Yesterday, Candice and Sabrina trained for a bicycle race by riding around an oval track. They both began riding at...

1. Translate the problem requirements: We need to find when Candice and Sabrina will be together at the starting point again. This happens when the total time elapsed is a multiple of both their individual lap times (42 seconds for Candice, 46 seconds for Sabrina).
2. Find the synchronization time: Calculate the least common multiple (LCM) of 42 and 46 seconds to determine when they'll both complete whole laps simultaneously.
3. Convert time to Candice's lap count: Divide the synchronization time by Candice's lap time to find how many laps she completed.
4. Verify the solution: Check that both cyclists complete whole numbers of laps at this time.

Execution of Strategic Approach

1. Translate the problem requirements

Let's think about what's actually happening here. Candice and Sabrina start at the same point and ride around a track at different speeds. Candice is faster, completing each lap in 42 seconds, while Sabrina takes 46 seconds per lap.

The question asks when they'll be together at the starting point again. This happens when both cyclists have completed whole numbers of laps at exactly the same moment in time.

In other words, we need to find a time when:

• Candice has finished some whole number of laps (let's call it C laps)
• Sabrina has finished some whole number of laps (let's call it S laps)
• They both finish these laps at exactly the same moment

This means we need: \(\mathrm{C} \times 42 \text{ seconds} = \mathrm{S} \times 46 \text{ seconds}\)

Process Skill: TRANSLATE - Converting the physical scenario into a mathematical timing problem

2. Find the synchronization time

To find when they'll both complete whole laps simultaneously, we need the smallest time that is a multiple of both 42 and 46 seconds. This is called the least common multiple (LCM).

Let's find the LCM of 42 and 46:

First, let's find the prime factorization of each number:

• \(42 = 2 \times 3 \times 7\)
• \(46 = 2 \times 23\)

For the LCM, we take the highest power of each prime factor:

• Highest power of 2: \(2^1 = 2\)
• Highest power of 3: \(3^1 = 3\)
• Highest power of 7: \(7^1 = 7\)
• Highest power of 23: \(23^1 = 23\)

Therefore: \(\mathrm{LCM} = 2 \times 3 \times 7 \times 23 = 966 \text{ seconds}\)

This means after 966 seconds, both cyclists will be at the starting point together for the first time since they began.

3. Convert time to Candice's lap count

Now we need to find how many laps Candice completed in 966 seconds.

Since Candice completes each lap in 42 seconds:
Number of laps = Total time ÷ Time per lap
Number of laps = \(966 \div 42 = 23\) laps

4. Verify the solution

Let's check our answer:

• Candice: \(23 \text{ laps} \times 42 \text{ seconds/lap} = 966 \text{ seconds}\)
• Sabrina: How many laps did she complete? \(966 \div 46 = 21\) laps
• Sabrina: \(21 \text{ laps} \times 46 \text{ seconds/lap} = 966 \text{ seconds}\) ✓

Both cyclists are indeed at the starting point after 966 seconds, confirming our solution.

Final Answer

Candice completed 23 laps around the track the next time she and Sabrina were together at the starting point.

The answer is B. 23

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misunderstanding what "together at the starting point" means
Many students think this means when one cyclist catches up to the other anywhere on the track, rather than specifically when both have completed whole numbers of laps and are simultaneously at the starting point. This leads them to set up incorrect equations about relative speeds or distances.

Faltering Point 2: Not recognizing this as an LCM problem
Students may try to solve this using relative speed concepts (like "Candice gains X seconds per lap on Sabrina") instead of recognizing that we need the least common multiple of the two lap times. This approach becomes much more complicated and error-prone.

Faltering Point 3: Setting up the wrong equation for synchronization
Some students might set up equations like \(42\mathrm{C} = 46\mathrm{S} + \text{(some offset)}\) thinking one person has a head start, when actually both start together and we simply need \(42\mathrm{C} = 46\mathrm{S}\) where both C and S are whole numbers.

Errors while executing the approach

Faltering Point 1: Incorrectly calculating prime factorizations
When finding LCM(42, 46), students commonly make errors like writing \(42 = 6 \times 7\) instead of \(42 = 2 \times 3 \times 7\), or missing that \(46 = 2 \times 23\) (not recognizing 23 as prime). These errors lead to wrong LCM calculations.

Faltering Point 2: Incorrect LCM calculation from prime factors
Even with correct prime factorizations, students might multiply incorrectly \((2 \times 3 \times 7 \times 23)\) or forget to include all unique prime factors, leading to wrong values like 462 or 1932 instead of 966.

Faltering Point 3: Division errors when converting time to laps
When calculating \(966 \div 42\), students may make arithmetic mistakes, potentially getting 22 or 24 instead of 23, especially if doing the division by hand or making mental calculation errors.

Errors while selecting the answer

Faltering Point 1: Selecting Sabrina's lap count instead of Candice's
Since the verification shows Sabrina completes 21 laps in the same time, students who calculated both values correctly might accidentally select answer choice A (21) instead of B (23), misreading which cyclist the question asks about.

Faltering Point 2: Choosing the lap time instead of lap count
Students might see 42 or 46 in the answer choices and select one of these, confusing the given lap times (42 seconds, 46 seconds) with the number of laps completed, leading them to pick C (42) or D (46).