A can complete a piece of work in 9 days and B can complete the same work in 12 days. If A and B work together from the start, in how many days will they finish the entire work?

Introduction / Context:
This is a straightforward time and work problem that involves two workers, A and B, who can complete a task individually in different numbers of days. The question asks for the combined time when they work together from the beginning. Such problems test your understanding of how to add work rates and then use the result to compute the time required for completion.

Given Data / Assumptions:

• A alone can finish the work in 9 days.
• B alone can finish the same work in 12 days.
• A and B work together on the entire job from start to finish.
• The work rate of each person is constant over time.
• Total work is assumed to be a single unit.

Concept / Approach:
We use the standard concept that if a worker finishes a job in N days, the worker's rate is 1/N of the job per day. For two workers, the combined rate is the sum of their individual rates. After we obtain the combined daily rate, we compute the total time as total work divided by this rate. This method always works for such basic work problems.

Step-by-Step Solution:
Step 1: Let total work = 1 unit. Step 2: A's daily rate = 1/9 of the work per day. Step 3: B's daily rate = 1/12 of the work per day. Step 4: Combined daily rate of A and B = 1/9 + 1/12. Step 5: Compute the sum using common denominator 36: 1/9 = 4/36 and 1/12 = 3/36, so combined rate = (4 + 3)/36 = 7/36. Step 6: Time to finish the entire work together = 1 / (7/36) = 36/7 days. Step 7: 36/7 days = 5 and 1/7 days, which is the required time.

Verification / Alternative check:
We can verify by approximate reasoning. A alone takes 9 days, and B alone takes 12 days. The combined time must be less than 9 days (the fastest individual time). The exact result 36/7 is approximately 5.14 days, which is comfortably less than 9 days. Also, if they worked for 5 1/7 days at the rate of 7/36 per day, they would complete exactly 1 unit of work, confirming the calculation.

Why Other Options Are Wrong:
5 2/7, 6 1/7, 6 2/7, and 7 days all correspond to times different from 36/7 days. Some of them are too large and would imply that working together is slower than one of them alone, which is impossible if both are contributing positively. None of these matches the precise reciprocal of the combined rate 7/36, so they cannot be correct.

Common Pitfalls:
Common mistakes include averaging the times (such as taking (9 + 12)/2 = 10.5 days) or adding the days directly without converting to rates, which is incorrect. Another frequent error is to invert the combined rate wrongly. Always remember: time = 1 / (sum of individual rates) when total work is taken as 1 unit.

Final Answer:
A and B together can complete the work in 5 1/7 days.