Question

A fraction becomes $\frac{4}{5}$ if one is added to both numerator and denominator. If 5 is subtracted from the 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
$8x-4y-20=0\dots \left(3\right)$
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 $\frac{7}{9}$.