Question

The numerator of a fraction is $4$ less than the denominator. If $1$ is added to both its numerator and denominator, it becomes $\frac{1}{2}$. Find the fraction.

Answer 1

Let the denominator of a fraction be $x$.

Then the numerator $=x-4$

Original fraction $=\frac{x}{x-4}$

When $1$ is added to both numerator and denominator, then

New fraction $=\frac{(x-4)+1}{x+1}=\frac{x-3}{x+1}$

It is given that the value of the new fraction is $\frac{1}{2}$.

$\therefore \frac{x-3}{x+1}=\frac{1}{2}$

$\Rightarrow 2(x-3)=1(x+1)$

$\Rightarrow 2x-6=x+1$

$\Rightarrow 2x-x=1+6$

$\Rightarrow x=7$

So, denominator $=x=7$

and, numerator $=x-4=7-4=3$

Hence, the original fraction is $\frac{3}{7}$