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Question

A fraction becomes 911 if 2 is added to both the numerator and the denominator. If 3 is added to both the numerator and the denominator, it becomes 56. Find the fraction.

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Solution

Let the fraction be xy, y0

Case1:

fraction becomes 911 if 2 is added to both the numerator and the denominator.

x+2y+2=911

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

11x+22=9y+18

11x9y+2218=0

11x9y+4=0 ---(1)

Case2:

fraction becomes 56 if 3 is added to both the numerator and the denominator.

x+3y+3=56

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

6x+18=5y+15

6x5y+1815=0

16x5y+3=0 ---(2)

Lets solve (1) and (2)

Multiply 11x9y+4=0 by 5

55x45y+20=0----(3)

Multiply 6x5y+3=0 by 9

54x45y+27=0 ---(4)

Lets do (3)-(4)

55x45y+20(54x45y+27)=0

55x45y+2054x+45y27)=0

x7=0

x=7

Substitute x=7 in 11x9y+4=0

11(7)9y+4=0

779y+4=0

819y=0

81=9y

819=y

9=y

The required fraction is 79.


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