Problem 51 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The corresponding growth or decay factor for an annual rate change of -0.1% is 1.001.

Step 1: Understand the concept of growth and decay factor.

Before proceeding to the calculation, it's essential to understand the concept of growth and decay factor. In the context of a percent change, the decay or growth factor represents how much is left after the change has taken place. Hence, a decrease corresponds to subtraction and an increase to addition.

Step 2: Convert percentage to decimal.

To use the rate in calculations, we need to convert the percentage form to a decimal. To convert -0.1\% to a decimal, divide by 100. Hence it becomes -0.001.

Step 3: Calculate the decay factor.

As this a decrease or decay, we subtract the rate from 1. Hence, we calculate the decay factor by the following formula, \[ Decay \ Factor = 1- Rate \] Therefore, Decay Factor = 1 -(-0.001) = 1.001. It's essential to note that even though the word 'decay' implies a decrease, the 'decay factor' is more than 1 because it's a negative percentage decrease. The decay factor thus indicates an increase.

Key Concepts

Percentage to Decimal Conversion

In mathematics, converting percentages into decimals is a fundamental step that simplifies various calculations. Imagine percentages as parts out of a hundred. This means, to convert any given percentage to its decimal form, you need to divide the percentage by 100.
For example, if you have a percentage of \( -0.1\% \), you divide \(-0.1\) by 100, which equals \(-0.001\).

• Percentage Division: \( -0.1 \div 100 = -0.001 \)
• Remember: Negative percentages convert just like positive ones, but retain their negative sign.

This conversion is valuable for further computations like finding growth or decay factors since it brings the percentage into a manageable form for mathematical operations.

Decay Factor Calculation

Once you have the decimal equivalent of your rate, you can calculate the decay factor. Decay factors show how much of a quantity remains after a percentage decrease. The formula to find a decay factor is:

• \( Decay \ Factor = 1 - Rate \)

In our example, the rate is \(-0.001\). Plug this into the formula as follows:

• \( Decay \ Factor = 1 - (-0.001) = 1 + 0.001 = 1.001 \)

It's interesting to note here: although the term 'decay' suggests a reduction, a negative rate means you're actually ending up with more than what you started with. Thus, a negative percentage decrease, when converted into a decay factor, results in a factor greater than 1.

Negative Percentage Change

Negative percentage changes can seem a bit counterintuitive but are fascinating once you understand them. A negative percentage decrease effectively acts like an increase in value. Let’s explore why:
When your rate of change is negative, it implies the opposite behavior of what it is traditionally tied to. Instead of reducing the quantity, you end up reversing the impact of decay in practice;

• For instance, with a \(-0.1\%\) change, it implies inverting a small reduction into a slight increase.

This is why when calculating for decay factor with this small, negative rate, the factor becomes greater than 1. Essentially, a negative percentage change resolves itself such that the actual measure "grows," reaffirming how unique such transformations are in real scenarios.