A fraction becomes $\frac{1}{3}$ if 1 is subtracted from both its numerator and denominator. If 1 is added to both the numerator and denominator, it becomes $\frac{1}{2}$. Find the fraction.

Given:

A fraction becomes $\frac{1}{3}$ if 1 is subtracted from both its numerator and denominator. If 1 is added to both the numerator and denominator, it becomes $\frac{1}{2}$.

 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 $\frac{1}{3}$ if 1 is subtracted from both its numerator and denominator. 

This implies,

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

According to the question,

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

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

$3x-3=y-1$

$y=3x-3+1$

$y=3x-2$.....(i)

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

This implies,

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

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

$2x+2=y+1$

$2x-y+2-1=0$

$2x-y+1=0$

$2x-(3x-2)+1=0$     (From (i))

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

$-x+3=0$

$x=3$

$\Rightarrow y=3(3)-2$

$y=9-2$

$y=7$

Therefore, the original fraction is $\frac{3}{7}$.  

Tutorialspoint

Updated on: 10-Oct-2022

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