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641 lines
24 KiB
HTML
641 lines
24 KiB
HTML
<!DOCTYPE html>
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<html lang="en"><!-- #BeginTemplate "/Templates/Main.dwt" --><!-- DW6 -->
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<title>Adding Fractions</title>
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<meta name="description" content="To add fractions there are Three Simple Steps: Step 1: Make sure the bottom numbers (the denominators) are the same, Step 2: Add the numerators, put that answer over the denominator, Step 3: Simplify the fraction (if needed)">
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<!-- #BeginEditable "Body" -->
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<div class="video">PGSG8gJWt1g</div>
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<h1 class="center">Adding Fractions</h1>
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<div class="def">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="numbers/images/fraction.svg" alt="fraction" height="107" width="320"></p>
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<p>A fraction like <span class="larger"><span class="intbl"> <em>3</em> <strong>4</strong> </span></span> says we have 3 out of the 4 parts the whole is divided into.</p>
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</div>
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<h3>To add fractions there are Three Simple Steps:</h3>
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<div class="bigul">
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<ul>
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<li>Step 1: Make sure the bottom numbers (the <b>denominators</b>) are the same</li>
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<li>Step 2: Add the top numbers (the <b>numerators</b>), put that answer over the <span class="denom">denominator</span></li>
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<li>Step 3: Simplify the fraction (if possible)</li>
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</ul>
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</div>
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<h2>Example:</h2>
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<div class="clear">
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<p class="center larger"><span class="intbl">
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<em>1</em>
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<strong>4</strong>
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</span> + <span class="intbl">
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<em>1</em>
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<strong>4</strong>
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</span></p>
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</div>
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<p><b>Step 1</b>. The bottom numbers (the denominators) are already the same. Go straight to step 2.</p>
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<p><b>Step 2</b>. Add the top numbers and put the answer over the same denominator:</p>
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<div class="clear">
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<p class="center larger"><span class="intbl">
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<em>1</em>
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<strong>4</strong>
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</span> + <span class="intbl">
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<em>1</em>
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<strong>4</strong>
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</span> = <span class="intbl">
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<em>1 + 1</em>
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<strong>4</strong>
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</span> = <span class="intbl">
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<em>2</em>
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<strong>4</strong>
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</span></p>
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</div>
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<p><b>Step 3</b>. Simplify the fraction:</p>
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<div class="clear">
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<p class="center larger"><span class="intbl">
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<em>2</em>
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<strong>4</strong>
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</span> = <span class="intbl">
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<em>1</em>
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<strong>2</strong>
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</span></p>
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<p>In picture form it looks like this:</p>
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</div>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr style="text-align:center;" class="larger">
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<td><span class="intbl">
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<em>1</em>
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<strong>4</strong>
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</span></td>
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<td style="width:20px;">+</td>
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<td><span class="intbl">
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<em>1</em>
