Problem 45 - 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 increase is bigger at 400% relative change.

Step 1: Calculating the Percentage Change in the First Sales Increase

To find the relative change in sales from \(100,000 to \)500,000, we first calculate the difference: \(500,000 - \)100,000 = $400,000. Next, we determine the percentage change by dividing this difference by the original amount and then multiplying by 100: \( \left( \frac{400,000}{100,000} \right) \times 100 = 400\% \).

Step 2: Calculating the Percentage Change in the Second Sales Increase

Now, we calculate the relative change in sales from \(20,000,000 to \)20,500,000. The change in this case is \(20,500,000 - \)20,000,000 = $500,000. To find the percentage change: \( \left( \frac{500,000}{20,000,000} \right) \times 100 = 2.5\% \).

Step 3: Comparing the Percentage Changes

The percentage change for the first sales increase is 400%, while the percentage change for the second sales increase is 2.5%. Although both involve an increase of $500,000, the relative change of the initial amounts shows that the first scenario has a much larger percentage increase.

Key Concepts

Understanding Relative Change

Relative change helps you understand the size of a change in relation to an original value. It's often expressed as a percentage. When evaluating changes like sales increases, relative change provides insight into the proportionality of that change. It answers the question: "How much has something changed in comparison to where it started?"

The formula to calculate relative change is:

• Find the difference between the new and the original value.
• Divide this difference by the original value.
• Multiply the result by 100 to convert it into a percentage.

Relative change is vital for comparing scenarios because it accounts for the size of the original value. For example, a $1,000 increase in sales is more impactful to a small business with $10,000 in total sales than to a corporation with millions in revenue. By using percentages, relative change levels the playing field, allowing fair comparisons between different-sized entities.

Interpreting Sales Increase

Sales increase is a tangible indicator of business growth and success. It reflects how well a company is doing in terms of generating revenue over a specific period.

Even though a sales increase might seem straightforward, evaluating its significance requires more than just a simple look at the absolute numbers. For instance, a $400,000 sales increase from $100,000 to $500,000 represents a significant change in the scale of this business, marked by a 400% increase.

Here's why that percentage matters:

• It helps identify growth patterns. Is this growth a one-time hype or part of a growing trend?
• It allows for performance benchmarking against industry standards or competitors.
• It aids in strategic decisions such as scaling operations or adjusting marketing efforts.

Whenever assessing a sales increase, it's not enough to consider only the final numbers. Delve into the percentage change to uncover its full impact on the business.

Magnitude Comparison in Percentages

Comparing magnitudes using percentages helps to assess which change is more significant, even if numerical changes appear similar. This is crucial when analyzing different financial or business scenarios like sales increases.

In the provided exercise, both scenarios show an increase of $500,000, yet their relative impacts are different due to the starting values. Employing percentage comparison offers clarity when evaluating such situations:

• The first scenario has a starting value of $100,000 and a percentage change of 400%.
• The second scenario starts at $20,000,000 and has a percentage change of 2.5%.
• Although both have the same absolute increase, the smaller initial amount in the first scenario results in a much larger percentage change.

This technique of comparing relative changes using percentages ensures accurate analysis of how substantial a change truly is, despite absolute numbers. You can thereby decide which change is more impactful relative to its starting point.