Question

Two numbers are in the ratio $2:3.$ If 2 is subtracted from the first number and 8 from the second number, the ratio becomes the reciprocal of the original ratio. Find the numbers.

• A. $4,6$
• B. $6,4$
• C. $2,3$
• D. $8,12$

Answer 1

The correct option is D $8,12$

Given, two numbers are in the ratio $2:3.$
Let the numbers be $2x,3x.$

Subtract 2 from the first number $\to 2x-2$
Subtract 8 from the second number $\to 3x-8$

Given, the new ratio is the reciprocal of $2:3.$
$\Rightarrow 2x-2:3x-8=3:2$
$\Rightarrow \frac{2x-2}{3x-8}=\frac{3}{2}$
$\Rightarrow 2(2x-2)=3(3x-8)$
$\Rightarrow 4x-4=9x-24$
$\Rightarrow 24-4=9x-4x$
$\Rightarrow 20=5x$
$\Rightarrow x=\frac{20}{5}$
$\Rightarrow x=4$

$\therefore 2x=2\times 4=8,3x=3\times 4=12$
Hence, the numbers are $8$ and $12.$