The rule just given does not work well when the factors are not commensurable. For example, if it be required to divide 5/7 by 2/9 we have by the above rule -

5 ÷ 2 = 5/2

7 ÷ 9 7/9 .

Producing fractional numerators and denominators for the resulting fraction, which require modification in order to reach those composed only of whole numbers. If the numerators, 5 and 7, of this compound fraction be multiplied by 9 (the denominator of the denominator fraction), or the compound fraction by 9, we shall have -

5/2/7/9x9 = 5x9/2/7x9/9

And, if these be again multiplied by 2 (the denominator of the numerator fraction), we shall have -

Like figures above and below in each fraction cancel each other (Art. 371), therefore, the result reduces to -

5 x 9

7x2'

in which we find the factors of the two original fractions. In one fraction - we have the factors in position as given, but in the other 2/9 they are inverted. The fraction in which the factors are inverted is the divisor. Hence, for division of fractions, we have this -

Rule. - Invert the factors of the divisor, and then, as in multiplication, multiply the numerators together for the numerator of the required fraction, and the denominators for the denominator of the required fraction.

Thus, as before, if 5/7 is required to be divided by 2/9, we have -

5/7x9/2 = 45/14

23 7 And, to divide - by -, we have -

47 9

23 x 9 = 207 47 x 7 329

Again, to divide 25/45 by 8/9, we have -

25 x 9 = 225 = 25 = 5 45 x 8 360 40 8

This last example has two factors, 9 and 45, one of which measures the other; also, the first fraction 25/45 is not in its lowest terms; when reduced it is 5/9. The question, there-fore, may be stated thus:

5x9 5;

9x8 = 8

for the two 9's cancel each other.