On opening day a ride on a roller coaster takes 30 seconds and the time required for unloading and loading...

1. Translate the problem requirements: Understand that each "cycle" consists of a 30-second ride plus 2 minutes (120 seconds) of loading/unloading time. We need to find how many people ride per cycle given that 768 people rode in 1 hour.
2. Calculate the total time per cycle: Add the ride time and loading/unloading time to determine how long each complete cycle takes.
3. Determine the number of cycles in one hour: Divide the total time available (1 hour) by the time per cycle to find how many complete cycles occurred.
4. Calculate average people per ride: Divide the total number of people (768) by the number of cycles to find the average number of people per ride.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what's happening with this roller coaster in simple terms. Think of it like this: every time people want to ride, there's a complete "cycle" that happens.

Each cycle has two parts:
• The actual ride time: 30 seconds
• The time to get people off and new people on: 2 minutes

We're told that in one full hour, exactly 768 people got to ride. We need to figure out how many people were on the roller coaster during each individual ride.

Process Skill: TRANSLATE - Converting the problem's everyday language into clear mathematical understanding

2. Calculate the total time per cycle

Now let's figure out how long each complete cycle takes. Remember, we need to add the ride time and the loading/unloading time.

Let me convert everything to the same units first. Since we have seconds and minutes, let's use seconds:
• Ride time: 30 seconds
• Loading/unloading time: 2 minutes = \(2 \times 60 = 120\) seconds

Total time per cycle = \(30 + 120 = 150\) seconds

So every single cycle (one complete ride from start to finish) takes exactly 150 seconds.

3. Determine the number of cycles in one hour

Now I need to figure out how many of these 150-second cycles can fit into one hour.

First, let's convert one hour to seconds:
1 hour = 60 minutes = \(60 \times 60 = 3,600\) seconds

Number of cycles = Total time available ÷ Time per cycle
Number of cycles = \(3,600 \div 150 = 24\)

So in that one hour, the roller coaster completed exactly 24 full cycles (24 separate rides).

4. Calculate average people per ride

This is the final step! We know:
• Total people who rode: 768
• Total number of rides (cycles): 24

Average people per ride = Total people ÷ Number of rides
Average people per ride = \(768 \div 24 = 32\)

Let me verify this makes sense: \(32 \times 24 = 768\) people ✓

Final Answer

The average number of people per ride is 32.

Looking at our answer choices, this matches option B. 32.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what constitutes one complete cycle

Students often think that the 30-second ride time IS the complete cycle, forgetting to include the 2-minute loading/unloading time. This fundamental misunderstanding leads them to calculate 120 cycles per hour instead of 24, resulting in a completely wrong final answer.

2. Confusion about the timing sequence

Some students incorrectly assume that loading/unloading happens simultaneously with the ride or only applies to some rides. They fail to recognize that EVERY ride must include both the 30-second ride AND the 2-minute loading/unloading as one complete cycle.

3. Misinterpreting the constraint 'operates nonstop'

Students might think 'nonstop' means there's no loading/unloading time, or they might add extra idle time thinking the roller coaster has breaks. The key insight is that 'nonstop' simply means the cycles repeat continuously without additional delays.

Errors while executing the approach

1. Unit conversion errors

Students frequently make mistakes when converting between seconds and minutes. Common errors include: converting 2 minutes as 200 seconds instead of 120 seconds, or forgetting to convert 1 hour to 3,600 seconds and instead using 60 or 100.

2. Arithmetic calculation mistakes

When dividing \(3,600 \div 150\), students might make computational errors and get 25 or 20 instead of 24. Similarly, when calculating \(768 \div 24\), they might get 30 or 36 instead of 32 due to rushed calculations.

Errors while selecting the answer

1. Selecting an intermediate calculation as the final answer

Students might correctly calculate that there are 24 cycles per hour but then mistakenly select 24 (choice A) as their final answer, forgetting that the question asks for people per ride, not number of rides.