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<strong>4</strong>
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</span></td>
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<td style="width:50px;">=</td>
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<td><span class="intbl">
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<em>2</em>
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<strong>4</strong>
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</span></td>
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<td style="width:50px;">=</td>
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<td><span class="intbl">
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<em>1</em>
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<strong>2</strong>
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</span></td>
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</tr>
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<tr style="text-align:center;">
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<td><img src="images/fractions/pizza-1-4.svg" alt="1/4" height="120" width="120"></td>
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<td class="larger"> </td>
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<td><img src="images/fractions/pizza-1-4.svg" alt="1/4" height="120" width="120"></td>
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<td class="larger"> </td>
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<td><img src="images/fractions/pizza-2-4.svg" alt="2/4" height="120" width="120"></td>
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<td class="larger"> </td>
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<td><img src="images/fractions/pizza-1-2.svg" alt="1/2" height="120" width="120"></td>
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</tr>
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</tbody></table>
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<p class="center">... and do you see how <span class="intbl">
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<em>2</em>
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<strong>4</strong>
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</span> is simpler as <span class="intbl">
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<em>1</em>
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<strong>2</strong>
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</span> ? (see <a href="equivalent_fractions.html">Equivalent Fractions</a>.)</p>
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<p> </p>
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<h2>Example:</h2>
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<div class="clear">
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<p class="center larger"><span class="intbl">
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<em>1</em>
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<strong>3</strong>
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</span> + <span class="intbl">
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<em>1</em>
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<strong>6</strong>
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</span></p>
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</div>
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<p><b>Step 1</b>: The bottom numbers are different. See how the slices are different sizes?</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr style="text-align:center;" class="larger">
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<td><span class="intbl">
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<em>1</em>
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<strong>3</strong>
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</span></td>
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<td style="width:20px;">+</td>
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<td><span class="intbl">
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<em>1</em>
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<strong>6</strong>
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</span></td>
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<td style="width:50px;">=</td>
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<td>?</td>
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<td style="width:50px;"> </td>
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<td> </td>
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</tr>
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<tr style="text-align:center;">
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<td><img src="images/fractions/pizza-1-3.svg" alt="1/3" height="120" width="120"></td>
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<td class="larger"> </td>
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<td><img src="images/fractions/pizza-1-6.svg" alt="1/6" height="120" width="120"></td>
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<td class="larger"> </td>
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<td><img src="images/fractions/pizza-huh.svg" alt="pie huh?" height="120" width="120"></td>
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<td class="larger"> </td>
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<td><img src="images/fractions/pizza-blank.svg" alt="empty" height="120" width="120"></td>
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</tr>
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</tbody></table>
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<p>We need to make them the same before we can continue, because we <b>can't</b> add them like that.</p>
