A survey was conducted to determine the popularity of 3 foods among students. The data collected from 75 students are...

1. Translate the problem requirements: We need to find students who like "none or only one" of the three foods (Pizza, Hoagies, Tacos). This means we want students who like exactly 0 foods OR exactly 1 food, excluding those who like 2 or 3 foods.
2. Calculate students who like exactly two foods: Use the given overlap data and subtract the "all three" group to find students who like exactly two foods but not the third.
3. Apply the inclusion-exclusion principle: Use the total count and all overlap information to find how many students like at least one food.
4. Find students liking exactly one food: Subtract those liking 2+ foods from those liking at least one food to get the "exactly one" group.
5. Calculate final answer: Add students who like none (total minus those liking at least one) to students who like exactly one food.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're looking for. The question asks for students who like "none or only one" of the three foods.

Think of it this way: imagine all 75 students standing in a room. Some students love multiple foods, some love just one food, and some might not like any of these three foods at all. We want to count only:

• Students who don't like Pizza, Hoagies, OR Tacos (none)
• Students who like exactly one food but not the other two (only one)

We do NOT want to count students who like 2 foods or all 3 foods.

Process Skill: TRANSLATE - Converting "none or only one" into clear mathematical categories

2. Calculate students who like exactly two foods

Let's figure out how many students like exactly 2 foods (but not all 3).

From the data:

• 28 students like both pizza and hoagies
• 37 students like both hoagies and tacos
• 40 students like both pizza and tacos
• 25 students like all three foods

Here's the key insight: when we say "28 like pizza and hoagies," this includes the 25 who like all three foods. So students who like EXACTLY pizza and hoagies (but not tacos) = \(28 - 25 = 3\).

Similarly:

• Exactly hoagies and tacos (but not pizza) = \(37 - 25 = 12\)
• Exactly pizza and tacos (but not hoagies) = \(40 - 25 = 15\)

Total students liking exactly 2 foods = \(3 + 12 + 15 = 30\)

3. Apply the inclusion-exclusion principle

Now let's find how many students like at least one food using inclusion-exclusion.

Start with individual preferences:

• 48 like Pizza + 45 like Hoagies + 58 like Tacos = \(151\)

But this counts overlaps multiple times, so subtract the two-food overlaps:

• \(151 - 28 - 37 - 40 = 46\)

But now we've subtracted the "all three" group too many times (we subtracted it 3 times but should only subtract it twice), so add it back:

• \(46 + 25 = 71\)

So 71 students like at least one food.

Process Skill: MANIPULATE - Systematically handling overlapping counts

4. Find students liking exactly one food

We know:

• 71 students like at least one food
• 25 students like all three foods
• 30 students like exactly two foods

Students who like exactly one food = \(71 - 25 - 30 = 16\)

5. Calculate final answer

Finally, we can answer the original question:

Students who like none of the foods = Total students - Students who like at least one
= \(75 - 71 = 4\)

Students who like exactly one food = 16 (from step 4)

Therefore, students who like "none or only one" = \(4 + 16 = 20\)

Final Answer

The number of students who like none or only one of the foods is 20.

This matches answer choice D.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "none or only one" as separate calculations: Students often think they need to find students who like "none" and students who like "only one" as completely separate problems, missing that the question asks for the sum of both groups.

2. Confusing "at least two" with "exactly two": When the problem states "28 like pizza and hoagies," students may incorrectly assume this means exactly pizza and hoagies (excluding those who like all three), rather than understanding this includes everyone who likes both pizza and hoagies, including those who also like tacos.

3. Attempting to solve without inclusion-exclusion principle: Students may try to directly count each category without recognizing this is a classic Venn diagram problem requiring systematic handling of overlapping sets.

Errors while executing the approach

1. Double-counting or under-counting in inclusion-exclusion: Students frequently make arithmetic errors when applying the inclusion-exclusion formula, particularly forgetting to add back the "all three" group after subtracting it too many times, or incorrectly handling the signs in the calculation.

2. Incorrect calculation of "exactly two" categories: Students often forget to subtract the "all three" group (25) from each pairwise intersection, leading to wrong values for students who like exactly two foods. For example, using 28 instead of \(28-25=3\) for those who like exactly pizza and hoagies.

3. Arithmetic errors in multi-step calculations: With multiple numbers to track (48, 45, 58, 28, 37, 40, 25), students commonly make basic arithmetic mistakes, especially when calculating \(48+45+58-28-37-40+25 = 71\).

Errors while selecting the answer

1. Providing partial answer instead of complete answer: Students may calculate either just the "none" group (4) or just the "only one" group (16) and select that as their final answer, forgetting that the question asks for students who like "none OR only one," requiring them to add both groups together.

2. Selecting intermediate calculation results: Students might select values they calculated during the solution process, such as 16 (exactly one food) or 30 (exactly two foods), instead of the final answer of 20.