A, B, and C can each complete a job working alone in 12 days, 18 days, and 36 days respectively. They all work together for 2 days, after which B stops working. How many additional days will A and C together take to finish the remaining part of the job?

Introduction / Context:
This time and work problem involves three workers A, B, and C who initially work together, and then one worker (B) leaves after some time. The goal is to compute how long the remaining workers A and C will take to complete the unfinished portion of the job. Such questions test understanding of work rates and partial completion of tasks over different time intervals.

Given Data / Assumptions:

• A alone completes the job in 12 days.
• B alone completes the job in 18 days.
• C alone completes the job in 36 days.
• All three work together for 2 days.
• After 2 days, B leaves and only A and C continue.
• Total work is taken as 1 complete job.

Concept / Approach:
The core approach is based on the concept of work rate. If a person completes a job in N days, the rate is 1/N job per day. We first calculate the combined rate of A, B, and C and find how much of the work they complete in 2 days. Then we compute the remaining work and divide it by the combined rate of A and C to find the required additional time. This stepwise method ensures clear handling of partial work.

Step-by-Step Solution:
Step 1: Let the total work be 1 job. Step 2: Rate of A = 1/12 job per day. Step 3: Rate of B = 1/18 job per day. Step 4: Rate of C = 1/36 job per day. Step 5: Combined rate of A, B, and C = 1/12 + 1/18 + 1/36. Step 6: Take LCM of 12, 18, and 36 which is 36. Step 7: 1/12 = 3/36, 1/18 = 2/36, 1/36 = 1/36, so combined rate = (3 + 2 + 1)/36 = 6/36 = 1/6 job per day. Step 8: In 2 days, the three together complete 2 * (1/6) = 1/3 of the job. Step 9: Remaining work = 1 - 1/3 = 2/3 of the job. Step 10: Now only A and C work, so their combined rate = 1/12 + 1/36. Step 11: Take LCM of 12 and 36 which is 36, so 1/12 = 3/36 and 1/36 = 1/36, giving combined rate = 4/36 = 1/9 job per day. Step 12: Time required by A and C to finish the remaining 2/3 job = (2/3) / (1/9) = (2/3) * 9 = 6 days.

Verification / Alternative check:
We can verify by calculating the effective total time if we track the work fractions carefully. The first 2 days contribute 1/3 of the job. The next 6 days with A and C at 1/9 job per day add 6 * (1/9) = 2/3 of the job. Together, 1/3 + 2/3 = 1 full job. Since all fractions add up perfectly, the answer of 6 days is consistent and correct.

Why Other Options Are Wrong:
3 days and 4 days are too short because at the rate of 1/9 job per day, they would complete only 1/3 or 4/9 of the remaining work, not 2/3. 8 days and 9 days are too long; they would cause A and C to do more work than required, exceeding the remaining 2/3 of the job. Only 6 days yields exactly the required remaining work.

Common Pitfalls:
One common mistake is to average the days taken by individuals instead of using work rates. Another frequent error is to forget that only the remaining work after the first phase must be considered when A and C continue. Some learners also mismanage fraction arithmetic while adding and dividing work rates, leading to incorrect results.

Final Answer:
A and C together will take 6 more days to finish the remaining part of the job after B leaves.