Compound Interest

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.

Compound interest formula: A = P(1 + r/n)^nt, where A=final amount, P=principal, r=annual interest rate, n=periods per year,...
Example: $2000 at 12%/yr compounded quarterly, 7 years. A = 2000(1 + 0.12/4)^(4·7) = $4575.86

Key Formula

A = P\left(1 + \frac{r}{n}\right)^{nt}

Where:

$A$ = The final amount (principal + all interest earned)
$P$ = The principal — the initial amount of money invested or borrowed
$r$ = The annual interest rate expressed as a decimal (e.g., 5% = 0.05)
$n$ = The number of times interest is compounded per year
$t$ = 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,\quad r = 0.06,\quad n = 12,\quad t = 5

Step 2: Substitute into the compound interest formula.

A = 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\left(1.005\right)^{60}

Step 4: Evaluate the exponent. Using a calculator, (1.005)^60 ≈ 1.34885.

A = 1000 \times 1.34885

Step 5: Multiply to find the final amount.

A \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,\quad r = 0.08,\quad n = 4,\quad t = 3

Step 2: Substitute into the compound interest formula.

A = 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 \times 1.26824 \approx 2536.48

Step 4: Now compute what simple interest would give: A = P(1 + rt).

A_{\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 = 56.48

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 Interest	Simple Interest
Definition	Interest computed on the principal plus all accumulated interest	Interest computed only on the original principal
Formula	A = P(1 + r/n)^(nt)	A = P(1 + rt)
Growth pattern	Exponential — growth accelerates over time	Linear — same amount of interest each period
When to use	Savings accounts, mortgages, credit cards, most real-world finance	Short-term loans, some bonds, introductory math problems
Result over long periods	Significantly larger final amount	Smaller 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.

Related Terms

Simple Interest — Interest on principal only; the main comparison term
Continuously Compounded Interest — Limiting case where compounding frequency is infinite
Doubling Time — Time for an investment to double under compounding
Interest — The general concept of a fee paid for borrowing money
Exponential Growth — The growth pattern that compound interest follows
Principal — The initial amount in the compound interest formula
Annual Percentage Rate (APR) — Standardized rate used to compare compound interest products