A can do a certain piece of work in 36 days and B can do the same work in 12 days. If they work together on the job, in how many days will they finish it?

Introduction / Context:
This is a straightforward time and work question where two workers have individual times to complete a job, and we are asked for the time when both work together. It tests the basic skill of converting individual times into rates, summing those rates, and then converting back to a combined time for finishing the work.

Given Data / Assumptions:
- A alone can finish the job in 36 days. - B alone can finish the same job in 12 days. - A and B work together at their constant rates. - Total work is taken as one complete job.

Concept / Approach:
If a worker completes a job in T days, their daily work rate is 1 / T of the job per day. When two workers collaborate, the combined rate equals the sum of their individual rates. Once we have the combined rate, the time taken to complete one whole job is the reciprocal of this combined rate. This is the core idea behind all such cooperation time-and-work problems.

Step-by-Step Solution:
Step 1: A's rate of work = 1 / 36 job per day. Step 2: B's rate of work = 1 / 12 job per day. Step 3: Combined daily rate of A and B working together = 1 / 36 + 1 / 12. Step 4: Convert to a common denominator 36: 1 / 12 = 3 / 36, so combined rate = 1 / 36 + 3 / 36 = 4 / 36. Step 5: Simplify 4 / 36 = 1 / 9 job per day. Step 6: Time taken to complete one full job at this rate = 1 / (1 / 9) = 9 days.

Verification / Alternative check:
We can check reasonableness by comparing the individual times. B is much faster, finishing in 12 days; A is slower at 36 days. When they work together, the combined time should be less than 12 days. Our answer, 9 days, is less than 12 and more than half of B's time, which is realistic. Also, if they work for 9 days at 1 / 9 job per day, they complete exactly 1 job, confirming the calculation.

Why Other Options Are Wrong:
8 or 6 days: These would imply a higher combined rate than 1 / 9 job per day and are not supported by the actual rates of A and B.
10 days: This is longer than 9 days and would correspond to a combined rate of only 1 / 10 job per day, which is less than the actual 1 / 9 job per day.
4 days: This is far too short and unrealistic given that even B alone takes 12 days.

Common Pitfalls:
Some students incorrectly average the times (36 and 12) instead of summing the rates. Others may make arithmetic mistakes when adding fractions with different denominators. Always convert times into rates (1 / T), add the rates, and then take the reciprocal of the sum to obtain the total time when both work together.

Final Answer:
A and B working together will complete the job in 9 days.