Men Working Hours Problem Solution

This problem falls under the category of work and time, specifically involving the concept that the total amount of work remains constant regardless of the number of men or hours worked, as long as the work itself is the same. We can use the formula:

Total Work = Number of Men × Number of Days × Hours per Day

Calculating Total Work in the First Scenario

In the first situation, we are given:

• Number of men (M\(_1\)) = 3
• Hours per day (H\(_1\)) = 7 hours
• Number of days (D\(_1\)) = 45 days

Using the formula, the total work done by 3 men working 7 hours a day for 45 days is:

\(\text{Work}_1 = M_1 \times D_1 \times H_1\)

\(\text{Work}_1 = 3 \times 45 \times 7\)

\(\text{Work}_1 = 135 \times 7\)

\(\text{Work}_1 = 945\) units of work

Calculating Days in the Second Scenario

In the second situation, we need to find the number of days for the same work. We are given:

• Number of men (M\(_2\)) = 9
• Hours per day (H\(_2\)) = 6 hours
• Number of days (D\(_2\)) = ? (Let this be \(D\))

Since the work is the same, we can equate the total work from both scenarios:

\(\text{Work}_1 = \text{Work}_2\)

\(M_1 \times D_1 \times H_1 = M_2 \times D_2 \times H_2\)

Now, substitute the known values into the equation:

\(3 \times 45 \times 7 = 9 \times D \times 6\)

Let's simplify both sides of the equation:

\(945 = 54 \times D\)

To find \(D\), we need to divide the total work by the product of the number of men and hours per day in the second scenario:

\(D = \frac{945}{54}\)

We can simplify this fraction. Both 945 and 54 are divisible by 9:

\(945 \div 9 = 105\)

\(54 \div 9 = 6\)

So, the equation becomes:

\(D = \frac{105}{6}\)

Both 105 and 6 are divisible by 3:

\(105 \div 3 = 35\)

\(6 \div 3 = 2\)

Therefore, the number of days \(D\) is:

\(D = \frac{35}{2}\)

This means 9 men working 6 hours a day will complete the same work in \(\frac{35}{2}\) days.

Final Answer Summary

The problem is a classic example of inverse proportionality in work and time. As the number of men and hours per day increase, the number of days required to complete the same work decreases. Our calculation shows that 9 men working 6 hours a day will complete the work in \(\frac{35}{2}\) days.