Example 2 Let $a, b$ be positive integers and satisfy
$$
\frac{2}{3}<\frac{a}{b}<\frac{5}{7} \text {. }
$$

When $b$ is the minimum value, $a+b=$ $\qquad$
(Fifth "Hope Cup" National Mathematics Invitational Competition Training Question)
Solution: According to the theorem, we have
$$
\frac{a}{b}=\frac{2+5}{3+7}=\frac{7}{10} \text {. }
$$

Thus, $a=7, b=10$. Therefore, $a+b=17$.
From the above theorem, it can be seen that although addition and subtraction is a rather clever and simple method, it is not suitable for the case where $b c-a d>1$. For example, the rational number with the smallest denominator between $\frac{2}{5}$ and $\frac{7}{9}$ is not $\frac{2+7}{5+9}=\frac{9}{14}$, but $\frac{1}{2}$. Therefore, the method of addition and subtraction does not have universal applicability.