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573 lines
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Solving Word Questions</h1>
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<p class="center"><i>With LOTS of examples!</i></p>
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<p>In Algebra we often have word questions like:</p>
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<div class="example">
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<h3>Example: Sam and Alex play tennis.</h3>
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<p>On the weekend Sam played 4 more games than Alex did, and together they played 12 games.</p>
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<p>How many games did Alex play?</p>
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</div>
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<p>How do we solve them?</p>
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<p>The trick is to break the solution into two parts:</p>
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<div class="center80">
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<p class="center large">Turn the English into Algebra.</p>
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<p class="center large">Then use Algebra to solve.</p>
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</div>
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<h2>Turning English into Algebra</h2>
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<p>To turn the English into Algebra it helps to:</p>
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<ul>
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<li>Read the whole thing first</li>
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<li>Do a <b>sketch</b> if possible</li>
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<li>Assign <b>letters</b> for the values</li>
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<li>Find or work out <b>formulas</b></li>
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</ul>
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<p>You should also write down <b>what is actually being asked for</b>, so you know where you are going and when you have arrived!</p>
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<p>Also look for key words:</p>
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<div class="simple">
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<table align="center" >
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<tr>
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<th >When you see </th>
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<th > </th>
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<th align="center" >Think</th>
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</tr>
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<tr>
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<td ><p>add, total, sum, increase, more, combined, together, plus, more than</p></td>
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<td > </td>
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<td align="center" class="huge">+</td>
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</tr>
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<tr>
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<td ><p>minus, less, difference, fewer, decreased, reduced</p></td>
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<td > </td>
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<td align="center" class="huge" >−</td>
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</tr>
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<tr>
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<td ><p>multiplied, times, of, product, factor</p></td>
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<td > </td>
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<td align="center" class="huge" >×</td>
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</tr>
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<tr>
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<td ><p>divided, quotient, per, out of, ratio, ratio, percent, rate</p></td>
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<td > </td>
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<td align="center" class="huge" >÷</td>
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</tr>
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<tr>
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<td >maximize or minimize</td>
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<td > </td>
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<td align="center" >geometry<br>
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formulas</td>
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</tr>
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<tr>
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<td >Rate, speed</td>
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<td > </td>
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<td align="center" >distance<br>
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formulas</td>
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</tr>
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<tr>
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<td >How long, days, hours, minutes, seconds</td>
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<td > </td>
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<td align="center" >time</td>
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</tr>
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</table>
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</div>
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<h2>Thinking Clearly</h2>
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<p>Some wording can be tricky, making it hard to think "the right way around", such as:</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px; font: italic bold 90px/90px Georgia; color:gold;">$</p>
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<h3>Example: Sam has 2 dollars less than Alex. How do we write this as an equation?</h3>
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<ul>
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<li>Let S = dollars Sam has</li>
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<li>Let A = dollars Alex has</li>
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</ul>
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<p class="center">Now ... is that: <span class="large">S − 2 = A</span></p>
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<p class="center">or should it be: <span class="large">S = A − 2</span></p>
