<h2>Rounding Whole Numbers</h2>
<p id="fs-id1333125">In \(2013,\) the U.S. Census Bureau reported the population of the state of New York as \(19,651,127\) people. It might be enough to say that the population is approximately \(20\) million. The word <em data-effect="italics">approximately</em> means that \(20\) million is not the exact population, but is close to the exact value.</p>
<p id="fs-id2368933">The process of approximating a number is called <strong>rounding</strong>. Numbers are rounded to a specific place value depending on how much accuracy is needed. \(20\) million was achieved by rounding to the millions place. Had we rounded to the one hundred thousands place, we would have \(19,700,000\) as a result. Had we rounded to the ten thousands place, we would have \(19,650,000\) as a result, and so on. The place value to which we round to depends on how we need to use the number.</p>
<p id="fs-id2132131">Using the number line can help you visualize and understand the rounding process. Look at the number line in the figure below. Suppose we want to round the number \(76\) to the nearest ten. Is \(76\) closer to \(70\) or \(80\) on the number line?</p>
<div class="wp-caption alignnone" id="attachment_99397" style="width: 800px"><a href="https://data.ulearngo.com/assets/38406db6-6f66-4379-8757-ca4a71b54a9f"><img alt="" class="wp-image-99397 size-full" height="36" src="https://data.ulearngo.com/assets/38406db6-6f66-4379-8757-ca4a71b54a9f" width="800"/></a><p class="wp-caption-text">We can see that \(76\) is closer to \(80\) than to \(70.\) So \(76\) rounded to the nearest ten is \(80.\) Image credit: OpenStax Prealgebra</p></div>
<p id="fs-id1885331">Now consider the number \(72.\) Find \(72\) in the figure below.</p>
<div class="wp-caption alignnone" id="attachment_99398" style="width: 800px"><a href="https://data.ulearngo.com/assets/665d52da-222d-4687-896a-58fd8345ad36"><img alt="" class="wp-image-99398 size-full" height="36" src="https://data.ulearngo.com/assets/665d52da-222d-4687-896a-58fd8345ad36" width="800"/></a><p class="wp-caption-text">We can see that \(72\) is closer to \(70,\) so \(72\) rounded to the nearest ten is \(70.\) Image credit: OpenStax Prealgebra</p></div>
<p>How do we round \(75\) to the nearest ten. Find \(75\) in the figure below.</p>
<div class="wp-caption alignnone" id="attachment_99399" style="width: 800px"><a href="https://data.ulearngo.com/assets/cb5ba485-4947-418f-853f-db0ecf763108"><img alt="" class="size-full wp-image-99399" height="36" src="https://data.ulearngo.com/assets/cb5ba485-4947-418f-853f-db0ecf763108" width="800"/></a><p class="wp-caption-text">The number \(75\) is exactly midway between \(70\) and \(80.\) Image credit: OpenStax Prealgebra</p></div>
<p id="fs-id1384953">So that everyone rounds the same way in cases like this, mathematicians have agreed to round to the higher number, \(80.\) So, \(75\) rounded to the nearest ten is \(80.\)</p>
<p id="fs-id1545266">Now that we have looked at this process on the number line, we can introduce a more general procedure. To round a number to a specific place, look at the number to the right of that place. If the number is less than \(5,\) round down. If it is greater than or equal to \(5,\) round up.</p>
<p id="fs-id1583269">So, for example, to round \(76\) to the nearest ten, we look at the digit in the ones place.</p>
<p><a href="https://data.ulearngo.com/assets/a621287a-6169-48cc-adf4-14f36b4b8548"><img alt="" class="size-full wp-image-99400 alignnone" height="124" src="https://data.ulearngo.com/assets/a621287a-6169-48cc-adf4-14f36b4b8548" width="210"/></a></p>
<p>The digit in the ones place is a \(6.\) Because \(6\) is greater than or equal to \(5,\) we increase the digit in the tens place by one. So the \(7\) in the tens place becomes an \(8.\) Now, replace any digits to the right of the \(8\) with zeros. So, \(76\) rounds to \(80.\)</p>
<p><a href="https://data.ulearngo.com/assets/e2fad5e3-0b82-4de3-8c4d-a3c341b9e10b"><img alt="" class="size-full wp-image-99401 alignnone" height="100" src="https://data.ulearngo.com/assets/e2fad5e3-0b82-4de3-8c4d-a3c341b9e10b" width="286"/></a></p>
<p id="fs-id2966405">Let’s look again at <strong><span class="no-emphasis" data-type="term">rounding </span></strong><span class="no-emphasis" data-type="term">\(72\) </span>to the nearest \(10.\) Again, we look to the ones place.</p>
<p><a href="https://data.ulearngo.com/assets/20a3534a-b13a-4aca-9527-173a084c8a54"><img alt="" class="size-full wp-image-99402 alignnone" height="124" src="https://data.ulearngo.com/assets/20a3534a-b13a-4aca-9527-173a084c8a54" width="179"/></a></p>
<p id="fs-id2700449">The digit in the ones place is \(2.\) Because \(2\) is less than \(5,\) we keep the digit in the tens place the same and replace the digits to the right of it with zero. So \(72\) rounded to the nearest ten is \(70.\)</p>
<p><a href="https://data.ulearngo.com/assets/c44c5505-b64f-462a-be84-b57365ce4647"><img alt="" class="size-full wp-image-99403 alignnone" height="66" src="https://data.ulearngo.com/assets/c44c5505-b64f-462a-be84-b57365ce4647" width="279"/></a></p>
<div class="ns-tutorial">
<div class="ns-school-defn2">
<h3>Round a Whole Number to a Specific Place Value</h3>
<ol>
<li>Locate the given place value. All digits to the left of that place value do not change.</li>
<li>Underline the digit to the right of the given place value.</li>
<li>Determine if this digit is greater than or equal to \(5.\)

