Compound Interest: Definition, Formula & Examples

Definitionnoun phrase

What is compound interest?

Compound interest is interest computed on the principal plus all the interest already earned, so the balance grows exponentially. The formula is A = P( 1 + rn )^nt, where P is the principal, r the annual rate, n the compoundings per year and t the years. It beats simple interest after the first period.

In Spanish: interés compuesto
Also called: compounding, interest on interest, compound interest formula
Grade level: Algebra 2

Grade 7Algebra 1Algebra 2CCSS HSA-SSE.B.3.cCCSS HSF-LE.A.2GEDSATACTASVABHiSETTABEACCUPLACER

Compound Interest in a picture

With compounding, each year’s 10% is taken of a bigger balance, so the yearly increase grows from 100 to about 146. Simple interest adds the same 100 every year.

Formal definition and notation

If a principal P earns an annual interest rate r (as a decimal), compounded n times per year, then after t years the balance is A = P( 1 + rn )^nt. Each period the balance is multiplied by 1 + rn, so A is an exponential function of t. As n grows without bound, the balance approaches continuous compounding, A = Pe^rt.

Symbol	Read it as	Meaning
P	P	the principal: the starting amount
rn	r over n	the interest rate for one compounding period
nt	n t	the total number of compounding periods
Pe^rt	P e to the r t	the balance with continuous compounding

Key facts

Compound interest pays interest on interest; simple interest, I = Prt, pays only on the principal.
More frequent compounding gives a slightly larger balance: monthly beats yearly at the same rate.
The growth factor per period is 1 + rn, and there are nt periods.
Continuous compounding uses euler's number: A = Pe^rt.
Loans and credit cards compound too, which is why unpaid debt grows quickly.
To find the time for a balance to reach a goal, solve for the exponent with a logarithm.

Compound interest formula

A = P( 1 + rn )^nt

A: the final balance (principal plus interest)
P: the principal
r: the annual interest rate as a decimal (6% = 0.06)
n: compoundings per year: 1 yearly, 4 quarterly, 12 monthly, 365 daily
t: the time in years

The interest earned is A − P. With yearly compounding (n = 1) the formula is A = P(1 + r)^t, the exponential growth model.

Try it: one year at 5%

Interactive

This machine multiplies by 1.05: one year of 5% interest compounded yearly. Starting from 1000 dollars it has run two years. Type the latest balance back in and press Run to add more years.

This interactive needs JavaScript. The worked examples below show the same ideas step by step.

Compound Interest examples, step by step

Example 1: Yearly compounding

Warm-up

You deposit 500 dollars at 4% interest compounded yearly. What is the balance after 3 years?

Yearly means n = 1: A = 500(1.04)³.
1.04³ = 1.124864.
500 × 1.124864 = 562.432.

Answer: The balance is about 562.43 dollars.

Example 2: Monthly compounding

Core

You invest 2000 dollars at 6% annual interest compounded monthly. Find the balance and the interest earned after 3 years.

Here P = 2000, r = 0.06, n = 12, t = 3.
Rate per month: 0.0612 = 0.005. Number of months: 12 × 3 = 36.
A = 2000(1.005)³⁶ ≈ 2000 × 1.19668 = 2393.36.
Interest: 2393.36 − 2000 = 393.36.

Answer: The balance is about 2393.36 dollars, so the interest earned is 393.36 dollars.

Example 3: Compound versus simple

Test-style

Maya puts 1000 dollars in an account paying 10% compounded yearly. Leo puts 1000 dollars in an account paying 10% simple interest. How much more does Maya have after 5 years?

Maya: 1000(1.1)⁵ = 1610.51.
Leo: simple interest I = Prt = 1000 × 0.1 × 5 = 500, so he has 1500.
Difference: 1610.51 − 1500 = 110.51.

Answer: Maya has 110.51 dollars more than Leo.

Common mistakes

✗ Mistake6% as r = 6: A = 2000(1 + 6)³.

✓ Insteadr = 0.06: A = 2000(1.06)³.

