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<!DOCTYPE html>
<html>
<head>
<meta charset="UTF-8">
<link href="shared/bookhub.css" rel="stylesheet" type="text/css">
<title>Essential Skills 2</title>
</head>
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<div id="book-content">
<div class="section" id="averill_1.0-ch03_s07" condition="start-of-chunk" version="5.0" lang="en">
<h2 class="title editable block">
<span class="title-prefix">3.7</span> Essential Skills 2</h2>
<div class="learning_objectives editable block" id="averill_1.0-ch03_s07_n01">
<h3 class="title">Topics</h3>
<ul class="itemizedlist" id="averill_1.0-ch03_s07_l01">
<li>Proportions</li>
<li>Percentages</li>
<li>Unit Conversions</li>
</ul>
</div>
<p class="para editable block" id="averill_1.0-ch03_s07_p01">In Essential Skills 1 in <a class="xref" href="averill_1.0-ch01#averill_1.0-ch01">Chapter 1 "Introduction to Chemistry"</a>, <a class="xref" href="averill_1.0-ch01_s09#averill_1.0-ch01_s09">Section 1.9 "Essential Skills 1"</a>, we introduced you to some of the fundamental mathematical operations you need to successfully manipulate mathematical equations in chemistry. Before proceeding to the problems in <a class="xref" href="averill_1.0-ch03#averill_1.0-ch03">Chapter 3 "Chemical Reactions"</a>, you should become familiar with the additional skills described in this section on proportions, percentages, and unit conversions.</p>
<div class="section" id="averill_1.0-ch03_s07_s01">
<h2 class="title editable block">Proportions</h2>
<p class="para editable block" id="averill_1.0-ch03_s07_s01_p01">We can solve many problems in general chemistry by using ratios, or proportions. For example, if the ratio of some quantity <em class="emphasis">A</em> to some quantity <em class="emphasis">B</em> is known, and the relationship between these quantities is known to be constant, then any change in <em class="emphasis">A</em> (from <em class="emphasis">A</em><sub class="subscript">1</sub> to <em class="emphasis">A</em><sub class="subscript">2</sub>) produces a proportional change in <em class="emphasis">B</em> (from <em class="emphasis">B</em><sub class="subscript">1</sub> to <em class="emphasis">B</em><sub class="subscript">2</sub>) and vice versa. The relationship between <em class="emphasis">A</em><sub class="subscript">1</sub>, <em class="emphasis">B</em><sub class="subscript">1</sub>, <em class="emphasis">A</em><sub class="subscript">2</sub>, and <em class="emphasis">B</em><sub class="subscript">2</sub> can be written as follows:</p>
<span class="informalequation block">
<math xml:id="averill_1.0-ch03_m089" display="block">
<semantics>
<mrow>
<mfrac>
<mrow>
<msub>
<mi>A</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>B</mi>
<mtext>1</mtext>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<msub>
<mi>A</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>B</mi>
<mn>2</mn>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mtext> constant</mtext>
</mrow>
</semantics>
</math>
</span>
<p class="para editable block" id="averill_1.0-ch03_s07_s01_p02">To solve this equation for <em class="emphasis">A</em><sub class="subscript">2</sub>, we multiply both sides of the equality by <em class="emphasis">B</em><sub class="subscript">2</sub>, thus canceling <em class="emphasis">B</em><sub class="subscript">2</sub> from the denominator:</p>
<span class="informalequation block">
<math xml:id="averill_1.0-ch03_m090" display="block">
<semantics>
<mtable>
<mtr>
<mtd>
<mrow>
<mo>(</mo>
<mrow>
<msub>
<mi>B</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>)</mo>
