Conditional Probability, Independence & Bayes · Hard problem

Three factories supply a store: factory $A$ makes $50\%$ of items, $B$ makes $30\%$, and $C$ makes $20\%$. Their defect rates are $1\%$, $2\%$, and $3\%$ respectively. A randomly chosen item is defective. Find the probability it came from factory $A$. Report $\dfrac{m}{n}$ and $m+n$.