Question

The ratio of two nubers 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.

• A. $2$ and $14$
• B. $8$ and $12$
• C. $13$ and $6$
• D. $4$ and $17$

Answer 1

The correct option is B $8$ and $12$

Let the ratio be $\frac{x}{y}=\frac{2}{3}$
$\Rightarrow 3x=2y$
$\Rightarrow 3x-2y=0$ --- (1)
Given, if $2$ is subtracted from $x$ and $8$ from $y$, then ratio $=\frac{x-2}{y-8}=\frac{3}{2}$

$\Rightarrow 2x-4=3y-24$
$\Rightarrow 2x-3y=-20$ --- (2)

Multiplying (1) with (2), we get,

$6x-4y=0$ ----(3)

Multiplying (2) with (3), we get,

$6x-9y=-60$ -----(4)

Subtracting (4) from (3), we get

$y=12$

Substituting $y=12$ in the (1), we get

$x=8$

Hence, the numbers are $8$ and $12$