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<p class="center">or should it be: <span class="large">S = 2 − A</span></p>
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<p class="center"> </p>
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<p class="center">The correct answer is <span class="hide">S = A − 2</span></p>
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<p>(<b>S − 2 = A</b> is a common mistake, as the question is written "Sam ... 2 less ... Alex")</p>
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</div>
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<div class="example">
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<h3>Example: on our street there are twice as many dogs as cats. How do we write this as an equation?</h3>
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<ul>
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<li>Let D = number of dogs</li>
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<li>Let C = number of cats</li>
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</ul>
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<p class="center">Now ... is that: <span class="large">2D = C</span></p>
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<p class="center">or should it be: <span class="large">D = 2C</span></p>
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<p>Think carefully now!</p>
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<p class="center">The correct answer is <span class="hide">D = 2C</span></p>
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<p>(<b>2D = C</b> is a common mistake, as the question is written "twice ... dogs ... cats")</p>
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</div>
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<h2>Examples</h2>
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<p>Let's start with a <b>really simple</b> example so we see how it's done:</p>
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<div class="example">
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<h3>Example: A rectangular garden is 12m by 5m, what is its <a href="../area.html">area</a>?</h3>
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<p> </p>
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<p><span class="center large">Turn the English into Algebra:</span></p>
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<p>Sketch:</p>
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<p class="center"><img src="images/garden-area.gif" width="198" height="87" alt="garden area 5m x 12m" /></p>
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<p>Letters:</p>
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<ul>
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<li>Use <b>w</b> for width of rectangle: <b>w = 12m</b></li>
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<li>Use <b>h</b> for height of rectangle: <b>h = 5m</b></li>
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</ul>
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<p><b>Formula</b> for <a href="../area.html">Area of a Rectangle</a>: <b>A = w × h</b></p>
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<p class="larger">We are being asked for the Area.</p>
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<p> </p>
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<p><span class="center large">Solve:</span></p>
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<p class="center"><b>A = w × h = 12 × 5 = 60 m<sup>2</sup></b></p>
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<p class="larger">The area is <b>60 square meters</b>.</p>
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</div>
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<p>Now let's try the example from the top of the page:</p>
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<div class="example">
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/tennis.jpg" width="119" height="89" alt="tennis" /></p>
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<h3>Example: Sam and Alex play Tennis. On the weekend Sam played 4 more games than Alex did, and together they played 12 games. How many games did Alex play?</h3>
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<p> </p>
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<p><span class="center large">Turn the English into Algebra:</span></p>
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<p>Letters:</p>
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<ul>
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<li>Use <b>S</b> for how many games Sam played</li>
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<li>Use <b>A</b> for how many games Alex played</li>
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</ul>
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<p>We know that Sam played 4 more games than Alex, so: <b>S = A + 4</b></p>
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<p>And we know that together they played 12 games: <b>S + A = 12</b></p>
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<p class="larger">We are being asked for how many games Alex played: <b>A</b></p>
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<p> </p>
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<p><span class="center large">Solve:</span></p>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right"><span class="center larger">S + A = 12</span></span></div>
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<div class="row"><span class="left"><b>S = A + 4</b>, so we can<br>
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substitute "A + 4" for S:</span><span class="right"><span class="larger"><span class="hilite">(A + 4)</span> + A = 12</span></span></div>
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<div class="row"><span class="left">Simplify:</span><span class="right"><span class="larger">2A + 4 = 12</span></span></div>
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<div class="row"><span class="left">Subtract 4 from both sides:</span><span class="right"><span class="larger">2A = 12 − 4</span></span></div>
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<div class="row"><span class="left">Simplify:</span><span class="right"><span class="larger">2A = 8</span></span></div>
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<div class="row"><span class="left">Divide both sides by 2:</span><span class="right"><span class="larger">A = 4</span></span></div>
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</div>
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<p><b>Which means that Alex played 4 games of tennis.</b></p>
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<p> </p>
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<p>Check: Sam played 4 more games than Alex, so Sam played 8 games.
