A can complete a piece of work in 21 days and B can complete the same work in 28 days. They start working together, but B leaves after 4 days. In how many additional days can A alone finish the remaining work?

Introduction / Context:
In this aptitude question on time and work, we are given the individual times taken by A and B to complete a job and are asked to determine how long A alone will take to finish the remaining work after both have worked together for some time. Such questions test understanding of work rates and the idea that work done is the product of rate and time.

Given Data / Assumptions:
- A can complete the entire work in 21 days. - B can complete the same work in 28 days. - A and B work together for the first 4 days. - After 4 days, B stops working and only A continues. - We assume they work at constant rates throughout.

Concept / Approach:
The key concept is that work rate is the reciprocal of time taken to finish one whole job. If someone finishes a job in T days, then the daily work rate is 1 / T. Total work is treated as 1 unit. When two people work together, their combined rate is the sum of their individual rates. First we find the work completed in the initial 4 days, then subtract from 1 to get the remaining work. Finally, we divide the remaining work by A's rate to find how many more days A will need to finish the job alone.

Step-by-Step Solution:
Step 1: Rate of A = 1 / 21 (job per day). Step 2: Rate of B = 1 / 28 (job per day). Step 3: Combined rate of A and B = 1 / 21 + 1 / 28. Step 4: Find a common denominator: 1 / 21 = 4 / 84 and 1 / 28 = 3 / 84, so combined rate = 7 / 84 = 1 / 12 job per day. Step 5: Work done in 4 days together = 4 * (1 / 12) = 4 / 12 = 1 / 3 of the total job. Step 6: Remaining work after 4 days = 1 - 1 / 3 = 2 / 3 of the job. Step 7: Now only A works, with rate 1 / 21 job per day. Step 8: Time taken by A to finish the remaining 2 / 3 job = (2 / 3) / (1 / 21) = (2 / 3) * 21 = 14 days.

Verification / Alternative check:
We can verify by computing the total work done explicitly. In 4 days together, A and B finish 1 / 3 of the work. A then works alone for 14 more days at a rate of 1 / 21 job per day, which yields 14 * (1 / 21) = 2 / 3 of the job. Adding 1 / 3 and 2 / 3 gives 1 full job. Thus, the answer 14 days is consistent and correct.

Why Other Options Are Wrong:
18 days: This would yield more than the remaining 2 / 3 of the work and is inconsistent with the rates.
21 days: This would suggest A is doing more than the remaining work and ignores the initial contribution by B.
16 days: This also overshoots the exact required time; 16 * (1 / 21) = 16 / 21, which does not match 2 / 3.
12 days: This would give only 12 / 21 = 4 / 7 of the work, less than 2 / 3, so the job would not be completed.

Common Pitfalls:
A common mistake is to subtract days directly instead of working with rates and fractions of work. Some students also forget to account for the work already done in the first 4 days, or they may add times instead of rates. Always remember that when people work together, you add their rates, not the days. Only after finding the remaining fraction of work should you divide by the appropriate rate to get the remaining time.

Final Answer:
A alone will take 14 additional days to complete the remaining work.