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 = a^2 + 2ab + b^2]`    

⇒ `(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`