<ol data-bullet-style="bullet" id="fs-id2486248" style="list-style-type: lower-alpha;">
<li>Yes—add \(1\) to the digit in the given place value.</li>
<li>No—do not change the digit in the given place value.</li>
</ol>
</li>
<li>Replace all digits to the right of the given place value with zeros.</li>
</ol>
</div>
</div>
<div class="ns-school-emp">
<h3>Example</h3>
<div class="ns-school-defn">
<p>Round \(843\) to the nearest ten.</p>
<h4>Solution</h4>
<table class="unnumbered unstyled" id="eip-659" summary="This table is two columns. The first column states locate the tens place. Next to this is the number 843 with the label “tens place” pointed at the 4 in 843. The next line down in the left column states “underline the digit to the right of the tens place”, followed by the number 843 with the 3 underlined. The next line states “since 3 is less than 5, do not change the digit in the tens place”, followed by the number 843 with the 3 remaining underlined. The next line states “replace all digits to the right of the tens place with zeros”, followed by the number 840 with the 0 underlined. The last line states “rounding 843 to the nearest ten gives 840.">
<tbody>
<tr>
<td data-valign="bottom">Locate the tens place.</td>
<td><a href="https://data.ulearngo.com/assets/1f6f259a-c2a6-4ccf-a653-c4742dbbbd45"><img alt="" class="size-full wp-image-99412 alignnone" height="78" src="https://data.ulearngo.com/assets/1f6f259a-c2a6-4ccf-a653-c4742dbbbd45" width="93"/></a></td>
</tr>
<tr>
<td>Underline the digit to the right of the tens place.</td>
<td><a href="https://data.ulearngo.com/assets/d0fd99aa-e8fb-45f2-89b1-5b7950daa619"><img alt="" class="size-full wp-image-99413 alignnone" height="20" src="https://data.ulearngo.com/assets/d0fd99aa-e8fb-45f2-89b1-5b7950daa619" width="93"/></a></td>
</tr>
<tr>
<td>Since 3 is less than 5, do not change the digit in the tens place.</td>
<td><a href="https://data.ulearngo.com/assets/e8b681c5-efb8-43a1-a3ce-36d163b9a86b"><img alt="" class="size-full wp-image-99414 alignnone" height="20" src="https://data.ulearngo.com/assets/e8b681c5-efb8-43a1-a3ce-36d163b9a86b" width="93"/></a></td>
</tr>
<tr>
<td>Replace all digits to the right of the tens place with zeros.</td>
<td><a href="https://data.ulearngo.com/assets/03c43823-e866-43cd-985f-7517b5fbccb2"><img alt="" class="size-full wp-image-99415 alignnone" height="23" src="https://data.ulearngo.com/assets/03c43823-e866-43cd-985f-7517b5fbccb2" width="93"/></a></td>
</tr>
<tr>
<td> </td>
<td>Rounding 843 to the nearest ten gives 840.</td>
</tr>
</tbody>
</table>
</div>
</div>
<div class="ns-school-emp">
<h3>Example</h3>
<div class="ns-school-defn">
<p id="fs-id1339512">Round each number to the nearest hundred:</p>
<ol class="circled" data-number-style="lower-alpha" id="eip-id1168287209769" style="list-style-type: lower-alpha;">
<li>\(23,658\)</li>
<li>\(3,978\)</li>
</ol>
<h4>Solution</h4>
<table>
<tbody>
<tr>
<td>(a)</td>
<td> </td>
</tr>
<tr>
<td data-valign="bottom">Locate the hundreds place.</td>
<td><a href="https://data.ulearngo.com/assets/01eb4852-3c02-45f7-be4a-57447b1781bd"><img alt="" class="size-full wp-image-99422 alignnone" height="79" src="https://data.ulearngo.com/assets/01eb4852-3c02-45f7-be4a-57447b1781bd" width="213"/></a></td>
</tr>
<tr>
<td>The digit of the right of the hundreds place is 5. Underline the digit to the right of the hundreds place.</td>
<td><a href="https://data.ulearngo.com/assets/555c47ac-a79f-4e21-a347-da2eccbb4b3e"><img alt="" class="size-full wp-image-99423 alignnone" height="24" src="https://data.ulearngo.com/assets/555c47ac-a79f-4e21-a347-da2eccbb4b3e" width="213"/></a></td>
</tr>
<tr>
<td>Since 5 is greater than or equal to 5, round up by adding 1 to the digit in the hundreds place. Then replace all digits to the right of the hundreds place with zeros.</td>
<td>
<div data-type="newline"><a href="https://data.ulearngo.com/assets/aa031b33-0135-4009-9933-2e04c441e2c1"><img alt="" class="size-full wp-image-99424 alignnone" height="140" src="https://data.ulearngo.com/assets/aa031b33-0135-4009-9933-2e04c441e2c1" width="213"/></a></div>

