Four men can repair a road in 7 hours. How many men are required to repair the same road in 2 hours, assuming all men work at the same constant rate?

Introduction / Context:
This is a standard inverse proportion time and work problem. You are given that four men can repair a road in a certain number of hours and asked how many men would be required to do the same work in fewer hours, assuming all men work at the same rate. It tests understanding that total work is proportional to the product of workers and time (man-hours).

Given Data / Assumptions:
Four men can repair the road in 7 hours. All men are identical in efficiency and work at a constant rate. We want to know how many men are needed to repair the same road in 2 hours. The total amount of work is fixed and does not change between scenarios.

Concept / Approach:
The total work required can be measured in man-hours. If W is the total work, then W = number of men * hours worked. Because the work is the same in both scenarios, we equate the man-hours for the first scenario with the man-hours for the second scenario. This gives a direct proportion that allows you to solve for the required number of men in the second scenario.

Step-by-Step Solution:
Step 1: Let the total work required to repair the road be W man-hours. Step 2: In the first scenario, four men take 7 hours. So W = 4 * 7 = 28 man-hours. Step 3: In the second scenario, let the required number of men be n, and the time is 2 hours. Step 4: The total work is the same, so W = n * 2 man-hours. Step 5: Equate the two expressions for W: 28 = 2n. Step 6: Solving for n gives n = 28 / 2 = 14. Step 7: Therefore, 14 men are required to complete the same work in 2 hours.

Verification / Alternative check:
We can check the result by computing the total man-hours in the second scenario. With 14 men working for 2 hours, total man-hours are 14 * 2 = 28 man-hours, which exactly matches the original 28 man-hours. Since the total work is unchanged, the solution is consistent and correct.

Why Other Options Are Wrong:
13 men would provide only 26 man-hours, which is not enough to complete the required 28 man-hours of work in 2 hours. 16 men would give 32 man-hours, implying more work than necessary, and similarly 17 and 12 men would give 34 and 24 man-hours respectively, which also do not match the required 28 man-hours. Only 14 men produce the correct amount of total work in the given time.

Common Pitfalls:
Some learners mistakenly treat the relation between men and time as additive instead of multiplicative. Others may attempt to average or directly scale, forgetting that doubling the number of workers halves the time needed if all else is constant, which is an inverse proportion. Always use the man-hours concept: work = men * time, and keep that product constant to handle such problems correctly.

Final Answer:
To repair the road in 2 hours, 14 men are required.