{"id":1060,"date":"2017-01-11T00:19:16","date_gmt":"2017-01-11T00:19:16","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/?post_type=chapter&#038;p=1060"},"modified":"2019-10-03T21:03:19","modified_gmt":"2019-10-03T21:03:19","slug":"introduction-systems-and-scales-of-measurement","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wmopen-mathforliberalarts\/chapter\/introduction-systems-and-scales-of-measurement\/","title":{"raw":"Systems and Scales of Measurement","rendered":"Systems and Scales of Measurement"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Describe the general relationship between the U.S. customary units and metric units of length, weight\/mass, and volume<\/li>\r\n \t<li>Define the metric prefixes and use them to perform basic conversions among metric units<\/li>\r\n \t<li>Solve application problems using metric units<\/li>\r\n<\/ul>\r\n<ul>\r\n \t<li>State the freezing and boiling points of water on the Celsius and Fahrenheit temperature scales.<\/li>\r\n \t<li>Convert from one temperature scale to the other, using conversion formulas<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the United States, both the <b>U.S. customary measurement system<\/b> and the <b>metric system<\/b> are used, especially in medical, scientific, and technical fields. In most other countries, the metric system is the primary system of measurement. If you travel to other countries, you will see that road signs list distances in kilometers and milk is sold in liters. People in many countries use words like \u0093kilometer,\u0094 \u0093liter,\u0094 and \u0093milligram\u0094 to measure the length, volume, and weight of different objects. These measurement units are part of the metric system.\r\n\r\nUnlike the U.S. customary system of measurement, the metric system is based on 10s. For example, a liter is 10 times larger than a deciliter, and a centigram is 10 times larger than a milligram. This idea of \u009310\u0094 is not present in the U.S. customary system\u0097there are 12 inches in a foot, and 3 feet in a yard and 5,280 feet in a mile!\r\n\r\nSo, what if you have to find out how many milligrams are in a decigram? Or, what if you want to convert meters to kilometers? Understanding how the metric system works is a good start.\r\n\r\nIn this section we will discover the basic units used in the metric system, and show how to convert between them. We will also explore temperature scales.\u00a0In the United States, temperatures are usually measured using the <b>Fahrenheit<\/b> scale, while most countries that use the metric system use the <b>Celsius<\/b> scale to record temperatures. Learning about the different scales\u0097, including how to convert between them\u0097 will help you figure out what the weather is going to be like, no matter which country you find yourself in.\r\n\r\n&nbsp;\r\n<h2>Metric System Basics<\/h2>\r\n<h3>What Is Metric?<\/h3>\r\nThe metric system uses units such as <b>meter<\/b>, <b>liter<\/b>, and <b>gram<\/b> to measure length, liquid volume, and mass, just as the U.S. customary system uses feet, quarts, and ounces to measure these.\r\n\r\nIn addition to the difference in the basic units, the metric system is based on 10s, and different measures for length include kilometer, meter, decimeter, centimeter, and millimeter. Notice that the word \u0093meter\u0094 is part of all of these units.\r\n\r\nThe metric system also applies the idea that units within the system get larger or smaller by a power of 10. This means that a meter is 100 times larger than a centimeter, and a kilogram is 1,000 times heavier than a gram. You will explore this idea a bit later. For now, notice how this idea of \u0093getting bigger or smaller by 10\u0094 is very different than the relationship between units in the U.S. customary system, where 3 feet equals 1 yard, and 16 ounces equals 1 pound.\r\n<h3>Length, Mass, and Volume<\/h3>\r\nThe table below shows the basic units of the metric system. Note that the names of all metric units follow from these three basic units.