{"id":14872,"date":"2018-06-15T20:38:47","date_gmt":"2018-06-15T20:38:47","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/chapter\/mathematical-treatment-of-measurement-results\/"},"modified":"2022-01-11T03:31:00","modified_gmt":"2022-01-11T03:31:00","slug":"mathematical-treatment-of-measurement-results","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/mathematical-treatment-of-measurement-results\/","title":{"raw":"Mathematical Treatment of Measurement Results","rendered":"Mathematical Treatment of Measurement Results"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nBy the end of this section, you will be able to:\r\n<ul>\r\n \t<li>Explain the dimensional analysis (factor label) approach to mathematical calculations involving quantities<\/li>\r\n \t<li>Use dimensional analysis to carry out unit conversions for a given property and computations involving two or more properties<\/li>\r\n<\/ul>\r\n<\/div>\r\nIt is often the case that a quantity of interest may not be easy (or even possible) to measure directly but instead must be calculated from other directly measured properties and appropriate mathematical relationships. For example, consider measuring the average speed of an athlete running sprints. This is typically accomplished by measuring the <em>time<\/em> required for the athlete to run from the starting line to the finish line, and the <em>distance<\/em> between these two lines, and then computing <em>speed<\/em> from the equation that relates these three properties:\r\n<p style=\"text-align: center;\">[latex]\\text{speed}=\\frac{\\text{distance}}{\\text{time}}[\/latex]<\/p>\r\nAn Olympic-quality sprinter can run 100 m in approximately 10 s, corresponding to an average speed of [latex]\\frac{\\text{100 m}}{\\text{10 s}}=\\text{10 m\/s}[\/latex].\r\n\r\nNote that this simple arithmetic involves dividing the numbers of each measured quantity to yield the number of the computed quantity (100\/10 = 10) <em>and likewise<\/em> dividing the units of each measured quantity to yield the unit of the computed quantity (m\/s = m\/s). Now, consider using this same relation to predict the time required for a person running at this speed to travel a distance of 25 m. The same relation between the three properties is used, but in this case, the two quantities provided are a speed (10 m\/s) and a distance (25 m). To yield the sought property, time, the equation must be rearranged appropriately:\r\n<p style=\"text-align: center;\">[latex]\\text{time}=\\frac{\\text{distance}}{\\text{speed}}[\/latex]<\/p>\r\nThe time can then be computed as [latex]\\frac{\\text{25 m}}{\\text{10 m\/s}}=\\text{2.5 s}[\/latex].\r\n\r\nAgain, arithmetic on the numbers (25\/10 = 2.5) was accompanied by the same arithmetic on the units (m\/m\/s = s) to yield the number and unit of the result, 2.5 s. Note that, just as for numbers, when a unit is divided by an identical unit (in this case, m\/m), the result is \u201c1\u201d\u2014or, as commonly phrased, the units \u201ccancel.\u201d\r\n\r\nThese calculations are examples of a versatile mathematical approach known as <strong>dimensional analysis <\/strong>(or the <strong>factor-label method<\/strong>). Dimensional analysis is based on this premise: <em>the units of quantities must be subjected to the same mathematical operations as their associated numbers<\/em>. This method can be applied to computations ranging from simple unit conversions to more complex, multi-step calculations involving several different quantities.\r\n<h2>Conversion Factors and Dimensional Analysis<\/h2>\r\nA ratio of two equivalent quantities expressed with different measurement units can be used as a <strong>unit conversion factor<\/strong>. For example, the lengths of 2.54 cm and 1 in. are equivalent (by definition), and so a unit conversion factor may be derived from the ratio,\r\n<p style=\"text-align: center;\">[latex]\\frac{\\text{2.54 cm}}{\\text{1 in.