Problem 85 - College Algebra Essentials

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 the north and south campuses have before the merger?

Step-by-Step Solution

Answer

Before the merger, the north campus had 600 students and the south campus had 400 students.

Step 1: Establish the system of equations

Let's say that the number of students in the north campus is \(N\) and the number of students in the south campus is \(S\). We know that \(N + S = 1000\), because the total number of students is 1000. Additionally, the percentage of music students in the north and south campuses when merged together is 42%, or 420 students. We can state this as \(0.10N + 0.90S = 420\). This gives us a second equation.

Step 2: Solve the system of equations

We can solve this system of equations either by substitution or elimination. However, since none of the equations have been solved for either variable, using elimination would be easier. Multiply the first equation by 0.10 and the second equation by 1 to eliminate \(N\): \n\n0.10N + 0.10S = 100 \n0.10N + 0.90S = 420 \n\nSubtracting these two equations gives: \n0.80S = 320 \n\nSo, \(S = 400\). Substituting \(S = 400\) into the equation \(N + S = 1000\) gives \(N = 600\)

Step 3: Conclusion

Therefore, before the merger, the north campus had 600 students, and the south campus had 400 students.

Key Concepts

Systems of Equations

When tackling an algebra word problem, especially one involving mixing percentages, systems of equations are a reliable method for finding the solution. A system of equations involves more than one equation that shares common variables. To solve the word problem about the performing arts school, two equations are established from the given data:

• The total number of students at both campuses is 1000, giving us the equation: \( N + S = 1000 \)
• The second equation comes from the percentage of music students in each campus: \( 0.10N + 0.90S = 420 \). This equation is derived from figuring out how the percentages at north and south contribute to the total music students.

To solve these equations, you can use methods such as substitution or elimination. In this scenario, elimination is used to simplify the process by eliminating one variable, in this case, \( N \), making it easier to find the values of \( S \) and subsequently \( N \). Understanding systems of equations will greatly enhance your problem-solving skills, particularly in algebraic word problems.

Percentages

Percentages are a critical aspect of this problem and many real-world scenarios. They represent a fraction of 100 and are used to express proportions and comparisons between different quantities. In this exercise, we deal with converting the percentage of music majors into actual numbers.

• The north campus has \(10\%\) music majors, meaning \(0.10 \times N\).
• The south campus has \(90\%\) music majors, represented as \(0.90 \times S\).

When the two campuses merged, understanding this percentage conversion allows you to set up the second equation in the system of equations: \(0.10N + 0.90S = 420\). This means that 42% of the overall students, specifically 420 out of 1000, are music majors. Knowing how to work with percentages enables you to apply it to mix proportions from different groups, thereby simplifying and solving complex algebra problems.

Mathematical Reasoning

Mathematical reasoning is the logical thought process that helps solve problems. In problems involving systems of equations, knowing how to translate word problems into mathematical expressions is crucial. With the problem at hand, step-by-step reasoning is needed to interpret the percentages and their effects.
Firstly, create clear variables for unknowns, such as \(N\) for north campus and \(S\) for south campus students. This helps in visualizing the problem. Then, understand the role of each percentage. The reasoning process allows us to deduce relationships between variables and set up the equations.
Using mathematical reasoning, you then determine the best way to solve the equations, choosing between substitution or elimination. This reasoning not only finds solutions but also helps verify that those solutions make sense (for instance, checking that 600 and 400 add up to 1000 confirms the calculations). Developing strong reasoning skills aids in methodically approaching any word problem, ensuring accurate solutions.