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<p>The number "6" is twice as big as "3", so to make the bottom numbers the same we can multiply the top and bottom of the first fraction by <b>2</b>, like this:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td class="larger" align="center">× 2</td>
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</tr>
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<tr>
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<td><img src="images/style/arrow-rt-over.svg" alt="right over arrow" height="30" width="79"></td>
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</tr>
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</tbody></table>
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<div class="clear">
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<table class="larger" align="center" cellspacing="5" cellpadding="0" border="0">
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<tbody>
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<tr style="text-align:center;" class="larger">
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<td><span class="intbl">
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<em>1</em>
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<strong>3</strong>
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</span></td>
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<td style="width:50px;"><b> = </b></td>
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<td><span class="intbl">
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<em>2</em>
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<strong>6</strong>
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</span></td>
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</tr>
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</tbody></table>
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</div>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td><img src="images/style/arrow-rt-under.svg" alt="right under arrow" height="30" width="79"></td>
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</tr>
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<tr>
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<td class="larger" align="center">× 2</td>
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</tr>
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</tbody></table>
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<p class="center large">Important: you multiply <b>both top and bottom</b> by the same amount,<br>
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to keep the value of the fraction the same</p>
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<p>Now the fractions have the same bottom number ("6"), and our question looks like this:</p>
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<div class="clear">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr style="text-align:center;" class="larger">
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<td class="larger"><span class="intbl">
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<em>2</em>
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<strong>6</strong>
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</span></td>
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<td class="larger">+</td>
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<td class="larger"><span class="intbl">
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<em>1</em>
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<strong>6</strong>
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</span></td>
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<td style="width:50px;"> </td>
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<td> </td>
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<td style="width:50px;"> </td>
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<td> </td>
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</tr>
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<tr style="text-align:center;">
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<td><img src="images/fractions/pizza-2-6.svg" alt="2/6" height="120" width="120"></td>
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<td class="larger"> </td>
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<td><img src="images/fractions/pizza-1-6.svg" alt="1/6" height="120" width="120"></td>
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<td class="larger"> </td>
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<td><img src="images/fractions/pizza-blank.svg" alt="empty" height="120" width="120"></td>
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<td class="larger"> </td>
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<td><img src="images/fractions/pizza-blank.svg" alt="empty" height="120" width="120"></td>
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</tr>
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</tbody></table>
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</div>
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<p>The bottom numbers are now the same, so we can go to step 2.</p>
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<p> </p>
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<p><b>Step 2</b>: Add the top numbers and put them over the same denominator:</p>
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<p class="center large"><span class="intbl">
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<em>2</em>
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<strong>6</strong>
