || | increasing its motion in triple proportion geometrical;| what distance would it move in the space of 30 seconds, | or half a minute Ans * mi. 3 fur. 29 po. | #. * . . . 34 yd. 1 ft. 1 in. 1 b. c.| POSITION. | Position is a rule by which true, or required numbers|| |are found by means of false, or supposed numbers. º SINGLE POSITION. | Single position is the method of resolving questions|| |by working with one false or supposed number. | | Those questions belong to this rule, in which the re-|| |sult or number found by the operations on the supposed.| |number, is to the proposed number, as the number given || | in the question, is to the true or required number, and || || the following rule is merely an application of the prin-1 iples of proportion. | Rule.—Suppose any number and perform such ope-j |rations on it as you would with the answer to prove it.} |Then say as this result is to the supposed number, so || |is the number in the question to the number required. || teacher being asked how many scholars he] had as many more as I now have, how many scholars had he 㺠&xiºść...&& |Suppose he had 16 Then as 46 : 16 :: 138 proof. | |As many more=16 % 16 48 || || 4 as many = 8 * 48 || || 3 as many = 4 828 24 || | } as many = 2 138 12 || | 46 46)2208(48 Ans. – | 184 138|| | 368 | 368 | 2. A, B, and C, talking of their ages, B said that his | age was once and a half the age of A; and C said that || #| his age was once and one tenth the age of both, and that || || the sum of all their ages was 105 years; what was the age of each Ans. A's 20, B's 30, and C’s 55 yrs. || 3. A person after spending one half, and one third of his money had 12 dollars left, how much had he at || first 7 Ans. $72. || 4. A man being asked his age, answered that if its || |fourth, its sixth, its ninth, and its twelfth were added to || |it, that the sum would be 58 years; what was his age t Ans. 36 yrs. || 5. A farmer being asked how many sheep he had, answered that he had them in four pastures, in the first! he had one third of his flock, in the second one fourth, | in the third one sixth, and in the fourth 49; how many | | | sheep had he Ans. 196. 6. A man being asked how many oxen, how many || cows, and how many sheep he had, answered that he had 3 times as many cows as oxen, and 9 times as many || sheep as cows, and that the whole nnmber was 341 ; } required the number of each sort º Ans. 11 oxen, 33 cows, and 297 sheep. ; DOUBLE POSITION. “. . . . . . . W2. Double position is the method of resolving questions|| by working with two false or supposed numbers. Those || | questions belong to this rule, in which the result found || |by the operations performed on the supposed numbers, is } º * not to the supposed number as the number given in the | | question is to the true number. | RULE.—Suppose any two numbers and perform such || |operations on each as is required by the conditions of || the question, noting the error in each case. || | Then multiply the first error by the last supposition, || |and the last error by the first supposition. || | Then if the errors are alike, that is both greater, or || |both less than the given number, take their difference || |for a divisor, and the difference of their products for a || dividend; but if the errors are unlike, that is one greater || |and the other less than the given number, take their sum || for a divisor, and the sum of their products for a divi-ſ son, the rule cannot be he aid of Algebra. 1. There is a number, which being increased by || |its third, its fourth, and 20 more will be doubled; what || is the number - Ans. 48. $3; & .2%. , , º, º . . . . . . ." .. 3 suppose 12 Suppose 24 3 properly shown | 3. * … # : - • = 6 20 20 *::: 39 58 12 × 2 = 24 24 × 2 = 48 # Error 15 Error 10 24 - 12 3. §: 60 30 * 1st error, 15 360 2d error, 10 EXCHANGE. 