If you run an average of 3 miles a day over 1 week and an average of 4 miles a day over the next 2 weeks, what is your average daily run distance over that 3-week period?

Before you solve this problem, explain why it makes sense that your average daily run distance over the 3-week period is not just the average of 3 and 4—namely, 3.5.

Should your average daily run distance over the 3 weeks be greater than 3.5 or less than 3.5?

Explain how to answer this without a precise calculation.

Now determine the exact average daily run distance over the 3-week period. Explain your solution.

### Reasoning Without Calculation

Let's first consider why the average daily run distance over the 3-week period should not simply be the average of 3 miles and 4 miles, which is 3.5 miles. 

When you average over different periods of time, you have to consider the number of days you are running at each average distance. In this case, you run 3 miles a day for 1 week (7 days) and 4 miles a day for 2 weeks (14 days). Since you spend more time running 4 miles per day (14 days) than you do running 3 miles per day (7 days), the overall average will be weighted more towards 4 miles than 3 miles. 

Therefore, without doing the precise calculation, we can reason that the average daily run distance over the 3-week period should be greater than 3.5 miles because you are running 4 miles a day for a longer period than you are running 3 miles a day.

### Exact Average Daily Run Distance Calculation

To find the exact average daily run distance over the entire 3-week period, we need to calculate the total number of miles run and divide that by the total number of days.

1. **Total Distance for the First Week:**
   \[
   3 \, \text{miles/day} \times 7 \, \text{days} = 21 \, \text{miles}
   \]

2. **Total Distance for the Next Two Weeks:**
   \[
   4 \, \text{miles/day} \times 14 \, \text{days} = 56 \, \text{miles}
   \]

3. **Total Distance Over 3 Weeks:**
   \[
   21 \, \text{miles} + 56 \, \text{miles} = 77 \, \text{miles}
   \]

4. **Total Number of Days:**
   \[
   7 \, \text{days} + 14 \, \text{days} = 21 \, \text{days}
   \]

5. **Average Daily Run Distance:**
   \[
   \text{Average daily distance} = \frac{77 \, \text{miles}}{21 \, \text{days}} = 3.67 \, \text{miles/day}
   \]

### Conclusion

The average daily run distance over the 3-week period is approximately **3.67 miles per day**. This is greater than 3.5 miles, which aligns with our initial reasoning since you spent more time running 4 miles per day.

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**Do you want more details or have any questions?**

### Relative Questions:
1. How would the average change if you ran 5 miles per day in the third week instead?
2. What if you only ran 2 miles per day in the first week? How would that affect the average?
3. How would the average daily distance be affected if the number of days in each week were different?
4. Can you generalize the formula to calculate the average daily run distance if the distances and weeks vary?
5. How would the average distance change if you ran 6 miles a day in the last week instead?

**Tip:** When calculating averages over different periods, always consider the duration each average applies to; longer periods weigh more heavily in the final average.

Learn how to calculate the average daily run distance over a 3-week period with detailed step-by-step explanation. Understand why the average is not simply the mean of different averages over distinct time periods. Suitable for Grades 6-8.

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