26. Multiply 2 3 4 5 6 and together. Here the threes, fours, fives, and sixes being omitted, the 203. When a numerator and a denominator can be divided by any (the same) number, draw a small line through the numbers, and use the quotients instead of them. Here 12 and 18 are divisible by 6, which goes twice in the former and 3 times in the latter; I therefore put 2 opposite 12, and 3 opposite 18; 12 and 18 are therefore cancelled: in like manner 13 cancels 26 by 2, and 3 cancels 6 by 2; the ones need not be put down; I therefore multiply 2, 2, 11, and 2 together for the numerator, and 27 and 7 for the denominator. 8 f The truth of this will appear by multiplying the given fractions together, and reducing the product to its lowest terms; thus, ex. 29. X 9 12 96 36 324 the rule. This method of cancelling, when it can be applied, saves much trouble. 1 3 4 1 2 34. Multiply 4- 2- and together. Prod. 7. 7 14 204. To divide one fraction by another. RULE. Invert the divisor, and then multiply the fractions (with the divisor so inverted) together, as in Art. 198. Mixed numbers must be reduced to improper fractions, and complex fractions to simple ones, previous to the operation". The reason of this rule will appear from the first example, where 3 is only one seventh part of 5, and therefore the quotient arising 20 from a divisor seven times too great, must be seven times less than it ought to be, and consequently must be multiplied by 7 to make it right; wherefore 3 5 (as in the example) is the true quotient of divided by ; 4 7 20 20 and this quotient arises by multiplying 3 by 7, and 4 by 5, or by inverting the divisor, as has been shewn. 205. When the numerator of the divisor will divide the numerator of the dividend, and the denominator divide the denominator, both without remainder; then (without inverting the divisor) divide numerator by numerator, and denominator by denominator, and the quotients will form a fraction, which (reduced if necessary) will be the answer. The truth of this process will appear by working the examples included under this rule by the former rule, as in both cases the same result will be produced; and it is recommended to prove every operation in this by the former rule. 206. To divide a whole number by a fraction. RULE. Invert the fraction, and multiply the numerator of the inverted fraction by the whole number, under the product place the denominator, and reduce this fraction (if necessary) for the answer i. i If I be put for a denominator to the whole number, it will become a fraction, and the divisor being inverted, this rule will coincide with the first rule in Division of Fractions, and is consequently founded on the same principles. |