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272 lines
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272 lines
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<article id="content" role="main"><!-- #BeginEditable "Body" -->
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<h1 align="center">Rounding Methods</h1>
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<p align="center">There are many ways to round numbers ...<br />
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</p>
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<h2>Firstly, what is "Rounding" ?</h2>
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<div class="clear"></div>
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<p>Rounding means making a number <b>simpler</b> but keeping its value close to what it was. <b>The result is less accurate, but easier to use. </b><br />
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</p>
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<div class="example">
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<h3>Example: 7.3 rounds to 7</h3>
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<p>Because 7.3 is closer to 7 than to 8</p>
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<p><i>(Note: we round to whole numbers in these examples, but we can <a href="#round-tens">round to tens, tenths</a>, etc)</i></p>
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</div>
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<p>But what about <b>7.5</b>? Is it closer to 7 or closer to 8? </p>
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<p class="center"><img src="images/round-5.gif" width="400" height="74" alt="round 5" /></p>
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<p class="center"><b>7.5</b> is halfway in between, so what should we do?</p>
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<h2>Half Round Up (the common method of rounding)</h2>
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<p>The <b>common method</b> of rounding is to make 0.5 go <b>up</b>, so 7.5 rounds up to 8</p>
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<p class="center larger">7.5 usually rounds up to 8</p>
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<p>But this is not a law or anything, it is just what people normally agree to do, and we get this:</p>
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<ul>
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<li>7.6 rounds up to 8</li>
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<li>7.5 rounds up to 8</li>
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<li>7.4 rounds down to 7</li>
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</ul>
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<p>Learn more about this method at <a href="../rounding-numbers.html">Rounding Numbers</a>.</p> <h2>Half Round Down</h2>
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<p>But <b> 5 can go down</b> if we want. In that case 7.5 rounds down to 7, and we get this:</p>
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<ul>
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<li>7.6 rounds up to 8</li>
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<li>7.5 rounds down to 7</li>
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<li>7.4 rounds down to 7</li>
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</ul>
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<p>But we should always let people know we are using "Half Round Down".</p>
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<p>Why make 0.5 go down? Maybe there are lots of 0.5's in our numbers and we want to see what rounding down does to our results.</p>
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<p class="center larger">Have a Play ... try different rounding methods on the <a href="rounding-tool.html">Rounding Tool</a>.</p>
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<h2>Negative Numbers</h2>
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<p>But what about <b>-7.5</b> ? </p>
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<ul>
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<li>Does it round to -8 (and is that going "up" or "down" ?), </li>
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<li>Or does it round to -7 ?</li>
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</ul>
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<p class="larger">Help! I am confused!</p>
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<p>In fact the whole world is confused about rounding negative numbers ... some computer programs round -7.5 to -8, others to -7</p>
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<p>But we can agree <b>here</b> that "up" means heading in a positive direction, like on this number line:</p>
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<p class="center"><img src="images/round-up.svg" alt="round up" /></p>
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<h2>Half Round Up (including negative numbers)</h2>
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<p>So we get this:</p>
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<ul>
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<li>7.6 rounds up to 8</li>
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<li>7.5 rounds up to 8</li>
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<li>7.4 rounds down to 7</li>
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<li></li>
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<li>-7.4 rounds up to -7</li>
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<li>-7.5 rounds up to -7</li>
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<li>-7.6 rounds down to -8</li>
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</ul>
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<h2>Half Round Down (including negative numbers)</h2>
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<p>When we <b>round 0.5 down</b> we get this:</p>
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<ul>
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<li>7.6 rounds up to 8</li>
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<li>7.5 rounds down to 7</li>
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<li>7.4 rounds down to 7</li>
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<li></li>
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<li>-7.4 rounds up to -7</li>
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<li>-7.5 rounds down to -8</li>
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<li>-7.6 rounds down to -8</li>
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</ul>
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<p> </p>
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<h2>"Symmetric" Rounding</h2>
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<p>But maybe you think "7.5 rounds up to 8, so -7.5 should go to -8", which is nice and <a href="../geometry/symmetry.html">symmetrical</a>.</p>
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<p> Well you are in luck because that is<b> rounding towards or away from zero</b>:</p>
