Problem 47 - Algebra 2

Question

For each annual rate of change, find the corresponding growth or decay factor. $$ +500 \% $$

Step-by-Step Solution

Answer

The growth factor corresponding to an annual rate of change of +500% is 6.

Step 1: Convert the rate of change to decimal

To do this, divide the rate of change by 100. In this case, we have \( \frac{500}{100} = 5 \). So the rate of change as a decimal is 5.

Step 2: Calculate the growth factor

The growth factor is calculated as 1 plus the rate of change (expressed as a decimal). So the growth factor is \( 1 + 5 = 6 \).

Key Concepts

Understanding Growth Factor

The growth factor is a crucial concept in understanding both exponential growth and decay. In simple terms, it tells us how much something increases or decreases from one period to the next. Let's break it down:

• A growth factor greater than 1 indicates growth. This is because each unit is growing compared to the previous period.
• A growth factor less than 1 indicates decay, meaning the value decreases each period.
• A growth factor of exactly 1 means no change; the value remains constant over time.

With the problem we're solving, we have a substantial increase as the growth factor is 6. This means that the quantity is six times larger than the original value, showing a 500% increase. Understanding this concept helps in subjects like finance, biology, and physics, where predicting growth or decay is crucial.

Exploring Rate of Change

The rate of change is often expressed as a percentage. It shows how much a quantity increases or decreases relative to the initial amount over a specific period. To work effectively with mathematics and real-world applications, converting this rate into a usable form is necessary.

• To find the growth or decay factor from a percentage, first convert the percentage into a decimal.
• For example, a rate of change of 500% is divided by 100 to become 5. This conversion is a foundational skill for working with percentages.
• The decimal conversion transforms a percentage into a more versatile form used for further calculations.

Understanding the rate of change assists in quickly identifying growth trends or predicting future conditions in various scenarios.

Mastering Decimal Conversion

Decimal conversion is pivotal when dealing with rates and percentages, especially in mathematical computations related to growth and decay. By transforming percentages into decimals, calculations become more straightforward and manageable.

• To convert a percentage into a decimal, divide the percentage by 100. This removes the percent symbol and transforms the number into a decimal format.
• For a rate of +500%, dividing by 100 gives us 5. This simple division sheds the percentage context, making it easier to work with in equations.
• A decimal provides a clear and concise numerical representation that can be easily used in multiplicative calculations.

Engaging in decimal conversion ensures accuracy in calculations and allows for seamless transitions between different mathematical operations, easing the analysis of data-driven patterns.