Question

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

Answer 1

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

Again, $\frac{x}{y-1}=\frac{1}{3}$
⇒ 3x = 1(y − 1)
⇒ 3x − y = −1 .....(ii)
On subtracting (i) from (ii), we get:
x = (−1 + 4) = 3
On substituting x = 3 in (i), we get:
2 × 3 − y = −4
⇒ 6 − y = −4
⇒ y = (6 + 4) = 10
∴ x = 3 and y = 10
Hence, the required fraction is $\frac{3}{10}$.