Question : The denominator of a fraction is 4 more than twice the numerator. When the numerator is increased by 3 and the denominator is decreased by 3, the fraction becomes $\frac{2}{3}$. What is the difference between the denominator and numerator of the original fraction?
Option 1: 13
Option 2: 10
Option 3: 12
Option 4: 11
Correct Answer: 11
Solution :
Let the original fraction be $\frac{x}{y}$
According to the question,
$y = 2x + 4$ .............(1)
When 3 is added to the numerator and 3 is subtracted from the denominator the fraction becomes $\frac{2}{3}$
⇒ new fraction = $\frac{x + 3}{y – 3} = \frac{2}{3}$
Put the value of y from eq. (1)
⇒ $\frac{x + 3}{2x + 4 – 3} = \frac23$
⇒ $\frac{x + 3}{2x + 1}= \frac{2}{3}$
⇒ $3x + 9 = 4x + 2$
⇒ $x = 7$
Put the value of x in eq. (1)
⇒ $y = 2x + 4$
= 2 × 7 + 4
= 18
The original fraction = $\frac{7}{18}$
⇒ Required difference = 18 – 7 = 11
Hence, the correct answer is 11.
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