A in $p$ days, B in $2p$ days. Together for some days, A leaves; B finishes the remaining work in $r$ days. Total time?

A's rate is $\frac1p$ and B's is $\frac1{2p}$, so together $\frac{3}{2p}$ per day. If they work $t$ days together then B works $r$ days alone: $\frac{3t}{2p} + \frac{r}{2p} = 1$, giving $t = \frac{2p-r}{3}$; total time $= t + r = \frac{2(p+r)}{3}$ days.