A company opens an account with an initial balance of $5,000.00. The account earns 4% annual interest compounded semi-annually, and...

1. TRANSLATE the problem information

• Given information:
  • Initial balance (P): $5,000
  • Annual interest rate: 4% = 0.04
  • Compounding: semi-annually (n = 2)
  • Balance after 1 year: $5,202

• What this tells us: We need an exponential function that matches these conditions.

2. INFER the testing strategy

• Since we know \(\mathrm{A(1) = 5{,}202}\), we can test each option by substituting \(\mathrm{t = 1}\)
• The correct equation should give us exactly $5,202 when \(\mathrm{t = 1}\)
• This is more efficient than deriving the formula from scratch

3. SIMPLIFY by testing each option

Option A: \(\mathrm{A(t) = 5{,}000(1.02)^{2t}}\)

• \(\mathrm{A(1) = 5{,}000(1.02)^{2×1} = 5{,}000(1.02)^2}\)
• \(\mathrm{(1.02)^2 = 1.0404}\) (use calculator)
• \(\mathrm{5{,}000 × 1.0404 = 5{,}202}\) ✓ This matches!

Option B: \(\mathrm{A(t) = 202(1.02)^{2t}}\)

• \(\mathrm{A(1) = 202(1.02)^2 = 202(1.0404) ≈ 210.36}\) ✗

Option C: \(\mathrm{A(t) = 5{,}000(0.02)^{2t}}\)

• \(\mathrm{A(1) = 5{,}000(0.02)^2 = 5{,}000(0.0004) = 2}\) ✗

Option D: \(\mathrm{A(t) = 5{,}000(1.04)^t}\)

• \(\mathrm{A(1) = 5{,}000(1.04)^1 = 5{,}200}\) ✗ (Close, but not exact)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse '4% compounded semi-annually' with simple annual compounding, leading them to think the growth factor should be 1.04 per year.

They reason: '4% annual interest means multiply by 1.04 each year, so \(\mathrm{A(t) = 5{,}000(1.04)^t}\) makes sense.'

This may lead them to select Choice D (\(\mathrm{5{,}000(1.04)^t}\)) without testing, since D looks like the 'obvious' compound interest formula.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students attempt to verify their choice but make calculation errors with \(\mathrm{(1.02)^2}\), getting approximately 1.04 instead of 1.0404.

This makes Option A seem to give \(\mathrm{A(1) ≈ 5{,}000(1.04) = 5{,}200}\), which doesn't match the given $5,202. They might then incorrectly eliminate the right answer and guess among the remaining choices.

The Bottom Line:

This problem tests whether students understand that semi-annual compounding splits the annual rate in half but compounds twice as often, and whether they can perform precise calculations to distinguish between very similar values ($5,200 vs $5,202).