A and B together can complete a job in 15 days, and A alone can complete the same job in 20 days. If B worked alone, how many days would he take to complete half of the job?

Introduction / Context:
This time and work problem focuses on finding the individual efficiency of worker B, given the combined time of A and B as well as A’s individual time. After determining B’s time for the whole job, we then compute the time required by B to complete only half of the job when working alone.

Given Data / Assumptions:
- A and B together can finish the job in 15 days.
- A alone can finish the job in 20 days.
- We need the time B alone would take to complete half the work.
- Total work is taken as 1 unit.

Concept / Approach:
First, we convert the given times into daily work rates. The combined rate of A and B is 1 / 15, and A’s rate is 1 / 20. The rate of B is the difference between these two. Once B’s rate is known, we can find his time for the full job by taking the reciprocal of his rate. Since the question asks for the time to do half the work, we take half of that full-job time.

Step-by-Step Solution:
Step 1: Let total work = 1 unit. Step 2: Combined rate of A and B = 1 / 15 work per day. Step 3: Rate of A alone = 1 / 20 work per day. Step 4: Rate of B = combined rate - rate of A. Step 5: Rate of B = 1 / 15 - 1 / 20. Step 6: Take LCM of 15 and 20 which is 60, so 1 / 15 = 4 / 60 and 1 / 20 = 3 / 60. Step 7: Rate of B = 4 / 60 - 3 / 60 = 1 / 60 work per day. Step 8: Time taken by B alone for the entire job = 1 / (1 / 60) = 60 days. Step 9: Time taken by B to complete half the work = 60 / 2 = 30 days.

Verification / Alternative check:
Check the combined rate: 1 / 20 + 1 / 60 = 3 / 60 + 1 / 60 = 4 / 60 = 1 / 15. This matches the given combined time of 15 days, so the rates are consistent. Half of 60 days is indeed 30 days, so the final answer is correct.

Why Other Options Are Wrong:
- 60: This is the time B needs for the entire job, not for half of it.
- 45 and 40: These do not correspond to any direct fraction of B’s full-job time and do not fit the derived rate of 1 / 60.
- 20: This would mean B is faster than he actually is, contradicting the given combined time of 15 days.

Common Pitfalls:
Typical mistakes include averaging times instead of subtracting rates to find the individual rate of B. Another common error is forgetting that half of the job requires half of the full time when working at a constant rate. Always distinguish between full-job time and partial-job time, especially when only a fraction of the work is asked for.

Final Answer:
B would take 30 days to complete half of the job when working alone.