Work Problem Overview

This problem is a typical work and time question. It requires understanding the relationship between the number of workers (people), the time taken (days), and the resources used (brooms) to complete a task.

Rate Determination

The key principle is that the total work ($W$) needed to sweep the floor is constant. The rate of work is influenced by the number of brooms ($B$) used. We assume that the number of people ($P$) is sufficient to operate the brooms effectively (i.e., $P \ge B$ for both scenarios). Therefore, the rate of work is directly proportional to the number of brooms ($B$).

We can express the rate as Rate $= k \times B$, where $k$ is a constant representing the work done per broom per day.

The general formula for work is $W = \text{Rate} \times \text{Time}$. Substituting our rate expression, we get $W = k \times B \times D$. Since the total work ($W$) and the constant $k$ are the same in both situations, we can establish a direct relationship between the number of brooms and the days:

$B_1 \times D_1 = B_2 \times D_2$

Calculation Steps

Identify the given values from the problem:

• Initial number of brooms ($B_1$): 12
• Initial number of days ($D_1$): 4
• Final number of brooms ($B_2$): 8
• Final number of people ($P_2$): 8. Since $P_2 = B_2$, our assumption that enough people are available holds true.
• We need to find the final number of days ($D_2$).

Using the derived formula $B_1 \times D_1 = B_2 \times D_2$:

$12 \times 4 = 8 \times D_2$

First, calculate the product on the left side:

$48 = 8 \times D_2$

Next, solve for $D_2$ by dividing both sides of the equation by 8:

$D_2 = \frac{48}{8}$

$D_2 = 6$

Result

It will take 6 days for 8 people to sweep the floor using 8 brooms.