Solution:

If $a > 1$ and $b < 2$, we can decrease the sum by decreasing $a$ and increasing $b$. You can follow a similar procedure if $c > 1$ and $b < 2$. Therefore, the sum is minimized when $b = 2$.

We can then cross-multiply the two fractions and see that we are trying to minimize
$$\frac{a + c + 4}{(a + 2)(c + 2)} = \frac{7}{(a + 2)(c + 2)}.$$
The product of two numbers with a fixed sum is maximized when those two numbers are equal, so $\frac{7}{(a + 2)(c + 2)}$ is minimized for $a = c = \frac{3}{2}$, which gives us an answer of $\frac{4}{7}$.