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<h4> Activity </h4>
<p>A family has $1,000 to invest and is considering two options: investing in government bonds that offer 2% simple
interest, or investing in a savings account at a bank, which charges a $20 fee to open an account and pays 2% compound
interest. Both options pay interest annually.</p>
<p>Here are two tables showing what they would earn in the first couple of years if they do not invest additional
amounts or withdraw any money.</p>
<div class="os-raise-d-flex os-raise-justify-content-between">
<table class="os-raise-skinnytable">
<caption>Bonds</caption>
<thead>
<tr>
<th scope="col">
years of investment
</th>
<th scope="col">
amount in dollars
</th>
</tr>
</thead>
<tbody>
<tr>
<td>
0
</td>
<td>
$1,000
</td>
</tr>
<tr>
<td>
1
</td>
<td>
$1,020
</td>
</tr>
<tr>
<td>
2
</td>
<td>
$1,040
</td>
</tr>
</tbody>
</table>
<br>
<table class="os-raise-skinnytable">
<caption>Savings Account</caption>
<thead>
<tr>
<th scope="col">
years of investment
<th scope="col">
amount in dollars
</th>
</th>
</tr>
</thead>
<tbody>
<tr>
<td>
0
</td>
<td>
$980
</td>
</tr>
<tr>
<td>
1
</td>
<td>
$999.60
</td>
</tr>
<tr>
<td>
2
</td>
<td>
$1,019.59
</td>
</tr>
</tbody>
</table>
</div>
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="617fa5af-b3e0-4803-900b-878a8384cf56" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent">
<li>Bonds: How does the investment grow with simple interest? </li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p> The investment grows linearly. It increases by $20 every year. <br>
</p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="20d8d821-ea4b-4915-ba05-f3d92d62fd62" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="2">
<li>Savings account: How are the amounts $999.60 and $1,019.59 calculated? </li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>$999.60 is 1.02 times $980, and $1,019.59 is 1.02 times $999.60. <br>
</p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="70433847-7edb-4227-b695-376148df0393" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="3">
<li> Write an equation to represent the relationship between the amount of money and the number of years of
investment for each option. </li>
</ol>
<ol class="os-raise-noindent" start="1" type="a">
<li> Write an equation for the data in the Bonds table. Use the “^” symbol to enter an exponent, if
needed. </li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>\(
b=1,000+20x\) where \(x\) is the years since the investments were made and \(b\) is the value of the
bonds.<br>
</p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="cb0cdf2a-9d54-411a-bd85-742a3b28cad2" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="2" type="a">
<li> Write an equation for the data in the Savings Account table. Use the “^” symbol to enter an
exponent, if needed. </li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>\(s=980 \cdot (1.02)^x\) where \(s\) represents the value of the savings account, in dollars, where and
\(x\) is the years since the investments were made<br>
</p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="4639fd85-403a-4bfa-aebe-3d51e560508d" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="4">
<li>Which investment option should the family choose? Use your equations or calculations to support your answer.
</li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<li>The family should choose bonds if they are investing for less than 12 years. After 12 years, the savings account
contains $1,242.88 because this is the value from \(980 \cdot (1.02)^{12} \). And, the bonds are worth $1,240
after 12 years because \(1,000 + 20(12)\) is the value.</li>
<li>The family should choose the savings account if they are saving for something long term like college or
retirement. This is because over time, the savings account will contain more money than the bonds have earned. For
instance, after 20 years, the savings account holds $1,456.23 from \([980 \cdot (1.02)^{20}]\) while the bonds
only yield $1,400 from \([ 1,000 + 20(20)]\).</li>
</div>
</div>
<!--Interaction End -->
<br>
<ol class="os-raise-noindent" start="5">
<li> Use the Desmos graphing tool to graph the two investment options and show how the money grows in each. </li>
</ol>
<div class="os-raise-ib-desmos-gc" data-schema-version="1.0"></div>
<br>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal1" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
</div>
<div class="os-raise-ib-cta-prompt">
<p>Then select the <strong>solution</strong> button to compare your work. </p>
</div>
</div>
<br>
<br>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal1">
<p>Compare your answer:</p>
<p><img alt=" GRAPH OF TWO FUNCTIONS, REPRESENTING TWO INVESTMENT OPTIONS. FUNCTION B IS LINEAR AND REPRESENTS THE BOND INVESTMENT. FUNCTION S IS EXPONENTIAL AND REPRESENTS THE SAVINGS INVESTMENT. THE X-AXIS IS LABELED YEARS OF INVESTMENT AND GOES FROM 0 TO 18. THE Y-AXIS IS LABELED AMOUNT IN DOLLARS AND GOES FROM 0 TO THIRTEEN HUNDRED. STARTING AT X EQUALS 12, THE EXPONENTIAL FUNCTION, S, SURPASSES THE LINEAR FUNCTION, B.
" height="431" src="https://k12.openstax.org/contents/raise/resources/37032751e6b08aa8171cbaf4854a58df5e28b164" width="300"></p>
</div>
<h4>Video: Using Tables to Compare Linear and Exponential Functions</h4>
<p>Watch the following video to learn more about comparing linear and exponential functions:</p>
<div class="os-raise-d-flex-nowrap os-raise-justify-content-center">
<div class="os-raise-video-container"><video controls="true" crossorigin="anonymous">
<source src="https://k12.openstax.org/contents/raise/resources/b2c38efd9bdabdfa5993319c5eb1d37b24042ed2">
<track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/cde8cdd3b62ea446c6bfc2f233327ae95b83d98f" srclang="en_us">
https://k12.openstax.org/contents/raise/resources/b2c38efd9bdabdfa5993319c5eb1d37b24042ed2
</video></div>
</div>