If a, b and c are positive integers such that $ab = 432$, $bc = 96$ and $c < 9$, then the smallest possible value of $a + b + c$ is Below are the steps I have done so far.
Find $a$ in terms of $c$
$$
ab = 432 \\
bc = 96 \\
\frac{ab}{bc} = \frac ac = \frac 92 \implies a = 4.5c.
$$
Find $b$ in terms of $c$
$$  
ab = 432 \\
\implies 4.5c \cdot b = 432 \\
\implies b = \frac{96}c.
$$
Substituting, we get
$$
a + b + c = 4.5c + \frac{96}c + c = 5.5c + \frac{96}c.
$$
How should I proceed from here?

The factors of $96$ that are  less than $9$ are $\{1,2,3,4,6,8\}$
Plug these values of $c$ into your equation above to find the set of $a+b+c$
