In a model of automobile performance, Car A travels exactly 10 kilometers per liter of fuel, and Car B travels...

Phase 1: Owning the Dataset

Visualization Selection

Since we're comparing two cars with different fuel efficiencies, let's use a comparison table:

Testing with Concrete Numbers

Let's test with N = 300 km to understand the relationship:

• Car A needs: \(\mathrm{300/10 = 30}\) liters
• Car B needs: \(\mathrm{300/15 = 20}\) liters
• Difference: \(\mathrm{30 - 20 = 10}\) liters

Phase 2: Understanding the Question

Breaking Down the Mathematical Relationship

The key statement tells us: "Car A requires exactly F more liters of fuel than Car B to travel exactly N kilometers."

This translates to:

• Fuel for Car A = \(\mathrm{N/10}\) liters
• Fuel for Car B = \(\mathrm{N/15}\) liters
• \(\mathrm{F = (N/10) - (N/15)}\)

Simplifying the Formula

Let's find a common denominator:
\(\mathrm{F = N/10 - N/15}\)
\(\mathrm{F = 3N/30 - 2N/30}\)
\(\mathrm{F = N/30}\)

Therefore: \(\mathrm{N = 30F}\)

Key Insight

We need to find values where one is exactly 30 times the other!

Phase 3: Finding the Answer

Systematic Check of F Values

Using our relationship \(\mathrm{N = 30F}\):

If \(\mathrm{F = 300}\) → N should be \(\mathrm{300 \times 30 = 9,000}\)
Is 9,000 in our choices? Yes! ✓
Stop here - we found our answer.

Verification

Let's verify with F = 300 and N = 9,000:

• Car A fuel for 9,000 km: \(\mathrm{9,000 ÷ 10 = 900}\) liters
• Car B fuel for 9,000 km: \(\mathrm{9,000 ÷ 15 = 600}\) liters (use calculator for precision)
• Difference: \(\mathrm{900 - 600 = 300}\) liters ✓

Phase 4: Solution

Our selections are:

• F = 300
• N = 9,000

These values satisfy the relationship where Car A needs exactly 300 more liters than Car B to travel exactly 9,000 kilometers.