1. There must be as many decimals in the dividend, as in both the divisor and quotient; therefore point off for decimals in the quotient so many figures, as the decimal places in the dividend exceed those in the divisor. 2. If the figures in the quotient are not so many as the rule requires, supply the defect by prefixing cyphers. 3. If the decimal places in the divisor be more than those in the dividend, add cyphers as decimals to the dividend, till the number of decimals in the dividend be equal to those in the divisor, and the quotient will be integers till all these decimals are used. And, in case of a remainder, after all the figures of the dividend are used, and more figures are wanted in the quotient, annex cyphers to the remainder, to continue the division as far as necessary. 4. The first figure of the quotient will possess the same place of integers or decimals, as that figure of the dividend, which stands over the units place of the first product. EXAMPLES. 1. Divide 3424.6056 by 43.6. 43.6)3424-6056(78.546 3052 3726 3488 2380 2180 2005 1744 2616 2616 2. Divide 3877875 by ⚫675. 3. Divide 0081892 by ⚫347. 4. Divide 7-13 by ⚫18. Ans. 5745000. Ans. 39. CONTRACTIONS. I. If the divisor be an integer with any number of cyphers at the end; cut them off, and remove the decimal point in the dividend so many places farther to the left, as there were cyphers cut off, prefixing cyphers, if need be; then proceed as before, Here I first divide by 3, and then by 7, because 3 times 7 is 21. 2. Divide 41020 by 32000. Ans. 1.281875. NOTE. Hence, if the divisor be 1 with cyphers, the quotient will be the same figures with the dividend, having the decimal point so many places farther to the left, as there are cyphers in the divisor. 217.3 EXAMPLES. 100=2.173. 5.16 by 1000='00516. 419 by 10=41.9. •21 by 1000=00021. II. When the number of figures in the divisor is great, the operation may be contracted, and the necessary number of decimal places obtained. RULE. 1. Having, by the 4th general rule, found what place of decimals or integers the first figure of the quotient will possess; consider how many figures of the quotient will serve the present purpose; then take the same number of figures on the left of the divisor, and as many of the dividend figures as will contain them less than ten times; by these find the first figure of the quotient. 2. And for each following figure, divide the last remainder by the divisor, wanting one figure to the right more than before, but observing what must be carried to the first product for such omitted figures, as in the contraction of Multiplication; and continue the operation till the divisor is exhausted. 3. When there are not so many figures in the divisor, as are required to be in the quotient, begin the division with all the figures as usual, and continue it till the number of figures in the divisor and those remaining to be found in the quotient be equal; after which use the contraction. EXAMPLES. 1. Divide 2508-928065051 by 92-41035, so as to have four decimals in the quotient.-In this case, the quotient will contain six figures. Hence 2. Divide 721-17562 by 2.257432, so that the quotient may contain three decimals. Ans. 319-467. 3. Divide 12.169825 by 3.14159, so that the quotient may contain five decimals. Ans. 3.87377. 4. Divide 87.076326 by 9.365407, and let the quotient contain seven decimals. Ans. 9-2976559. REDUCTION OF DECIMALS. CASE 1. To reduce a vulgar fraction to its equivalent decimal. RULE.* Divide the numerator by the denominator, annexing as many cyphers as are necessary; and the quotient will be the decimal required. EXAMPLES. 1. Reduce to a decimal. 4)5.000000 6)1.250000 •208333, &c: 2. Required the equivalent decimal expressions for,, and 3. Ans. 25, 5, and 75. Let the vulgar fraction, whose decimal expression is required, be Now since every decimal fraction has 10, 100, 1000, &c. for its denominator; and, if two fractions be equal, it will be, as the denominator of one is to its numerator, so is the denominator of the other to its numerator; therefore 13: 7x1000, &c. 7:: 1000, &c.: 13 7000, &c. =53846, the nume rator of the decimal required; and is the same as by the rule. |