Problem 77 - Algebra and Trigonometry

Question

At the north campus of a performing arts school, 10% of the students are music majors. At the south campus, 90% of the students are music majors. The campuses are merged into one east campus. If 42% of the 1000 students at the east campus are music majors, how many students did each of the north and south campuses have before the merger?

Step-by-Step Solution

Answer

The North campus had 600 students and the South campus had 400 students before the merger.

Step 1: Define the Variables

Let \( N \) represent the number of students at the North campus and \( S \) represent the number of students at the South campus. According to the problem, it is known that \( N + S = 1000 \), which is the total number of students.

Step 2: Formulate the Weighted Average

The fact that 42% of the total students on the East campus are music majors implies a weighted average of the proportions of music majors in the North and South campuses. This leads to the following equation: \( 0.10N + 0.90S = 0.42 * 1000 \).

Step 3: Solve the Equations

Now there are two equations that can be solved simultaneously. The first one is \( N + S = 1000 \) and the second one is \( 0.10N + 0.90S = 420 \). Multiplying the first equation by 0.90 gives \( 0.90N + 0.90S = 900 \). By subtracting this from the second equation we get \( -0.80N = -480 \), which leads to \( N = 600 \) and \( S = 400 \) by substituting \( N = 600 \) into the first equation.

Key Concepts

Percentage Problems

Percentage problems often involve figuring out both past and present scenarios using percentages to connect different stages of a problem. When working with percentages, it is essential to understand that percentages represent a part of a whole. In this exercise, we see that 10% and 90% of the students at the respective campuses were music majors, indicating their share of the total student population in those places.

To solve a percentage problem, it involves:

• Finding out what percentage represents in real numbers.
• Applying it across given data to find unknown variables, such as in this problem to determine student numbers from given percentages.

Percentage problems often require a keen eye on what 'part' and 'whole' represent in the specific context of the problem, allowing you to derive meaningful solutions through logical deductions.

Systems of Equations

Systems of equations are sets of equations with multiple variables that you solve together, as they are all interconnected. They are crucial in finding out unknowns in various mathematical problems, much like in our exercise. Here, two equations were essential: one giving the total number of students and another giving the composition based on music majors.

To tackle systems of equations:

• Identify the equations involved from the problem's narrative.
• Use methods like substitution or elimination to solve for one variable at a time.

In this exercise, after forming the equations from the problem, we use elimination to reduce the problem to a single variable equation, making it easier to compute. Mastery in forming and solving systems of equations opens doors to solving more complex algebraic problems effectively.

Weighted Averages

Weighted averages allow for more nuanced averaging and balancing out different quantities or sets. They often account for differing importance or impact when averaging, as not all parts carry the same weight. In this given exercise, we identified how the weighted average of music majors across two campuses was calculated.

Key steps to handle weighted averages include:

• Recognizing each component's weight, e.g., the percentage of music majors per campus.
• Calculating a combined average reflecting those weights to find an accurate average fitting all factors.

By sorting through the campus percentages, we determined how the overall 42% was achieved, helping to retroactively calculate the campus distribution. Understanding weighted averages is crucial, as it helps accurately depict situations with varying impacts, driving more realistic interpretations and solutions.