Problem 19 - 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. There are \(3,200\) students at our school. If \(52 \%\) of them are female, how many female students are there at our school?

Step-by-Step Solution

Answer

There are 1,664 female students.

Step 1: Understand the Problem

We know that there are a total of \(3,200\) students, and we are given that \(52\%\) of these students are female. We need to calculate the number of female students.

Step 2: Setting Up the Equation

The problem can be restated as the basic percent problem formula: "What is \(52\%\) of \(3,200\)?" This translates to the equation \(x = \frac{52}{100} \times 3,200\), where \(x\) is the number of female students.

Step 3: Solve the Equation

First, convert the percentage to a decimal by dividing by \(100\), thus \(52\% = 0.52\). Then multiply by the total number of students: \(x = 0.52 \times 3,200\).

Step 4: Calculate the Result

Perform the multiplication: \(x = 0.52 \times 3,200 = 1,664\).

Step 5: Conclusion

Thus, there are \(1,664\) female students at the school.

Key Concepts

Basic Percent Calculation

Understanding basic percent calculation is foundational for solving many mathematical problems. Percent means "per hundred," and it is represented by the symbol \(\%\). To convert a percentage into a decimal, which is often necessary for calculations, divide the percentage by 100. For instance, to convert \(52\%\) into a decimal, you would compute \(52 \div 100 = 0.52\).
This step is crucial because it allows you to perform further multiplication or division with actual numbers rather than with percentages.
It's important to note that percent calculations are frequently used to determine parts of a whole or evaluate percentage changes. For example, determining what percentage of a number represents or identifying the percentage increase or decrease in values over time.

Percent of a Quantity

To find the percent of a quantity, you'll often see phrases like "What is \(x\%\) of \(y\)?" This is a direct application of basic percent calculation.
Here you need to multiply the percentage (in its decimal form) by the total quantity. For example, if you want to find \(52\%\) of \(3,200\), you would set it up as \(0.52 \times 3,200\). The result of this multiplication will give you the part of the total quantity that corresponds to the given percentage.

• First, convert the percentage to a decimal: \(52\% = 0.52\).
• Next, multiply the decimal by the total: \(0.52 \times 3,200\).
• Finally, perform the calculation to find the result: \(1,664\).

This method is straightforward and applies to various scenarios, whether calculating discounts, interest rates, or any other percentage problems.

Percentage Equations

Percentage equations involve setting up expressions to find unknown quantities in percent problems. These equations often start with a form like: "What is \(part\) of \(whole\)\%?" and translate into linear equations.
The general formula used is \(part = \frac{percentage}{100} \times whole\). This is what you apply when you determine how many females are in the school population by setting the equation \(x = 0.52 \times 3,200\). Solving it gives you the number of female students, \(1,664\), by finding the specific part of the total defined by the percentage.

Such equations are helpful not only in educational settings but also in real-life situations like budgeting, calculating tips, or analyzing data. Understanding how to manipulate percentage equations enables you to solve complex, practical problems effectively.