3

189

1149

that is, by multiplying the denominator by 20 (54), which gives. Whence £4 15s. 9d. is equal to £413, or £10. 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 £21, or £177; and £, reduced £_33_ to shillings by multiplying the numerator by 20, or dividing the denominator by this number, gives s. or 2s. or 2s. 9d. 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 denomination, and so on through all the terms, and the last sum will be the fraction requiredt.

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.

366

6h. is of a day, and being added to one day, or d. gives d. the denominator of which being multiplied by 7, it becomes 3w. and being added to 2 weeks or twice w. gives w. If we now multiply the denominator of this by 4, we shall have 3 of a month, as an equivalent expression for 2w. 1d. 6h. To find the value of & of a mile in furlongs, poles, &c.

+ 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 4l. 15s. 9d. being 1149d.. is equal to 1149. because 1d. is al

240

01

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

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

Reduce 7oz. 4pwt. to the fraction of a pound, Troy. Ans. §.

What part of a mile is 6fur. 16pls. ?

What part of a hogshead is 9 gallons?

Ans..

Ans. 4.

Ans. 84
Ans..

Ans. 40.

What part of a cwt. is of a pound, Avoirdupois ? Ans. 597.

What part of a pound is of a farthing?

What is the value of
What is the value of

What is the value of
What is the value of

Ans. 7740.

of a pound, Troy? Ans. 7oz. 4dwt. of a pound, Avoirdupois ?

Ans. 9oz. 23dr. of a cwt.? Ans. Sqrs. 3lb. 1oz. 12‡dr. of a mile ? Ans. 1fur. 16pls. 2yds. 1ft. 9 in. Ans. 12h. 55' 231"

What is the value of 7 of day?

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 decimals parts, and dividing the numerator instead of multiplying the denominator by the numbers successively employed in raising them to the required denomination.

1575

900

If we take the sum already used, namely, £4 15s. 9d. the pence, 9, may be written 8, or 988. the numerator of which admits of being divided by 12 without a remainder. It is thus reduced to shillings and becomes 7s. or 0,758. which added to the 15s. makes 15,75s. or reducing the 15 to the same denomination, or 157508; and this is reduced to pounds, by dividing it by 20, the result of which is 7875, or 0.7875. 4l. 15s. 9d. therefore may be expressed in one denomination, thus, 4,7875l. 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.

0,1875
20

2,7500

12

9,0000

The product therefore of 4l. 15s. 9d. by 37 is 177l. 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.

000

Proceeding as before, we reduce the farthings, 3, considered as 3004. to hundredths of a penny by dividing by the figure on the left, 4, and place the quotient, 75, as a decimal on the right of the pence; we then take this sum, considered as 375d. or $750d. that is, annexing as many ciphers as may be necessary, and divide it by 12, which brings it into decimals of a shilling. Lastly, the shillings and parts of a shilling, 19,3125s. considered as 19313500 s. are reduced to decimals of a pound by dividing by 20, which gives the result above found.

1000000

We may proceed in a similar manner with other denominations of money and with those of the several weights and measOne example in these will suffice as an illustration of the

ures.

method.

To reduce 17pls. 1ft. 6in. to the decimal of a mile.

The decimal in this, as in many other cases, becomes periodical (97).

From what has been said, the following rules are sufficiently evident. To reduce a number from a lower denomination to the decimal of a higher, we first change it, or suppose it to be changed into a fraction, having 10, or some multiple of 10, for its denominator, and divide the numerator by so many as make one of this higher denomination, and the quotient is the required decimal; which, together with the whole number of this denomination, may again be converted into a fraction, having 10 or a multiple of 10 for its denominator, and thus by division be reduced to a still higher name, and so on.

Also, to reduce a decimal of a higher denomination to a lower, we multiply it by so many as one makes of this lower, and those figures which remain on the left of the comma, when the proper number is separated for decimals (91), will constitute the whole number of this denomination, the decimal part of which may be still further reduced, if there be lower denominations, by multiplying it by the number which one makes of the next denomination, and so on.

15

It may be proper to add in this place, that shillings, pence and farthings may readily be converted into the fraction of a pound, and the fraction of a pound reduced to shillings, pence and farthings, without having recourse to the above rules. As shillings are so many twentieths of a pound, by dividing any given number of shillings by 2, we convert them into decimals of a pound, thus, 15s. which may be written 18. or 1881. being divided by 2 give 75 hundredths, or 0,75 of a pound. Also, as farthings are so many 960ths of a pound, one pound being equal to 960 farthings, the pence converted into farthings and united with those of this denomination, may be written as so many 960ths of a pound. If now we increase the numerator and denominator one twenty fourth part, we shall convert the denominator into thousandths, and the numerator will become a decimal.

Whence, to convert shillings, pence and farthings, into the decimal of a pound, divide the shillings by 2, adding a cipher when necessary, and let the quotient occupy the first place, or first and second, if there be two figures, and let the farthings, contained in the pence and farthings, be considered as so many thousandths, increasing the number by one, when the number is nearer 24 than 0, and by 2, when it is nearer 48 than 24, and so on.

Thus, to reduce 15s. 9d. to the decimal of a pound, we have,

0,75
37

0,787

This result, it will be remarked, is not exactly the same as that obtained by the other method; the reason is, that we have increased the number of farthings, S6, by only one, whereas, allowing one for every 24, we ought to have increased it one and a half. Adding, therefore, a half, or 5 units of the next lower order, we shall have 0,7875, as before.

On the other hand, the decimal of a pound is converted into the lower denominations, or its value is found in shillings, pence and farthings, by doubling the first figure for shillings, increasing it by one, when the second figure is 5, or more than 5, and considering what remains in the second and third places, as farthings, after having diminished them one for every 24.