Problem 13 - 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 \(\$ 135\) to \(\$ 120\)

Step-by-Step Solution

Answer

The percent change is an 11.1% decrease.

Step 1: Find the change in value

Calculate the difference between the original and changed values: \[ \text{Change} = \text{Original Value} - \text{New Value} = 135 - 120 = 15 \]

Step 2: Determine if it's an increase or decrease

Compare the original value to the new value. Since \( 135 > 120 \), it indicates a decrease.

Step 3: Calculate the percent of change

Use the formula for percent of change, which is: \[ \text{Percent Change} = \left( \frac{\text{Change in Value}}{\text{Original Value}} \right) \times 100 \]Substitute the known values: \[ \text{Percent Change} = \left( \frac{15}{135} \right) \times 100 \approx 11.1\% \]

Step 4: Interpret the result

Since the percent change is calculated as \( 11.1\% \), and the original value decreased, the interpretation of the change is an \( 11.1\% \) decrease.

Key Concepts

Understanding Percent Increase

A percent increase occurs when the value of an item goes up over time. To find it, calculate the difference between the new value and the original value, ensuring the new value is greater. Then, divide this difference by the original value and multiply by 100.

Here's a basic breakdown of the steps:

• Identify the new and original values.
• Subtract the original value from the new value to find the increase.
• Divide the increase by the original value.
• Multiply the result by 100 to convert to a percentage.

This will give you the percent increase. Imagine buying a stock at \(50 and later selling it for \)60. The increase is $10, so you'd find the percent increase as: \( \frac{{10}}{{50}} \times 100 = 20\% \). The value increased by 20%.

Exploring Percent Decrease

When a value decreases, it is known as a percent decrease. Similar to a percent increase, this involves comparing the original and new values.

Follow these steps to calculate percent decrease:

• Identify both the original and new values.
• Subtract the new value from the original value to find the decrease.
• Divide this decrease by the original value.
• Multiply the result by 100 to get a percentage.

Consider an item originally priced at \(135, now reduced to \)120. Here, the decrease is $15. Thus, the percent decrease is \( \frac{{15}}{{135}} \times 100 \approx 11.1\% \). So the price decreased by 11.1%.

How to Calculate Percent Change

Calculating percent change involves a straightforward process whether the value is increasing or decreasing. The concept remains the same: determine the difference between the old and new values, divide by the original, and convert to a percentage.

To calculate percent change:

• Find the difference: subtract the smaller value from the larger one, keeping track of whether it’s an increase or decrease.
• Divide this difference by the original value.
• Convert this ratio into a percentage by multiplying by 100.

For illustration, if the value changes from \(135 to \)120, the steps are:

• Difference: \(135 - \)120 = $15
• Calculation: \( \frac{{15}}{{135}} \times 100 \approx 11.1\% \)

This shows a change of 11.1%.

Interpreting Percent Change

Understanding what a percent change indicates is crucial. It's more than just a number; it tells the story of how values shift over time.

When you calculate a positive percentage, it points to a percent increase. Conversely, a negative percentage indicates a percent decrease. It’s essential for interpreting the direction and magnitude of change.

In real-life applications, interpreting percent change can answer questions like:

• How much has the cost of a product increased or decreased?
• Is an investment growing in value?
• Has there been a decrease in energy consumption over time?

In the exercise given, the result was an 11.1% decrease, helping you understand that the price or value reduced by that amount.