A, B and C together can finish a task in 12 days. A is twice as productive as B, and C alone can finish the task in 36 days. If C goes on leave and only A and B work, in how many days will A and B together complete the task?

Introduction / Context:
This problem mixes information about a group working together with relative efficiency information and a single individual time. We know how long A, B and C take together, we know that A is twice as productive as B, and we know how long C alone takes. We are asked how long A and B together would take if C is not available. This kind of question checks your skill in setting up and solving equations for work rates.

Given Data / Assumptions:

• A, B and C together can finish the task in 12 days.
• A is twice as productive as B.
• C alone can finish the task in 36 days.
• All three work at constant rates.
• We are asked for the time taken by A and B together when C is absent.

Concept / Approach:
We treat the task as 1 unit of work. Let B's rate be a base variable and express A's rate in terms of B's rate using the efficiency ratio. C's rate comes directly from the single-worker time. We then use the total time for A + B + C to form an equation for the sum of their rates. Solving this gives B's rate, and from there A's rate. Adding A and B's rates yields the combined rate when they work without C, and its reciprocal gives the required time.

Step-by-Step Solution:
Step 1: Let total work = 1 unit. Step 2: Let B's daily rate = b units per day. Step 3: A is twice as productive as B, so A's rate = 2b units per day. Step 4: C alone finishes the work in 36 days, so C's rate = 1/36 per day. Step 5: A + B + C together finish in 12 days, so 2b + b + 1/36 = 1/12. Step 6: This becomes 3b + 1/36 = 1/12. Step 7: Subtract 1/36 from both sides: 3b = 1/12 − 1/36 = 2/36 = 1/18. Step 8: Therefore, b = 1/54 and A's rate = 2b = 1/27. Step 9: A + B together have rate = 1/27 + 1/54 = 1/18 of the work per day. Step 10: Time taken by A and B together = 1 / (1/18) = 18 days.

Verification / Alternative check:
We can verify by recomputing the joint rate of A, B and C. Using A's rate 1/27, B's rate 1/54, and C's rate 1/36, we have 1/27 + 1/54 + 1/36 = 2/54 + 1/54 + 1/36. Converting to denominator 108 gives 4/108 + 2/108 + 3/108 = 9/108 = 1/12, matching the given completion time of 12 days. This confirms that the computed rates are correct and that 18 days for A and B together is consistent.

Why Other Options Are Wrong:
10 or 15 days are too small and would require a higher combined rate than 1/18, which contradicts the derived individual rates. 20 and 24 days are too large and would imply that A and B together are slower than they actually are. Only 18 days matches the reciprocal of the combined rate obtained from the given information.

Common Pitfalls:
Students often misinterpret "twice as productive" and mistakenly think A takes twice the time of B instead of half. Another mistake is to forget C's contribution when using the 12-day completion data. Using symbols for the rates and carefully setting up the equation ensures that each piece of information is used correctly and avoids sign or fraction errors.

Final Answer:
A and B together will complete the task in 18 days when C is absent.