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 two numbers be x and y.

Given, ratio of two numbers is $\dfrac{2}{3}$.

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

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

$\therefore \dfrac{x - 2}{y - 8} = \dfrac{3}{2}$

Substituting value of x from (i) in above equation,

$\Rightarrow \dfrac{\dfrac{2y}{3} - 2}{y - 8} = \dfrac{3}{2} \\[1em] \Rightarrow \dfrac{\dfrac{2y - 6}{3}}{y - 8} = \dfrac{3}{2} \\[1em] \Rightarrow \dfrac{2y - 6}{3(y - 8)} = \dfrac{3}{2} \\[1em] \Rightarrow 2(2y - 6) = 3(y - 8) \times 3 \\[1em] \Rightarrow 4y - 12 = 9y - 72 \\[1em] \Rightarrow 9y - 4y = 72 - 12 \\[1em] \Rightarrow 5y = 60 \\[1em] \Rightarrow y = \dfrac{60}{5} \\[1em] \Rightarrow y = 12.$

Substituting value of y in eq. (i),

$x = \dfrac{2y}{3} = \dfrac{2 \times 12}{3}$ = 8.

Hence, the numbers are 8 and 12.