Two numbers are in the ratio 4 : 5. If 30 is subtracted from each of the numbers, the ratio becomes 1 : 2. Find the numbers.

Let two numbers be x and y.

Given,

Numbers are in the ratio 4 : 5.

$\Rightarrow \dfrac{x}{y} = \dfrac{4}{5} \\[1em] \Rightarrow x = \dfrac{4y}{5} …….(1)$

Given,

If 30 is subtracted from each of the numbers, the ratio becomes 1 : 2.

$\Rightarrow \dfrac{x - 30}{y - 30} = \dfrac{1}{2} \\[1em] \Rightarrow 2(x - 30) = y - 30 \\[1em] \Rightarrow 2x - 60 = y - 30 \\[1em] \Rightarrow 2x = y - 30 + 60 \\[1em] \Rightarrow 2x = y + 30 \\[1em] \Rightarrow x = \dfrac{y + 30}{2} …….(2)$

From (1) and (2), we get :

$\Rightarrow \dfrac{4y}{5} = \dfrac{y + 30}{2} \\[1em] \Rightarrow 4y \times 2 = 5(y + 30) \\[1em] \Rightarrow 8y = 5y + 150 \\[1em] \Rightarrow 8y - 5y = 150 \\[1em] \Rightarrow 3y = 150 \\[1em] \Rightarrow y = \dfrac{150}{3} \\[1em] \Rightarrow y = 50.$

Substituting value of y in equation (1), we get :

$\Rightarrow x = \dfrac{4 \times 50}{5} \\[1em] = \dfrac{200}{5} \\[1em] = 40.$

Hence, numbers are 40 and 50.