Question
Kevin runs on his treadmill every day for 5 days. He records his distance each day and rounds that distance to the nearest whole mile. The median distance he ran is 3 miles and the range is 2 miles.
Enter the correct values in each box to make the statement true.
The longest possible distance Kevin could have run one day during this week is mile(s).
The shortest possible distance Kevin could have run one day during this week is mile(s).
Enter the correct values in each box to make the statement true.
The longest possible distance Kevin could have run one day during this week is mile(s).
The shortest possible distance Kevin could have run one day during this week is mile(s).
Answers
GPT-4o mini
To solve for the longest and shortest possible distances Kevin could have run, given that the median distance is 3 miles and the range is 2 miles, we can analyze the information step by step.
1. **Understanding median**: The median of a dataset of 5 values means that the third value, when arranged in order, is the median. Thus, when the distances Kevin ran are arranged in ascending order: \( a_1, a_2, a_3, a_4, a_5 \), we know:
\[
a_3 = 3
\]
2. **Understanding range**: The range is the difference between the maximum and minimum values. Given that the range is 2 miles, we have:
\[
a_5 - a_1 = 2
\]
3. **Expressing the distances**: If we let the shortest distance be \( a_1 \) and the longest distance be \( a_5 \), we can express \( a_5 \) in terms of \( a_1 \):
\[
a_5 = a_1 + 2
\]
4. **Finding the values**: Since \( a_3 = 3 \) must also be between \( a_1 \) and \( a_5 \), we can deduce:
- \( a_1 \) must be less than or equal to 3 (because \( a_3 \) is the middle value).
- \( a_5 \) must be greater than or equal to 3.
5. **Determining shortest distance \( a_1 \)**:
The maximum \( a_1 \) can be is 3, hence it cannot exceed 3:
- If \( a_1 = 3 \), then \( a_5 = 3 + 2 = 5 \), which allows for \( a_2 \) and \( a_4 \) also as 3.
- If \( a_1 = 2 \), then \( a_5 = 2 + 2 = 4 \), which is valid.
- If \( a_1 = 1 \), then \( a_5 = 1 + 2 = 3 \), which again is valid.
Thus, \( a_1 \) must be **at least 1** and **at most 3**.
6. **Finding longest distance \( a_5 \)**:
From \( a_1 \) being lowest at 1,
\[
a_5 = 1 + 2 = 3
\]
or if \( a_1 = 2 \), then
\[
a_5 = 2 + 2 = 4,
\]
or if \( a_1 = 3 \), then
\[
a_5 = 3 + 2 = 5.
\]
Therefore, the longest possible distance Kevin could have run in a day is **5 miles**, and the shortest possible distance is **1 mile**.
**Final values to enter:**
- Longest possible distance: **5 miles**
- Shortest possible distance: **1 mile**
1. **Understanding median**: The median of a dataset of 5 values means that the third value, when arranged in order, is the median. Thus, when the distances Kevin ran are arranged in ascending order: \( a_1, a_2, a_3, a_4, a_5 \), we know:
\[
a_3 = 3
\]
2. **Understanding range**: The range is the difference between the maximum and minimum values. Given that the range is 2 miles, we have:
\[
a_5 - a_1 = 2
\]
3. **Expressing the distances**: If we let the shortest distance be \( a_1 \) and the longest distance be \( a_5 \), we can express \( a_5 \) in terms of \( a_1 \):
\[
a_5 = a_1 + 2
\]
4. **Finding the values**: Since \( a_3 = 3 \) must also be between \( a_1 \) and \( a_5 \), we can deduce:
- \( a_1 \) must be less than or equal to 3 (because \( a_3 \) is the middle value).
- \( a_5 \) must be greater than or equal to 3.
5. **Determining shortest distance \( a_1 \)**:
The maximum \( a_1 \) can be is 3, hence it cannot exceed 3:
- If \( a_1 = 3 \), then \( a_5 = 3 + 2 = 5 \), which allows for \( a_2 \) and \( a_4 \) also as 3.
- If \( a_1 = 2 \), then \( a_5 = 2 + 2 = 4 \), which is valid.
- If \( a_1 = 1 \), then \( a_5 = 1 + 2 = 3 \), which again is valid.
Thus, \( a_1 \) must be **at least 1** and **at most 3**.
6. **Finding longest distance \( a_5 \)**:
From \( a_1 \) being lowest at 1,
\[
a_5 = 1 + 2 = 3
\]
or if \( a_1 = 2 \), then
\[
a_5 = 2 + 2 = 4,
\]
or if \( a_1 = 3 \), then
\[
a_5 = 3 + 2 = 5.
\]
Therefore, the longest possible distance Kevin could have run in a day is **5 miles**, and the shortest possible distance is **1 mile**.
**Final values to enter:**
- Longest possible distance: **5 miles**
- Shortest possible distance: **1 mile**