Work and Time Problem Solution

This problem is a classic example of a work and time problem where the total work done is directly proportional to the number of men, the hours they work per day, and the number of days they work. We can use a generalized formula to solve such problems efficiently.

Understanding the Work-Time Formula

The fundamental principle for work and time problems, especially when different groups of people work for different durations and complete varying amounts of work, is based on the idea that the "Man-Hours-Days" equivalent for a unit of work remains constant. The formula used is:

\[ \frac{M_1 \times D_1 \times H_1}{W_1} = \frac{M_2 \times D_2 \times H_2}{W_2} \]

Where:

• \(M\) represents the number of men (or workers).
• \(D\) represents the number of days.
• \(H\) represents the number of hours worked per day.
• \(W\) represents the amount of work completed.
• Subscripts \(1\) and \(2\) refer to the two different scenarios described in the question.

Analyzing the Given Data

Let's break down the information provided for both scenarios:

Scenario 1: Initial Work Parameters

In the first situation, we have a group of men working to complete the entire work.

• Number of men (\(M_1\)): 10 men
• Hours worked per day (\(H_1\)): 8 hours/day
• Number of days taken (\(D_1\)): 28 days
• Amount of work (\(W_1\)): 1 (representing the complete work, or 100%)

Scenario 2: New Work Parameters and Goal

In the second situation, a different group of men working with different hours aims to complete 50% of that original work.

• Number of men (\(M_2\)): 8 men
• Hours worked per day (\(H_2\)): 5 hours/day
• Number of days to find (\(D_2\)): ? (This is what we need to calculate)
• Amount of work (\(W_2\)): 0.5 (representing 50% of the complete work)

Calculating the Days to Complete 50% Work

Now, let's substitute these values into our work-time formula:

\[ \frac{M_1 \times D_1 \times H_1}{W_1} = \frac{M_2 \times D_2 \times H_2}{W_2} \]

Substituting the values:

\[ \frac{10 \times 28 \times 8}{1} = \frac{8 \times D_2 \times 5}{0.5} \]

Let's simplify both sides of the equation:

Left side:

\[ 10 \times 28 \times 8 = 280 \times 8 = 2240 \]

So, the equation becomes:

\[ 2240 = \frac{8 \times D_2 \times 5}{0.5} \]

Right side:

First, calculate the product in the numerator:

\[ 8 \times D_2 \times 5 = 40 \times D_2 \]

Now, divide by 0.5 (which is the same as multiplying by 2):

\[ \frac{40 \times D_2}{0.5} = 40 \times D_2 \times 2 = 80 \times D_2 \]

So the full equation is:

\[ 2240 = 80 \times D_2 \]

To find \(D_2\), divide both sides by 80:

\[ D_2 = \frac{2240}{80} \]

\[ D_2 = 28 \]

Therefore, 8 men working 5 hours a day will complete 50% of that work in 28 days.

Final Answer Conclusion

Based on the calculations using the men-days-hours-work formula, 8 men working 5 hours a day will take 28 days to complete 50% of the work.