At a certain university, the ratio of the number of teaching assistants to the number of students in any course...

1. Translate the problem requirements: The ratio of teaching assistants to students must be greater than 3:80, meaning \(\mathrm{TA/Students} > \mathrm{3/80}\). We need to find the maximum number of students when there are 5 TAs.
2. Set up the inequality constraint: Convert the ratio requirement into a mathematical inequality using the given number of teaching assistants (5).
3. Solve for the maximum student count: Rearrange the inequality to find the upper bound for the number of students that satisfies the constraint.
4. Determine the largest integer solution: Since we need a whole number of students, identify the largest integer that still satisfies the strict inequality requirement.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what this problem is really asking. We have a university rule that says: "The ratio of teaching assistants to students must always be greater than 3:80."

Think of this like a minimum staffing requirement. If a typical ratio might be 3 teaching assistants for every 80 students, this university wants to do better than that - they want more teaching assistants per student to ensure better support.

We're told there are 5 teaching assistants in our course, and we need to find the maximum number of students allowed while still satisfying this "better than 3:80" requirement.

In mathematical terms, this means:
Number of TAs / Number of Students > 3/80

For our specific case with 5 TAs:
\(\mathrm{5 / Number\ of\ Students} > \mathrm{3/80}\)

Process Skill: TRANSLATE - Converting the ratio constraint language into a mathematical inequality

2. Set up the inequality constraint

Now we have our inequality: \(\mathrm{5/Students} > \mathrm{3/80}\)

Let's call the number of students "S" to make this easier to work with:
\(\mathrm{5/S} > \mathrm{3/80}\)

We want to find the largest possible value of S that still makes this inequality true. To do this, we need to rearrange the inequality to get S by itself.

3. Solve for the maximum student count

Starting with: \(\mathrm{5/S} > \mathrm{3/80}\)

To solve for S, we can cross-multiply. When we do this, we need to be careful about the direction of the inequality sign.

Cross-multiplying gives us:
\(\mathrm{5 \times 80} > \mathrm{3 \times S}\)
\(\mathrm{400} > \mathrm{3S}\)

Now divide both sides by 3:
\(\mathrm{400/3} > \mathrm{S}\)

Let's calculate \(\mathrm{400/3}\):
400 ÷ 3 = 133.333...

So we have: \(\mathrm{S} < \mathrm{133.333...}\)

This means the number of students must be less than 133.333...

Process Skill: MANIPULATE - Carefully rearranging the inequality while preserving the constraint

4. Determine the largest integer solution

Since we can't have a fraction of a student, we need the largest whole number that is still less than 133.333...

The largest integer less than 133.333... is 133.

Let's verify this works by checking our original constraint:

• With 133 students: \(\mathrm{5/133} \approx \mathrm{0.0376}\) and \(\mathrm{3/80} = \mathrm{0.0375}\)
• Since 0.0376 > 0.0375, the constraint \(\mathrm{5/133} > \mathrm{3/80}\) is satisfied ✓

Let's also check that 134 students would violate the constraint:

• With 134 students: \(\mathrm{5/134} \approx \mathrm{0.0373}\) and \(\mathrm{3/80} = \mathrm{0.0375}\)
• Since 0.0373 < 0.0375, the constraint would be violated ✗

Process Skill: APPLY CONSTRAINTS - Ensuring we find the maximum value that still satisfies the strict inequality

Final Answer

The maximum number of students possible in a course with 5 teaching assistants is 133.

The answer is D. 133

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the direction of the inequality constraint

Students often misread "greater than 3:80" and set up the inequality as \(\mathrm{5/S} < \mathrm{3/80}\) instead of \(\mathrm{5/S} > \mathrm{3/80}\). This happens because they focus on the numbers rather than carefully reading that the ratio must be "greater than" the given benchmark, meaning more teaching assistants per student, not fewer.

2. Confusing which ratio to compare

Some students mistakenly think they need to compare students to teaching assistants (S:5) with the given ratio (80:3), setting up \(\mathrm{S/5}\) compared to \(\mathrm{80/3}\). However, the problem states the ratio of teaching assistants to students, so the correct setup should compare \(\mathrm{5/S}\) with \(\mathrm{3/80}\).

3. Misunderstanding what "maximum number of students" means in context

Students may not realize they're looking for the largest whole number that still satisfies the strict inequality. They might think they can use the boundary value or round up instead of recognizing that they need the largest integer strictly less than the calculated limit.

Errors while executing the approach

1. Sign error when cross-multiplying the inequality

When cross-multiplying \(\mathrm{5/S} > \mathrm{3/80}\), students sometimes incorrectly get \(\mathrm{5 \times 80} < \mathrm{3 \times S}\) instead of \(\mathrm{5 \times 80} > \mathrm{3 \times S}\). This fundamental algebraic error leads to \(\mathrm{S} > \mathrm{400/3}\) instead of \(\mathrm{S} < \mathrm{400/3}\), giving a completely wrong direction for the constraint.

2. Arithmetic mistakes in calculating 400/3

Students may incorrectly compute 400 ÷ 3, getting values like 132.something or 134.something instead of 133.333.... This calculation error directly impacts which integer they identify as the maximum allowable value.

3. Forgetting to flip inequality when solving for S

Some students correctly cross-multiply to get 400 > 3S but then incorrectly write \(\mathrm{S} > \mathrm{400/3}\) when dividing by 3, instead of maintaining \(\mathrm{S} < \mathrm{400/3}\). This leads them to think there's a minimum rather than a maximum constraint on student numbers.

Errors while selecting the answer

1. Using the boundary value instead of the largest integer below it

Since \(\mathrm{400/3} = \mathrm{133.333...}\), students might incorrectly round this to 133 and then think the answer is 134 (one more than 133), not realizing that 133 itself is the maximum integer that satisfies \(\mathrm{S} < \mathrm{133.333...}\).

2. Not verifying the final answer against the original constraint

Students may arrive at an answer but fail to check it by substituting back into \(\mathrm{5/S} > \mathrm{3/80}\). This verification step would catch errors and confirm that their chosen number of students actually satisfies the university's requirement while a higher number would violate it.