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202. When a numerator and denominator are the same, both may be omitted in the operation, a small line being drawn through the figures omitted; this operation is called cancelling".
3 24. Multiply A and : together.
3 3 Thus, + x # = 7 the product. Here we left out the 4,
* The numerator of a fraction may be considered as a multiplier, and the denominator as a divisor; it is plain, that if any number be both multiplied and divided by any (the same) number, the result will be the same as the given number; wherefore such multiplication and division (as they mutually destroy the effect of each other) may be omitted: which is what the rule directs. -
It may be further observed, that one numerator cancels only one equal denominator, and vice versa; but it will cancel two or more when it is equal to their product: thus, in ex. 26. the numerator 4 cancels the denominator 4 only, but the denominator 6 cancels both the numerators 2 and 3, because it equals 2 × 3.
Here the threes, fours, fives, and sizes 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'.
OPERATION. - Erplanation. 2 4 - Here 8 and 36 both divide by 4; I dash ? 8 out these numbers, and write the quotients Thus, $ X ** = + prod. 2 and 9 opposite its respective number: 9 2% 27 also 12 and 9 divide by 3; I cancel these, 3 9 and write down the quotients 4 and 3; then
I multiply 2 by 4 and 3 by 9 for the answer.
Erplanation. 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.
f The truth of this will appear by multiplying the given fractions together, and reducing the product to its lowest terms; thus, ex. 29. 8 X * = 26. 9 36 324
which reduced to its sowest terms is 8, the same with the answer obtained by 27
the rule. This method of cancelling, when it can be applied, saves much trouble.
2 251 36. Multiply of +. *; and # together. Prod. His
DIVISION OF VULGAR FRACTIONS.
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 # is re
quired to be divided by 5. . First, suppose 3 were to be divided by 5 only, 7 4
divisor # is only one seventh part of 5, and therefore the quotient # , arising
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 x 7
20 and this quotient arises by multiplying 3 by 7, and 4 by 5, or by inverting the divisor, as has been shewn.
2I T 20 (as in the example) is the true quotient of- divided by-#3
205. When the numerator of the divisor will divide the numerator of the dividend, and the denominator divide the denominator, both without remainder; them (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".
OPERATIon. - Earplanation.
12 4 3 I put the sign of division beThus, H -- -- = + quotient. tween the fractions, then divide
35 5 7 12 by 4, and 35 by 5.
* 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.