A does half as much work as B, and C does half as much work as A and B together. If C alone can finish the entire work in 40 days, in how many days will A, B and C together finish the work?

Introduction / Context:
This time and work question involves comparative efficiencies. The problem describes how much work A does compared to B, and how much work C does compared to A and B together. We are also told how long C alone takes to complete the work. Using these relationships, we must determine how much time the three of them will take to finish the work together. This tests understanding of relative work rates and algebraic manipulation.

Given Data / Assumptions:
- A does half as much work as B. - C does half as much work as A and B together. - C alone can finish the work in 40 days. - Work rates remain constant throughout. - Total work is assumed to be one complete unit.

Concept / Approach:
We use variables to represent work rates. Let the daily work rate of B be b units. Then A's rate is b / 2. The combined rate of A and B is 3b / 2. Since C does half as much as A and B together, C's rate is (1 / 2) * (3b / 2) = 3b / 4. Using the information that C alone completes the work in 40 days, we find b and thus obtain the individual rates of A, B and C. Finally, we add these rates to get the combined rate of all three working together and invert it to find the time needed.

Step-by-Step Solution:
Step 1: Let B's work rate = b jobs per day. Step 2: A does half as much as B, so A's rate = b / 2 jobs per day. Step 3: A and B together have rate = b / 2 + b = 3b / 2 jobs per day. Step 4: C does half as much as A and B together, so C's rate = (1 / 2) * (3b / 2) = 3b / 4 jobs per day. Step 5: C alone takes 40 days to complete one job, so C's rate is 1 / 40 job per day. Step 6: Set 3b / 4 = 1 / 40, leading to b = (1 / 40) * (4 / 3) = 1 / 30 job per day. Step 7: Therefore, B's rate = 1 / 30, A's rate = 1 / 60, and C's rate = 1 / 40. Step 8: Combined rate of A, B and C = 1 / 60 + 1 / 30 + 1 / 40. Step 9: Use common denominator 120: 1 / 60 = 2 / 120, 1 / 30 = 4 / 120, 1 / 40 = 3 / 120, so total rate = 9 / 120 = 3 / 40 job per day. Step 10: Time to complete whole work together = 1 / (3 / 40) = 40 / 3 days.

Verification / Alternative check:
We can verify by checking the time C alone takes with the calculated rate: C's rate from our algebra is 1 / 40 job per day, which indeed corresponds to 40 days to finish the job, matching the given condition. The combined rate 3 / 40 job per day is greater than each individual rate, which makes sense because three people working together should be faster than any one of them alone. Thus, 40 / 3 days is a reasonable and verified answer.

Why Other Options Are Wrong:
30 days or 16 days: These times are too small compared with C's 40 days alone and do not match the combined rate of 3 / 40 job per day.
15 days: This would correspond to a combined rate of 1 / 15 job per day, which is larger than the rate we computed using the given relationships and so does not fit the data.
20 days: This is still not equal to 40 / 3 and also contradicts the derived combined rate.

Common Pitfalls:
Many students misread “half as much work” and confuse it with “twice as much work”. Another frequent mistake is to treat the times directly instead of the rates, which leads to incorrect ratios. It is very important to translate verbal descriptions of efficiency into algebraic relations between rates, solve for the unknowns, and then sum the rates when workers collaborate on the same task.

Final Answer:
A, B and C together will complete the work in 40/3 days.