148. Suppose that the given fraction has ciphers at the right of its denominator; for example, ³¹/₂₅₀₀. Then annexing a cipher to the numerator is the same thing as taking one away from the denominator; for, (108) ³¹⁰/₂₅₀₀ is the same thing as ³¹/₂₅₀, and ³¹⁰/₂₅₀ as ³¹/₂₅. The rule, therefore, is in this case: Take away the ciphers from the denominator. EXERCISES. Reduce the following fractions to decimal fractions: 1 36 297 1 ---, ----, ----, and ---. 800 1250 64 128 _Answer_, ·00125, ·0288, 4·640625, and ·0078125. Find decimals of 6 places very near to the following fractions: 27 156 22 194 2637 1 1 3 --, ---, -----, ---, ----, ----, ---, and ---. 49 33 37000 13 9907 2908 466 277 _Answer_, ·551020, 4·727272, ·000594, 14·923076, ·266175, ·000343, ·002145, and ·010830. 149. From (121) it appears, that if two fractions have the same denominator, the first may be divided by the second by dividing the numerator of the first by the numerator of the second. Suppose it required to divide 17·762 by 6·25. These fractions (138), when reduced to a common denominator, are 17·762 and 6·250, or ¹⁷⁷⁶²/₁₀₀₀ and ⁶²⁵⁰/₁₀₀₀. Their quotient is therefore ¹⁷⁷⁶²/₆₂₅₀, which must now be reduced to a decimal fraction by the last rule. The process at full length is as follows: Leave out the cipher in the denominator, and annex ciphers to the numerator, or, which will do as well, to the remainders, when it becomes necessary, and divide as in (145). 625)17762(284192 1250 ----- 5262 5000 ----- 2620 2500 ----- 1200 625 ----- 5750 5625 ----- 1250 1250 ---- 0 Here four ciphers have been annexed to the numerator, and one has been taken from the denominator. Make five decimal places in the quotient, which then becomes 2·84192, and this is the quotient of 17·762 divided by 6·25. 150. The rule for division of one decimal by another is as follows: Equalise the number of decimal places in the dividend and divisor, by annexing ciphers to that which has fewest places. Then, further, annex as many ciphers to the dividend[18] as it is required to have decimal places, throw away the decimal point, and operate as in common division. Make the required number of decimal places in the quotient. [18] Or remove ciphers from the divisor; or make up the number of ciphers partly by removing from the divisor and annexing to the dividend, if there be not a sufficient number in the divisor.