Problem 15 - Prealgebra

Question

Solve each of the following problems by first restating it as one of the three basic percent problems of Section 7.2 . In each case, be sure to show the equation. Number of Students If \(48 \%\) of the students in a certain college are female and there are \(1,440\) female students, what is the total number of students in the college?

Step-by-Step Solution

Answer

The total number of students in the college is 3,000.

Step 1: Identify the type of percentage problem

The problem involves finding the whole when a part and the percentage are known. Here, the number of female students is given, which is the part, and the percentage of female students in the college is given as well.

Step 2: Restate as a percentage equation

The problem asks for the total number of students, which is the whole. We can write the equation: \( 0.48 \times \text{Total number of students} = 1,440 \). Here, \(0.48\) is the decimal form of \(48\%\), and \(1,440\) is the number of female students.

Step 3: Solve for the total number of students

Rearrange the equation to find the total number of students: \( \text{Total number of students} = \frac{1,440}{0.48} \). Calculate the result: \( \text{Total number of students} = 3,000 \).

Key Concepts

Percentage Equation

A percentage equation is a fundamental concept that helps solve percentage-related problems by representing parts and wholes in an equation format. It is a mathematical expression that relates a part of a quantity, the percentage of that part, and the total quantity (the whole). In a usual setup, the percentage is expressed as a decimal. For instance, in the given exercise, we know that 48% of the students are female, and this percentage can be translated into a decimal as 0.48. This conversion is crucial because it turns the percentage into a format that we can use in calculations to find either the whole, the part, or the percentage itself.

To set up a percentage equation, identify the known values and place them in the relation:

• Part = Percentage (as a decimal) × Whole

This equation allows you to effortlessly solve for any of the unknowns, provided you have the others. Here, the equation becomes:

• 0.48 × Total number of students = 1,440

Finding the Whole

In percentage problems, finding the whole is a common task where you determine the total quantity when given a part and its percentage. Given the equation set up with the percentage equation, finding the whole becomes straightforward by isolating the whole on one side of the equation. The method is simple: divide the known part by the percentage in its decimal form.

In the context of our exercise, you have:

• Part = 1,440 (female students)
• Percentage of the part = 0.48

The formula to find the whole becomes:

• Total = Part ÷ Percentage (as a decimal)
• Total = 1,440 ÷ 0.48

By performing this division, you find that the total number of students is 3,000. This division separates the part from the percentage to reveal the entire population or quantity in question.

Prealgebra Percent Problems

Prealgebra percent problems introduce students to solving real-life scenarios involving percentages before delving into more advanced topics in algebra. These problems typically involve three key scenarios: finding the part, finding the percentage, and finding the whole when dealing with percentages. Each scenario utilizes the percentage equation but may require rearranging or different operations depending on the unknown.

When tackling a percentage problem in prealgebra, it is essential to:

• Identify what you know: the part, the percentage, or the whole.
• Determine what you need to find: is it the whole, part, or percentage?
• Translate the problem into a percentage equation.
• Perform mathematical operations like multiplication or division to solve the equation.

This structured approach helps learners understand how each component interacts, building a foundational skill set that will aid in later mathematical topics. Strengthening your understanding of these foundational problems enhances your ability to solve more complex problems involving percentages as you progress in your studies.