Understanding the Work and Time Problem

This problem involves calculating the time required to complete a task based on the number of students, the hours they work per day, and the number of days taken. The core concept here is that the total amount of work done is proportional to the product of these three factors: the number of workers, the rate at which they work (hours per day), and the duration they work (days).

We can express this relationship. Let:

• $S$ be the number of students (workers).
• $H$ be the number of hours worked per day.
• $D$ be the number of days required to complete the work.

The total work ($W$) can be considered proportional to the product $S \times H \times D$. Since the task remains the same in both scenarios, the total work required is constant.

Breaking Down the Information

Let's list the information provided for the two scenarios:

Scenario 1:

• Number of students ($S_1$): 40
• Hours worked per day ($H_1$): 9 h
• Days to complete the work ($D_1$): 5 days

Scenario 2:

• Number of students ($S_2$): 45
• Hours worked per day ($H_2$): 10 h
• Days to complete the work ($D_2$): ? (This is what we need to find)

Calculating the Required Time

The principle of work problems states that the total work done is constant if the task is the same. Therefore, we can set the work done in Scenario 1 equal to the work done in Scenario 2:

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

Now, let's substitute the known values into the equation:

$ 40 \text{ students} \times 9 \text{ h/day} \times 5 \text{ days} = 45 \text{ students} \times 10 \text{ h/day} \times D_2 \text{ days} $

First, calculate the total 'student-hours' required in the first scenario:

Total Work (Scenario 1) = $40 \times 9 \times 5$

Total Work (Scenario 1) = $40 \times 45$

Total Work (Scenario 1) = $1800$ student-hours

Now, set this equal to the work done in the second scenario:

$ 1800 = 45 \times 10 \times D_2 $

Simplify the right side:

$ 1800 = 450 \times D_2 $

To find $D_2$, we rearrange the equation:

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

Now, perform the division:

$ D_2 = \frac{180}{45} $

$ D_2 = 4 $

So, the time required for 45 students working 10 hours a day to complete the same task is 4 days.