Problem 32 - 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 \(27-42\) 8 is \(40 \%\) of what?

Step-by-Step Solution

Answer

B = 20. So, 8 is 40% of 20.

Step 1: Understand the Given Values

In this problem, we know that A = 8 (which is 40% of B) and P = 40% = 0.40 (convert the percentage to a decimal by dividing by 100). We need to find B.

Step 2: Substitute the Given Values into the Formula

Substitute A = 8 and P = 0.40 into the formula. It will look like this: 8 = 0.40*B.

Step 3: Calculate the Unknown

Now, we solve the equation for B. We do this by dividing both sides of the equation by 0.40. This gives: B = 8 / 0.40

Key Concepts

Percent Problems

Understanding percent problems is crucial for mathematical literacy, as they frequently arise in various real-world contexts like finance, sales, and statistical data interpretation. Percent means 'per hundred' and is symbolized by the '%' sign. When faced with a percent problem, such as '8 is 40% of what?', it is essential to identify the base value (B) being sought, the percentage (P), and the part of the base value (A) represented by this percentage.

In the given exercise, we aim to find the value that represents 100% when 8 is known to be 40% of it. The core of these problems lies in setting up the correct proportion or equation that will help solve for the unknown variable by applying algebraic techniques.

Algebraic Equations

Algebraic equations are mathematical statements that express the equality between two algebraic expressions, often involving unknown variables. They form the backbone of solving percent problems algebraically. When we convert the statement '8 is 40% of what?' into an algebraic equation, we translate the word 'is' into '=', the percentage into a decimal, and the 'what' into a variable, commonly 'B' in percent formulas.

This transformation allows us to work with the equation algebraically to find the unknown variable. In our example, the equation takes the form of 8 = 0.40*B, which can be manipulated using arithmetic operations to solve for B, the base value, providing us with a clear solution to the problem.

Mathematical Problem-Solving

Mathematical problem-solving encompasses understanding the problem, devising a plan, implementing the plan, and reviewing the result. In percent problems, this translates to recognizing which value corresponds to the part, percent, or whole; determining the right equation or strategy to use; solving for the unknown; and then checking if the solution makes sense within the context of the problem.

Using the percent formula, we substitute known values into the equation and apply arithmetic operations to solve for the unknown. The process involves logical thinking and manipulation of algebraic expressions, always keeping in mind the relationship within the percent formula, and ensuring that the final answer is consistent with the given information.

Converting Percentages to Decimals

Converting percentages to decimals is a vital step in solving percent problems within algebra. This is done by dividing the percentage by 100, essentially moving the decimal point two places to the left. For example, to convert 40% into a decimal, we divide 40 by 100, yielding 0.40. This conversion is necessary because it transforms the percent into a form that can be used in algebraic equations.

Once the percent is converted into a decimal, it can be multiplied with other numbers or used to express proportions, as seen in the example where P = 40% becomes P = 0.40. This allows us to set up the percent equation in a straightforward, solvable manner, enabling us to find the unknown base (B) in our problem.