Learning goals
- Calculate simple interest and the total amount owed or earned
- Calculate compound interest for a given rate, time, and compounding frequency
- Explain why the same rate produces different results under simple vs compound interest
Why the distinction matters
Interest is the cost of borrowing money — or the reward for lending and investing it. The rate alone does not tell the full story: you also need the rule for how interest is applied over time.
Simple interest charges (or pays) interest only on the original principal. Compound interest charges (or pays) interest on the principal plus interest that has already been added — “interest on interest.”
Short-term consumer loans and some notes may use simple interest. Savings accounts, credit cards, mortgages, and most long-term investments use compounding. Always read which method applies before comparing offers.
Simple interest
I = P × r × t
P is principal, r is the annual interest rate as a decimal (for example 6% → 0.06), and t is time in years. The total amount is A = P + I = P(1 + r t). Interest each year is the same dollar amount if P, r, and the year length stay fixed.
Compound interest (periodic)
A = P(1 + r/n)^(n t)
A is the ending amount, n is compounding periods per year (12 for monthly, 4 for quarterly, 1 for annual), r is the nominal annual rate as a decimal, and t is years. Interest earned is A − P. More frequent compounding (larger n) increases A for the same r and t.
Effective annual rate (EAR)
EAR = (1 + r/n)^n − 1
EAR converts a nominal rate with compounding frequency n into a true yearly growth factor. Two loans with the same APR but different compounding can have different EARs — use EAR (or APY on deposits) for fairer comparisons. See also Quantarizm’s APR vs APY guide.
Example 1 — Simple interest on a loan
Given
You borrow $2,000.00 at 8% simple interest for 3 years. How much interest do you pay, and what is the total repayment?
Steps
- Identify P = 2000, r = 0.08, t = 3.
- Interest I = 2000 × 0.08 × 3 = 480.
- Total amount A = 2000 + 480 = 2480.
Answer
Interest is $480.00; total repayment is $2,480.00.
Tip: Each year adds the same $160.00 of interest (2000 × 0.08) under simple interest.
Example 2 — Same inputs with annual compounding
Given
Same $2,000.00 at 8% for 3 years, but interest compounds once per year. What is the ending amount?
Steps
- Use A = P(1 + r/n)^(n t) with n = 1: A = 2000(1.08)^3.
- 1.08² = 1.1664; 1.1664 × 1.08 = 1.259712.
- A = 2000 × 1.259712 = 2519.424.
Answer
Ending amount is about $2,519.42 (interest ≈ $519.42).
Tip: Compared with simple interest’s $480.00 compounding adds about $39.42 more over three years at this rate.
Example 3 — Monthly compounding
Given
Invest $5,000.00 at 6% nominal annual rate, compounded monthly, for 2 years. Find the future value.
Steps
- P = 5000, r = 0.06, n = 12, t = 2.
- Periodic rate r/n = 0.06/12 = 0.005; periods n t = 24.
- A = 5000(1.005)^24 ≈ 5000 × 1.1271597 ≈ 5635.80.
Answer
Future value is about $5,635.80.
Tip: With annual compounding instead (n = 1), A = 5000(1.06)^2 = 5618 — slightly less than monthly compounding.
Example 4 — Choosing the right model
Given
A store offers “10% simple interest for 6 months” on a $900.00 purchase financed in-house. What do you owe at the end if no payments are made until then?
Steps
- Use simple interest: t = 6/12 = 0.5 year.
- I = 900 × 0.10 × 0.5 = 45.
- Amount due = 900 + 45 = 945.
Answer
You would owe $945.00 under that simple-interest quote.
Tip: If the fine print said “10% APR compounded monthly,” the balance would be higher — always match the formula to the contract language.
Using percent instead of a decimal
Common mistake
I = 2000 × 8 × 3 = 48,000.
Better approach
Convert 8% to 0.08 first: I = 2000 × 0.08 × 3 = 480. The same rule applies inside compound formulas (r/n must use the decimal rate).
Treating APR like APY
Common mistake
Assuming “6% APR compounded monthly” grows money by exactly 6% in one year.
Better approach
Monthly compounding at 6% APR yields EAR = (1 + 0.06/12)^12 − 1 ≈ 6.17%. The headline APR is not always the effective yearly growth.
Check your understanding
1.Find simple interest on $1,500.00 at 5% for 4 years.
Answer: I = 1500 × 0.05 × 4 = $300.00.
2.Find A for $1,000.00 at 5% compounded annually for 2 years.
Answer: A = 1000(1.05)^2 = $1,102.50.
3.For the same P, r, and t > 0 with n ≥ 1, which is larger: simple interest total or compound total?
Answer: Compound total is larger (or equal only in degenerate edge cases such as t = 0). Interest on interest adds extra growth.
4.What is n for quarterly compounding?
Answer: n = 4 (four times per year).
Key takeaways
- Simple interest: I = P r t and A = P(1 + r t) — interest does not earn interest.
- Compound interest: A = P(1 + r/n)^(n t) — balances grow on principal plus prior interest.
- Same nominal rate can produce different results depending on compounding frequency; EAR/APY make comparisons fairer.
- Always match the formula to the product’s rules before comparing “10%” offers.