A fraction becomes 9/11 if 2 is added to both numerator and the denominator.

Question:

A fraction becomes 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 5/6. Find the fraction.

Solution:

Let the numerator and denominator of the fraction be $x$ and $y$ respectively. Then the fraction is $\frac{x}{y}$

If 2 is added to both numerator and the denominator, the fraction becomes $\frac{9}{11}$. Thus, we have

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

$\Rightarrow 11(x+2)=9(y+2)$

$\Rightarrow 11 x+22=9 y+18$

$\Rightarrow 11 x-9 y=18-22$

$\Rightarrow 11 x-9 y+4=0$

If 3 is added to both numerator and the denominator, the fraction becomes $\frac{5}{6}$. Thus, we have

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

$\Rightarrow 6(x+3)=5(y+3)$

$\Rightarrow 6 x+18=5 y+15$

$\Rightarrow 6 x-5 y=15-18$

$\Rightarrow 6 x-5 y+3=0$

So, we have two equations

$11 x-9 y+4=0$

 

$6 x-5 y+3=0$

Here x and y are unknowns. We have to solve the above equations for x and y.

By using cross-multiplication, we have

$\frac{x}{-9 \times 3-(-5) \times 4}=\frac{-y}{11 \times 3-6 \times 4}=\frac{1}{11 \times(-5)-6 \times(-9)}$

$\Rightarrow \frac{x}{-27+20}=\frac{-y}{33-24}=\frac{1}{-55+54}$

$\Rightarrow \frac{x}{-7}=\frac{-y}{9}=\frac{1}{-1}$

$\Rightarrow \frac{x}{7}=\frac{y}{9}=1$

$\Rightarrow x=7, y=9$

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

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