Example Card: The Power of Compounding

Let's see how changing the compounding frequency, a key idea of compound interest, can supercharge an investment's growth.

Example Problem $6,000 is invested at $5\%$ compounded quarterly. Find the value of the investment after 8 years.

Step by Step 1. We will use the compound interest formula: $$A = P\left(1 + \frac{r}{n}\right)^{nt}$$ 2. The principal $P$ is $6,000$. The rate $r$ is $5\%$ or $0.05$. The time $t$ is $8$ years. Since interest is compounded quarterly, $n=4$. 3. Substitute the values into the formula: $$ A = 6000\left(1 + \frac{0.05}{4}\right)^{4(8)} $$ 4. Now, use the order of operations to simplify. First, handle the terms inside the parentheses and the exponent. $$ A = 6000(1.0125)^{32} $$ 5. Next, simplify the power. It's best to use a calculator for this and not round the intermediate result. $$ A \approx 6000(1.48813082) $$ 6. Finally, multiply to find the total amount and round to the nearest cent. $$ A \approx 8928.78 $$ 7. The value of the investment will be $8,928.78$ dollars.