\r\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\r\n<tbody>\r\n<tr>\r\n<td><b>Length<\/b><\/td>\r\n<td><b>Mass<\/b><\/td>\r\n<td><b>Volume<\/b><\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"3\"><i>basic units<\/i><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>meter<\/td>\r\n<td>gram<\/td>\r\n<td>liter<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"3\"><i>other units you may see<\/i><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>kilometer<\/td>\r\n<td>kilogram<\/td>\r\n<td>dekaliter<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>centimeter<\/td>\r\n<td>centigram<\/td>\r\n<td>centiliter<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>millimeter<\/td>\r\n<td>milligram<\/td>\r\n<td>milliliter<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn the metric system, the basic unit of length is the meter. A meter is slightly larger than a yardstick, or just over three feet.\r\n\r\nThe basic metric unit of mass is the gram. A regular-sized paperclip has a mass of about 1 gram.\r\n\r\nAmong scientists, one gram is defined as the mass of water that would fill a 1-centimeter cube. You may notice that the word \u0093mass\u0094 is used here instead of \u0093weight.\u0094 In the sciences and technical fields, a distinction is made between weight and mass. Weight is a measure of the pull of gravity on an object. For this reason, an object\u0092s weight would be different if it was weighed on Earth or on the moon because of the difference in the gravitational forces. However, the object\u0092s mass would remain the same in both places because mass measures the amount of substance in an object. As long as you are planning on only measuring objects on Earth, you can use mass\/weight fairly interchangeably\u0097but it is worth noting that there is a difference!\r\n\r\nFinally, the basic metric unit of volume is the liter. A liter is slightly larger than a quart.\r\n<table border=\"1\" width=\"602\" cellspacing=\"0\" cellpadding=\"0\">\r\n<tbody>\r\n<tr>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/30201020\/image106.jpg\" width=\"162\" height=\"109\" \/><\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/30201021\/image107.jpg\" width=\"148\" height=\"104\" \/><\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/30201022\/image108.jpg\" width=\"94\" height=\"117\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The handle of a shovel is about 1 meter.<\/td>\r\n<td>A paperclip weighs about 1 gram.<\/td>\r\n<td>A medium-sized container of milk is about 1 liter.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThough it is rarely necessary to convert between the customary and metric systems, sometimes it helps to have a mental image of how large or small some units are. The table below shows the relationship between some common units in both systems.\r\n<table border=\"1\" width=\"507\" cellspacing=\"0\" cellpadding=\"0\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td><b>Common Measurements in Customary and Metric Systems<\/b><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><i>Length<\/i><\/td>\r\n<td>1 centimeter is a little less than half an inch.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>1.6 kilometers is about 1 mile.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>1 meter is about 3 inches longer than 1 yard.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><i>Mass<\/i><\/td>\r\n<td>1 kilogram is a little more than 2 pounds.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><i>\u00a0<\/i><\/td>\r\n<td>28 grams is about the same as 1 ounce.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><i>Volume<\/i><\/td>\r\n<td>1 liter is a little more than 1 quart.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><i>\u00a0<\/i><\/td>\r\n<td>4 liters is a little more than 1 gallon.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h3>Prefixes in the Metric System<\/h3>\r\nThe metric system is a base 10 system. This means that each successive unit is 10 times larger than the previous one.