Question

If we add$1$ to the numerator and subtract $1$from the denominator a fraction reduces to$1$. It becomes $\frac{1}{2}$, if we only add $1$to the denominator. What is the fraction?

Answer 1

Solve for the required fraction:

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

Add $1$ to the numerator and subtract $1$ from the denominator.

$$\begin{gathered} \Rightarrow \frac{x+1}{y-1}=1 \\ \Rightarrow x+1=y-1 \\ \Rightarrow x=y-2......\left(1\right) \end{gathered}$$

Add $1$to the denominator

$$\begin{gathered} \Rightarrow \frac{x}{y+1}=\frac{1}{2} \\ \Rightarrow 2x=y+1......\left(2\right) \end{gathered}$$

Substituting the equation $\left(1\right)$ in $\left(2\right)$, we get

$2\left(y-2\right)=y+1$

$$\begin{gathered} 2y-4=y+1 \\ 2y-y=1+4 \end{gathered}$$

$y=5$

Substituting this in the equation $\left(1\right)$ we get

$$\begin{gathered} x=5-2 \\ x=3 \end{gathered}$$

Hence, the required fraction is $\frac{3}{5}$.