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Compound Interest Formulas & Examples Cheat Sheet
A printable reference covering compound interest formulas, compounding frequency, APY, continuous compounding, and growth examples for grades 8-12.
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Compound interest is the way money grows when interest is added to the balance and then earns more interest over time. This cheat sheet helps students compare savings accounts, loans, investments, and credit card balances using the same core ideas. It is useful because small changes in rate, time, or compounding frequency can create large differences in final value. Students in grades 8-12 can use these formulas to solve real financial problems and make better money decisions. The main compound interest formula is A = P(1 + r/n)^(nt), where P is the starting amount, r is the annual interest rate as a decimal, n is the number of compounding periods per year, and t is time in years. When interest compounds once per year, the formula becomes A = P(1 + r)^t. Continuous compounding uses A = Pe^(rt), which models growth when interest is added constantly. Effective annual yield, also called APY, helps compare accounts with different compounding schedules.
Key Facts
- The compound interest formula is A = P(1 + r/n)^(nt), where A is the final amount, P is the principal, r is the annual rate as a decimal, n is compounds per year, and t is years.
- For annual compounding, use A = P(1 + r)^t because n = 1.
- The interest earned is I = A - P, which subtracts the original principal from the final amount.
- Convert a percent rate to a decimal by dividing by 100, so 6% becomes r = 0.06.
- More frequent compounding usually produces a larger final amount when the principal, rate, and time stay the same.
- The effective annual yield is APY = (1 + r/n)^n - 1, which gives the true one-year growth rate with compounding.
- For continuous compounding, use A = Pe^(rt), where e is approximately 2.718.
- To solve for time with compound interest, rearrangement often requires logarithms, such as t = ln(A/P) / (n ln(1 + r/n)).
Vocabulary
- Principal
- The principal is the starting amount of money invested, saved, borrowed, or owed.
- Interest
- Interest is the extra money earned on savings or charged on borrowing.
- Compound interest
- Compound interest is interest calculated on both the original principal and the interest already added.
- Compounding frequency
- Compounding frequency is how often interest is added to the account balance during one year.
- Annual percentage rate
- Annual percentage rate, or APR, is the stated yearly interest rate before adjusting for compounding.
- Annual percentage yield
- Annual percentage yield, or APY, is the actual yearly growth rate after compounding is included.
Common Mistakes to Avoid
- Using the percent instead of the decimal rate is wrong because the formula needs r as a decimal. For example, use 0.05 for 5%, not 5.
- Forgetting to multiply n by t in the exponent is wrong because the exponent must count the total number of compounding periods. Monthly compounding for 3 years has nt = 12 x 3 = 36 periods.
- Confusing APR and APY is wrong because APR is the stated annual rate, while APY includes the effect of compounding. Two accounts with the same APR can have different APYs.
- Subtracting interest each year instead of adding it to the balance is wrong for savings and investments because compound interest grows from the new balance. The next period's interest is based on the updated amount.
- Rounding too early is wrong because small rounding errors can grow over many periods. Keep several decimal places during calculations and round only the final answer.
Practice Questions
- 1 A student deposits 500 dollars in an account earning 4% annual interest compounded yearly for 6 years. What is the final amount?
- 2 An account starts with 1,200 dollars and earns 5% annual interest compounded monthly for 3 years. Use A = P(1 + r/n)^(nt) to find the balance.
- 3 Find the APY for an account with an APR of 6% compounded quarterly. Use APY = (1 + r/n)^n - 1.
- 4 Two savings accounts both advertise 5% APR, but one compounds annually and the other compounds monthly. Which account will have the larger balance after one year, and why?