Conditional Probability/ Bayes' Theorem puzzle

Your numerator should be those people that get sick AND test positive, or $0.99\times 0.01x$.

In some sense, your universe is those people who test positive, which is $0.01x$. That includes healthy people with false positives, and sick people with true positives. In this problem those two populations are of equal size, hence $\frac{1}{2}$.