Calculate compound interest over whole-number periods

Compound interest is modeled by $A=P(1+r)^n$ , where $P$ is the principal, $r$ is the interest rate per compounding period written as a decimal, $n$ is a whole number of periods, and $A$ is the resulting balance; the interest earned is $A-P$ . The learner understands that each period’s interest is added to the balance and itself earns interest, distinguishing exponential growth from simple interest, and can calculate or represent successive balances for familiar rates and periods; fractional periods, changing rates, continuous compounding, and solving with logarithms are beyond this scope.

Detailed Explanation: 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.

Learn by doing: Calculate compound interest over whole-number periods

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