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Together they played 8 + 4 = 12 games. Yes!</p>
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</div>
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<p>A slightly harder example:</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/table.jpg" width="200" height="111" alt="table" /></p>
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<h3>Example: Alex and Sam also build tables.<br>
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Together they make 10 tables in 12 days.<br>
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<br>
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Alex working alone can make 10 in 30 days.<br>
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<br>
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How long would it take Sam working alone to make 10 tables?</h3>
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<p> </p>
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<p><span class="center large">Turn the English into Algebra:</span></p>
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<p>Letters:</p>
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<ul>
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<li>Use <b>a</b> for Alex's work rate</li>
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<li>Use <b>s</b> for Sam's work rate</li>
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</ul>
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<p>12 days of Alex and Sam is 10 tables, so: <b>12a + 12s = 10</b></p>
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<p>30 days of Alex alone is also 10 tables: <b>30a = 10</b></p>
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<p class="larger">We are being asked how long it would take Sam to make 10 tables.</p>
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<p><span class="center large">Solve:</span></p>
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<p><b>30a = 10</b>, so Alex's rate (tables per day) is: <b>a = 10/30 = 1/3</b></p>
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<div class="tbl">
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<div class="row"><span class="left">Start with: </span><span class="right"><b>12a + 12s = 10</b></span></div>
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<div class="row"><span class="left">Put "1/3" for a:</span><span class="right"><b>12<span class="hilite">(1/3)</span> + 12s = 10</b></span></div>
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<div class="row"><span class="left">Simplify:</span><span class="right"><b>4 + 12s = 10</b></span></div>
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<div class="row"><span class="left">Subtract 4 from both sides:</span><span class="right"><b>12s = 6</b></span></div>
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<div class="row"><span class="left">Divide both sides by 12:</span><span class="right"><b>s = 6/12</b></span></div>
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<div class="row"><span class="left">Simplify:</span><span class="right"><b>s = 1/2</b></span></div>
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</div>
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<p>Which means that Sam's rate is half a table a day (faster than Alex!)</p>
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<p>So 10 tables would take Sam just 20 days.</p>
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<p>Should Sam be paid more I wonder?</p>
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</div>
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<p>And another "substitution" example:</p>
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<div class="example">
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/track.jpg" width="100" height="137" alt="track" /></p>
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<h3>Example: Jenna is training hard to qualify for the National Games.<br>
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<br>
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She has a regular weekly routine, training for five hours a day on some days and 3 hours a day on the other days.<br>
