Question

The sum of a numerator and denominator of a fraction is $18$. If the denominator increased by $2$, the fraction 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,

$x+y=18and\frac{x}{y+2}=\frac{1}{3}or,x+y=18=>y=18-x....\left(i\right)Also,\frac{x}{y+2}=\frac{1}{3}=>3x=y+2=>3x-y-2=\mathrm{0....}\left(ii\right)$

Solving by substitution method,

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

$3x-y-2=0=>3x-18+x-2=0=>4x=20=>x=5$

Putting $x=5$ in equation (i), we get,

$y=18-x=>y=18-5=>y=13$

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