A fraction becomes $\frac{9}{11}$ if 2 is added to both numerator and the denominator. If 3 is added to both the numerator and the denominator, it becomes $\frac{5}{6}$. Find the fraction.

Given:

A fraction becomes $\frac{9}{11}$ if 2 is added to both numerator and the denominator. If 3 is added to both the numerator and the denominator, it becomes $\frac{5}{6}$.

 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 $\frac{9}{11}$ if 2 is added to both numerator and the denominator.

This implies,

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

According to the question,

$\frac{x+2}{y+2}=\frac{9}{11}$

$11(x+2)=9(y+2)$    (On cross multiplication)

$11x+22=9y+18$

$11x=9y+18-22$

$11x=9y-4$

$x=\frac{9y-4}{11}$.....(i)

When 3 is added to both the numerator and the denominator the original fraction becomes $\frac{5}{6}$.

This implies,

$\frac{x+3}{y+3}=\frac{5}{6}$

$6(x+3)=5(y+3)$    (On cross multiplication)

$6x+18=5y+15$

$6x-5y+18-15=0$

$6x-5y+3=0$

$6(\frac{9y-4}{11})-5y+3=0$

$\frac{6(9y-4)-11(5y)+11(3)}{11}=0$

$54y-24-55y+33=0(11)$

$-y+9=0$

$y=9$

$\Rightarrow x=\frac{9(9)-4}{11}$

$x=\frac{81-4}{11}$$

$x=\frac{77}{11}$

$x=7$

Therefore, the original fraction is $\frac{7}{9}$. 

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Updated on: 10-Oct-2022

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