**The graph below shows the number of notepads that each location of a business used last year. The number of departments per location are presented in the table below. Four of the locations used the same number of notepads per department, but the Northwest and Central offices used less notepads per department. The graph shows that the Northwest location used the same number of notepads as the Midtown and Southeast locations overall. If the Northwest location used less notepads per department, which of the following could be the number of departments for the Northwest location?**


To solve this problem, we'll use the information provided and reason through the possible number of departments for the Northwest location.

### Given Information:
1. **Four locations** used the same number of notepads per department.
2. The **Northwest** and **Central** locations used fewer notepads per department.
3. The graph indicates that the **Northwest location** used the same total number of notepads as the **Midtown** and **Southeast** locations.

### Approach:
- Let’s denote the number of notepads used by the Northwest location as \( N_{NW} \), and similarly for the other locations.
- Let \( d_{NW} \), \( d_{Mid} \), and \( d_{SE} \) represent the number of departments in the Northwest, Midtown, and Southeast locations, respectively.
- Assume the four locations (excluding Northwest and Central) use \( n \) notepads per department. The total number of notepads for each of these locations would be \( n \times d \), where \( d \) is the number of departments at that location.

### Step 1: Express the total number of notepads used by the Northwest location.
Since the Northwest location used fewer notepads per department, let’s say it used \( m \) notepads per department, where \( m < n \).

Thus, the total number of notepads used by the Northwest location is:

\[
N_{NW} = m \times d_{NW}
\]

### Step 2: Use the condition that the total number of notepads used by Northwest equals that of Midtown and Southeast.
This implies:

\[
m \times d_{NW} = n \times d_{Mid} = n \times d_{SE}
\]

### Step 3: Analyze the relationship.
Given \( m < n \), for \( m \times d_{NW} \) to equal \( n \times d_{Mid} \) or \( n \times d_{SE} \), the number of departments in the Northwest location, \( d_{NW} \), must be greater than the number of departments in the Midtown or Southeast locations.

This leads us to infer that:

\[
d_{NW} > d_{Mid}, d_{SE}
\]

### Conclusion:
- The correct number of departments for the Northwest location is a value greater than the number of departments in the Midtown and Southeast locations.
  
Since the problem mentions that "four of the locations used the same number of notepads per department," we can rule out the Central office as one of these locations. If we knew the specific number of departments for Midtown and Southeast, we would compare those numbers to potential options for the Northwest location. 

Without exact numbers for the departments, the key takeaway is that **the number of departments for the Northwest location must be greater than the number of departments in the Midtown and Southeast locations** to maintain the same total number of notepads with fewer notepads per department.

Would you like more detailed calculations or do you have any questions?

### Related Questions:
1. How would you calculate the total number of notepads used by the Central location?
2. If the Central location used a specific number of notepads, how could we find the notepads per department?
3. What if the Northwest location had exactly one more department than another location; how would this change the total calculation?
4. How could the number of notepads used per department be calculated if the total notepads are known?
5. What is the implication if two locations have the same number of departments but different total notepad usage?
6. How could you verify the consistency of the data using ratios of notepads per department?
7. If the Southeast location had 6 departments, what might be the total number of notepads?
8. How would the problem change if the notepads used per department were equal across all locations?

### Tip:
When dealing with comparative problems like this, breaking down the problem into manageable steps and setting up equations based on relationships is crucial for finding the correct answer.

Explore how to solve a problem involving notepad usage across multiple locations using mathematical reasoning. Understand the relationships between notepads used per department and total notepads used in different locations. Suitable for students in Grades 6-8.

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