Question

A fraction becomes $\frac{4}{5}$ if 1 is added to each of the numerator and denominator. However, if we subtract 5 from each, the fraction becomes $\frac{1}{2}$. Find the fraction.

Answer 1

Let the required fraction be $\frac{x}{y}$ where $x$ be the numerator and $y$ be the denominator.
First Case :
According to the question,
$\frac{x+1}{y+1}=\frac{4}{5}$
$\Rightarrow$ $5x+5=4y+4$
$\Rightarrow$ $5x-4y=-1$
Second Case : 5 is subtracted from x and y
So, $\frac{x-5}{y-5}=\frac{1}{2}$
$\Rightarrow$ $2x-10=y-5$
$\Rightarrow$ $2x-y=5$ ..... (2)
Multiplying equation (2) by 4 and equation (1) by 1, we get
$5x-4y=-1$ ..... (3)
$8x-4y=20$ .... (4)
Subtracting (4) from (3), we get
$-3x=-21$
$\Rightarrow$ $x=7$
Substituting the value of x in (2) we get
$2\times 7-y=5$
$\Rightarrow$ $y=9$
So, $\frac{x}{y}=\frac{7}{9}$
Hence, the reqired fraction is $\frac{7}{9}$.