Mary passed a certain gas station on a highway while traveling west at a constant speed of 50 miles per...

1. Translate the problem requirements: Mary has a 15-minute head start at 50 mph, Paul travels at 60 mph. We need to find how long after Paul passes the gas station does he catch up to Mary.
2. Calculate Mary's head start distance: Determine how far ahead Mary is when Paul starts from the gas station.
3. Determine the closing rate: Find how much faster Paul gains on Mary each hour since he travels faster.
4. Calculate catch-up time: Divide Mary's head start distance by Paul's rate of gaining ground to find when they meet.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what's happening in plain English:

Mary drives past a gas station first, traveling at 50 mph. Then, 15 minutes later, Paul drives past the same gas station at 60 mph. Both are heading in the same direction (west). We want to know how long after Paul passes the gas station will he catch up to Mary.

Think of it like a race where one person gets a head start. Mary gets a 15-minute head start, but Paul is faster. The question is: how long will it take Paul to make up for that head start?

Process Skill: TRANSLATE - Converting the story into a clear mathematical scenario

2. Calculate Mary's head start distance

When Paul reaches the gas station, Mary has already been driving for 15 minutes. Let's figure out how far ahead she is at that moment.

Mary's speed is 50 miles per hour. In 15 minutes (which is 1/4 of an hour), she travels:

Distance = Speed × Time

Distance Mary is ahead = \(50 \text{ mph} \times \frac{1}{4} \text{ hour} = 12.5 \text{ miles}\)

So when Paul starts from the gas station, Mary is already 12.5 miles ahead of him.

3. Determine the closing rate

Now here's the key insight: since Paul is traveling faster than Mary, he gains ground on her every hour.

Paul travels at 60 mph while Mary travels at 50 mph. This means that every hour:

• Paul covers 60 miles
• Mary covers 50 miles
• So Paul gains \(60 - 50 = 10 \text{ miles}\) on Mary every hour

We call this the "closing rate" or "rate of gaining ground" = 10 miles per hour.

4. Calculate catch-up time

Now we can answer the main question using simple logic:

Mary has a 12.5-mile head start, and Paul gains 10 miles on her every hour.

Time for Paul to catch up = Head start distance ÷ Closing rate

Time = \(12.5 \text{ miles} \div 10 \text{ miles per hour} = 1.25 \text{ hours}\)

1.25 hours = 1 hour + 0.25 hours = 1 hour + 15 minutes = 1 hr 15 min

Final Answer

Paul will catch up with Mary 1 hour and 15 minutes after he passes the gas station.

Let's verify: After 1 hr 15 min (1.25 hours) from the gas station:

• Paul will have traveled: \(60 \times 1.25 = 75 \text{ miles}\)
• Mary will have traveled: \(50 \times (1.25 + 0.25) = 50 \times 1.5 = 75 \text{ miles}\)

(Note: Mary gets the extra 0.25 hours because of her head start)

Both have traveled 75 miles from the gas station, confirming they meet at this point.

Answer: D. 1 hr 15 min

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the timing reference point
Students often get confused about what time reference to use. The question asks "how long after he passed the gas station did Paul catch up with Mary" but students might calculate from when Mary passed the gas station instead. This leads to adding the 15-minute head start to their final answer, getting 1 hr 30 min instead of 1 hr 15 min.

2. Setting up the problem as a meeting point instead of a catch-up scenario
Some students treat this like two people starting from different locations moving toward each other, rather than recognizing it's a chase problem where both people are moving in the same direction with one person having a head start.

3. Failing to recognize the need to calculate Mary's head start distance
Students might jump straight into relative speed calculations without first determining how far ahead Mary is when Paul starts from the gas station. This crucial step of converting the 15-minute time advantage into a 12.5-mile distance advantage is often overlooked.

Errors while executing the approach

1. Time unit conversion errors
The most common error is incorrectly converting 15 minutes to hours. Students might use \(\frac{15}{100} = 0.15 \text{ hours}\) instead of \(\frac{15}{60} = 0.25 \text{ hours}\). This would give Mary a head start of \(50 \times 0.15 = 7.5 \text{ miles}\) instead of 12.5 miles, leading to an incorrect final answer.

2. Arithmetic errors in the final division
When calculating \(12.5 \div 10 = 1.25 \text{ hours}\), students might make computational errors or incorrectly convert 1.25 hours back to hours and minutes, potentially getting 1 hour 25 minutes instead of 1 hour 15 minutes.

Errors while selecting the answer

1. Converting from the wrong reference point
Even if students calculate correctly that it takes 1.25 hours, they might mistakenly think this is the time from when Mary started, leading them to subtract the 15-minute head start and select 1 hour (choice C) instead of 1 hr 15 min (choice D).