Question

The denominator of a fraction is 3 more than its numerator. The sum of the fraction and its reciprocal is `2 9/10`. Find the fraction. 

Answer 1

Let the numerator be x. 

∴ Denominator = x + 3 

∴ Original number = `x/(x + 3)` 

According to the question:

`x/(x + 3) + 1/(x/(x + 3)) = 2 9/10` 

⇒ `x/(x + 3) + (x + 3)/x = 29/10`

⇒ `(x^2 + (x + 3)^2)/(x(x + 3)) = 29/10`   ...[(a + b)² = a² + 2ab + b²]    

⇒ `(x^2 + (x)^2 + 2 xx x xx 3 + (3)^2)/(x(x + 3)) = 29/10`

⇒ `(x^2 + x^2 + 6x + 9)/(x^2 + 3x) = 29/10`

⇒ `(2x^2 + 6x + 9)/(x^2 + 3x) = 29/10`

⇒ 29(x² + 3x) = 10(2x² + 6x + 9)

⇒ 29x² + 87x = 20x² + 60x + 90

⇒ 29x² – 20x² + 87x – 60x – 90 = 0

⇒ 9x² + 27x – 90 = 0 

⇒ 9(x² + 3x – 10) = 0

⇒ (x² + 3x – 10) =  9 × 0

⇒ x² + 3x – 10 = 0

⇒ x² + 5x – 2x – 10 = 0

⇒ x(x + 5) – 2(x + 5) = 0

⇒ (x – 2)(x + 5) = 0

⇒ x – 2 = 0 or x + 5 = 0

⇒ x = 2 or  x = –5 (rejected)

So, number = x = 2 

Denominator = x + 3

= 2 + 3

= 5

So, required fraction = `2/5`