REDUCTION OF FRACTIONS TO LOWER TERMS.

115. If the numerator and denominator of a fraction be both multiplied or both divided by the same number, the value of the fraction is not altered.

Thus, if the line AB be divided into 5 equal parts at the points C, D, E, and F, then AF is of AB.

Now, if each of the parts be sub-divided into 3 equal parts, AB will contain 15 of these sub-divisions, and AF 12 of these sub-divisions. Therefore AF is 1 of AB.

Since AF is of AB and also 13 of AB, it follows that 1. But is obtained from 1 by dividing both numerator and denominator by 3. Therefore, .

To reduce a fraction to lower terms,

Divide the numerator and denominator by any common factor.

A fraction is expressed in its lowest terms when both the numerator and denominator are divided by the greatest common divisor.

The common factors cancelled are 4, 4, and 7.

Reduce to its lowest terms.

Since no common factor can readily be detected, we find the G.C.M.

259) 333 (1

259

74) 259 (3

222

37) 74(2

74

Divide 259 and 333 each by 37, their G.C.M. Then 353 = 25.

Reduce to lowest terms by the method of inspection or

by the method of finding the G.C.M.

NOTE. In the answers to all examples, fractions should be left in

their lowest terms.

If three like quantities are taken 7 times, the result will be 7 times 3 of the same quantities.

15 means of 15, which equals 9. Hence,

117. To find the product of a whole number and a fraction,

Find the product of the numerator and whole number, and divide the result by the denominator.

A factor common to the whole number and the denominator of the fraction may be cancelled. For, cancelling a factor common to the whole number and the denominator of the fraction before the multiplication, is evidently equivalent to dividing the numerator and denominator of the resulting fraction by that factor after the multiplication. Which may be done by ? 115.

If the line AB be divided into 5 equal parts at the points C, D, E, and F, AF will be of AB. Now, if each part be sub-divided into three equal parts, there will evidently be 15 such parts in the whole line, and each part will be of the line.

That is, of is of the whole. will be

of

And of

Therefore,

+++, or will be twice; that is,

of the whole.

of the whole.

To multiply a fraction by a fraction,

Find the product of the numerators for the required numerator, and of the denominators for the required denominator.

Mixed numbers must first be reduced to improper fractions.

Any factor common to a numerator and denominator should be cancelled before the multiplication.

Reducing the mixed number to an improper fraction, we