The value of a car is currently $25,000, and the value is estimated to decrease each year by 15% from...

1. TRANSLATE the problem information

• Given information:
  • Current car value: \(\$25,000\) (initial value)
  • Value decreases by 15% each year
  • Target value: \(\$17,000\) (final value)
  • Need equation for time t

2. INFER the mathematical model

• This describes exponential decay since the value decreases by a fixed percentage each year
• For exponential decay: \(\mathrm{Final~Value} = \mathrm{Initial~Value} \times (\mathrm{decay~factor})^{\mathrm{time}}\)
• Key insight: If value decreases by 15% each year, then 85% remains each year
• Therefore: \(\mathrm{decay~factor} = 1 - 0.15 = 0.85\)

3. TRANSLATE into equation form

• Substitute the values:
  • Final Value = \(\$17,000\)
  • Initial Value = \(\$25,000\)
  • Decay factor = 0.85
• Equation: \(17,000 = 25,000(0.85)^t\)

4. Match with answer choices

• Looking at the options, this matches choice (C)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students confuse the decay rate with the decay factor, thinking that a 15% decrease means multiplying by 0.15 instead of 0.85.

They incorrectly reason: "The car loses 15% each year, so I multiply by 0.15." This fundamental misunderstanding of exponential decay leads them to set up the equation as \(17,000 = 25,000(0.15)^t\).

This may lead them to select Choice B (\(17,000 = 25,000(0.15)^t\)).

Second Most Common Error:

Poor TRANSLATE execution: Students reverse the initial and final values in the equation setup, placing \(\$25,000\) on the left and \(\$17,000\) on the right.

They might correctly identify the decay factor as 0.85 but incorrectly think "start with the target value and work backward." This creates the equation \(25,000 = 17,000(0.85)^t\).

This may lead them to select Choice D (\(25,000 = 17,000(0.85)^t\)).

The Bottom Line:

This problem tests whether students understand that in exponential decay, you multiply by what remains (100% - 15% = 85%) rather than what's lost (15%). The key insight is recognizing that 0.85 represents the decay factor, not 0.15.