Question

A fraction becomes $\frac{1}{3}$, if 2 is added to both of its numerator and denominator. If 3 is added to both of its numerator and denominator, then it becomes $\frac{2}{5}$. Find the fraction.

Answer 1

Let the required fraction be $\frac{x}{y}$.
Then, we have:
$\frac{x+2}{y+2}=\frac{1}{3}$
⇒ 3(x + 2) = 1(y + 2)
⇒ 3x + 6 = y + 2
⇒ 3x − y = −4 ....(i)

Again, $\frac{x+3}{y+3}=\frac{2}{5}$
⇒ 5(x + 3) = 2(y + 3)
⇒ 5x + 15 = 2y + 6
⇒ 5x − 2y = −9 ....(ii)
On multiplying (i) by 2, we get:
6x − 2y = −8 ....(iii)
On subtracting (ii) from (iii), we get:
x = (−8 + 9) = 1
On substituting x = 1 in (i), we get:
3 × 1 − y = −4
⇒ 3 − y = −4
⇒ y = (3 + 4) = 7
∴ x = 1 and y = 7
Hence, the required fraction is $\frac{1}{7}$.