Staggered start in work: A alone can finish a job in 15 days, and B alone can do it in 10 days. A starts the work and B joins after 5 days. For how many total days did the work last until completion?

Introduction / Context:
This is a partial-duration collaboration problem. A works alone initially, then B joins. We track the fraction of work completed by A in the first phase and then compute the time for the remainder at the combined rate in the second phase.

Given Data / Assumptions:

• A’s time = 15 days ⇒ rate_A = 1/15.
• B’s time = 10 days ⇒ rate_B = 1/10.
• A works alone for 5 days, then A and B work together until completion.

Concept / Approach:
Work done in first 5 days = 5 * (1/15) = 1/3. Remaining work = 2/3. Combined rate = 1/15 + 1/10 = 1/6 job/day. Time for remainder = (2/3) / (1/6). Add the initial 5 days to get total duration.

Step-by-Step Solution:
Initial work by A in 5 days = 5/15 = 1/3. Remaining = 1 − 1/3 = 2/3. Combined rate = 1/15 + 1/10 = (2 + 3)/30 = 1/6. Time for remainder = (2/3) / (1/6) = (2/3)*6 = 4 days. Total duration = 5 + 4 = 9 days.

Verification / Alternative check:
Rate check: In the last 4 days at 1/6 per day, they do 4/6 = 2/3, which matches the remaining portion.

Why Other Options Are Wrong:
8 or 5 days do not account for the correct split; 4 days is only the second phase, not the total.

Common Pitfalls:
Forgetting to add the initial solo days to the subsequent joint days; mixing up fractions when switching phases.

Final Answer:
9