Simple and compound interest are two ways money can grow when saved or cost more when borrowed. This cheat sheet helps students compare how each method works, when each formula is used, and why compounding can make a big difference over time. It is useful for understanding savings accounts, loans, credit cards, investments, and financial choices.
Key Facts
- Simple interest is calculated with I = P × r × t, where P is principal, r is annual interest rate as a decimal, and t is time in years.
- The total amount with simple interest is A = P + I or A = P(1 + rt).
- Compound interest is calculated with A = P(1 + r/n)^(nt), where n is the number of compounding periods per year.
- For annual compounding, the compound interest formula becomes A = P(1 + r)^t.
- Interest earned from compounding increases over time because interest is added to both the original principal and earlier interest.
- A higher compounding frequency, such as monthly instead of yearly, usually gives a slightly larger final amount when the rate and time are the same.
- The Rule of 72 estimates doubling time with years to double ≈ 72 ÷ annual interest rate percent.
- When comparing financial options, use the same principal, rate format, time unit, and compounding frequency before deciding which option costs or earns more.
Vocabulary
- Principal
- The original amount of money saved, invested, or borrowed before interest is added.
- Interest
- The extra money earned on savings or paid as the cost of borrowing money.
- Simple Interest
- Interest calculated only on the original principal for the entire time period.
- Compound Interest
- Interest calculated on the principal plus any interest that has already been added.
- Compounding Frequency
- How often interest is added to the account balance, such as yearly, quarterly, monthly, or daily.
- Annual Percentage Rate
- The yearly interest rate written as a percent, usually used to compare loans or savings products.
Common Mistakes to Avoid
- Using the percent instead of the decimal rate, such as using 6 instead of 0.06, gives an answer that is 100 times too large.
- Forgetting to match time units with the rate is wrong because an annual rate must use time measured in years unless it is converted.
- Using the simple interest formula for a compound interest problem misses interest earned on earlier interest and usually underestimates growth.
- Ignoring compounding frequency is wrong because monthly, quarterly, and yearly compounding can produce different final balances.
- Comparing loans or accounts only by the interest rate can be misleading because fees, time length, and compounding rules also affect the total cost or earnings.
Practice Questions
- 1 A student deposits $500 at 4% simple interest for 3 years. How much interest is earned, and what is the final amount?
- 2 A savings account has $800 at 5% interest compounded annually for 2 years. What is the final amount?
- 3 Which earns more after 4 years: 1,000 at 6% compounded annually? Find both amounts.
- 4 Explain why compound interest is helpful for long-term saving but can be risky for unpaid credit card debt.
Understanding Simple vs Compound Interest Compared
A useful way to see the difference is to track the balance year by year. Suppose 500 dollars earns six percent for five years. With simple interest, the interest amount stays at 30 dollars each year because it is based only on the original 500 dollars.
The total interest after five years is 150 dollars. With yearly compounding, the first year still adds 30 dollars. In the second year, the rate applies to 530 dollars, not 500 dollars.
The increase may seem small at first. Over many years, each new interest amount becomes larger than the one before it.
Compounding frequency changes how often a balance is updated. A stated annual rate does not always show the full story. A bank may list an annual percentage rate, often called APR, which is the named yearly rate.
It may list annual percentage yield, often called APY, which includes the effect of compounding during the year. For a savings account, APY is usually more useful when comparing the growth of deposits.
Monthly or daily compounding gives money more chances to earn interest within one year. The difference is often small for a short period, but it becomes more noticeable with a large balance or a long time period.
Borrowing creates the reverse effect. Interest is a cost, so faster compounding can make a debt grow faster. Credit cards commonly calculate interest from the unpaid balance each day.
If a cardholder pays the full statement balance by the due date, a grace period may prevent interest on many purchases. If part of the balance remains unpaid, interest can begin building on that remaining amount. A minimum payment may keep an account current, but it can leave most of the principal unpaid.
Then future interest is charged on a balance that falls very slowly. Installment loans work differently because scheduled payments gradually reduce the principal. Paying extra toward principal early can reduce later interest charges.
When solving problems, build a timeline before using numbers. Mark the starting balance, the rate period, the number of updates, and any deposits or payments. Convert a percent to a decimal correctly.
Six percent becomes zero point zero six. Match the time unit to the rate. A yearly rate needs years unless the calculation has been adjusted for months or days.
Do not round each step heavily because small rounding errors can grow. The Rule of 72 gives a quick estimate of doubling time by dividing 72 by the interest rate stated as a percent.
It works best for moderate rates and long-term compound growth. It is an estimate, not a replacement for a full calculation.