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<p class="center"><img src="images/round-away.svg" alt="round away" /></p>
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<h2>Round Half Away From 0</h2>
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<p>For this method, 0.5 rounds the number so it is <b>further away from zero</b>, like this:</p>
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<ul>
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<li>7.6 rounds away to 8</li>
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<li>7.5 rounds away to 8</li>
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<li>7.4 rounds to 7</li>
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<li></li>
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<li>-7.4 rounds to -7</li>
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<li>-7.5 rounds away to -8</li>
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<li>-7.6 rounds away to -8</li>
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</ul>
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<h2>Round Half Towards 0</h2>
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<p>Or we can have 0.5 round the number closer to zero, like this:</p>
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<ul>
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<li>7.6 rounds away to 8</li>
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<li>7.5 rounds to 7</li>
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<li>7.4 rounds to 7</li>
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<li></li>
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<li>-7.4 rounds to -7</li>
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<li>-7.5 rounds to -7</li>
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<li>-7.6 rounds away to -8</li>
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</ul>
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<h2>But Being Consistent Can Be Bad</h2>
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<p>Choosing any of those methods <b>can</b> be bad, though! </p>
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<p>Imagine you are adding up a long list of numbers. You decide to round each number to make it quicker. If there are a lot of 0.5s they all get rounded up and your answer will have a <b>bias</b>.</p>
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<div class="example">
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<h3>Example: Add these numbers before and after rounding: 5.5, 7.5, 6.5, 9.5</h3>
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<p>Before rounding: <b>5.5 + 7.5 + 6.5 + 9.5 = 29</b></p>
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<p>After rounding: <b>6 + 8 + 7 + 10 = 31</b></p>
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<p>The calculation was a lot easier, but the answer drifted <b>up</b> a lot!</p>
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</div>
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<p>How can we stop the rounding being all one direction? </p>
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<p>We can decide to round <b>towards even (or odd) numbers</b>, or we can just choose <b>randomly</b>.</p>
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<h2>Round to Even (Banker's Rounding)</h2>
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<p>We round 0.5 to the nearest <b>even</b> digit</p>
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<div class="example">
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<p>Example:</p>
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<p>7.5 rounds <b>up</b> to <b>8</b> (because 8 is an even number)</p>
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<p>but 6.5 rounds <b>down</b> to <b>6</b> (because 6 is an even number) </p>
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</div>
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<p>Other numbers (not ending in 0.5) round to nearest as usual, so: </p>
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<ul>
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<li>7.6 rounds up to 8</li>
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<li>7.5 rounds <b>up</b> to <b>8</b> (because 8 is an even number)</li>
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<li>7.4 rounds down to 7</li>
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<li>6.6 rounds up to 7</li>
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<li>6.5 rounds <b>down</b> to <b>6</b> (because 6 is an even number) </li>
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<li>6.4 rounds down to 6</li>
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<li>etc </li>
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</ul>
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<h2>Round to Odd</h2>
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<p>Just like "Round To Even", but 0.5 heads towards odd numbers</p>
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<div class="example">
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<p>Example:</p>
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<p>7.5 rounds down to<b> 7</b> (because 7 is an odd number)</p>
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<p>but 6.5 rounds <b>up</b> to <b>7</b> (because 7 is an odd number) </p>
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</div>
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<h2>Round Randomly</h2>
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<p>We could also choose to round 0.5 up or down randomly, but how? By tossing a coin? Or a computer function? </p>
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<p>With a large list of numbers this can give good results, but also gives a different answer each time (unless we use a fixed list of random choices).</p>
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<h2>Floor and Ceiling</h2>
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<p>There are two other methods that don't even consider 0.5. They are called <a href="../sets/function-floor-ceiling.html">Floor and Ceiling</a>.</p>
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<p>Floor gives us the <b>nearest integer down</b> (and ceiling goes up).</p>
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<div class="example">
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<h3>Example: What is the floor and ceiling of 2.31?</h3>
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<p align="center"><img src="../sets/images/function-floor-ceiling.svg" alt="Floor and Ceiling function" /></p>
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<p align="center" class="larger">The Floor of 2.31 is <b>2</b> <br />
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The Ceiling of 2.31 is <b>3</b></p>
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</div>
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<h2>Floor</h2>
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<p>Using "floor", all digits go down, no matter what the dropped digit is:</p>
