From 2010 through 2020, the value of a certain investment fund increased each year by approximately 8% of its value...

1. TRANSLATE the problem information

• Given information:
  • Investment fund increases by 8% each year from 2010-2020
  • Initial value in 2010: \(\$25,000\)
  • Need to find V = value after t years

2. TRANSLATE the growth language into mathematics

• "Increases by 8%" means:
  • Original value + 8% of original value
  • Original value + \(0.08 \times \mathrm{Original\ value}\)
  • Original value \(\times (1 + 0.08)\)
  • Original value \(\times 1.08\)

3. INFER the pattern for repeated growth

• After 1 year: \(25,000 \times 1.08\)
• After 2 years: \((25,000 \times 1.08) \times 1.08 = 25,000 \times (1.08)^2\)
• After t years: \(25,000 \times (1.08)^\mathrm{t}\)

4. Match with answer choices

• \(\mathrm{V} = 25,000(1.08)^\mathrm{t}\) corresponds to choice C

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students interpret "increases by 8%" as "multiply by 8%"

They think: "8% growth means I multiply by 0.08 each year"
This creates the model \(\mathrm{V} = 25,000(0.08)^\mathrm{t}\)

This leads them to select Choice A (\(\mathrm{V} = 25,000(0.08)^\mathrm{t}\))

Second Most Common Error:

Conceptual confusion about growth vs. decay: Students think "increases by 8%" means "8% is lost each year"

They reason: "If it grows by 8%, then 8% is removed from the original, leaving 92%"
This creates the model \(\mathrm{V} = 25,000(0.92)^\mathrm{t}\)

This leads them to select Choice B (\(\mathrm{V} = 25,000(0.92)^\mathrm{t}\))

The Bottom Line:

The key challenge is correctly translating percentage growth language into mathematical operations. Students must understand that "increases by 8%" means the new amount is 108% of the original, not 8% of the original.