Question

The difference of denominator and numerator of a fraction is 4. If 5 is added to both numerator and denominator, the fraction is increased by `1/15`. Find the fraction.

Answer 1

Let the numerator be x.

Then the denominator is x + 4.

Original fraction:

`x/(x+4)`

New fraction after adding 5 to numerator and denominator

`(x+5)/(x+9)`

The fraction is increased by `1/15`

`(x+5)/(x+9) - x/(x+4) = 1/15`

Take LCM:

`((x+5)(x+4) - x(x+9))/((x+9) (x+4)) = 1/15`

(x + 5) (x + 4) = x² + 9x + 20

(x² + 9x + 20) − (x² + 9x) = 20

`20/((x+9)(x+4)) = 1/15`

(x + 9) (x + 4) = 300

x² + 13x + 36 − 300 = 0

x² + 13x − 264 = 0

(x − 11) (x + 24) = 0

x = 11 or x = −24

The required fraction is `11/15`