Two numbers are in the ratio 8 : 13. If 14 is added to each of the numbers, the ratio becomes 2 : 3. Find the numbers.

Let the numbers be 8x and 13x.

Given,

If 14 is added to each of the numbers, the ratio becomes 2 : 3.

$\Rightarrow \dfrac{8x + 14}{13x + 14} = \dfrac{2}{3} \\[1em] \Rightarrow (8x + 14) \times 3 = (13x + 14) \times 2 \\[1em] \Rightarrow 24x + 42 = 26x + 28 \\[1em] \Rightarrow 26x - 24x = 42 - 28 \\[1em] \Rightarrow 2x = 14 \\[1em] \Rightarrow x = \dfrac{14}{2} \\[1em] \Rightarrow x = 7.$

The two numbers are 8x and 13x

= 8 × 7, 13 × 7

= 56, 91.

Hence, the numbers are 56 and 91.