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2. Divide 72117562 by 2°257432, so that the quotient may contain three decimals. Ans. 319'467. 3. Divide 12.169825 by 3°14159, so that the quotient may contain five decimals. Ans. 3873774. Divide 87076326 by 9°365407, and let the quotient contain seven decimals. Ans. 9'2976554

REDUCTION of DECIMALS.

CASE I.

To reduce a vulgar fraction to its quivalent decimal,

RULE.*

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Divide the numerator by the denominator, annexing as many cyphers as are necessary; and the quotient will be the decimal required.

EXAMPLES

* Let the given vulgar fraction, whose decimal expression is required, be Now since every decimal fraction has 10, 100, 1000, &c. for its denominator; and, if two fractions be equal,

it

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1. Write the given numbers perpendicularly under each other for dividends, proceeding orderly from the least to, the greatest.

2. Opposite to each dividend, on the left hand, place such a number for a divisor, as will bring it to the next superior name, and draw a line between them,

3. Begin

it will be, as the denominator of one is to its numerator, so is the denominator of the other to its numerator; therefore 13:7:: 1000, '&c.: 7X 1000, &c. ___0000, &c.

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=53846, the numerator of

the decimal required; and is the same as by the rule.

975

The reason of the rule may be explained from the first example; thus, three farthings is of a penny, which brought to a decimal is 75; consequently 93d. may be expressed 9'75d. but 9.75 is 35 of a penny of a shilling, which brought to a decimal is 8125; and therefore 15s. 94d. may be expressed 15.8125s. In like manner 15-8125s. is 58125 of a shilling 158125 of a pound, by bringing it to a decimal, 7906251. as by the rule.

10000

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3. Begin with the highest, and write the quotient of each division, as decimal parts, on the right hand of the dividend next below it; and so on, till they be all used, and the last quotient will be the decimal sought.

EXAMPLES.

1. Reduce 15s. 94d. to the decimal of a pound.

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790625 the decimal required.

2. Reduce 9s. to the decimal of a pound.

3. Reduce 19s. 5d. to the decimal of a pound.

Ans. 45

Ans. 972916.

4. Reduce 10oz. 18dwt. 16gr. to the decimal of a lb. Troy.

Ans. 911111, &c.

5. Reduce 2qrs. 14lb. to the decimal of a cwt.

Ans. 625, &c.

6. Reduce 17yd. ift. 6in. to the decimal of a mile.

Ans. 00994318, &c. 7. Reduce 3qrs. 2nls. to the decimal of a yard. Ans. 875. 8. Reduce igal. of wine to the decimal of a hhd.

Ans. 015873.

9. Reduce 3bu. ipe. to the decimal of a quarter.

Ans. 40625.

10. Reduce 10w. 2d. to the decimal of a year.

CASE III.

Ans. 1972602, &c,

To find the decimal of any number of shillings, pence and farthings, by inspection.

RULE.*

Write half the greatest even number of shillings for the first decimal figure, and let the farthings in the given pence

and

*The invention of the rule is as follows as shillings are so many 20ths of a pound, half of them must be so many 10ths, and

consequently

and farthings possess the second and third places; observing to increase the second place by 5, if the shillings be odd; and the third place by 1, when the farthings exceed 12; and by 2, when they exceed 37.

EXAMPLES.

Find the decimal of 15s. 84d. by inspection.

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2. Find by inspection the decimal expression of 16s. 4žd. and 135. 10d. Ans. 819 and '694.

3. Value the following sums by inspection, and find their total, viz. 19s. 114d. + 6s. 2d. † 12s. 8d. + 1s. 10d. + 2d. + 14d. Ans. 2043 the total.

CASE

consequently take the place of 10ths in the decimal; but when they are odd, their half will always consist of two figures, the first of which will be half the even number, next less, and the second a 5; and this confirms the rule as far as it respects shillings.

Again, farthings are so many 96oths of a pound; and had it happened, that 1000, instead of 960, had made a pound, it is plain any number of farthings would have made so many thousandths, and might have taken their place in the decimal accordingly. But 960, increased by part of itself, is = 1000; con. sequently any number of farthings, increased by their part, will be an exact decimal expression for them. Whence, if the number of farthings be more than 12, a part is greater than, and therefore I must be added; and when the number of farthings is more than 37, a part is greater than 14, for which 2 must be added; and thus the rule is shewn to be right.

CASE IV.

To find the value of any given decimal in terms of the integer.

RULE.

1. Multiply the decimal by the number of parts in the next less denomination, and cut off as many places for a remainder on the right hand, as there are places in the given decimal.

2. Multiply the remainder by the parts in the next inferior denomination, and cut off for a remainder, as before. 3. Proceed in this manner through all the parts of the integer, and the several denominations, standing on the left hand, make the answer,

EXAMPLES.

1. Find the value of 37623 of a pound.

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67 of a league ?

4. What is the value of 6725cwt.? Ans. 2qrs. 19lb. 5oz.

5. What is the value of

Ans. 2mls. 3pls. 1yd. 3in. 1b.c.

6. What is the value of 61 of a tun of wine?

Ans. 2hhd. 27gal. 2qt. Ipt.

7. What is the value of 461 of a chaldron of coals?

Ans. 16bu. 2pe.

3. What is the value of 42857 of a month?

Ans. 1w. 4d. 23h. 59′ 56′′.

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CASE

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