Two cyclists, 108 miles apart, start riding toward each other at the same time. One cycles 3 times as fast as the other. If they meet 3 hours later, what is the speed (in mi/h) of the faster cyclist? a. Write an equation using the information as it is given above that can be solved to answer this problem. Use the variable $$ r $$ to represent the speed of the slower cyclist. b. What is the speed of the faster cyclist? mi/hr

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Let's solve the problem step by step.

### **a. Writing the Equation**

Let $$ r $$ represent the speed of the slower cyclist in miles per hour (mi/h).

- **Speed of the slower cyclist:** $$ r $$ mi/h
- **Speed of the faster cyclist:** $$ 3r $$ mi/h (since one cycles 3 times as fast as the other)

Both cyclists are moving toward each other and meet after 3 hours. The total distance covered by both cyclists together will equal the initial distance between them, which is 108 miles.

**Distance covered by the slower cyclist:** 
$$ \text{Distance} = \text{Speed} \times \text{Time} = r \times 3 = 3r \text{ miles} $$

**Distance covered by the faster cyclist:** 
$$ \text{Distance} = \text{Speed} \times \text{Time} = 3r \times 3 = 9r \text{ miles} $$

**Total distance covered by both cyclists:**
$$ 3r + 9r = 12r $$

Since this total distance equals the initial distance of 108 miles:
$$ 12r = 108 $$

### **b. Finding the Speed of the Faster Cyclist**

Now, solve for $$ r $$:
$$ 12r = 108 $$
$$ r = \frac{108}{12} $$
$$ r = 9 \text{ mi/h} $$

**Speed of the faster cyclist:**
$$ 3r = 3 \times 9 = 27 \text{ mi/h} $$

### **Answer:**  
The speed of the faster cyclist is **27 mi/h**.
The faster cyclist is traveling at 27 miles per hour.

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