# If we add 1 to the numerator and subtract 1 from the denominator, a fraction becomes 1.

Question:

If we add 1 to the numerator and subtract 1 from the denominator, a fraction becomes 1. It also becomes 1/2 if we only add 1 to the denominator. What is the fraction?

Solution:

Let the numerator and denominator of the fraction be $x$ and $y$ respectively. Then the fraction is $\frac{x}{y}$

If 1 is added to the numerator and 1 is subtracted from the denominator, the fraction becomes. Thus, we have

$\frac{x+1}{y-1}=1$

$\Rightarrow x+1=y-1$

$\Rightarrow x+1-y+1=0$

$\Rightarrow x-y+2=0$

If 1 is added to the denominator, the fraction becomes $\frac{1}{2}$. Thus, we have

$\frac{x}{y+1}=\frac{1}{2}$

$\Rightarrow 2 x=y+1$

$\Rightarrow 2 x-y-1=0$

So, we have two equations

$x-y+2=0$

$2 x-y-1=0$

Here x and y are unknowns. We have to solve the above equations for x and y.

By using cross-multiplication, we have

$\frac{x}{(-1) \times(-1)-(-1) \times 2}=\frac{-y}{1 \times(-1)-2 \times 2}=\frac{1}{1 \times(-1)-2 \times(-1)}$

$\Rightarrow \frac{x}{1+2}=\frac{-y}{-1-4}=\frac{1}{-1+2}$

$\Rightarrow \frac{x}{3}=\frac{-y}{-5}=\frac{1}{1}$

$\Rightarrow \frac{x}{3}=\frac{y}{5}=1$

$\Rightarrow x=3, y=5$

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