To solve this problem, we need to determine Vanna's average speed over a 15-minute period, where she walked at two different rates for different durations.

Step-by-step solution:

1. Break down the two intervals:

   • Vanna walked at a rate of 2 miles per hour for 10 minutes.
   • Vanna walked at a rate of 3 miles per hour for 5 minutes.

2. Convert the time into hours:

   • 10 minutes is $\frac{10}{60} = \frac{1}{6}$ hours.
   • 5 minutes is $\frac{5}{60} = \frac{1}{12}$ hours.

3. Calculate the distance for each segment:

   • Distance = rate × time.
   • For the first segment (2 mph for $\frac{1}{6}$ hours): $\text{Distance}_1 = 2 \times \frac{1}{6} = \frac{2}{6} = \frac{1}{3} \text{ miles}.$
   • For the second segment (3 mph for $\frac{1}{12}$ hours): $\text{Distance}_2 = 3 \times \frac{1}{12} = \frac{3}{12} = \frac{1}{4} \text{ miles}.$

4. Total distance: $\text{Total distance} = \frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12} \text{ miles}.$

5. Total time:

   • The total time is $15$ minutes, which is $\frac{15}{60} = \frac{1}{4}$ hours.

6. Average speed:

   • The average speed is the total distance divided by the total time. $\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} = \frac{\frac{7}{12}}{\frac{1}{4}} = \frac{7}{12} \times \frac{4}{1} = \frac{28}{12} = \frac{7}{3} \text{ miles per hour}.$

Thus, the correct answer is Choice G: $\frac{7}{3}$.

Let me know if you'd like further details or have any questions!

Related Questions:

1. How do you find the average speed when traveling at multiple speeds over different times?
2. How can unit conversion (minutes to hours) impact your calculation of speed?
3. What is the difference between average speed and average velocity?
4. How would the answer change if Vanna walked for 20 minutes instead of 15?
5. How can we check our work to ensure the distances add up to the correct total time?

Tip:

When working with time-based problems, always convert units (e.g., minutes to hours) consistently to avoid errors!