Question

When the numerator of a fraction is increased by 2 and the denominator by 1, the fraction becomes equal to `5/8`, and if the numerator and denominator 8 are each diminished by 1, the fraction becomes `1/2`. Find the fraction.

Answer 1

Here, let the required fraction be `x/y`.

According to the given condition, 

(1) When the numerator is increased by 2 and the denominator by 1, the fraction becomes `5/8`:

`(x + 2)/(y + 1) = 5/8`

Cross-multiplying the equation,

8(x + 2) = 5(y + 1)

8x + 16 = 5y + 5

8x − 5y = −11     ...(i)

(2) When the numerator and denominator are each diminished by 1, the fraction becomes `1/2`:

`(x - 1)/(y - 1) = 1/2`

Cross-multiplying the equation,

2(x − 1) = (y − 1)

2x − 2 = y − 1

2x − y = 1     ...(ii)

Solving equation (ii):

y = 2x − 1     ...(iii)

Substituting equation (iii) into equation (i):

8x − 5(2x − 1) = −11

8x − 10x + 5 = −11

−2x + 5 = −11

−2x = −16

x = `(-16)/-2`

∴ x = 8

Now, substitute x = 8 into equation (iii):

y = 2x − 1

y = 2(8) − 1

y = 16 − 1

∴ y = 15

Hence, the fraction is `8/15`.