A can complete a piece of work in 6 days working 8 hours per day, while B can complete the same work in 4 days working 10 hours per day. If A and B work together and the work must be finished in 5 days, how many hours per day should they work together?

Introduction / Context:
This problem combines time and work with variable daily working hours. Workers A and B have different daily schedules and total days when working alone. We are asked to determine how many hours per day they should work together in order to complete the job in a specified number of days. This type of problem is very relevant in project planning where both number of days and daily effort can change.

Given Data / Assumptions:

• A completes the work in 6 days when working 8 hours per day.
• B completes the work in 4 days when working 10 hours per day.
• The work must be completed in 5 days if A and B work together.
• We must find the required number of working hours per day when they work together.
• Total work is the same in all cases and is measured in effective man hours.

Concept / Approach:
We first compute the total amount of work in man hours using each person data. This gives a consistent measure for the entire job. Next, we compute A and B work rates per hour. When they work together, the combined rate is the sum of their hourly rates. To finish the job in 5 days, the total work must equal the product of days, hours per day, and the combined hourly rate. Solving this relation yields the required number of hours per day.

Step-by-Step Solution:
Step 1: Compute total work using A schedule. Step 2: A works 6 days at 8 hours per day, so total man hours = 6 * 8 = 48 man hours. Step 3: Therefore, the entire job equals 48 effective man hours. Step 4: A hourly work rate = 1 job / 48 hours = 1/48 job per hour. Step 5: Using B schedule, B works 4 days at 10 hours per day which is 40 hours to complete the same job. Step 6: B hourly work rate = 1 job / 40 hours = 1/40 job per hour. Step 7: Combined hourly rate of A and B working together = 1/48 + 1/40. Step 8: LCM of 48 and 40 is 240, so 1/48 = 5/240 and 1/40 = 6/240. Step 9: Combined rate = (5/240 + 6/240) = 11/240 job per hour. Step 10: Let the required working hours per day when they work together be H. Step 11: They need to finish the job in 5 days, so total work = 5 * H * (11/240). Step 12: Since total work is 1 job, we get 5 * H * (11/240) = 1. Step 13: Solve for H: H = 1 / (5 * 11/240) = 240 / 55 = 48/11 hours per day.

Verification / Alternative check:
To verify, multiply the found H by the time and combined rate. With H = 48/11 hours per day, total hours worked over 5 days = 5 * 48/11 = 240/11 hours. Total work done = (combined rate) * total hours = (11/240) * (240/11) = 1 job. This confirms that in 5 days, working 48/11 hours per day at their combined rate, A and B will complete exactly one full job.

Why Other Options Are Wrong:
Any value other than 48/11 hours per day will lead to total work different from 1 job when multiplied by 5 days and the combined rate of 11/240 job per hour. Values such as 59/11, 70/11, 65/11, or simply 6 hours per day either overshoot or undershoot the required total work. Therefore, these options are inconsistent with the given data and combined rate.

Common Pitfalls:
Learners may incorrectly assume that the number of hours per day is simply an average of 8 and 10, which ignores different completion times and rates. Another mistake is confusing rate per day with rate per hour, which leads to wrong combined rates. It is essential to convert everything to a common measure such as job per hour, then apply the formula for total work carefully.

Final Answer:
A and B must work together for 48/11 hours per day to complete the work in 5 days.