</mrow>
<mfrac>
<mrow>
<msub>
<mi>A</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>B</mi>
<mtext>1</mtext>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mo stretchy="false">(</mo>
<menclose notation="updiagonalstrike">
<mrow>
<msub>
<mi>B</mi>
<mn>2</mn>
</msub>
</mrow>
</menclose>
<mo stretchy="false">)</mo>
<mfrac>
<mrow>
<msub>
<mi>A</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<menclose notation="updiagonalstrike">
<mrow>
<msub>
<mi>B</mi>
<mn>2</mn>
</msub>
</mrow>
</menclose>
</mrow>
</mfrac>
</mtd>
</mtr>
<mtr>
<mtd>
<mfrac>
<mrow>
<msub>
<mi>B</mi>
<mn>2</mn>
</msub>
<msub>
<mi>A</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>B</mi>
<mn>1</mn>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<msub>
<mi>A</mi>
<mn>2</mn>
</msub>
</mtd>
</mtr>
</mtable>
</semantics>
</math>
</span>
<p class="para editable block" id="averill_1.0-ch03_s07_s01_p03">Similarly, we can solve for <em class="emphasis">B</em><sub class="subscript">2</sub> by multiplying both sides of the equality by 1/<em class="emphasis">A</em><sub class="subscript">2</sub>, thus canceling <em class="emphasis">A</em><sub class="subscript">2</sub> from the numerator:</p>
<span class="informalequation block">
<math xml:id="averill_1.0-ch03_m091" display="block">
<semantics>
<mtable>
<mtr>
<mtd>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msub>
<mi>A</mi>
<mn>2</mn>
</msub>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mfrac>
<mrow>
<msub>
<mi>A</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>B</mi>
<mtext>1</mtext>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<menclose notation="updiagonalstrike">
<mrow>
<msub>
<mi>A</mi>
<mn>2</mn>
</msub>
</mrow>
</menclose>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mfrac>
<mrow>
<menclose notation="updiagonalstrike">
<mrow>
<msub>
<mi>A</mi>
<mn>2</mn>
</msub>
</mrow>
</menclose>
</mrow>
<mrow>
<msub>
<mi>B</mi>
<mn>2</mn>
</msub>
</mrow>
</mfrac>
</mtd>
</mtr>
<mtr>
<mtd>
<mfrac>
<mrow>
<msub>
<mi>A</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>A</mi>
<mn>2</mn>
</msub>
<msub>
<mi>B</mi>
<mn>1</mn>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msub>
<mi>B</mi>
<mn>2</mn>
</msub>
</mrow>
</mfrac>
</mtd>
</mtr>
</mtable>
</semantics>
</math>
</span>
<p class="para editable block" id="averill_1.0-ch03_s07_s01_p04">If the values of <em class="emphasis">A</em><sub class="subscript">1</sub>, <em class="emphasis">A</em><sub class="subscript">2</sub>, and <em class="emphasis">B</em><sub class="subscript">1</sub> are known, then we can solve the left side of the equation and invert the answer to obtain <em class="emphasis">B</em><sub class="subscript">2</sub>:</p>
<span class="informalequation block">
<math xml:id="averill_1.0-ch03_m092" display="block">
<semantics>
<mtable>
<mtr>
<mtd>
<mfrac>
<mn>1</mn>
<mrow>
<msub>
<mi>B</mi>
<mn>2</mn>
</msub>
</mrow>
</mfrac>
<mo>=</mo>
<mtext> numerical value</mtext>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>B</mi>
<mn>2</mn>
</msub>
<mo>=</mo>
<mfrac>
<mtext>1</mtext>
<mrow>
<mtext>numerical value</mtext>
</mrow>
</mfrac>
</mtd>
</mtr>
</mtable>
</semantics>
</math>
</span>
<p class="para editable block" id="averill_1.0-ch03_s07_s01_p05">If the value of <em class="emphasis">A</em><sub class="subscript">1</sub>, <em class="emphasis">A</em><sub class="subscript">2</sub>, or <em class="emphasis">B</em><sub class="subscript">1</sub> is unknown, however, we can solve for <em class="emphasis">B</em><sub class="subscript">2</sub> by inverting both sides of the equality:</p>
