A can complete a piece of work in 12 days when working 8 hours per day. B can complete the same work in 8 days when working 10 hours per day. If both A and B work together, each working 8 hours per day, in how many days will they complete the entire work together?

Introduction / Context:
This Time and Work problem introduces an additional twist where A and B initially work for different hours per day, but later they are asked to work the same number of hours per day. We must convert their given information into hourly work rates so that we can compare them on a common basis and then compute the time required when both work together for 8 hours each day.

Given Data / Assumptions:

• A finishes the work in 12 days working 8 hours per day.
• B finishes the same work in 8 days working 10 hours per day.
• When they work together, both work 8 hours per day.
• Total work is considered as 1 job.
• Each person's hourly rate is constant, regardless of the schedule.

Concept / Approach:
We first convert the given information into total person-hours needed to complete the job for A and B individually. From that we determine their hourly work rates. Once we know their hourly rates, we can sum them to get the combined hourly rate when they work together. Since they work 8 hours a day, we convert this combined hourly rate into a daily rate, and finally obtain the total days required to finish the task together.

Step-by-Step Solution:
Let total work W = 1 unit. Total hours A takes = 12 days * 8 hours/day = 96 hours. So, A's hourly rate = 1 / 96 units per hour. Total hours B takes = 8 days * 10 hours/day = 80 hours. So, B's hourly rate = 1 / 80 units per hour. When both work together, combined hourly rate = 1 / 96 + 1 / 80. Compute: 1 / 96 + 1 / 80 = (5 + 6) / 480 = 11 / 480 units per hour. They work 8 hours per day together, so combined daily rate = 8 * (11 / 480) = 88 / 480 = 11 / 60 units per day. Time required = total work / daily rate = 1 / (11 / 60) = 60 / 11 days.

Verification / Alternative check:
As a check, note that in 60 / 11 days, the combined work done is (11 / 60) * (60 / 11) = 1 unit, which perfectly matches the total work. Also, A alone would have taken 12 days and B alone 8 days in their original schedules, so a joint time of a little more than 5 days (60 / 11 ≈ 5.45 days) is reasonable because working together at 8 hours per day should be faster than either working alone at their respective patterns.

Why Other Options Are Wrong:
Option A (51/24 days) is approximately 2.125 days, which is unrealistically small compared to the individual times of A and B. Option B (87/5 days) and Option C (57/12 days) either overestimate or underestimate the time such that the total work computed using the combined daily rate does not equal 1. Only 60 / 11 days maintains consistency with the derived hourly rates and the total work equation.

Common Pitfalls:
A frequent error is to treat 12 days and 8 days as if A and B worked the same hours per day, directly using 1 / 12 and 1 / 8 as daily rates. This ignores the difference in their daily working hours and leads to incorrect results. It is crucial to convert everything to a common measure, such as person-hours, before summing or comparing rates. Another mistake is mishandling fraction arithmetic when adding 1 / 96 and 1 / 80.

Final Answer:
Working together for 8 hours each day, A and B can complete the job in 60/11 days (approximately 5.45 days).