A car rental company charges a flat fee of $45 per day. In addition, there is a charge of $0.25...

1. TRANSLATE the problem information

• Given information:
  • Flat fee: $45 per day
  • Per-mile charge: $0.25 per mile
  • Total charge: $98
  • Need to find: number of miles driven

• What this tells us: The total cost has two parts - a fixed amount plus a variable amount based on miles.

2. TRANSLATE into a mathematical equation

• Let \(\mathrm{m}\) = number of miles driven
• Total charge = Fixed fee + Variable fee
• \(\mathrm{98 = 45 + 0.25m}\)

3. SIMPLIFY by solving for m

• Subtract the flat fee from both sides:
  \(\mathrm{98 - 45 = 0.25m}\)
  \(\mathrm{53 = 0.25m}\)

• Divide both sides by 0.25:
  \(\mathrm{m = 53 ÷ 0.25}\)

• To divide by 0.25, multiply by 4 (since \(\mathrm{0.25 = 1/4}\)):
  \(\mathrm{m = 53 × 4 = 212}\)

Answer: 212

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret the cost structure and subtract the flat fee from the total, thinking the remaining $53 represents the miles directly.

They calculate: \(\mathrm{98 - 45 = 53}\), then select Choice B (53) thinking this is the number of miles, forgetting that $53 represents the total variable cost, not the miles themselves.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{98 = 45 + 0.25m}\) and get to \(\mathrm{53 = 0.25m}\), but then incorrectly compute \(\mathrm{53 ÷ 0.25}\).

They might calculate \(\mathrm{53 ÷ 0.25}\) as \(\mathrm{53 ÷ 0.25 ≈ 21}\) (confusing the division) or make other arithmetic errors, leading them to select Choice A (21).

The Bottom Line:

This problem tests whether students can distinguish between the dollar amount of variable costs ($53) and the quantity that generates those costs (212 miles). The division by 0.25 is also a common computational stumbling block.