Problem 39 - Applied Calculus

Question

Find the relative, or percent, change. \(B\) changes from 12,000 to 15,000

Step-by-Step Solution

Answer

The percent change is 25%.

Step 1: Calculate the Change in Value

First, find the change in value by subtracting the initial value from the final value. In this case, subtract 12,000 from 15,000: \[ \text{Change in Value} = 15,000 - 12,000 = 3,000 \]

Step 2: Find the Relative Change

To find the relative change, divide the change in value by the initial value (12,000) and convert it into a fraction: \[ \frac{3,000}{12,000} \]

Step 3: Convert to a Percentage

Convert the fraction to a percentage by multiplying it by 100. This gives the relative, or percent, change: \[ \left(\frac{3,000}{12,000}\right) \times 100 = 25\% \]

Key Concepts

Relative Change

The concept of relative change helps us understand how much a quantity has increased or decreased in relation to its original value. It's often used to express changes in financial data, scientific measurements, and everyday comparisons.
To calculate the relative change, follow these straightforward steps:

• Identify the initial value and the final value. In our example, the initial value is 12,000, and the final value is 15,000.
• Calculate the change in value by subtracting the initial value from the final value. This difference is 3,000.
• Take this change and divide it by the initial value. Here, it’s dividing 3,000 by 12,000.

The result is a fraction that represents the relative change. This method gives a clear proportional view of how the change compares to the starting point.

Percent Change

Percent change is a way to express the relative change as a percentage, making it easier to understand and compare different changes. Percentage is a familiar concept that resonates well in many contexts, from discounts to interest rates.
Once you've determined the relative change, converting it to a percentage is simple:

• Multiply the relative change (which we found by dividing 3,000 by the initial 12,000) by 100.
• This operation effectively scales the relative change to a scale we are more accustomed to using—percentages.
• In this example, multiplying the fraction by 100 gives a percent change of 25%.

The use of the percentage format provides a straightforward way to communicate the scale of change, making it accessible and relatable for audiences at all levels.

Basic Arithmetic

Understanding basic arithmetic operations is crucial for calculating both relative change and percent change. These operations include addition, subtraction, multiplication, and division—tools that form the backbone of any mathematical calculation.
In the context of calculating relative or percent change:

• Subtraction is used to find the change in value. For example, subtract 12,000 from 15,000 to get 3,000.
• Division is key for calculating the relative change, dividing the change by the initial value (3,000 ÷ 12,000).
• Multiplication turns the relative change into percent change by scaling it with 100.

These operations, though simple, are powerful. They enable us to make informed decisions based on data comparisons. It's the proper application of these basic skills that allows us to capture and communicate complex data changes in an easily digestible form.