Question

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

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.

Answer 1

Let the numerator and denominator of the fraction be x and y respectively. Then the fraction is `x/y`

If 2 is added to both numerator and the denominator, the fraction becomes `9/11` Thus, we have

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

`⇒ 11(x+2)=9(y+2)`

`⇒ 11x + 22=9y +18 `

` ⇒ 11x - 9y =18 -22`

`⇒ 11x -9y = - 4`        ...(i)

If 3 is added to both numerator and the denominator, the fraction becomes `5/6`. Thus, we have

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

` ⇒ 6(x+3)=5(y+3)`

`⇒ 6x+18 =5y+15`

`⇒ 6x -5y =15 -18`

`⇒ 6x -5y = - 3 `            ...(ii)

From equation (ii)

6x – 5y = -3

⇒ 6x + 3 = 5y

⇒ y = `(6x + 3)/5`

Now on putting the value of y in equation (i)

11x – 9y = -4

⇒ `11x - 9((6x + 3)/5) = -4`

⇒ 55x – 54x – 27 = -20

⇒ x = 27 - 20

⇒ x = 7

Now putting x = 7 in equation (ii)

⇒ y = `(6(7) + 3)/5`

⇒ y = `(42 + 3)/5`

⇒ y = `45/5`

⇒ y = 9

Hence, the numerator is 7 and the denominator is 9.

Hence, the required fraction is `7/9`.