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<div class="example">
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<h3>Example: 7.8 goes down to 7</h3>
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<p>so does 7.2, 7.5, 7.9, etc.</p>
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<p>And 7 goes to 7, too.</p>
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</div>
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<h2>Ceiling</h2>
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<p>And "ceiling" goes up:</p>
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<div class="example">
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<h3>Example: 7.1 goes up to 8</h3>
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<p>so does 7.2, 7.5, 7.8, etc.</p>
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<p>But <b>7 stays at 7</b>.</p>
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</div>
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<p> </p>
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<h2>Summary </h2>
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<table align="center">
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<tr>
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<th width="60">Number</th> <th width="60" align="right">Half<br />
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Up</th> <th width="60" align="right">Half<br />
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Down</th> <th width="60" align="right">Half<br />Away 0</th> <th width="60" align="right">Half<br />To 0</th> <th width="60" align="right">Half<br />
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Even</th> <th width="60" align="right">Half<br />
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Odd</th> <th width="60" align="right">Floor</th> <th width="60" align="right">Ceiling</th>
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</tr><tr><td> 8</td> <td width="50" align="right">8</td> <td width="50" align="right">8</td> <td width="50" align="right">8</td> <td width="50" align="right">8</td> <td width="50" align="right">8</td> <td width="50" align="right">8</td> <td width="50" align="right">8</td> <td width="50" align="right">8</td>
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</tr><tr><td> 7.6</td> <td width="50" align="right">8</td> <td width="50" align="right">8</td> <td width="50" align="right">8</td> <td width="50" align="right">8</td> <td width="50" align="right">8</td> <td width="50" align="right">8</td> <td width="50" align="right">7</td> <td width="50" align="right">8</td>
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</tr><tr><td> 7.5</td> <td width="50" align="right">8</td> <td width="50" align="right">7</td> <td width="50" align="right">8</td> <td width="50" align="right">7</td> <td width="50" align="right">8</td> <td width="50" align="right">7</td> <td width="50" align="right">7</td> <td width="50" align="right">8</td>
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</tr><tr><td> 7.4</td> <td width="50" align="right">7</td> <td width="50" align="right">7</td> <td width="50" align="right">7</td> <td width="50" align="right">7</td> <td width="50" align="right">7</td> <td width="50" align="right">7</td> <td width="50" align="right">7</td> <td width="50" align="right">8</td>
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</tr><tr>
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<td> 7</td> <td width="50" align="right">7</td> <td width="50" align="right">7</td> <td width="50" align="right">7</td> <td width="50" align="right">7</td> <td width="50" align="right">7</td> <td width="50" align="right">7</td> <td width="50" align="right">7</td> <td width="50" align="right">7</td>
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</tr><tr><td>-7</td> <td width="50" align="right">-7</td> <td width="50" align="right">-7</td> <td width="50" align="right">-7</td> <td width="50" align="right">-7</td> <td width="50" align="right">-7</td> <td width="50" align="right">-7</td> <td width="50" align="right">-7</td> <td width="50" align="right">-7</td>
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</tr><tr><td>-7.4</td> <td width="50" align="right">-7</td> <td width="50" align="right">-7</td> <td width="50" align="right">-7</td> <td width="50" align="right">-7</td> <td width="50" align="right">-7</td> <td width="50" align="right">-7</td> <td width="50" align="right">-8</td> <td width="50" align="right">-7</td>
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</tr><tr><td>-7.5</td> <td width="50" align="right">-7</td> <td width="50" align="right">-8</td> <td width="50" align="right">-8</td> <td width="50" align="right">-7</td> <td width="50" align="right">-8</td> <td width="50" align="right">-7</td> <td width="50" align="right">-8</td> <td width="50" align="right">-7</td>
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</tr><tr><td>-7.6</td> <td width="50" align="right">-8</td> <td width="50" align="right">-8</td> <td width="50" align="right">-8</td> <td width="50" align="right">-8</td> <td width="50" align="right">-8</td> <td width="50" align="right">-8</td> <td width="50" align="right">-8</td> <td width="50" align="right">-7</td>
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</tr><tr><td>-8</td> <td width="50" align="right">-8</td> <td width="50" align="right">-8</td> <td width="50" align="right">-8</td> <td width="50" align="right">-8</td> <td width="50" align="right">-8</td> <td width="50" align="right">-8</td> <td width="50" align="right">-8</td> <td width="50" align="right">-8</td>
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</tr></table>
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<p> </p>
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<h2><a id="round-tens"></a>Rounding to Tens, Tenths, Whatever ...</h2>
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<p>In our examples we rounded to whole numbers, but you can round to tens, or tenths, etc:</p>
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<div class="example">
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<h3>Example: "Half Round Up" to <b>tens</b> (nearest 10):</h3>
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<p>25 rounds up to 30</p>
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<p>24.97 rounds down to 20</p>
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</div>
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<div class="example">
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<h3>Example: "Half Round Up" to <b>hundredths</b> (nearest 1/100):</h3>
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<p>0.5168 rounds up to 0.52</p>
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<p>1.41119 rounds down to 1.41</p>
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</div>
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<p> </p>
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<div class="related"><a href="../rounding-numbers.html">Rounding Numbers</a> <a href="rounding-tool.html">Rounding Tool</a> <a href="index.html">Numbers Index</a></div>
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