The denominator of a fraction is 3 more its numerator. If the numerator is increased by 7 and the denominator is decreased by 2, the fraction is 2. Find the sum of the numerator and the denominator of the fraction.

Let the numerator of the fraction be x.

The denominator of a fraction is 3 more than its numerator.

So, denominator = x + 3

If the numerator is increased by 7 and the denominator is decreased by 2, the fraction is 2.

$⇒ \dfrac{numerator + 7}{denominator - 2} = 2\\[1em] ⇒ \dfrac{x + 7}{(x + 3) - 2} = 2\\[1em] ⇒ \dfrac{x + 7}{x + 3 - 2} = 2\\[1em] ⇒ \dfrac{x + 7}{x + 1} = 2\\[1em]$

By cross multiplying, we get :

⇒ (x + 7) = 2(x + 1)

⇒ x + 7 = 2x + 2

⇒ 2x - x = 7 - 2

⇒ x = 5

Denominator = (x + 3)

= 5 + 3

= 8

The fraction is $\dfrac{5}{8}$.

The sum of the numerator and the denominator of the fraction = 5 + 8 = 13.

Hence, the sum of the numerator and the denominator of the fraction is 13.