One leaves after partial teamwork: P and Q together can finish a work in 30 days. They work together for 10 days and then Q leaves. If P finishes the remaining work alone in 20 more days, how long would P take to complete the whole work alone?

Introduction / Context:
This is a standard “dropout” work problem. Use the team phase to find how much work remains, then use P’s solo phase duration to deduce P’s solo rate. Finally invert to find P’s solo time for the entire job from start to finish.

Given Data / Assumptions:

• (P + Q) time = 30 days ⇒ rate(P+Q) = 1/30 job/day.
• Together phase = 10 days ⇒ work done = 10 * 1/30 = 1/3.
• Remaining work = 2/3 completed by P alone in 20 days.

Concept / Approach:
P’s solo rate = (remaining work) / (time taken) = (2/3) / 20 = 1/30 job/day. Therefore P’s solo time for 1 job is 30 days. This also implies that Q’s contribution during the first 10 days matched P’s, since the total combined rate was 1/30 and P’s rate is 1/30 as well, making Q’s rate effectively zero across the later phase (as Q left).

Step-by-Step Solution:

Verification / Alternative check:
If P’s rate is 1/30, then (P+Q) = 1/30 ⇒ Q’s rate must be 0 across those 10 days on average. Practically, this signals that the data force P’s solo rate to equal the combined rate, leaving Q’s effective rate negligible or indicating the team rate was driven by P alone—a common stylized outcome in such problems.

Why Other Options Are Wrong:
20, 50, 40, and 60 days do not reproduce the observed split of 1/3 and 2/3 with the given calendar durations.

Common Pitfalls:
Subtracting times instead of rates or miscomputing the fraction of work done in the team phase. Always track work fractions precisely.

Final Answer:
30 days