Question

A fraction becomes $\frac{4}{5}$ if 4 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. $\frac{8}{11}$
• C. $\frac{-8}{11}$
• D. None of the above

Answer 1

The correct option is B $\frac{8}{11}$

Let x be the numerator and y be the denominator.
Then, it is given that $\frac{(x+4)}{(y+4)}=\frac{4}{5};\frac{(x-5)}{(y-5)}=\frac{1}{2}$
$\frac{(x+4)}{(y+4)}=\frac{4}{5}\Rightarrow 5x+20=4y+16\Rightarrow 5x-4y+4=0\dots .\left(1\right)$
$\frac{(x-5)}{(y-5)}=\frac{1}{2}\Rightarrow 2x-10=y-5\Rightarrow 2x-y-5=0\dots .\left(2\right)$
Multiplying (2) by 4, we get 8x - 4y - 20 = 0 …(3)
Subtracting (3) from (1), we get -3x + 24 = 0 $\Rightarrow x=8$
Putting x = 8 in (2), we get $2\times 8-y-5$ = 0 $\Rightarrow y=11$.
Therefore, the fraction is $\frac{8}{11}$.