A bike rental shop charges each customer a fixed safety fee and a constant hourly rate for a bike rental....

1. TRANSLATE the problem information

• Given information:
  • Fixed safety fee (same for all customers) + hourly rate × hours = total cost
  • Customer 1: 2 hours, paid $22.10 total
  • Customer 2: 5 hours, paid $43.40 total
  • Need to find: hourly rate

2. TRANSLATE into mathematical equations

• Let \(\mathrm{f}\) = fixed safety fee (dollars)
• Let \(\mathrm{r}\) = hourly rate (dollars per hour)
• Customer 1: \(\mathrm{f + 2r = 22.10}\)
• Customer 2: \(\mathrm{f + 5r = 43.40}\)

3. INFER the solution approach

• This creates a system of two linear equations with two unknowns
• Since we only need the hourly rate \(\mathrm{r}\), elimination method will work well
• We can eliminate \(\mathrm{f}\) by subtracting one equation from the other

4. SIMPLIFY using elimination

• Subtract the first equation from the second equation:
  \(\mathrm{(f + 5r) - (f + 2r) = 43.40 - 22.10}\)
• This gives us: \(\mathrm{3r = 21.30}\)
• Therefore: \(\mathrm{r = 21.30 \div 3 = 7.10}\)

5. Verify the answer

• If \(\mathrm{r = 7.10}\), then \(\mathrm{f = 22.10 - 2(7.10) = 7.90}\)
• Check: \(\mathrm{7.90 + 5(7.10) = 7.90 + 35.50 = 43.40}\) ✓

Answer: B. $7.10

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may struggle to set up the correct equations from the word problem. They might confuse which values represent the total cost versus the components (fixed fee + hourly charges), or they might try to work with just one customer's information instead of recognizing they need both to create a system.

This leads to incomplete setup and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the system but make algebraic errors during elimination. Common mistakes include sign errors when subtracting equations, or arithmetic errors when dividing 21.30 by 3.

This may lead them to select Choice A (6.85) or Choice D (7.30) due to calculation mistakes.

The Bottom Line:

This problem tests whether students can recognize that rental pricing (fixed + variable costs) creates a linear system, then execute the elimination method accurately. The key insight is that comparing two different customers gives you exactly the information needed to solve for the hourly rate.