Cruz is training for a triathlon. He left his house at 6:00 and ran until 7:30. Then he rode his bike until 9:45. He covered a total distance of 51 miles. His speed while biking was 1.6 times his speed while running. What were cruz's biking and running speeds? 

Given:
1. Cruz runs from 6:00 to 7:30, which is 1.5 hours.
2. He bikes from 7:30 to 9:45, which is 2.25 hours.
3. Total distance = 51 miles.
4. His biking speed is 1.6 times his running speed.

Let's break this down:

### Let:
- Cruz's running speed = \( r \) miles per hour.
- His biking speed = \( 1.6r \) miles per hour.

### Step-by-step solution:

1. **Calculate distances covered:**  
   - **Distance running** = Speed × Time = \( r \times 1.5 \) miles.
   - **Distance biking** = Speed × Time = \( 1.6r \times 2.25 \) miles.

2. **Total distance equation:**  
   The sum of the distances covered while running and biking equals the total distance:
   \[
   1.5r + 1.6r \times 2.25 = 51
   \]

3. **Simplify the equation:**  
   \[
   1.5r + 3.6r = 51
   \]
   \[
   5.1r = 51
   \]

4. **Solve for \( r \):**  
   \[
   r = \frac{51}{5.1} = 10
   \]

### Conclusion:
- Cruz's running speed, \( r \), is 10 miles per hour.
- His biking speed, \( 1.6r \), is:
  \[
  1.6 \times 10 = 16 \text{ miles per hour}
  \]

### Final Answer:
- **Running speed:** 10 miles per hour.
- **Biking speed:** 16 miles per hour.

Would you like further details or have any questions?

### Related Questions:
1. How would Cruz's running and biking speeds change if the total distance was 60 miles instead of 51?
2. If Cruz increased his running speed by 2 miles per hour, how would that affect his total time?
3. What would Cruz's biking speed need to be if his running speed remained the same but he wanted to complete 51 miles in 3.5 hours?
4. How much time would Cruz save if his biking speed was twice his running speed?
5. If Cruz trained to maintain a constant speed throughout the triathlon, what should that speed be?

### Tip:
To solve problems involving different speeds and times, break them into smaller parts, setting up equations based on distance = speed × time.

Cruz is training for a triathlon. He left his house at 6:00 and ran until 7:30. Then he rode his bike until 9:45. He covered a total distance of 51 miles. His speed while biking was 1.6 times his speed while running. What were Cruz's biking and running speeds?

Learn how to solve a distance-time problem involving Cruz's triathlon training. Cruz ran for 1.5 hours and biked for 2.25 hours, covering 51 miles total. His biking speed was 1.6 times his running speed. This detailed step-by-step solution demonstrates how to find Cruz's biking and running speeds using algebraic equations. Suitable for students in Grades 8-10.

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