Question

The sum of numerator and denominator of a fraction is 3 less than twice the denominator. If each of the numerator and denominator is decreased by 1, the fraction becomes $\frac{1}{2}$. Find the fraction.

Answer 1

Let the required fraction be $\frac{x}{y}$.
Then, we have:
x + y = 2y − 3
⇒ x − y = −3 ...(i)
And, $\frac{x-1}{y-1}=\frac{1}{2}$
⇒ 2(x − 1) = 1(y − 1)
⇒ 2x − 2 = y − 1
⇒ 2x − y = 1 ...(ii)
On subtracting (i) from (ii), we get:
x = (1 + 3) = 4
On substituting x = 4 in (i), we get:
4 − y = −3
⇒ y = (4 + 3) = 7
∴​ ​x = 4 and y = 7
Hence, the fraction is $\frac{4}{7}$.