\r\n\r\nThe names of metric units are formed by adding a prefix to the basic unit of measurement. To tell how large or small a unit is, you look at the <b>prefix<\/b>. To tell whether the unit is measuring length, mass, or volume, you look at the base.\r\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\r\n<tbody>\r\n<tr>\r\n<td colspan=\"7\"><b>Prefixes in the Metric System<\/b><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><i>kilo-<\/i><\/td>\r\n<td><i>hecto-<\/i><\/td>\r\n<td><i>deka-<\/i><\/td>\r\n<td>meter\r\n\r\ngram\r\n\r\nliter<\/td>\r\n<td><i>deci-<\/i><\/td>\r\n<td><i>centi-<\/i><\/td>\r\n<td><i>milli-<\/i><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1,000 times <b>larger<\/b> than base unit<\/td>\r\n<td>100 times <b>larger<\/b> than base unit<\/td>\r\n<td>10 times <b>larger<\/b> than base unit<\/td>\r\n<td>base units<\/td>\r\n<td>10 times <b>smaller<\/b> than base unit<\/td>\r\n<td>100 times <b>smaller<\/b> than base unit<\/td>\r\n<td>1,000 times <b>smaller<\/b> than base unit<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nUsing this table as a reference, you can see the following:\r\n<ul>\r\n \t<li>A kilogram is 1,000 times larger than one gram (so 1 kilogram = 1,000 grams).<\/li>\r\n \t<li>A centimeter is 100 times smaller than one meter (so 1 meter = 100 centimeters).<\/li>\r\n \t<li>A dekaliter is 10 times larger than one liter (so 1 dekaliter = 10 liters).<\/li>\r\n<\/ul>\r\nHere is a similar table that just shows the metric units of measurement for mass, along with their size relative to 1 gram (the base unit). The common abbreviations for these metric units have been included as well.\r\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\r\n<tbody>\r\n<tr>\r\n<td colspan=\"7\"><b>Measuring Mass in the Metric System<\/b><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>kilogram\r\n\r\n(kg)<\/td>\r\n<td>hectogram\r\n\r\n(hg)<\/td>\r\n<td>dekagram\r\n\r\n(dag)<\/td>\r\n<td>gram\r\n\r\n(g)<\/td>\r\n<td>decigram\r\n\r\n(dg)<\/td>\r\n<td>centigram\r\n\r\n(cg)<\/td>\r\n<td>milligram\r\n\r\n(mg)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1,000 grams<\/td>\r\n<td>100 grams<\/td>\r\n<td>10 grams<\/td>\r\n<td>gram<\/td>\r\n<td>0.1 gram<\/td>\r\n<td>0.01 gram<\/td>\r\n<td>0.001 gram<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSince the prefixes remain constant through the metric system, you could create similar charts for length and volume. The prefixes have the same meanings whether they are attached to the units of length (meter), mass (gram), or volume (liter).\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nWhich of the following sets of three units are all metric measurements of <strong>length<\/strong>?\r\n\r\nA) inch, foot, yard\r\n\r\nB) kilometer, centimeter, millimeter\r\n\r\nC) kilogram, gram, centigram\r\n\r\nD) kilometer, foot, decimeter\r\n\r\n[reveal-answer q=\"728320\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"728320\"]\r\n\r\nB) kilometer, centimeter, millimeter\r\n\r\nAll of these measurements are from the metric system. You can tell they are measurements of length because they all contain the word \u0093meter.\u0094\r\n\r\n[\/hidden-answer]\r\n\r\n<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=126793&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"200\"><\/iframe>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=126794&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"200\"><\/iframe>\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=126795&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"200\"><\/iframe>\r\n\r\n<\/div>\r\n<h3>Converting Units Up and Down the Metric Scale<\/h3>\r\nConverting between metric units of measure requires knowledge of the metric prefixes and an understanding of the decimal system\u0097that\u0092s about it.\r\n\r\nFor instance, you can figure out how many centigrams are in one dekagram by using the table above. One dekagram is larger than one centigram, so you expect that one dekagram will equal many centigrams.\r\n\r\nIn the table, each unit is 10 times larger than the one to its immediate right. This means that 1 dekagram = 10 grams; 10 grams = 100 decigrams; and 100 decigrams = 1,000 centigrams. So, 1 dekagram = 1,000 centigrams.