A and B together can complete a piece of work in 24 days. B alone completes one-third of the same work in 12 days. In how many days will A alone complete the remaining two-thirds of the work?

Introduction / Context:
This time and work question compares the individual efficiency of two workers, A and B, using information about how long they take together and how much of the work B can do alone. Questions of this type are very common in aptitude exams because they test your ability to convert verbal statements into work rates and then combine or separate those rates correctly.

Given Data / Assumptions:

• A and B together complete the entire work in 24 days.
• B alone completes one-third of the same work in 12 days.
• Work rate is assumed to be constant over time.
• We are asked to find the number of days A alone will take to complete the remaining two-thirds of the work.

Concept / Approach:
The main ideas are: total work is treated as one unit, each person has a steady work rate, and time is calculated as work divided by rate. First we convert the information about B into a daily work rate. Then we compute the combined rate of A and B using their joint time. Finally we subtract B's rate from the combined rate to get A's rate, and use that to find how long A takes to finish the remaining part of the work.

Step-by-Step Solution:
Step 1: Let total work = 1 unit. Step 2: B completes one-third of the work in 12 days, so B's daily rate = (1/3) / 12 = 1/36 of the work per day. Step 3: A and B together complete the entire work in 24 days, so their combined daily rate = 1 / 24. Step 4: A's rate = (A + B) rate − B's rate = 1/24 − 1/36. Step 5: Compute A's rate: 1/24 − 1/36 = (3 − 2) / 72 = 1/72 of the work per day. Step 6: After B finishes one-third of the work, the remaining work = 1 − 1/3 = 2/3 of the work. Step 7: Time for A to finish the remaining 2/3 = (2/3) / (1/72) = (2/3) × 72 = 48 days.

Verification / Alternative check:
We can check consistency by working backward. If A's rate is 1/72 per day, in 48 days A completes (1/72) × 48 = 48/72 = 2/3 of the work. B had already done 1/3, so together that makes 1 full work. The given data and computed rates are consistent, so the answer 48 days is correct.

Why Other Options Are Wrong:
36 days: With A's rate of 1/72, in 36 days A would complete only 36/72 = 1/2 of the work, not two-thirds. 54 days: This would give (1/72) × 54 = 3/4 of the work, which is too much. 60 and 72 days would both make A extremely slow relative to B and contradict the given combined time of 24 days. Therefore, only 48 days fits all the given information.

Common Pitfalls:
A frequent mistake is to assume that since B does one-third in 12 days, the remaining two-thirds should be done in simply double that time. However, the remaining work is done by A alone, whose rate is different from B's rate. Another common error is to treat 24 days as if A and B each took 24 days individually, which is not stated. Always convert everything into daily rates and use proper addition or subtraction of those rates.

Final Answer:
A alone will take 48 days to complete the remaining two-thirds of the work.