A student deposits $5{,}000 into a college savings account that earns 4% annual interest, compounded quarterly. Which of the following...

1. TRANSLATE the problem information

• Given information:
  • Principal amount: $5,000
  • Annual interest rate: 4%
  • Compounding frequency: quarterly
  • Time period: n years

• What this tells us:
  • This involves compound interest since money grows multiple times per year
  • Quarterly means 4 compounding periods per year
  • We need the compound interest formula

2. INFER the approach

• This is a compound interest problem because the account "earns interest, compounded quarterly"
• We need the compound interest formula: \(\mathrm{A = P(1 + r/k)^{(kt)}}\)
• The key insight: quarterly compounding means the annual rate gets divided by 4, but we compound 4 times as often

3. TRANSLATE the given values into formula variables

• \(\mathrm{P}\) (principal) = 5,000
• \(\mathrm{r}\) (annual rate) = 4% = 0.04
• \(\mathrm{k}\) (compounding frequency) = 4 (quarterly)
• \(\mathrm{t}\) (time) = n years

4. SIMPLIFY by substituting into the formula

• \(\mathrm{A = P(1 + r/k)^{(kt)}}\)
• \(\mathrm{A = 5,000(1 + 0.04/4)^{(4n)}}\)
• \(\mathrm{A = 5,000(1 + 0.01)^{(4n)}}\)
• \(\mathrm{A = 5,000(1.01)^{(4n)}}\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse "4% annual interest" with the growth factor per compounding period. They think since it's 4% annually, the base should be (1.04), not realizing that quarterly compounding means each quarter earns 1% \(\mathrm{(0.04/4 = 0.01)}\).

This leads them to incorrectly use 1.04 as the base and may lead them to select Choice C \(\mathrm{(A = 5,000(1.04)^n)}\) or Choice D \(\mathrm{(A = 5,000(1.04)^{(4n)})}\).

Second Most Common Error:

Poor INFER reasoning: Students recognize they need 1.01 as the base (correctly dividing the annual rate by 4), but don't understand that quarterly compounding over n years means 4n total compounding periods. They think the exponent should just be n.

This may lead them to select Choice A \(\mathrm{(A = 5,000(1.01)^n)}\).

The Bottom Line:

Compound interest problems require careful attention to both the rate per period AND the number of periods. The annual rate must be divided by the compounding frequency, and the time must be multiplied by that same frequency.