Question

A fraction becomes $\frac{4}{5}$ when 1 is added to each of the numerator and denominator. However, if we subtract 5 from each then it becomes $\frac{1}{2}$. The fraction is

• A. $\frac{5}{8}$
• B. $\frac{5}{6}$
• C. $\frac{7}{9}$
• D. $\frac{13}{16}$

Answer 1

The correct option is C $\frac{7}{9}$

Let the fraction be $\frac{x}{y}$

$\Rightarrow \frac{x+1}{y+1}=\frac{4}{5}$

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

$\Rightarrow \frac{x-5}{y-5}=\frac{1}{2}$

$\Rightarrow 2x-10=y-5\Rightarrow 2x-y=5.....\left(2\right)$

$\Rightarrow y=2x-5$

substitute the value of y in equation (1)

$\Rightarrow 5x-4(2x-5)=-1$

$\Rightarrow 5x+20-8x=-1\Rightarrow x=7$

Put the value of $x$ in eqn (2)

$\Rightarrow y=2\times 7-y=5\Rightarrow y=9$

$\therefore \frac{x}{y}=\frac{7}{9}$