The reason of this may be easily explained-The decimal 0.25 is 25-100 ths, for which we take first 25 times, and then the 100 th part of the product; hence we derive the directions of the latter part of the rule, because to allow for the decimal places in the multiplier, is actually to divide by 10, by 100, &c. according to the number of those places, or the denominator of the decimal fraction. EXAMPLE 4. To multiply £42.641 by 3.27. £ 42.641 3.27 298487 85282 127923 £139.43607 There being 3 places of decimals in the multiplicand, and 2 in the multiplier, we cut off 5 places in the product; we cut off 3 places, because if the multiplier were a whole number 13943.607 would be the value of the product, on the principle of Addition; and we cut off 2 places more, because by using the multiplier as a whole number, viz. 327, while its real value is only the 100 th part, or 3.27, we have the product, at first, 100 times its proper amount; and therefore to correct it, we take the 100 th part of the 1 st product by removing the decimal point 2 places more to the left. There being 1 place of decimals in the multiplicand, and 1 in the multiplier, we must have 2 places of decimals in the product; which is effected by placing one 0 before the 6, and then prefixing the decimal point. The operation is to take 2-10 ths of 3-10 ths of a £; now twice 3 tenths are 6 tenths, and the tenth part of 6 tenths is 6-100 ths. EXAMPLE 6. To multiply £0.0024 by 0.24. £ 0.0024 0.24 £ 0.000576 There being 6 places of decimals in the two factors, we make 6 places of decimals in the product, by prefixing 3 ciphers to the 576 before we place the decimal point. DECIMAL DIVISION. Rule. Divide as in Division of whole numbers, paying no regard in the operation to any distinction between whole numbers and decimals. For the valuation of the quotient, it is to be observed, that when the divisor is a whole number, the value of each quotient figure will be the same as the lowest dividend figure used to procure that quotient figure; and when the divisor is not a whole number, it may be made such, by cancelling the decimal point in the divisor, and correcting the dividend by removing the decimal point so many places to the right, as there are places in the given divisor.* EXAMPLE 1. To divide £ 3764 by 10, by 100, and by 1000. £ 3.764 To divide by 10 we remove the decimal point one place to the left, by 100 two places, and by 1000 three places. EXAMPLE 2. To divide 37.64 by 7. 7) 37.64 5.37714 &c. The number of places of decimals to which the quotient should extend must depend upon the purpose for which it is made.Three places are sufficient in the division of Pounds of money, when the quotient is to be valued to the nearest penny or farthing; for other purposes, and particularly when there are whole numbers, 5 places are the most that are usually necessary. The ciphers for continuing the division may be used, but they are not necessary. The quotient may otherwise be valued by cutting off so many places on the right of the quotient, as the decimal places in the dividend, including the ciphers used to continue the operation, exceed those in the divisor. The divisor not being contained in the 37 units, there are not any units in the quotient; and the same being the case with the 376 tenths, there are not any tenths in the quotient; the first quotient figure, 4 hundredths, is produced from the 3764 hundredths. It may here be observed, that in the division of a decimal quantity by a whole number, after the whole number in the quotient has been obtained, instead of extending the quotient decimally, the remainder is sometimes required to be reduced and divided according to the ordinary form of division; and care must then be taken, that the whole of the remainder, including the decimal, is multiplied by the reducing multiplier. EXAMPLE 4. To divide £ 872 by 0.4. £ 4.) 8720. £ 2180. As there is 1 decimal place in the divisor, and none in the dividend, the dividend is corrected by annexing a cipher, and then removing the decimal place (which in decimal calculations is always considered to be on the right of all whole numbers) one place to the right; the divisor is then considered a whole number. The reason of this removal of the decimal place in the dividend and then calling the divisor a whole number, is, evidently, to multiply both terms by the denominator of the decimal, and thus to substitute an equivalent but easier operation, producing the same result; thus, as in the above, instead of dividing £ 872 by 4-10 ths we divide £ 8720 by 4; and, as in the next example, instead of dividing £ 837.64 by 6.17 we take 100 times each term, and divide £ 83764 by 617. |