<span class="informalequation block">
<math xml:id="averill_1.0-ch03_m093" display="block">
<semantics>
<mrow>
<msub>
<mi>B</mi>
<mn>2</mn>
</msub>
<mo>=</mo>
<mfrac>
<mrow>
<msub>
<mi>A</mi>
<mn>2</mn>
</msub>
<msub>
<mi>B</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>A</mi>
<mn>1</mn>
</msub>
</mrow>
</mfrac>
</mrow>
</semantics>
</math>
</span>
<p class="para editable block" id="averill_1.0-ch03_s07_s01_p06">When you manipulate equations, remember that <em class="emphasis">any operation carried out on one side of the equality must be carried out on the other</em>.</p>
<p class="para editable block" id="averill_1.0-ch03_s07_s01_p07">Skill Builder ES1 illustrates how to find the value of an unknown by using proportions.</p>
<div class="exercises block" id="averill_1.0-ch03_s07_s01_n01">
<h3 class="title">Skill Builder ES1</h3>
<p class="para" id="averill_1.0-ch03_s07_s01_p08">If 38.4 g of element A are needed to combine with 17.8 g of element B, then how many grams of element A are needed to combine with 52.3 g of element B?</p>
<p class="simpara">Solution</p>
<p class="para" id="averill_1.0-ch03_s07_s01_p09">We set up the proportions as follows:</p>
<span class="informalequation">
<math xml:id="averill_1.0-ch03_m094" display="block">
<semantics>
<mtable columnalign="left">
<mtr columnalign="left">
<mtd columnalign="left">
<msub>
<mi>A</mi>
<mn>1</mn>
</msub>
</mtd>
<mtd columnalign="left">
<mo>=</mo>
</mtd>
<mtd>
<mtext> 38</mtext>
<mtext>.4 g</mtext>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>B</mi>
<mn>1</mn>
</msub>
</mtd>
<mtd columnalign="left">
<mo>=</mo>
</mtd>
<mtd>
<mtext> 17</mtext>
<mtext>.8 g</mtext>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>A</mi>
<mn>2</mn>
</msub>
</mtd>
<mtd columnalign="left">
<mo>=</mo>
</mtd>
<mtd>
<mtext> ?</mtext>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>B</mi>
<mn>2</mn>
</msub>
</mtd>
<mtd columnalign="left">
<mo>=</mo>
</mtd>
<mtd>
<mtext> 52</mtext>
<mtext>.3 g</mtext>
</mtd>
</mtr>
<mtr>
<mtd>
<mfrac>
<mrow>
<msub>
<mi>A</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>B</mi>
<mn>1</mn>
</msub>
</mrow>
</mfrac>
</mtd>
<mtd columnalign="left">
<mo>=</mo>
</mtd>
<mtd>
<mfrac>
<mrow>
<msub>
<mi>A</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>B</mi>
<mn>2</mn>
</msub>
</mrow>
</mfrac>
</mtd>
</mtr>
<mtr>
<mtd>
<mfrac>
<mrow>
<mtext>38</mtext>
<mtext>.4 g</mtext>
</mrow>
<mrow>
<mtext>17</mtext>
<mtext>.8 g</mtext>
</mrow>
</mfrac>
</mtd>
<mtd columnalign="left">
<mo>=</mo>
</mtd>
<mtd>
<mfrac>
<mrow>
<msub>
<mi>A</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<mtext>52</mtext>
<mtext>.3 g</mtext>
</mrow>
</mfrac>
</mtd>
</mtr>
</mtable>
</semantics>
</math>
</span>
<p class="para" id="averill_1.0-ch03_s07_s01_p10">Multiplying both sides of the equation by 52.3 g gives</p>
<span class="informalequation">
<math xml:id="averill_1.0-ch03_m095" display="block">
<semantics>
<mtable columnalign="left">
<mtr columnalign="left">
<mtd columnalign="left">
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mn>38.4</mn>
<mtext> g)(52</mtext>
<mtext>.3 g)</mtext>
</mrow>
<mrow>
<mtext>17</mtext>
<mtext>.8 g</mtext>
</mrow>
</mfrac>
</mtd>
<mtd columnalign="left">
<mo>=</mo>
</mtd>
<mtd>
<mfrac>
<mrow>
<msub>
<mi>A</mi>
<mn>2</mn>
</msub>
<menclose notation="updiagonalstrike">
<mrow>
<mo stretchy="false">(</mo>
<mn>52.3</mn>
<mtext> g)</mtext>
</mrow>
</menclose>
</mrow>
<mrow>
<menclose notation="updiagonalstrike">
<mrow>
<mo stretchy="false">(</mo>
<mn>52.3</mn>
<mtext> g)</mtext>
</mrow>
</menclose>
</mrow>
</mfrac>