189 | 2. A and B commenced trade with equal sums of mo- || |ney, A gained a sum equal to one fifth of his stock, and || |B lost $200, then A's money was double that of B; | what was the stock of each Ans. $500. || 3. A and B talking of their ages, says A to B, four fifths of my age is equal to seven eighths of yours, and the sum of our ages is 67 years; what were their ages 7 | Ans. A’s 35, and B's 32 yrs. 4. A man has two silver cups of unequal weight, hav- ing one cover to both weighing 5 oz. now if the cover| be put on the less cup they will be double the weight of the greater, but if the cover be put on the greater cup, they will be three times the weight of the less; what is | he weight of each 2 Ans. 4 oz. and 3 oz. || 5. There is a fish whose head is 10 feet long, his tail is as long as his head and half his body, and his body is | as long as his head and tail; what is the whole length || of the fish : Ans. 80 feet. || 6. A laborer hired for 40 days, upon the condition that || he should receive 20 cts. for every day he worked, and forfeit 10 cts. for every day he was idle; at settlement || |he received 5 dollars, how many days did he work, and || how many days was he idle 2 . || Ans. He wrought 30 days, and was idle 10.| EXCHANGE. The object of exchange is to find how much of the money of one country is equivalent to a given sum of the money of another. x . || By the par of eachange between two countries is | meant the intrinsic value of the one, compared with the other: it is estimated by the weight and fineness of the coins. sº. " & || | The course of eachange at any time, is the sum of the | money of one country which at that time is given for a || | certain sum of the money of another country. The course : of exchange varies according to the circumstances of trade. |All the calculations in exchange can be performed by the 90 promiscuous exercises. w Ex. 1. If 142 sterling is worth $4,44 cts. 4 ms. ; what || is 65.9 sterling worth Ans. $288,86. | 2. What is the value of $500 in English money, at || |$4,44.4, per £ sterling Ans. 1124° 10s. 24d. || | 3. What is the value of 1250 T.s. at $4,444, per £H |sterling ! Ans. $557,05,5. 4. What is the value of $1000, in English money, at || ||$4,444, per £ sterling Ans. 225& 53d. | | 1. A merchant had 1000 dollars in bank; he drew out| |at one time $237.50, at another time, $116.09, and at lollars, and at another time $750.50; how much || d he in bank after making the last deposite º | 2 Sold 8 bales of linen, 4 of which contained 9 piecesſ |each, and in each piece was 35 yards; the other 4 bales || yards; how many pieces and how many yards were in || | all º Ams. 84 pieces, 2556 yards. || thus—to his wife #, to his elder son 4 of the remainder, i. | and to his other son the rest; what is the share of each ' || | Elder son's $2440.87%. Other son's $1627.25. |cent? Ans. $54.41.1; | 5 If a tower is 384 feet high from the foundation, a | part under water, how much is visible above the water? | §§ Ans. 272 feet. a yard that is 20 feet square ? Ans. 1600. PROMISCUOUS EXERCISES. , $241.06% : after which he deposited at one timeſ | Ans. $2655.84%.] |contained 12 pieces each, and in each piece was 27| 3. If a man leave 6509 dollars to his wife and two sons, Wife's share $2440.874. || Ans. | 4 What is the commission on $2176.50, at 24 per || sixth part of which is under the earth, and an eighth || w many bricks 9 inches long and 4 inches wide, | he value of a slab of marble, the length of fighes, and the breadih iſ ſº ſo; PROMISCUOUS EXERCISES. 191 || *&º | 2 feet 9 inches in breadth, and 3 feet 4 inches in depth; | how many solid feet does it contain? Ans. 41 ft. 3 in. || ||| 9 A line 35 yards long will exactly reach from the top of a fort, standing on the brink of a river, to the op-|| |posite bank, known to be 27 yards from the foot of the | wall; what is the height of the wall? | x Ans. 22 yards 3% inches—H 10 The account of a certain school is as follows, viz. # of the boys learn geometry, 4 learn grammar, ºr || learn arithmetic, ºr learn to write, and 9 learn to read: || what number is there of each? Ans, 3 wholearn geometry, 80 grammar, 24 || arithmetic, 12 