# Multiplication of a Whole Number by a Fraction

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

Now let us learn the multiplication of fractional numbers.

Let us suppose 6 is multiplied by $$\frac{1}{3}$$

represent 1 ................                       represent 6 ................

$$\frac{1}{3}$$ of 6 means 2

or, $$\frac{1}{3}$$ of 6 = $$\frac{1}{3}$$ + $$\frac{1}{3}$$ + $$\frac{1}{3}$$ + $$\frac{1}{3}$$ + $$\frac{1}{3}$$ + $$\frac{1}{3}$$

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

= $$\frac{6}{3}$$

= 2

$$\frac{1}{3}$$ of 6 = $$\frac{1}{3}$$ × 6 = 2, the shaded portion.

Fraction of a whole number = $$\frac{\textbf{Numerator of fraction}}{\textbf{Denominator of fraction}}$$ ∙ $$\frac{\textbf{Whole number}}{1}$$

Hence we conclude that, to multiply a fractional number by a whole number we multiply the numerator of the fractional number by the whole number and the denominator of the fractional number by 1. The first product thus obtained is the numerator and the second product is the denominator of the required product.

Solved examples on multiplication of a whole number by a fraction:

1. Multiply the following:

(i) $$\frac{17}{21}$$ by 7.

= $$\frac{17}{21}$$ × 7

= $$\frac{17 × 7}{21 × 1}$$

= $$\frac{17 × 1}{3 × 1}$$

= $$\frac{17}{3}$$

= 5$$\frac{2}{3}$$

(ii) $$\frac{2}{9}$$ by 3

= $$\frac{2}{9}$$ × 3

= $$\frac{2 × 3}{9 × 1}$$

= $$\frac{2 × 1}{3 × 1}$$

= $$\frac{2}{3}$$

2. Find the product:

(i) $$\frac{2}{3}$$ × 5

= $$\frac{2 × 5}{3 × 1}$$

= $$\frac{10}{3}$$

= 3$$\frac{1}{3}$$

(ii) 1$$\frac{2}{9}$$ × 5

= (1 + $$\frac{2}{9}$$) × 5

= $$\frac{9 + 2}{9}$$ × 5

= $$\frac{11}{9}$$ × 5

= $$\frac{11 × 5}{9 × 1}$$

= $$\frac{55}{9}$$

= 6$$\frac{1}{9}$$

(iii) 3$$\frac{5}{6}$$ × 4

= $$\frac{23}{6}$$ × 4

= $$\frac{23 × 4}{6 × 1}$$

= $$\frac{23 × 2}{3 × 1}$$

= $$\frac{46}{3}$$

= 15$$\frac{1}{3}$$