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By dividing 13853 by 24, and the quotient thence arising by 20, and this second quotient by 12, we shall evidently obtain the number of pounds, ounces, pennyweights, and grains in 13853 grains. The operation may be seen below.

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To reduce a compound number to the lowest denomination contained in it, multiply the highest by so many as one of this denomination makes of the next lower, and to the product add the number belonging to the next lower; proceed with each succeeding denomination in a similar manner, and the last sum will be the number required.

To reduce a number from a lower denomination to a higher, divide by so many as it takes of this lower denomination to make one of the higher, and the quotient will be the number of the higher; which may be further reduced in the same manner if there are still higher denominations, and the last quotient together with the several remainders will be equivalent to the number to be reduced.

Examples for practice.

In 59lb. 13dwt. 5gr. how many grains?
In 8012131 grains how many pounds, &c. ?

Ans. 340157.

Ans. 1390lb. 11oz. 18dwt. 19gr.

In 121. Os. 9 d. how many half pence?
In 58099 half pence how many pounds &c.?

Ans. 58099.

Ans. 121l. Os. 91d.

In 48 guineas at 28s. each how many 4 pence?

Ans. 3584.

In one year of 365d. 5h. 48′ 48′′ how many seconds?

Ans. 31556928.

102. When we have occasion to make use of a number consisting of several denominations as an abstract number, instead of reducing the several parts to the lowest denomination contained in it, we may reduce all the lower denominations to a fraction of the highest. Taking the sum before used, namely, 41. 15s. 9d. we reduce the lower denominations to the higher, as in the last article by division. The number of pence 9, or, is divided by 12, by multiplying the denominator by this number (54), we have thus, s. which being added to 15s. or 10s. the whole number being reduced to the form of a fraction of the same denominator, we have 1 and 2, which being added, make

189

180

12

18

1. This is further reduced to pounds by dividing it by 20, that is, by multiplying the denominator by 20 (54), which

240

gives. Whence £4. 15s. 9d. is equal to £4, or £. This may now be used like any other fraction, and the value of the result found in the different denominations. If we multiply it by 37, we shall have £43, or £177; and £3, reduced to shillings by multiplying the numerator by 20, or dividing the denominator by this number, gives s. or 2s. or 2s. 9d.

2

33

From the above example we may deduce the following general rules, namely,

To reduce the several parts of a compound number to a fraction of the highest denomination contained in it, make the lowest term the numerator of a fraction, having for its denominator the number which it takes of this denomination to make one of the next higher, and add to this the next term reduced to a fraction of the same denomination, then multiply the denominator of this sum by so many as make one of the next denonination, and so on through all the terms, and the last sum will be the fraction required.†

To find the value of a fraction of a higher denomination in terms of a lower, multiply the numerator of the fraction by so many as make one of the lower denomination, and divide the product by the denominator, and the quotient will be the entire number of this denomination, the fractional part of which may be still further reduced in the same manner.

30

To reduce 2w. 1d. 6h. to the fraction of a month.

36

6h. is of a day, and being added to one day, or 4d. gives d. the denominator of which being multiplied by 7, it becomes W. and being added to 2 weeks or twice 18w. gives 8w. If we now multiply the denominator of this by 4, we shall have of a month, as an equivalent expression for 2w. 1d. 6h.

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It will often be found more convenient to reduce the several parts of the compound number to the lowest denomination, as by the preceding article for a numerator, and to take for the denominator so many of this denomination as it takes to make one of that, to which the expression is to be reduced; thus 41. 15s. 9d. being 1149d. is equal to 1491. because 1d. is 1.

240

240

To find the value of of a mile in furlongs, poles, &c.

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Reduce 13s. 6d. 2q. to the fraction of a pound.

Ans. £, or £.

Reduce 6fur. 26pls. 3yds. 2ft. to the fraction of a mile.

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Reduce 7oz. 4pwt. to the fraction of a pound, Troy.
What part of a mile is 6fur. 16pls.?

What part of a hogshead is 9 gallons?

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Ans. 3.

Ans..

Ans..

Ans..

Ans..

What part of a penny is

What part of a cwt. is

of a pound?
of a pound, Avoirdupois?

What part of a pound is of a farthing?

What is the value of 3 of a pound, Troy?

What is the value of 4 of a pound, Avoirdupois ?

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Ans. 7oz. 4dwt.

Ans. 9oz. 2 dr.

Ans. 3qrs. 3lb. 1oz. 123dr.

Ans. 1fur. 16pls. 2yds. 1ft. 9 in.

What is the value of of a day?

900

Ans. 12h. 55' 23"

The several parts of a compound number may also be reduced to the form of a decimal fraction of the highest denomination contained in it, by first finding the value of the expression in a vulgar fraction, as in the last article, and then reducing this to a decimal, or more conveniently by changing the terms to be reduced into decimal parts, and dividing the numerator instead of multiplying the denominator by the numbers successively employed in raising them to the required denomination. If we take the sum already used, namely, £4. 15s. 9d. the pence, 9, may be written, or 8, the numerator of which admits of being divided by 12 without a remainder. It is thus reduced to shillings and becomes s. or 0,75s. which added to the 15s. makes 15,75s. or reducing the 15 to the same denomination, 1575 or 1575; and this is reduced to pounds, by dividing it by 20, the result of which is 7, or 0,7875. 4l. 15s. 9d. therefore may be expressed in one denomination, thus, 4,78751. and in this state it may be used like any other number consisting of an entire and fractional part. If it be multiplied by 37, we shall have for the product 177,1375l. This decimal of a pound may be reduced to shillings and pence, by reversing the above process, or by multiplying successively by 20 and then by 12.

10000

75

то

0,1375
20

7875

2,7500

12

9,0000

The product therefore of 4l. 15s. 9d. by 37 is 1777. 2s. 9d. as before obtained.

The operation, just explained, admits of a more convenient disposition, as in the following example.

To reduce 19s. 3d. 3q. to the decimal of a pound.

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