Understanding Inverse Proportion in Construction

Imagine a scenario where 15 men can effortlessly build a wall in just 10 hours. But what if the workforce shrinks to only 4 men? How long would it take them to complete the same task? This question delves into the fascinating world of inverse proportion and man-hours in construction. By exploring these concepts, we aim to provide a clear, step-by-step solution to this problem.

Inverse Proportion and the Formula

Let's start by understanding the inverse relationship between the number of men and the time taken to build a wall. When more men are working, the job gets done in less time. Conversely, fewer men mean more time required. This inverse relationship can be expressed mathematically as:

Where (t) represents the time (in hours), (m) is the number of men, and (k) is the constant of proportionality.

Calculating the Constant of Proportionality

Given that 15 men can build the wall in 10 hours, we can find the constant (k). Using the formula:

t  k/m
10 hrs  k/15 men

Multiplying both sides by 15, we get:

15 men × 10 hrs  k
k  150 men-hrs

Applying the Formula to 4 Men

Now, we can use this constant (k 150) to find out how long it would take 4 men to build the wall. Using the formula again:

t  k/m
 150 men-hrs / 4 men  37.5 hrs

Hence, it would take 4 men 37.5 hours to build the same wall.

Alternative Method Using Man-Hours

Alternatively, we can use the concept of man-hours to solve this problem. The total man-hours required to build the wall is 15 men × 10 hours 150 man-hours. If we have only 4 men, we need to find the time (t) it would take for them to complete the same work:

4 men × t hours  150 man-hours

Solving for (t), we get:

t  150 man-hours / 4 men  37.5 hours

As expected, this method confirms our earlier solution.

Additional Scenarios

Let's explore some additional scenarios to solidify our understanding of inverse proportion and man-hours:

If it takes 10 men 10 hours to build a wall, how long would it take 5 men?

Using the same method as before:

5 men × t hours  150 man-hours

Solving for (t), we get:

t  150 man-hours / 5 men  30 hours

Therefore, it would take 5 men 30 hours to build the wall.

Additionally, consider a scenario with 10 men working for 10 hours. The total man-hours are (10 men × 10 hours 100 man-hours). With 4 men:

4 men × t hours  100 man-hours

Solving for (t), we get:

t  100 man-hours / 4 men  25 hours

So, 4 men would need 25 hours to complete the work.

Understanding these principles is crucial in construction and project management, allowing for efficient workforce planning and resource allocation.