A farmer grows red and white flowers. In his warehouse, $\frac{1}{12}$ of the white flowers are roses. $\frac{2}{3}$ of the red flowers are roses. $25\%$ of all the flowers are roses. The rest are lily flowers. You randomly chose a flower from the warehouse. What is the probability that the flower is red? What is the probability that he flower is red if it is a rose? Given that the number of roses is $300$, what is the total number of flowers in the warehouse?
We worked out that $\frac{2}{7}$ of the flowers are red. But then we got stuck.
Hint:
If there are $w$ white flowers and $r$ red flowers, then we have $$\frac{w}{12} + \frac{2r}{3} = \frac{w+r}{4} \implies \frac wr =\frac 52$$ Now, $$P(\text{red |rose})=\frac{P(\text{red and rose})}{P(\text{rose})} =\frac{\frac{2r}{3}}{\frac{2r}{3} + \frac{w}{12}} = \frac{\frac 23}{\frac 23 + \frac{1}{12}\cdot \frac wr}$$ $$\cdots$$