Problem 1 - Prealgebra

Question

Find the percent of change. Round to the nearest tenth, if necessary. Then state whether the percent of change is a percent of increase or a percent of decrease. from \(\$ 50\) to \(\$ 67\)

Step-by-Step Solution

Answer

Increase of 34.0%

Step 1: Determine the amount of change

To find the percent change, first determine how much the initial value has changed. Subtract the original amount from the new amount: Change = New amount - Original amount = 67 - 50 = 17.

Step 2: Calculate the percent of change

The formula for the percent of change is: \[ \text{Percent of Change} = \left( \frac{\text{Change}}{\text{Original amount}} \right) \times 100 \% \]Substitute the numbers into the formula:\[ \text{Percent of Change} = \left( \frac{17}{50} \right) \times 100 \% \]

Key Concepts

Understanding Percent Increase

When we talk about percent increase, we refer to how much a quantity has grown compared to its original value. This is an essential mathematical concept that helps us measure growth in various contexts, like finance, population, or even sports statistics.

To calculate a percent increase:

• Find the difference between the new and the original value. This is often called the "increase amount."
• Use the formula: \[ \text{Percent Increase} = \left( \frac{\text{Increase Amount}}{\text{Original Value}} \right) \times 100 \% \]
• This will give you your increase expressed as a percentage.

Understanding percent increase is crucial for interpreting changes accurately and making informed decisions. You can see these calculations everywhere, from rising stock prices to increased population counts.

Understanding Percent Decrease

A percent decrease tells us how much a quantity has reduced, relative to its original size. This measure is useful for assessing reductions in areas like costs, quantities, or greenhouse gas emissions.

Here's how to determine the percent decrease:

• Identify the difference, or "decrease amount," by subtracting the new value from the original value.
• Apply the formula: \[ \text{Percent Decrease} = \left( \frac{\text{Decrease Amount}}{\text{Original Value}} \right) \times 100 \% \]
• This calculation provides the decrease as a percentage of the original amount.

Understanding percent decrease helps in evaluating how significant a reduction is, whether in a business budget cut or a decrease in pollution levels.

The Role of Mathematical Formulas

Formulas are like mathematical recipes, allowing us to solve problems systematically. For percent change, the main formula is: \[ \text{Percent of Change} = \left( \frac{\text{Change}}{\text{Original Amount}} \right) \times 100 \% \]This fundamental formula aids in both percent increase and decrease calculations.

Key aspects of using formulas include:

• A clear understanding of each part of the formula—what it represents and how it functions.
• Substituting the correct values into the formula accurately.
• Performing arithmetic operations with precision to avoid miscalculations.

Applying these mathematical formulas enables us to handle complex data sets and interpret percentage changes accurately.

Effective Problem-Solving Techniques

Problem-solving is all about applying mathematical concepts to find solutions. When dealing with percent change problems, effective strategies can simplify the process.

Consider these techniques:

• Understand the problem: Clearly define whether you are calculating an increase or decrease.
• Break it down: Simplify the problem by calculating in steps, such as determining the change and then the percentage.
• Practice makes perfect: Regular practice with various examples enhances understanding and efficiency.
• Check your work: Re-evaluate your calculations to confirm their accuracy and validity.

By honing problem-solving skills, you can approach similar mathematical challenges with confidence and ease.