Jim began a 

233233​-mile 

bicycle trip to build up stamina for a triathlete competition.​ Unfortunately, his bicycle chain​ broke, so he finished the trip walking. The whole trip took 

66 

hours. If Jim walks at a rate of 

44 

miles per hour and rides at 

4242 

miles per​ hour, find the amount of time he spent on the bicycle.

Question content area bottom

Part 1

Jim rode for 

enter your response here 

hours.

We can solve this problem by setting up a system of equations.

### Let:
- \( t_b \) be the time Jim spent riding the bicycle.
- \( t_w \) be the time Jim spent walking.

We know the total time is 66 hours:
\[
t_b + t_w = 66 \quad \text{(Equation 1)}
\]

Next, we know the distances covered by walking and riding. Jim's total trip was 233 miles, so the distance covered riding and walking together is:
\[
r_b \cdot t_b + r_w \cdot t_w = 233 \quad \text{(Equation 2)}
\]
Where:
- \( r_b = 42 \) miles per hour (riding speed),
- \( r_w = 4 \) miles per hour (walking speed).

Now, we substitute these values into Equation 2:
\[
42 \cdot t_b + 4 \cdot t_w = 233
\]

We now have two equations:
1. \( t_b + t_w = 66 \)
2. \( 42t_b + 4t_w = 233 \)

### Solving the system of equations:

1. From Equation 1, solve for \( t_w \):
\[
t_w = 66 - t_b
\]

2. Substitute \( t_w = 66 - t_b \) into Equation 2:
\[
42t_b + 4(66 - t_b) = 233
\]
\[
42t_b + 264 - 4t_b = 233
\]
\[
38t_b = 233 - 264
\]
\[
38t_b = -31
\]
\[
t_b = \frac{-31}{38} \approx -0.82
\]

There seems to be an error in either calculation or setup. Let me rework the setup.It appears there's a mistake with the provided values or setup, as the solution for \( t_b \) (the time Jim spent on the bicycle) is yielding a negative value, which isn't physically meaningful. Let me verify the question's inputs (like speeds) or constraints for any potential errors or typos to correct this.

Can you please confirm the speeds mentioned in the problem (walking and riding)? Specifically, if it's 42 mph or something else.

Jim began a 233-mile bicycle trip to build up stamina for a triathlete competition. Unfortunately, his bicycle chain broke, so he finished the trip walking. The whole trip took 66 hours. If Jim walks at a rate of 4 miles per hour and rides at 42 miles per hour, find the amount of time he spent on the bicycle.

In this problem, Jim undertakes a 233-mile journey over 66 hours. With his bicycle speed at 42 miles per hour and walking speed at 4 miles per hour, we use a system of equations to find how much time he spent biking. This problem is suitable for Grades 9-11 and covers algebraic concepts and solving linear equations.

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