To find out how fast Yuki ran in miles per hour, we can use the formula for speed:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]
First, let's convert the distance Yuki ran, which is \(2 \frac{3}{4}\) miles, to an improper fraction:
\[ 2 \frac{3}{4} = \frac{2 \times 4 + 3}{4} = \frac{8 + 3}{4} = \frac{11}{4} \text{ miles} \]
Yuki ran this distance in \( \frac{2}{5} \) hour.
Now we can calculate the speed:
\[ \text{Speed} = \frac{\frac{11}{4} \text{ miles}}{\frac{2}{5} \text{ hour}} \]
Dividing by a fraction is the same as multiplying by its reciprocal:
\[ \text{Speed} = \frac{11}{4} \times \frac{5}{2} \]
Now, we can perform the multiplication:
\[ \text{Speed} = \frac{11 \times 5}{4 \times 2} = \frac{55}{8} \text{ miles per hour} \]
Finally, we can express \( \frac{55}{8} \) as a mixed number if desired:
\[ \frac{55}{8} = 6 \frac{7}{8} \text{ miles per hour} \]
Therefore, Yuki ran at a speed of:
\[ \frac{55}{8} \text{ miles per hour} \quad \text{or} \quad 6 \frac{7}{8} \text{ miles per hour}. \]