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There being 11 decimal figures in the dividend, and none in the divisor, 11 figures are to be cut off in the quotient; but as the quotient itself consists of but 10 figures, prefix to them a cipher to complete that number.

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52 48 48 48 0 t Because the number of decimal figures in the divisor and dividend are alike, the quotient will be integers. Divide 2 by 3.1416 3.1416)2.0000,000.636618-H = quotient. 1 88496 115040 94248 207920 188496

194240
188496

57440
3.1416

260240

: 251328

8912–H In this example there are four decimal figures in the divisor, and none in the dividend; therefore, according to the rule, four ciphers are annexed to the dividend, which, in this condition, is yet less than the divisor. A cipher must then be put in the quotient in the place of integers, and other ciphers annexed to the dividend; and the division being now performed, the decimal figures of the quotient are obtained.

Divide 7234.5 by 6.5 Quotient = 1113.
Divide 476.520 by .423 — = 1126.5-H
Divide .45695 by 12.5 = .0365-H
Divide 2.3 by 96 — = .02395-H
Divide 87446071 by .004387 — = 1993.3000000.
Divide .624672 by 482 = .001296.

REDUCTION OF DECIMALS.

RULE I.

To reduce a Vulgar Fraction to a Decimal of the same value. Having annexed a sufficient number of ciphers, as decimals, to the numerator of the vulgar fractions, divide by the denominator; and the quotient thence arising will be the decimal fraction required.

EXAMPLE.

Reduce # to a decimal fraction. 4)3.00

.75=decimal required. For ; of one acre, mile, yard, or any thing, is equal to 1 of 3 acres, miles, yards, &c.; therefore if 3 be divided by 4, the quotient is the answer required.

Reduce 3 to a decimal fraction. Answer .4
Reduce 43 - - - - - - .48
Reduce # - - - - - - . 1146789
Reduce # - - - - - - .7777–H
Reduce #4 - - - - - - .91.30434–H

Reduce 3, #, 3, 4, and so on to ar, to their corresponding decimal fractions; and in this operation the various modes of

interminate decimals may be easily observed.

RULE II.

To reduce Quantities of the same, or of different Denominations, to Decimal Fractions of higher Denominations.

If the given quantity consist of one denomination only, write it as the numerator of a vulgar fraction; then consider how many of this make one of the higher denomination, mentioned in the question, and write this latter number under the former, as the denominator of a vulgar fraction. When this has been done, divide the numerator by the denominator, as directed in the foregoing rule, and the quotient resulting will be the decimal fraction required.

But if the given quantity contain several denominations, reduce them to the lowest term for the numerator; reduce likewise that quantity whose fraction is sought to the same denomination, for the denominator of a vulgar fraction; then divide as before directed.

EXAMPLES. Reduce 9 inches to the decimal of a foot. The foot being equal to 12 inches, the vulgar fraction will be or ; then 12)9.00 .75=decimal fraction required. Reduce 8 inches to the decimal of a yard. 8 inches

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72 80 72 8 Reduce 5 furlongs 00 perches to the decimal of a mile. 1 mile 5 furlongs 8 40 * 8 fur. 200 40 = vulgar fraction. - 320 320 per. 320)200.00.625 = decimal sought. 1920 800 640 1600 1600

Reduce 21 minutes 54 seconds to the decimal of a degree. Ans. .365. - Reduce .056 of a pole to the decimal of an acre. Ans. .00035. Reduce 13 cents to the decimal of an eagle. Ans. .013. Reduce 14 minutes to the decimal of a day. Ans. .00972+ Reduce 3 hours 46 minutes to the decimal of a week. Ans.

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RULE III.

To find the value of Decimal Fractions in terms of the lower denominations.

Multiply the given decimal by the number of the next lower denomination which makes an integer of the present, and point off as many places at the right-hand of the product, for a remainder, as there are figures in the given decimal. Multiply this remainder by the number of the next inferior denomination, and point off a remainder as before. Proceed in this manner through all the parts of the integer, and the several denominations standing on the left-hand are the value required.

EXAMPLES.

Required the value of .3375 of an acre.
4 = number of roods in an acre.

1.3500
40 = number of perches in a rood.

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.* 14.0000
The value, therefore, is 1 rood 14 perches.

- What is the value of .6875 of a yard?
- 3 = number of feet in a yard.

2.0625
12 = number of inches in a foot.

.7500
12 = number of lines in an inch. .

9.0000
The answer here is 2 feet 9 lines.

What is the value of .084 of a furlong? Ans.3 per 1 yd.

2 ft. 11 in.
What is the value of .683 of a degree? Ans. 40 m. 58 sec.
48 thirds.
What is the value of .0053 of a mile? Ans, 1 per. 3 yds.
2 ft. 5 in.-H.

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PROPORTION IN DECIMAL FRACTIONS.

Having reduced all the fractional parts in the given quantities to their corresponding decimals, and having stated the three known terms, so that the fourth, or required quantity, may be as much greater or less than the third as the second term is greater or less than the first, then multiply the second and third terms together, and divide the product by the first term, and the quotient will be the answer;-in the same denomination with the third term.

EXAMPLES,

If 3 acres 3 roods of land can be purchased for 93 dollars 60 cents, how much will 15 acres 1 rood cost at that rate 2 3 acs. 3 rds. = 3.75 acres. 15 acs. 1 rq. = 15.25 acres. $93, 60 cents = $93.60 o

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If a clock gain 14 seconds in 5 days 6 hours, how much will it gain in 17 days 15 hours? Ans. 47 seconds.

If 187 dollars 85 cents gain 12 dollars 33 cents interest in a year, at what rate per cent. is this interest? Ans. 6.56+

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