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

Given:

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

 To do:

We have to find the original fraction.

Solution:

Let the numerator and denominator of the original fraction be $x$ and $y$ respectively.

The original fraction$=\frac{x}{y}$

The fraction becomes $1$ if 1 is added to the numerator and 1 is subtracted from the denominator. 

This implies,

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

According to the question,

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

$x+1=1(y-1)$    (On cross multiplication)

$x+1=y-1$

$y=x+1+1$

$y=x+2$.....(i)

When 1 is added to only the denominator, it becomes $\frac{1}{2}$.

This implies,

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

$2(x)=1(y+1)$    (On cross multiplication)

$2x=y+1$

$2x-y-1=0$

$2x-(x+2)-1=0$     (From (i))

$2x-x-2-1=0$

$x=3$

$\Rightarrow y=x+2$

$y=3+2$

$y=5$

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

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Updated on: 10-Oct-2022

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