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?
Understanding Compound Interest Formulas & Examples
Compounding works in steps. At each step, a bank or lender finds the interest for that period by using the balance currently recorded. That new interest is placed into the account or added to the debt.
The next calculation starts from this updated balance. For monthly compounding, the yearly rate is split into twelve smaller rates.
A stated annual rate of six percent becomes one half of one percent per month. The monthly rate applies repeatedly, so it is not correct to simply multiply the starting amount by six percent for every year after the first.
Time has a powerful effect because the later interest payments are larger than the earlier ones. Consider two people who each save the same amount at the same rate. The person who starts earlier can finish with more money even if they stop adding money sooner.
This is why saving regularly as a teenager can matter. The same pattern makes unpaid debt dangerous. A credit card balance that remains unpaid can grow each month, while new purchases, late fees, and payments change the balance being used for the next interest charge.
APY is useful because advertised rates do not always tell the full story. Two accounts may list the same annual rate, yet one may calculate interest daily while the other calculates it yearly. APY converts each schedule into a one year result, making the comparison fairer.
Students should check whether a rate is an APY, an annual percentage rate, or a monthly rate. For savings, a higher APY usually means faster growth when account rules are similar.
For borrowing, a higher annual percentage rate usually means a higher cost. Fees, minimum balance rules, and introductory rates can still change which account is better.
Continuous compounding is mainly a mathematical model. It treats interest as being added in extremely tiny intervals rather than once per day or month. It is useful in some advanced finance and science problems, but most real bank accounts use a stated schedule such as daily or monthly.
The difference between daily and continuous compounding is usually small at ordinary rates. When solving problems, write down the starting balance, the rate in decimal form, the number of compounding periods, and the time unit before calculating. Keep years consistent with the rate.
Round only at the end, since early rounding can change the final cents. For loans with regular payments, the simple growth formula alone is not enough because each payment reduces the balance before later interest is charged.