Person A can write 32 pages in 6 hours, while person B can write 40 pages in 5 hours. If they both write together at their respective constant speeds, in how many hours will they together be able to write a total of 110 pages?

Introduction / Context:
This problem belongs to the Time and Work category, framed in terms of writing pages. We are given the individual writing speeds of A and B, and we must find the total time taken when they work together to produce a specific number of pages. It involves converting total pages written in a given time into rates (pages per hour), adding those rates, and then applying the formula time = work / rate.

Given Data / Assumptions:

• A writes 32 pages in 6 hours.
• B writes 40 pages in 5 hours.
• We need time to write 110 pages when A and B work together.
• Both A and B write at constant speeds.
• They start and work together without any break.

Concept / Approach:
First, we calculate the individual rates of A and B in pages per hour. Next, we add these rates to find the combined rate when they write together. Finally, we use the basic relation: Time = Required pages / Combined rate. The answer then needs to be converted from hours into hours and minutes so that it matches one of the given options exactly.

Step-by-Step Solution:
Rate of A = 32 pages / 6 hours = 32 / 6 = 16 / 3 pages per hour. Rate of B = 40 pages / 5 hours = 8 pages per hour. Combined rate of A and B = 16 / 3 + 8. Express 8 as 24 / 3, so combined rate = 16 / 3 + 24 / 3 = 40 / 3 pages per hour. Required work = 110 pages. Time taken together = 110 / (40 / 3) hours. This is equal to 110 * 3 / 40 = 330 / 40 = 33 / 4 hours. 33 / 4 hours = 8.25 hours. 0.25 hours = 0.25 * 60 minutes = 15 minutes. Therefore, total time = 8 hours 15 minutes.

Verification / Alternative check:
We can verify by computing how many pages they write in 8 hours and then in an additional 15 minutes. In 8 hours, pages written = (40 / 3) * 8 = 320 / 3 ≈ 106.67 pages. In 0.25 hours (15 minutes), they write (40 / 3) * 0.25 = 10 / 3 ≈ 3.33 pages. Adding these gives approximately 110 pages, confirming the correctness of the result.

Why Other Options Are Wrong:
Option A (7 hours), Option B (6 hours 10 minutes) and Option C (5 hours 25 minutes) all correspond to lesser times. If we multiply the combined rate of 40 / 3 pages per hour by any of these times, the total pages produced will be less than 110 pages. Only 8 hours 15 minutes gives very close to 110 pages, exactly matching the required total when calculated precisely with fractions.

Common Pitfalls:
A common mistake is to average the times 6 hours and 5 hours instead of working with pages per hour. Another frequent error is in converting mixed fractions like 33 / 4 into hours and minutes, or forgetting to divide by the combined rate correctly. Always remember that rates add directly, and use fractions accurately to avoid rounding errors.

Final Answer:
Working together, A and B will write 110 pages in 8 hours 15 minutes.