}}\\text{(2.54 cm}=\\text{1 in.) or 2.54}\\frac{\\text{cm}}{\\text{in.}}[\/latex]<\/p>\r\nSeveral other commonly used conversion factors are given in Table 1.\r\n<table id=\"fs-idm222237232\" class=\"span-all\" summary=\"This table is divided into 3 columns. They are titled length, volume, and mass. The following units are under the length column: 1 meter is equal to 1.0936 yards, 1 inch is equal to 2.54 cm 1 kilometer is equal to 0.62137 miles, 1 mile is equal to 1609.3 meters. The following units are under the volume column: 1 liter is equal to 1.0567 quarts, 1 quart is equal to 0.94635 meters, one cubic foot is equal to 28.317 liters, 1 tablespoon is equal to 14.787 milliliters. The following units are under the mass column: 1 kilogram is equal to 2.2046 pounds, 1 pound is equal to 453.59 grams, 1 avoirdupois ounce is equal to 28.349 grams, 1 troy ounce is equal to 31.103 grams.\">\r\n<thead>\r\n<tr>\r\n<th colspan=\"3\">Table 1. Common Conversion Factors<\/th>\r\n<\/tr>\r\n<tr>\r\n<th>Length<\/th>\r\n<th>Volume<\/th>\r\n<th>Mass<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>1 m = 1.0936 yd<\/td>\r\n<td>1 L = 1.0567 qt<\/td>\r\n<td>1 kg = 2.2046 lb<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1 in. = 2.54 cm (exact)<\/td>\r\n<td>1 qt = 0.94635 L<\/td>\r\n<td>1 lb = 453.59 g<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1 km = 0.62137 mi<\/td>\r\n<td>1 ft<sup>3<\/sup> = 28.317 L<\/td>\r\n<td>1 (avoirdupois) oz = 28.349 g<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1 mi = 1609.3 m<\/td>\r\n<td>1 tbsp = 14.787 mL<\/td>\r\n<td>1 (troy) oz = 31.103 g<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWhen we multiply a quantity (such as distance given in inches) by an appropriate unit conversion factor, we convert the quantity to an equivalent value with different units (such as distance in centimeters). For example, a basketball player\u2019s vertical jump of 34 inches can be converted to centimeters by:\r\n<p style=\"text-align: center;\">[latex]\\text{34 in.}\\times \\frac{\\text{2.54 cm}}{1\\cancel{\\text{in.}}}=\\text{86 cm}[\/latex]<\/p>\r\nSince this simple arithmetic involves <em>quantities<\/em>, the premise of dimensional analysis requires that we multiply both <em>numbers and units<\/em>. The numbers of these two quantities are multiplied to yield the number of the product quantity, 86, whereas the units are multiplied to yield [latex]\\frac{\\text{in.}\\times \\text{cm}}{\\text{in.}}[\/latex] . Just as for numbers, a ratio of identical units is also numerically equal to one, [latex]\\frac{\\text{in.}}{\\text{in.}}=\\text{1,}[\/latex] and the unit product thus simplifies to <em>cm<\/em>. (When identical units divide to yield a factor of 1, they are said to \u201ccancel.\u201d) Using dimensional analysis, we can determine that a unit conversion factor has been set up correctly by checking to confirm that the original unit will cancel, and the result will contain the sought (converted) unit.\r\n<div class=\"textbox examples\">\r\n<h3>Example 1: <strong>Using a Unit Conversion Factor<\/strong><\/h3>\r\nThe mass of a competition frisbee is 125 g. Convert its mass to ounces using the unit conversion factor derived from the relationship 1 oz = 28.349 g (Table 1).\r\n\r\n[reveal-answer q=\"798792\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"798792\"]\r\n\r\nIf we have the conversion factor, we can determine the mass in kilograms using an equation similar the one used for converting length from inches to centimeters.\r\n<p style=\"text-align: center;\">[latex]x\\text{ oz}=\\text{125 g}\\times \\text{unit conversion factor}[\/latex]<\/p>\r\nWe write the unit conversion factor in its two forms:\r\n<p style=\"text-align: center;\">[latex]\\frac{\\text{1 oz}}{\\text{28.35 g}}\\text{ and }\\frac{\\text{28.349 g}}{\\text{1 oz}}[\/latex]<\/p>\r\nThe correct unit conversion factor is the ratio that cancels the units of grams and leaves ounces.