Question

The numerator of a fraction is 3 less than its denominator. If 1 is added to the denominator, the fraction is decreased by `1/15`. Find the fraction.

Answer 1

Let the denominator of the required fraction be x.
Numerator of the required fraction = x - 3 

∴Original fraction = `(x - 3)/x` 

If 1 is added to the denominator, then the new fraction obtained is `(x - 3)/(x + 1)`  

According to the given condition, 

`(x - 3)/(x + 1) = (x - 3)/x - 1/15` 

⇒ `(x - 3)/x - (x - 3)/(x + 1) = 1/15` 

⇒ `((x - 3)(x + 1) - x(x - 3))/(x(x + 1)) = 1/15` 

⇒ `(x^2 - 2x - 3 - x^2 + 3x)/(x^2 + x) = 1/15`

⇒ `(x - 3)/(x^2 + x) = 1/15` 

⇒ x² + x = 15x − 45

⇒ x² − 14x + 45 = 0

⇒ x² − 9x − 5x + 45 = 0

⇒ x(x − 9) − 5(x − 9) = 0

⇒ (x − 5)(x − 9) = 0

⇒ x − 5 = 0 or x − 9 = 0

⇒ x = 5 or x = 9

When x = 5

`(x - 3)/x = (5 - 3)/5 = 2/5` 

When x = 9

`(x-  3)/x = (9 - 3)/9 = 6/9 = 2/3` (This fraction is neglected because this does not satisfies the given  condition.) 

Hence, the required fraction is `2/5`.