Let's go through the problem in the image step by step.

1. Yu Yan competes in a sprint triathlon: The race has three parts: a 0.5-mile swim, a 12.4-mile bike ride, and a 3.1-mile run.

   • Part a: Yu Yan completes the swim in 25 minutes. We need to calculate her average swimming speed in miles per minute:
     • The distance is 0.5 miles, and the time is 25 minutes.
     • The speed is $\frac{0.5 \text{ miles}}{25 \text{ minutes}} = 0.02 \text{ miles per minute}$.

   However, it seems the solution on the paper is incorrect. Let's fix this.

   $\text{Speed} = \frac{0.5 \text{ miles}}{25 \text{ minutes}} = 0.02 \text{ miles per minute}.$

   • Part b: Convert this to miles per hour. Since 1 hour = 60 minutes:

     • $0.02 \text{ miles per minute} \times 60 = 1.2 \text{ miles per hour}$.

   • Part c: We need to write an equation relating the total distance $d$ Yu Yan swims in miles to the number of minutes $m$ she swims. The general form of the equation is:

     $d = 0.02m.$

   • Part d: Write an equation relating the total distance $d$ Yu Yan swims in miles to the number of hours $h$ she swims. Since her swimming speed is 1.2 miles per hour:

     $d = 1.2h.$

   • Part e: The length of the pier is 0.2 miles. Yu Yan swims at her average speed of 1.2 miles per hour. How many minutes does it take her to swim this distance?

     • First, calculate the time in hours: $\frac{0.2}{1.2} = \frac{1}{6} \text{ hours}$.
     • Convert this to minutes: $\frac{1}{6} \times 60 = 10 \text{ minutes}$.

   • Part f: Yu Yan completes the 12.4-mile bike ride in 48 minutes. What is her average biking speed in miles per hour?

     • Convert 48 minutes to hours: $48 \text{ minutes} = \frac{48}{60} = 0.8 \text{ hours}$.
     • Speed is $\frac{12.4 \text{ miles}}{0.8 \text{ hours}} = 15.5 \text{ miles per hour}$.

   • Part g: Write an equation relating the total distance $d$ Yu Yan bikes to the time in hours $h$. Her biking speed is 15.5 miles per hour, so the equation is:

     $d = 15.5h.$

If you'd like, I can provide a more detailed explanation for each part. Do you have any questions or need further clarification on any of the steps?

Follow-up questions:

1. How is speed calculated when given time and distance?
2. Why is unit conversion (minutes to hours) important in this context?
3. How would you write an equation for running if her speed was known?
4. How would you calculate the total time for all three parts of the triathlon?
5. What if Yu Yan swam faster or slower? How would that affect the equations?

Tip:

When dealing with different units (e.g., minutes vs. hours), always ensure to convert them properly before calculating speeds or distances.