Question

Let $\frac{p}{q}$ be a fraction expressed in the lowest form. If the numerator is increased by 2 and the denominator is increased by 1, the resulting fraction equals $\frac{1}{2}$. If, however, the numerator is increased by 1 and the denominator is decreased by 2, the resulting fraction equals $\frac{3}{5}$. The value of $(p+q)$ equals ($p,q$ are natural number )

• A. $7$
• B. $9$
• C. $12$
• D. $13$

Answer 1

Answer: (B)

$9$

The given fraction is $\frac{p}{q}$
then, $\frac{p+2}{q+1}=\frac{1}{2}$

$2p+4=q+1$
$q-2p=3$.............. (1)
$\frac{p+1}{q-2}=\frac{3}{5}$

$5p+5=3q-6$
$3q-5p=11$ ..............(2)
Multiply equation(1) by 3 and subtract from equation(2), we get
$3q-6p-3q+5p=9-11$
$-p=-2$
$p=2$
and, $q=7$
Hence, $p+q=2+7=9$