Problem 74 - Elementary Algebra

Question

The ratio of male students to female students at a certain university is 5 to 4 . If there is a total of 6975 students, find the number of male students and the number of female students.

Step-by-Step Solution

Answer

There are 3875 male students and 3100 female students.

Step 1: Understanding the Problem

We have a total of 6975 students with the ratio of male to female students given as 5 to 4. We need to find how many students are male and how many are female based on this ratio.

Step 2: Defining Variables

Let the number of male students be represented by \( 5x \) and the number of female students by \( 4x \). The total number of students is the sum of male and female students, given by: \[ 5x + 4x = 6975. \]

Step 3: Solving the Equation

Combine the terms in the equation: \[ 9x = 6975. \] Solve for \( x \) by dividing both sides by 9: \[ x = \frac{6975}{9} = 775. \]

Step 4: Calculating the Number of Male Students

The number of male students is given by \( 5x \): \[ 5x = 5 \times 775 = 3875. \]

Step 5: Calculating the Number of Female Students

The number of female students is given by \( 4x \): \[ 4x = 4 \times 775 = 3100. \]

Key Concepts

Male to Female Ratio

In mathematics, a ratio compares two quantities, showing how many times one value contains or is contained within the other. In this example, we have a male to female ratio of 5 to 4. This means for every 5 male students, there are 4 female students. Ratios are often used in everyday scenarios to give us an understanding of proportion and relationship between quantities.

When dealing with ratios, it helps to think of them as parts of a whole. For instance, in our problem, the total number of parts is 5 + 4 = 9. Understanding this is key to setting up the algebraic expression that will help find the actual counts of male and female students.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. Here, we use the variables to represent unknown quantities. In this exercise, the expressions for male and female students are represented as \( 5x \) and \( 4x \), respectively. The variable \( x \) is used to denote a common factor that when multiplied by these numbers gives us the actual counts.

In the context of this problem, the algebraic expression \( 5x + 4x \) sums the male and female students to provide the total number, 6975. This expression is critical to solving the problem as it allows us to set up and solve an equation to find \( x \), which will then help us find the specific numbers of male and female students.

The power of algebraic expressions lies in their ability to transform a word problem into a solvable mathematical equation.

Word Problems in Algebra

Word problems in algebra involve translating a narrative into a mathematical form. They require recognizing how the story told by words converts into mathematical components like expressions and equations. Often, word problems can be challenging because they involve not just math skills, but also reading comprehension and logical thinking.

In this exercise, the challenge is understanding the relationship between the male and female students expressed through the ratio, then using that understanding to form a math problem with the expressions \( 5x \) and \( 4x \) that predictably relate to the word problem's conditions.

The key to solving word problems is a systematic approach:

• Carefully read and understand each part of the problem.
• Define variables and form algebraic expressions that relate to those variables.
• Set up equations based on the conditions described in the problem.
• Solve the equations to find the desired quantities.

Mastering word problems is a valuable skill that enhances problem-solving abilities and mathematical understanding.