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccc}x\\text{ oz}\\hfill &amp; \\text{=}\\hfill &amp; 125\\cancel{\\text{g}}\\times \\frac{\\text{1 oz}}{\\text{28.349}\\cancel{\\text{g}}}\\hfill \\\\ \\hfill &amp; =\\hfill &amp; \\left(\\frac{125}{\\text{28.349}}\\right)\\text{oz}\\hfill \\\\ \\hfill &amp; =\\hfill &amp; \\text{4.41 oz (three significant figures)}\\hfill \\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n<h4><strong>Check Your Learning<\/strong><\/h4>\r\nConvert a volume of 9.345 qt to liters.\r\n\r\n[reveal-answer q=\"798791\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"798791\"]8.844 L[\/hidden-answer]\r\n\r\n<\/div>\r\nBeyond simple unit conversions, the factor-label method can be used to solve more complex problems involving computations. Regardless of the details, the basic approach is the same\u2014all the <em>factors<\/em> involved in the calculation must be appropriately oriented to insure that their <em>labels<\/em> (units) will appropriately cancel and\/or combine to yield the desired unit in the result. This is why it is referred to as the factor-label method. As your study of chemistry continues, you will encounter many opportunities to apply this approach.\r\n<div class=\"textbox examples\">\r\n<h3>Example 2: <strong>Computing Quantities from Measurement Results and Known Mathematical Relations<\/strong><\/h3>\r\nWhat is the density of common antifreeze in units of g\/mL? A 4.00-qt sample of the antifreeze weighs 9.26 lb.\r\n\r\n[reveal-answer q=\"944637\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"944637\"]\r\n\r\nSince [latex]\\text{density}=\\frac{\\text{mass}}{\\text{volume}}[\/latex] , we need to divide the mass in grams by the volume in milliliters. In general: the number of units of B = the number of units of A \u00d7 unit conversion factor. The necessary conversion factors are given in Table 1.6: 1 lb = 453.59 g; 1 L = 1.0567 qt; 1 L = 1,000 mL. We can convert mass from pounds to grams in one step:\r\n<p style=\"text-align: center;\">[latex]\\text{9.26}\\cancel{\\text{lb}}\\times \\frac{\\text{453.59 g}}{1\\cancel{\\text{lb}}}=4.20\\times {10}^{3}\\text{g}[\/latex]<\/p>\r\nWe need to use two steps to convert volume from quarts to milliliters.\r\n<ol>\r\n \t<li><strong>Convert quarts to liters: <\/strong>[latex]\\text{4.00}\\cancel{\\text{qt}}\\times \\frac{\\text{1 L}}{\\text{1.0567}\\cancel{\\text{qt}}}=\\text{3.78 L}[\/latex]<\/li>\r\n \t<li><strong>Convert liters to milliliters: <\/strong>[latex]\\text{3.78}\\cancel{\\text{L}}\\times \\frac{\\text{1000 mL}}{1\\cancel{\\text{L}}}=3.78\\times {10}^{3}\\text{mL}[\/latex]<\/li>\r\n<\/ol>\r\nThen, [latex]\\text{density}=\\frac{\\text{4.20}\\times {10}^{3}\\text{g}}{3.78\\times {10}^{3}\\text{mL}}=\\text{1.11 g\/mL}[\/latex].\r\n\r\nAlternatively, the calculation could be set up in a way that uses three unit conversion factors sequentially as follows:\r\n<p style=\"text-align: center;\">[latex]\\frac{\\text{9.26}\\cancel{\\text{lb}}}{\\text{4.00}\\cancel{\\text{qt}}}\\times \\frac{\\text{453.59 g}}{1\\cancel{\\text{lb}}}\\times \\frac{\\text{1.0567}\\cancel{\\text{qt}}}{1\\cancel{\\text{L}}}\\times \\frac{1\\cancel{\\text{L}}}{\\text{1000 mL}}=\\text{1.11 g\/mL}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n<h4><strong>Check Your Learning<\/strong><\/h4>\r\nWhat is the volume in liters of 1.000 oz, given that 1 L = 1.0567 qt and 1 qt = 32 oz (exactly)?\r\n\r\n[reveal-answer q=\"99428\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"99428\"][latex]2.956\\times {10}^{-2}\\text{L}[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 3: <strong>Computing Quantities from Measurement Results and Known Mathematical Relations<\/strong><\/h3>\r\nWhile being driven from Philadelphia to Atlanta, a distance of about 1250 km, a 2014 Lamborghini Aventador Roadster uses 213 L gasoline.