If five persons together can make five mats in 5 hours, then how many mats will ten persons make in 10 hours, assuming everyone works at the same constant rate?

Introduction / Context:
This is a classic work and productivity problem where we use the idea of “person-hours” or “man-hours”. We are told what five persons can produce in a certain number of hours and are asked to generalize the result to a different number of persons and hours. The key goal is to understand that total work done is proportional to the product of the number of workers and the number of hours, provided the rate is constant.

Given Data / Assumptions:
- Five persons together can make five mats in 5 hours. - We assume each person works at the same constant rate. - We need to find the number of mats that ten persons can make in 10 hours. - The type of work and rate per person remain unchanged.

Concept / Approach:
The core idea is that Work = Rate * Time. For multiple workers with identical efficiency, total work equals the per person rate multiplied by the number of persons and the time worked. First, we determine the combined work rate from the initial situation. From this, we can find the work rate per person per hour. Then we apply that rate to a new situation with ten persons working for 10 hours to compute the total number of mats produced.

Step-by-Step Solution:
Step 1: Total work in the first scenario is 5 mats. Step 2: Person-hours used = 5 persons * 5 hours = 25 person-hours. Step 3: Rate of work per person-hour = total mats / total person-hours = 5 / 25 = 1 / 5 mat per person-hour. Step 4: In the new scenario, ten persons work for 10 hours. Step 5: Person-hours in the new scenario = 10 persons * 10 hours = 100 person-hours. Step 6: Total mats produced = person-hours * rate per person-hour = 100 * (1 / 5) = 20 mats.

Verification / Alternative check:
We can think proportionally: doubling the number of persons from 5 to 10 doubles the output per hour, and doubling the time from 5 hours to 10 hours doubles the output again. So, overall, the output increases by a factor of 4. Originally, the group produced 5 mats; four times that is 20 mats, which matches our detailed calculation.

Why Other Options Are Wrong:
10 mats: This corresponds only to doubling either persons or hours, not both, so it is too small.
15 mats: This does not match the factor-of-four increase in total person-hours and fails the proportionality check.
5 mats: This would mean no change in output despite more workers and more time, which is clearly incorrect.
25 mats: This is larger than the value obtained by strict proportional reasoning and would require a higher rate than given in the original situation.

Common Pitfalls:
Students may wrongly assume that doubling persons and doubling time just adds, rather than multiplies, the effect on work done. Another common mistake is to forget to convert the first scenario into a per person per hour rate. Remember that as long as all workers have equal efficiency, total work scales linearly with both the number of workers and the time worked.

Final Answer:
Ten persons working for 10 hours will make 20 mats.