A car was purchased for $24,000 in 2020. Each year from 2020 to 2025, the car's value decreased by 8%...

1. TRANSLATE the problem information

• Given information:
  • Initial value: \(\$24,000\) in 2020
  • Value decreases by \(8\%\) each year
  • Need function v(t) where t = years after 2020

• What "decreases by 8%" means: The car retains \(100\% - 8\% = 92\%\) of its value each year

2. INFER the mathematical pattern

• Since the car retains \(92\% = 0.92\) of its value each year, we multiply by 0.92 repeatedly
• This creates an exponential decay pattern
• After t years: value = (initial value) × (retention rate)^t

3. Write the function

• Starting value: \(\$24,000\)
• Retention rate: 0.92 per year
• Function: \(\mathrm{v(t)} = 24,000(0.92)^\mathrm{t}\)

4. Verify the answer

• Check after 1 year: \(\mathrm{v(1)} = 24,000(0.92) = \$22,080\)
• Decrease: \(\$24,000 - \$22,080 = \$1,920\)
• Percentage check: \(\$1,920 \div \$24,000 = 0.08 = 8\%\) ✓

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students focus on the "8% decrease" and use 0.08 as the base in their exponential function, thinking the decreasing amount should be the multiplier rather than the remaining amount.

Their reasoning: "It decreases by 8% each year, so I multiply by 0.08 each time."

This leads them to select Choice A (\(\mathrm{v(t)} = 24,000(0.08)^\mathrm{t}\))

Second Most Common Error:

Poor TRANSLATE reasoning: Students confuse decrease with increase and use 1.08 as the base, thinking that 8% means adding 8% to get 108% of the original value.

Their reasoning: "8% change means multiply by 1.08."

This leads them to select Choice C (\(\mathrm{v(t)} = 24,000(1.08)^\mathrm{t}\))

The Bottom Line:

The key insight is recognizing that a percentage decrease requires using the retention rate (what's left over) as the base in exponential decay, not the decrease rate itself. The phrase "decreases by 8%" must be translated to "retains 92%" for the correct exponential function.