Problem 11 - 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 68 min to 51 min

Step-by-Step Solution

Answer

The percent of change is a 25.0% decrease.

Step 1: Determine the Amount of Change

First, find the difference between the original amount (68 min) and the new amount (51 min). Calculate: \[68 - 51 = 17 \text{ min}\] So, the amount of change is 17 min.

Step 2: Calculate the Percent Change

Next, use the formula for percent change:\[\text{Percent Change} = \left( \frac{\text{Amount of Change}}{\text{Original Amount}} \right) \times 100\]Substitute the known values into the formula:\[\text{Percent Change} = \left( \frac{17}{68} \right) \times 100\]Then, calculate the result:\[\text{Percent Change} \approx 25.0\%\] Thus, the percent change is approximately 25.0%.

Step 3: Identify Increase or Decrease

Since the original amount (68 min) decreased to a new amount (51 min), this is a percent of decrease. Therefore, we have a 25.0% decrease.

Key Concepts

Understanding Percent Increase and Decrease

When we talk about percent increase and decrease, we are referring to how much a certain amount has grown or shrunk as a percentage. It's a way to express change relative to the original value, making it easier to comprehend the shift in size or quantity.

To find a percent change:

• Calculate the difference between the new value and the original value. This is called the amount of change.
• Divide this amount of change by the original value.
• Multiply the result by 100 to convert it to a percentage.

A **percent increase** occurs when the new value is larger than the original value. Conversely, a **percent decrease** happens when the new value is smaller. These calculations help in analyzing real-life situations such as discounts, interest rates, and population growth.

Solving Mathematical Word Problems

Mathematical word problems can seem daunting, but breaking them down into manageable steps makes them much easier to tackle. They involve reading, interpreting, and solving mathematical equations based on the description given in words.

Here’s a method to approach them:

• Read the problem carefully to understand what is being asked.
• Identify the known values and what you need to find out.
• Translate the problem from words into a mathematical formula or equation.
• Solve the equation using appropriate mathematical operations.
• Make sure to interpret your answer within the context of the problem, checking units and whether it is a percent increase or decrease.

Practicing these steps can make solving any mathematical word problem more straightforward and less stressful.

Employing Basic Arithmetic Operations

Basic arithmetic operations are the foundation of all math problems. They include addition, subtraction, multiplication, and division. These operations are crucial for calculating percent changes and solving mathematical word problems.

Let's break down these operations:

• **Addition** involves combining two or more numbers. For instance, if you get extra minutes added to a project, you use addition to determine the new total.
• **Subtraction** is used when you are taking one number away from another. In our example, the minutes reduced from 68 to 51 involves subtracting 51 from 68 to find the amount of change.
• **Multiplication** helps find the total when you have groups of the same size. When converting a fraction to a percentage, you multiply by 100.
• **Division** is needed when you distribute a total evenly into parts or when calculating something per unit or to find the fraction of a whole, such as \((\frac{17}{68})\).

Fluency in these basic operations is essential in solving more complex problems and understanding how percentages work.