<p><a href="https://www.prepswift.com/quizzes/quiz/prepswift-compound-interest" target="_blank">Compound Interest Exercise</a></p>
<p><strong>Note: </strong>about 2 minutes and 40 seconds into the video, Greg incorrectly says that the amount obtained using simple interest after years 300 and 400 are $\$1300$ and $\$1700$ respectively. Clearly, we're missing a zero - it should be $\$13000$ and $\$17000$ respectively.</p><p>If you're earning <span style="color:#27ae60;">Compound Interest</span> on some money you deposited in the bank, that means the interest you receive each year gets bigger...and bigger...and bigger...and bigger each year. For that reason, the algebraic equation that expresses compound interest is an exponential equation. </p>
<p><strong><span style="color:#8e44ad;">Intuitive Understanding</span></strong></p>
<p>Before we dive into the formula, let's figure out compound interest intuitively. If you're receiving $5\%$ interest each year and you deposit $\$100$, you would have $(100)(1.05) = \$105$ after the first year. To determine the amount of money after year $2$, we calculate $5\%$ of $105$ and add that on:</p>
<p style="text-align: center;">after year $2 = 105(1.05)=110.25$</p>
<p>We just repeat the process:</p>
<p style="text-align: center;">after year $3 = 110.25(1.05)=115.7625$</p>
<p>So, the number of completed years determines how many times we multiply by $1.05$. That's why the formula looks like this, where $n$ is the number of years:</p>
<p>$$total = 100(1.05)^n$$</p>
<p>The <strong><span style="color:#8e44ad;">general formula</span></strong> for compound annual interest is below, where $t$ is the total, $p$ is the principal, $r$ is the rate as a percent, and $n$ is the number of years.</p>
<p>$$t = p(1 + \frac{r}{100})^n$$</p>
<p><strong><span style="color:#8e44ad;">Example</span></strong></p>
<p>If we deposit $\$500$ in an account earning $7\%$ compound annual interest, we would have this much money have $20$ years:</p>
<p>$$total = 500(1.07)^{20}= 1934.84$$</p>
<p> </p>