Question

If $2$ is added to each one of the two given number,then their ratio becomes $1:2$ However if $4$ is subtracted from each of the given number,the ratio becomes $5:11$.Find the number.

Answer 1

Let, the two numbers are $x$ and $y$.

According to the first conditions,

$\frac{x+2}{y+2}=\frac{1}{2}\Rightarrow 2x-y=-2⟶\left(1\right)$

and according to the second condition,

$\frac{x-4}{y-4}=\frac{5}{11}\Rightarrow 11x-5y=24⟶\left(2\right)$

Multiplying equation $\left(1\right)$ by 5 we get,

$10x-5y=-10⟶\left(3\right)$

Subtracting equation $\left(2\right)$ from equation $\left(3\right)$,

$\Rightarrow 11x-5y-10x+5y=24+10$

$\Rightarrow x=34$

Putting the value of $x$ in equation $\left(1\right)$,

$\Rightarrow 2\times 34-y=-2\Rightarrow 68-y=-2\Rightarrow y=70$

Hence, the required numbers are $34,70.$