A, B, and C together can complete a job in 9 days, while C alone can complete the same job in 36 days. If only A and B work together, in how many days will they complete 50 percent of the job?

Introduction / Context:

This time and work problem links the combined efficiency of three people with the individual efficiency of one of them, and then asks about the time required when only the other two work together on half of the job. It tests understanding of how to separate one worker's contribution from a known combined rate, and how to use the remaining rate to compute partial completion time.

Given Data / Assumptions:

• A, B, and C together finish the job in 9 days.
• C alone finishes the job in 36 days.
• We are asked for the time A and B together would take to finish 50 percent of the job.
• Work is taken as one unit and all rates are constant.

Concept / Approach:

We begin by converting completion times into daily work rates. The combined rate of A, B, and C is 1/9 of the job per day, and C's rate alone is 1/36 of the job per day. Subtracting C's rate from the combined rate gives the joint rate of A and B. Once we know the rate at which A and B work together, we can compute how long they will take to complete half of the job by dividing the required work by their daily rate.

Step-by-Step Solution:

Verification / Alternative Check:

If A and B work together for 6 days at a rate of 1/12 per day, they complete 6 * 1/12 = 6/12 = 1/2 of the job, which is exactly the required fraction. Also, if we consider A, B, and C together, with C's rate of 1/36, the sum 1/12 + 1/36 is 4/36 = 1/9, matching the given combined completion time of 9 days. This confirms that the derived rates and time for A and B are consistent.

Why Other Options Are Wrong:

Options like 9 or 12 days would mean A and B together work for longer, producing 9 * 1/12 = 3/4 or 12 * 1/12 = 1 unit of work, which would exceed the 50 percent requirement. Option 15 days is even further off. Only 6 days corresponds exactly to half of the job being completed by A and B together at their calculated rate.

Common Pitfalls:

One common error is to compute the time for A, B, and C together and then try to scale it directly without isolating C's contribution. Another mistake is miscalculating the difference between 1/9 and 1/36, which leads to an incorrect rate for A and B. Correctly performing the rate subtraction and remembering that the question asks for half the job are essential to avoid these errors.

Final Answer:

Working together, A and B will complete 50 percent of the job in 6 days.