A percent must be written as a decimal in the formula. Using 6 would multiply the money by 7 every year.

✗ MistakeMonthly at 6% for 3 years: 2000(1 + 0.06)³⁶.

✓ Instead2000(1 + 0.005)³⁶.

When you count months in the exponent, you must also use the monthly rate 0.0612. Rate and periods must match.

✗ MistakeReporting A when the question asks for the interest.

✓ InsteadInterest = A − P.

A is the whole balance. The interest is only the part added on top of the principal.

Not to be confused with

Compound Interestvs.Simple interest

Simple interest I = Prt is paid only on the principal, so it grows linearly. Compound interest is paid on the growing balance, so it grows exponentially. 1000 dollars at 10% for 5 years: 1500 simple, 1610.51 compound.

Compound Interestvs.Annual percentage yield

The stated rate r is the nominal yearly rate. The yield is the actual yearly growth after compounding: 6% compounded monthly gives 1.005¹² − 1 ≈ 6.17%.

Quick check: 3 questions

Question 1 of 3

Which expression gives the balance after 4 years on 800 dollars at 5% compounded quarterly?

Show answer and explanation

Answer: B. Quarterly: rate per quarter 0.054 = 0.0125 and 4 × 4 = 16 quarters. 800(1.05)⁴ is yearly compounding, 800(1.0125)⁴ counts only 4 quarters, and 1.2 multiplies the rate instead of dividing.

Question 2 of 3

What is 1500 dollars at 8% compounded yearly worth after 2 years?

Show answer and explanation

Answer: C. 1500(1.08)² = 1500 × 1.1664 = 1749.60. 1740 is simple interest, 1620 is one year, and 1524 uses 0.8% instead of 8%.

Question 3 of 3

Which account ends with the most money after 10 years, all starting with 1000 dollars at 6%?

Show answer and explanation

Answer: C. At the same rate, more compounding periods give more interest on interest: monthly (about 1819.40) beats quarterly (about 1814.02), yearly (about 1790.85) and simple interest (1600).

Compound Interest: frequently asked questions

What is the formula for compound interest?

The formula is A = P( 1 + rn )^nt. P is the principal, r is the annual rate as a decimal, n is the number of times interest is added per year and t is the number of years. A is the final balance; subtract P to get just the interest. For continuous compounding use A = Pe^rt.

How is compound interest different from simple interest?

Simple interest is figured only on the original principal, so you earn the same amount every year: I = Prt. Compound interest is figured on the current balance, which includes earlier interest, so each year’s interest is bigger than the last. Over long periods the gap becomes large, which is why compounding matters for savings and for debt.

What does compounded monthly mean?

It means interest is calculated and added to the balance 12 times a year. Each month the bank applies one twelfth of the annual rate. So 6% compounded monthly adds 0.5% per month, and over one year the balance is multiplied by 1.005¹² ≈ 1.0617, slightly more than the 6% you would get with yearly compounding.

Practice and tools

Compound Interest CalculatorFind the balance for any rate, time and compounding period Investment CalculatorProject savings with regular deposits Savings Goal CalculatorWork backward from a target balance

Sources

Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSA-SSE.B.3.c, CCSS HSF-LE.A.2

Cite this entry

Effortless Math Team. “Compound Interest.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/compound-interest/

Flashcard

TermCompound InterestA = P( 1 + rn )^nt

DefinitionInterest earned on the principal and on earlier interest. 1000 dollars at 10% compounded yearly: 1100, then 1210, then 1331.

Study the full deck

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Compound Interest

What is compound interest?

Compound Interest in a picture

Formal definition and notation

Key facts

Compound interest formula

Try it: one year at 5%

Compound Interest examples, step by step

Example 1: Yearly compounding

Example 2: Monthly compounding

Example 3: Compound versus simple

Common mistakes

Not to be confused with

Quick check: 3 questions

Compound Interest: frequently asked questions

What is the formula for compound interest?

How is compound interest different from simple interest?

What does compounded monthly mean?

Related terms

Practice and tools

Sources

Flashcard

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