Problem 33 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The number is 170.

Step 1: Identify known values

Identify the known values in the problem: \(A = 40.8\) and \(P = 24% = 0.24\).

Step 2: Substitute known values into percent formula

Substitute the known values into the percent formula \(A=P \cdot B\) giving you \(40.8 = 0.24 \cdot B\).

Step 3: Solve for unknown

Solve for \(B\) by dividing both sides of the equation by 0.24. This gives \(B = 40.8 / 0.24 \) which simplifies to \(B = 170\).

Key Concepts

Algebra

Algebra is a branch of mathematics that involves variables and numerical relationships. It allows us to create and manipulate mathematical expressions and equations. In this exercise, we are tasked with solving a problem using the percent formula. The formula given is \( A = P \cdot B \), which connects three variables: \(A\), \(P\), and \(B\), where \(A\) is a part of a whole represented by \(B\), and \(P\), the percentage, reflects the relationship between them.

In algebra, manipulation of equations involves performing operations equally on both sides to maintain the equation's balance. Here, we are solving for \(B\), which involves isolating \(B\) from the equation \(40.8 = 0.24 \cdot B\). By reorganizing the equation to solve for the unknown \(B\), you strengthen your understanding of algebraic manipulation. This requires dividing both sides of the equation by 0.24 to isolate \(B\), leading to \(B = 170\). When we understand how to isolate variables and solve equations, we unlock a powerful tool in algebra for understanding relationships within mathematical problems.

Problem Solving

Problem solving involves a methodical process that guides you from understanding a problem to finding its solution. The problem at hand asks us to determine "24% of what number is 40.8?" Each step in this process requires you to identify, substitute, and solve.

Start by breaking down the problem:

• Identify what you know: \(A = 40.8\) (the part) and \(P = 24\% = 0.24\) (the percentage).
• Substitute these values into the percent formula: \(40.8 = 0.24 \cdot B\).
• Finally, solve for the unknown \(B\) by isolating it in your equation, which results in \(B = 170\).

This structured approach ensures that each component is carefully considered and addressed, improving accuracy and confidence in problem solving. Persistence and practice in problem solving help develop critical thinking skills applicable to diverse situations. Make sure to validate your final answer by plugging it back into the context of the problem to ensure it makes sense.

Mathematical Formulas

Mathematical formulas are crucial tools that provide a definitive method to solve problems involving specific relationships. In this exercise, the percent formula \(A = P \cdot B\) is the key formula. It succinctly relates three elements: the part \(A\), the whole \(B\), and the percentage \(P\).

Understanding how to use this formula involves recognizing how percentages function as representations of proportions. For example, 24% is expressed as a decimal, 0.24, when used in calculations. This conversion is necessary because mathematical operations require numerical representations of percentages. The formula simplifies the process of discovering one value when others are known.

Mathematical formulas serve as efficient tools for organizing and processing information. By mastering them, you can tackle different mathematical problems systematically. It reflects a universal language that has diverse applications in not just math but also in fields like science, economics, and engineering. Hence, familiarity with such formulas extends your ability to interact with complex problems across various disciplines.