A can build a wall in 16 days, while B can destroy the same wall in 8 days. A works alone for 5 days and then B joins A for the next 2 days. After that B stops, and only A works. In how many more days will A build the remaining part of the wall?

Introduction / Context:
This question combines constructive and destructive work. A builds a wall while B destroys it. They work alone and together in different periods. We need to track the net progress on the wall over time and then determine how long A will take to finish what is left. Such questions test understanding of positive and negative work rates, as well as careful tracking of partial completion and partial destruction of the job.

Given Data / Assumptions:

• A can build the wall alone in 16 days.
• B can destroy the entire wall alone in 8 days.
• A works alone for the first 5 days.
• A and B then work together for the next 2 days.
• After that, B stops and only A continues to work.
• All work rates are constant and linear with time.

Concept / Approach:
We treat building the wall as positive work and destroying it as negative work. A's building rate is positive, while B's rate represents negative progress. When they work together, the effective rate is A's rate minus B's rate. We compute how much of the wall is built after the first 5 days, then how the situation changes during the 2 days when B joins and destroys part of the work. After that we find the remaining fraction of the wall to be built and divide it by A's building rate to get the additional time required.

Step-by-Step Solution:
Step 1: A's building rate is 1 / 16 of the wall per day. Step 2: B can destroy the wall in 8 days, so B's destructive rate is 1 / 8 of the wall per day in the negative direction. Step 3: In the first 5 days, only A works, so A builds 5 times 1/16 = 5/16 of the wall. Step 4: For the next 2 days, A and B work together. Net rate is 1/16 minus 1/8 = 1/16 minus 2/16 = -1/16 of the wall per day. Step 5: Over these 2 days, the wall is reduced by 2 times 1/16 = 1/8. Thus the built portion after 7 days is 5/16 minus 1/8 = 5/16 minus 2/16 = 3/16 of the wall. Step 6: Remaining work equals 1 - 3/16 = 13/16 of the wall. Step 7: A works alone afterwards at a rate of 1/16 per day, so time required is (13/16) divided by (1/16) = 13 days.

Verification / Alternative check:
Consider the wall as 16 equal parts. A builds 1 part per day and B destroys 2 parts per day. In 5 days, A builds 5 parts. In the next 2 days, A builds 2 more parts but B destroys 4 parts, resulting in a net loss of 2 parts. So after 7 days, 5 + 2 - 4 = 3 parts are standing. The remaining parts are 13. Since A builds 1 part per day, he needs 13 more days to finish the wall. This unit based reasoning matches the earlier fraction based calculation.

Why Other Options Are Wrong:

• 12 days would leave 1 part of the wall unfinished, since 12 is less than the required 13 days at A's rate.
• 14 days and 16 days both exceed the necessary time and imply extra work beyond the required completion.
• Only 13 days exactly match the remaining 13 parts at the rate of 1 part per day.

Common Pitfalls:
A common mistake is to add the times 16 and 8 directly or to add the rates without considering the negative sign for destruction. Others may incorrectly assume that work can never go backward, which is not true in problems involving repair and damage. It is important to treat B's contribution as negative work and carefully track net progress over each time segment.

Final Answer:
After B stops, A will need 13 days to build the remaining part of the wall.