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Compound Interest — Definition, Formula & Examples

Compound Interest

A method of computing interest in which interest is computed from the up-to-date balance. That is, interest is earned on the interest and not just on original balance.

Formula: A=P(1+rn)nt\displaystyle A = P\left(1 + \frac{r}{n}\right)^{nt}
AA = final amount
PP = principal, or original amount
rr = rate of interest per year
nn = number of periods per year
tt = time, in years
Example: $2000 is deposited at 12% per year, compounded quarterly. Find the balance after 7 years.
Solution: That means P=2000,P = 2000\text{,} r=0.12,r = 0.12\text{,} n=4,n = 4\text{,} and t=7,t = 7\text{,} soA=2000(1+0.124)4⋅7≈$4575.86A = 2000\left(1 + \frac{0.12}{4}\right)^{4\cdot 7} \approx \text{\textdollar}4575.86

See also

Simple interest, continuously compounded interest, doubling time

Key Formula

A=P(1+rn)ntA = P\left(1 + \frac{r}{n}\right)^{nt}
Where:
  • AA = The final amount (principal + all interest earned)
  • PP = The principal — the initial amount of money invested or borrowed
  • rr = The annual interest rate expressed as a decimal (e.g., 5% = 0.05)
  • nn = The number of times interest is compounded per year
  • tt = The number of years the money is invested or borrowed

Worked Example

Problem: You invest $1,000 in a savings account that earns 6% annual interest, compounded monthly. How much will you have after 5 years?
Step 1: Identify the values: P = 1000, r = 0.06, n = 12 (monthly), t = 5.
P=1000,r=0.06,n=12,t=5P = 1000,\quad r = 0.06,\quad n = 12,\quad t = 5
Step 2: Substitute into the compound interest formula.
A=1000(1+0.0612)12×5A = 1000\left(1 + \frac{0.06}{12}\right)^{12 \times 5}
Step 3: Simplify inside the parentheses. Dividing 0.06 by 12 gives 0.005.
A=1000(1.005)60A = 1000\left(1.005\right)^{60}
Step 4: Evaluate the exponent. Using a calculator, (1.005)^60 ≈ 1.34885.
A=1000×1.34885A = 1000 \times 1.34885
Step 5: Multiply to find the final amount.
A≈1348.85A \approx 1348.85
Answer: After 5 years, your investment grows to approximately $1,348.85. Of that, $348.85 is interest earned.

Another Example

This example uses quarterly compounding instead of monthly, applies the formula to a loan rather than an investment, and directly compares the result to simple interest.

Problem: You borrow $2,000 at 8% annual interest, compounded quarterly. How much do you owe after 3 years, and how much more is that compared to simple interest?
Step 1: Identify the values: P = 2000, r = 0.08, n = 4 (quarterly), t = 3.
P=2000,r=0.08,n=4,t=3P = 2000,\quad r = 0.08,\quad n = 4,\quad t = 3
Step 2: Substitute into the compound interest formula.
A=2000(1+0.084)4×3=2000(1.02)12A = 2000\left(1 + \frac{0.08}{4}\right)^{4 \times 3} = 2000\left(1.02\right)^{12}
Step 3: Evaluate: (1.02)^12 ≈ 1.26824.
A=2000×1.26824≈2536.48A = 2000 \times 1.26824 \approx 2536.48
Step 4: Now compute what simple interest would give: A = P(1 + rt).
Asimple=2000(1+0.08×3)=2000(1.24)=2480A_{\text{simple}} = 2000(1 + 0.08 \times 3) = 2000(1.24) = 2480
Step 5: Find the difference between compound and simple interest amounts.
2536.48−2480=56.482536.48 - 2480 = 56.48
Answer: With compound interest you owe $2,536.48, which is $56.48 more than the $2,480 you would owe under simple interest.

Frequently Asked Questions

What is the difference between compound interest and simple interest?
Simple interest is calculated only on the original principal, so the interest earned each period stays the same. Compound interest is calculated on the principal plus all previously earned interest, so the amount of interest grows each period. Over time, compound interest produces a larger total than simple interest at the same rate.
How does compounding frequency affect compound interest?
The more frequently interest is compounded, the more total interest you earn (or owe). For example, monthly compounding yields more than annual compounding at the same rate, because interest is added to the balance more often and begins earning its own interest sooner. In the extreme case, interest is compounded continuously using the formula A = Pe^(rt).
How do you find just the interest earned, not the total amount?
Subtract the original principal from the final amount. The compound interest earned is I = A − P, where A is the result of the compound interest formula. For instance, if you invest $1,000 and end with $1,348.85, the interest earned is $348.85.

Compound Interest vs. Simple Interest

Compound InterestSimple Interest
DefinitionInterest computed on the principal plus all accumulated interestInterest computed only on the original principal
FormulaA = P(1 + r/n)^(nt)A = P(1 + rt)
Growth patternExponential — growth accelerates over timeLinear — same amount of interest each period
When to useSavings accounts, mortgages, credit cards, most real-world financeShort-term loans, some bonds, introductory math problems
Result over long periodsSignificantly larger final amountSmaller final amount at the same rate and time

Why It Matters

Compound interest is the standard way banks, credit cards, and investment accounts calculate growth, so understanding it is essential for personal finance. In math courses, it appears in algebra, precalculus, and statistics whenever you study exponential growth or geometric sequences. Recognizing how compounding frequency and time dramatically affect outcomes helps you make smarter decisions about saving, investing, and borrowing.

Common Mistakes

Mistake: Forgetting to convert the percentage rate to a decimal before substituting into the formula.
Correction: Always divide the percentage by 100 first. For example, 6% must be entered as r = 0.06, not r = 6.
Mistake: Using the number of years for the exponent instead of n × t when interest compounds more than once per year.
Correction: The exponent must be the total number of compounding periods: n × t. If interest compounds monthly for 5 years, the exponent is 12 × 5 = 60, not 5.

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