Problem 52 - Algebra for College Students

Question

Set up an algebraic equation and solve each problem. The ratio of male students to female students at a certain university is 5 to 7 . If there is a total of 16,200 students, find the number of male students and the number of female students.

Step-by-Step Solution

Answer

There are 6,750 male students and 9,450 female students.

Step 1: Understanding the Ratio

The ratio of male to female students is given as 5:7. This means for every 5 male students, there are 7 female students.

Step 2: Defining Variables

Let the number of male students be represented by \( 5x \) and the number of female students by \( 7x \). Here, \( x \) is the multiplier for the ratio components.

Step 3: Setting Up the Equation

Since the total number of students is 16,200, we can write the equation: \[ 5x + 7x = 16,200 \]. This equation represents the total number of students as the sum of male and female students.

Step 4: Solving for x

Combine like terms in the equation: \[ 12x = 16,200 \]. Then, solve for \( x \) by dividing both sides by 12: \[ x = \frac{16,200}{12} \]. Calculate \( x \): \( x = 1,350 \).

Step 5: Calculating Male and Female Students

Substitute \( x = 1,350 \) back into the expressions for male and female students: \( 5x = 5 \times 1,350 = 6,750 \) for the number of male students and \( 7x = 7 \times 1,350 = 9,450 \) for the number of female students.

Key Concepts

Ratios and Proportions

Understanding ratios and proportions is crucial, especially in situations where we compare quantities. A ratio is a comparison between two numbers showing how many times the first number contains the second. For instance, in the original problem, a university has a student ratio of 5 males to 7 females. This means that for every 5 male students, there are 7 female students.
A proportion, on the other hand, states that two ratios are equal. If we have a total of 16,200 students, a proportion helps us determine the actual counts of male and female students while maintaining this ratio. The beauty of proportions lies in this ability to scale the numbers up or down while preserving the relationship.

Solving Equations

Solving algebraic equations involves finding the value of unknown variables. In the provided problem, equations help us find the actual number of male and female students from the given ratio and total count. Start by expressing the ratio in equation form. Let the number of male students be represented by \(5x\) and female students by \(7x\). Here, \(x\) is a common multiplier.
To find \(x\), gather the total as an equation: \(5x + 7x = 16,200\). Simplify to get \(12x = 16,200\). By dividing both sides by 12, we solve for \(x\). Our solution shows \(x = 1,350\), allowing us to determine that there are 6,750 males and 9,450 females.
This step-by-step approach illustrates the power of equations in breaking down complex problems into manageable parts.

College-Level Algebra

College-level algebra typically tackles more sophisticated problems compared to high school algebra. It delves into complex equations, functions, and logical reasoning. The ability to set up and solve equations, such as those involving ratios and proportions, is a fundamental skill in college algebra. It extends to real-world applications like analyzing community demographics or even financial calculations.
Understanding relationships between quantities, as seen in the ratio problem, showcases analytical thinking, which is essential in various academic disciplines. College students often face such problems in statistics, economics, and science courses.

Student Demographics Analysis

Student demographics analysis plays a critical role in understanding and planning for student populations. By analyzing the demographic ratios—such as male to female ratios in our example—we can derive insights about the student body. Such analyses are essential for universities aiming to balance representation and allocation of resources.
In the given problem, analyzing the male to female student ratio allows insights into gender representation within the university. With a total of 16,200 students, the precise breakdown into 6,750 males and 9,450 females gives administrators data to inform decisions on facilities, recruitment, and policy adjustments to enhance diversity and equality.