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<h2 data-type="title">Translating Words to Algebraic Expressions</h2>
<p>In the previous section, we listed many operation symbols that are used in algebra, and then we translated expressions and equations into word phrases and sentences. Now we’ll reverse the process and translate word phrases into algebraic expressions. The symbols and variables we’ve talked about will help us do that. They are summarized in the table below.</p>
<table summary=".">
<thead>
<tr>
<th>Operation</th>
<th>Phrase</th>
<th>Expression</th>
</tr>
</thead>
<tbody>
<tr>
<td data-valign="top"><strong>Addition</strong></td>
<td>\(a\) plus \(b\)

<div> </div>

the sum of \(a\) and \(b\)

<div> </div>

\(a\) increased by \(b\)

<div> </div>

\(b\) more than \(a\)

<div> </div>

the total of \(a\) and \(b\)

<div> </div>

\(b\) added to \(a\)</td>
<td data-valign="top">\(a+b\)</td>
</tr>
<tr>
<td data-valign="top"><strong>Subtraction</strong></td>
<td>\(a\) minus \(b\)

<div> </div>

the difference of \(a\) and \(b\)

<div> </div>

\(b\) subtracted from \(a\)

<div> </div>

\(a\) decreased by \(b\)

<div> </div>

\(b\) less than \(a\)</td>
<td data-valign="top">\(a-b\)</td>
</tr>
<tr>
<td data-valign="top"><strong>Multiplication</strong></td>
<td>\(a\) times \(b\)

<div> </div>

the product of \(a\) and \(b\)</td>
<td data-valign="top">\(a\cdot b\), \(ab\), \(a\left(b\right)\), \(\left(a\right)\left(b\right)\)</td>
</tr>
<tr>
<td data-valign="top"><strong>Division</strong></td>
<td>\(a\) divided by \(b\)

<div> </div>

the quotient of \(a\) and \(b\)

<div> </div>

the ratio of \(a\) and \(b\)

<div> </div>

\(b\) divided into \(a\)</td>
<td data-valign="top">\(a÷b\), \(a/b\), \(\frac{a}{b}\), \(b\overline{)a}\)</td>
</tr>
</tbody>
</table>
<p>Look closely at these phrases using the four operations:</p>
<ul data-bullet-style="bullet">
<li>the sum <em>of</em> \(a\) <em>and</em> \(b\)</li>
<li>the difference <em>of</em> \(a\) <em>and</em> \(b\)</li>
<li>the product <em>of</em> \(a\) <em>and</em> \(b\)</li>
<li>the quotient <em>of</em> \(a\) <em>and</em> \(b\)</li>
</ul>
<p>Each phrase tells you to operate on two numbers. Look for the words <strong><em>of</em></strong> and <strong><em>and</em></strong> to find the numbers.</p>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<p>Translate each word phrase into an algebraic expression:</p>
<ol type="1">
<li>the difference of \(20\) and \(4\)</li>
<li>the quotient of \(10x\) and \(3\)</li>
</ol>
</div>
<div>
<h3 data-type="title">Solution</h3>
<p>The key word is <em>difference</em>, which tells us the operation is subtraction. Look for the words <em>of</em> and <em>and</em> to find the numbers to subtract.</p>
<p>\(\begin{array}{}\\ \text{the difference}\phantom{\rule{0.2em}{0ex}}\text{of}\phantom{\rule{0.2em}{0ex}}20\phantom{\rule{0.2em}{0ex}}and\phantom{\rule{0.2em}{0ex}}4\hfill \\ 20\phantom{\rule{0.2em}{0ex}}\text{minus}\phantom{\rule{0.2em}{0ex}}4\hfill \\ 20-4\hfill \end{array}\)</p>
<p>The key word is <em>quotient</em>, which tells us the operation is division.</p>
<p>\(\begin{array}{}\\ \text{the quotient of}\phantom{\rule{0.2em}{0ex}}10x\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}3\hfill \\ \text{divide}\phantom{\rule{0.2em}{0ex}}10x\phantom{\rule{0.2em}{0ex}}\text{by}\phantom{\rule{0.2em}{0ex}}3\hfill \\ 10x÷3\hfill \end{array}\)</p>
<p>This can also be written as \(\begin{array}{l}10x/3\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.4em}{0ex}}\frac{10x}{3}\hfill \end{array}\)</p>
</div>
</div>
</div>
</div>
<p>How old will you be in eight years? What age is eight more years than your age now? Did you add \(8\) to your present age? Eight <em>more than</em> means eight added to your present age.</p>
<p>How old were you seven years ago? This is seven years less than your age now. You subtract \(7\) from your present age. Seven <em>less than</em> means seven subtracted from your present age.</p>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<p>Translate each word phrase into an algebraic expression:</p>
<ol type="1">
<li>Eight more than \(y\)</li>
<li>Seven less than \(9z\)</li>
</ol>
</div>
<div>
<h3 data-type="title">Solution</h3>
<p>The key words are <em>more than</em>. They tell us the operation is addition. <em>More than</em> means “added to”.</p>
<p>\(\begin{array}{l}\text{Eight more than}\phantom{\rule{0.2em}{0ex}}y\\ \text{Eight added to}\phantom{\rule{0.2em}{0ex}}y\\ y+8\end{array}\)</p>
<p>The key words are <em>less than</em>. They tell us the operation is subtraction. <em>Less than</em> means “subtracted from”.</p>
<p>\(\begin{array}{l}\text{Seven less than}\phantom{\rule{0.2em}{0ex}}9z\\ \text{Seven subtracted from}\phantom{\rule{0.2em}{0ex}}9z\\ 9z-7\end{array}\)</p>
</div>
</div>
</div>
</div>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<p>Translate each word phrase into an algebraic expression:</p>
<ol type="1">
<li>five times the sum of \(m\) and \(n\)</li>
<li>the sum of five times \(m\) and \(n\)</li>
</ol>
</div>
<div>
<h3 data-type="title">Solution</h3>
<p>There are two operation words: <em>times</em> tells us to multiply and <em>sum</em> tells us to add. Because we are multiplying \(5\) times the sum, we need parentheses around the sum of \(m\) and \(n.\)</p>
<p>five times the sum of \(m\) and \(n\)</p>

