Workers A, B and C can complete a job alone in 9 days, 12 days and 36 days respectively. They all start working together, but A leaves after 3 days. For how many additional days must B and C work together to finish the remaining portion of the job?

Introduction / Context:
This time and work problem describes three workers with different individual completion times. They start a job together, but one worker leaves after a certain period. We must work out how much of the job remains and then determine how long the remaining two workers need to finish the job working at their combined rate.

Given Data / Assumptions:

• A alone completes the job in 9 days.
• B alone completes the job in 12 days.
• C alone completes the job in 36 days.
• A, B and C start together.
• A leaves after 3 days of joint work.
• B and C continue to work together thereafter at constant rates.

Concept / Approach:
We convert each solo completion time into a daily work rate. Then we calculate how much of the job is completed by all three workers during the first 3 days. The remaining fraction of the job is then completed by B and C together. We find the combined rate of B and C and divide the remaining work by this rate to determine the number of additional days required.

Step-by-Step Solution:
Assume total work = 1 job.Rates: A = 1 / 9 per day, B = 1 / 12 per day, C = 1 / 36 per day.Combined rate of A, B and C = 1 / 9 + 1 / 12 + 1 / 36.Convert to denominator 36: 1 / 9 = 4 / 36, 1 / 12 = 3 / 36, 1 / 36 = 1 / 36, so total = 8 / 36 = 2 / 9 per day.Work done in first 3 days by all three = 3 * 2 / 9 = 6 / 9 = 2 / 3.Remaining work = 1 - 2 / 3 = 1 / 3.After A leaves, only B and C work. Their combined rate = 1 / 12 + 1 / 36.Compute: 1 / 12 = 3 / 36, 1 / 36 = 1 / 36, so B + C rate = 4 / 36 = 1 / 9 per day.Time taken by B and C to finish remaining 1 / 3 = (1 / 3) / (1 / 9) = 3 days.

Verification / Alternative check:
Check the total work: first 3 days at 2 / 9 per day give 2 / 3 of the job. Next 3 days at 1 / 9 per day from B and C give 3 * 1 / 9 = 1 / 3. Adding these fractions, 2 / 3 + 1 / 3 = 1, which shows the job is exactly completed and confirms that B and C need 3 additional days after A leaves.

Why Other Options Are Wrong:
If B and C worked for 4, 5 or 6 extra days, they would complete more than the remaining 1 / 3 of the work, which is impossible under a constant positive rate. Each extra day adds another 1 / 9 of the job, so any time beyond 3 days overshoots the total work. Therefore, those options are inconsistent with the required amount of remaining work.

Common Pitfalls:
Some learners mistakenly think that the total time includes the first 3 days and then choose 6 instead of 3. However, the question specifically asks for the days after A leaves. Others miscalculate the combined rates by adding times instead of inverse times. Always remember that rates add, not raw times, when combining workers.

Final Answer:
B and C together will take an additional 3 days to complete the remaining portion of the job after A leaves.