Problem 3 - 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 45 in. to 18 in.

Step-by-Step Solution

Answer

The percent of change is 60.0% and it is a percent of decrease.

Step 1: Identify Initial and Final Values

The initial value, also known as the original amount, is 45 inches. The final value, or the new amount, is 18 inches. We will use these values to calculate the amount of change.

Step 2: Calculate the Amount of Change

Subtract the final value from the initial value to find the amount of change. \( \text{Change} = 45 - 18 = 27 \) inches.

Step 3: Calculate the Percent of Change

Divide the amount of change by the initial value and then multiply by 100 to convert it to a percentage.\[ \text{Percent of Change} = \left( \frac{27}{45} \right) \times 100 \approx 60.0\% \]

Step 4: Determine Type of Change

Since the final value is less than the initial value, the percent of change represents a decrease.

Key Concepts

Understanding Percent of Increase

Percent of increase is a common concept in mathematics that helps to quantify how much a value has grown in comparison to its original starting point. Imagine seeing a sale where prices are being marked up. The percent of increase tells you by what percentage the price has risen. To calculate the percent of increase:

• First, subtract the original value from the new value. This gives you the amount that has increased.
• Next, divide this increase by the original value to find how much the original has grown in relative terms.
• Finally, multiply this result by 100 to convert it into a percentage.

For example, if a product's price went from $50 to $65, first find the difference, which is $15. Then, divide 15 by 50, which equals 0.3. Multiply by 100, and you get a 30% increase. An increase in percent represents growth, a crucial concept for understanding economic trends, budgeting, and personal finance.

Explaining Percent of Decrease

Percent of decrease is used when a value reduces from its original amount. It's the tool we often use to detect how much something has declined. This is particularly useful in sales, discounts, and understanding reductions in size or volume. Calculating the percent of decrease involves:

• Subtracting the new value from the original value to find the decrease.
• Then divide the amount decreased by the original value to express this decrease relative to the starting point.
• Lastly, multiply by 100 to express it as a percentage.

For instance, if the original size of a pillow was 30 inches and now it measures 21 inches, the decrease is 9 inches. Dividing 9 by 30 gives 0.3, and converting it to a percentage results in a 30% decrease. A decrease indicates a reduction, crucial in various contexts such as analyzing investments, expenses, and environmental changes.

Calculation Steps for Percent of Change

Accurately calculating the percent of change involves a series of steps that include basic arithmetic operations. Start by identifying the initial and final values, which form the basis of the calculation. In the given exercise, the initial value is 45 inches, and the final one is 18 inches.Steps to calculate the percent of change:

• Determine the amount of change by subtracting the final value from the initial value. Here, it’s 45 - 18 = 27 inches.
• To find the percent of change, divide this change by the initial value and multiply the result by 100 to get a percentage.
• In our case, \[ \left( \frac{27}{45} \right) \times 100 = 60\% \] which signifies a considerable transformation.

After acquiring this percentage, identify whether it signifies an increase or a decrease. For the example in question, since the final figure is lower than the initial, it represents a 60% decrease. Knowing these steps helps in analyzing changes in value, such as financial statements or inventory levels in business.