\r\n<ol>\r\n \t<li>What (average) fuel economy, in miles per gallon, did the Roadster get during this trip?<\/li>\r\n \t<li>If gasoline costs $3.80 per gallon, what was the fuel cost for this trip?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"520673\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"520673\"]\r\n<h4>Part 1<\/h4>\r\nWe first convert distance from kilometers to miles: [latex]\\text{1250 km}\\times \\frac{\\text{0.62137 mi}}{\\text{1 km}}=\\text{777 mi}[\/latex]\r\n\r\nThen we convert volume from liters to gallons: [latex]213\\cancel{\\text{L}}\\times \\frac{\\text{1.0567}\\cancel{\\text{qt}}}{1\\cancel{\\text{L}}}\\times \\frac{\\text{1 gal}}{4\\cancel{\\text{qt}}}=\\text{56.3 gal}[\/latex]\r\n\r\nThen,\r\n<p style=\"text-align: center;\">[latex]\\text{(average) mileage}=\\frac{\\text{777 mi}}{\\text{56.3 gal}}=\\text{13.8 miles\/gallon}=\\text{13.8 mpg}[\/latex]<\/p>\r\nAlternatively, the calculation could be set up in a way that uses all the conversion factors sequentially, as follows:\r\n<p style=\"text-align: center;\">[latex]\\frac{1250\\cancel{\\text{km}}}{213\\cancel{\\text{L}}}\\times \\frac{\\text{0.62137 mi}}{1\\cancel{\\text{km}}}\\times \\frac{1\\cancel{\\text{L}}}{\\text{1.0567}\\cancel{\\text{qt}}}\\times \\frac{4\\cancel{\\text{qt}}}{\\text{1 gal}}=\\text{13.8 mpg}[\/latex]<\/p>\r\n\r\n<h4>Part 2<\/h4>\r\nUsing the previously calculated volume in gallons, we find: [latex]56.3\\text{ gal}\\times\\frac{\\$3.80}{1\\text{ gal}}=\\$214[\/latex]\r\n\r\n[\/hidden-answer]\r\n<h4><strong>Check Your Learning<\/strong><\/h4>\r\nA Toyota Prius Hybrid uses 59.7 L gasoline to drive from San Francisco to Seattle, a distance of 1300 km (two significant digits).\r\n<ol>\r\n \t<li>What (average) fuel economy, in miles per gallon, did the Prius get during this trip?<\/li>\r\n \t<li>If gasoline costs $3.90 per gallon, what was the fuel cost for this trip?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"676222\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"676222\"]\r\n<ol>\r\n \t<li>51 mpg<\/li>\r\n \t<li>$62<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Atomic Bombs and Dimensional Analysis<\/h3>\r\nHow did a magazine photograph help a physicist crack one of the secrets of the atom bomb?\r\n\r\nhttps:\/\/www.youtube.com\/watch?t=50&amp;v=_gaCAFcW6OY\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\"><\/div>\r\n<h2>Glossary<\/h2>\r\n<strong>dimensional analysis:<\/strong> (also, factor-label method) versatile mathematical approach that can be applied to computations ranging from simple unit conversions to more complex, multi-step calculations involving several different quantities\r\n\r\n<strong>unit conversion factor:<\/strong> ratio of equivalent quantities expressed with different units; used to convert from one unit to a different unit","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to:<\/p>\n<ul>\n<li>Explain the dimensional analysis (factor label) approach to mathematical calculations involving quantities<\/li>\n<li>Use dimensional analysis to carry out unit conversions for a given property and computations involving two or more properties<\/li>\n<\/ul>\n<\/div>\n<p>It is often the case that a quantity of interest may not be easy (or even possible) to measure directly but instead must be calculated from other directly measured properties and appropriate mathematical relationships. For example, consider measuring the average speed of an athlete running sprints. This is typically accomplished by measuring the <em>time<\/em> required for the athlete to run from the starting line to the finish line, and the <em>distance<\/em> between these two lines, and then computing <em>speed<\/em> from the equation that relates these three properties:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{speed}=\\frac{\\text{distance}}{\\text{time}}[\/latex]<\/p>\n<p>An Olympic-quality sprinter can run 100 m in approximately 10 s, corresponding to an average speed of [latex]\\frac{\\text{100 m}}{\\text{10 s}}=\\text{10 m\/s}[\/latex].