What is Compound Interest?
Compound interest is interest calculated on both the original principal and the interest already earned, so a balance grows faster the longer it compounds. It's the engine behind long-term savings growth and credit card debt alike.
Compound interest is interest earned on both the principal and previously accumulated interest, calculated as A = P(1 + r/n)^(nt), where the balance grows exponentially over time.
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Step-by-step worked examples
You deposit $2,000 at 4% annual interest, compounded annually (n=1), for 5 years. Find the final amount.
A = P(1+r/n)^(nt) A = 2000×(1+0.04/1)^(1×5) A = 2000×(1.04)^5 (1.04)^5 ≈ 1.2167 A ≈ $2,433.31
$1,000 is invested at 6% annual interest compounded monthly (n=12) for 3 years. Find the final amount.
A = P(1+r/n)^(nt) A = 1000×(1+0.06/12)^(12×3) A = 1000×(1.005)^36 (1.005)^36 ≈ 1.1967 A ≈ $1,196.65
Compare $10,000 at 5% compounded annually (n=1) vs. compounded quarterly (n=4) for 10 years. Which grows more, and what's the difference?
Annual: A = 10000×(1.05)^10 ≈ 10000×1.62889 = $16,288.95 Quarterly: A = 10000×(1+0.05/4)^(4×10) = 10000×(1.0125)^40 ≈ 10000×1.64362 = $16,436.19 Quarterly compounding grows more by about $147.25
Flashcards
Quick quiz
Q1.$1,000 at 10% compounded annually for 2 years. What is A?
Q2.In A=P(1+r/n)^nt, what does 'n' represent?
Q3.Which grows a balance faster at the same annual rate?
Q4.What makes compound interest grow exponentially rather than linearly?
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Common mistakes
Using the annual rate directly when compounding monthly. — Correct: Divide the annual rate by n (e.g. r/12 for monthly) before applying the formula.
Confusing simple interest (P×r×t) with compound interest. — Correct: Simple interest grows linearly; compound interest grows exponentially because interest earns interest.
Forgetting to multiply n×t in the exponent. — Correct: The exponent is the TOTAL number of compounding periods, n times t, not just t.
Assuming more compounding periods always doubles your money faster. — Correct: More frequent compounding helps, but the effect shrinks as n grows very large (it approaches continuous compounding).
FAQ
What is compound interest?
Interest calculated on the principal plus all interest accumulated so far, leading to exponential balance growth.
What is the compound interest formula?
A = P(1 + r/n)^(nt), where P is the principal, r the annual rate, n the compounding frequency, and t the time in years.
How do you calculate compound interest?
Plug principal, annual rate, compounding frequency and time into A = P(1+r/n)^(nt), then subtract P to find the interest earned.
What are examples of compound interest?
Savings accounts, certificates of deposit, and credit card balances all typically use compound interest.