Of the 200 students at College T majoring in one or more of the sciences, 130 are majoring in chemistry...

1. Translate the problem requirements: We need to clarify what "majoring in one or more of the sciences" means - these 200 students can major in chemistry, biology, both, or other sciences (but at least 30 are in other sciences only). We need to find the possible range for students majoring in both chemistry and biology.
2. Set up the constraint relationships: Establish the total counts and identify what "at least 30 not in chemistry or biology" tells us about the maximum number who can be in chemistry or biology.
3. Find the minimum overlap scenario: Determine the smallest possible number of students majoring in both subjects by maximizing the number studying chemistry or biology exclusively.
4. Find the maximum overlap scenario: Determine the largest possible number of students majoring in both subjects by minimizing the number studying only one subject while respecting all constraints.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're working with in everyday terms. We have 200 students at College T who are majoring in "one or more of the sciences." This means these students could be:

• Majoring in chemistry only
• Majoring in biology only
• Majoring in both chemistry and biology
• Majoring in other sciences (like physics, mathematics, etc.) but not chemistry or biology

We're told that:

• 130 students are majoring in chemistry (some might also do biology)
• 150 students are majoring in biology (some might also do chemistry)
• At least 30 students are not majoring in either chemistry or biology (they're doing other sciences)

Our goal is to find the possible range for how many students are majoring in both chemistry and biology.

Process Skill: TRANSLATE - Converting the problem's language into clear mathematical relationships

2. Set up the constraint relationships

Now let's think about the constraints. If at least 30 students are not majoring in chemistry or biology, then at most 170 students (\(200 - 30 = 170\)) can be majoring in chemistry or biology or both.

This gives us our key constraint: The total number of students in chemistry or biology cannot exceed 170.

We can think of this visually as two overlapping circles (chemistry and biology) that together can contain at most 170 students.

Process Skill: INFER - Drawing the non-obvious conclusion about the maximum students in chemistry/biology

3. Find the minimum overlap scenario

To find the minimum number of students majoring in both subjects, we want to maximize how many students are in chemistry or biology while minimizing the overlap.

Let's call the number of students majoring in both chemistry and biology "x".

Students majoring in chemistry only: \(130 - x\)

Students majoring in biology only: \(150 - x\)

Students majoring in both: \(x\)

The total students in chemistry or biology = \((130 - x) + (150 - x) + x = 280 - x\)

Since this total cannot exceed 170:

\(280 - x \leq 170\)

\(280 - 170 \leq x\)

\(110 \leq x\)

So the minimum overlap is 110 students.

Process Skill: APPLY CONSTRAINTS - Using the maximum limit to find minimum overlap

4. Find the maximum overlap scenario

For the maximum overlap, we need to consider what limits how many students can be majoring in both subjects.

The maximum number of students who can major in both chemistry and biology is limited by whichever group is smaller. Since we have 130 chemistry majors and 150 biology majors, the maximum overlap cannot exceed 130 (you can't have more students doing both than are doing chemistry).

Let's verify this works with our constraint: If \(x = 130\), then:

Students in chemistry or biology = \(280 - 130 = 150\)

Students not in chemistry or biology = \(200 - 150 = 50\)

Since \(50 \geq 30\), this satisfies our "at least 30" requirement.

Therefore, the maximum overlap is 130 students.

Process Skill: CONSIDER ALL CASES - Checking both the logical maximum and constraint satisfaction

5. Final Answer

The number of students majoring in both chemistry and biology can range from 110 to 130.

Looking at our answer choices, this corresponds to option D: 110 to 130.

Verification:

• Minimum case (110): Total in chem/bio = \(280 - 110 = 170\), leaving exactly 30 in other sciences ✓
• Maximum case (130): Total in chem/bio = \(280 - 130 = 150\), leaving 50 in other sciences \((\geq 30)\) ✓

The answer is D.

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misinterpreting "one or more of the sciences"
Students often assume that all 200 students are majoring in either chemistry or biology only, missing that some students could be majoring in other sciences like physics or mathematics. This leads them to incorrectly set up the constraint that all 200 students must be accounted for within chemistry and biology circles only.

Faltering Point 2: Misunderstanding the "at least 30" constraint
Students frequently misinterpret "at least 30 students are not majoring in either chemistry or biology" as "exactly 30 students" rather than "30 or more students." This error prevents them from correctly establishing that at most 170 students can be in chemistry or biology, which is crucial for finding the minimum overlap.

Faltering Point 3: Confusing the constraint direction
Students sometimes think the constraint means "at least 30 students ARE majoring in chemistry or biology" instead of "at least 30 students are NOT majoring in either." This fundamental misreading completely reverses the constraint and leads to incorrect range calculations.

Errors while executing the approach

Faltering Point 1: Incorrect application of the inclusion-exclusion principle
When setting up the equation for total students in chemistry or biology, students often write it as \((130 + 150)\) instead of \((130 + 150 - x)\), forgetting to subtract the overlap. This leads to the wrong constraint equation and incorrect minimum value calculation.

Faltering Point 2: Sign errors in inequality manipulation
When solving \(280 - x \leq 170\), students frequently make algebraic errors, such as getting \(x \leq 110\) instead of \(x \geq 110\), or incorrectly rearranging to get \(280 \leq 170 + x\). These sign and manipulation errors lead to wrong minimum values.

Errors while selecting the answer

Faltering Point 1: Selecting based on partial analysis
Students often calculate either just the minimum or just the maximum value correctly, then select an answer choice that contains only one of their calculated endpoints without verifying that their range is complete. For example, they might find the minimum is 110 and select choice E (110 to 150) without properly calculating the maximum should be 130.