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Compound Interest Formula: Complete Guide & Calculator

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Master compound interest formulas with step-by-step examples, calculator, and practice problems. Essential for SAT, AP, IB exams. Free interactive tools included.

Compound Interest Formula: Complete Guide with Examples

Master compound interest calculations for exams, investments, and financial planning

A = P(1+r/n)nt

Standard Formula

5 Types

Compounding Frequencies

A = Pert

Continuous Compounding

What is Compound Interest?

Compound interest is the interest calculated on both the initial principal and the accumulated interest from previous periods. Unlike simple interest, which only calculates interest on the principal amount, compound interest creates exponential growth by earning "interest on interest." This powerful financial concept is fundamental for students studying mathematics, economics, and business, and appears frequently on standardized tests including SAT, AP, IB, GCSE, and A-Level examinations.

Key Difference: If you invest $1,000 at 5% for 20 years, simple interest earns $1,000 total, while compound interest earns $1,653.30—a difference of $653.30 purely from compounding.

Core Compound Interest Formulas

Standard Compound Interest Formula

General Compound Interest Formula
\[ A = P\left(1 + \frac{r}{n}\right)^{nt} \]

A = Final amount (future value including principal and interest)

P = Principal amount (initial investment or loan)

r = Annual interest rate (in decimal form: 5% = 0.05)

n = Number of times interest is compounded per year

t = Time period in years

The compound interest (CI) alone can be calculated by subtracting the principal from the final amount:

Compound Interest Only
\[ CI = A - P = P\left(1 + \frac{r}{n}\right)^{nt} - P \]

Annual Compounding Formula

When interest is compounded once per year (n = 1), the formula simplifies to:

\[ A = P(1 + r)^t \]

Use this formula when the problem states "compounded annually" or doesn't specify a compounding frequency.

Continuous Compounding Formula

Continuous compounding represents the theoretical maximum growth rate when interest is compounded infinitely often. This formula uses Euler's number (e ≈ 2.71828):

Continuous Compounding
\[ A = Pe^{rt} \]

e = Euler's number (approximately 2.71828)

r = Annual interest rate (decimal)

t = Time in years

Use this formula when the problem explicitly states "compounded continuously."

Compounding Frequency Variations

The value of "n" (compounding frequency) dramatically affects the final amount. Here are the most common compounding periods:

Compounding TypeValue of nFormulaCommon Use
Annuallyn = 1\( A = P(1 + r)^t \)Bonds, some savings accounts
Semi-annuallyn = 2\( A = P(1 + r/2)^{2t} \)Corporate bonds, some loans
Quarterlyn = 4\( A = P(1 + r/4)^{4t} \)Most savings accounts, CDs
Monthlyn = 12\( A = P(1 + r/12)^{12t} \)Mortgages, credit cards
Weeklyn = 52\( A = P(1 + r/52)^{52t} \)Some short-term investments
Dailyn = 365\( A = P(1 + r/365)^{365t} \)High-yield savings, some banks
Continuousn → ∞\( A = Pe^{rt} \)Theoretical maximum, some investments

Step-by-Step Methodology

How to Solve Compound Interest Problems

Step 1: Identify the Variables

Read the problem carefully and extract: Principal (P), Interest rate (r), Time period (t), and Compounding frequency (n).

Step 2: Convert the Interest Rate

Convert the percentage rate to decimal form by dividing by 100. For example, 8% becomes 0.08.

Step 3: Select the Appropriate Formula

Choose the formula based on compounding frequency: standard formula for periodic compounding, or continuous formula if specified.

Step 4: Substitute and Calculate

Plug the values into the formula and solve. Use a scientific calculator for exponential calculations.

Step 5: Interpret the Result

Remember that A represents the total amount. Subtract P to find only the interest earned (CI).

Worked Example 1: Quarterly Compounding

Problem: Calculate the compound interest on $10,000 invested at 8% per annum for 3 years, compounded quarterly.

Solution:

Given: P = $10,000, r = 8% = 0.08, t = 3 years, n = 4 (quarterly)

Using the formula:

\[ A = 10000\left(1 + \frac{0.08}{4}\right)^{4 \times 3} \]

\[ A = 10000(1 + 0.02)^{12} \]

\[ A = 10000(1.02)^{12} \]

\[ A = 10000 \times 1.26824 \]

\[ A = \$12,682.40 \]

Compound Interest: CI = A - P = $12,682.40 - $10,000 = $2,682.40

Worked Example 2: Continuous Compounding

Problem: If $5,000 is invested at 6% annual interest compounded continuously for 10 years, what is the final amount?

