Problem 45 - 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 increase from 5 to 10 has a larger relative change (100%) compared to the increase from 30 to 50 (66.67%).

Step 1: Understand the Problem

We need to compare the relative changes in class sizes and determine which one is bigger in magnitude. There are two scenarios: increasing class size from 5 to 10, and from 30 to 50.

Step 2: Calculate the Relative Change for the First Scenario

For an increase in class size from 5 to 10, the formula for relative change is: \[\text{Relative Change} = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100\%\]Substitute the values:\[\text{Relative Change} = \frac{10 - 5}{5} \times 100\% = 100\%\]

Step 3: Calculate the Relative Change for the Second Scenario

For an increase in class size from 30 to 50, using the same relative change formula:\[\text{Relative Change} = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100\%\]Substitute the values:\[\text{Relative Change} = \frac{50 - 30}{30} \times 100\% = \frac{20}{30} \times 100\% = 66.67\%\]

Step 4: Compare the Relative Changes

The relative change from 5 to 10 is 100%, whereas the relative change from 30 to 50 is 66.67%. Since 100% is greater than 66.67%, the relative change from 5 to 10 is bigger in magnitude.

Key Concepts

Class Size

Class size refers to the number of students in a class or group. It can impact the quality of education, student engagement, and classroom management. In mathematical contexts, class size is a numeric value that can be analyzed to assess changes over time or between different classes.

When dealing with problems involving changes in class size, it's important to understand not just the total number of students but also what changes in this number represent. For instance, comparing two different scenarios in class size can involve calculating relative changes, which helps in understanding which change had a larger impact.

In this particular exercise, two class sizes are assessed: one increases from 5 students to 10, while the other from 30 to 50. Both changes reflect an increase, but the magnitude and significance of these changes need to be compared to determine which is larger.

Magnitude Comparison

Magnitude comparison involves determining which of two values represents a larger change or impact. In the realm of mathematics, this often involves using relative change formulas to ensure a fair comparison, especially when the base values differ.

Relative change is calculated as the percentage change from the original value, which allows comparisons on equal footing, regardless of scale. This is important because it considers the initial size, not just the absolute difference.

• For an increase from 5 to 10 students, the relative change is calculated as follows:
  o \(\frac{10 - 5}{5} \times 100\% = 100\%\)
• For an increase from 30 to 50 students, the relative change is calculated as follows:
  o \(\frac{50 - 30}{30} \times 100\% = 66.67\%\)

When comparing these percentages, 100% is clearly larger than 66.67%, indicating a greater relative increase in the first scenario.

Mathematics Problem-Solving

Problem-solving in mathematics involves a methodical approach comprised of understanding the problem, carrying out calculations, and drawing conclusions based on the results. Each step is critical in ensuring accurate solutions and understanding.

Here, the problem involves calculating relative changes to compare class size increases. The steps employed are:

• Understanding the problem: Identifying the variables, in this case, the initial and final class sizes.
• Calculating relative changes using the formula: \(\text{Relative Change} = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100\%\)
• Comparing results to determine which change is larger.

This structured method not only solves the problem at hand but also enhances comprehension of how comparative analysis is performed in real-world scenarios. It underscores the importance of relative changes over absolute ones, particularly when initial sizes vary.