Question

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

Answer 1

let the first number be x and second be y

$\frac{x}{y}=\frac{2}{3}$ $\Rightarrow 3x=2y...\left(1\right)$

$\Rightarrow \frac{x-2}{y-8}=\frac{3}{2}$

$\Rightarrow 2x-4=3y-24$

$\Rightarrow 2x-3y=-20$

$\Rightarrow x=\frac{3y-20}{2}$

put the value of x in equation (1)

$3(3y-20)=4y$

$9y-60=4y$

$5y=60$

$y=12$

put $y=12$ in equation (1)

$x=8$

Numbers are $8$ and $12$.