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R57 • Lesson 87 of 105

Compound Interest

Understanding compound interest and exponential growth in finance

Reserve & Extensions • K-12

Prerequisites: H38, M17

Key Concepts

  • compound interest
  • exponential growth
  • compounding periods
  • financial applications

Compound Interest

Simple interest is calculated only on the original amount. Compound interest is calculated on the original amount plus any interest already earned. This "interest on interest" effect causes money to grow much faster over time -- and it is how most real-world savings, loans, and investments actually work.

The Formula

A = P(1 + r/n)nt
VariableMeaning
AFinal amount (principal + interest)
PPrincipal (initial amount)
rAnnual interest rate (as a decimal)
nNumber of times interest compounds per year
tTime in years

Compounding Periods

Compounding Frequencyn value
Annually1
Semi-annually2
Quarterly4
Monthly12
Daily365

Worked Example 1: Annual Compounding

You invest $1,000 at 6% interest compounded annually for 3 years.

  1. P = 1,000, r = 0.06, n = 1, t = 3
  2. A = 1,000(1 + 0.06/1)1×3 = 1,000(1.06)3
  3. A = 1,000 × 1.191016 = $1,191.02
  4. Interest earned: $1,191.02 - $1,000 = $191.02

Compare with simple interest: I = 1,000 × 0.06 × 3 = $180. Compounding earned an extra $11.02.

Worked Example 2: Monthly Compounding

You deposit $5,000 at 4.8% interest compounded monthly for 5 years.

  1. P = 5,000, r = 0.048, n = 12, t = 5
  2. A = 5,000(1 + 0.048/12)12×5 = 5,000(1 + 0.004)60
  3. A = 5,000(1.004)60 = 5,000 × 1.27048 = $6,352.40
  4. Interest earned: $1,352.40

Compound vs. Simple Interest Over Time

The difference between compound and simple interest grows dramatically with time. Here is $1,000 at 8%:

YearsSimple Interest TotalCompound Interest TotalDifference
5$1,400$1,469$69
10$1,800$2,159$359
20$2,600$4,661$2,061
30$3,400$10,063$6,663

The Rule of 72

To estimate how long it takes for money to double with compound interest, divide 72 by the interest rate (as a whole number):

Doubling Time ≈ 72 / rate

At 6% interest: 72 / 6 = 12 years to double. At 9%: 72 / 9 = 8 years. This is a quick mental math tool, not an exact calculation.

Worked Example 3: Rule of 72

At 4% annual interest, approximately how long to double your money?

  1. 72 / 4 = 18 years
  2. Check: $1,000 at 4% for 18 years: A = 1,000(1.04)18 = $2,025.82. Very close to double.

Compound Interest on Debt

Compound interest works against you on loans and credit cards. A $5,000 credit card balance at 18% APR compounded monthly, left unpaid for 5 years, grows to over $12,000. The same power that builds wealth in savings destroys it in debt.

Practice Problems

1. Calculate the final amount: $2,000 at 5% compounded annually for 4 years.

Show Solution

A = 2,000(1.05)4 = 2,000 × 1.21551 = $2,431.01

2. You invest $3,000 at 6% compounded semi-annually for 3 years. How much do you have?

Show Solution

A = 3,000(1 + 0.06/2)2×3 = 3,000(1.03)6 = 3,000 × 1.19405 = $3,582.16

3. Using the Rule of 72, estimate the doubling time at 12% interest.

Show Solution

72 / 12 = 6 years

4. You invest $1,000 at 5% for 10 years. How much more do you earn with monthly compounding vs. simple interest?

Show Solution

Simple: I = 1,000 × 0.05 × 10 = $500. Total = $1,500. Compound: A = 1,000(1 + 0.05/12)120 = 1,000(1.004167)120 = $1,647.01. Difference: $1,647.01 - $1,500 = $147.01.

5. A credit card has an 18% APR compounded monthly. If you owe $2,000 and make no payments, how much do you owe after 2 years?

Show Solution

A = 2,000(1 + 0.18/12)24 = 2,000(1.015)24 = 2,000 × 1.42950 = $2,859.01

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