Problem 52 - Algebra 2

Question

For each annual rate of change, find the corresponding growth or decay factor. $$ +0.1 \% $$

Step-by-Step Solution

Answer

The growth factor for an annual rate of change of +0.1% is 1.001.

Step 1: Convert Percentage to Decimal

The rate of change is given in percentage terms, but we need to convert this to a decimal to calculate the growth factor. The conversion is done by dividing the percentage by 100. So, \( 0.1\% = 0.1/100 = 0.001 \) That is, +0.1% is equal to 0.001 in decimal terms.

Step 2: Calculate the Growth Factor

The formula for the growth factor is 1 plus the rate of growth (in decimal form). So, the growth factor would be \(1 + 0.001 = 1.001\)

Key Concepts

Growth Factor

When we talk about a growth factor, we're discussing how much something increases over a period of time. In simple terms, it's a way to represent growth, like the growth of money or population, on a year-to-year basis.
To find the growth factor, you first need to work with the rate of change expressed as a decimal. This involves adding this decimal to 1.

• If you have a positive rate of change, add it to 1 to get the growth factor.
• For example, a rate of +0.1% becomes a growth factor of 1.001 once converted and added to one.

This tells you that for every 1 unit, you now have a bit more than 1 after applying the growth factor — 1.001 units, in this instance. It's a simple yet powerful tool to project future growth over time.

Decimal Conversion

Decimal conversion is the crucial first step in working with percentages, particularly when finding growth or decay factors.
To convert a percentage into a decimal, remember this basic rule: Divide the percentage value by 100. This is necessary because percentages are based on a scale of 100, but decimals are not.

• For instance, converting 0.1% into a decimal is done by calculating \(0.1 \div 100 = 0.001\).
• This is a small conversion but essential for further calculations.

Doing this conversion accurately is key. Once you have the decimal form, you can easily use it to find changes in the value over time.

Percentage Change

Understanding percentage change is about grasping how much a quantity increases or decreases in percentage terms over a certain period.
This measure is useful for looking at trends and making comparisons. If you have an initial value and a final value, the formula to find percentage change is: \[\text{Percentage Change} = \left( \frac{\text{Final Value} - \text{Initial Value}}{\text{Initial Value}} \right) \times 100\%\]

• Positive results indicate growth, a negative result would mean decay or decrease.
• This calculation helps understand not just the direction, but also the relative size of the change.

In our example, with a slight increase of 0.1%, though it might seem tiny, it's measurable and can be significant over longer terms or larger volumes. Recognizing this subtle change in percentage terms helps in many real-world applications, from finance to science.