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<br>
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She trains altogether 27 hours in a seven day week. On how many days does she train for five hours?</h3>
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<p>Letters:</p>
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<ul>
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<li>The number of "5 hour" days: d</li>
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<li>The number of "3 hour" days: e</li>
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</ul>
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<p>We know there are seven days in the week, so: <b>d + e = 7</b></p>
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<p>And she trains 27 hours in a week, with <b>d</b> 5 hour days and <b>e</b> 3 hour days: <b>5d + 3e = 27</b></p>
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<p class="larger">We are being asked for how many days she trains for 5 hours: d</p>
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<p> </p>
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<p class="large">Solve:</p>
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<div class="tbl">
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<div class="row"><span class="left"> </span><span class="right"><b>d + e = 7</b></span></div>
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<div class="row"><span class="left">So:</span><span class="right"><b> e = 7 − d</b></span></div>
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<div class="row"><span class="left">Put that in <b>5d + 3e = 27</b></span><span class="right"><b>5d + 3(7−d) = 27</b></span></div>
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<div class="row"><span class="left">Simplify:</span><span class="right"><b>5d + 21 − 3d = 27</b></span></div>
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<div class="row"><span class="left">Subtract 21 from both sides:</span><span class="right"><b>5d − 3d = 6</b></span></div>
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<div class="row"><span class="left">Simplify:</span><span class="right"><b>2d = 6</b></span></div>
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<div class="row"><span class="left">Divide both sides by 2:</span><span class="right"><b>d = 3</b></span></div>
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</div>
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<p>The number of "5 hour" days is 3</p>
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<p><strong>Check</strong>: She trains for 5 hours on 3 days a week, so she must train for 3 hours a day on the other 4 days of the week.</p>
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<p>3 × 5 hours = 15 hours, plus 4 × 3 hours = 12 hours gives a total of 27 hours</p>
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</div>
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<p>Some examples from Geometry:</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><span class="center"><img src="../geometry/images/circle-area-pie-are-squared.svg" alt="circle area: pie are squared" /></span></p>
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<h3>Example: A <a href="../geometry/circle.html">circle</a> has an area of 12 mm<sup>2</sup>, what is its radius?</h3>
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<p>Letters:</p>
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<ul>
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<li>Use <b>A</b> for Area: <b>A = 12 mm<sup>2</sup></b></li>
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<li>Use <b>r</b> for radius</li>
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</ul>
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<p>And the formula for Area is: <b>A = <span class="times">π</span> r<sup>2</sup></b></p>
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<p class="larger">We are being asked for the radius.</p>
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|
||
<p class="large">Solve:</p>