</mtd>
</mtr>
<mtr>
<mtd>
<mtext> </mtext>
<msub>
<mi>A</mi>
<mn>2</mn>
</msub>
</mtd>
<mtd columnalign="left">
<mo>=</mo>
</mtd>
<mtd>
<mtext> 113 g</mtext>
</mtd>
</mtr>
</mtable>
</semantics>
</math>
</span>
<p class="para" id="averill_1.0-ch03_s07_s01_p11">Notice that grams cancel to leave us with an answer that is in the correct units. <em class="emphasis">Always check to make sure that your answer has the correct units.</em></p>
</div>
<div class="exercises block" id="averill_1.0-ch03_s07_s01_n02">
<h3 class="title">Skill Builder ES2</h3>
<p class="para" id="averill_1.0-ch03_s07_s01_p12">Solve to find the indicated variable.</p>
<ol class="orderedlist" id="averill_1.0-ch03_s07_s01_l01" numeration="loweralpha">
<li>
<span class="inlineequation">
<math xml:id="averill_1.0-ch03_m096" display="inline">
<semantics>
<mrow>
<mfrac>
<mrow>
<mtext>16</mtext>
<mtext>.4 g</mtext>
</mrow>
<mrow>
<mtext>41</mtext>
<mtext>.2 g</mtext>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mi>x</mi>
<mrow>
<mn>18.</mn>
<mtext>3 g</mtext>
</mrow>
</mfrac>
</mrow>
</semantics>
</math>
</span>
</li>
<li>
<span class="inlineequation">
<math xml:id="averill_1.0-ch03_m097" display="inline">
<semantics>
<mrow>
<mfrac>
<mrow>
<mtext>2</mtext>
<mtext>.65 m</mtext>
</mrow>
<mrow>
<mtext>4</mtext>
<mtext>.02 m</mtext>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mtext>3</mtext>
<mtext>.28 m</mtext>
</mrow>
<mi>y</mi>
</mfrac>
</mrow>
</semantics>
</math>
</span>
</li>
<li>
<span class="inlineequation">
<math xml:id="averill_1.0-ch03_m098" display="inline">
<semantics>
<mrow>
<mfrac>
<mrow>
<mtext>3</mtext>
<mtext>.27</mtext>
<mo>×</mo>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</msup>
<mtext> g</mtext>
</mrow>
<mi>x</mi>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mtext>5</mtext>
<mtext>.0</mtext>
<mo>×</mo>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
<mtext> g</mtext>
</mrow>
<mrow>
<mn>3.2</mn>
<mtext> g</mtext>
</mrow>
</mfrac>
</mrow>
</semantics>
</math>
</span>
</li>
<li>Solve for <em class="emphasis">V</em><sub class="subscript">1</sub>: <span class="inlineequation"><math xml:id="averill_1.0-ch03_m099" display="inline"><semantics><mrow><mfrac><mrow><msub><mi>P</mi><mn>1</mn></msub></mrow><mrow><msub><mi>P</mi><mn>2</mn></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi>V</mi><mn>2</mn></msub></mrow><mrow><msub><mi>V</mi><mn>1</mn></msub></mrow></mfrac></mrow></semantics></math></span>
</li>
<li>Solve for <em class="emphasis">T</em><sub class="subscript">1</sub>: <span class="inlineequation"><math xml:id="averill_1.0-ch03_m100" display="inline"><semantics><mrow><mfrac><mrow><msub><mi>P</mi><mn>1</mn></msub><msub><mi>V</mi><mn>1</mn></msub></mrow><mrow><msub><mi>T</mi><mn>1</mn></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi>P</mi><mn>2</mn></msub><msub><mi>V</mi><mn>2</mn></msub></mrow><mrow><msub><mi>T</mi><mn>2</mn></msub></mrow></mfrac></mrow></semantics></math></span>
</li>
</ol>
<p class="simpara">Solution</p>
<ol class="orderedlist" id="averill_1.0-ch03_s07_s01_l02" numeration="loweralpha">
<li>
<p class="para">Multiply both sides of the equality by 18.3 g to remove this measurement from the denominator:</p>
<span class="informalequation">
<math xml:id="averill_1.0-ch03_m101" display="block">
<semantics>
<mtable columnalign="left">
<mtr>
<mtd>
<mtext>(18</mtext>
<mtext>.3 g)</mtext>
<mfrac>
<mrow>
<mtext>16</mtext>
<mtext>.4 </mtext>
<menclose notation="downdiagonalstrike">
<mtext>g</mtext>
</menclose>
</mrow>
<mrow>
<mtext>41</mtext>
<mtext>.2 </mtext>
