Problem 33 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the percent formula, \(A=P B: A\) is \(P\) percent of \(B,\) to solve Exercises \(27-42\) \(24 \%\) of what number is \(40.8 ?\)

Step-by-Step Solution

Answer

The number is 170.

Step 1: Identify and Rename the Variables

Identify \(P\), \(A\), and \(B\) from the problem. Here, \(P= 24 \%\), \(A= 40.8\), and \(B\) is what is to be found.

Step 2: Write the Percentage Equation

Write down the equation using the given inputs, which is \(40.8 = 24\% \cdot B\). But, to ease calculations, it's better to convert the percentage to its decimal equivalent. Hence, the equation becomes \(40.8 = 0.24 \cdot B\).

Step 3: Solve for B

Manipulate the equation to solve for \(B\). This can be done by dividing both sides of the equation by \(0.24\), which gives us \(B = \frac{40.8}{0.24}\).

Step 4: Calculate

Calculate the value to get \(B = 170\).

Key Concepts

Solving Percentage Problems

Understanding how to solve percentage problems is essential in algebra and everyday life. When dealing with such problems, it's important to clearly identify what you're looking for. Are you finding the percentage of a number, the total amount when given a percentage, or the percentage itself given a part and a whole?

Let's simplify the approach: First, identify the components involved in the percentage formula, which as per the standard notation is \( A = P\% \times B \), where \( A \) is the part, \( P\% \) is the percentage, and \( B \) is the whole. Then, translate the percentage problem into an algebraic equation. If necessary, convert percentages to decimals for easier calculation. Finally, manipulate the equation to isolate the variable you're solving for and perform the arithmetic to find your answer. This structured approach will help students breakdown percentage problems into manageable steps.

Percentage to Decimal Conversion

When solving percentage problems in algebra, converting percentages to decimals simplifies the process significantly. A percentage represents a number out of 100. To convert a percentage to a decimal, divide it by 100. This is done by moving the decimal point two places to the left.

For instance, to convert \( 24\% \) to a decimal, divide \( 24 \) by \( 100 \) or simply move the decimal point two places left to get \( 0.24 \). Remembering this will help students quickly make conversions without having to perform long division each time. It's a straightforward process that ensures that percentage problems remain within reach for learners of all levels.

Algebraic Equations

Algebraic equations are the backbone of algebra. They are mathematical statements that show the equality of two expressions. To solve an equation, you need to find the value of the variable that makes the statement true.

In the case of percentage problems, they often take the form \( A = P\% \times B \). To solve these equations, follow algebraic principles: keep the equation balanced by performing the same operations on both sides. When the percentage \( P\% \) is converted to a decimal, the equation looks more like classical algebra and can be tackled using familiar methods like addition, subtraction, multiplication, and division. Clearing away distractions and focusing on these foundational concepts will help any student navigate the world of algebraic equations with confidence.