This cheat sheet covers how money grows with compound interest and how values decrease through depreciation. Students need these tools for finance, savings, loans, car values, technology prices, and real-world exponential models. It helps organize the main formulas so students can choose the correct model quickly.
The focus is on recognizing growth, decay, rates, time, and compounding periods.
Key Facts
- Compound interest with periodic compounding is modeled by , where is principal, is the annual rate as a decimal, is compounds per year, and is time in years.
- Annual compound growth is modeled by when interest is compounded once per year.
- Depreciation is modeled by , where is the annual depreciation rate as a decimal.
- The growth factor for a rate is , and the decay factor for a rate is .
- Continuous compounding is modeled by , where is approximately .
- The interest earned is , where is the final amount and is the starting amount.
- To convert a percent rate to a decimal, divide by , so and .
- If a quantity is multiplied by the same factor each time period, the situation is exponential and can often be written as .
Vocabulary
- Principal
- The principal is the starting amount of money or the original value, represented by .
- Compound Interest
- Compound interest is interest calculated on both the original principal and the interest already earned.
- Depreciation
- Depreciation is a decrease in value over time, often modeled by multiplying by a decay factor such as .
- Interest Rate
- The interest rate is the percent increase or decrease per time period, written as a decimal in formulas.
- Compounding Period
- A compounding period is how often interest is added, such as monthly with or quarterly with .
- Growth Factor
- A growth factor is the multiplier used when a quantity increases by rate each period.
Common Mistakes to Avoid
- Using instead of for is wrong because formulas require the rate as a decimal, not a whole percent number.
- Using for depreciation is wrong because depreciation means the value decreases, so the multiplier should be .
- Forgetting the exponent in is wrong because the number of compounding periods depends on both frequency and time.
- Treating compound interest as simple interest is wrong because compound interest repeatedly multiplies the amount, while simple interest adds the same amount each period.
- Rounding too early is wrong because small rounding errors can grow after repeated compounding or depreciation.
Practice Questions
- 1 A savings account starts with dollars and earns annual interest compounded yearly. Find the value after years using .
- 2 A laptop costs dollars and depreciates by per year. Find its value after years using .
- 3 An account has dollars at annual interest compounded monthly. Use to find the balance after years.
- 4 Explain how you can tell whether a situation should use a growth factor or a decay factor without doing a calculation.
Understanding Compound Interest and Depreciation Reference
The important idea is that each new change is based on the current amount, not the original amount. A savings balance that earns interest grows by a slightly larger dollar amount each period because earlier interest remains in the account. This is why compound growth curves upward over time.
A fixed yearly increase behaves differently. Adding the same number of dollars every year is linear, while multiplying by the same factor every year is exponential. Students should read wording carefully for clues such as increases by a fixed percent, loses a fixed percent, or is compounded monthly.
The rate and the time unit must match. If a rate is given per year but changes happen monthly, the annual rate is split into twelve equal periodic rates. The number of growth periods is then twelve times the number of years.
A common error is to divide the rate by twelve but leave the exponent as the number of years. That mixes monthly information with yearly information.
More frequent compounding gives a larger final balance when the stated annual rate is positive. The difference may look small over one year, yet it can matter over many years or for large amounts of money.
Annual percentage rate, often called APR, describes the stated yearly rate before the effect of repeated compounding is included. The effective annual rate tells how much the balance truly grows in one year after all compounding periods. For example, a bank can state one APR while compound interest monthly.
The effective yearly increase is then a little greater than the APR. This distinction helps when comparing savings accounts, credit cards, and loans.
For borrowing, compounding works against the borrower because interest is charged on unpaid interest. Paying more than the required minimum can reduce the time that this process has to build.
Depreciation needs careful interpretation because it usually describes a percentage loss from the current value. A car that loses ten percent each year does not lose the same dollar amount every year. Its first loss is based on the purchase price.
Later losses are based on a lower value. This model can estimate resale value, but it does not guarantee a market price. Mileage, condition, repairs, demand, and new product releases can change actual values.
In school problems, identify the starting value, convert every percent to a decimal, and decide whether the requested result is final value or total change. Continuous compounding is a limiting model where growth is treated as happening at every instant. It appears in finance and in science models, though most everyday accounts use daily, monthly, or yearly compounding.