A can do one-third of a job in 3 days and B can do half of the same job in 9 days. If A and B work together, in how many days can they finish half of the job?

Introduction / Context:
This time and work problem gives partial work done by A and B in specified times and then asks how long they will take together to finish only half of the job. Rather than being told total time for each person, we are told how much of the job each can complete in a certain number of days. The question tests the ability to convert these partial completion statements into daily work rates and then combine them.

Given Data / Assumptions:
- A can do one-third of the job in 3 days. - B can do half of the job in 9 days. - A and B work together at their constant respective rates. - We are asked for the time needed to complete half of the job, not the whole job. - Total job is considered as one unit.

Concept / Approach:
If a worker does a fraction of the job in a certain number of days, we can convert that to a rate by dividing the fraction by the time. Once we know the daily work rates of A and B, we add them to get the combined rate. Since the question only asks about half of the job, we then compute the time needed by dividing the required fraction (1 / 2) by the combined rate. This avoids extra steps of first finding time for a full job.

Step-by-Step Solution:
Step 1: A does 1 / 3 of the job in 3 days. Step 2: A's daily work rate = (1 / 3) / 3 = 1 / 9 of the job per day. Step 3: B does 1 / 2 of the job in 9 days. Step 4: B's daily work rate = (1 / 2) / 9 = 1 / 18 of the job per day. Step 5: Combined daily work rate of A and B = 1 / 9 + 1 / 18. Step 6: Use common denominator 18: 1 / 9 = 2 / 18, so combined rate = 2 / 18 + 1 / 18 = 3 / 18 = 1 / 6 of the job per day. Step 7: We need time to complete half of the job, that is 1 / 2 of the work. Step 8: Time = required work / combined rate = (1 / 2) / (1 / 6) = (1 / 2) * 6 = 3 days.

Verification / Alternative check:
We can verify by calculating how much work they would do in 3 days at the combined rate. At 1 / 6 job per day, in 3 days they complete 3 * (1 / 6) = 3 / 6 = 1 / 2 of the job, which matches the requirement. Since the calculations are straightforward and consistent, 3 days is correct.

Why Other Options Are Wrong:
4, 5 or 6 days: At 1 / 6 job per day, these would correspond to 4 / 6, 5 / 6 or 6 / 6 of the job, which are larger than half of the job except when the whole job is completed.
2 days: This would give only 2 / 6 = 1 / 3 of the job, which is less than the required half.

Common Pitfalls:
Students sometimes mistakenly treat the given fractions as whole jobs or fail to convert them correctly into daily rates. Another error is to compute the time for the entire job and then try to adjust for half the job, which can lead to confusion. Always convert partial completions into per day rates, sum the rates for combined work, and then directly compute the time for the required fraction of the job.

Final Answer:
Working together, A and B will finish half of the job in 3 days.