Question

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

Answer 1

Let the fraction be $\frac{p}{q}$

$\frac{p+1}{q+1}=\frac{4}{5}$

$\Rightarrow 5(p+1)=4(q+1)$

$\Rightarrow 5p+5=4q+4$

$\Rightarrow 5p+1=4q$

$\Rightarrow q=\frac{5p+1}{4}$

$\frac{p-5}{q-5}=\frac{1}{2}$

$\Rightarrow 2p-10=q-5$

$\Rightarrow 2p-10=\frac{5p+1-20}{4}$

$\Rightarrow 8p-40=5p-19$

$\Rightarrow 3p=21$

$p=7$

$q=\frac{36}{4}=9$

Fraction $=\frac{7}{9}$.