|
||
<p>We need to rearrange the formula to find the area</p>
|
||
<div class="tbl">
|
||
<div class="row"><span class="left">Start with:</span><span class="right"><b>A = <span class="times">π</span> r<sup>2</sup></b></span></div>
|
||
<div class="row"><span class="left">Swap sides:</span><span class="right"><b> <span class="times">π</span> r<sup>2</sup> = A</b></span></div>
|
||
<div class="row"><span class="left">Divide both sides by <b><span class="times">π</span></b>:</span><span class="right"><b> r<sup>2</sup> = A / <b><span class="times">π</span></b></b></span></div>
|
||
<div class="row"><span class="left">Take square root of both sides:</span><span class="right"><b>r = √(A / <b><span class="times">π)</span></b></b></span></div>
|
||
<div class="row"><span class="left">Now we can use the formula:</span><span class="right"><b>r = √(12 / <b><span class="times">π)</span></b></b></span></div>
|
||
<div class="row"><span class="left">And we get:</span><span class="right"><b>r = 1.954</b> (to 3 places)</span></div>
|
||
</div>
|
||
</div>
|
||
<div class="example">
|
||
<h3>Example: A <a href="../geometry/hexahedron.html">cube</a> has a volume of 125 mm<sup>3</sup>, what is its surface area?</h3>
|
||
<p>Make a quick sketch:</p>
|
||
<p class="center"><img src="images/cube-area-volume.gif" width="321" height="104" alt="cube area=6s^2, volume=s^3" /></p>
|
||
<p>Letters:</p>
|
||
<ul>
|
||
<li>Use <b>V</b> for Volume</li>
|
||
<li>Use <b>A</b> for Area</li>
|
||
<li>Use <b>s</b> for side length of cube</li>
|
||
</ul>
|
||
<p>Formulas:</p>
|
||
<ul>
|
||
<li>Volume of a cube: <b>V = s<sup>3</sup></b></li>
|
||
<li>Surface area of a cube: <b>A = 6s<sup>2</sup></b></li>
|
||
</ul>
|
||
<p class="larger">We are being asked for the surface area.</p>
|
||
|
||
<p class="large">Solve:</p>
|
||
<p>First work out <b>s</b> using the volume formula:</p>
|
||
<div class="tbl">
|
||
<div class="row"><span class="left">Start with:</span><span class="right"><b>V = s<sup>3</sup></b></span></div>
|
||
<div class="row"><span class="left">Swap sides:</span><span class="right"><b> s<sup>3</sup> = V</b></span></div>
|
||
<div class="row"><span class="left">Take cube root of both sides:</span><span class="right"><b>s = ∛(V<b><span class="times">)</span></b></b></span></div>
|
||
<div class="row"><span class="left">And we get:</span><span class="right"><b>s = ∛(125<b>) = 5</b></b></span></div>
|
||
</div>
|
||
<p>Now we can calculate surface area:</p>
|
||
<div class="tbl">
|
||
<div class="row"><span class="left">Start with:</span><span class="right"><b>A = 6s<sup>2</sup></b></span></div>
|
||
<div class="row"><span class="left">And we get:</span><span class="right"><b>A = 6(5)<sup>2</sup></b></span></div>
|
||
<div class="row"><span class="left"> </span><span class="right"><b>A = 6 × 25 = 150 mm<sup>2</sup></b></span></div>
|
||
</div>
|
||
</div>
|
||
<p>An example about Money:</p>
|
||
<div class="example">
|
||
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/pizza.jpg" width="150" height="86" alt="pizza" /></p>
|
||
<h3>Example: Joel works at the local pizza parlor. When he works overtime he earns 1¼ times the normal rate.<br>
|
||
<br>
|
||
One week Joel worked for 40 hours at the normal rate of pay and also worked 12 hours overtime. If Joel earned $660 altogether in that week, what is his normal rate of pay?</h3>
|
||
<p> </p>
|
||
<p>Letters:</p>
|
||
<ul>
|
||
<li>Joel's normal rate of pay: $N per hour</li>
|
||
</ul>
|
||
<p>Formulas:</p>
|
||
<ul>
|
||
<li>Joel works for 40 hours at $N per hour = $40N</li>
|
||
<li>When Joel does overtime he earns 1¼ times the normal rate = $1.25N per hour</li>
|
||
<li>Joel works for 12 hours at $1.25N per hour = $(12 × 1¼N) = $15N</li>
|
||
<li>And together he earned $660, so:</li>
|
||
</ul>
|
||
<p class="center"><b>$40N + $(12 × 1¼N) = $660</b></p>
|
||
<p class="larger">We are being asked for Joel's normal rate of pay $N.</p>
|
||
<p> </p>
|
||
<p class="large">Solve:</p>
|
||
<div class="tbl">
|
||
<div class="row"><span class="left">Start with<b></b></span><span class="right"><b>$40N + $(12 × 1¼N) = $660</b></span></div>
|
||
