Question

A fraction becomes $\frac{4}{5}$ if one is added to both numerator and denominator. If, however, 5 is subtracted from numerator and denominator, the resultant fraction becomes $\frac{1}{2}$. Find the fraction.

• A. $\frac{-7}{9}$
• B. $\frac{7}{10}$
• C. $\frac{-7}{10}$
• D. $\frac{7}{9}$

Answer 1

The correct option is D

$\frac{7}{9}$

Let $x$ be the numerator and $y$ be the denominator.
Then, it is given that $\frac{(x+1)}{(y+1)}=\frac{4}{5};\frac{(x-5)}{(y-5)}=\frac{1}{2}$

$\frac{(x+1)}{(y+1)}=\frac{4}{5}$
$\Rightarrow 5x+5=4y+4$
$\Rightarrow 5x-4y+1=0\dots .\left(1\right)$

$\frac{(x-5)}{(y-5)}=\frac{1}{2}$
$\Rightarrow 2x-10=y-5$
$\Rightarrow 2x-y-5=0\dots .\left(2\right)$

Multiplying (2) by 4,
we get
$8x-4y-20=0$ …(3)

Subtracting (3) from (1),
we get $-3x+21=0$
$\Rightarrow x=7$

Putting $x=7$ in (2),
we get
$2\times 7-y-5$ = 0
$\Rightarrow y=9$.

Therefore, the fraction is $\frac{7}{9}$.