# Multiplication of a Fraction by a Fraction

We will discuss here about the multiplication of a fraction by a fraction.

$$\frac{1}{2}$$ is multiplied by $$\frac{1}{3}$$ or, $$\frac{1}{3}$$ of $$\frac{1}{2}$$

 Suppose this is whole (1)
 The whole figure has been divided into two halves.
 For showing $$\frac{1}{3}$$ of $$\frac{1}{2}$$, it is further sub divided half of the figure into 3 equal parts.
 Whole figure is divided into 6 equal parts. Here the double shaded portion is $$\frac{1}{3}$$ of $$\frac{1}{2}$$ parts.
 Now $$\frac{1}{3}$$ of $$\frac{1}{2}$$ is $$\frac{1}{6}$$ of the whole figure Therefore, $$\frac{1}{3}$$ × $$\frac{1}{2}$$ = $$\frac{1}{6}$$ or, $$\frac{1}{3}$$ × $$\frac{1}{2}$$ = $$\frac{1 × 1}{3 × 2}$$ =  $$\frac{1}{6}$$

Hence we conclude that, when we multiply a fractional number, multiply the numerator of the first fraction by the numerator of the second fraction and the denominator of the first fraction by the denominator of the second fraction. The first product is the numerator and the second product is the denominator of the required product.

The following rules are given below for the multiplication of a fractional number by a fractional number:

(a) Change mixed fraction into improper fraction.

(b) Product of two fractions = (Product of numerators)/(Product of denominators).

(c) Reduce numerator and denominator to the lowest terms.

(d) The answer should be a whole number, a mixed fraction or a proper fraction and never an improper fraction.

[The same rule can be applied for multiplying any number or fraction].

Solved examples on multiplication of a fraction by a fraction:

1. $$\frac{1}{2}$$ × $$\frac{1}{3}$$

= $$\frac{1 × 1}{2 × 3}$$

= $$\frac{1}{6}$$

2. 2$$\frac{1}{2}$$ × $$\frac{1}{3}$$

= $$\frac{2 × 2 + 1}{2}$$ × $$\frac{1}{3}$$

= $$\frac{5}{2}$$ × $$\frac{1}{3}$$

= $$\frac{5 × 1}{2 × 3}$$

= $$\frac{5}{6}$$

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3. 4$$\frac{1}{3}$$ × 2$$\frac{1}{5}$$

= $$\frac{4 × 3 + 1}{3}$$ × $$\frac{2 × 5 + 1}{5}$$

= $$\frac{13}{3}$$ × $$\frac{11}{5}$$

= $$\frac{13 × 11}{3 × 5}$$

= $$\frac{143}{15}$$

= 9$$\frac{8}{15}$$

4. $$\frac{11}{3}$$ × $$\frac{12}{55}$$

= $$\frac{11 × 12}{3 × 55}$$

[Reducing numerator and denominator to the lowest terms]

= $$\frac{4}{5}$$

5. Find the product:

(a) $$\frac{4}{3}$$ × $$\frac{7}{9}$$

=
$$\frac{4 × 7}{3 × 9}$$

= $$\frac{28}{27}$$

(b) 5$$\frac{1}{3}$$ × $$\frac{2}{5}$$

= $$\frac{5 × 3 + 1}{3}$$ × $$\frac{2}{5}$$

= $$\frac{16}{3}$$ × $$\frac{2}{5}$$

= $$\frac{16 × 2}{3 × 5}$$

= $$\frac{32}{15}$$

= 2$$\frac{2}{15}$$