In a marketing survey, 60 people were asked to rank three flavors of ice cream, chocolate, vanilla, and strawberry, in...

1. Translate the problem requirements: Convert the given fractions into actual numbers of people and clarify what "vanilla before chocolate" and "vanilla before strawberry" mean in terms of ranking positions
2. Identify the ranking constraints: Determine how many people fall into each category based on the given information about vanilla's position relative to other flavors
3. Apply logical deduction to find overlaps: Since there are only 3 possible positions for vanilla (first, second, last), use the constraint that people ranking vanilla first must rank it before both other flavors
4. Calculate vanilla-first rankings: Find the number of people who rank vanilla before both chocolate and strawberry, which equals those ranking vanilla first

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're dealing with in plain English. We have 60 people ranking three ice cream flavors: chocolate, vanilla, and strawberry. Each person puts these three flavors in order from first choice to third choice, with no ties allowed.

The problem gives us three key pieces of information about vanilla's position:

• \(\frac{3}{5}\) of people ranked vanilla last (in 3rd position)
• \(\frac{1}{10}\) of people ranked vanilla before chocolate
• \(\frac{1}{3}\) of people ranked vanilla before strawberry

Let's convert these fractions to actual numbers of people:

• People who ranked vanilla last: \(\frac{3}{5} \times 60 = 36\) people
• People who ranked vanilla before chocolate: \(\frac{1}{10} \times 60 = 6\) people
• People who ranked vanilla before strawberry: \(\frac{1}{3} \times 60 = 20\) people

Process Skill: TRANSLATE - Converting the fraction language into concrete numbers makes the relationships much clearer to work with.

2. Identify the ranking constraints

Now let's think about what "vanilla before chocolate" and "vanilla before strawberry" actually mean in terms of positions.

Since there are only 3 positions (1st, 2nd, 3rd), vanilla can only be in one of these three spots for any person. We already know that 36 people put vanilla in 3rd place (last).

This means the remaining \(60 - 36 = 24\) people put vanilla in either 1st or 2nd position.

When someone ranks "vanilla before chocolate," vanilla must be in a higher position (lower number) than chocolate. Similarly for "vanilla before strawberry."

3. Apply logical deduction to find overlaps

Here's the key insight: if someone ranks vanilla FIRST, then vanilla automatically comes before both chocolate and strawberry.

Let's think about this step by step:

• 6 people rank vanilla before chocolate
• 20 people rank vanilla before strawberry
• Some people might rank vanilla before both flavors

Since vanilla can only be in position 1, 2, or 3, and we know 36 people put it in position 3, we need to figure out how the remaining 24 people split between positions 1 and 2.

If vanilla is in position 1 (first place), it's automatically before both other flavors.
If vanilla is in position 2 (second place), it can only be before ONE of the two remaining flavors (the one in 3rd place).

Process Skill: INFER - The critical insight is recognizing that ranking vanilla first means it comes before both other flavors simultaneously.

4. Calculate vanilla-first rankings

Let's call the number of people who rank vanilla first as 'x'. These x people contribute to both:

• People who rank vanilla before chocolate (6 total)
• People who rank vanilla before strawberry (20 total)

The remaining people who rank vanilla in 2nd position can rank it before only one other flavor each.

People with vanilla in 2nd position = \(24 - x\)

For vanilla to be before chocolate in exactly 6 cases total:
\(x + \text{(people with vanilla 2nd AND chocolate 3rd)} = 6\)

For vanilla to be before strawberry in exactly 20 cases total:
\(x + \text{(people with vanilla 2nd AND strawberry 3rd)} = 20\)

Since people with vanilla in 2nd position total \((24 - x)\), and they split between having chocolate last and strawberry last:
\((6 - x) + (20 - x) = 24 - x\)
\(26 - 2x = 24 - x\)
\(26 - 24 = 2x - x\)
\(2 = x\)

Therefore, 2 people ranked vanilla first.

Final Answer

The answer is 2 people ranked vanilla first, which corresponds to choice A.

We can verify: If 2 people rank vanilla first, then 22 people rank vanilla second. Of these 22 people, 4 rank chocolate last (so vanilla comes before chocolate), and 18 rank strawberry last (so vanilla comes before strawberry). This gives us \(2 + 4 = 6\) people with vanilla before chocolate, and \(2 + 18 = 20\) people with vanilla before strawberry, matching our requirements exactly.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "before" relationships

Students often confuse what "vanilla before chocolate" means in ranking context. They might think this means vanilla is ranked higher in preference (better liked) rather than understanding it means vanilla appears in an earlier position (1st or 2nd) compared to chocolate's position in the individual's ranking list.

2. Missing the constraint that rankings must be complete

Students may overlook that each person must rank all three flavors in positions 1, 2, and 3 with no ties. This constraint is crucial because it means if 36 people put vanilla last, then exactly 24 people must put vanilla in either 1st or 2nd position - there are no other possibilities.

3. Failing to recognize the overlap significance

Many students don't immediately see that people who rank vanilla first automatically satisfy both "vanilla before chocolate" AND "vanilla before strawberry" conditions simultaneously. They may try to treat these as completely separate, non-overlapping groups.

Errors while executing the approach

1. Incorrect setup of the constraint equation

When setting up the equation \((6 - x) + (20 - x) = 24 - x\), students often make algebraic errors. They might forget that the people ranking vanilla second must split between those who put chocolate last and those who put strawberry last, leading to incorrect equation formulation.

2. Arithmetic mistakes in fraction-to-number conversion

Students frequently make basic calculation errors when converting fractions to actual people counts: \(\frac{3}{5} \times 60\), \(\frac{1}{10} \times 60\), and \(\frac{1}{3} \times 60\). These early arithmetic errors cascade through the entire solution.

3. Mishandling the logical deduction about position constraints

Students may incorrectly reason about how many people can have vanilla in 2nd position while satisfying the "before" constraints. They might not properly account for the fact that someone with vanilla in 2nd position can only have it "before" one other flavor (whichever one they ranked 3rd).

Errors while selecting the answer

1. Confusing the question target

After solving correctly, students might accidentally select the number representing people who ranked vanilla second (22) or the total people who ranked vanilla in 1st or 2nd position (24) instead of specifically those who ranked vanilla first (2).

2. Failing to verify the solution

Students may arrive at \(x = 2\) but not double-check that this value actually satisfies all the original constraints. Without verification, they might second-guess their correct answer and change it to a different choice that "looks more reasonable" given the larger numbers in the problem.