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13. 1,490+, &c. REDUCTION OF DENOMINATE DECIMALS.
174. We have seen i 71. a: a denominate number is one is ach sector aris denomirated or expressed
A denominate decimal is a decimal fraction in which the kind of cost tha: has been vided is ex. pressed. Toss 5 of a £. and 67 of a sing, are denomina:e decima's. Toe ei tha: was divided in the first fraction being £1, azi na: in the second 1 shilling.
CASE I. 175. To reduce a denominate number to an equivalent decimal of a given and higher denomination.
RILE. Dicide the giren number by so many as make one of the next higher denomination, annezing ciphers if necessary. Then diride this quotient by so many as make one of the denomination still higher, and so on, till you reach the denomination to which the number is to be reduced: the last quotient will be the decimal required. Er. 1. Reduce 9d to the decimal of a £.
£,0375 Ans. £,0375 2. Reduce 7 drams to the decimal of a lb. avoirdupois.
3. Reduce 26d to the decimal of a £.
Ans. ,1083333+. 4. Reduce ,056 poles to the decimal of an acre.
Ans. ,00035A. 5. Reduce 14 minutes to the decimal of a day.
Ans. ,0097222da. + 6. Reduce ,21 pints to the decimal of a peck.
§ 176. To reduce denominate numbers of different denominations to an equivalent decimal of a given : denomination.
Ex. Reduce £1 4s 9d to the denomination of a £. d=,75d, hence, 9d=9,75d. But, 9,75d=9,75 of 1 of a shilling=275=,81258. Hence, 4s 9d=4,8125s=4,8125 of b of a £= 4.3125=,240625£. Therefore, £1 4s 9 d=
£1,240625. Hence we have the following
Divide the lowest denomination named, by that number which makes one of the denomination next higher, annexing ciphers if necessary: then annex this quotient to the next higher denomination, and divide as before : proceed in the same manner through all the denominations to the last : the last result will be the answer sought. Ex. 1. Reduce £19 17s 31d to the decimal of a £.
Ans. £19,863+. 2. Reduce 15s 6d to the decimal of a £.
3. Reduce 74d. to the denomination of shillings.
Ans. ,625s. 4. Reduce 21b. 5oz. 12pwt. 16gr. Troy, to the de: cimal of a lb.
Ans. 2,46944411. t.
CASE III. 177. To find the value of a denominate decimal in terms of integers of inferior denominations.
RULE. I. Multiply the given decimal by that number which makes one of the next less denomination, and cut off from the right so many figures as there are places in the given decimal.
II. Multiply the figures so cut off by the number which makes one of the next less denomination, and cut off as before, and proceed in the same way to the lowest denomination : the figures to the left will form the answer sought. Ex. 1. What is the value of ,832296 of a £.
Ans. 16s. 7d. 3far. 2. What is the value of ,00208416. Troy?
Ans. 12,00384gr. 3. What is the value of ,625 of a cwt.
Ans. 2qr. 1416. 1. What is the value of ,625 of a gallon?
Ans. 2qt. lpt.
5. What is the value of £,3375?
Ans. 6s. 9d. 6. What is the value of ,3375 of a ton?
Ans. 6cwt 3qr. 7. What is the value of ,05 of an acre ?
QUESTIONS. 173. To what is the value of a vulgar fraction equal ? How do you reduce a vulgar fraction to its equivalent deci. mal ?
$ 174. What is a denominate number? What is a donominate decimal ?
$ 175. How do you reduce a denominate number to a de. cimal of a given denomination ?
§ 176. How do you reduce denominate numbers of diffe. rent denominations to decimals of a given denomination ?
$ 177. How do you find the value of a denominate decimal in terms of integers of inferior denominations ?
REDUCTION OF CURRENCIES. $ 178. Previous to the act of Congress which established a uniform currency throughout the United States, in dollars, dimes, cents, and mills, each state had its own particular currency. This circumstance has caused the dollar to be divided differently in the different states, though the real value of it is the same in all.
Thus 1 dollar is reckoned in England at 48 6d, called English, or sterling money. Canada and
} 58 Nova Scotia
New York Currenc
The New England States, Vir.
New England Currency: ginia, Kentucky and Tennessee. New Jersey, Pennsylvania,
78 6d Delaware and
CASE I. $ 179. To reduce a sum in either currency to Federal Money.
The reduction of these currencies to Federal money consists in this: having given any sum expressed in pounds, shillings and pence, to find its value in dollars, cents, and mills. Take for example the English Currency.
$1=4s 6d=54d. Hence, £l : $1 :: 240 : 54, .
or $1= 14 of a £=% of a £. The number of dollars in any sum will therefore be expressed by the number of times which that sum contains the fraction to
As similar reasoning may be employed for each of the currencies we have for them all the following
RULE. 1. Reduce the given sum to the decimal of a £. as in ø 175.
II. Then find the value of 1 dollar in the fraction of a pound and divide the given sum, reduced as above,
this fraction. The quotient will be the answer
180. To divide by a fraction we invert