So 23,658 rounded to the nearest hundred is 23,700.</td>
</tr>
</tbody>
</table>
<table>
<tbody>
<tr>
<td>(b)</td>
<td> </td>
</tr>
<tr>
<td data-valign="bottom">Locate the hundreds place.</td>
<td><a href="https://data.ulearngo.com/assets/e18c5d00-eba9-4b49-9621-291a254f110f"><img alt="" class="size-full wp-image-99425 alignnone" height="79" src="https://data.ulearngo.com/assets/e18c5d00-eba9-4b49-9621-291a254f110f" width="357"/></a></td>
</tr>
<tr>
<td>Underline the digit to the right of the hundreds place.</td>
<td><a href="https://data.ulearngo.com/assets/e8d8c2bb-6459-420a-8d70-5b5db35d0e3f"><img alt="" class="size-full wp-image-99426 alignnone" height="25" src="https://data.ulearngo.com/assets/e8d8c2bb-6459-420a-8d70-5b5db35d0e3f" width="357"/></a></td>
</tr>
<tr>
<td>The digit to the right of the hundreds place is 7. Since 7 is greater than or equal to 5, round up by added 1 to the 9. Then place all digits to the right of the hundreds place with zeros.</td>
<td>
<div data-type="newline"><a href="https://data.ulearngo.com/assets/d7f20a6d-f81d-4c49-9bed-11be1980eaaa"><img alt="" class="size-full wp-image-99427 alignnone" height="145" src="https://data.ulearngo.com/assets/d7f20a6d-f81d-4c49-9bed-11be1980eaaa" width="357"/></a> </div>

So 3,978 rounded to the nearest hundred is 4,000.</td>
</tr>
</tbody>
</table>
</div>
</div>
<div class="ns-school-emp">
<h3>Example</h3>
<div class="ns-school-defn">
<p id="fs-id1339512">Round each number to the nearest thousand:</p>
<ol class="circled" data-number-style="lower-alpha" id="eip-id1168287209769" style="list-style-type: lower-alpha;">
<li>\(147,032\)</li>
<li>\(29,504\)</li>
</ol>
<h4>Solution</h4>
<table class="unnumbered unstyled" data-label="" id="eip-695" summary="This table is set in two columns. The left column is annotated and the right column shows numbers. In the left column, it states, the number 147,032 with the label “thousands place” pointed at the 7 in 147,032. The next line in the left column states “the digit to the right of the thousands place is 0. Since 0 is less than 5, we do not change the digit in the thousands place” followed by the number 147,032 with the 0 underlined. the next line states “we then replace all digits to the right of the thousands place with zeros” followed by the number 147,000. The last line states “so 147,032 rounded to the nearest thousand is 147,000.">
<tbody>
<tr>
<td>(a)</td>
<td> </td>
</tr>
<tr>
<td data-valign="bottom">Locate the thousands place. Underline the digit to the right of the thousands place.</td>
<td><a href="https://data.ulearngo.com/assets/4db9d0b0-1cd3-4e6d-800e-500ff699d875"><img alt="" class="size-full wp-image-99444 alignnone" height="79" src="https://data.ulearngo.com/assets/4db9d0b0-1cd3-4e6d-800e-500ff699d875" width="155"/></a></td>
</tr>
<tr>
<td>The digit to the right of the thousands place is 0. Since 0 is less than 5, we do not change the digit in the thousands place.</td>
<td><a href="https://data.ulearngo.com/assets/1dfd0139-77b4-4fea-b70a-bb028ad5c38f"><img alt="" class="size-full wp-image-99445 alignnone" height="25" src="https://data.ulearngo.com/assets/1dfd0139-77b4-4fea-b70a-bb028ad5c38f" width="155"/></a></td>
</tr>
<tr>
<td>We then replace all digits to the right of the thousands pace with zeros.</td>
<td>
<div data-type="newline"><a href="https://data.ulearngo.com/assets/cf36827a-dee0-4ae9-92bd-6ec5963c0d3d"><img alt="" class="size-full wp-image-99446 alignnone" height="25" src="https://data.ulearngo.com/assets/cf36827a-dee0-4ae9-92bd-6ec5963c0d3d" width="155"/></a></div>