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</span> + <span class="intbl">
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<em>1</em>
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<strong>6</strong>
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</span> = <span class="intbl">
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<em>2 + 1</em>
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<strong>6</strong>
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</span> = <span class="intbl">
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<em>3</em>
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<strong>6</strong>
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</span></p>
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<p>In picture form it looks like this:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr style="text-align:center;" class="larger">
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<td class="larger"><span class="intbl">
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<em>2</em>
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<strong>6</strong>
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</span></td>
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<td class="larger">+</td>
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<td class="larger"><span class="intbl">
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<em>1</em>
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<strong>6</strong>
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</span></td>
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<td>=</td>
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<td><span class="intbl">
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<em>3</em>
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<strong>6</strong>
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</span></td>
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<td> </td>
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<td> </td>
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</tr>
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<tr style="text-align:center;">
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<td><img src="images/fractions/pizza-2-6.svg" alt="2/6" height="120" width="120"></td>
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<td class="larger" width="20"> </td>
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<td><img src="images/fractions/pizza-1-6.svg" alt="1/6" height="120" width="120"></td>
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<td class="larger" width="50"> </td>
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<td><img src="images/fractions/pizza-3-6.svg" alt="3/6" height="120" width="120"></td>
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<td class="larger" width="50"> </td>
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<td><img src="images/fractions/pizza-blank.svg" alt="empty" height="120" width="120"></td>
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</tr>
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</tbody></table>
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<p><br></p>
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<p><b>Step 3</b>: Simplify the fraction:</p>
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<p class="center larger"><span class="intbl">
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<em>3</em>
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<strong>6</strong>
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</span> = <span class="intbl">
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<em>1</em>
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<strong>2</strong>
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</span></p>
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<p> </p>
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<p>In picture form the whole answer looks like this:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr style="text-align:center;" class="larger">
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<td class="larger"><span class="intbl">
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<em>2</em>
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<strong>6</strong>
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</span></td>
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<td class="larger">+</td>
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<td class="larger"><span class="intbl">
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<em>1</em>
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<strong>6</strong>
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</span></td>
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<td>=</td>
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<td><span class="intbl">
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<em>3</em>
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<strong>6</strong>
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</span></td>
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<td>=</td>
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<td><span class="intbl">
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<em>1</em>
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<strong>2</strong>
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</span></td>
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</tr>
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<tr style="text-align:center;">
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<td><img src="images/fractions/pizza-2-6.svg" alt="2/6" height="120" width="120"></td>
