Problem 32 - Introductory Algebra for College Students

Question

Use the percent formula, \(A=P B: A\) is \(P\) percent of \(B,\) to solve. 8 is \(40 \%\) of what?

Step-by-Step Solution

Answer

The value of \(B\) is 20.

Step 1: Identify Given Values

From the problem, the given values are \(A = 8\) and \(P = 40\% = 0.4\).

Step 2: Rearrange the Formula

To make \(B\) the subject of the formula, the percentage formula should be rearranged as follows: \(B = A / P\).

Step 3: Substitute Given Values

Now, substitute \(A = 8\) and \(P = 0.4\) into the rearranged formula so it becomes \(B = 8 / 0.4\).

Step 4: Calculate B

By calculating the above, we find that \(B = 20\).

Key Concepts

Understanding Algebra in Percent Problems

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. When working with percent problems in algebra, it's helpful to know how to manipulate these symbolic equations to find the unknown variables. For percent problems like the one in the exercise, you use the percent formula, which is represented as:

• \( A = P \times B \) where \( A \) is \( P \) percent of \( B \).

Sometimes in algebra, you need to rearrange the formula to solve for different variables. In our case, we need to solve for \( B \). To simplify, you can isolate \( B \) by dividing both sides of the equation by \( P \), resulting in:

• \( B = \frac{A}{P} \)

This manipulation of the formula is a vital step in many algebra problems, especially those involving percentages. Understanding these manipulations can assist in forming a straightforward path to solve the problem.

Problem Solving with the Percent Formula

Problem-solving is a critical skill in mathematics, and solving percent problems often requires a structured approach. Here, we'll break down how to successfully use the percent formula in a problem-solving context. The problem given is to find what number, \( B \), 8 is 40% of.
The steps involved in solving this exercise are straightforward:

• First, identify the known values. The problem tells you that \( A = 8 \) and the percent \( P = 40\% \), which you should convert to a decimal \( 0.4 \).
• Next, rearrange the percent formula to solve for \( B \), which involves dividing \( A \) by \( P \): \( B = \frac{A}{P} \).
• Substitute the known values into the rearranged formula: \( B = \frac{8}{0.4} \).
• Finally, perform the division to find \( B \). In this case, \( B = 20 \).

Practicing this logical sequence in problem-solving helps improve mathematical reasoning skills, making it easier to solve similar math problems in the future.

Mathematics Education: Mastering Percentages

In mathematics education, understanding percentages is crucial, as they are widely used in various applications in real life. Teaching students about percentages often involves explaining their practical uses, such as in shopping discounts, finance interests, and science data analysis.
In the classroom, emphasis is placed on ensuring students can:

• Comprehend the concept of percentages as a portion of 100, allowing for easy comparison and measurement.
• Convert between percentages, decimals, and fractions, as this is essential for calculations and problem-solving.
• Apply the percent formula \( A = P \times B \) to real-world problems, as seen in the exercise.

By mastering these foundational skills, students build confidence in their abilities to tackle diverse mathematical challenges. The percent problem in this exercise is a practical example of how these principles are put into use, demonstrating how algebra can be used to solve everyday mathematical puzzles.