Which of the following describe(s) two quantities that vary inversely? The price per gallon of a certain home heating oil...

1. Translate the problem requirements: Clarify what "vary inversely" means - when one quantity increases, the other decreases proportionally such that their product remains constant
2. Establish the inverse variation test: For each scenario, determine if the relationship follows the pattern where \(\mathrm{quantity_1} \times \mathrm{quantity_2} = \mathrm{constant}\)
3. Analyze each scenario systematically: Apply the inverse variation test to scenarios I, II, and III to see which ones satisfy the constant product requirement
4. Match findings to answer choices: Identify which scenarios demonstrate inverse variation and select the corresponding answer choice

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what "vary inversely" means in everyday terms.

Imagine you have a fixed budget of $100 to buy apples. If apples cost $2 per pound, you can buy 50 pounds. But if the price goes up to $5 per pound, you can only buy 20 pounds. Notice that as the price per pound increases, the number of pounds you can buy decreases.

This is inverse variation: when one quantity increases, the other decreases in such a way that their product stays the same. In our apple example: \(\$2 \times 50 \text{ pounds} = \$100\), and \(\$5 \times 20 \text{ pounds} = \$100\).

So two quantities vary inversely when: \(\mathrm{quantity_1} \times \mathrm{quantity_2} = \mathrm{constant\ value}\)

Process Skill: TRANSLATE - Converting the mathematical concept into concrete, relatable examples

2. Establish the inverse variation test

Now we have our test for inverse variation. For any two quantities to vary inversely, their product must always equal the same constant value, no matter how the individual quantities change.

Our test question for each scenario will be: "If I multiply these two quantities together, do I always get the same number?"

If yes → inverse variation
If no → not inverse variation

3. Analyze each scenario systematically

Let's apply our test to each scenario:

Scenario I: Price per gallon × Number of gallons for $600

This is exactly like our apple example! You have a fixed budget of $600.
- If oil costs $3 per gallon: you can buy 200 gallons (\(\$3 \times 200 = \$600\))
- If oil costs $6 per gallon: you can buy 100 gallons (\(\$6 \times 100 = \$600\))
- If oil costs $10 per gallon: you can buy 60 gallons (\(\$10 \times 60 = \$600\))

The product is always $600. ✓ This is inverse variation.

Scenario II: Revenue and number of tickets sold at $6 each

Let's think about this: if you sell more tickets, do you make less money? That doesn't make sense!
- Sell 10 tickets: revenue = \(\$6 \times 10 = \$60\)
- Sell 20 tickets: revenue = \(\$6 \times 20 = \$120\)
- Sell 30 tickets: revenue = \(\$6 \times 30 = \$180\)

As tickets sold increases, revenue increases too. The product (revenue × tickets) keeps getting bigger: \(60 \times 10 = 600\), then \(120 \times 20 = 2,400\), then \(180 \times 30 = 5,400\).

This is direct variation (both go up together), not inverse variation. ✗

Scenario III: Speed × Time for a 60-mile trip

This makes intuitive sense: the faster you drive, the less time the trip takes.
- Drive at 30 mph: takes 2 hours (\(30 \times 2 = 60 \text{ miles}\))
- Drive at 60 mph: takes 1 hour (\(60 \times 1 = 60 \text{ miles}\))
- Drive at 20 mph: takes 3 hours (\(20 \times 3 = 60 \text{ miles}\))

The product is always 60 miles. ✓ This is inverse variation.

Process Skill: APPLY CONSTRAINTS - Using the fixed total ($600, 60 miles) to identify inverse relationships

4. Match findings to answer choices

Our analysis shows:
- Scenario I: Inverse variation ✓
- Scenario II: Not inverse variation ✗
- Scenario III: Inverse variation ✓

So scenarios I and III demonstrate inverse variation.

Looking at our answer choices:

1. I only
2. II only
3. III only
4. I and III ←This matches our findings
5. II and III

Final Answer

The answer is D. I and III.

Both scenario I (price per gallon and gallons purchased with fixed budget) and scenario III (speed and time for fixed distance) demonstrate inverse variation because in each case, the product of the two quantities remains constant.

Common Faltering Points

Errors while devising the approach

1. Confusing inverse variation with direct variation
Students often mix up the definitions. They might think "vary inversely" means both quantities change in the same direction (both increase or both decrease together), when it actually means they change in opposite directions. This fundamental misunderstanding leads to selecting scenarios where quantities increase together, like revenue and tickets sold.

2. Missing the "fixed constraint" requirement
Students may not recognize that inverse variation requires a constant product, which only happens when there's a fixed constraint (like $600 budget or 60-mile distance). They might analyze scenarios without identifying what stays constant, leading to incorrect classification of relationships.

3. Focusing on individual quantity behavior instead of their relationship
Students might get distracted by how each quantity changes individually rather than examining how the two quantities relate to each other. For example, in Scenario II, they might focus on "more tickets sold" or "higher revenue" separately instead of testing whether their product remains constant.

Errors while executing the approach

1. Incorrect setup of the multiplication test
When testing if two quantities vary inversely, students might multiply the wrong elements together. For instance, in Scenario I, they might multiply "price per gallon × $600" instead of "price per gallon × number of gallons," leading to incorrect conclusions about whether the product stays constant.

2. Using inconsistent or unrealistic test values
Students may choose test values that don't properly demonstrate the relationship, or they might change multiple variables at once instead of letting one determine the other. This makes it difficult to see whether the product truly remains constant across different scenarios.

Errors while selecting the answer

1. Mismatching analysis results to answer choices
After correctly identifying that Scenarios I and III show inverse variation, students might still select the wrong answer choice due to careless reading. They might pick "A. I only" or "C. III only" instead of "D. I and III," especially if they're rushing or not carefully checking their work against the available options.