The binary nature of the "parameter space" $\{\text{ill},\text{well}\}$ makes this simple formuation possible:
$\frac{\Pr(\text{ill})}{\Pr(\text{well})} \cdot \frac{\Pr(\text{positive} \mid \text{ill})}{\Pr(\text{positive} \mid \text{well})} = \frac{\Pr(\text{ill} \mid \text{positive})}{\Pr(\text{well}\mid\text{positive})}.$
The numerator and denominator of the first fraction above are "prior probabilities"; those of the last are "posterior probabilities".
One may write it thus: $ \operatorname{logit} \Pr(\text{ill}) + \log\frac{\Pr(\text{positive} \mid \text{ill})}{\Pr(\text{positive} \mid \text{well})} = \operatorname{logit} \Pr(\text{ill} \mid \text{positive}). $
where $ \operatorname{logit} p = \log \frac{p}{1-p}. $ (Google the word "logit" and you'll see it.)
So $\dfrac{\Pr(\text{ill})}{\Pr(\text{well})}= \dfrac{5}{95}=\dfrac{1}{19}$ and $\dfrac{\Pr(\text{positive} \mid \text{ill})}{\Pr(\text{positive} \mid \text{well})} = \dfrac{99}{10}$. Plug in the numbers.