<\/p>\n<p>Note that this simple arithmetic involves dividing the numbers of each measured quantity to yield the number of the computed quantity (100\/10 = 10) <em>and likewise<\/em> dividing the units of each measured quantity to yield the unit of the computed quantity (m\/s = m\/s). Now, consider using this same relation to predict the time required for a person running at this speed to travel a distance of 25 m. The same relation between the three properties is used, but in this case, the two quantities provided are a speed (10 m\/s) and a distance (25 m). To yield the sought property, time, the equation must be rearranged appropriately:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{time}=\\frac{\\text{distance}}{\\text{speed}}[\/latex]<\/p>\n<p>The time can then be computed as [latex]\\frac{\\text{25 m}}{\\text{10 m\/s}}=\\text{2.5 s}[\/latex].<\/p>\n<p>Again, arithmetic on the numbers (25\/10 = 2.5) was accompanied by the same arithmetic on the units (m\/m\/s = s) to yield the number and unit of the result, 2.5 s. Note that, just as for numbers, when a unit is divided by an identical unit (in this case, m\/m), the result is \u201c1\u201d\u2014or, as commonly phrased, the units \u201ccancel.\u201d<\/p>\n<p>These calculations are examples of a versatile mathematical approach known as <strong>dimensional analysis <\/strong>(or the <strong>factor-label method<\/strong>). Dimensional analysis is based on this premise: <em>the units of quantities must be subjected to the same mathematical operations as their associated numbers<\/em>. This method can be applied to computations ranging from simple unit conversions to more complex, multi-step calculations involving several different quantities.<\/p>\n<h2>Conversion Factors and Dimensional Analysis<\/h2>\n<p>A ratio of two equivalent quantities expressed with different measurement units can be used as a <strong>unit conversion factor<\/strong>. For example, the lengths of 2.54 cm and 1 in. are equivalent (by definition), and so a unit conversion factor may be derived from the ratio,<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{\\text{2.54 cm}}{\\text{1 in.}}\\text{(2.54 cm}=\\text{1 in.) or 2.54}\\frac{\\text{cm}}{\\text{in.}}[\/latex]<\/p>\n<p>Several other commonly used conversion factors are given in Table 1.<\/p>\n<table id=\"fs-idm222237232\" class=\"span-all\" summary=\"This table is divided into 3 columns. They are titled length, volume, and mass. The following units are under the length column: 1 meter is equal to 1.0936 yards, 1 inch is equal to 2.54 cm 1 kilometer is equal to 0.62137 miles, 1 mile is equal to 1609.3 meters. The following units are under the volume column: 1 liter is equal to 1.0567 quarts, 1 quart is equal to 0.94635 meters, one cubic foot is equal to 28.317 liters, 1 tablespoon is equal to 14.787 milliliters. The following units are under the mass column: 1 kilogram is equal to 2.2046 pounds, 1 pound is equal to 453.59 grams, 1 avoirdupois ounce is equal to 28.349 grams, 1 troy ounce is equal to 31.103 grams.\">\n<thead>\n<tr>\n<th colspan=\"3\">Table 1. Common Conversion Factors<\/th>\n<\/tr>\n<tr>\n<th>Length<\/th>\n<th>Volume<\/th>\n<th>Mass<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1 m = 1.0936 yd<\/td>\n<td>1 L = 1.0567 qt<\/td>\n<td>1 kg = 2.2046 lb<\/td>\n<\/tr>\n<tr>\n<td>1 in. = 2.54 cm (exact)<\/td>\n<td>1 qt = 0.94635 L<\/td>\n<td>1 lb = 453.59 g<\/td>\n<\/tr>\n<tr>\n<td>1 km = 0.62137 mi<\/td>\n<td>1 ft<sup>3<\/sup> = 28.317 L<\/td>\n<td>1 (avoirdupois) oz = 28.349 g<\/td>\n<\/tr>\n<tr>\n<td>1 mi = 1609.3 m<\/td>\n<td>1 tbsp = 14.787 mL<\/td>\n<td>1 (troy) oz = 31.103 g<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>When we multiply a quantity (such as distance given in inches) by an appropriate unit conversion factor, we convert the quantity to an equivalent value with different units (such as distance in centimeters). For example, a basketball player\u2019s vertical jump of 34 inches can be converted to centimeters by:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{34 in.