Question

The sum of a fraction and its reciprocal is `13/6`. Find the fraction if its numerator is 1 less than the denominator.

Answer 1

Let the denominator be x.

Then the numerator is x − 1.

Fraction:

`(x-1)/x`

`x/(x-1)`

Use the given condition

`(x-1)/x + x/(x-1) = 13/6`

Take LCM x(x − 1)

`((x-1)^2 + x^2)/(x(x-1)) = 13/6`

(x−1)² + x² = (x² − 2x + 1) + x² = 2x² − 2x + 1

`(2x^2 - 2x + 1)/(x(x-1)) = 13/6`

6(2x² − 2x + 1) = 13x(x − 1)

12x² − 12x + 6 = 13x² − 13x

13x² − 13x − 12x² + 12x − 6 = 0

x² − x − 6 = 0

(x − 3) (x + 2) = 0

x = 3 or x = −2

Reject the negative value.

The required fraction is `2/3`