P can complete a task in 6 days, Q can destroy it in 4 days. P works for 3 days, then Q joins for 2 days. In how many days will P finish the remaining work alone?

The correct option is 4 days.

Let's solve the problem step-by-step to find how many days P will take to finish the remaining work alone.

Step 1: Determine work rates

Let total work be represented as 1 unit.

P's 1 day work (building) = $\frac{1}{6}$

Q's 1 day work (destroying) = $-\frac{1}{4}$

Step 2: Calculate work done in the first 3 days

P works alone for the first 3 days.

Work done by P in 3 days = $3\times \frac{1}{6}=\frac{1}{2}$

Step 3: Calculate work done when Q joins P for 2 days

Combined 1 day work of P and Q = $\frac{1}{6}-\frac{1}{4}=\frac{2-3}{12}=-\frac{1}{12}$

Work done by P and Q together in 2 days = $2\times \left(-\frac{1}{12}\right)=-\frac{1}{6}$

Step 4: Calculate total work completed so far and remaining work

Total work completed after 5 days (3 days of P + 2 days of P and Q) = $\frac{1}{2}-\frac{1}{6}=\frac{3-1}{6}=\frac{2}{6}=\frac{1}{3}$

Remaining work to be done = $1-\frac{1}{3}=\frac{2}{3}$

Step 5: Calculate days required for P to complete remaining work alone

Time taken by P = $\frac{\text{Remaining Work}}{\text{P's 1 day work}}=\frac{\frac{2}{3}}{\frac{1}{6}}=\frac{2}{3}\times 6=4\text{days}$

Therefore, P will finish the remaining work alone in 4 days.