Why do the denominators of two fractions with numerators $1$ add up to a third fraction that have the special things below?

Question

I found out that the denominators of two fractions with numerators $1$ add up to a third fraction that has the sum in the numerator and the product in the denominator. For example, ${1\over 5}+{1\over 6}={11\over 30}$ because one, $0.2+0.1\overline6=0.3\overline6$, which convert to the main fractions, and two, the denominators with numerators $1$ add up to a third fraction having a sum in the numerator and evaluates to the product in the denominator. Why does this always happen? This is mind-blown for me!

Answer 1

Let $m, n$ be non-zero integers. Then the sum of the unit fractions $1/m$ and $1/n$ is $$\frac{1}{m} + \frac{1}{n} = \frac{1}{m} \cdot \frac{n}{n} + \frac{1}{n} \cdot \frac{m}{m} = \frac{n}{mn} + \frac{m}{mn} = \frac{m + n}{mn}$$