Question

If $2$ is added to the numerator of a fraction, it reduces to $\frac{1}{2}$ and if $1$ is subtracted from denominator, it reduces to $\frac{1}{3}$. Find the fraction.

Answer 1

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

From the given information, we have,

$\frac{x+2}{y}=\frac{1}{2}and\frac{x}{y-1}=\frac{1}{3}or,\frac{x+2}{y}=\frac{1}{2}=>2(x+2)=y=>y=2x+\mathrm{4....}\left(i\right)Also,\frac{x}{y-1}=\frac{1}{3}=>3x=y-\mathrm{1....}\left(ii\right)$

Using substitution method,

Substituting equation (i) in equation (ii), we get,

$3x=2x+4-1=>x=3$

Substituting $x=3$ in equation (i), we get,

$y=2x+4=>y=2\left(3\right)+4=>y=6+4=>y=10$

Thus, the original fraction is $\frac{x}{y}=\frac{3}{10}$