Marla starts running around a circular track at the same time Nick starts walking around the same circular track. Marla...

1. Translate the problem requirements: We need to find when Marla will be exactly 4 complete laps ahead of Nick. Marla runs at \(\mathrm{32\,laps/hour}\) and Nick walks at \(\mathrm{12\,laps/hour}\), both starting simultaneously.
2. Calculate the relative advantage rate: Determine how much faster Marla gains laps compared to Nick per unit time.
3. Set up the gap-closing equation: Use the relative rate to find when the 4-lap difference will be achieved.
4. Convert to required time units: Transform the result from hours to minutes as requested in the problem.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what's happening in plain English. Imagine Marla and Nick are both moving around the same circular track at the same time, but Marla is running much faster than Nick is walking.

Marla completes \(\mathrm{32}\) laps every hour, while Nick only completes \(\mathrm{12}\) laps in that same hour. We want to know: at what point in time will Marla have completed exactly 4 more complete laps than Nick?

Think of it like this: both start at the same starting line at the same time. As time passes, Marla pulls ahead because she's faster. We're looking for the specific moment when Marla has gone around the track exactly 4 more times than Nick has.

Process Skill: TRANSLATE - Converting the word problem into a clear mathematical relationship

2. Calculate the relative advantage rate

Now let's figure out how much faster Marla gains on Nick. Since they're both moving forward, what matters is the difference in their speeds.

Marla's speed: \(\mathrm{32\,laps\,per\,hour}\)
Nick's speed: \(\mathrm{12\,laps\,per\,hour}\)

The key insight is that Marla gains on Nick at a rate equal to the difference in their speeds:
Relative rate = \(\mathrm{32 - 12 = 20\,laps\,per\,hour}\)

This means that every hour, Marla completes \(\mathrm{20}\) more laps than Nick does. In other words, the gap between them grows by \(\mathrm{20}\) laps every hour.

3. Set up the gap-closing equation

Now we can use simple reasoning: if the gap grows by \(\mathrm{20}\) laps every hour, how long does it take for the gap to reach exactly \(\mathrm{4}\) laps?

We can think of this as a proportion:

• \(\mathrm{20}\) laps gained in \(\mathrm{1}\) hour
• \(\mathrm{4}\) laps gained in ? hours

Using basic proportional thinking:
Time needed = \(\mathrm{(4\,laps) ÷ (20\,laps\,per\,hour) = \frac{4}{20} = \frac{1}{5}\,hour}\)

So it takes \(\mathrm{\frac{1}{5}}\) of an hour for Marla to get exactly 4 laps ahead.

4. Convert to required time units

The problem asks for the answer in minutes, not hours, so we need to convert:

\(\mathrm{\frac{1}{5}\,hour = \frac{1}{5} × 60\,minutes = 12\,minutes}\)

Let's verify this makes sense: In \(\mathrm{12}\) minutes (which is \(\mathrm{\frac{1}{5}}\) of an hour):

• Marla completes: \(\mathrm{32 × \frac{1}{5} = 6.4\,laps}\)
• Nick completes: \(\mathrm{12 × \frac{1}{5} = 2.4\,laps}\)
• Difference: \(\mathrm{6.4 - 2.4 = 4\,laps}\) ✓

Final Answer

After \(\mathrm{12}\) minutes, Marla will have completed exactly 4 more laps than Nick. The answer is (C) 12.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding what "4 more laps" means

Students often confuse the relative difference with absolute positions. They might think they need to find when Marla has completed exactly 4 laps total, rather than understanding that they need to find when the gap between Marla's lap count and Nick's lap count equals 4. This leads them to set up equations like "\(\mathrm{32t = 4}\)" instead of the correct "\(\mathrm{32t - 12t = 4}\)".

2. Incorrectly modeling the simultaneous motion

Some students struggle with the concept that both people are moving at the same time from the same starting point. They might attempt to solve this as if Nick is stationary while Marla moves, or think that one person needs to complete a full lap before the other starts moving. This misunderstanding leads to using only Marla's speed in their calculations.

3. Confusing the relative speed concept

Students may not recognize that this is a relative speed problem. Instead of finding the difference in speeds (\(\mathrm{20\,laps\,per\,hour}\)), they might try to work with the individual speeds separately, leading to more complex and incorrect approaches like setting up separate time equations for each person.

Errors while executing the approach

1. Arithmetic errors in unit conversion

Even when students correctly identify that they need \(\mathrm{\frac{1}{5}}\) of an hour, they often make mistakes converting to minutes. Common errors include: \(\mathrm{\frac{1}{5} × 60 = 15\,minutes}\) (forgetting the fraction), or converting incorrectly by doing \(\mathrm{60 ÷ 5 = 15}\) instead of the correct \(\mathrm{\frac{1}{5} × 60 = 12\,minutes}\).

2. Decimal confusion in lap calculations

When verifying their answer, students sometimes get confused by the decimal lap counts (\(\mathrm{6.4}\) laps for Marla, \(\mathrm{2.4}\) laps for Nick). They may round these incorrectly or think that partial laps don't count, leading them to doubt their correct answer and recalculate unnecessarily.

Errors while selecting the answer

No likely faltering points - Once students have correctly calculated \(\mathrm{12}\) minutes, the answer selection is straightforward as it directly matches option (C) without requiring additional interpretation or conversion.