<menclose notation="downdiagonalstrike">
<mtext>g</mtext>
</menclose>
</mrow>
</mfrac>
<mo>=</mo>
<menclose notation="downdiagonalstrike">
<mrow>
<mtext>(18</mtext>
<mtext>.3 g)</mtext>
</mrow>
</menclose>
<mfrac>
<mi>x</mi>
<mrow>
<menclose notation="downdiagonalstrike">
<mrow>
<mn>18.</mn>
<mtext>3 g</mtext>
</mrow>
</menclose>
</mrow>
</mfrac>
</mtd>
</mtr>
<mtr>
<mtd>
<mtext> 7</mtext>
<mtext>.28 g </mtext>
<mo>=</mo>
<mi>x</mi>
</mtd>
</mtr>
</mtable>
</semantics>
</math>
</span>
</li>
<li>
<p class="para">Multiply both sides of the equality by 1/3.28 m, solve the left side of the equation, and then invert to solve for <em class="emphasis">y</em>:</p>
<span class="informalequation">
<math xml:id="averill_1.0-ch03_m102" display="block">
<semantics>
<mtable>
<mtr>
<mtd>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mn>3.28</mn>
<mtext> m</mtext>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mfrac>
<mrow>
<mtext>2</mtext>
<mtext>.65 m</mtext>
</mrow>
<mrow>
<mtext>4</mtext>
<mtext>.02 m</mtext>
</mrow>
</mfrac>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<menclose notation="downdiagonalstrike">
<mrow>
<mn>3.</mn>
<mtext>28 m</mtext>
</mrow>
</menclose>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mfrac>
<mrow>
<menclose notation="downdiagonalstrike">
<mrow>
<mn>3.</mn>
<mtext>28 m</mtext>
</mrow>
</menclose>
</mrow>
<mi>y</mi>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>y</mi>
</mfrac>
</mtd>
</mtr>
<mtr>
<mtd>
<mtext> </mtext>
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mn>4.02</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mn>3.28</mn>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mn>2.65</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mn>4.98</mn>
<mtext> m</mtext>
</mtd>
</mtr>
</mtable>
</semantics>
</math>
</span>
</li>
<li>
<p class="para">Multiply both sides of the equality by 1/3.27 × 10<sup class="superscript">−3</sup> g, solve the right side of the equation, and then invert to find <em class="emphasis">x</em>:</p>
<span class="informalequation">
<math xml:id="averill_1.0-ch03_m103" display="block">
<semantics>
<mtable>
<mtr>
<mtd>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<menclose notation="downdiagonalstrike">
<mrow>
<mtext>3</mtext>
<mtext>.27</mtext>
<mo>×</mo>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</msup>
<mtext> g</mtext>
</mrow>
</menclose>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mfrac>
<mrow>
<menclose notation="downdiagonalstrike">
<mrow>
<mtext>3</mtext>
<mtext>.27</mtext>
<mo>×</mo>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</msup>
<mtext> g</mtext>
</mrow>
</menclose>
</mrow>
<mi>x</mi>
</mfrac>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mtext>3</mtext>
<mtext>.27</mtext>
<mo>×</mo>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</msup>
<mtext> g</mtext>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mfrac>
<mrow>
<mtext>5</mtext>
<mtext>.0</mtext>
<mo>×</mo>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
<mtext> </mtext>
<menclose notation="downdiagonalstrike">
<mtext>g</mtext>
</menclose>
</mrow>
<mrow>
<mtext>3</mtext>
<mtext>.2 </mtext>
<menclose notation="downdiagonalstrike">
<mtext>g</mtext>
</menclose>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>x</mi>
</mfrac>
</mtd>
</mtr>
<mtr>
<mtd>
<mtext> </mtext>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mn>3.2</mn>
<mtext> g)(3</mtext>
<mtext>.27</mtext>
<mo>×</mo>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</msup>
<mtext> g)</mtext>
</mrow>
<mrow>
<mtext>5</mtext>
<mtext>.0</mtext>
<mo>×</mo>