Problem 44 - Applied Calculus

Question

Which relative change is bigger in magnitude? Justify your answer. An increase in class size from 5 to 10 ; an increase in class size from 30 to 50 .

Step-by-Step Solution

Answer

The relative change from 5 to 10 is larger, as it is 1 compared to \(\frac{2}{3}\).

Step 1: Identify the Initial and Final Values

For the first change, the initial class size is 5 and the final class size is 10. For the second change, the initial class size is 30 and the final class size is 50.

Step 2: Calculate the Absolute Change

The absolute change is calculated by subtracting the initial value from the final value. For the first change: \[\text{Absolute Change} = 10 - 5 = 5\]For the second change: \[\text{Absolute Change} = 50 - 30 = 20\]

Step 3: Determine the Relative Change

The relative change is calculated as the absolute change divided by the initial value. This gives us the magnitude of change relative to the starting point.For the first change: \[\text{Relative Change} = \frac{10 - 5}{5} = \frac{5}{5} = 1\]For the second change: \[\text{Relative Change} = \frac{50 - 30}{30} = \frac{20}{30} = \frac{2}{3}\]

Step 4: Compare the Relative Changes

We compare the relative changes calculated in the previous step. The relative change for the increase from 5 to 10 is 1, while the relative change for the increase from 30 to 50 is \(\frac{2}{3}\). Since 1 is greater than \(\frac{2}{3}\), the relative change in the first scenario is larger in magnitude.

Key Concepts

Absolute Change

Absolute change is a straightforward concept in mathematics, often used to quantify how much one value has increased or decreased compared to another. It is simply the difference between the final value and the initial value. When you have two values and you move from one to the other, absolute change measures that shift without considering proportions or percentages.

For example, if a class size increases from 5 to 10, the absolute change is:

• Start by subtracting the initial value from the final value: \[\text{Absolute Change} = 10 - 5 = 5\]

This calculation tells you that the class size increased by 5 students.

Similarly, if another class size increases from 30 to 50, the absolute change would be:

• \[\text{Absolute Change} = 50 - 30 = 20\]

This means 20 more students joined. Understanding absolute change is crucial as it lays the groundwork for more complex comparisons.

Initial and Final Values

Initial and final values are the starting and ending points in any change scenario. They are foundational for calculating both absolute and relative changes. To grasp the concept, always start by identifying these values for the parameter you're examining.

Consider our examples:

• First situation: The initial class size is 5 and it expands to a final size of 10.
• Second situation: Here, the class initially starts at 30 and grows to a final size of 50.

These differences give context and meaning to calculations. Without them, it wouldn't be possible to say how much the class size has changed or whether the increase is significant. By acknowledging these values, you can proceed to determine both absolute and relative changes.

Magnitude Comparison

Magnitude comparison comes into play when you need to evaluate which change is more significant. Simply put, it compares the sizes of changes not directly by their differences (absolute change) but in relation to their starting points (initial values).

To understand which relative change is larger, compute the relative change for each scenario:

• For the first class size change: \[\text{Relative Change} = \frac{10 - 5}{5} = 1\]
• For the second class size change: \[\text{Relative Change} = \frac{50 - 30}{30} = \frac{2}{3}\]

After calculating these, compare the results.While the absolute change in the second situation (20) is larger than the first (5), the relative change tells us another story. The change from 5 to 10 is proportionally larger (as a whole number) than the change from 30 to 50, allowing you to see which change is more impactful proportionally.