Worker A can finish a piece of work in 15 days, and worker B can complete the same work in 12 days. B works on the job alone for 8 days and then leaves. For how many additional days must A work alone to finish the remaining part of the work?

Introduction / Context:
This is a straightforward time and work problem involving two workers with different individual completion times. One worker, B, works for a certain period and then leaves, and we must find how long the other worker, A, will take to complete the remaining portion of the job.

Given Data / Assumptions:

• Worker A can finish the work in 15 days working alone.
• Worker B can finish the work in 12 days working alone.
• B works alone for 8 days and then stops working.
• A finishes the remaining work alone at a constant rate.

Concept / Approach:
We interpret the total work as a fixed quantity that can be divided into units. It is often convenient to take the total work as the least common multiple of the given times so that daily work units become integers. We then compute how much of the work B completes in 8 days and subtract that from the total. The remainder is completed by A, and we divide the remaining work by A's daily rate to get the number of days A needs.

Step-by-Step Solution:
Let the total work be 60 units, which is the least common multiple of 12 and 15. A's rate = 60 / 15 = 4 units per day. B's rate = 60 / 12 = 5 units per day. B works alone for 8 days, so work done by B = 8 * 5 = 40 units. Remaining work = 60 - 40 = 20 units. A now works alone at 4 units per day. Time required for A to finish = 20 / 4 = 5 days. Therefore, A will need 5 additional days to complete the work.

Verification / Alternative check:
You can view the fractions directly: B's daily rate is 1 / 12 of the work. In 8 days B does 8 / 12 = 2 / 3 of the work. The remaining work is 1 - 2 / 3 = 1 / 3 of the job. A completes 1 / 15 of the job per day, so the time for A is (1 / 3) / (1 / 15) = 5 days, consistent with the unit based approach.

Why Other Options Are Wrong:
6 days would imply that A completes 6 / 15 = 2 / 5 of the job, which is more than the remaining 1 / 3. 4 days would give only 4 / 15 of the job, which is less than the 1 / 3 needed. 3 days or 7 days also do not produce the fraction 1 / 3 of the remaining work when multiplied by A's daily rate.

Common Pitfalls:
Some learners mistakenly average the times or mix up which worker continues and which stops. Others forget to compute how much work B has already completed before bringing in A's contribution. Always track the fraction of work done at each stage and ensure that the sum of completed portions equals the total work.

Final Answer:
A alone will take 5 days to finish the remaining work.