Two cyclists, 96 miles apart, start riding toward each other at the same time. One cycles 3 times as fast as the other. If they meet 4 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  to represent the speed of the slower cyclist.

To solve this problem, let's define the variables and set up an equation.

### Step-by-step Solution:

1. **Define the Variables:**
   Let \( x \) be the speed of the slower cyclist in miles per hour (mi/h).  
   The speed of the faster cyclist is 3 times that of the slower cyclist, so it is \( 3x \) mi/h.

2. **Understand the Situation:**
   The cyclists are 96 miles apart and are riding toward each other. They meet after 4 hours.

3. **Calculate the Distance Each Cyclist Travels:**
   The distance covered by each cyclist is equal to their speed multiplied by the time they ride.

   - Distance covered by the slower cyclist: \( 4x \) miles
   - Distance covered by the faster cyclist: \( 4(3x) = 12x \) miles

4. **Set Up the Equation:**
   The total distance they cover when they meet is 96 miles (the initial distance between them).

   Therefore, the equation becomes:
   \[
   4x + 12x = 96
   \]

### Final Equation:
\[
16x = 96
\]

This is the equation that can be solved to find the speed of the slower cyclist \( x \), and subsequently, the speed of the faster cyclist \( 3x \). 

Would you like to see the solution for this equation, or do you have any other questions? 

### Related Questions:
1. What is the speed of the slower cyclist in this scenario?
2. How much distance does each cyclist cover before they meet?
3. What would be the time taken for them to meet if the slower cyclist's speed was doubled?
4. If the distance between them were increased to 120 miles, how would that affect their meeting time?
5. How would the equation change if the faster cyclist was only 2 times faster than the slower one?

**Tip:** When solving problems involving two objects moving toward each other, always consider the combined rate of movement and how the distances add up.

Two cyclists, 96 miles apart, start riding toward each other at the same time. One cycles 3 times as fast as the other. If they meet 4 hours later, what is the speed (in mi/h) of the faster cyclist?

Learn how to solve the problem of two cyclists riding toward each other, one 3 times faster than the other, and meeting after 4 hours. This example demonstrates the use of algebraic equations and distance-speed-time relationships, suitable for students in Grades 8-10.

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