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<td class="larger" width="20"> </td>
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<td><img src="images/fractions/pizza-1-6.svg" alt="1/6" height="120" width="120"></td>
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<td class="larger" width="50"> </td>
|
||
<td><img src="images/fractions/pizza-3-6.svg" alt="3/6" height="120" width="120"></td>
|
||
<td class="larger" width="50"> </td>
|
||
<td><img src="images/fractions/pizza-1-2.svg" alt="1/2" height="120" width="120"></td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p> </p>
|
||
|
||
<h3>With Pen and Paper</h3>
|
||
<p>And here is how to do it with a pen and paper (press the play button):</p>
|
||
<script>playerMain("numbers/images/fractions-add-a.txt",610,200,'');</script>
|
||
<p> </p>
|
||
<div class="simple">
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td><a href="numbers/fractions-addition-animation.html"><img src="numbers/images/adding-fractions-thumb.gif" alt="adding fractions thumb" height="114" width="150"></a></td>
|
||
<td> </td>
|
||
<td>
|
||
|
||
<h3>Play with it!</h3>
|
||
<p>Try <a href="numbers/fractions-addition-animation.html">Adding Fractions Illustrated</a>.</p></td>
|
||
</tr>
|
||
</tbody></table> </div>
|
||
|
||
<h2>A Rhyme To Help You Remember</h2>
|
||
<p class="center large"><i><span class="times">♫</span> "If adding or subtracting is your aim,<br>
|
||
The bottom numbers must be the same!</i><i><br>
|
||
<span class="times">♫</span> "Change the bottom using multiply or divide,<br>
|
||
But the same to the top must be applied,</i><i><br>
|
||
<span class="times">♫</span> "And don't forget to simplify,<br>
|
||
Before its time to say good bye"</i></p>
|
||
|
||
<h2>Example:</h2>
|
||
<p class="center larger"><span class="intbl">
|
||
<em>1</em>
|
||
<strong>3</strong>
|
||
</span> + <span class="intbl">
|
||
<em>1</em>
|
||
<strong>5</strong>
|
||
</span></p>
|
||
<p>Again, the bottom numbers are different (the slices are different sizes)!</p>
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr style="text-align:center;" class="larger">
|
||
<td><span class="intbl">
|
||
<em>1</em>
|
||
<strong>3</strong>
|
||
</span></td>
|
||
<td style="width:20px;">+</td>
|
||
<td><span class="intbl">
|
||
<em>1</em>
|
||
<strong>5</strong>
|
||
</span></td>
|
||
<td style="width:50px;">=</td>
|
||
<td>?</td>
|
||
<td style="width:50px;"> </td>
|
||
<td> </td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td><img src="images/fractions/pizza-1-3.svg" alt="1/3" height="120" width="120"></td>
|
||
<td class="larger"> </td>
|
||
<td><img src="images/fractions/pizza-1-5.svg" alt="1/5" height="120" width="120"></td>
|
||
<td class="larger"> </td>
|
||
<td><img src="images/fractions/pizza-huh.svg" alt="pie huh?" height="120" width="120"></td>
|
||
<td class="larger"> </td>
|
||
<td><img src="images/fractions/pizza-blank.svg" alt="empty" height="120" width="120"></td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p>But let us try dividing them into smaller sizes that <b>will each be the same</b>:</p>
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr style="text-align:center;" class="larger">
|
||
<td><span class="intbl">
|
||
<em>5</em>
|
||
<strong>15</strong>
|
||
</span></td>
|
||
<td>+</td>
|
||
<td><span class="intbl">
|
||
<em>3</em>
|
||
<strong>15</strong>
|
||
</span></td>
|
||
<td> </td>
|
||
<td> </td>
|
||
<td> </td>
|
||
<td> </td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td><img src="images/fractions/pizza-5-15.svg" alt="5/15" height="120" width="120"></td>
|
||
<td class="larger" width="20"> </td>
|
||
<td><img src="images/fractions/pizza-3-15.svg" alt="3/15" height="120" width="120"></td>
|
||
<td class="larger" width="50"> </td>
|
||
<td><img src="images/fractions/pizza-blank.svg" alt="empty" height="120" width="120"></td>
|
||
<td class="larger" width="50"> </td>
|
||
<td><img src="images/fractions/pizza-blank.svg" alt="empty" height="120" width="120"></td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p>The first fraction: by multiplying the top and bottom by 5 we ended up with <span class="intbl">
|
||
<em>5</em>
|
||
<strong>15</strong>
|
||
</span> :</p>
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td class="larger" align="center">× 5</td>
|
||
</tr>
|
||
<tr>
|
||
<td><img src="images/style/arrow-rt-over.svg" alt="right over arrow" height="30" width="79"></td>
|
||
</tr>
|
||
</tbody></table>
|
||
<div class="clear">
|
||
<table class="larger" align="center" cellspacing="5" cellpadding="0" border="0">
|
||
<tbody>
|
||
<tr style="text-align:center;" class="larger">
|
||
<td><span class="intbl">
|
||
<em>1</em>
|
||
<strong>3</strong>
|
||
</span></td>
|
||
<td><b> = </b></td>
|
||
<td><span class="intbl">
|
||
<em>5</em>
|
||
<strong>15</strong>
|
||
</span></td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td><img src="images/style/arrow-rt-under.svg" alt="right under arrow" height="30" width="79"></td>
|
||
</tr>
|
||
<tr>
|
||
<td class="larger" align="center">× 5</td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p>The second fraction: by multiplying the top and bottom by 3 we ended up with <span class="intbl">
|
||
<em>3</em>
|
||
<strong>15</strong>
|
||
</span> :</p>
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td class="larger" align="center">× 3</td>
|
||
</tr>
|
||
<tr>
|
||
<td><img src="images/style/arrow-rt-over.svg" alt="right over arrow" height="30" width="79"></td>
|
||
</tr>
|
||
</tbody></table>
|
||
<div class="clear">
|
||
<table class="larger" align="center" cellspacing="5" cellpadding="0" border="0">
|
||
<tbody>
|
||
<tr style="text-align:center;" class="larger">
|
||
<td><span class="intbl">
