City A and City B are located 610 kilometers (km) apart from each other on Highway X. At 8:00 a.m.,...

1. Translate the problem requirements: Alonzo starts at 8:00 AM from City A driving toward City B at 90 km/h. Ryan starts at 9:30 AM from City B driving toward City A at 100 km/h. We need to find how long Ryan drives before they meet.
2. Calculate Alonzo's head start advantage: Determine how far Alonzo travels during the 1.5 hour period before Ryan starts driving.
3. Set up the meeting point scenario: From 9:30 AM onward, both drivers are moving toward each other, so we can use their combined speed to determine when they meet.
4. Calculate Ryan's driving time: Find how long it takes from Ryan's start time until they meet, using the remaining distance and their relative speeds.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what's happening in everyday language first. We have two people driving toward each other on the same highway. Think of it like two friends walking toward each other in a long hallway - eventually they'll meet somewhere in the middle.

Here's what we know:

• Alonzo starts his journey at 8:00 AM from City A, heading toward City B
• Alonzo drives at 90 km/h
• Ryan starts later at 9:30 AM from City B, heading toward City A
• Ryan drives at 100 km/h
• The cities are 610 km apart
• We need to find how long Ryan drives before they meet

Process Skill: TRANSLATE - Converting the problem setup into clear, manageable pieces

2. Calculate Alonzo's head start advantage

Since Alonzo gets a head start, let's figure out how much of a advantage this gives him. From 8:00 AM to 9:30 AM, that's 1.5 hours where Alonzo is driving alone.

In plain English: If Alonzo drives for 1.5 hours at 90 km/h, how far does he travel?
Distance = Speed × Time = \(90 \text{ km/h} \times 1.5 \text{ h} = 135 \text{ km}\)

So by the time Ryan starts driving at 9:30 AM, Alonzo has already covered 135 km of the 610 km journey. This means there are \(610 - 135 = 475 \text{ km}\) remaining between them when Ryan starts.

3. Set up the meeting point scenario

Now here's the key insight: Starting from 9:30 AM, both drivers are moving toward each other. Think of it this way - every hour that passes, they get closer to each other by the sum of their speeds.

Alonzo continues at 90 km/h toward Ryan
Ryan drives at 100 km/h toward Alonzo
Together, they're closing the gap at \(90 + 100 = 190 \text{ km/h}\)

This is like saying the distance between them shrinks by 190 km every hour.

4. Calculate Ryan's driving time

Now we can find when they meet. We know:

• Starting distance between them (at 9:30 AM): 475 km
• Combined speed of approach: 190 km/h

Time until they meet = Distance ÷ Combined Speed
Time = \(475 \text{ km} \div 190 \text{ km/h} = 2.5 \text{ hours}\)

So Ryan drives for 2.5 hours = 2 hours and 30 minutes before meeting Alonzo.

Process Skill: VISUALIZE - Seeing the problem as two objects moving toward each other helps simplify the calculation

Final Answer

Ryan drives for 2 hours and 30 minutes before meeting Alonzo.

This matches answer choice A: "2 hours 30 minutes"

Verification: At 9:30 AM + 2.5 hours = 12:00 PM, Ryan will have traveled \(100 \times 2.5 = 250 \text{ km}\) from City B, and Alonzo will have traveled a total of \(90 \times 4 = 360 \text{ km}\) from City A (since he started at 8:00 AM). Indeed, \(250 + 360 = 610 \text{ km}\), confirming they meet exactly when the total distance is covered.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the timing sequence

Students often fail to recognize that Alonzo gets a 1.5-hour head start before Ryan begins driving. They might set up the problem as if both drivers start simultaneously at 8:00 AM, completely missing that Ryan starts at 9:30 AM. This leads to incorrect distance calculations and wrong meeting times.

2. Confusion about what distance to use for the meeting calculation

Students may incorrectly use the original 610 km distance for their meeting calculation instead of recognizing that they need to first subtract Alonzo's head start distance. They fail to understand that by 9:30 AM, the effective distance between the two drivers is reduced to 475 km, not the original 610 km.

3. Setting up individual position equations instead of using relative speed

Students might attempt to create separate distance equations for each driver and solve for when their positions are equal, making the problem unnecessarily complex. They don't recognize that this is a classic "closing gap" scenario where using combined speeds (190 km/h) provides a much simpler solution path.

Errors while executing the approach

1. Arithmetic errors in basic calculations

Students make computational mistakes such as incorrectly calculating 90 × 1.5 = 135 km for Alonzo's head start distance, or getting 610 - 135 = 475 km wrong. They might also err when adding the speeds: 90 + 100 = 190 km/h, or when dividing 475 ÷ 190 = 2.5 hours.

2. Time conversion errors

Students may struggle with converting between different time formats. For example, they might incorrectly calculate the 1.5-hour head start (from 8:00 AM to 9:30 AM) as 1 hour or 2 hours, or fail to properly convert 2.5 hours into "2 hours and 30 minutes."

Errors while selecting the answer

1. Answering the wrong question

Students calculate the correct 2.5 hours but then select an answer choice that represents Alonzo's total driving time (4 hours) instead of Ryan's driving time. They get confused about whose driving time the question is asking for, especially after working with both drivers' distances in their calculations.