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 denominator it becomes `5/6` . Find the fraction.

#### Solution

Let Numerator = x

Denominator = y

Fraction will = x/y

A fraction becomes 9/11, if 2 is added to both the numerator and the denominator `(x + 2)/(y+2) = 9/11`

By Cross multiplication, we get

11x + 22 = 9y + 18

Subtracting 22 both side, we get

11x = 9y – 4

Dividing by 11, we get

x = 9y – 4/11 … (i)

Given that 3 is added to both the numerator and the denominator it becomes 5/6.

If, 3 is added to both the numerator and the denominator it becomes 5/6

(x+3)/y +3 = 5/6 … (ii)

By Cross multiplication, we get

6x + 18 = 5y + 15

Subtracting the value of x, we get

6(9y – 4 )/11 + 18 = 5y + 15

Subtract 18 both side we get

6(9y – 4 )/11 = 5y - 3

54 – 24 = 55y - 33

-y = -9

y = 9

Putting this value of y in equation (i), we get

x = 9y – 4

11 … (i)

x = (81 – 4)/77

x = 77/11

x = 7

Hence our fraction is 7/9