A new car was purchased for $32,000. The value of the car is estimated to decrease by 20% each year...

1. TRANSLATE the problem information

• Given information:
  • Initial car value: \(\$32,000\)
  • Value decreases by 20% each year
  • Need equation for value V after t years
• What this tells us: We're dealing with compound percentage decrease over time

2. INFER the mathematical model needed

• Since the car loses a percentage of its value each year (not a fixed dollar amount), this is exponential decay
• General exponential formula: \(\mathrm{V = P(b)^t}\)
• We need to identify P (initial value) and b (decay factor)

3. TRANSLATE the decay information

• \(\mathrm{P = 32,000}\) (given initial value)
• If the car decreases by 20% each year, it retains 80% of its value each year
• Retention factor: \(\mathrm{b = 1 - 0.20 = 0.80}\)

4. Assemble the equation

• Substituting into \(\mathrm{V = P(b)^t}\):
• \(\mathrm{V = 32,000(0.8)^t}\)

5. Verify the equation

• After 1 year: \(\mathrm{V = 32,000(0.8)^1 = 25,600}\) (use calculator)
• Decrease: \(\$32,000 - \$25,600 = \$6,400\)
• Check: 20% of \(\$32,000 = \mathrm{0.20 × 32,000 = \$6,400}\) ✓

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often confuse the decay rate (20% = 0.2) with the retention factor (80% = 0.8). They see "decreases by 20%" and immediately think the factor should be 0.2, not realizing they need the portion that remains after the decrease.

This leads them to select Choice A (\(\mathrm{V = 32,000(0.2)^t}\))

Second Most Common Error:

Poor INFER reasoning: Students may recognize they need 0.8 as the factor but fail to distinguish between exponential and linear models. They might think the decrease is 0.2t (meaning 20% times t years) rather than compound decay.

This may lead them to select Choice D (\(\mathrm{V = 32,000(1 - 0.2t)}\))

The Bottom Line:

The key challenge is understanding that "decreases by 20%" means you keep 80%, not that 20% is your multiplier. Plus, compound percentage changes require exponential models, not linear ones.