<div class="row"><span class="left">Simplify:</span><span class="right"><b>$40N + $15N = $660</b></span></div>
|
||
<div class="row"><span class="left">Simplify more:</span><span class="right"><b>$55N = $660</b></span></div>
|
||
<div class="row"><span class="left">Divide both sides by 55:</span><span class="right"><b>$N = $12</b></span></div>
|
||
</div>
|
||
<p>So Joel’s normal rate of pay is $12 per hour</p>
|
||
|
||
<p><strong>Check</strong></p>
|
||
<p>Joel’s normal rate of pay is $12 per hour, so his overtime rate is 1¼ × $12 per hour = $15 per hour. So his normal pay of 40 × $12 = $480, plus his overtime pay of 12 × $15 = $180 gives us a total of $660</p>
|
||
</div>
|
||
<p>More about Money, with these two examples involving Compound Interest</p>
|
||
<div class="example">
|
||
<h3>Example: Alex puts $2000 in the bank at an annual <a href="../money/compound-interest.html">compound interest</a> of 11%. How much will it be worth in 3 years?</h3>
|
||
<p>This is the compound interest formula:</p>
|
||
<p class="center"><img src="../money/images/fv-formula.svg" alt="PV(1+r)^n = FV" /></p>
|
||
<p>So we will use these letters:</p>
|
||
<ul>
|
||
<li>Present Value <b>PV = $2,000</b></li>
|
||
<li>Interest Rate (as a decimal): <b>r = 0.11</b></li>
|
||
<li>Number of Periods: <b>n = 3</b></li>
|
||
<li>Future Value (the value we want): <b>FV</b></li>
|
||
</ul>
|
||
<p class="larger">We are being asked for the Future Value: <b>FV</b></p>
|
||
<p> </p>
|
||
<p class="large">Solve:</p>
|
||
<div class="tbl">
|
||
<div class="row"><span class="left">Start with:</span><span class="right"><b>FV = PV × (1+r)<sup>n</sup></b></span></div>
|
||
<div class="row"><span class="left">Put in what we know:</span><span class="right"><b>FV = $2000 × (1+0.11)<sup>3</sup></b></span></div>
|
||
<div class="row"><span class="left">Calculate:</span><span class="right"><b>FV = $2000 × 1.367631</b></span></div>
|
||
<div class="row"><span class="left">Calculate:</span><span class="right"><b>FV = $2735.26</b> (to nearest cent)</span></div>
|
||
</div>
|
||
</div>
|
||
<div class="example">
|
||
<h3>Example: Roger deposited $1,000 into a savings account. The money earned interest compounded annually at the same rate. After nine years Roger's deposit has grown to $1,551.33<br>
|
||
<br>
|
||
What was the annual rate of interest for the savings account?</h3>
|
||
|
||
<p>The <a href="../money/compound-interest.html">compound interest</a> formula:</p>
|
||
<p class="center"><img src="../money/images/fv-formula.svg" alt="PV(1+r)^n = FV" /></p>
|
||
<p>With:</p>
|
||
<ul>
|
||
<li>Present Value PV = $1,000</li>
|
||
<li>Interest Rate (the value we want): r</li>
|
||
<li>Number of Periods: n = 9</li>
|
||
<li>Future Value: FV = $1,551.33</li>
|
||
</ul>
|
||
<p class="larger">We are being asked for the Interest Rate: r</p>
|
||
<p> </p>
|
||
<p class="large">Solve:</p>
|
||
<div class="tbl">
|
||
<div class="row"><span class="left">Start with:</span><span class="right"><b>FV = PV × (1+r)<sup>n</sup></b></span></div>
|
||
<div class="row"><span class="left">Put in what we know:</span><span class="right"><b>$1,551.33 = $1000 × (1+r)<sup>9</sup></b></span></div>
|
||
<div class="row"><span class="left">Swap sides:</span><span class="right"><b> $1000 × (1+r)<sup>9 </sup>= $1,551.33</b></span></div>
|
||
<div class="row"><span class="left">Divide both sides by 1000:</span><span class="right"><b> (1+r)<sup>9 </sup>= $1,551.33 / $1,000</b></span></div>
|
||
<div class="row"><span class="left">Simplify:</span><span class="right"><b> (1+r)<sup>9 </sup>= 1.55133</b></span></div>
|
||
<div class="row"><span class="left">9th root:</span><span class="right"><b> 1+r = 1.55133<sup>(1/9) </sup></b></span></div>
|
||
<div class="row"><span class="left">Calculate:</span><span class="right"><b> 1+r = 1.05</b></span></div>
|
||
<div class="row"><span class="left">Calculate:</span><span class="right"><b>r = 0.05 = 5%</b></span></div>
|
||
</div>
|
||
<p>So the annual rate of interest is 5%</p>
|
||
|
||
<p><strong>Check</strong>: $1,000 × (1.05)<sup>9</sup> = $1,000 × 1.55133 = $1,551.33</p>
|
||
</div>
|
||
