A financial advisor notes that a particular investment fund's value triples every 5 years. In 2010, an investor's account in...

1. TRANSLATE the problem information

• Given information:
  • Investment fund value triples every 5 years
  • Account worth $4,500 in 2010
  • Need to find when it first reaches or exceeds $40,000

• What this tells us: We have exponential growth with a factor of 3 every 5-year period

2. INFER the mathematical approach

• This is an exponential growth problem where we need to solve an inequality
• Strategy: Set up \(\mathrm{Value = Initial \times (Growth\ Factor)^{(Number\ of\ periods)} \geq Target}\)
• We need the smallest number of periods that satisfies this condition

3. TRANSLATE into mathematical equation

Set up the inequality: \(\$4,500 \times 3^n \geq \$40,000\)
(where n = number of 5-year periods)

4. SIMPLIFY to solve for n

• Divide both sides by $4,500:
  \(3^n \geq \$40,000 \div \$4,500 = 8.89\) (use calculator for division)

• Find the smallest n where \(3^n \geq 8.89\):
  • \(3^1 = 3\) (too small)
  • \(3^2 = 9\) (this works since \(9 \gt 8.89\))

• Therefore: \(n = 2\) periods

5. INFER the final answer

• Total time needed: \(2\ \mathrm{periods} \times 5\ \mathrm{years/period} = 10\ \mathrm{years}\)
• Target year: \(2010 + 10 = 2020\)

Answer: B (2020)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret 'triples every 5 years' as adding $4,500 every 5 years instead of multiplying by 3.

They might think: 2015 → $9,000, 2020 → $13,500, 2025 → $18,000, etc. Since none of these reach $40,000 by 2025, they get confused and may select Choice D (2025) as the latest option, or abandon systematic solution and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(3^n \geq 8.89\) but make arithmetic errors when evaluating powers of 3 or calculating \(40,000 \div 4,500\).

For example, if they miscalculate and think \(3^3\) is needed, this leads to \(n = 3\) periods = 15 years, giving \(2010 + 15 = 2025\). This may lead them to select Choice D (2025).

The Bottom Line:

This problem tests whether students can correctly identify and set up exponential growth versus linear growth, then systematically solve the resulting inequality without computational errors.