}\\times \\frac{\\text{2.54 cm}}{1\\cancel{\\text{in.}}}=\\text{86 cm}[\/latex]<\/p>\n<p>Since this simple arithmetic involves <em>quantities<\/em>, the premise of dimensional analysis requires that we multiply both <em>numbers and units<\/em>. The numbers of these two quantities are multiplied to yield the number of the product quantity, 86, whereas the units are multiplied to yield [latex]\\frac{\\text{in.}\\times \\text{cm}}{\\text{in.}}[\/latex] . Just as for numbers, a ratio of identical units is also numerically equal to one, [latex]\\frac{\\text{in.}}{\\text{in.}}=\\text{1,}[\/latex] and the unit product thus simplifies to <em>cm<\/em>. (When identical units divide to yield a factor of 1, they are said to \u201ccancel.\u201d) Using dimensional analysis, we can determine that a unit conversion factor has been set up correctly by checking to confirm that the original unit will cancel, and the result will contain the sought (converted) unit.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 1: <strong>Using a Unit Conversion Factor<\/strong><\/h3>\n<p>The mass of a competition frisbee is 125 g. Convert its mass to ounces using the unit conversion factor derived from the relationship 1 oz = 28.349 g (Table 1).<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q798792\">Show Answer<\/span><\/p>\n<div id=\"q798792\" class=\"hidden-answer\" style=\"display: none\">\n<p>If we have the conversion factor, we can determine the mass in kilograms using an equation similar the one used for converting length from inches to centimeters.<\/p>\n<p style=\"text-align: center;\">[latex]x\\text{ oz}=\\text{125 g}\\times \\text{unit conversion factor}[\/latex]<\/p>\n<p>We write the unit conversion factor in its two forms:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{\\text{1 oz}}{\\text{28.35 g}}\\text{ and }\\frac{\\text{28.349 g}}{\\text{1 oz}}[\/latex]<\/p>\n<p>The correct unit conversion factor is the ratio that cancels the units of grams and leaves ounces.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccc}x\\text{ oz}\\hfill & \\text{=}\\hfill & 125\\cancel{\\text{g}}\\times \\frac{\\text{1 oz}}{\\text{28.349}\\cancel{\\text{g}}}\\hfill \\\\ \\hfill & =\\hfill & \\left(\\frac{125}{\\text{28.349}}\\right)\\text{oz}\\hfill \\\\ \\hfill & =\\hfill & \\text{4.41 oz (three significant figures)}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<h4><strong>Check Your Learning<\/strong><\/h4>\n<p>Convert a volume of 9.345 qt to liters.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q798791\">Show Answer<\/span><\/p>\n<div id=\"q798791\" class=\"hidden-answer\" style=\"display: none\">8.844 L<\/div>\n<\/div>\n<\/div>\n<p>Beyond simple unit conversions, the factor-label method can be used to solve more complex problems involving computations. Regardless of the details, the basic approach is the same\u2014all the <em>factors<\/em> involved in the calculation must be appropriately oriented to insure that their <em>labels<\/em> (units) will appropriately cancel and\/or combine to yield the desired unit in the result. This is why it is referred to as the factor-label method. As your study of chemistry continues, you will encounter many opportunities to apply this approach.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 2: <strong>Computing Quantities from Measurement Results and Known Mathematical Relations<\/strong><\/h3>\n<p>What is the density of common antifreeze in units of g\/mL? A 4.00-qt sample of the antifreeze weighs 9.26 lb.