Solving the Work and Time Problem

This question involves calculating the time required to complete a task when the number of workers and the hours they work per day change. We can solve this using the concept of combined work, which relates the number of people, the time spent, and the total work done.

Understanding the Concept

The total amount of work done is directly proportional to the number of students, the number of hours they work per day, and the number of days they work. We can express this relationship as:

Work = Students $\times$ Hours per day $\times$ Days

Let the initial conditions be $S_1$ students, $H_1$ hours per day, and $D_1$ days. Let the new conditions be $S_2$ students, $H_2$ hours per day, and $D_2$ days.

The total work done is constant. Therefore:

$S_1 \times H_1 \times D_1 = S_2 \times H_2 \times D_2$

Initial Conditions Analysis

From the question, we have:

• Number of students ($S_1$) = 40
• Hours worked per day ($H_1$) = 9 h
• Number of days ($D_1$) = 5 days

Calculating Total Work

First, let's calculate the total work required in terms of 'student-hours'.

Total Work = $S_1 \times H_1 \times D_1$

Total Work = $40 \text{ students} \times 9 \text{ h/day} \times 5 \text{ days}$

Total Work = $1800$ student-hours

This means the task requires a total effort equivalent to 1800 hours if only one student were working, or other combinations adding up to this total effort.

New Conditions Analysis

Next, we look at the new conditions given:

• Number of students ($S_2$) = 45
• Hours worked per day ($H_2$) = 10 h
• Number of days ($D_2$) = ? (This is what we need to find)

Calculating New Time Required

We can use the relationship derived earlier:

$S_1 \times H_1 \times D_1 = S_2 \times H_2 \times D_2$

We already know the total work is 1800 student-hours. So, we need to find $D_2$ such that the new group completes this work.

$1800 \text{ student-hours} = 45 \text{ students} \times 10 \text{ h/day} \times D_2 \text{ days}$

$1800 = 450 \times D_2$

To find $D_2$, we rearrange the equation:

$D_2 = \frac{1800}{450}$

$D_2 = 4$

Conclusion

Therefore, if 45 students work 10 hours per day, they will require 4 days to complete the same task.