Problem 4 - 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 \(80 \mathrm{cm}\) to \(55 \mathrm{cm}\)

Step-by-Step Solution

Answer

The percent of change is a 31.3% decrease.

Step 1: Calculate the Change in Value

First, find the difference between the original value and the new value. The original value is 80 cm and the new value is 55 cm. Calculate the change: \(80 - 55 = 25\). This shows a decrease by 25 cm.

Step 2: Determine the Original Value Percentage

Now calculate how much 25 cm decrease is as a percentage of the original value, which is 80 cm. Use the formula: \(\frac{\text{Change}}{\text{Original Value}} \times 100\). Substitute the known values: \(\frac{25}{80} \times 100\).

Step 3: Calculate the Percent Change

Perform the calculation: \(\frac{25}{80} \approx 0.3125\), and multiply by 100 to convert it to a percentage: \(0.3125 \times 100 = 31.25\%\).

Step 4: Round to the Nearest Tenth

Round 31.25% to the nearest tenth. The digit in the tenths place is 2 and the digit in the hundredths place is 5, which means rounding up. Thus, 31.25% rounds to 31.3%.

Step 5: Determine the Type of Percent Change

Since the original value decreased from 80 cm to 55 cm, the percent change represents a decrease. Therefore, the percent change is a decrease of 31.3%.

Key Concepts

Percent Increase

Percent increase is a concept used to describe how much a quantity has grown, in terms of percentage, compared to its original value. To calculate percent increase, follow these simple steps:

• First, find the difference between the new value and the original value. This difference is the amount of increase.
• Next, divide the increase by the original value to find out how much larger the new value is, relative to the original.
• Finally, multiply the result by 100 to convert it into a percentage. This will give you the percent increase.

For example, if a book originally costs $50 and now costs $65, the increase is $15. Dividing $15 by $50 and then multiplying by 100 gives a 30% increase.

Percent Decrease

Percent decrease tells us how much a quantity has reduced, expressed as a percentage of its original value. Calculating it involves the following steps:

• First, find the difference between the original value and the new value. This is the amount of decrease.
• Then, divide the decrease by the original value to measure how much smaller the new value is in comparison to the original.
• Multiply this result by 100 to convert it to a percentage. This result shows the percent decrease.

In our original exercise, the length decreases from 80 cm to 55 cm, a reduction of 25 cm. By dividing 25 cm by 80 cm and multiplying by 100, we find a 31.25% decrease. This is then rounded to 31.3%.

Rounding Decimals

Rounding decimals is a fundamental concept in mathematics, where a number is approximated to a specified degree of precision. This is often necessary when describing percentages or other decimal-based figures. To round a number to the nearest tenth:

• Identify the number in the tenths place. This is the first digit to the right of the decimal point.
• Look at the number immediately to the right of the tenths place: the hundredths place.
• If the hundredths digit is 5 or greater, round up by increasing the tenths digit by one.
• If the hundredths digit is less than 5, keep the tenths digit unchanged.

For example, rounding 31.25 to the nearest tenth involves looking at 5 in the hundredths place. Since it is 5, increase 2 in the tenths place, resulting in 31.3.

Calculating Percentages

Understanding how to calculate percentages is a versatile skill in math. A percentage represents a dimensionless ratio, meaning it compares numbers without units. Here's a simple method to compute percentages:

• Convert the part of interest (e.g., increase or decrease) into a fraction of the total or original amount.
• Multiply the fraction by 100 to turn it into a percentage.

For instance, in the exercise above, we calculated the percentage decrease from 80 cm to 55 cm by dividing the decrease (25 cm) by the initial amount (80 cm) and then multiplying by 100. Calculating percentages helps in expressing changes clearly and is especially useful in understanding growth trends, discounts, or differences quantitatively.