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urés, &c.) for a numerator ; then reduce the integral part to the same term, for a denominator ? place the former over the latter, and they will be the fraction required, wbich, if it be not in its lowest terms, must be reduced by Problem 1.
1. Reduce 13s. 6d. 2qrs. to the fraction of a pound.
s. d. qrs.
20 shill. integral part. 13 6 2 given sum.
960 Denominator. 650 Numerator.
Ans. 448=4£.. 2. What part of a hundred weight are 3qrs. 141. ?
Ans. 912=. 3. What part of a yard are 3qrs. 3na. ? 4. What part of a pound sterling are 13s. 4d.
Ans. . 5. What part of a civil year are 3 weeks, 4 days?
Ans. = 6. What part of a mile are 6fur. 26pol. 3yds. 2f..
fur. pol. yds. ft. feet.
a mile=5280 Denom. Ans. 448=. 7. Reduce 7oz. 4dwt. to the fraction of a tt. Troy.
Ans. 8. What part of an acre are 2 roods, 20 perches ?
Ans. š. 9. Reduce 54 gallons to the fraction of a hhd. of wine.
Ans. &. 10. What part of a hogshead of wine are 9 gallons ?
Ans. 11. What part of a pound Troy are 10oz. 10dwt. 10gr.?
Ans. 4. 12. What part of a dollar are 80cts. ?
DECIMAL FRACTIONS. A DECIMAL is a fraction whose denominator is a unit, with as many ciphers annexed to it as the numerator has places; and is usually expressed by writing the numerator only with a point before it called the separatrix. Thus Bo, o 2836, are decimal fractions, and are expressed by ,5 ,25 ,236 respectively.
The place of a figure in decimals, as in whole numbers, determines its relative value : That in the first place next the separatrix is 10th parts; that in the second, 100th parts, &c. decreasing in the same tenfold proportion to the right hand, as whole numbers increase decimally from units to the left hand.
Ciphers placed at the right hand of decimals, make no alteration in their value ; for ,5 ,50 ,500,&c. are decimals of the same value, being each equal to 1; but if placed at the left hand, the value of the fraction is decreased in a tenfold proportion for every cipher prefixed; thus ,5 ,05,005, &c. are 5 tenth parts, 5 hundredth parts, and 5 thousandth parts respectively. In the following Table, the doctrine is exemplified at large.
ADDITION OF DECIMALS. RULE.-Set the numbers so that the decimal points may stand directly under each other, then add as in whole numbers, carrying one for every ten, and place the decis mal point, in the sum, directly under the decimal points of the numbers which have been added.
In this first example, the sum is 455 integers, or pounds, and 38.40 parts of a unit or pound; or it is 455 units, and 5 tenth parts, 8 hundredth parts, and 4 thousandth parts of a unit, or 1 ; the cipher at the right of the decimal places does not affect the value of the other figures, and it is, therefore, thrown away.
Hence we may observe, that decimals, and Federal Money, are subject to one and the same law of notation, and, consequently, of operation.
For since 1 dollar is the money unit, and a dime being the tenth, a cent the hundredth, and a mill the thousandth part of a dollar, it is evident that any number of dollars, dimes, cents, and mills, is simply the expression of dollars, and decimal parts of a dollar: thus, 11 dollars, 6 diņes, 5 cents=11,65 or 1195 dol. &c.
3. What is the sum of 276, +39,213+72014,9+417,+ 5032, and +2214,298 acres ? Ans. 79993,411 acres.
4. What is the sum of ,014+,9816+,32+,15914+ ,72913+,0047 gallons !
Ans. 2,20857gal. 5. What is the sum of 27,148+918,73+14016,+294304, +7138, and +221,7 bushels ? Ans. 316625,578bus.
6. Add the following sums of dollars together, viz. $12,34565+7,891+2,34+14,+0011.
Ans. #36,57775 or $36, 5di. 7cts. 770 mills. 7. To 9,999999 miles add one millionth part of a mile.
SUBTRACTION OF DECIMALS. RULE.-Set the less number under the greater in the same manner as in addition; then subtract as in whole numbers, and place the decimal point in the remainder directly under the other points.
4. From ,9173ft. subtract ,2138. Ans. ,7035 foot. 5. From $2,73 subtract $1,9185. Ans. $,8115. 6. Subtract 91,713acres from 407. Ans. 315,287ac.
7. What is the difference between 67tons and ,92 of a ton ?
Ans. 66,08 tons. 8. From 1 league subtract the millionth part of itself.
Ans. ,999999 league.
MULTIPLICATION OF DECIMALS. Rule.-Place the factors, (whether mixed numbers, or pure decimals,) and multiply them, as in whole numbers ; and from the product, towards the right hand, point off as many figures for decimals as there are decimal places in the factors. But if there be not so many figures in the product, prefix ciphers to supply the defect.
3. Multiply 31,72rods by 65,3. Prod. 2071,316rods. 4. Multiply ,62foot by ,04.
Prod. ,0248foot. 5. Multiply 51,6yards by 21. Prod. 1083,6yards. 6. Multiply ,051£ by ,0091. Prod. ,0004641£.
7 What cost 6,21 yards of cloth, at 2dols. 32cts. 5 mills per yard ?
Ans. $14, 4d. 3c. 823 m. 8. Multiply 7,02 dollars by 5,27 dollars.
Ans. 36,9954dols. or $36,99cts. 54m. 9. Multiply 41dols. 25cts. by 120dols. Ans. $4950. 10. Multiply 3dols. 45cts. by 16cts.
Ans. ,5520 or 55cts. 2m. 11. Multiply 65cts. by ,09 or 9cts.
Ans. ,0585=5cts. 84m. 12. Multiply 10dols. by 10cts.
Ans. $1. 13. Multiply 341,45dols. by ,007 or 7 mills.
Ans. $2,39+ Note.—To multiply Decimal Fractions by 10, 100, 1000, &c. is only to remove the separatrix so many places towards the right hand, as there are ciphers in the multiplier.
EXAMPLES. Multiply $64,674 by $10. Multiply $3,2158 by 1000 cords.
DIVISION OF DECIMALS. RULE.—Divide as in whole numbers; and observe the following rules for pointing off in the quotient.
1. Point off for decimals in the quotient so many figures, as the decimal places in the dividend exceed those in the divisor.
2. If the figures in the quotient are not so many as the rule requires, supply the defect by prefixing ciphers.
3. If the decimal places in the divisor be more than those in the dividend, add ciphers as decimals to the dividend, until the number of decimals in the dividend be equal to those in the divisor, and the quotient will be integers until all these decimals are used. And in case of a remainder, after all the figures of the dividend are used, and more figures are wanted in the quotient, annex ci