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nHow many milligrams are in one decigram?\r\n\r\n[reveal-answer q=\"363102\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"363102\"]\r\n\r\nIdentify locations of milligrams and decigrams.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>kg<\/td>\r\n<td>hg<\/td>\r\n<td>dag<\/td>\r\n<td>g<\/td>\r\n<td>dg<\/td>\r\n<td>cg<\/td>\r\n<td>mg<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>^<\/td>\r\n<td>^<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nDecigrams (dg) are larger than milligrams (mg), so you expect there to be many mg in one dg.\r\n\r\nDg is 10 times larger than a cg, and a cg is 10 times larger than a mg.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\times10[\/latex]<\/td>\r\n<td>[latex]\\times10[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>kg<\/td>\r\n<td>hg<\/td>\r\n<td>dag<\/td>\r\n<td>g<\/td>\r\n<td>dg<\/td>\r\n<td>cg<\/td>\r\n<td>mg<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\downarrow[\/latex]<\/td>\r\n<td>[latex]\\uparrow[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">[latex]\\rightarrow[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSince you are going from a larger unit to a smaller unit, multiply.\r\n\r\n<i>Multiply: 1 \u00b7 10 \u00b7 10, to find the number of milligrams in one decigram.\u00a0<\/i>\r\n<p style=\"text-align: center;\">[latex]1\\text{ dg}\\cdot10\\cdot10=100\\text{ mg}[\/latex]<\/p>\r\nThere are 100 milligrams (mg) in 1 decigram (dg).\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nConvert 3,085 milligrams to grams.\r\n\r\n[reveal-answer q=\"353889\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"353889\"]\r\n\r\nOne gram is 1,000 times larger than a milligram, so you can move the decimal point in 3,085 three places to the left.\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom15\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=1000&amp;theme=oea&amp;iframe_resize_id=mom15\" width=\"100%\" height=\"250\"><\/iframe>\r\n<iframe id=\"mom100\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=1001&amp;theme=oea&amp;iframe_resize_id=mom100\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<iframe id=\"mom13\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=1005&amp;theme=oea&amp;iframe_resize_id=mom13\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nConvert 1 centimeter to kilometers.\r\n\r\n[reveal-answer q=\"4330\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"4330\"]\r\n\r\nIdentify locations of kilometers and centimeters.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>km<\/td>\r\n<td>hm<\/td>\r\n<td>dam<\/td>\r\n<td>m<\/td>\r\n<td>dm<\/td>\r\n<td>cm<\/td>\r\n<td>mm<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>^<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<td>^<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nKilometers (km) are larger than centimeters (cm), so you expect there to be less than one km in a cm.\r\n\r\nCm is 10 times smaller than a dm; a dm is 10 times smaller than a m, etc.\r\n\r\nSince you are going from a smaller unit to a larger unit, divide.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\div10[\/latex]<\/td>\r\n<td>[latex]\\div10[\/latex]<\/td>\r\n<td>[latex]\\div10[\/latex]<\/td>\r\n<td>[latex]\\div10[\/latex]<\/td>\r\n<td>[latex]\\div10[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>km<\/td>\r\n<td>hm<\/td>\r\n<td>dam<\/td>\r\n<td>m<\/td>\r\n<td>dm<\/td>\r\n<td>cm<\/td>\r\n<td>mm<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>^<\/td>\r\n<td>[latex]\\leftarrow[\/latex]<\/td>\r\n<td>[latex]\\leftarrow[\/latex]<\/td>\r\n<td>[latex]\\leftarrow[\/latex]<\/td>\r\n<td>[latex]\\leftarrow[\/latex]<\/td>\r\n<td>^<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nDivide: [latex]1\\div10\\div10\\div10\\div10\\div10[\/latex], to find the number of kilometers in one centimeter.<em>\u00a0<\/em>\r\n<p style=\"text-align: center;\">[latex]1\\text{ cm}\\div10\\div10\\div10\\div10\\div10=0.00001\\text{ km}[\/latex]<\/p>\r\n1 centimeter (cm) = 0.00001 kilometers (km).