Question

If the numerator of a fraction is multiplied by $2$ and its denominator is increasing by $1$, it becomes $1$.However if the numerator is increasing by $4$ and denominator is multiplied by $2$, the fraction becomes $\frac{1}{2}$. Find the fraction.

Answer 1

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

case 1 $\frac{x\times 2}{y+1}=1$

$\Rightarrow 2x=y+1$

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

case 2

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

$\Rightarrow x+4=y$

$\Rightarrow y=x-4$

from eq 1 &2

$x+4=2x-1$

$x=5$

from eq 2

$y=x+4=5+4=9$

Hence the required fraction is $\frac{5}{9}$