Question

Find the fraction which becomes $\frac{1}{2}$ when the denominator is increased by $4$, and $\frac{1}{8}$ when the numerator is diminished by $5$

Answer 1

Let the fraction be $\frac{x}{y}$.

The fraction becomes $\frac{1}{2}$ when denominator is increased by $4$.

Therefore, $\frac{x}{y+4}=\frac{1}{2}$

$\Rightarrow 2x=y+4$

$2x-y=4$ ....(1)

Now, the fraction is equal to$\frac{1}{8}$ when numerator is diminished by $5$.

Therefore, $\frac{x-5}{y}=\frac{1}{8}$

$\Rightarrow 8x-40=y$

$\Rightarrow 8x-y=40$ ....(2)

Subtracting equations (1) and (2), we get

$6x=36$

$\Rightarrow x=6$

Putting this value in equation (1), we get

$2\left(6\right)-y=4$

$\Rightarrow 12-y=4$

$\Rightarrow y=8$

Therefore, the fraction is $\frac{6}{8}$.