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, then in how many days will A, B, and C working together complete the work?

Introduction / Context:
This time and work question tests your ability to interpret relative workloads and convert them into rates of working. You are told how much work A does compared to B and how much work C does compared to A and B together. Given the time taken by C alone to finish the work, you must determine how long all three together will take to complete the same task.

Given Data / Assumptions:
A does half as much work as B in the same time, meaning A is half as efficient as B. C does half as much work as A and B together, which means the efficiency of C is half of the combined efficiency of A and B. C alone can finish the entire work in 40 days. The total work is taken as one complete unit, and all workers maintain constant efficiency over time.

Concept / Approach:
We interpret the statements in terms of rates of work. We let B’s daily work be the base rate, express A’s and C’s rates in terms of B’s rate, and then use the given information about C’s time to solve for the actual rates. Once we determine the individual rates of A, B, and C, we sum them to get the combined rate of all three working together. The required time is the reciprocal of this combined rate. Using fractional arithmetic keeps the solution exact and avoids rounding errors.

Step-by-Step Solution:
Step 1: Let B’s rate of work be b units of work per day. Step 2: A does half as much work as B, so A’s rate is b / 2 units per day. Step 3: Together, A and B have a combined rate of b / 2 + b = 3b / 2 units per day. Step 4: C does half as much work as A and B together, so C’s rate is (1 / 2) * (3b / 2) = 3b / 4 units per day. Step 5: We are told that C alone can finish the entire work in 40 days. Therefore, C’s rate is also equal to 1 / 40 work per day. Step 6: Equate the two expressions for C’s rate: 3b / 4 = 1 / 40. Step 7: Solving for b, we get b = (1 / 40) * (4 / 3) = 1 / 30. This is B’s rate, so B alone takes 30 days. Step 8: A’s rate is b / 2 = (1 / 30) / 2 = 1 / 60, so A alone would take 60 days. Step 9: Check C’s rate: 3b / 4 = 3 * (1 / 30) / 4 = 1 / 40, which matches the given data. Step 10: Combined rate of A, B, and C is 1 / 60 + 1 / 30 + 1 / 40. Step 11: Take the common denominator 120: 1 / 60 = 2 / 120, 1 / 30 = 4 / 120, and 1 / 40 = 3 / 120. Step 12: Sum of the rates is (2 + 4 + 3) / 120 = 9 / 120 = 3 / 40 work per day. Step 13: Time taken together is 1 / (3 / 40) = 40 / 3 days, which is 13 1/3 days.

Verification / Alternative check:
As a check, compute how much work is done in 13 1/3 days. In fractional form, 13 1/3 days is 40 / 3 days. At a combined rate of 3 / 40 per day, total work done is (3 / 40) * (40 / 3) = 1 unit of work, which confirms that the job is completed exactly. The logic connecting the relative work shares and the final combined rate is therefore correct.

Why Other Options Are Wrong:
17 4/7 days, 16 days, and 20 days are all larger than 13 1/3 and correspond to smaller combined rates than what we calculated. Meanwhile, 15 3/2 days does not simplify to a reasonable fraction for this setup and would not match the derived rates based on the given information. Only 13 1/3 days is perfectly consistent with the defined relationships and the given time for C alone.

Common Pitfalls:
A common error is misinterpreting phrases like “does half as much work” and incorrectly assigning time values rather than rate values. Another pitfall is to assume that the time ratios are directly equal to the verbal ratios instead of translating into work rates first. Always remember that work comparisons are most easily handled by defining a base rate and building every other rate from that before converting to time.

Final Answer:
All three together, A, B, and C will complete the work in 13 1/3 days.