Question

$P$ and $Q$ can do a piece of work in 20 days and 30 days respectively. They started the work together and Q left after some days of work and P finished the remaining work in 5 days. After how many days from the start did Q leave?

Answer 1

P's 1 day's work $=\frac{1}{20}$ and Q's 1 day's work $=\frac{1}{30}$ P's work for 5 days $=\frac{1}{20}\times 5=\frac{5}{20}=\frac{1}{4}$
Therefore, the remaining work $=1-\frac{1}{4}=\frac{3}{4}$ (as the total work is always 1 ). This remaining work was done by both P and Q.
Work done by $P$ and $Q$ in a day $=\frac{1}{20}+\frac{1}{30}=\frac{5}{60}=\frac{1}{12}$
Therefore, the number of days they worked together $=\frac{3/4}{1/12}=\frac{3}{4}\times \frac{12}{1}=9$ days.
So, Q left after 9 days from the day the work started.