writing, and 9 reading. || 11 A merchant, in bartering with a farmer for wood || at $5 per cord, rated his molasses at $25 per hd., which || was worth no more than $20; what price ought the far- || mer to have asked for his wood to be equal to the mer- || chant's bartering price? Ans. $6,25. || 12 A and B dissolve partnership, and equally divide || their gain: A's share, which was $332 50 cts., lay for | 21 months; B's for 9 months only: the adventure of B. is required. Ans. $775 834 cts. || 13. If a water-hogshead holds 110 gals. and the pipe || which fills the hogshead discharges 15 gal, in 3 minutes, and the tap will discharge 20 gal. in 5 minutes, and these || were both left running one hour, how many gallons || | would the hogshead then contain; and if the tap was || then stopped, in what time would the hogshead be filled? || . ~ Ans. 60 gal, and filled in 10 min. || | 14 A has B's note for $50075cts, with 9 months in-|| |terest, at 6 per cent., due on it, for which B gave him || 5064 feet of boards, at 24 cts per foot, with 140 pounds || | of tallow, at 13 cts. per pound, and is to pay the rest in || |flax seed, at 924 cts, per bushel; how many bushels of flax | | seed must A receive, to balance the note? | || 15 A, B, and C, in company, had put in $5762; A's | money was in 5 months, B's 7, and C's 9 months: they || |gained $780, which was so divided, that of A's was , || |of B's, and # of B's was of C's: but B, having received | & | 192 PROMISCUOUS EXERCISES. | | $2087, absconded: what did each gain, and put in; and || | what did A and C gain or lose by B's misconduct? | | A's stock $2494,887 gain 260 Ans.< B’s do $2227,577 do 325 ‘T C's do $1039,536 do 195 A and C would gain $465,577 | 16 When 100 boxes of prunes cost 2 dollars 10 cents | each, and by selling them at 3 dollars 50 cents per cwt. || the gain is 25 percent, the weight of each box, one with | another, is required. Ans. 84 lb. | 17 There are two columns, in the ruins of Persepo- |lis, left standing upright; one is 64 feet above the plain, || the other 50. Between these, in a right line stands an |ancient statue, the head whereof is 97 feet from the sum-|| |mit of the higher, and 86 feet from the top of the lower | column, and the distance between the lower column and || || the centre of the statue's base, is 76 feet; the distance || | between the top of the columns is required. Ans. 157+ft. || | 18 If I see the flash of a cannon, fired from a fort on || the other side of a river, and hear the report 47 seconds || | afterwards, what distance was the fort from where Iſ | stood? . Ans. 53674 feet. || | NotE.-Sound, if not interrupted, will move at the rate of about || ||1142 feet in a second of time. - º | 19 What is the difference between the interest of || |$1000 at 6 per cent. for 8 years, and the discount of the | same sum at the same rate, and for the same time? º Ans. The interest exceeds the discount by | - $155 67 cts. 5 m. | | 20 If a tower be built in the following manner, ºr || | of its height of stone, 27 feet of brick, and 3 of its height| | of wood, what was the height of the tower? || | Ans. 113 feet 4 inches.| 21 A captain, 2 lieutenants, and 30 seamen, take a || prize worth $7002, which they divide into 100 shares, |of which the captain takes 12, the two lieutenants each || |5, and the remainder is to be divided equally among the #º º how much will each man receive? Ans. Cap-H $40,24, each lieutenant's, $350,10, and || A P R A CTIC AI, SYSTEM O F BOOK – R.E.E.PING, FOR |MECHANICS AND RETAILERS. || Book-KEEPING is the method of recording a system- | |atic account of business transactions. - | It is of two kinds—Single and Double Entry. The |former, only, will be noticed in this work. On account || | of the simplicity of Single Entry, it is, perhaps, the best || | which can be recommended to farmers, mechanics, and | retailers. It consists of two principal books—the Day | Book, or Waste Book, and the Leger, and one auxiliary || | book, the Cash-Book. Q-3 THE DAY BOOK. This book is ruled with a column on the left hand || | for the date, and three columns on the right, the first, | for the folio or page of the Leger, to which the account || | is transferred; and the last two for dollars and cents. This book exhibits a minute history of business trans- || | actions in the order of time in which they occur, with | every circumstance, necessary to render the transaction | plain and intelligible. : 17 4 CINCINNAT1, 1833. 