Compound interest describes growth that builds on itself over time. Instead of earning interest only on the starting amount, you earn interest on the principal plus the interest already added. This makes savings, investments, and debts grow faster than simple interest.
Understanding compound interest helps you compare financial choices and recognize the long-term effect of time.
Key Facts
- Compound interest formula: A = P(1 + r/n)^(nt)
- A is the final amount, P is the principal, r is the annual interest rate as a decimal, n is the number of compounding periods per year, and t is time in years.
- Interest earned is I = A - P.
- More frequent compounding gives a larger final amount when P, r, and t are the same.
- Continuous compounding formula: A = Pe^(rt)
- Rule of 72 estimate: doubling time in years ≈ 72 / annual percent rate
Vocabulary
- Principal
- The principal is the original amount of money invested, saved, or borrowed.
- Interest
- Interest is the extra money earned on an investment or charged on a loan.
- Compound interest
- Compound interest is interest calculated on both the original principal and previously earned interest.
- Compounding frequency
- Compounding frequency is how many times per year interest is added to the account balance.
- Continuous compounding
- Continuous compounding is the limiting case where interest is added constantly, modeled by A = Pe^(rt).
Common Mistakes to Avoid
- Using the percent rate directly in the formula, such as 6 instead of 0.06, is wrong because r must be written as a decimal.
- Forgetting to multiply time by the compounding frequency in the exponent is wrong because the exponent nt counts the total number of compounding periods.
- Confusing final amount with interest earned is wrong because A includes the original principal, while the interest earned is A - P.
- Assuming monthly compounding doubles the rate is wrong because compounding frequency changes how often interest is added, not the stated annual rate itself.
Practice Questions
- 1 A student deposits $800 at 5% annual interest compounded quarterly for 6 years. Find the final amount using A = P(1 + r/n)^(nt).
- 2 Compare $1,500 invested at 4.8% annual interest for 10 years with annual compounding and with continuous compounding. Find both final amounts.
- 3 Two accounts have the same principal and annual interest rate, but one compounds yearly and one compounds daily. Explain which account grows more and why the difference becomes more noticeable over long times.