Question

The denominator of a fraction is 2 more than its numerator. If 1 is subtracted from both the numerator and denominator, the fraction is decreased by `1/21.` Find the fraction.

Answer 1

Let the numerator be x.

Then the denominator is x + 2.

Original fraction:

`x/(x+2)`

Fraction after subtracting 1 from numerator and denominator

`(x-1)/(x+1)`

The fraction is decreased by `1/21`

`x/(x+2) - (x-1)/(x+1) = 1/21`

Take LCM (x + 2)

`(x(x+1) - (x-1) (x+2))/((x+2) (x+1)) = 1/21`

Expand:

x(x + 1) = x² + x

(x − 1) (x + 2) = x² + x − 2

(x² + x) − (x² + x − 2) = 2

`2/((x+2) (x+1)) = 1/21`

(x + 2) (x + 1) = 42

x² + 3x − 40 = 0

(x + 8) (x − 5) = 0

x = 5 or x = −8

Reject the negative value.

The required fraction is `5/7`