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Proceeding as before, we reduce the farthings, 3, considered as q. 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 3758d. 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 193125.00s. are reduced to decimals of a pound by dividing by 20, which gives the result above found.

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.

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0,00531531 &c.

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.

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 giving number of shillings by 2, we convert them into decimals of a pound, thus, 15s. which may be written 17. or 187. 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, 36, 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.

In addition to the rules that have been given, it may be observed, that in those cases, where it is required to reduce a number from one denomination to another, when the two denominations are not commensurable or when one will not exactly divide the other, it will be found most convenient, as a general rule, to reduce the one, or both, when it is necessary, to parts so small, that a certain number of the one will exactly make a unit of the other. If it were required, for instance, to reduce pounds to dollars, as a pound does not contain an exact number of dollars without a fraction, we first convert the pounds into shillings, and then, as a certain number of shillings make a dollar, by dividing the shillings by this number, we shall find the number of dollars required. A similar method may be pursued in other cases of a like nature, as may be seen in the following examples. In 178 guineas at 28s. each, how many crowns at 6s. 8d.?

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Ans. 747 crowns and 4 shillingst.

In this case, I reduce both the guineas and the crown to pence, and then divide the former result by the latter. In dividing by 30, I first separate one figure on the right of the dividend for a decimal, which is the same as dividing it by 10, and then divide the figures on the left, or the quotient, by 8 (47), joining what remains as tens to the figures separated, to form the entire remainder, which is reduced back to the original denomination.

↑ Questions of this kind may often be conveniently performed by fractions; thus, 178 guineas, or 4984s. divided by 6s. 8d. or 63s. or, reducing the whole number to the form of a fraction, 20s. becomes 4984 multiplied by (74), or 14952, or 1495.2, which is equal to 74712; and, 12, or 3, of 6s. 8d. is 3 times of 80d. or 48d. or 4s.

3 20

To reduce 137 five franc pieces to pounds, shillings, &c. the franc being valued at $0,1866.

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Reduce 7s. 93d. (o the decimal of a pound. Ans. 0,390625.
Reduce 3qrs. 2na. to the decimal of a yard. Ans. 0,875.
Find the value of 0,852517. in shillings, pence, &c.

Ans. 17s. Od. 21q. nearly.

Reduce 2417. 18s. 9d. to federal money. Ans. $806,4583 &c.
Find the value of 0,42857 of a month.

Ans. 1w. 4d. 23h. 59′ 56". Required the circumference of the earth in English statute miles, a degree being estimated at 57008 toises.

Ans. 24855,488.

We have given rules for reducing a compound number from one denomination to another, as we shall have frequent occasion in what follows for making these reductions. They are not, however, necessary, except in particular cases, previously to performing the fundamental operations. The several denominations of a compound number may be regarded like the different orders of units in a simple one, that is, the number or numbers of

† A toise or French fa thom is equal to 6 French feet, and a French foot is equal to 12,7893 English inches.

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each denomination may be made the subject of a distinct operation, the result of which, being reduced when necessary, may be united to the next, and so on through all the denominations.

Addition of Compound Numbers.

103. THE addition of compound numbers depends on the same principles as that of simple numbers, the object being simply to unite parts of the same denomination, and when a number of these are found, sufficient to form one, or more than one of a higher, these last are retained to be united to others of the same denomination in the given numbers; as in simple addition the tens are carried from one column to the next column on the left. We must, then, place the compound numbers, that are to be added, in such a manner, that their units, or parts of the same name, may stand under each other; we must then find separately the sum of each column, always recollecting how many parts of each denomination it takes to make one of the next higher, See the following example in pounds, shillings, and pence.

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First, adding together the pence, because they are the parts of the least value, and taking together both the units and tens of this denomination, we find 29; but as 12 pence make a shilling, this sum amounts to 2 shillings and 5 pence; we then write down only the 5 pence, and retain the shillings in order to unite them to the column to which they belong.

Next, we add separately the units and the tens of the next denomination; the first give, by joining to them the 2 shillings reserved from the pence, 22; we write down only the two units and retain the two tens for the next column, the sum of which, by this

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