Ratio compounded

Since $9a = 7b$ , we know that $b$ must be a multiple of 9. Since $12c = 7b$ we know that $b$ must be a multiple of 12. Thus, $b$ must be a multiple of their lowest common multiple, which is $2^ 2 \times 3^2 = 36$ . Thus, the smallest possible value of $b$ is 36. With this, we obtain $a = \frac {36}{9} \times 7 = 28$ , and $c = \frac {36}{3} \times 7 = 21$ , which are both integers.

We are trying to find equivalent ratios for $a:b$ and $b:c$ so that the value of $b$ is equal for those ratios. We do this by finding the least common multiple (LCM) between $b = 9$ and $b = 12$ , which is $36$ . Since the smallest value of $a$ is concerned, we are going to focus on $a:b = 7:9$ . Since $b = 36$ , we can apply proportion: $7:9 = a:36$ . This implies that $a$ $=$ $\frac{36 \times 7}{9}$ $=$ $\boxed{28}$ .