\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom200\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=998&amp;theme=oea&amp;iframe_resize_id=mom200\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\nOnce you begin to understand the metric system, you can use a shortcut to convert among different metric units. The size of metric units increases tenfold as you go up the metric scale. The decimal system works the same way: a tenth is 10 times larger than a hundredth; a hundredth is 10 times larger than a thousandth, etc. By applying what you know about decimals to the metric system, converting among units is as simple as moving decimal points.\r\n\r\nHere is the first problem from above: How many milligrams are in one decigram? You can recreate the order of the metric units as shown below:\r\n<p style=\"text-align: center;\">[latex] \\displaystyle kg\\quad hg\\quad dag\\quad g\\quad d\\underbrace{g\\quad c}_{1}\\underbrace{g\\quad m}_{2}g[\/latex]<\/p>\r\nThis question asks you to start with 1 decigram and convert that to milligrams. As shown above, milligrams is two places to the right of decigrams. You can just move the decimal point two places to the right to convert decigrams to milligrams: [latex] \\displaystyle 1\\ dg=1\\underbrace{0}_{1}\\underbrace{0}_{2}.\\ mg[\/latex].\r\n\r\nThe same method works when you are converting from a smaller to a larger unit, as in the problem: Convert 1 centimeter to kilometers.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle k\\underbrace{m\\quad h}_{5}\\underbrace{m\\quad d}_{4}\\underbrace{am\\quad }_{3}\\underbrace{m\\quad d}_{2}\\underbrace{m\\quad c}_{1}m\\quad mm[\/latex]<\/p>\r\nNote that instead of moving to the right, you are now moving to the left\u0097so the decimal point must do the same:\r\n<p style=\"text-align: center;\">[latex] \\displaystyle 1\\ cm=0.\\underbrace{0}_{5}\\underbrace{0}_{4}\\underbrace{0}_{3}\\underbrace{0}_{2}\\underbrace{1}_{1}\\ km[\/latex].<\/p>\r\n\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nHow many milliliters are in 1 liter?\r\n\r\n[reveal-answer q=\"95548\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"95548\"]\r\n\r\nThere are 10 milliliters in a centiliter, 10 centiliters in a deciliter, and 10 deciliters in a liter. Multiply: [latex]10\\cdot10\\cdot10[\/latex], to find the number of milliliters in a liter, 1,000.\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom10\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=999&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<h3>Factor Label Method<\/h3>\r\nThere is yet another method that you can use to convert metric measurements\u0097the <b>factor label method<\/b>. You used this method when you were converting measurement units within the U.S. customary system.\r\n\r\nThe factor label method works the same in the metric system; it relies on the use of unit fractions and the cancelling of intermediate units. The table below shows some of the <b>unit equivalents<\/b> and <b>unit fractions<\/b> for length in the metric system. (You should notice that all of the unit fractions contain a factor of 10. Remember that the metric system is based on the notion that each unit is 10 times larger than the one that came before it.)\r\n\r\nAlso, notice that two new prefixes have been added here: mega- (which is very big) and micro- (which is very small).\r\n<table border=\"1\" cellspacing=\"0\" cellpadding=\"0\">\r\n<tbody>\r\n<tr>\r\n<td><b>Unit Equivalents<\/b><\/td>\r\n<td colspan=\"2\"><b>Conversion Factors<\/b><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1 meter = 1,000,000 micrometers<\/td>\r\n<td>[latex] \\displaystyle \\frac{1\\ m}{1,000,000\\ \\mu m}[\/latex]<\/td>\r\n<td>[latex] \\displaystyle \\frac{1,000,000\\ \\mu m}{1\\ m}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1 meter = 1,000 millimeters<\/td>\r\n<td>[latex] \\displaystyle \\frac{1\\ m}{1,000\\ mm}[\/latex]<\/td>\r\n<td>[latex] \\displaystyle \\frac{1,000\\ mm}{1\\ m}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1 