To multiply fractions we must make the product of their numerators and denominators and simplify the resulting fraction, if possible. The product of the fractions results in one that is less than the fractions that were multiplied.

To divide fractions we multiply the first by the reciprocal of the second, in other words, we multiply the dividend by the inverted divisor. Then, if necessary, we simplify the resulting fraction.

**Multiplication of fractions**

**General rules for multiplying fractions**

- The terms of the fractions we want to multiply are decomposed into factors (as much as possible).
- It is simplified by removing the common factors in the numerators and denominators.
- The numerators are multiplied together and the result is written as the numerator of the resulting fraction.
- The denominators are multiplied together and the result is written as the denominator of the resulting fraction.
- The resulting fraction is simplified, if possible.

**Example**: multiply the following fractions: 6/5; 7/4

6/5 × 7/4 = (6 × 7) / (5 × 4) = 42/20

We simplify the fraction:

42/20 = 21/10

**Division of fractions**

**General rules for dividing fractions**

- The terms of the fractions we want to divide are factorized (as much as possible).
- It is simplified by removing the common factors in the numerators and denominators.
- The first fraction is multiplied by the reciprocal of the second.
- The resulting fraction is simplified, if possible.

**Example**: divide the following fractions: 15/7; 9/4

The reciprocal of the second fraction is: 4/9

Thus:

7/15 ÷ 9/4 = 15/7 × 4/9 = (15 × 4) / (7 × 9) = 60/63

We simplify the fraction:

60/63 = 20/21

**Exercises**

Perform the following operations:

- [(2nd
^{2}) / 3b] × [(6b^{2}) / 4a]

[(2nd^{2}) / 3b] × [(6b^{2}) / 4a] = [(2a^{2}) × (6b^{2})] / [(3b) × (4a)] = (12a^{2}b^{2}) / 12ab = ab

- [(x
^{3 }- x) / (2x^{2 }+ 6x)] ÷ [(5x^{2 }- 5x) / (2x + 6)]

[(x^{3 }- x) / (2x^{2 }+ 6x)] ÷ [(5x^{2 }- 5x) / (2x + 6)] = [(x^{3 }- x) / (2x^{2 }+ 6x)] × [(2x + 6) / (5x^{2 }- 5x)] =

= [x (x^{2} - 1) / 2x (x + 3)] × [2 (x + 3) / 5x (x - 1)] = = [2 (x^{2 }- 1) (x + 3)] / [10 (x + 3) (x - 1)] =

Like x^{2 }- 1 = (x + 1) (x - 1)

= [2 (x^{2 }- 1) (x + 3)] / [10 (x + 3) (x - 1)] = (x + 1) / 5