A box contains some black balls and 30 white balls. If the probability of drawing a black ball is two-fifths of a white ball; find the number of black balls in the box.

Let the box contain x black balls.

Total number of balls = x + 30

∴ No. of possible outcomes = x + 30.

No. of favourable outcomes (for drawing a black ball) = x

P(drawing a black ball) = $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{x}{x + 30}$

No. of favourable outcomes (for drawing a white ball) = 30

P(drawing a white ball) = $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{30}{x + 30}$

Given,

Probability of drawing a black ball is two-fifths of a white ball,

$\therefore \dfrac{x}{x + 30} = \dfrac{2}{5} \times \dfrac{30}{x + 30} \\[1em] \Rightarrow \dfrac{x}{x + 30} = \dfrac{12}{x + 30} \\[1em] \Rightarrow x = \dfrac{12}{x + 30} \times (x + 30) \\[1em] \Rightarrow x = 12.$

Hence, the number of black balls = 12.