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q944637\">Show Answer<\/span><\/p>\n<div id=\"q944637\" class=\"hidden-answer\" style=\"display: none\">\n<p>Since [latex]\\text{density}=\\frac{\\text{mass}}{\\text{volume}}[\/latex] , we need to divide the mass in grams by the volume in milliliters. In general: the number of units of B = the number of units of A \u00d7 unit conversion factor. The necessary conversion factors are given in Table 1.6: 1 lb = 453.59 g; 1 L = 1.0567 qt; 1 L = 1,000 mL. We can convert mass from pounds to grams in one step:<\/p>\n<p style=\"text-align: center;\">[latex]\\text{9.26}\\cancel{\\text{lb}}\\times \\frac{\\text{453.59 g}}{1\\cancel{\\text{lb}}}=4.20\\times {10}^{3}\\text{g}[\/latex]<\/p>\n<p>We need to use two steps to convert volume from quarts to milliliters.<\/p>\n<ol>\n<li><strong>Convert quarts to liters: <\/strong>[latex]\\text{4.00}\\cancel{\\text{qt}}\\times \\frac{\\text{1 L}}{\\text{1.0567}\\cancel{\\text{qt}}}=\\text{3.78 L}[\/latex]<\/li>\n<li><strong>Convert liters to milliliters: <\/strong>[latex]\\text{3.78}\\cancel{\\text{L}}\\times \\frac{\\text{1000 mL}}{1\\cancel{\\text{L}}}=3.78\\times {10}^{3}\\text{mL}[\/latex]<\/li>\n<\/ol>\n<p>Then, [latex]\\text{density}=\\frac{\\text{4.20}\\times {10}^{3}\\text{g}}{3.78\\times {10}^{3}\\text{mL}}=\\text{1.11 g\/mL}[\/latex].<\/p>\n<p>Alternatively, the calculation could be set up in a way that uses three unit conversion factors sequentially as follows:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{\\text{9.26}\\cancel{\\text{lb}}}{\\text{4.00}\\cancel{\\text{qt}}}\\times \\frac{\\text{453.59 g}}{1\\cancel{\\text{lb}}}\\times \\frac{\\text{1.0567}\\cancel{\\text{qt}}}{1\\cancel{\\text{L}}}\\times \\frac{1\\cancel{\\text{L}}}{\\text{1000 mL}}=\\text{1.11 g\/mL}[\/latex]<\/p>\n<\/div>\n<\/div>\n<h4><strong>Check Your Learning<\/strong><\/h4>\n<p>What is the volume in liters of 1.000 oz, given that 1 L = 1.0567 qt and 1 qt = 32 oz (exactly)?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q99428\">Show Answer<\/span><\/p>\n<div id=\"q99428\" class=\"hidden-answer\" style=\"display: none\">[latex]2.956\\times {10}^{-2}\\text{L}[\/latex]<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 3: <strong>Computing Quantities from Measurement Results and Known Mathematical Relations<\/strong><\/h3>\n<p>While being driven from Philadelphia to Atlanta, a distance of about 1250 km, a 2014 Lamborghini Aventador Roadster uses 213 L gasoline.<\/p>\n<ol>\n<li>What (average) fuel economy, in miles per gallon, did the Roadster get during this trip?<\/li>\n<li>If gasoline costs $3.80 per gallon, what was the fuel cost for this trip?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q520673\">Show Answer<\/span><\/p>\n<div id=\"q520673\" class=\"hidden-answer\" style=\"display: none\">\n<h4>Part 1<\/h4>\n<p>We first convert distance from kilometers to miles: [latex]\\text{1250 km}\\times \\frac{\\text{0.62137 mi}}{\\text{1 km}}=\\text{777 mi}[\/latex]<\/p>\n<p>Then we convert volume from liters to gallons: [latex]213\\cancel{\\text{L}}\\times \\frac{\\text{1.0567}\\cancel{\\text{qt}}}{1\\cancel{\\text{L}}}\\times \\frac{\\text{1 gal}}{4\\cancel{\\text{qt}}}=\\text{56.3 gal}[\/latex]<\/p>\n<p>Then,<\/p>\n<p style=\"text-align: center;\">[latex]\\text{(average) mileage}=\\frac{\\text{777 mi}}{\\text{56.3 gal}}=\\text{13.8 miles\/gallon}=\\text{13.8 mpg}[\/latex]<\/p>\n<p>Alternatively, the calculation could be set up in a way that uses all the conversion factors sequentially, as follows:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{1250\\cancel{\\text{km}}}{213\\cancel{\\text{L}}}\\times \\frac{\\text{0.62137 mi}}{1\\cancel{\\text{km}}}\\times \\frac{1\\cancel{\\text{L}}}{\\text{1.0567}\\cancel{\\text{qt}}}\\times \\frac{4\\cancel{\\text{qt}}}{\\text{1 gal}}=\\text{13.8 mpg}[\/latex]<\/p>\n<h4>Part 2<\/h4>\n<p>Using the previously calculated volume in gallons, we find: [latex]56.3\\text{ gal}\\times\\frac{\\$3.80}{1\\text{ gal}}=\\$214[\/latex]<\/p>\n<\/div>\n<\/div>\n<h4><strong>Check Your Learning<\/strong><\/h4>\n<p>A Toyota Prius Hybrid uses 59.7 L gasoline to drive from San Francisco to Seattle, a distance of 1300 km (two significant digits).