How do I model this expression maximization problem as a Knapsack problem?
One approach (not based on the knapsack problem): suppose you are given a threshold $\tau$ and want to check whether you can find a subset such that

$$\frac{a_{i_1} + a_{i_2} + a_{i_3}+ \dots + a_{i_k}}{b_{i_1} + b_{i_2} + b_{i_3}+ \dots + b_{i_k}} \ge \tau.$$

Could you solve that problem?  If you could, you could use binary search on $\tau$.

Note that the above condition is equivalent to

$$a_{i_1} + a_{i_2} + a_{i_3}+ \dots + a_{i_k} \ge \tau \cdot (b_{i_1} + b_{i_2} + b_{i_3}+ \dots + b_{i_k}),$$

or equivalently,

$$a_{i_1} - \tau b_{i_1} + a_{i_2} - \tau b_{i_2} + a_{i_3} - \tau b_{i_3} + \dots + a_{i_k} - \tau b_{i_k} \ge 0.$$

And that enables us to find a simple way to check for the existence of that subset.

See if that gives you any ideas.