Question
Mitchell has the following data:
7 4 4 m 4 5 5 6
If the range is 3, which number could m be?
7 4 4 m 4 5 5 6
If the range is 3, which number could m be?
Answers
GPT-4o mini
The range of a set of numbers is calculated as \( \text{Range} = \text{Maximum} - \text{Minimum} \).
In Mitchell's data, we need to determine what number \( m \) can be such that the range is 3. Thus, we have:
\[
\text{Maximum} - \text{Minimum} = 3
\]
The current numbers we have (excluding \( m \)) are 4, 4, 5, 5, and 6. Let's find the maximum and minimum values based on the values we have:
- **Current Minimum**: 4
- **Current Maximum**: 6
Using the current minimum as 4, for the range to be 3, we set the maximum to:
\[
4 + 3 = 7
\]
This implies that the maximum of the set must be 7 if the minimum remains 4. Therefore, for \( m \) to be 7, it must be included in our dataset for the maximum to increase to 7.
Now let's check if \( m \) can actually be any other value:
If we keep the current maximum at 6 and look for a possible minimum of 3 (since 6 - 3 = 3), then \( m \) should be a number less than or equal to 3.
Now we can check our options:
1. **If \( m = 3 \)**:
- New set: \( 3, 4, 4, 4, 5, 5, 6 \)
- Maximum = 6, Minimum = 3 → Range = \( 6 - 3 = 3 \) (this is valid).
2. **If \( m = 7 \)**:
- New set: \( 4, 4, 4, 5, 5, 6, 7 \)
- Maximum = 7, Minimum = 4 → Range = \( 7 - 4 = 3 \) (this is also valid).
Therefore, \( m \) could be either 3 or 7. No other values would maintain a range of 3, specifically:
- If \( m \) is less than 3, the minimum would be less than 4, causing the range to exceed 3.
- If \( m \) is greater than 7, the maximum would exceed 7, also increasing the range.
Thus, the only possibilities for \( m \) are **3 or 7**.
In Mitchell's data, we need to determine what number \( m \) can be such that the range is 3. Thus, we have:
\[
\text{Maximum} - \text{Minimum} = 3
\]
The current numbers we have (excluding \( m \)) are 4, 4, 5, 5, and 6. Let's find the maximum and minimum values based on the values we have:
- **Current Minimum**: 4
- **Current Maximum**: 6
Using the current minimum as 4, for the range to be 3, we set the maximum to:
\[
4 + 3 = 7
\]
This implies that the maximum of the set must be 7 if the minimum remains 4. Therefore, for \( m \) to be 7, it must be included in our dataset for the maximum to increase to 7.
Now let's check if \( m \) can actually be any other value:
If we keep the current maximum at 6 and look for a possible minimum of 3 (since 6 - 3 = 3), then \( m \) should be a number less than or equal to 3.
Now we can check our options:
1. **If \( m = 3 \)**:
- New set: \( 3, 4, 4, 4, 5, 5, 6 \)
- Maximum = 6, Minimum = 3 → Range = \( 6 - 3 = 3 \) (this is valid).
2. **If \( m = 7 \)**:
- New set: \( 4, 4, 4, 5, 5, 6, 7 \)
- Maximum = 7, Minimum = 4 → Range = \( 7 - 4 = 3 \) (this is also valid).
Therefore, \( m \) could be either 3 or 7. No other values would maintain a range of 3, specifically:
- If \( m \) is less than 3, the minimum would be less than 4, causing the range to exceed 3.
- If \( m \) is greater than 7, the maximum would exceed 7, also increasing the range.
Thus, the only possibilities for \( m \) are **3 or 7**.