|
||
<em>1</em>
|
||
<strong>5</strong>
|
||
</span></td>
|
||
<td><b> = </b></td>
|
||
<td><span class="intbl">
|
||
<em>3</em>
|
||
<strong>15</strong>
|
||
</span></td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td><img src="images/style/arrow-rt-under.svg" alt="right under arrow" height="30" width="79"></td>
|
||
</tr>
|
||
<tr>
|
||
<td class="larger" align="center">× 3</td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p>The bottom numbers are now the same, so we can go ahead and add the top numbers:</p>
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr style="text-align:center;" class="larger">
|
||
<td><span class="intbl">
|
||
<em>5</em>
|
||
<strong>15</strong>
|
||
</span></td>
|
||
<td>+</td>
|
||
<td><span class="intbl">
|
||
<em>3</em>
|
||
<strong>15</strong>
|
||
</span></td>
|
||
<td>=</td>
|
||
<td><span class="intbl">
|
||
<em>8</em>
|
||
<strong>15</strong>
|
||
</span></td>
|
||
<td> </td>
|
||
<td> </td>
|
||
</tr>
|
||
<tr style="text-align:center;">
|
||
<td><img src="images/fractions/pizza-5-15.svg" alt="5/15" height="120" width="120"></td>
|
||
<td class="larger" width="20"> </td>
|
||
<td><img src="images/fractions/pizza-3-15.svg" alt="3/15" height="120" width="120"></td>
|
||
<td class="larger" width="50"> </td>
|
||
<td><img src="images/fractions/pizza-8-15.svg" alt="8/15" height="120" width="120"></td>
|
||
<td class="larger" width="50"> </td>
|
||
<td><img src="images/fractions/pizza-blank.svg" alt="empty" height="120" width="120"></td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p>The result is already as simple as it can be, so that is the answer: </p>
|
||
<p class="center larger"><span class="intbl">
|
||
<em>1</em>
|
||
<strong>3</strong>
|
||
</span> + <span class="intbl">
|
||
<em>1</em>
|
||
<strong>5</strong>
|
||
</span> = <span class="intbl">
|
||
<em>8</em>
|
||
<strong>15</strong>
|
||
</span></p>
|
||
|
||
<h2>Making the Denominators the Same</h2>
|
||
<p>In the previous example how did we know to cut them into <span class="frac"><sup>1</sup>/<sub>15</sub></span>ths to make the denominators the same? We simply multiplied the two denominators together (3 × 5 = 15).</p>
|
||
<p>Read about the two main ways to make the denominators the same here:</p>
|
||
<ul>
|
||
<li><a href="numbers/common-denominator.html">Common Denominator Method</a>, or the</li>
|
||
<li><a href="least-common-denominator.html">Least Common Denominator Method</a></li>
|
||
</ul>
|
||
<p>They both work, use which one you prefer!</p>
|
||
<div class="example">
|
||
<p style="float:right; margin: 10px;"><img src="images/cupcakes.jpg" alt="cupcakes" height="117" width="185"></p>
|
||
|
||
<h3>Example: Cupcakes</h3>
|
||
<p>You want to make and sell cupcakes:</p>
|
||
<ul>
|
||
<li>A friend can supply the ingredients, if you give them <sup>1</sup>/<sub>3</sub> of sales</li>
|
||
<li>And a market stall costs <sup>1</sup>/<sub>4</sub> of sales</li>
|
||
</ul>
|
||
<p>How much is that altogether?</p>
|
||
<p> </p>
|
||
<p>We need to add <sup>1</sup>/<sub>3</sub> and <sup>1</sup>/<sub>4</sub></p>
|
||
<div class="center larger"><span class="intbl"><em>1</em><strong>3</strong></span> + <span class="intbl"><em>1</em><strong>4</strong></span> = <span class="intbl"><em>?</em><strong>?</strong></span></div>
|
||
<!-- 1/3 + 1/4 = 1/1 -->
|
||
<p><b>First</b> make the bottom numbers (the denominators) the same.</p>
|
||
<p>Multiply top and bottom of <sup>1</sup>/<sub>3</sub> by <b>4</b>:</p>
|
||
<div class="center larger"><span class="intbl"><em>1×4</em><strong>3×4</strong></span> + <span class="intbl"><em>1</em><strong>4</strong></span> = <span class="intbl"><em>?</em><strong>?</strong></span></div>
|
||
<!-- 1*4/3*4 + 1/4 = 1/1 -->
|
||
<p>And multiply top and bottom of <sup>1</sup>/<sub>4</sub> by <b>3</b>:</p>
|
||
<div class="center larger"><span class="intbl"><em>1×4</em><strong>3×4</strong></span> + <span class="intbl"><em>1×3</em><strong>4×3</strong></span> = <span class="intbl"><em>?</em><strong>?</strong></span></div>
|
||
<!-- 1*4/3*4 + 1*3/4*3 = 1/1 -->
|
||
<p><b>Now</b> do the calculations:</p>
|
||
<div class="center larger"><span class="intbl"><em>4</em><strong>12</strong></span> + <span class="intbl"><em>3</em><strong>12</strong></span> = <span class="intbl"><em>4+3</em><strong>12</strong></span> = <span class="intbl"><em>7</em><strong>12</strong></span></div>
|
||
<!-- 4/12 + 3/12 = 4+3/12 = 7/12 -->
|
||
<p> </p>
|
||
<p>Answer: <span class="intbl">
|
||
<em>7</em>
|
||
<strong>12</strong>
|
||
</span> of sales go in ingredients and market costs.</p>
|
||
</div>
|
||
<p> </p>
|
||
|
||
<h2>Adding Mixed Fractions</h2>
|
||
<p>We have a special (more advanced) page on <a href="numbers/fractions-mixed-addition.html">Adding Mixed Fractions</a>.</p>
|
||
<p> </p>
|
||
<a id="q"></a>
|
||
<div class="questions">930,931, 1399,932, 1400,933, 1401, 1402, 3564, 3565</div>
|
||
|
||
<div class="related">
|
||
<a href="fractions.html">Introduction to Fractions</a>
|
||
<a href="simplifying-fractions.html">Simplifying Fractions</a>
|
||
<a href="equivalent_fractions.html">Equivalent Fractions</a>
|
||
<a href="least-common-multiple.html">Least Common Multiple</a>
|
||
<a href="least-common-multiple-tool.html">Least Common Multiple Tool</a>
|
||
<a href="least-common-denominator.html">Least Common Denominator</a>
|
||
<a href="fractions_subtraction.html">Subtracting Fractions</a>
|
||
<a href="fractions_multiplication.html">Multiplying Fractions</a>
|
||
<a href="fractions_division.html">Dividing Fractions</a>
|
||
<a href="fractions-menu.html">Fractions Index</a>
|
||
</div>
|
||
<!-- #EndEditable -->
|
||
|
||
</article>
|
||
|
||
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