Solution:

Given: P = $5,000, r = 6% = 0.06, t = 10 years

Using the continuous compounding formula:

\[ A = 5000 \times e^{0.06 \times 10} \]

\[ A = 5000 \times e^{0.6} \]

\[ A = 5000 \times 1.8221 \]

\[ A = \$9,110.50 \]

Compound Interest: CI = $9,110.50 - $5,000 = $4,110.50

Compound Interest vs Simple Interest

FeatureSimple InterestCompound Interest
Calculation BasisOnly on principalOn principal + accumulated interest
Formula\( SI = \frac{P \times r \times t}{100} \)\( A = P(1 + r/n)^{nt} \)
Growth PatternLinear (constant)Exponential (accelerating)
Interest AmountSame each periodIncreases each period
Best for BorrowersYes (lower total interest)No (higher total interest)
Best for SaversNo (lower returns)Yes (higher returns)
Common ApplicationsAuto loans, short-term loansSavings accounts, investments, credit cards

Interactive Compound Interest Calculator

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Calculation Results

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Frequently Asked Questions (FAQs)

What is compound interest in simple terms? +
Compound interest is interest calculated on both your initial investment (principal) and all the interest you've already earned. It's often called "interest on interest." For example, if you invest $100 at 10% annual interest, you'll have $110 after year one. In year two, you earn interest on $110 (not just $100), giving you $121. This creates exponential growth over time, making compound interest a powerful tool for long-term savings and investments.
How do I calculate compound interest manually? +
To calculate compound interest manually: (1) Convert the interest rate from percentage to decimal by dividing by 100, (2) Identify the compounding frequency (n) - annually=1, quarterly=4, monthly=12, (3) Use the formula A = P(1 + r/n)^(nt), (4) Calculate the value inside parentheses first (1 + r/n), (5) Raise it to the power of (n×t), (6) Multiply by the principal P, (7) Subtract P from the result to get interest alone. For continuous compounding, use A = Pe^(rt) where e ≈ 2.71828.
What's the difference between annual and continuous compounding? +
Annual compounding calculates interest once per year using the formula A = P(1 + r)^t, while continuous compounding calculates interest infinitely often using A = Pe^(rt). Continuous compounding represents the theoretical maximum growth rate. For practical purposes, the difference is small but increases with higher rates and longer periods. For example, $1,000 at 10% for 5 years gives $1,610.51 with annual compounding versus $1,648.72 with continuous compounding—a difference of $38.21.
Does more frequent compounding always mean more interest? +
Yes, more frequent compounding always results in more interest earned, though the difference diminishes as frequency increases. The progression from annual to continuous compounding shows increasing returns, but the jump from monthly to daily compounding produces much smaller gains than from annual to monthly. For savers and investors, this means seeking accounts with more frequent compounding. For borrowers, less frequent compounding is preferable as it results in lower total interest paid on loans.
How is compound interest different from simple interest? +
Simple interest is calculated only on the principal amount using the formula SI = (P × r × t)/100, resulting in linear growth where you earn the same amount each period. Compound interest is calculated on the principal plus accumulated interest, creating exponential growth. Over 20 years, $1,000 at 5% simple interest earns $1,000 total, while compound interest (annual) earns $1,653.30—a 65% difference. Simple interest is common in car loans and short-term loans, while compound interest is standard for savings accounts, investments, and credit cards.
Why is compound interest important for students to learn? +
Compound interest is crucial for students because it appears on major standardized tests (SAT, AP, IB, GCSE, A-Levels) and forms the foundation of financial literacy. Understanding compound interest helps students make informed decisions about student loans, credit cards, savings accounts, and future investments. It demonstrates the power of starting to save early—investing $1,000 at age 20 versus age 30 can result in tens of thousands of dollars difference by retirement. The mathematical concepts also reinforce exponential functions and logarithms taught in algebra and precalculus courses.
What is the Rule of 72 and how does it relate to compound interest? +
The Rule of 72 is a quick mental math formula to estimate how long it takes for an investment to double with compound interest. Simply divide 72 by the annual interest rate (as a whole number, not decimal). For example, at 6% interest, your money doubles in approximately 72 ÷ 6 = 12 years. At 9%, it doubles in 72 ÷ 9 = 8 years. This rule works best for interest rates between 6% and 10% and assumes annual compounding. It's a valuable shortcut for quick financial planning and appears on many standardized tests.
Can compound interest work against me? +
Yes, compound interest works against you when you're a borrower, especially with credit cards. Credit card companies typically compound interest daily, causing debt to grow exponentially if not paid off quickly. A $5,000 credit card balance at 18% APR compounded daily grows to $6,006.88 in just one year if no payments are made. This is why financial advisors stress paying off high-interest debt before investing—the interest you avoid paying on debt often exceeds the returns you could earn from investments. Understanding this dual nature of compound interest is critical for financial health.

Key Formulas Summary

Formula TypeMathematical ExpressionWhen to Use
General Compound Interest\( A = P(1 + r/n)^{nt} \)Any periodic compounding frequency
Annual Compounding\( A = P(1 + r)^t \)When n = 1 (once per year)
Continuous Compounding\( A = Pe^{rt} \)When compounding is continuous
Interest Only\( CI = A - P \)To find interest earned/paid
Present Value\( P = \frac{A}{(1 + r/n)^{nt}} \)To find initial investment needed

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Disclaimer: This educational resource is designed to help students understand compound interest formulas for academic purposes, including preparation for SAT, AP, IB, GCSE, and A-Level examinations. While the mathematical formulas and examples are accurate, this content should not be considered professional financial advice. For specific investment or loan decisions, please consult with a qualified financial advisor. All examples use simplified scenarios for educational clarity.

Last Updated: January 25, 2026 | © RevisionTown

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