10.3 Financial Literacy: Simple and Compound Interest

Quick Answer: Simple interest is I = Prt with total A = P(1 + rt) — it adds the same interest each period, so it grows linearly. Compound interest is A = P(1 + r/n)^(nt) — it earns interest on prior interest, so it grows exponentially. Over time, compound interest always out-earns simple interest when rates and time are positive.

Florida benchmark MA.912.FL.3.2 requires solving real-world problems involving simple and compound interest. These problems connect the Non-Linear Relationships reporting category to financial literacy and appear alongside the exponential-function items in the third domain (31-38% of the exam).

Simple Interest

The simple interest formula is I = Prt, where P is the principal (initial amount), r is the annual interest rate as a decimal, and t is the time in years. The interest earned is a constant amount each year, so the total value grows linearly.

The total amount after t years is A = P + Prt = P(1 + rt).

Worked example: $2,000 invested at 5% simple interest for 6 years. Interest I = 2,000 × 0.05 × 6 = $600. Total A = 2,000 + 600 = $2,600. Equivalently, A = 2,000(1 + 0.05 × 6) = 2,000 × 1.30 = $2,600. Each year adds $100 in interest, so the growth is linear.

If the time is given in months, convert: t = months / 12. For 18 months, t = 1.5.

Compound Interest

The compound interest formula is A = P(1 + r/n)^(nt), where:

• P is the principal
• r is the annual interest rate as a decimal
• n is the number of compounding periods per year
• t is the number of years
• A is the total amount after t years

The interest earned is A - P.

Common compounding frequencies:

Worked example: $2,000 invested at 5% interest compounded quarterly for 6 years. Here r = 0.05, n = 4, t = 6. A = 2,000(1 + 0.05/4)^(4 × 6) = 2,000(1.0125)^24. Compute 1.0125^24 ≈ 1.3482, so A ≈ 2,000 × 1.3482 = $2,696.40. The interest earned is $696.40 — about $96 more than simple interest produced for the same principal, rate, and time.

Comparing Simple and Compound Interest

The critical insight: simple interest grows linearly because each period adds Pr (a constant). Compound interest grows exponentially because each period multiplies the current balance by (1 + r/n), so prior interest itself earns interest. The longer the time and the higher the compounding frequency, the bigger the gap.

For a $1,000 investment at 8% for 10 years:

• Simple: A = 1,000(1 + 0.08 × 10) = 1,000 × 1.8 = $1,800.
• Compounded annually: A = 1,000(1.08)^10 ≈ 1,000 × 2.1589 = $2,158.92.
• Compounded monthly: A = 1,000(1 + 0.08/12)^(12 × 10) = 1,000(1.006667)^120 ≈ $2,219.64.

The monthly compounding case earns about $420 more than simple interest over the same 10-year period — a 23% larger balance, driven purely by compounding frequency.

Continuous Compounding (Context Only)

The limit as n → ∞ gives A = Pe^(rt). Florida's Algebra 1 EOC does not test continuous compounding, but you may see the phrase "compounded continuously" in a word problem setup that is solved by substitution. If you do encounter it, the model is A = Pe^(rt) with base e ≈ 2.71828.

Solving for Other Variables

The compound formula can be rearranged to solve for P, r, or t, but on the Algebra 1 EOC most items give P, r, n, and t and ask for A, or ask which of two investment options pays more. To compare options, compute A for each and subtract.

Worked comparison: Option A pays 6% simple interest for 4 years on $5,000. Option B pays 6% compounded monthly for 4 years on $5,000. Which pays more interest?

• Option A: I = 5,000 × 0.06 × 4 = $1,200; A = $6,200.
• Option B: A = 5,000(1 + 0.06/12)^(12 × 4) = 5,000(1.005)^48 ≈ 5,000 × 1.2705 ≈ $6,352.50. Interest = $1,352.50.
• Option B pays about $152.50 more.

Unit Conversion Traps

Florida items frequently embed unit traps:

• Rate given as a percent (5%) must be converted to decimal (0.05) before substitution.
• Time in months must be converted to years (t = months/12).
• A 6-month CD at 4% uses t = 0.5, not 6, when the rate is annual.
• Quarterly compounding means n = 4, so the per-period rate r/n is the annual rate divided by 4.

A common mistake is to substitute the rate as 5 (instead of 0.05) or to substitute t = 6 when the time is 6 months at an annual rate — both produce wildly wrong answers.

EOC Trap Summary

1. Forgetting to divide the annual rate by n before exponentiating.
2. Forgetting to multiply n by t in the exponent (using (1 + r/n)^t instead of (1 + r/n)^(nt)).
3. Confusing interest earned (A - P) with total amount A.
4. Mixing simple and compound formulas when the question specifies one model.
5. Leaving the rate as a percentage (5) instead of a decimal (0.05).

Financial interest items are pure substitution once you identify the model. Read the compounding keyword (annually, quarterly, monthly) to set n, convert the rate to a decimal, convert time to years, and substitute carefully.