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?
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 $\frac{3}{5}$.
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