Question

Find a fraction which becomes $\left(\frac{1}{2}\right)$ when 1 is subtracted from the numerator and 2 is added to the denominator, and the fraction becomes $\left(\frac{1}{3}\right)$ when 7 is subtracted from the numerator and 2 is subtracted from the denominator.

Answer 1

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

Again, $\frac{x-7}{y-2}=\frac{1}{3}$
⇒ 3(x − 7) = 1(y − 2)
⇒ 3x − 21 = y − 2
⇒ 3x − y = 19 ....(ii)
On subtracting (i) from (ii), we get:
x = (19 − 4) = 15
On substituting x = 15 in (i), we get:
2 × 15 − y = 4
⇒ 30 − y = 4
⇒ y = 26
∴​ ​x = 15 and y = 26
Hence, the given fraction is $\frac{15}{26}$.