4 6 David Judkins, T}r. To 10 lbs, coffee at 17 cts.51 70 “ 25 lbs. sugar, at 10 cts. 250 Timothy W. Coolidge, Dr. To I bl. sugar, weighing 135 lbs. neat, at 84 cts. $11.47% “ 1 bag coffee, 98 lbs. at 15 cents, 14. 70} | Geo. H. Eaton, Dr. To 1 bl. flour, $3.87, “I lb. Y. H. Tea, ! #24 “I keg lard, neat weight 60 lbs., at 64 cts. 375) David Judkins, Cr. 8|By cash on account, James Wilson, Dr. To 3 weeks boarding, at 2 dollar per week, - Hiram Ames, Dr. To i2 yards broadcloth at 6 doi. lars per yard, $7200 | “30 yds. muslin at 14 cts. 420 Cr. By an order on J. Jones, for Gro- ceries, $51 00 | “Cash, - 20 00 ||76 Timothy W. Coolidge, Cr. * I}|25 4 By a bill of carpenter work, | James Wilson, Dr. 5|To 1 bl. vinegar, $3 25 “l keg 10d. nails, weight “ 111 lbs. at 8 cts. 26 71 's ss 2 cts. 20 174 || 75 || 00 00 20 #: 00 13 || ſ CINCINNAT1, 1833. 3 * | $ cts. * George Hamilton, Cr. | Jan. 19|By 1 set of Fancy chairs, 3||25 || 00 % George H. Eaton, Cr. “ 21|By cash, to balance account, } || 8 || 75 Robert Young, Dr. “ 25|To cash on account, $300 “ 10 lbs. N. O. sugar, at “ 10 cents, } 00 “ 12 lbs. coffee at 165 cts, 2 00 “ 4 lb. Y. H. Tea, 624 || 3 || 6′ 62% Jackson Moore, Dr. « 29|To 21 lbs. Ham, at 10 cts. $210 “ 1 box soap, 30 lbs. at 5 : cents, 1 50 3|| 3 60 David Judkins, Dr. « 30|To 12 bls, apples at 75 cents, 1||| 9 00 Cr. . By 47 bush. corn, at 20 cents, 1||| 9 40 º James Wilson, Cr. “ 31|By cash on account, 2|| 15 00 - Thomas Hilton, Dr. #| Feb. 3 To cash, $500 #. “ 1 lb. Y. H. Tea, I 06 “ 16 lbs. Rice, at 64 cts. I 00 2 || 7 || 06 Cr. By 13 days labor, at 874 cts. 2||11| 373 George Hamilton, Dr. “ 10 To 1 canister Imperial Tea, 14 lbs. at $187.4 per lb. 3||26. 25 : Hiram Ames, Cr. “ 13|By order on E. Disney for goods, 2010 00 100 CINCINNAT1, 1833. \ ||$|cts. Robert Young, Dr. Feb. 15|To 1 box sperm candles, “25 lbs., at 30 cts. per lb. $750 “2 bu. dried fruit, at $1 º “25 per bushel, 2 50 3||10 00 James Wilson, Dr. “ 21|To 10 gals. molasses, at 40 cents, $400 “ 4 lbs. Old Hyson Tea, at 93 cents, 3 72| 2 || 7 | 72 Robert Young, Cr. “ 27|By 12 cords of wood at $2.25 Ames & Smith, Dr. March 6 To 18 lbs. sole leather, at | “25 cents, $450 “I side upper leather, 275 “3 calfskins, $125, 375 Hiram Ames, Dr. “ 6|To 500 feet white pine boards, at “$1250 per M. $6.25 “25 bu. potatoes, at 50 cts. 12 50. “I ton of Hay, 10 00 | David Judkins, Dr.] “ 8|To 200 lbs. flour, at $175 Thomas Hilton, Dr. “ 8|To cash paid his order to William | - - George Hamilton, Dr. “ 15|To 1 copy Whelpley’s Com- g pend, $125 | “I ream letter-paper, 450 “I doz. Spelling books, 3|| 6 75 *#& 5 CINCINNAT1, 1833. {{ {& Mar. 46 23 26 28 3 0 31 Theodore P. Letton, Dr. To sharpening his plough, 31,00 A “Shoeing his horse, 1,624 “Repairing chain, 25 Timothy W. Coolidge, Dr. To 2 qrs. tuition of himself at evening school, at $3 per qr. Thomas Hilton, Cr. By the hire of his horse 10 days, at 624 cts. per day, Ames and Smith, Cr. 5|By 1 hlid. sugar, weight 1317 lb. neat, at 74 cts. T. P. Letton, Dr. To I ream wrapping paper, $1,624 “ 1 beaver hat, 5,00 “ 1 set silver tea-spoons, 6,00 Henry C. Sanxay, Cr. By my order on him in favor of Jno. Torrence for stationary, Ames and Smith, Dr. To 2000 ft. clear pine boards, at $20 per M. $40,00 “500 common do. at $8, 4,00 “5000 shingles, at $2.25 11.25 “Cash to balance account, 32,52 Jackson Moore, Cr. By painting my house, 2 98 12 87 21 Ct9. 87 624 874 77 º CINCINNAT1, 1833. 