The denominator of a positive fraction is 3 more than its numerator. The fraction and its reciprocal add up to $\frac{29}{10}$. What is the fraction?

Explanation

1. 01

   Let the fraction be $\frac{n}{n + 3}$: $\frac{n}{n + 3} + \frac{n + 3}{n} = \frac{29}{10}$.

2. 02

   Combine: $\frac{n^2 + (n + 3)^2}{n(n + 3)} = \frac{29}{10}$, so $10(2n^2 + 6n + 9) = 29(n^2 + 3n)$.

3. 03

   $9n^2 + 27n - 90 = 0$, i.e. $n^2 + 3n - 10 = 0$, so $(n + 5)(n - 2) = 0$.

4. 04

   $n = 2$: the fraction is $\frac{2}{5}$. Check: $\frac{2}{5} + \frac{5}{2} = \frac{4 + 25}{10} = \frac{29}{10}$.