Problem 46 - Applied Calculus

Question

Which relative change is bigger in magnitude? Justify your answer. An increase in sales from \(\$ 100,000\) to \(\$ 500,000 ;\) an increase in sales from \(\$ 20,000,000\) to \(\$ 20,500,000\)

Step-by-Step Solution

Answer

The first scenario's change is larger in magnitude with a 400% increase compared to 2.5% in the second scenario.

Step 1: Calculate the Relative Change for the First Scenario

First, we calculate the relative change from \(100,000\) to \(500,000\). The formula for relative change is \( \frac{{\text{New Value} - \text{Old Value}}}{{\text{Old Value}}} \). Using the given values we have: \[ \frac{{500,000 - 100,000}}{{100,000}} = \frac{{400,000}}{{100,000}} = 4 \] The relative change is 400%.

Step 2: Calculate the Relative Change for the Second Scenario

Now, let's calculate the relative change from \(20,000,000\) to \(20,500,000\) using the same formula: \[ \frac{{20,500,000 - 20,000,000}}{{20,000,000}} = \frac{{500,000}}{{20,000,000}} = 0.025 \] The relative change is 2.5%.

Step 3: Compare the Magnitude of Relative Changes

Now we compare the relative changes calculated in Steps 1 and 2. The first scenario has a relative change of 400%, whereas the second scenario has a relative change of 2.5%. Clearly, the first scenario has a larger magnitude of relative change.

Key Concepts

Percentage Change

Understanding percentage change is essential when analyzing differences between two values, whether that's sales, prices, or other quantities. Percentage change is a metric that shows the amount of change relative to the initial value, expressed as a percentage. The basic formula to calculate it is:

• \( \frac{{\text{New Value} - \text{Old Value}}}{{\text{Old Value}}} \times 100 \% \)

A positive percentage indicates an increase, while a negative percentage indicates a decrease.
In the provided example, calculating the percentage change in sales from \(100,000 to \)500,000 results in a 400% increase. Meanwhile, going from \(20,000,000 to \)20,500,000 results in a smaller increase of 2.5%.
Notice how even though the amount of money increased by $500,000 in both cases, the percentage change is vastly different due to the initial amounts involved. This highlights why percentage change is more insightful than just examining raw differences.

Sales Increase

Sales increase is a critical aspect for businesses to track and analyze. Understanding the increase in sales involves looking beyond just the nominal numbers. To accurately gauge business growth or market share increase over time, assessing relative changes or percentage increases provides clarity.

• Nominal Increase: Simply the difference between the new and the old sales values, like \\(400,000 in the example.
• Relative Increase: The percentage increase calculated from the nominal difference related to the original value.

In the first scenario provided, a jump from \)100,000 to \(500,000 marks a sharp increase of 400%.
Whereas, in the second scenario, \)20,000,000 climbing to $20,500,000 is just a 2.5% bump.
This difference illustrates that a substantial nominal increase may have a different business impact depending on the context of the starting number.

Magnitude Comparison

Magnitude comparison of relative changes is a critical analysis that helps determine which change is more significant or impactful. The magnitude is essentially the size or extent of change regardless of whether it is an increase or decrease.

• When comparing percentages, the larger percentage signifies a larger relative change, regardless of the nominal figures involved.
• This analysis is important because it allows businesses to prioritize areas with higher growth potential.

For example, the given problem shows that comparing a 400% increase to a 2.5% increase quickly reveals which change is larger in magnitude.
This type of comparison is particularly useful when evaluating results across different scales and is a standard approach when assessing financial performance or investment returns.