Problem 46 - Algebra 2

Question

For each annual rate of change, find the corresponding growth or decay factor. $$ +70 \% $$

Step-by-Step Solution

Answer

The corresponding growth factor for an annual rate change of +70% is 1.70

Step 1: Byte Conversion

Firstly, convert the percentage value to a decimal. An annual rate change of +70% can be written as 0.70 in decimal form.

Step 2: Decision on Growth or Decay

Identify whether the rate is for growth or decay. Since the rate is positive, we are dealing with a growth rate.

Step 3: Calculation of Growth Factor

Now, calculate the growth factor which equals 1 plus the growth rate in decimal form. So, the growth factor is \(1 + 0.70 = 1.70\)

Key Concepts

Rate of Change

A rate of change is a numerical value that represents how much a quantity increases or decreases over time. In many contexts, such as finance and demographics, it helps to understand the growth or decay of a variable. For instance, a rate of change could indicate how fast a company's profits grow or how quickly a population declines. When you see a percentage like +70%, it signifies the quantity is growing, and each year it is increasing by 70% of its original size. To determine whether it is a growth or decay - a positive percentage represents growth, while a negative one represents decay.
Understanding the rate of change is crucial because it impacts how values evolve over time. If you know the rate of change, you can calculate other important metrics such as growth factors, which help in projecting future values.

Decimal Conversion

Decimal conversion is the process of turning a percentage into a decimal format. This conversion is essential for many mathematical calculations, especially when determining growth factors. To perform this conversion, you divide the percentage by 100. For example, converting 70% into a decimal requires dividing 70 by 100, resulting in 0.70. This makes it easier to use in further calculations, such as those involving multiplication for growth projections.
Decimal conversions simplify comparisons and computations when working with percentages. Remember, a percentage is a part per hundred, and by converting it to decimal form, you unify its representation alongside other numeric forms, making calculations more straightforward.

Percentage to Decimal

Converting a percentage to a decimal is a simple yet fundamental task in various mathematical scenarios. To convert a percentage to decimal, simply take the percentage number and divide it by 100. This is because the term "percent" means per hundred. So, for a percentage like 70%, you perform the calculation:
\[ \frac{70}{100} = 0.70 \]
In plain terms, this means shifting the decimal point two places to the left. This conversion is particularly useful when performing operations that involve fractions or continuous growth calculations, as decimals are more universally applicable in these operations.

• Step-by-step: Simply take the number, remove the percent sign, and divide by 100.
• Reason: Makes it compatible with arithmetic calculations involving multiplication or division.

Using decimals instead of percentages ensures mathematical expressions are conducted accurately, maintaining numerical consistency in formulas.