Problem 49 - Algebra 2

Question

For each annual rate of change, find the corresponding growth or decay factor. $$ -55 \% $$

Step-by-Step Solution

Answer

The decay factor corresponding to a -55% annual rate of change is 0.45.

Step 1: Convert Percentage to Decimal

To convert the percentage to a decimal, divide the percentage by 100. Hence, the decimal form of -55% is \(-\frac{55}{100}\) or -0.55.

Step 2: Calculate Decay Factor

The decay factor is found by adding 1 to the rate (in decimal form). As we're having a decay (negative rate), it means we subtract this value from 1. Therefore, the decay factor would be \(1 -0.55\).

Key Concepts

Growth Factor

The growth factor is a measure used to determine how much a quantity increases over time. In cases where the rate of change is positive, the growth factor helps describe this increase as a multiplier. It's relevant when you're looking at situations like population growth, investments, or any scenario where an increase occurs.

To find the growth factor, follow this straightforward process:

• First, convert your percentage rate of change into a decimal. This is crucial because it allows you to work with more precise figures.
• Add the decimal to 1. This step adds the original value into the equation since a growth factor naturally means the original plus the increase.

For example, if the rate of change is 20%, convert it to decimal form as 0.20, and add it to 1, resulting in a growth factor of 1.20. The value of 1.20 implies that the quantity is now 120% of its original value.

Decay Factor

Decay factor measures how much a quantity decreases over time. It's the counterpart to the growth factor, applicable when the rate of change is negative. This is common in scenarios like depreciation of assets or resources that are depleted.

How do you find the decay factor? Here's a simple method:

• Start by converting the percentage rate of change into a decimal.
• If you have a decay, subtract this value from 1. This accounts for the loss because you're reducing the original quantity.

In the given exercise, a rate of -55% was provided. After converting to a decimal, it becomes -0.55. The decay factor is calculated as 1 - 0.55, which equals 0.45. This means the remaining quantity is 45% of its original amount.

Percentage to Decimal Conversion

Converting percentages to decimals is a fundamental process that simplifies calculations involving growth or decay factors. This conversion is essential in many mathematical applications, making it easier to handle and understand data.

Here's how to convert a percentage to a decimal:

• Take the percentage value you have. Let's say it's 25%.
• Divide it by 100. This means you are literally moving the decimal point two places to the left.

Following our example, 25% becomes 0.25 once converted into a decimal. This conversion is crucial because all subsequent calculations use these decimal forms to ensure accuracy. Remember, working with decimals rather than percentages provides precision and makes mathematical operations more coherent. Whether it's for determining a growth factor, decay factor, or any other use, ensuring accurate decimal conversion is key.