A can complete a piece of work alone in 36 days, whereas B can complete the same work alone in 12 days. If A and B work together at their usual constant rates, in how many days will they finish the entire work?

Introduction / Context:
This is a basic time and work question where you know how long each person, A and B, takes to complete a job individually. You are asked how long they will take if they work together. It is a direct application of adding work rates and then computing the total time.

Given Data / Assumptions:

A alone can complete the work in 36 days.
B alone can complete the same work in 12 days.
Both A and B work together without interruption at constant efficiencies.
The total work is considered as 1 unit of job.

Concept / Approach:
If a worker finishes a job in T days, the rate of work is 1/T of the job per day. When multiple workers cooperate, their rates add up. After finding the combined rate of A and B, we simply take the reciprocal of the combined rate to find how many days they need to complete one full unit of work.

Step-by-Step Solution:
Let total work = 1 unit. Daily rate of A = 1/36 of the work per day. Daily rate of B = 1/12 of the work per day. Combined daily rate of A and B = 1/36 + 1/12. Take the LCM of 36 and 12, which is 36. Convert fractions: 1/36 stays 1/36, and 1/12 = 3/36. Combined rate = 1/36 + 3/36 = 4/36. Simplify 4/36 to 1/9 of the work per day. Time taken to finish the whole work together = 1 / (1/9) = 9 days.

Verification / Alternative check:
Think of the job as 36 equal parts. A completes 1 part per day and B completes 3 parts per day. Together they complete 4 parts per day. Total parts = 36, so days needed = 36 / 4 = 9 days, which matches the earlier calculation. This consistency confirms that the answer is correct.

Why Other Options Are Wrong:
Option 8 days would require a combined rate of 1/8 per day which is higher than 1/9 and not supported by the given individual rates.
Option 6 days is even faster and would imply a combined rate of 1/6 per day, which does not equal 1/36 + 1/12.
Option 10 days would correspond to a combined rate of 1/10 per day; substituting back shows it does not match the sum of individual rates.
Option 12 days is simply B's individual time and ignores the help provided by A, so it is clearly incorrect.

Common Pitfalls:
Some students mistakenly average the times (36 and 12) to get 24 days, which is wrong because you must add rates, not times. Others mis-handle the fraction addition or forget to simplify. Always convert times to rates, add those rates and then invert to get the total time when everyone works together.

Final Answer:
Therefore, A and B working together can complete the work in 9 days, so the correct option is 9 days.