Compound Interest

Functions
principle

Also known as: compounding, compound growth

Grade 9-12

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Interest calculated on both the initial principal and the accumulated interest from previous periods. Compound interest governs savings accounts, loans, mortgages, and investments.

This concept is covered in depth in our exponential modeling guide, with worked examples, practice problems, and common mistakes.

Definition

Interest calculated on both the initial principal and the accumulated interest from previous periods. The formula A = P\left(1 + \frac{r}{n}\right)^{nt} gives the amount after t years, and A = Pe^{rt} gives the continuously compounded amount.

💡 Intuition

Simple interest pays you only on your original deposit. Compound interest pays you interest on your interest—your money earns money on the money it already earned. The more frequently you compound, the more you earn, because each tiny interest payment starts earning its own interest sooner. The ultimate limit of compounding more and more frequently is continuous compounding: A = Pe^{rt}.

🎯 Core Idea

Compound interest is exponential growth applied to money. The key insight is that as compounding frequency n increases, the amount approaches the continuous limit Pe^{rt}—this is exactly how Euler's number e was discovered.

Example

Invest \$1000 at 6% for 10 years:
- Annual compounding: A = 1000(1.06)^{10} = \1790.85$
- Monthly: A = 1000\left(1 + \frac{0.06}{12}\right)^{120} = \1819.40$
- Continuous: A = 1000e^{0.6} = \1822.12$

Formula

A = P\left(1 + \frac{r}{n}\right)^{nt}
A = Pe^{rt} \quad \text{(continuous compounding)}
where P = principal, r = annual rate, n = compounding periods per year, t = years.

Notation

P = principal (initial amount), r = annual interest rate (as a decimal), n = number of compounding periods per year, t = time in years, A = final amount.

🌟 Why It Matters

Compound interest governs savings accounts, loans, mortgages, and investments. Understanding it is critical for personal finance. Einstein (apocryphally) called it 'the eighth wonder of the world.' The continuous form Pe^{rt} connects finance to calculus and exponential growth models.

💭 Hint When Stuck

Label every variable in the formula before plugging in: P = initial amount, r = rate as decimal, n = times per year, t = years. Then substitute carefully.

Formal View

A = P\!\left(1 + \frac{r}{n}\right)^{\!nt}; continuous limit: A = Pe^{rt} = \lim_{n \to \infty} P\!\left(1 + \frac{r}{n}\right)^{\!nt}

🚧 Common Stuck Point

The rate r must be a decimal, not a percentage: 6% means r = 0.06, not r = 6. Also, r and n must use the same time unit—if r is annual, n is compoundings per year.

⚠️ Common Mistakes

  • Using r = 6 instead of r = 0.06 for a 6% rate—always convert the percentage to a decimal.
  • Confusing the number of compounding periods: monthly means n = 12, not n = 1/12. Quarterly means n = 4.
  • Thinking continuous compounding gives dramatically more than daily compounding—the difference is usually small. The real power of compound interest comes from TIME, not compounding frequency.

Frequently Asked Questions

What is Compound Interest in Math?

Interest calculated on both the initial principal and the accumulated interest from previous periods. The formula A = P\left(1 + \frac{r}{n}\right)^{nt} gives the amount after t years, and A = Pe^{rt} gives the continuously compounded amount.

Why is Compound Interest important?

Compound interest governs savings accounts, loans, mortgages, and investments. Understanding it is critical for personal finance. Einstein (apocryphally) called it 'the eighth wonder of the world.' The continuous form Pe^{rt} connects finance to calculus and exponential growth models.

What do students usually get wrong about Compound Interest?

The rate r must be a decimal, not a percentage: 6% means r = 0.06, not r = 6. Also, r and n must use the same time unit—if r is annual, n is compoundings per year.

What should I learn before Compound Interest?

Before studying Compound Interest, you should understand: exponential function, e.

Prerequisites

exponential functione

How Compound Interest Connects to Other Ideas

To understand compound interest, you should first be comfortable with exponential function and e. Once you have a solid grasp of compound interest, you can move on to annuities and present future value.

Want the Full Guide?

This concept is explained step by step in our complete guide:

Exponents and Logarithms: Rules, Proofs, and Applications →
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