<p>And an example of a <a href="../numbers/ratio.html">Ratio</a> question:</p>
|
||
|
||
<div class="example">
|
||
<h3>Example: At the start of the year the ratio of boys to girls in a class is 2 : 1<br>
|
||
<br>
|
||
But now, half a year later, four boys have left the class and there are two new girls. The ratio of boys to girls is now 4 : 3<br>
|
||
<br>
|
||
How many students are there altogether now?</h3>
|
||
|
||
<p>Letters:</p>
|
||
<ul>
|
||
<li>Number of boys now: <b>b</b></li>
|
||
<li>Number of girls now: <b>g</b></li>
|
||
</ul>
|
||
<p>The current ratio is 4 : 3</p>
|
||
<p class="center large"><span class="intbl"><em>b</em><strong>g</strong></span> = <span class="intbl"><em>4</em><strong>3</strong></span></p>
|
||
<p>Which can be rearranged to <b>3b = 4g</b></p>
|
||
<p>At the start of the year there was <b>(b + 4)</b> boys and <b>(g − 2)</b> girls, and the ratio was 2 : 1</p>
|
||
<p class="center large"><span class="intbl"><em>b + 4</em><strong>g − 2</strong></span> = <span class="intbl"><em>2</em><strong>1</strong></span></p>
|
||
<p>Which can be rearranged to <b>b + 4</b> = <b>2(g − 2)</b></p>
|
||
<p class="larger">We are being asked for how many students there are altogether now: <b>b + g</b></p>
|
||
|
||
|
||
<p class="large">Solve:</p>
|
||
<div class="tbl">
|
||
<div class="row"><span class="left">Start with:</span><span class="right"><b>b + 4</b> = <b>2(g − 2)</b></span></div>
|
||
<div class="row"><span class="left">Simplify:</span><span class="right"><b>b + 4 </b>= <b>2g − 4</b></span></div>
|
||
<div class="row"><span class="left">Subtract 4 from both sides:</span><span class="right"><b>b</b> = <b>2g − 8</b></span></div>
|
||
<div class="row"><span class="left">Multiply both sides by 3 (so we get 3b):</span><span class="right"><b>3b</b> = <b>6g − 24</b></span></div>
|
||
<div class="row"><span class="left"> Remember <b>3b = 4g</b>:</span><span class="right"><b class="hilite">4g</b> = <b>6g − 24</b></span></div>
|
||
<div class="row"><span class="left"> Subtract <b>6g</b> from both sides<b>:</b></span><span class="right"><b>−2g</b> = <b>− 24</b></span></div>
|
||
<div class="row"><span class="left"> Divide both sides by −2:</span><span class="right"><b>g</b> = <b>12</b></span></div>
|
||
</div>
|
||
<p>There are <b>12 girls</b>!</p>
|
||
<p>And <b>3b = 4g</b>, so <b>b = 4g/3 = 4 × 12 / 3 = 16</b>, so there are <b>16 boys</b></p>
|
||
<p>So there are now 12 girls and 16 boys in the class, making <b>28 students altogether</b>.</p>
|
||
|
||
<p><strong>Check</strong></p>
|
||
<p>There are now 16 boys and 12 girls, so the ratio of boys to girls is 16 : 12 = 4 : 3<br>
|
||
At the start of the year there were 20 boys and 10 girls, so the ratio was 20 : 10 = 2 : 1</p>
|
||
</div>
|
||
<p>And now for some <a href="quadratic-equation.html">Quadratic Equations</a>:</p>
|
||
<div class="example">
|
||
<h3>Example: The product of two consecutive even integers is 168. What are the integers?</h3>
|
||
<p><b>Consecutive</b> means one after the other. And they are <b>even</b>, so they could be 2 and 4, or 4 and 6, etc.</p>
|
||
<p>We will call the smaller integer <b>n</b>, and so the larger integer must be <b>n+2</b></p>
|
||
<p>And we are told the product (what we get after multiplying) is 168, so we know:</p>
|
||
<p class="center"><b>n(n + 2) = 168</b></p>
|
||
<p class="larger">We are being asked for the integers</p>
|
||
|
||
<p class="large">Solve:</p>
|
||
<div class="tbl">
|
||
<div class="row"><span class="left">Start with:</span><span class="right"><b>n(n + 2) = 168</b></span></div>
|
||
<div class="row"><span class="left">Expand:</span><span class="right"><b>n<sup>2</sup> + 2n = 168</b></span></div>
|
||
<div class="row"><span class="left">Subtract 168 from both sides:</span><span class="right"><b>n<sup>2</sup> + 2n − 168 = 0</b></span></div>
|
||
</div>
|
||
<p>That is a <a href="quadratic-equation.html">Quadratic Equation</a>, and there are many ways to solve it. Using the <a href="../quadratic-equation-solver.html">Quadratic Equation Solver</a> we get −14 and 12.</p>
|
||
<p><b>Check −14:</b> −14(−14 + 2) = (−14)×(−12) = 168 YES</p>
|
||
<p><b>Check 12:</b> 12(12 + 2) = 12×14 = 168 YES</p>
