Problem 74 - Introductory and Intermediate Algebra for College Students 4th

Question

What does the percent formula, \(A=P B,\) describe? Give an example of how the formula is used.

Step-by-Step Solution

Answer

The percent formula \(A=P \times B\) is used to calculate the amount or total (A) for a particular percentage (P) of a base number (B). For instance, to figure out a student's test percentage, you could set it up like 84 = \(P\) x 100. Solving this equation, you find that \(P\) = 0.84 or 84%.

Step 1: Explain the Percent Formula

The percent formula is \(A=P \times B\), where:\n- \(A\) is the amount, which is the result of applying the percentage to the base.\n- \(P\) is the percentage we want to apply. It's important to remember to express this as a decimal in the calculations (for example, 20% becomes 0.20).\n- \(B\) is the base, which is the number we want to apply the percentage to.

Step 2: Give a Practical Example

As an illustration, let's say that a student scored 84 out of 100 on a test. To find out what percent the student scored, you could set \(A\) as 84 (the score), the base \(B\) as 100 (the total points possible) and solve for \(P\). This leads to the equation 84 = \(P\) x 100.

Step 3: Solve the Equation

In order to find the value of \(P\), you divide the amount 84 by the base 100. The calculation goes as follows: \(P\) = 84 / 100 = 0.84. So, the percentage score of the student is 0.84 or 84%.

Key Concepts

Percentage Calculation

Understanding percentage calculation is important in many aspects of everyday life.
Whether you're calculating discounts, determining interest on savings, or analyzing statistics, percentages are incredibly useful.
The percent formula is a primary mathematical tool used for these calculations. It is represented as:

• \(A = P \times B\)

In this formula:

• \(A\) is the amount, or the result of the percentage applied to the base.
• \(P\) is the percentage, expressed as a decimal for calculation purposes.
• \(B\) is the base, which is the preliminary value to which the percentage is applied.

For example, if you want to find 20% of 50, you convert the 20% to its decimal form, resulting in 0.20.
Then multiply 0.20 by 50 to find that \(A = 0.20 \times 50 = 10\).
So, 20% of 50 is 10.

Mathematical Formula

The mathematical formula for calculating percentages, \(A = P \times B\), is simple but powerful.
Here’s a quick breakdown of how you can use this formula:

• Identify the numbers in your problem as the amount \(A\), percentage \(P\), and base \(B\).
• Convert the percentage to a decimal form by dividing it by 100.
• Insert these numbers into the formula and solve to find the missing value.

This formula is versatile. It can be used to calculate increases or decreases in values by understanding the role each part plays in the equation.
Simply substituting different values lets you adapt this formula to a wide range of scenarios.

Practical Examples

Using practical examples can significantly help in understanding the percent formula.
For instance, consider a scenario where you're shopping and find a shirt that costs \(40 but is on a 25% discount.
You can calculate the discount amount by setting \(P = 0.25\) and \(B = 40\).The calculation would look like:

• \(A = 0.25 \times 40 = 10\)

Hence, the discount on the shirt is \)10.
Subtract this from the original price to find that you'll pay \(30.Another example is determining how much tax is added to a bill.
Suppose your restaurant bill is \)50, and the tax rate is 8%.

• Convert 8% to a decimal: \(P = 0.08\)
• Calculate the tax: \(A = 0.08 \times 50 = 4\)

Thus, \(4 is added to your bill for tax, totaling a \)54 payment.
These examples show the value of understanding percentage calculations in real-world situations.