Compound Interest

Learn how compound interest grows your money faster by earning interest on interest.

Intermediate30 minLesson
Printable worksheet

Definition

Compound interest is interest calculated on both the initial principal AND the accumulated interest from previous periods. Unlike simple interest, your money grows exponentially because you earn "interest on interest."

The Compound Interest Formula

A=P(1+r)tA = P(1 + r)^t
Where:
  • AA = Final amount (principal + interest)
  • PP = Principal (starting amount)
  • rr = Annual interest rate (as a decimal)
  • tt = Time (in years)

Finding Just the Interest

To find only the interest earned:
I=A−P=P(1+r)t−PI = A - P = P(1 + r)^t - P

Why It Grows Faster

With compound interest, each year's interest is added to the principal, so the next year you earn interest on a larger amount. This creates exponential growth!

Try it now

In the compound interest formula A=P(1+r)tA = P(1+r)^t, what does the exponent tt represent?

Worked Examples

You invest 1000 dollars at 5% compound interest for 3 years. How much will you have?
1

Identify the values

P=1000P = 1000, r=5%=0.05r = 5\% = 0.05, t=3t = 3 → Values identified
2

Write the formula

A=P(1+r)tA = P(1 + r)^t → Formula ready
3

Substitute values

A=1000(1+0.05)3=1000(1.05)3A = 1000(1 + 0.05)^3 = 1000(1.05)^3 → Values substituted
4

Calculate the power

(1.05)3=1.05×1.05×1.05=1.157625(1.05)^3 = 1.05 \times 1.05 \times 1.05 = 1.157625 → (1.05)3=1.157625(1.05)^3 = 1.157625
5

Multiply by principal

A=1000×1.157625=1157.63A = 1000 \times 1.157625 = 1157.63 → A=1157.63A = 1157.63

Common Mistakes

Forgetting to add 1 to the rate inside the parentheses

Why it's wrong: Writing (0.05)3(0.05)^3 instead of (1.05)3(1.05)^3 gives a tiny number, not growth.

Correct: Always use (1+r)t(1 + r)^t. The 1 represents keeping your original money!

Using simple interest formula for compound problems

Why it's wrong: Simple interest is I=PrtI = Prt. Compound interest uses exponents: A=P(1+r)tA = P(1+r)^t.

Correct: Look for keywords: "compounded" means use the exponential formula.

Not converting percentage to decimal

Why it's wrong: Using r=5r = 5 instead of r=0.05r = 0.05 will give astronomical results.

Correct: Always divide percentage by 100: 5%=0.055\% = 0.05

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Formula
Simple Interest:I = P × r × t = 2000 × 0.06 × 5 = $600.00
Compound Interest:A = P(1 + r/n)^(nt)A = 2000 · (1 + 0.06)^5 = $2,676.45

Simple Interest

Interest Earned:$600.00
Total Amount:$2,600.00

Compound Interest

Interest Earned:$676.45
Total Amount:$2,676.45

Difference: $76.45 more with compound interest

Same money, different compounding

CompoundingnTotal AmountInterest Earned
Annually1$2,676.45$676.45
Monthly12$2,697.70$697.70
Daily365$2,699.65$699.65
Continuously∞$2,699.72$699.72

Growth Over Time

yearsAmountSimpleCompound

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Practice Problems

15 problems
Problem 1 of 15
Easy
In the compound interest formula A=P(1+r)tA = P(1+r)^t, what does the exponent tt represent?

Why It Matters

Compound interest is one of the most powerful concepts in finance:
  • Savings accounts: Your money grows faster over time
  • Retirement funds: Small contributions become large sums over decades
  • Investments: The stock market uses compound returns
  • Credit cards: Debt can grow quickly if you only pay minimums
  • Student loans: Interest compounds, increasing what you owe
Albert Einstein reportedly called compound interest "the eighth wonder of the world." Understanding it helps you build wealth and avoid debt traps!

Real World Applications

Retirement Savings

Retirement accounts use compound interest to grow small contributions into large nest eggs over 30-40 years.

Example:

Investing 100 dollars monthly at 7% for 40 years can grow to over 260000 dollars!

1Try It Yourself

You invest 2000 dollars in a retirement account earning 6% compound interest.

How much will you have after 20 years?

Step 1: Write the mathematical expression

Use the formula A=P(1+r)tA = P(1+r)^t:

Credit Card Debt

Credit cards compound interest monthly. High rates can quickly multiply what you owe.

Example:

A 1000 dollar balance at 20% APR becomes about 1220 dollars after one year if unpaid.

2Try It Yourself

You have 500 dollars in credit card debt at 18% annual interest, compounded yearly.

How much will you owe after 3 years if you make no payments?

Step 1: Write the mathematical expression

Calculate the debt growth:

College Savings

529 plans and education savings accounts use compound interest to help families prepare for college costs.

Example:

Starting a 529 plan when your child is born gives 18 years of compound growth!

Key Takeaways

  • 1Compound interest formula: A=P(1+r)tA = P(1 + r)^t
  • 2Interest is calculated on principal PLUS previously earned interest
  • 3Always add 1 to the rate: use (1+r)(1 + r), not just rr
  • 4Convert percentage to decimal before calculating
  • 5Compound interest grows faster than simple interest over time
  • 6The longer the time, the bigger the difference becomes

Frequently Asked Questions

With simple interest, you only earn interest on the original principal. With compound interest, you earn interest on interest. Over time, this creates exponential growth that far exceeds simple interest.
With simple interest, you only earn interest on the original principal. With compound interest, you earn interest on interest. Over time, this creates exponential growth that far exceeds simple interest.
It means interest is calculated and added to your account once per year. Interest can also be compounded monthly, daily, or even continuously - more frequent compounding leads to slightly more growth.
For monthly compounding, adjust the formula: A=P(1+r12)12tA = P(1 + \frac{r}{12})^{12t} where you divide the rate by 12 and multiply time by 12.

Glossary

Compound Interest
Interest calculated on the initial principal and all accumulated interest
Principal
The original amount of money invested or borrowed
Exponential Growth
Growth that increases at an ever-faster rate, like compound interest
Compounding Period
How often interest is calculated and added (annually, monthly, daily)
Growth Factor
The value (1+r)t(1 + r)^t that multiplies the principal to give final amount
Today's challengeOne problem, three tries

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