Problem 8 - Applied Calculus

Question

Find the relative, or percent, change. \(W\) changes from \(0.3\) to \(0.05\)

Step-by-Step Solution

Answer

The percent change is -83.33%.

Step 1: Identify Initial and Final Values

The initial value of \(W\) is \(0.3\) and the final value of \(W\) is \(0.05\). These are the starting and ending points for the calculation of the relative, or percent, change.

Step 2: Calculate the Change in Value

The change in value is determined by subtracting the initial value from the final value. This gives us: \( \text{Change in } W = 0.05 - 0.3 = -0.25 \).

Step 3: Calculate the Absolute Change

Since the result from Step 2 is negative, it indicates a decrease. The absolute change is \(0.25\) (without considering the negative sign).

Step 4: Calculate the Relative Change

The relative, or percent, change is calculated by dividing the change in value by the initial value, and then multiplying by 100 to get a percentage. \[ \text{Relative change} = \left( \frac{-0.25}{0.3} \right) \times 100 = -83.33\% \]This shows a decrease by approximately 83.33%.

Key Concepts

Percent Change

Percent change is a way to express the difference between two values in percentage terms. This is especially helpful in comparing how much something has increased or decreased in relation to its original state. For example, when we say something has increased by 50%, it means that the final value is half again as large as the original. Percent change is used in various fields, from finance to science, to easily convey the significance of changes in data.

The formula to calculate percent change is:

• Calculate the difference between the final and initial values, which gives you the change.
• Divide this difference by the initial value.
• Multiply by 100 to get the percent change.

This formula gives us a clear picture of the relative change expressed as a percentage.

Absolute Change

Absolute change is the straightforward difference between the final and initial values. It shows how much the value has actually changed in numerical terms, without any context of size or relative proportion. This is important to understand because it gives us the actual quantity of change, which may be useful for understanding the scale of the change without any bias or adjustment for proportionality.

For example, if a stock price moves from \(10 to \)15, the absolute change is $5. This tells us exactly how much money is involved in the change, which can be more informative for absolute gain or loss assessments.

• Absolute change = Final value - Initial value.

In our original problem with values changing from 0.3 to 0.05, the absolute change was calculated as \(-0.25\). Dropping the sign, it becomes a positive 0.25, providing a numeric value of this decrement.

Initial and Final Values

Identifying initial and final values is the first crucial step in calculating change. These values serve as reference points, with the initial value representing the starting position and the final value indicating where you ended up. This step is important because all calculations of change revolve around these two numbers.

For our given exercise, the initial value of \(W\) was 0.3, which is where we started. The final value was 0.05, which indicates the endpoint of our evaluation. Knowing these two values allows us to correctly ascertain both absolute and percent change, each painting a different aspect of how much change has occurred.

• Initial value: the starting point.
• Final value: the ending point.

Calculation of Change

Calculating change involves figuring out both the magnitude and direction of the change that occurred between the initial and final values. This includes both absolute and percent change calculations. Understanding these calculations allows you to measure change effectively, making it easier to measure growth or reduction across different contexts.

The steps to calculate change are:

• Find the difference between final and initial values to get the absolute change.
• Convert that difference into a percent change by dividing the absolute change by the initial value and multiplying by 100.

In our example, the change in \(W\) was \(-0.25\), and when expressed as a percent change, it turned out to be \(-83.33\%\). This highlights an impressive decrease, emphasizing how crucial it is to understand both forms of measurement for comprehensive data analysis.