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 $\frac{x}{y}$.

Then, according to the given conditions, we have

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

$\Rightarrow 5x+5=4y+4$ and $2x-10=y-5$

$\Rightarrow 5x-4y+1=0$ and $2x-y-5=0$

By using cross-multiplication, we have

$\frac{x}{-4\times -5-(-1)\times 1}=\frac{-y}{5\times -5-2\times 1}=\frac{1}{5\times -1-2\times -4}$

$\Rightarrow \frac{x}{20+1}=\frac{y}{25+2}=\frac{1}{-5+8}$

$\Rightarrow \frac{x}{21}=\frac{y}{27}=\frac{1}{3}\Rightarrow x=\frac{21}{3}=7$ and $y=\frac{27}{3}=9$.

$\therefore$ fraction is $\frac{x}{y}=\frac{7}{9}$