Worker A can complete a piece of work alone in 24 days. The time taken by A to complete one-third of the work is equal to the time taken by worker B to complete one-half of the same work. If both A and B start working together from the beginning, in how many days will they complete the entire work?

Introduction / Context:
This question connects the partial work done by two workers, A and B, and uses that relation to deduce B's efficiency. A's time to complete one-third of the work equals B's time to complete one-half of the work. Once we know the individual times of A and B for the full job, we can combine their rates to find how long they take to complete the job together.

Given Data / Assumptions:

• A can complete the full work in 24 days.
• Time taken by A to complete one-third of the work equals the time taken by B to complete one-half of the work.
• Both A and B work together on the entire job.
• Total work is 1 unit.
• Work rates remain constant throughout.

Concept / Approach:
We use the relationship between the partial portions (one-third and one-half) to find B's full-time for the job. Time equals work divided by rate. Therefore, we equate the time expressions and solve for B's rate. After obtaining individual rates for both A and B, we add them to get the combined rate. Finally, we take the reciprocal of that rate to find the total time for completing the work together.

Step-by-Step Solution:
Step 1: Let total work be 1 unit. Step 2: A completes the work in 24 days, so A's rate = 1/24 units per day. Step 3: Time taken by A to finish one-third of the work = (1/3) / (1/24) = 8 days. Step 4: This same amount of time, 8 days, is the time B takes to finish one-half of the work. Step 5: So, for B, (1/2) / (rate of B) = 8 days. Let B's rate be r units per day. Then (1/2) / r = 8, so r = (1/2) / 8 = 1/16 units per day. Step 6: Therefore, B alone would take 16 days to complete the full job. Step 7: Combined rate of A and B = 1/24 + 1/16. Step 8: Use the least common multiple of 24 and 16, which is 48. Then 1/24 = 2/48 and 1/16 = 3/48. Step 9: Combined rate = 2/48 + 3/48 = 5/48 units per day. Step 10: Time taken together = 1 / (5/48) = 48/5 days.

Verification / Alternative check:
We can check by computing the work completed in 48/5 days (which is 9.6 days). At rate 5/48 units per day, the work done in 48/5 days is (5/48) * (48/5) = 1 unit, confirming that the job is completed. The time is less than both 24 days and 16 days, which is expected when two workers cooperate.

Why Other Options Are Wrong:

• 21/3 days: This simplifies to 7 days, which is unrealistically small given that A alone needs 24 days and B needs 16 days.
• 48 days: This is the time it would take a far slower worker, not the combined time of A and B.
• 40 days: Also too large, and it does not correspond to the combined rate derived from the data.

Common Pitfalls:
A common mistake is to mix up the fractions and equate the wrong parts of the work, for example confusing one-third and one-half. Some students also incorrectly handle the step of computing B's rate. It is important to clearly set up the equation for time as work divided by rate and solve carefully. Additionally, avoid simply averaging 24 and 16 days, which does not give the correct combined time.

Final Answer:
A and B together will complete the work in 48/5 days.