If 2 is added to the numerator of a fraction, it reduces to $\frac{1}{2}$ and if 1 is subtracted from the denominator, it 1 reduces to $\frac{1}{3}$. Find the fraction.

Given:

If 2 is added to the numerator of a fraction, it reduces to $\frac{1}{2}$ and if 1 is subtracted from the denominator, it 1 reduces to $\frac{1}{3}$.

 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 $\frac{1}{2}$ if 2 is added to the numerator. 

This implies,

New fraction$=\frac{x+2}{y}$

According to the question,

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

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

$2x+4=y$

$y=2x+4$.....(i)

When 1 is subtracted from the denominator, it 1 reduces to $\frac{1}{3}$.

This implies,

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

$3(x)=1(y-1)$    (On cross multiplication)

$3x=y-1$

$3x-y+1=0$

$3x-(2x+4)+1=0$     (From (i))

$3x-2x-4+1=0$

$x-3=0$

$x=3$

$\Rightarrow y=2(3)+4$

$y=6+4$

$y=10$

Therefore, the original fraction is $\frac{3}{10}$.   

Tutorialspoint

Updated on: 10-Oct-2022

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