To determine how many miles Francis can run in one hour, we start by calculating his speed in miles per hour.
Francis ran \(2 \frac{3}{5}\) miles in \(\frac{7}{12}\) of an hour. First, we convert \(2 \frac{3}{5}\) into an improper fraction.
\[ 2 \frac{3}{5} = 2 + \frac{3}{5} = \frac{2 \cdot 5}{5} + \frac{3}{5} = \frac{10}{5} + \frac{3}{5} = \frac{13}{5} \text{ miles} \]
Now we have that he ran \(\frac{13}{5}\) miles in \(\frac{7}{12}\) of an hour.
Next, to find the speed in miles per hour, we can set up the following calculation:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{\frac{13}{5}}{\frac{7}{12}} \]
When dividing fractions, we multiply by the reciprocal:
\[ \text{Speed} = \frac{13}{5} \times \frac{12}{7} \]
Next, we multiply the numerators and the denominators:
\[ \text{Speed} = \frac{13 \times 12}{5 \times 7} = \frac{156}{35} \]
Finally, we simplify the fraction \(\frac{156}{35}\). First, find the GCD of 156 and 35. The factors of 156 are \(2 \times 2 \times 3 \times 13\) and of 35 are \(5 \times 7\); they have no common factors other than 1, so the fraction is already in simplest form.
To express this as a decimal, we can perform the division:
\[ \frac{156}{35} \approx 4.4571 \]
Thus, Francis can run approximately \(4.4571\) miles in one hour. To be precise, he can run exactly \(\frac{156}{35}\) miles in one hour, or around \(4.46\) miles when rounded to two decimal places.
Therefore, the answer is:
\[ \boxed{\frac{156}{35}} \text{ miles per hour} \quad \text{(or approximately 4.46 miles per hour)} \]
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