Let's determine James' speed in miles per hour step by step.
First, we need to understand the distances and time given in the question.
1. Distance Conversion
James runs [tex]\(4 \frac{2}{3}\)[/tex] miles:
- Convert the mixed number to an improper fraction:
[tex]\[
4 \frac{2}{3} = 4 + \frac{2}{3} = \frac{12}{3} + \frac{2}{3} = \frac{14}{3}
\][/tex]
- As a decimal, [tex]\(4 \frac{2}{3}\)[/tex] is approximately [tex]\(4.66667\)[/tex] miles.
2. Time Conversion
The time taken is [tex]\(\frac{3}{4}\)[/tex] hour.
3. Speed Calculation
Speed is calculated using the formula:
[tex]\[
\text{Speed} = \frac{\text{Distance}}{\text{Time}}
\][/tex]
Substituting our values in:
[tex]\[
\text{Speed} = \frac{4.66667}{0.75} \approx 6.22222 \text{ miles per hour}
\][/tex]
Based on our calculations, James' speed in miles per hour is approximately [tex]\(6.22222\)[/tex], which as a mixed number is closest to [tex]\(6 \frac{2}{9}\)[/tex].
Thus, James' speed in miles per hour is:
[tex]\[
\boxed{6 \frac{2}{9}}
\][/tex]
