$P$ and $Q$ can do a project in $150$ and $100$ days respectively. In how many days can they complete $60\%$ of the project if they work together?

Let the total work be the LCM of $150$ and $100$, which is $300\text{ units}$.Efficiency of $P=\frac{300}{150}=2\text{ units/day}$ Efficiency of $Q=\frac{300}{100}=3\text{ units/day}$ Combined efficiency of $P$ and $Q=2+3=5\text{ units/day}$ We need to find the time taken to complete $60\%$ of the project:
$$\text{Required work}=60\%\text{ of }300=\frac{60}{100}\times 300=180\text{ units}$$
$$\text{Time taken}=\frac{\text{Required work}}{\text{Combined efficiency}}=\frac{180}{5}=36\text{ days}$$