Question

A fraction becomes $\frac{1}{3}$, if $1$ is subtracted from both its numerator and denominator. If $3$ is added to both the numerator and the denominator, it becomes $\frac{1}{2}$. Find the fraction.

Answer 1

Let the numerator be $x$ and the denominator be $y$.

From the given information, we have,

$\frac{x-1}{y-1}=\frac{1}{3}and\frac{x+3}{y+3}=\frac{1}{2}or,3(x-1)=1(y-1)=>3x-3=y-1=>3x-y-2=0Also,2(x+3)=1(y+3)=>2x+6=y+3=>2x-y+3=0$\\

Using cross-multiplication method,

$\frac{x}{(-1)\left(3\right)-\left(-1\right)(-2)}=\frac{y}{(-2)\left(2\right)-\left(3\right)\left(3\right)}=\frac{1}{\left(3\right)\left(-1\right)-\left(2\right)(-1)}\frac{x}{-3-2}=\frac{y}{-4-9}=\frac{1}{-3+2}\frac{x}{-5}=\frac{y}{-13}=\frac{1}{-1}=>\frac{x}{-5}=\frac{1}{-1}or,x=5=>\frac{y}{-13}=\frac{1}{-1}or,y=13$

Thus, the original fraction is $\frac{x}{y}=\frac{5}{13}$