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.? 5980,8 80 6s. 8d. 178 59808 Ans. 747 crowns and 4 shillings. 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 80, 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. To reduce 137 five franc pieces to pounds, shillings, &c. the franc being valued at $0,1796. + 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, s. becomes 4984 multiplied by (74), or 14952, or 1425,2, which is equal to 74712; and 13, or 3, of 6s. 8d. is 3 times of 80d. or 48d. or 4s. 1 3 20 Reduce 7s. 94d. to the decimal of a pound. Ans. 0,390625. Reduce Sqrs. 2na. to the decimal of a yard. Ans. 0,875. Find the value of 0,85251l. in shillings, pence, &c. Ans. 17s. Od. 21q. nearly. Reduce 2411. 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 toisest. 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 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. † A toise or French fathom is equal to 6 French feet, and a French foot is equal to 12,7893 English inches. 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, 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 means, amounts to 5 tens, but as the pound, made up of 20 shil. lings, contains 2 tens, we obtain the number of pounds resulting from the shillings, by dividing the tens of these last by 2; the quotient is 2, and the remainder 1, which last is written under the column to which it belongs, while the pounds are reserved for the next column on the left; as this column is the last the operation is performed as in simple numbers, and the whole sum is found to be 27561. 12s. 5d. The method of proving the addition of compound numbers is derived from the same principles, as that for simple numbers, and is performed in the same manner, care being taken in passing from one denomination to another, to substitute instead of the decimal ratio, the value of each part in the terms of that, which follows it on the right. Let there be, for example, The operation on the pounds is performed according to the rule of article 19; then we change the two pounds into tens of shillings, and obtain 4 of these tens, which, joined to that written under the column, makes 5, from which we subtract the 3 units of this column, and place the remainder, 2, underneath, counting it as tens with regard to the next column. There still remain 2 shillings, which must be reduced to pence; adding the result, 24 pence, to the 5 that are written, we have a total of 29, which must be again obtained by the addition of all the pence, as these are the parts of the lowest denomination in the question. This really happens, and proves the operation to be right. SUBTRACTION OF COMPOUND NUMBERS. 104. THIS operation is performed in the same way as the subtraction of simple numbers, except with regard to the number which it is necessary to borrow from the higher denominations, in order to perform the partial subtractions, when the lower number exceeds the upper. For instance, £ S. d. from 795 3 0 take 684 17 4 Difference 110 5 8 |