Lesson 5.4

Compound Interest

Interest that earns interest — that's the magic of compounding. The formula captures how money grows when interest is reinvested.

Introduction

Simple interest only earns on the principal. Compound interest earns on principal + previous interest. Over long periods, this snowball effect is enormous — Einstein reportedly called it "the eighth wonder of the world."

Past Knowledge

Exponential functions (5.2), growth factor .

Today's Goal

Apply the compound interest formula with different compounding periods.

Future Success

5.5 extends this to continuous compounding using .

Key Concepts

The Formula

A = final amount

P = principal (initial deposit)

r = annual interest rate (decimal)

n = compounds per year

t = time in years

Common n Values

Compounding	n
Annually	1
Quarterly	4
Monthly	12
Weekly	52
Daily	365

Worked Examples

Example 1: Monthly Compounding

Basic

$5,000 at 6% compounded monthly for 10 years.

Identify values

Substitute

Calculate

Nearly doubled! Interest earned: ≈ $4,097

Example 2: Comparing Compounding Frequencies

Intermediate

$1,000 at 8% for 5 years. Compare annually vs. quarterly.

Annually (n = 1)

Quarterly (n = 4)

Quarterly earns $16.62 more — more compounding = more growth

Example 3: Solving for Time

Advanced

How long for $3,000 to grow to $5,000 at 4% compounded monthly?

Substitute

Divide and take ln

Solve

About 12 years and 9 months

Common Pitfalls

Rate as Percentage

6% must be written as , not 6. Forgetting to convert gives absurd results.

Confusing n and t

is how often interest compounds per year. is the total years. The exponent is , not just .

Real-Life Applications

Every bank account, credit card, student loan, and retirement fund uses this formula. Understanding it is one of the most financially impactful math skills you can learn.