So 147,032 rounded to the nearest thousand is 147,000.</td>
</tr>
</tbody>
</table>
<table class="unnumbered unstyled" data-label="" id="eip-596" summary="This table is set in two columns. The left column is annotated and the right column shows numbers. In the left column, it states, locate the thousands place followed by the number 29,504 with the label “thousands place” pointed at the 9 in 29,504. The next line states “underline the digit to the right of the thousands place” followed by the number 29,504 with the 5 underlined. The next line states the digit to the right of the thousands place is 5. Since 5 is greater than or equal to 5, round up by adding 1 to the 9. Then replace all digits to the right of the thousands place with zeros”, followed by the number 29,504 with the label “add 1, 9 +1 = 10, write 0 in the thousands place, add 1 to the ten thousands place” pointed at the 9 and the label “replace with 0s” pointed at the 5, 0, and 4. This item points downward at the last item, which is the number 30,000. The last line states “so 29,504 rounded to the nearest thousand is 30,000.">
<tbody>
<tr>
<td style="width: 555px;">(b)</td>
<td style="width: 382px;"> </td>
</tr>
<tr>
<td data-valign="bottom" style="width: 555px;">Locate the thousands place.</td>
<td style="width: 382px;"><a href="https://data.ulearngo.com/assets/fbca0e30-71ed-43a5-845e-b0318d5fb3f9"><img alt="" class="size-full wp-image-99447 alignnone" height="80" src="https://data.ulearngo.com/assets/fbca0e30-71ed-43a5-845e-b0318d5fb3f9" width="352"/></a></td>
</tr>
<tr>
<td style="width: 555px;">Underline the digit to the right of the thousands place.</td>
<td style="width: 382px;"><a href="https://data.ulearngo.com/assets/70567d0c-7e5f-434a-bebf-3212dfbddd12"><img alt="" class="size-full wp-image-99448 alignnone" height="25" src="https://data.ulearngo.com/assets/70567d0c-7e5f-434a-bebf-3212dfbddd12" width="358"/></a></td>
</tr>
<tr>
<td style="width: 555px;">The digit to the right of the thousands place is 5. Since 5 is greater than or equal to 5, round up by adding 1 to the 9. Then replace all digits to the right of the thousands place with zeros.</td>
<td style="width: 382px;">
<div data-type="newline"><a href="https://data.ulearngo.com/assets/7d0540ff-0c65-4bb2-bbcc-d5915f388213"><img alt="" class="size-full wp-image-99449 alignnone" height="150" src="https://data.ulearngo.com/assets/7d0540ff-0c65-4bb2-bbcc-d5915f388213" width="352"/></a></div>

So 29,504 rounded to the nearest thousand is 30,000.</td>
</tr>
</tbody>
</table>
<p id="fs-id1751923">Notice that in part (b), when we add \(1\) thousand to the \(9\) thousands, the total is \(10\) thousands. We regroup this as \(1\) ten thousand and \(0\) thousands. We add the \(1\) ten thousand to the \(3\) ten thousands and put a \(0\) in the thousands place.</p>
</div>
</div>
<div class="ns-school-emp">
<h2>Discussion</h2>
<div class="ns-school-defn">
<p>Give an example from your everyday life where it helps to round numbers. You can share your example in the comments section below.</p>
</div>
</div>

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Round Whole Numbers

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Round Whole Numbers

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