A can do a piece of work in 18 days. He works alone for 12 days, and then B finishes the remaining work in 8 days. In how many days can B alone complete the entire work?

Introduction / Context:
This question focuses on determining the efficiency of worker B based on partial work completed by A and the remaining work done by B. A works alone initially, and B completes whatever is left. From this, we infer how fast B works and then compute the total time B alone would need to finish the whole job.

Given Data / Assumptions:
- A alone can complete the work in 18 days.
- A works alone for 12 days.
- B completes the remaining work in 8 days.
- Total work is treated as 1 unit.

Concept / Approach:
We first compute how much work A finishes in 12 days. Subtracting that from the total gives the remaining work. Since B finishes that remaining portion in 8 days, we can deduce B’s daily work rate. Finally, we calculate the time B alone would take to complete the entire 1 unit of work by taking the reciprocal of B’s rate.

Step-by-Step Solution:
Step 1: Let total work = 1 unit. Step 2: Rate of A = 1 / 18 work per day. Step 3: Work done by A in 12 days = 12 * (1 / 18) = 12 / 18 = 2 / 3 of the work. Step 4: Remaining work after A finishes = 1 - 2 / 3 = 1 / 3 of the work. Step 5: B completes this remaining 1 / 3 of the work in 8 days. Step 6: Rate of B = (1 / 3) / 8 = 1 / 24 work per day. Step 7: Time taken by B alone to complete the full work = 1 / (1 / 24) = 24 days.

Verification / Alternative check:
We can verify by recomputing the work distribution. In 12 days, A completes 12 / 18 = 2 / 3 of the job. B’s rate is 1 / 24, so in 8 days B finishes 8 * 1 / 24 = 8 / 24 = 1 / 3, which exactly matches the remaining work. This confirms that both the interim calculations and B’s total time of 24 days are consistent.

Why Other Options Are Wrong:
- 16, 20 days: These imply that B works faster, which would result in more work being done in 8 days than the remaining 1 / 3, contradicting the data.
- 28, 29 days: These imply B is slower than 24 days, which would mean that in 8 days B would complete less than the required 1 / 3 of the work.

Common Pitfalls:
Learners sometimes forget to convert days into work fractions and instead try to average times. Another common mistake is to assume that A and B share the remaining work together, while in this problem A stops after 12 days. Correct interpretation of who works when is crucial for setting up accurate calculations.

Final Answer:
B alone can complete the entire work in 24 days.