Problem 7 - Applied Calculus

Question

Find the relative, or percent, change. \(R\) changes from 50 to 47

Step-by-Step Solution

Answer

-6% change.

Step 1: Identify Initial and Final Values

First, identify the initial value and the final value. In this exercise, the initial value is 50, and the final value is 47.

Step 2: Calculate Absolute Change

Next, calculate the absolute change by subtracting the initial value from the final value: \[\text{Absolute Change} = 47 - 50 = -3\]

Step 3: Calculate Relative Change

Find the relative, or percent, change by dividing the absolute change by the initial value and then multiplying by 100 to achieve a percentage:\[\text{Relative Change} = \left( \frac{-3}{50} \right) \times 100\]

Step 4: Simplify the Expression and Solve

Simplify the expression by calculating the division and multiplication:\[\frac{-3}{50} = -0.06\]Then multiply by 100:\[-0.06 \times 100 = -6\%\]

Key Concepts

Percent Change

Understanding percent change is crucial in determining how a quantity has increased or decreased over time relative to its original value. To calculate the percent change, you first need the absolute change, which is the difference between the final and the initial values. Then, you divide the absolute change by the initial value. Finally, you multiply the result by 100 to convert it to a percentage. This gives you the percent change, which can be positive or negative. A negative percent change means the quantity has decreased, as seen in our example where the change is \(-6\%\), indicating a decline.

Absolute Change

Absolute change simply measures the pure difference between the initial and final values, without any regard to their proportional relationship. It is calculated by subtracting the initial value from the final value: \( \text{Absolute Change} = \text{final value} - \text{initial value} \).
In the given exercise, \( R \) changes from 50 to 47. Here, the absolute change is \( 47 - 50 = -3 \). This means that the quantity decreased by 3 units. Absolute change is represented as a simple number rather than as a percentage, which makes it straightforward but not helpful for comparison across different scales or contexts.

Initial and Final Values

Identifying the initial and final values is the first critical step in computing both absolute and percent changes. The initial value is what you start with, and the final value is what you end up with. In many real-world scenarios, it's essential to compare these two to understand growth or decline.

• The **initial value** in our exercise is 50.
• The **final value** is 47.

These values are crucial because they form the basis of any change calculations. Without accurately identifying these numbers, your calculations could lead to incorrect conclusions. Understanding context is essential too, as any change might be interpreted differently based on external factors or goals.