44 {{ 31 31 31 31 To 4 bu. wheat, at $125, $500 “ 1 bl. mess pork, 900 “ 2 bu. salt, at 50 cts. 1 00 Thomas Hilton, Dr. | “8 lbs. brown sugar, 11 cts. 88 George Hamilton, Dr. To 12 cedar posts, at 25c. $300 “ 1 plough, 937; “ 1 scythe, 1.62% Jackson Moore, Dr. To repairing his wagon and plough, Thomas Hilton, Cr. By 1 pair shoes, $150 “I mahogany table, 12 50 “ cash, 650 - George Hamilton, Cr. By an order on J. Hulse, $500 6 END OF THE DAY BOOK. 15 2 14 3||11 00 00 00 50 i º tºº §§§ºº T H E L E G E R . THIs book is used to collect the scattered accounts of the Day Book, and to arrange all that relates to each individual, into one separate statement. The business of transferring the accounts from the Day Book to the Leger, is called posting. The Leger is ruled with a double line in the middle of the page, to sepa- rate the debits from the credits. Each side has two columns for dollars and cents, one for the page of the Day Book, from which the particular item is brought, and a column for the date. . When an account is posted, the page of the Leger on which this account is kept, is written in the column for that purpose in the Day Book, and also the page of the Day Book from which the account was posted, is written in the 2d column of the Leger. In posting, begin with the first account in the Day Book, which you will perceive is the name of David Judkins. Enter his name in the first page of the Leger, in a large, fair hand, with Dr. on the left and Cr. on the right.— As there are several articles charged to D. Judkins on the 4th of January, instead of specifying each article in the Leger, we merely say, For Sundries, and enter the amount in the proper columns—see Leger, page 1. The Leger has an index or alphabet, in which the names of persons are arranged under their initial letters, with the page in the Leger where the account may be found. ALPHABET TO THE LEGER. A I J R Ames, Hiram - 2 Judkins, David - 1. Ames & Smith - 4 B K S Balance -º- * 5 Sanxay, H. C. - 4 C L T Coolidge. T. W. . 1 |Letton, T. P. - 4 D M U Moore, Jackson - 3 E N V Eaton, George H. 1. F. O W Wilson, Jas. & 2 G P X *~. Y H Q. Young, Robt. * Hilton, Thomas - 2 > Hamilton, Geo. - 3 Z # - * |TDr. David Judkins, Tcz. T | 1533T | \ | |Jan. 4 To sundries|2|4|20 ||Jan. 8 By cash |2|3|00 || | “30 “ Apples |3||| 900 || “ 30 “ Corn 3 9|40 | Mar.7 “ Flour |4|| 350 “ Bal. 4||30 | | | ||1670 16||70 | Apl. 1.To balance | of account - bro't down, || 430 | Nore.—The Dr. on the left hand side of the page signifies debtor, and that || || the sums entered on that side of the page, are those for articles sold to others, | and for which they owe you. The Cr. on the right hand side signifies credit, or || that the sums entercq on that side of the page are for articles received of the | person under whose account they stand, and for which you owe him. #. Timothy W. Coolidge. |jan. To #. 3|2|Jan By, work fºllº |Mar. 20 “ Tunion 5|| 600 |Apl. 4: Balance||7|174 X-3:----- - 32 174| |2|17, | | Apl. 1|To balance | || 7|174 | | Nore.—The Dr. side of this account shows the amount of articles Mr. Cool- || | idge has received of me, and the Cr. side shows what I have received of him. H | It appears that the total amount of my account against him is $32,174, from ºf which deduct the $25 00 (which stands to his credit on the right of the account) || | and there will remain a balance of $7174 due me. ºš: | George H. Eaton. |Jan. G|To sundries *| sº Iſan:21 By cash º, 3 iº #º &#º # * | Note:-This account presents equal sums on both sides; hence it is evident || || that I owe J. H. Eaton nothing, and that he owes me nothing. The account || s fully closed. -- > º ** . * Dr. Hiram Ames. Cr. 2 Jan 1 iTo sund.2|| 7020 Jan. II. By sund. |2|| 1 ||00 Mar.6 “ Corn |4|| 28|75||Feb. 13| “Order |3|| 1000 Apl. 1| “ Bal. 23|95 Apl. 1. To Bal. 23.95 Thomas Hilton. Feb. 3 To sund.[3| 706 ||Feb. 3. By labor,4||11374 Mar 8 “ Cash. |4||1875 ||Mar:23 “ hire of “ 31|| “ do. 6||15|SS horse, 5|| 625 << 31|| “ Sund. |6||1400 Apl. 1 “ Balj ||10|064 Apl. 1|To Ball 1006, James Wilson. Jan. I To boar- || || ||Jan. 31. By cash 3||1500 - ding, |2|| 600|April 1 “ Bal. '085 “ 15|“Sunds. |2||12||13 - Feb. 21 “ .do. 4|| 7 |72 –– || 2585 |2585 April 1. To bal- ance bro’t down, 1085 Nork.