Raghu can complete a job in 12 days working alone, while Sam can finish the same job in 15 days working alone. A third person, Aru, has an efficiency equal to two thirds of the combined efficiency of Raghu and Sam together. In how many days can Aru alone complete the job?

Introduction / Context:
This question tests the concept of combining work rates and then scaling them to find the time for one worker whose rate is a given fraction of a combined rate. Raghu and Sam have known individual completion times, and Aru's efficiency is defined relative to their combined efficiency.

Given Data / Assumptions:

• Raghu alone can complete the work in 12 days.
• Sam alone can complete the same work in 15 days.
• Aru's efficiency is two thirds of the combined efficiency of Raghu and Sam working together.
• All work at constant rates and there is only one job to be completed.

Concept / Approach:
We first calculate the combined rate of Raghu and Sam by adding their individual rates. Then we take two thirds of this combined rate to get Aru's rate. Since time is the reciprocal of rate for a fixed amount of work, we can convert Aru's rate back into the number of days Aru would need to complete the job alone.

Step-by-Step Solution:
Raghu's rate = 1 / 12 jobs per day. Sam's rate = 1 / 15 jobs per day. Combined rate of Raghu and Sam = 1 / 12 + 1 / 15. Compute the sum: the least common multiple of 12 and 15 is 60, so 1 / 12 = 5 / 60 and 1 / 15 = 4 / 60. Therefore, combined rate = (5 + 4) / 60 = 9 / 60 = 3 / 20 jobs per day. Aru's rate is two thirds of this, so Aru's rate = (2 / 3) * (3 / 20) = 2 / 20 = 1 / 10 jobs per day. Hence, Aru alone would take 10 days to complete one job.

Verification / Alternative check:
As a check, note that Raghu and Sam together finish in 20 / 3 days, which is about 6.67 days. Aru's efficiency is two thirds of theirs, so his time should be 3 / 2 times their time: (3 / 2) * (20 / 3) = 10 days. This matches the computed time and confirms the result.

Why Other Options Are Wrong:
12 days would correspond to a rate of 1 / 12, which is slower than Aru's defined efficiency. 13 days and 15 days would give even lower rates and are inconsistent with Aru being two thirds as efficient as the pair. 9 days would give a slightly higher rate than 1 / 10 and does not match exactly two thirds of the combined rate.

Common Pitfalls:
Students sometimes misinterpret "two thirds of the efficiency" as "two thirds of the time," which is incorrect because time and efficiency are inversely related. Another frequent error is adding the times instead of the rates. Always convert completion times into rates before combining or scaling efficiencies.

Final Answer:
Aru alone would complete the work in 10 days.