\(\begin{array}{}\\ \phantom{\rule{4em}{0ex}}5\left(m+n\right)\hfill \end{array}\)

<p>To take a sum, we look for the words <em>of</em> and <em>and</em> to see what is being added. Here we are taking the sum <em>of</em> five times \(m\) and \(n.\)</p>
<p>the sum of five times \(m\) and \(n\)</p>

\(\begin{array}{}\\ \phantom{\rule{4em}{0ex}}5m+n\hfill \end{array}\)

<p>Notice how the use of parentheses changes the result. In part , we add first and in part , we multiply first.</p>
</div>
</div>
</div>
</div>
<p>Later in this course, we’ll apply our skills in algebra to solving equations. We’ll usually start by translating a word phrase to an . We’ll need to be clear about what the expression will represent. We’ll see how to do this in the next two examples.</p>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<p>The height of a rectangular window is \(6\) inches less than the width. Let \(w\) represent the width of the window. Write an expression for the height of the window.</p>
</div>
<div>
<h3 data-type="title">Solution</h3>
<table data-label="" summary=".">
<tbody>
<tr>
<td>Write a phrase about the height.</td>
<td>\(6\) less than the width</td>
</tr>
<tr>
<td>Substitute \(w\) for the width.</td>
<td>\(6\) less than \(w\)</td>
</tr>
<tr>
<td>Rewrite 'less than' as 'subtracted from'.</td>
<td>\(6\) subtracted from \(w\)</td>
</tr>
<tr>
<td>Translate the phrase into algebra.</td>
<td>\(w-6\)</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</div>
<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
<div>
<p>Blanca has dimes and quarters in her purse. The number of dimes is \(2\) less than \(5\) times the number of quarters. Let \(q\) represent the number of quarters. Write an expression for the number of dimes.</p>
</div>
<div>
<h3 data-type="title">Solution</h3>
<table data-label="" summary=".">
<tbody>
<tr>
<td>Write a phrase about the number of dimes.</td>
<td>two less than five times the number of quarters</td>
</tr>
<tr>
<td>Substitute \(q\) for the number of quarters.</td>
<td>\(2\) less than five times \(q\)</td>
</tr>
<tr>
<td>Translate \(5\) <em>times</em> \(q\).</td>
<td>\(2\) less than \(5q\)</td>
</tr>
<tr>
<td>Translate the phrase into algebra.</td>
<td>\(5q-2\)</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</div>
<h2 data-type="title">Optional Video: Algebraic Expression Vocabulary</h2>
<p><iframe allowfullscreen="allowfullscreen" frameborder="0" height="360" src="https://www.youtube.com/embed/E6lK-by__rA?rel=0&amp;modestbranding=1&amp;cc_lang_pref=en&amp;cc_load_policy=1" width="640"><span class="mce_SELRES_start" data-mce-type="bookmark" style="display: inline-block; width: 0px; overflow: hidden; line-height: 0;">﻿</span></iframe></p>

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Translating Words to Algebraic Expressions

Mathematics studies measurement, relationships, and properties of quantities and sets, using numbers and symbols. Arithmetic, algebra, geometry, and calculus are popular topics in mathematics.

Mathematics

Learn to identify and simplify algebraic expressions, evaluate expressions, and translate word problems into algebraic equations in this comprehensive tutorial on the language of algebra.


Translating Words to Algebraic Expressions

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