Question

The numerator of the given fraction is $4$ less than the denominator. If $5$ is added to both numerator and denominator, the resulting fraction is equal to $\frac{5}{7}$ . What is the original fraction?

Answer 1

Finding the original fraction:

Let the denominator of the fraction be $x$.

The given fraction is $\frac{x-4}{x}$.

$5$ is added to both numerator and denominator the resulting fraction is equal to $\frac{5}{7}$

$$\begin{gathered} \Rightarrow \frac{x-4+5}{x+5}=\frac{5}{7} \\ \Rightarrow \frac{x+1}{x+5}=\frac{5}{7} \\ \Rightarrow 7\left(x+1\right)=5(x+5) \\ \Rightarrow 7x+7=5x+25 \\ \Rightarrow 7x-5x=25-7 \\ \Rightarrow 2x=18 \\ \Rightarrow x=9 \end{gathered}$$

The given fraction is

$$\begin{gathered} \frac{x-4}{x}=\frac{9-4}{9} \\ =\frac{5}{9} \end{gathered}$$

Hence, the given fraction is $\frac{5}{9}$.