The ratio of two numbers is $\dfrac{2}{3}$. If 2 is subtracted from the first and 8 from the second, the ratio becomes the reciprocal of the original ratio. Find the numbers.

Let the numbers be x and y.

According to question ratio of numbers = $\dfrac{2}{3}$,

$\Rightarrow \dfrac{x}{y} = \dfrac{2}{3} \\[1em] \Rightarrow x = \dfrac{2y}{3} ……..(1)$

Given,

If 2 is subtracted from the first and 8 from the second, the ratio becomes the reciprocal of the original ratio.

$\Rightarrow \dfrac{x - 2}{y - 8} = \dfrac{3}{2} \\[1em] \Rightarrow 2(x - 2) = 3(y - 8) \\[1em] \Rightarrow 2x - 4 = 3y - 24 \\[1em]$

Substituting value of x from equation (1) in above equation, we get :

$\Rightarrow 2 \times \dfrac{2y}{3} - 4 = 3y - 24 \\[1em] \Rightarrow \dfrac{4y}{3} - 4 = 3y - 24 \\[1em] \Rightarrow -4 + 24 = 3y - \dfrac{4y}{3} \\[1em] \Rightarrow 20 = \dfrac{9y - 4y}{3} \\[1em] \Rightarrow 20 = \dfrac{5y}{3} \\[1em] \Rightarrow y = \dfrac{20 \times 3}{5} \\[1em] \Rightarrow y = 12.$

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

$\Rightarrow x = \dfrac{2 \times 12}{3} \\[1em] = \dfrac{24}{3} \\[1em] = 8.$

Hence, numbers are 8 and 12.