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Form the pair of linear equations for the following problems and find their solution by substitution method. (v)  A fraction becomes 9 / 11, if 2 is added to both the numerator and the denominator. If, 3 is added to both the numerator and the

Q3.    Form the pair of linear equations for the following problems and find their solution by substitution method.

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

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Let the numerator of the fraction be x and denominator of the fraction is y 

Now According to the question,

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

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

\Rightarrow 11x+22=9y+18

\Rightarrow 11x-9y=-4...........(1)

Also,

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

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

\Rightarrow 6x+18=5y+15

\Rightarrow 6x-5y=-3...........(2)

Now, From (1) we have

y=\frac{11x+4}{9}.............(3)

Substituting this value of y in (2)

6x-5\left ( \frac{11x+4}{9} \right )=-3

\Rightarrow 54x-55x-20=-27

\Rightarrow -x=20-27

\Rightarrow x=7

Substituting this value of x in (3)

y=\frac{11x+4}{9}=\frac{11(7)+4}{9}=\frac{81}{9}=9

Hence the required fraction is

 \frac{x}{y}=\frac{7}{9}.

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