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It is sometimes convenient to reduce compound numbers to a vulgar fraction. This is done by reducing the given number to the lowest denomination mentioned, for a numerator, and placing that number which it takes of this denomination to make one of the next higher, for a denominator.

24

480

In the example before given, 2 lb. 6 oz. 16 pwt. 12 gr. if you wish to reduce it to a vulgar fraction, reduce it to grains, which, placed for a numerator over 24, will form the fraction 14796. If you wish to reduce this to the fraction of a pennyweight, multiply the denominator by 20, which gives 14796; this is dividing the fraction by 20. (8. Con. III.) You may further reduce it by dividing the fraction by the number which it takes of the denomination which the fraction is in, to make one of the next higher denomination. This rule is not much needed, except to render the nature of reducing a number to a decimal more easy.

Reduce 2 fur. 26 pol. 3 yds. 2 ft. to the fraction of a mile. Ans. 4408, or §.

5280

Reduce 6 fur. 16 pol. to the fraction of a mile.

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Pup. These numbers are now expressed by vulgar fractions, and can be reduced to decimals; is it not much better to reduce them to decimals than it is to have them expressed in vulgar fractions?

Tut. It is better, and there is a shorter way to do it than to reduce them to vulgar fractions as taught above.

If for instance you wish to reduce 6 oz. 16 pwt. 12 grs. to decimals, you must consider each of these denominations to have a denominator, viz., 16, 14. You have been taught (20. Con. III.) how to reduce a vulgar to a decimal fraction. In the expression, 16, 12, 12 represent a fraction of a pennyweight, and if you reduce 12 to a decimal, you obtain 0,5 which is 5 tenths of a pwt. so that you may now consider the expression to be 6 oz. 16,5 pwt. Now to reduce 16 to a decimal, you will divide 16 by 20, which will give 0,8; but as 16 has be

come 16,5 the decimal will be 0,825.

reduced to a

decimal is 0,5; but 6 has become 6,825; and this divided by 12, gives 0,56875.

Hence, to reduce numbers of different denominations, as of weights, measures, &c. to decimals, we have the following rule.

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

Opposite to each dividend, on the left hand, place such a number for a divisor as will bring it to the next superior denomination, and draw a line perpendicularly between them. Begin with the upper number, 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 are all used, and the last quotient will be the decimal sought.

The example which has been given will stand as follows

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It is sometimes necessary to reduce a decimal back to the terms of an integer, for which we have the following rule.

Multiply the decimal by that number which it takes of the next less denomination to make one of that denomination in which the decimal is given; cut off from the right of the product, as many figures as there are figures in the given decimal. Then multiply those figures which you cut off, by that number which it takes of the next less denomination to make one of this, and again cut off, proceeding in this manner till you have multiplied by all the denominations below that in which the decimal is given; then the number standing on the left will be the answer.

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In the decimal 0,629 you have a decimal expression of a hundred weight. Now a unit in this decimal is one thousandth of a hundred weight, consequently, if you multiply this by 4, you obtain a number 4 times as great as this number is, a unit of which will be one quarter of one thousandth of a hundred weight. In this example the product of,629 by 4 is 2516; which is 516 of qr. of cwt. which, reduced to a mixed number, is 200 qr. so that you now have 0,1516 of a quarter to reduce to the terms of an integer; this, multiplied by 28, the number of pounds it takes to make a quarter, gives 14448, which is 14448 of of of a hundred weight; and 14448 duced to a mixed number, is 14-448 448 lb. 0,448 must now

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be reduced in the manner the preceding have been, and

the same for reducing any decimal to the terms of an integer.

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What is the value of 0,397 of a yard? Ans. 1 qr. 2 nls.

What is the value of 0,8593 of a pound Troy?

Ans. 10 oz. 6 pwt. 5 grs.

Reduce 12 pwt. 14 grs. to the decimal of an ounce.
Ans. 0,6291.

Reduce 3 qrs. 2 nails to the decimal of a yard.

Ans. 0,875.

Previous to the adoption of the Federal Currency, there existed various currencies in the United States, which exist in some places to the present day. As you may have occasion to transact business in those places, I will give you rules by which you may reduce these currencies to Federal Currency. It will be necessary for you to learn the following tables.

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The first denomination of Federal Money is an imag. inary coin, there being none of that kind made. second, viz. cents, is a copper coin, dimes and dollars are silver, and eagles are gold.

The following is the rule for reducing New England and Virginia; New York, North Carolina and Ohio currencies to Federal Money.*

* The currencies of N. England and Virginia are the same, and will be called New England Currency. The currencies of New York, North Carolina, and Ohio are the same, and will be called New York Currency; those of New Jersey, Pennsylvania, Delaware, and Maryland are the same, and will be called Pennsylvania Currency.

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Write the pounds, and at their right hand write half the greatest even number of shillings. Then find the number of farthings in the given pence and farthings, and if it exceed 12, increase it by 1, or if it exceed 36, increase it by 2, which sum place at the right of half the greatest even number of shillings, and if the shillings be an odd number, increase the left hand figure of the farthings, or the figure next to shillings, by 5. Multiply this number, thus produced, by 10, and divide the product by 3 if it be NewEngland Currency, and by 4 if it be New-York Currency; cut off the three right hand figures of the quotient, which will be cents and mills; the rest will be dollars.

If there be no shillings, or only 1 shilling, in the given sum, put a cipher in the place of half the number of shillings; then pro. ceed as before.

If pounds only are given to be reduced, multiply by 10, and divide as before, the quotient will be dollars. If there be a remainder, annex three ciphers and divide as before, the quotient will be cents and mills.

When pounds and an even number of shillings are given, annex half the even number of shillings to the pounds, as before, then divide, and the quotient will be dollars; if there be a remainder, ciphers must be annexed, and the quotient will be cents and mills.

Reduce £. 763 N. E. and N. Y. Currencies to Federal Currency.

3)7630

$2543,333 N. E.

4)7630

$1907,50 N. Y.

Reduce £. 50 7s. 8d. to Federal Money.

3)503830

$107,943 N. E.

4)50383

$ 125,957 N. Y.

Reduce £. 10 4d. 2qr. to Federal Money.

Ans. $33,396 N. E. $25,047 N. Y.

The reasons of this rule are as follows; 1 shilling is of a pound, and if 10 shillings had made a pound, 1 shilling would have been an exact decimal of a pound. But as 20 shillings make a pound, 2 shillings will be an exact decimal of a pound. Wherefore the reason of writing half the even number of shillings.

Again, 960 farthings make a pound. Now if this number be increased by part of itself, it will be an exact

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