Skill: Calculate compound interest over whole-number periods

Compound interest means that each period’s interest is added to the balance. In the next period, you earn interest on the new, larger balance.

Use the formula

$$A=P(1+r)^n$$

where:

• $P$ is the starting amount, or principal
• $r$ is the interest rate per period written as a decimal
• $n$ is the number of whole periods
• $A$ is the final balance

Example

You deposit 500$in an account that earns$4%$ interest per year. How much money will be in the account after 3 years? How much interest will you earn?

Step 1: Identify the values.

$$P=500,\qquad r=0.04,\qquad n=3$$

Convert $4\%$ to a decimal:

$$4\%=\frac{4}{100}=0.04$$

Step 2: Substitute into the formula.

$$A=500(1+0.04)^3$$ $$A=500(1.04)^3$$

Step 3: Calculate.

$$A=500(1.124864)=562.432$$

Round to the nearest cent:

$$A\approx \$562.43$$

So, the balance after 3 years is 562.43$.

Step 4: Find the interest earned.

Subtract the original deposit from the final balance:

$$\text{Interest}=A-P$$ $$\text{Interest}=\$562.43-\$500=\$62.43$$

The account earns 62.43$ in interest.

You can also see the growth period by period:

• After year 1: 500(1.04)=$520$
• After year 2: 520(1.04)=$540.80$
• After year 3: 540.80(1.04)=$562.43$

The amount grows by more than 20$ each year because the interest is added to the balance and then earns interest too.