The denominator of a fraction is 1 more than its numerator. The sum of fraction and its reciprocal is $2\dfrac{1}{2}$, find the fraction.

Let numerator be x and denominator be y.

Given,

The denominator of a fraction is 1 more than its numerator.

⇒ y = x + 1 ………(1)

The sum of fraction and its reciprocal is $2\dfrac{1}{2}$.

$\therefore \dfrac{x}{y} + \dfrac{y}{x} = 2\dfrac{1}{2} \\[1em] \Rightarrow \dfrac{x^2 + y^2}{xy} = \dfrac{5}{2} \\[1em] \Rightarrow 2(x^2 + y^2) = 5xy \\[1em] \Rightarrow 2x^2 + 2y^2 = 5xy$

Substituting value of y from equation (1) in above equation, we get :

⇒ 2x² + 2(x + 1)² = 5x(x + 1)

⇒ 2x² + 2(x² + 1 + 2x) = 5x² + 5x

⇒ 2x² + 2x² + 2 + 4x = 5x² + 5x

⇒ 4x² + 2 + 4x = 5x² + 5x

⇒ 5x² - 4x² + 5x - 4x - 2 = 0

⇒ x² + x - 2 = 0

⇒ x² + 2x - x - 2 = 0

⇒ x(x + 2) - 1(x + 2) = 0

⇒ (x - 1)(x + 2) = 0

⇒ x - 1 = 0 or x + 2 = 0

⇒ x = 1 or x = -2.

Let x = 1, y = x + 1 = 2.

Fraction = $\dfrac{x}{y} = \dfrac{1}{2}$.

Let x = -2, y = -2 + 1 = -1.

Fraction : $\dfrac{x}{y} = \dfrac{-2}{-1}$ = 2.

Hence, fraction = 2 or $\dfrac{1}{2}$.