Problem 42 - 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 \(27-42\) If 8 is decreased to \(6,\) the decrease is what percent of the original number?

Step-by-Step Solution

Answer

The decrease from 8 to 6 is 25% of the original number.

Step 1: Compute the Decrease

First calculate the amount of decrease. This can be done by subtracting the new number from the original number. So the decrease is \(8 - 6 = 2\).

Step 2: Use the Percentage Formula

Next, use the formula for finding percentages, which is \(A = P*B\), where \(A\) is \(P\) percent of \(B\). In this case, the number 2 (the decrease) represents a certain percentage (\(P\)) of the original number (\(B = 8\)). Therefore, rearranging the formula to solve for \(P\) gives \(P = \frac{A}{B}\). Substitute \(A = 2\) and \(B = 8\) then calculate \(P\).

Step 3: Change Decimal to Percentage

After calculating \(P\), convert the decimal to a percentage by multiplying it by 100.

Key Concepts

Percentage Calculation

When tackling problems involving percents, it is essential to grasp how to calculate a percentage. It starts with the basic percent formula: \( A = P \times B \), where:

• \( A \) is the part or amount which represents the percentage of the whole.
• \( P \) is the percent, expressed as a decimal.
• \( B \) is the entire or original amount.

To find the percentage, you often need to rearrange the formula to isolate \( P \) if it is unknown. For instance, \( P = \frac{A}{B} \). This lets you divide the part by the whole, resulting in a decimal that represents the percentage when multiplied by 100.

Always remember to convert the decimal into a readable percentage by multiplying it by 100. This provides the clear percent value, which is how we typically understand percentages in everyday contexts.

Percent Decrease

Understanding percent decrease is key to solving many real-world problems, such as determining discount amounts or loss in value. In the given exercise, you are required to find the decrease from 8 to 6 as a percentage of 8, the original number.

First, identify the decrease by subtracting the new value (6) from the original value (8). Here, the decrease is 2. This value represents the decrease amount, \( A \), in the percent formula, relative to the original number, \( B \), which is 8.

• The percent decrease formula derived is \( P = \frac{A}{B} \times 100 \).

Inserting the known values, \( A = 2 \), and \( B = 8 \), we find \( P = \frac{2}{8} \times 100 = 25\% \). This reveals that a 25% decrease has occurred.

This calculation is extremely useful in both academic and everyday contexts, making it quicker and easier to understand changes in values, such as price reductions or comparing quantities.

Percentage Formula Application

The percentage formula \( A = P \times B \) is more than just a mathematical equation; it is a useful tool for countless real-life applications. Being able to interpret it flexibly allows you to handle a variety of problems.

For instance, beyond just solving percent increase or decrease, the formula helps in adjusting recipes, calculating tips, or understanding statistics. Whenever you deal with parts of a whole, percentages provide a clear representation.

• Suppose you're asked to find what percent a particular value is of another; rearranging the formula \( P = \frac{A}{B} \times 100 \) is your go-to step.
• If needing to find the resulting part after applying a percentage, you directly apply \( A = P \times B \).

Remember, practicing this formula in different scenarios sharpens the skill, ensuring ease of application in diverse situations. It simplifies complex proportional relationships into understandable numbers, enhancing both efficiency and accuracy in everyday math tasks.