Problem 5 - Applied Calculus

Question

Find the relative, or percent, change. \(S\) changes from 400 to 450

Step-by-Step Solution

Answer

The percent change is 12.5%.

Step 1: Identify Initial and Final Values

First, identify the initial and final values of \(S\). The initial value is 400, and the final value is 450.

Step 2: Calculate the Absolute Change

Subtract the initial value from the final value to find the absolute change. \[\text{Absolute Change} = 450 - 400 = 50\]

Step 3: Calculate the Relative Change

Divide the absolute change by the initial value to get the relative change as a fraction. \[\text{Relative Change} = \frac{50}{400} = 0.125\]

Step 4: Convert to Percentage

Convert the relative change into a percent by multiplying by 100. \[\text{Percent Change} = 0.125 \times 100 = 12.5\%\]

Key Concepts

Percent Change

Calculating the percent change is a useful method to determine how much a quantity has increased or decreased compared to its original value. This concept is often applied in finance, science, and economics. To find the percent change, follow these steps:

• Calculate the absolute change, which is the difference between the final and initial values.
• Divide this absolute change by the initial value to get the relative change.
• Finally, multiply the relative change by 100 to convert it to a percentage.

For example, if something increases from 400 to 450, the absolute change is 50. Dividing 50 by the initial value of 400 gives a relative change of 0.125. Multiplied by 100, the percent change is 12.5%. This means there was a 12.5% increase from the original amount. Understanding this process helps in comparing changes in different contexts.

Absolute Change

The concept of absolute change refers to the straightforward difference between final and initial values. It simply tells us how much the value has increased or decreased.

• To find the absolute change, subtract the initial value from the final value.

In the exercise, with values changing from 400 to 450, the absolute change is calculated as:\[ 450 - 400 = 50 \]This tells us that the quantity has increased by 50 units. Absolute change gives a clear picture of the magnitude of change without considering the scale of the values involved. It’s a fundamental step in further calculations such as finding the relative or percent change.

Initial and Final Values

Understanding initial and final values is crucial when determining changes in quantities. The initial value is the starting point or original amount before any change occurs, while the final value is the amount after the change. For instance, in the example where a quantity changes from 400 to 450:

• The initial value is 400.
• The final value is 450.

Identifying these correctly is vital, as they are the foundation for calculating both absolute and relative changes. Mistaking initial or final values can lead to incorrect results, so always ensure these values are accurately identified before proceeding with any calculations.