Question

A fraction becomes $\frac{4}{5}$ if $1$ is added to both numerator and denominator. If, however, $5$ is subtracted from both numerator and denominator, the fraction becomes $\frac{1}{2}$. What is the fraction?

Answer 1

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

$\Rightarrow \frac{x+1}{y+1}=\frac{4}{5}$

$\Rightarrow 5x+5=4y+4$

$\Rightarrow y=\frac{5x+1}{4}$

$\Rightarrow \frac{x-5}{y-5}=\frac{1}{2}$

$\Rightarrow \frac{x-5}{\frac{5x+1}{4}-5}=\frac{1}{2}$

$\Rightarrow \frac{4(x-5)}{5x-19}=\frac{1}{2}\Rightarrow 8x-40=5x-19\Rightarrow x=7\Rightarrow y=\frac{5\times 7+1}{4}=9$

Fraction is $\frac{7}{9}$