Question

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

Answer 1

Calculating the numbers :

Let us consider the two numbers as $x$ and $y$ respectively.

$$\begin{gathered} \Rightarrow \frac{x}{y}=\frac{2}{3} \\ \Rightarrow 3x=2y \end{gathered}$$

$\Rightarrow 3x-2y=0...............\left(1\right)$

Given, if $2$ is subtracted from $x$ and $8$ from $y$,

$\begin{array}{rcl}& \Rightarrow & \frac{x-2}{y-8}=\frac{3}{2}\\ & \Rightarrow & 2x-4=3y-24\end{array}$

$\begin{array}{rcl}& \Rightarrow & 2x-3y=-20............\left(2\right)\end{array}$

Multiplying $\left(1\right)$ with $\left(2\right)$, we get

$6x-4y=0............\left(3\right)$

Multiplying $\left(2\right)$ with $\left(3\right)$, we get

$6x-9y=-60.............\left(4\right)$

Step 3: Solve the equations

On solving these equations we get

$5y=60$

$y=12$

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

$\Rightarrow 3x-2y=0$

$\begin{array}{rcl}& \Rightarrow & 3x-(2\times 12)=0\end{array}$

$\begin{array}{rcl}& \Rightarrow & 3x-24=0\end{array}$

$\begin{array}{rcl}& \Rightarrow & 3x=24\end{array}$

$\begin{array}{rcl}& \Rightarrow & x=8\end{array}$

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