Question

There are 35.000 students at State University, Each student takes four different courses each term. State University offers 999 courses each term. The largest classroom on campus holds 135 students. Is this a problem? If so, what is the problem?

Short answer

Yes, the average class size exceeds classroom capacity, indicating a problem.

Step-by-step solution

Understanding the Problem

State University has 35,000 students, each taking 4 courses. The university offers a total of 999 courses, and each classroom can hold a maximum of 135 students.

Calculate Total Course Enrollments

First, calculate the total course enrollments by multiplying the number of students by the number of courses each student takes. This is 35,000 students times 4 courses per student.\[ 35,000 \times 4 = 140,000 \]

Calculate Average Students per Course

To find the average number of students per course, divide the total course enrollments by the number of courses offered:\[ \frac{140,000}{999} \approx 140.14 \]This means, on average, about 140 students enroll in each course.

Compare Average Enrollment to Classroom Capacity

The largest classroom capacity is 135 students, while the average number of students per course is approximately 140.14. Since 140.14 is greater than 135, the average class size cannot be accommodated in the largest classroom.

Identify the Problem

The problem here is that the average course enrollment exceeds the classroom capacity, indicating that the university may not have classrooms large enough to accommodate the typical class without splitting it into smaller sections.

Key concepts

Combinatorics

Combinatorics is a branch of mathematics dealing with counting, arrangement, and combination of objects. In this problem, it helps us to systematically calculate scenarios involving large numbers. Here, we have 35,000 students, each taking four courses, contributing to a grand total of 140,000 course enrollments.

• Single Student's Courses: For one student, selecting 4 courses out of 999 cannot directly use combination methods since each student independently selects courses.
• Course Distribution: We instead multiply the total number of students by their courses. Combinatorics in deeper scenarios helps when different arrangements or selections from a larger pool of courses might be necessary.

Understanding combinatorics ensures we accurately form scenarios like this and predict outcomes based on given data.

Problem-Solving

Effective problem-solving starts with clear understanding. Here, we identify key figures: 35,000 students, each taking 4 courses, and 999 available courses. Next, calculate total enrollments (140,000), recognizing constraints like classroom capacity.

• Step-by-Step Approach: By analyzing each element, we systematically assess whether logistics match reality, providing a way to identify potential issues early on.
• Meeting Constraints: The main challenge is the mismatch between classroom size and average enrollment. Clearly identifying such mismatches is crucial in problem-solving, allowing further steps to adapt physical space or altering course sizes.

Breaking problems into smaller, manageable parts ensures no factor is overlooked, making complex tasks simpler.

Average Calculation

Average calculation is a fundamental skill in mathematics, crucial for fair representation of data. It takes total values and divides them among group members to find a comparative middle value. Here, we calculate the average students per course.

• Enrolling Average: With 140,000 total enrollments and 999 courses, we divide enrollments by courses, giving us an average of about 140 students per course.
• Capacity Comparison: This average surpasses the classroom maximum of 135, highlighting any discrepancy between expectations and real logistical planning.

Average calculations bring visibility to what might not be immediately obvious in numbers, guiding institutions in planning and decision-making.