Question

If 5 is added to the numerator and denominator it becomes $\frac{1}{2}$ and if $2$ is subtracted from the numerator and denominator it becomes $\frac{2}{3}$. Find the fractions.

• A. $\frac{-12}{-19}$
• B. $\frac{-11}{-19}$
• C. $\frac{-10}{-19}$
• D. $\frac{-12}{-18}$

Answer 1

Answer: (A)

$\frac{-12}{-19}$

Let the fraction be $\frac{x}{y}$.
If 5 is added to the numerator and denominator fraction becomes $\frac{1}{2}$ so, we get
$\frac{(x+5)}{(y+5)}=\frac{1}{2}$
By cross multiplying,
$2(x+5)=(y+5)$
$2x+10=y+5$
$2x-y=-5$ ---- (1)
If 2 is subtracted from numerator and denominator the fraction becomes $\frac{2}{3}$ so, we get
$\frac{(x-2)}{(y-2)}=\frac{2}{3}$
$3(x-2)=2(y-2)$
$3x-6=2y-4$
$3x-2y=2$ ----- (2)
Multiplying equation (1) by 2 and then subtracting equations 1 and 2,
$4x-2y=-10$
$3x-2y=2$
_____________
$x=-12$
Substituting the value of x in equation (2)
$3(-12)-2y=2$
$-36-2y=2$
$2y=-38$
$y=-19$ Therefore, fraction becomes $\frac{-12}{-19}$