|
||
<p>So there are two solutions: −14 and −12 is one, 12 and 14 is the other.</p>
|
||
|
||
<p> </p>
|
||
<p>Note: we could have also tried "guess and check":</p>
|
||
<ul>
|
||
<li>We could try, say, n=10: <b>10(12) = 120 NO</b> (too small)</li>
|
||
<li>Next we could try n=12: <b>12(14) = 168 YES</b></li>
|
||
</ul>
|
||
<p>But unless we remember that <a href="../multiplying-negatives.html">multiplying two negatives make a positive</a> we might overlook the other solution of (−14)×(−12).</p>
|
||
</div>
|
||
<p>And:</p>
|
||
<div class="example">
|
||
<h3>Example: You are an Architect. Your client wants a room twice as long as it is wide. They also want a 3m wide veranda along the long side.<br>
|
||
<br>
|
||
Your client has 56 square meters of beautiful marble tiles to cover the whole area.<br>
|
||
<br>
|
||
What should the length of the room be?</h3>
|
||
<p>Let's first make a sketch so we get things right!:</p>
|
||
<p class="center"><img src="images/room-veranda.gif" width="199" height="168" alt="room veranda twice as long as wide" /></p>
|
||
|
||
<p>Letters:</p>
|
||
<ul>
|
||
<li>the length of the room: <b>L </b></li>
|
||
<li>the width of the room: <b>W</b></li>
|
||
<li>the total Area including veranda: <b>A</b></li>
|
||
</ul>
|
||
|
||
<p>We know:</p>
|
||
<ul>
|
||
<li>the width of the room is half its length: <b>W = ½L</b></li>
|
||
<li>the total area is the (room width + 3) times the length: <b>A = (W+3) × L = 56</b></li>
|
||
</ul>
|
||
<p class="larger">We are being asked for the length of the room: L</p>
|
||
|
||
|
||
<p class="large">Solve:</p>
|
||
<div class="tbl">
|
||
<div class="row"><span class="left">Start with:</span><span class="right"><b>(W + 3) × L = 56</b></span></div>
|
||
<div class="row"><span class="left">Substitute <b>W = ½L</b>:</span><span class="right"><b>(<span class="hilite">½L</span> + 3)</b><b> × L = 56</b></span></div>
|
||
<div class="row"><span class="left">Simplify:</span><span class="right"><b>½L<sup>2</sup> + 3L = 56</b></span></div>
|
||
<div class="row"><span class="left">Multiply all terms by 2:</span><span class="right"><b>L<sup>2</sup> + 6L = 112</b></span></div>
|
||
<div class="row"><span class="left"> Subtract <b>112</b> from both sides<b>:</b></span><span class="right"><b>L<sup>2</sup> + 6L − 112 = 0</b></span></div>
|
||
</div>
|
||
<p>This is a <strong>quadratic equation</strong>, there are many ways to solve it, this time let's use <a href="factoring-quadratics.html">factoring</a>:</p>
|
||
<div class="tbl">
|
||
<div class="row"><span class="left">Start with:</span><span class="right"><b>L<sup>2</sup> + 6L − 112 = 0</b></span></div>
|
||
<div class="row"><span class="left">Two numbers that multiply to give ac=−112,<br>
|
||
and
|
||
add to give b=6 are 14 and −8:</span><span class="right"><b>L<sup>2</sup> + 14L − 8L − 112 = 0</b></span></div>
|
||
<div class="row"><span class="left">Group:</span><span class="right"><b>L(L +14) − 8(L + 14) = 0</b></span></div>
|
||
<div class="row"><span class="left">Group:</span><span class="right"><b>(L − 8)(L + 14) = 0</b></span></div>
|
||
</div>
|
||
<p>And so L = 8 or −14</p>
|
||
<p>There are two solutions to the quadratic equation, but only one of them is possible since the length of the room cannot be negative!</p>
|
||
<p>So the length of the room is <b>8 m </b></p>
|
||
|
||
<p><strong>Check</strong></p>
|
||
<p>L = 8, so W = ½L = 4</p>
|
||
<p>So the area of the rectangle = (W+3) × L = 7 × 8 = 56</p>
|
||
</div>
|
||
<p>There we are ...</p>
|
||
<p>... I hope these examples will help you get the idea of how to handle word questions. Now how about some practice?</p>
|
||
<div class="questions">
|
||
<script>getQ(189, 213, 7183, 170, 179, 284, 7192, 7194, 7197, 7196);</script>
|
||
</div>
|
||
|
||
<div class="related">
|
||
<a href="mathematical-models.html">Mathematical Models</a>
|
||
<a href="inequality-questions-solving.html">Solving Inequality Word Questions</a>
|
||
<a href="index.html">Algebra Index</a>
|
||
<a href="../numbers/index.html">Number Index</a>
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||
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