<\/p>\n<ol>\n<li>What (average) fuel economy, in miles per gallon, did the Prius get during this trip?<\/li>\n<li>If gasoline costs $3.90 per gallon, what was the fuel cost for this trip?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q676222\">Show Answer<\/span><\/p>\n<div id=\"q676222\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>51 mpg<\/li>\n<li>$62<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Atomic Bombs and Dimensional Analysis<\/h3>\n<p>How did a magazine photograph help a physicist crack one of the secrets of the atom bomb?<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Atom Bombs and Dimensional Analysis - Sixty Symbols\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/_gaCAFcW6OY?start=50&#38;feature=oembed\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\"><\/div>\n<h2>Glossary<\/h2>\n<p><strong>dimensional analysis:<\/strong> (also, factor-label method) versatile mathematical approach that can be applied to computations ranging from simple unit conversions to more complex, multi-step calculations involving several different quantities<\/p>\n<p><strong>unit conversion factor:<\/strong> ratio of equivalent quantities expressed with different units; used to convert from one unit to a different unit<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-14872\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Chemistry. <strong>Provided by<\/strong>: OpenStax College. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/openstaxcollege.org\">http:\/\/openstaxcollege.org<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at https:\/\/openstaxcollege.org\/textbooks\/chemistry\/get<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">All rights reserved content<\/div><ul class=\"citation-list\"><li>Atom Bombs and Dimensional Analysis - Sixty Symbols. <strong>Authored by<\/strong>: Sixty Symbols. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/_gaCAFcW6OY\">https:\/\/youtu.be\/_gaCAFcW6OY<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":23485,"menu_order":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Chemistry\",\"author\":\"\",\"organization\":\"OpenStax College\",\"url\":\"http:\/\/openstaxcollege.org\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at https:\/\/openstaxcollege.org\/textbooks\/chemistry\/get\"},{\"type\":\"copyrighted_video\",\"description\":\"Atom Bombs and Dimensional Analysis - Sixty Symbols\",\"author\":\"Sixty Symbols\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/_gaCAFcW6OY\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-14872","chapter","type-chapter","status-publish","hentry"],"part":14880,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/14872","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/users\/23485"}],"version-history":[{"count":4,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/14872\/revisions"}],"predecessor-version":[{"id":16724,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/14872\/revisions\/16724"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/parts\/14880"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/14872\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/media?parent=14872"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapter-type?post=14872"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/contributor?post=14872"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/license?post=14872"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}