--When an account is settled only, and not fully paid, as in the above, and several preceding accounts, the balance, whether it be in your ſº or ; against you, is brought down and placed distinctly by itself, and serves for the beginning of a new account, as you perceive has been done in the above exami-l * ple, the balance being $1085. - - - *...º.º. § º * George Hamilton. “ Sund. ſ £º wº 47|00 * * * *ś, ź Wºº |April 1 . To Ball 1050 To Tea, 1326|25||Jan. 19 Bychairs, 3 6||75||Mar 31|| “Sunds. “ do. 61400|Apl. 1 Balance, Cr. 6 23|00 11|50 Robert Young. | Jan. 27.To sund- Feb.27|By wood, | ries, 3| 6,624 | Feb. 15 “ do. 4|10|00. | Apl. 1| “ Bal. 374 | 00 wº * 2. Apl. 1. By Bal. 2700 | Norg-In the ubove account the difference between the Dr. and Cr, side is | $10.37%, by which I perceive that the balance against me, in favor of Robert foung, is s10374. % º Hºº Jackson Moore, dries, 3 * Sunds.[6 - 60 Ing, Mar.31 pl. 1. “ Bal. 00 40 1|00|| Apl. By bal. ance bro’t Jan. 29 To Sun- Mar.31|By paint- down, &iº ióño || : Dr. Ames & Smith. * cr al |Mar. G|To Sundries(4:1100|Mar.25 By sugar, 5||98|77| * 30 “ do. 5||87 |77 98|77|| 98|77|| NotE.-This account, like the one on the first page, is fully closed, the amount on the Dr. side being just equal to that on the Cr. T. P. Letton. |Marzoto Sundries; 21sº Ap. IVBy Bal. ||5|50| | << 26|| “ do. 5||12|624 15|50 || 15|50 || April 1|To balance || ||1550 NotE.-In the above account there is no sum on the Cr. side, and the infer- - ence is, that T. P. Letton owes me $15 50 for sundry articles expressed in de- tail, in the Day Book, page 5. Henry C. Sanaray. Mar.28|By my - Apl 1|To balance, |12|874 order, 5||1287| *** * 12|874 12|874| *ºtºtºx wº Apl. 1|By bal. 1257, | | | Norg-In this account it will be perceived that as there is no amount char. ged to H. C. Sanxay, on the Dr. side, I owe him $12.87%. *. Dr. — Balance. * ril 1. To D. Judkins, 1||4|30 ||April 1] By R. Young, 3|| “ T. W. Coolidge, 1 “ J. Moore, * H. Ames, w “ H. Sanxay, | “ T. Hilton, 2|10|06}|| “ J. Wilson, w “ G. Hamilton, “ T. P. Letton, & # $º *ś Notr.—This account exhibits the exact state of your books. It is made from the preceding accounts in the Leger. The Dr. side is an exhibit of the amounts due to you by others, and the Cr. side the amounts due by you to oth. ers. It is not strictly necessary that this account should be introduced in the Leger in single entry: it will be found convenient, however, to balance the book at stated intervals, and transfer the balances to the new accounts below, ºf as in the preceding Leger, and when that is done, a balance account like the #: above, will be found convenient, as presenting, at one view, the exact state ºf your Leger. FORM OF A BILL FROM THE PRECEDING. Mr. David Judkins, ..º % ; ; 3. % To Edward Thomson, Dr. anuary 4 || To 10 lbs. coffee at 17 cts. - & # & $1 #| º:----- . 25 lbs, sugar at 10 cts. - - - - 2 50 | . * --> 420 | * , 30 12 bbls, apples at 75 cts. & #: ś, ź. 900. March 8 200 lbs, flour at $175, - - - - 3|50|| . Cr. § 16 70|| anuary 8 | By Cash, & §: % § & # & $3 00 £3. º 30 47 bushels corn at 20 cts. - & & § 9 40 Errors excepted. Balance due, Rec'd payment in full, EDWARD THOMSON. . THE CASH BOOK. | The Cash Book is used to record the daily receipts and payments of mo- || y. It is ruled nearly the same as the Leger; the Dr. side exhibits the unt of money received, and the Cr. side, the amount paid out. Subtract || sum of the Cr. from that of the Dr. and the balance will always be equal || e amount of cash on hand. 3. º FoRM of A CASH Book, To cash on hand, 73|81|Jan. 1|By rent of house paid | !Cash rec'd of J. Young. 1640 | | T. P. Letton, 1 * “ H. Sanxay. Paid note to R. Hand, 500 1. “ “ D. Judkins | “ 1jFamily expenses, ...; * 1: By cash on hand, # 1 By cash paid Ames & Jan. 2 Smith, |By cash on hand, of T. Coolidge, Cash found on Main St. ** 2