Question

If the numerator of a fraction is increased by $2$ and the denominator is decreased by $4$ then it becomes $2$. If the numerator is decreased by $1$ and the denominator is increased by $2$, then it becomes $\frac{1}{3}$. Find the sum of the numerator and denominator of the fraction.

• A. $9$
• B. $10$
• C. $11$
• D. $12$

Answer 1

Answer: (C)

$11$

Let the fraction be $\frac{x}{y}$
As per the given statements, $\frac{x+2}{y-4}=2$
$\Rightarrow x+2=2y-8$
$\Rightarrow x-2y=-10$ --- $\left(1\right)$
Also, $\frac{x-1}{y+2}=\frac{1}{3}$
$\Rightarrow 3x-3=y+2$
$\Rightarrow 3x-y=5$ --- $\left(2\right)$

Multiplying eqn $\left(2\right)$ by $2$ and subtracting eqn $\left(1\right)$ from it, we get

$2(3x-y)-(x-2y)=2\left(5\right)-(-10)$

$\Rightarrow 5x=20$

$\Rightarrow x=4$

Substituting this value of $x$ in eqn $\left(2\right)$, we get

$3\left(4\right)-y=5$

$\Rightarrow y=12-5=7$
$\therefore x=4;y=7$
So, $x+y=4+7=11$