At a certain college last year, a total of 260 students took College Algebra in either the fall semester or...

1. Translate the problem requirements: We need to understand that 260 students took College Algebra in at least one semester, 40 took it in both semesters (overlap), and fall enrollment is twice the spring enrollment. We're looking for the fall semester enrollment.
2. Set up variables for the unknown quantities: Define spring semester students as our base variable since fall is described in terms of spring, keeping our algebra simple.
3. Apply the overlap counting principle: Use the logical relationship that total unique students equals fall students plus spring students minus the overlap (students counted in both semesters).
4. Solve the resulting equation: Substitute the given relationships and solve for our variables to find the fall semester enrollment.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what the problem is telling us in plain English:

• A total of 260 students took College Algebra last year
• Some took it in fall only, some in spring only, and some in both semesters
• 40 students took it in both fall AND spring (these are the overlapping students)
• Fall enrollment was twice as large as spring enrollment

We need to find how many students took College Algebra in the fall semester.

2. Set up variables for the unknown quantities

Since we know that fall enrollment is twice the spring enrollment, let's use spring as our base variable:

Let \(\mathrm{S} = \text{number of students who took College Algebra in spring semester}\)

Then \(2\mathrm{S} = \text{number of students who took College Algebra in fall semester}\)

We also know:

• 40 students took it in both semesters
• 260 students took it in at least one semester

3. Apply the overlap counting principle

Here's the key insight: when we count students, we need to avoid double-counting those who took the class in both semesters.

Think of it this way: if we simply add fall students + spring students, we count the 40 'both semester' students twice. So we need to subtract them once to get the correct total.

In plain English:

Total unique students = Fall students + Spring students - Students counted twice

Substituting our known values:

\(260 = 2\mathrm{S} + \mathrm{S} - 40\)

4. Solve the resulting equation

Now we solve our equation step by step:

\(260 = 2\mathrm{S} + \mathrm{S} - 40\)

\(260 = 3\mathrm{S} - 40\)

\(260 + 40 = 3\mathrm{S}\)

\(300 = 3\mathrm{S}\)

\(\mathrm{S} = 100\)

So spring enrollment was 100 students.

Since fall enrollment is twice the spring enrollment:

\(\text{Fall enrollment} = 2\mathrm{S} = 2(100) = 200 \text{ students}\)

Let's verify: \(\text{Fall}(200) + \text{Spring}(100) - \text{Both}(40) = 200 + 100 - 40 = 260\) ✓

Final Answer

The number of students who took College Algebra in the fall semester was 200.

This matches answer choice E.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "twice as many" relationship

Students often confuse which quantity should be the base variable. They might set \(\text{Fall} = \mathrm{S}\) and \(\text{Spring} = 2\mathrm{S}\) instead of the correct \(\text{Spring} = \mathrm{S}\) and \(\text{Fall} = 2\mathrm{S}\). This happens because they don't carefully read that "twice as many took it in fall as in spring," which means fall is the larger quantity.

2. Forgetting to account for the overlap

Students may try to solve this as a simple addition problem: \(\text{Fall} + \text{Spring} = 260\), without realizing that the 40 students who took both semesters would be double-counted. They miss that the 260 represents unique students, not total enrollments.

3. Misunderstanding what the total represents

Some students might think the 260 represents total course enrollments (counting each semester separately) rather than unique students. This leads them to set up the equation as \(\text{Fall} + \text{Spring} = 260\) instead of using the overlap formula.

Errors while executing the approach

1. Sign error in the overlap equation

When setting up the equation \(\text{Total} = \text{Fall} + \text{Spring} - \text{Both}\), students sometimes write it as \(\text{Total} = \text{Fall} + \text{Spring} + \text{Both}\), incorrectly adding the overlap instead of subtracting it. This fundamental error in the inclusion-exclusion principle leads to wrong calculations.

2. Arithmetic mistakes in solving the linear equation

Students may make basic algebraic errors when solving \(260 = 3\mathrm{S} - 40\), such as incorrectly moving terms across the equals sign or making mistakes in division (getting \(\mathrm{S} = 90\) instead of \(\mathrm{S} = 100\)).

Errors while selecting the answer

1. Selecting spring enrollment instead of fall enrollment

After correctly calculating \(\mathrm{S} = 100\) (spring enrollment), students might select answer choice B (100) instead of recognizing that the question asks for fall enrollment, which is \(2\mathrm{S} = 200\). This happens when they lose track of what the question is actually asking for.