meter = 100 centimeters<\/td>\r\n<td>[latex] \\displaystyle \\frac{1\\ m}{100\\ cm}[\/latex]<\/td>\r\n<td>[latex] \\displaystyle \\frac{100\\ cm}{1\\ m}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1 meter = 10 decimeters<\/td>\r\n<td>[latex] \\displaystyle \\frac{1\\ m}{10\\ dm}[\/latex]<\/td>\r\n<td>[latex] \\displaystyle \\frac{10\\ dm}{1\\ m}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1 dekameter = 10 meters<\/td>\r\n<td>[latex] \\displaystyle \\frac{1\\ dam}{10\\ m}[\/latex]<\/td>\r\n<td>[latex] \\displaystyle \\frac{10\\ m}{1\\ dam}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1 hectometer = 100 meters<\/td>\r\n<td>[latex] \\displaystyle \\frac{1\\ hm}{100\\ m}[\/latex]<\/td>\r\n<td>[latex] \\displaystyle \\frac{100\\ m}{1\\ hm}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1 kilometer = 1,000 meters<\/td>\r\n<td>[latex] \\displaystyle \\frac{1\\ km}{1,000\\ m}[\/latex]<\/td>\r\n<td>[latex] \\displaystyle \\frac{1,000\\ m}{1\\ km}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1 megameter = 1,000,000 meters<\/td>\r\n<td>[latex] \\displaystyle \\frac{1\\ Mm}{1,000,000\\ m}[\/latex]<\/td>\r\n<td>[latex] \\displaystyle \\frac{1,000,000\\ m}{1\\ Mm}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWhen applying the factor label method in the metric system, be sure to check that you are not skipping over any intermediate units of measurement!\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nConvert 7,225 centimeters to meters.\r\n\r\n[reveal-answer q=\"461145\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"461145\"]\r\n\r\nMeters is larger than centimeters, so you expect your answer to be less than 7,225.\r\n<p style=\"text-align: center;\">[latex]7,225\\text{ cm}=\\text{___ m}[\/latex]<\/p>\r\nUsing the factor label method, write 7,225 cm as a fraction and use unit fractions to convert it to m.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\frac{7,225\\ cm}{1}\\cdot \\frac{1\\ m}{100\\ cm}=\\_\\_\\_ m[\/latex]<\/p>\r\nCancel similar units, multiply, and simplify.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\frac{7,225\\ \\cancel{cm}}{1}\\cdot \\frac{1\\text{ m}}{100\\ \\cancel{\\text{cm}}}=\\_\\_\\_m[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\frac{7,225}{1}\\cdot \\frac{1\\text{ m}}{100}=\\frac{7,225}{100}\\text{m}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\frac{7,225\\text{ m}}{100}=72.25\\text{ m}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]7,225\\text{ centimeters}=72.25\\text{ meters}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nConvert 32.5 kilometers to meters.\r\n\r\n[reveal-answer q=\"574914\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"574914\"]\r\n\r\n32,500 meters\r\n\r\n[latex] \\displaystyle \\frac{32.5\\text{ km}}{1}\\cdot \\frac{1,000\\text{ m}}{1\\text{ km}}=\\frac{32,500\\text{ m}}{1}[\/latex].\r\n\r\nThe km units cancel, leaving the answer in m.\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom500\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=18877&amp;theme=oea&amp;iframe_resize_id=mom500\" width=\"100%\" height=\"200\"><\/iframe>\r\n\r\n<\/div>\r\nNow that you have seen how to convert among metric measurements in multiple ways, let\u0092's revisit the problem posed earlier.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nIf you have a prescription for 5,000 mg of medicine, and upon getting it filled, the dosage reads 5g of medicine, did the pharmacist make a mistake?\r\n\r\n[reveal-answer q=\"600572\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"600572\"]\r\n\r\nConvert mg to g.\r\n<p style=\"text-align: center;\">[latex]5,000\\text{ mg}=\\text{___ g}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\frac{5,000\\text{ mg}}{1}\\cdot \\frac{1\\text{ g}}{1,000\\text{ mg}}=\\text{ g}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\frac{5,000\\cancel{\\text{mg}}}{1}\\cdot \\frac{1\\text{ g}}{1,000\\ \\cancel{\\text{mg}}}=\\text{ g}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\frac{5,000\\cdot 1\\text{ g}}{1\\cdot 1,000}=\\frac{5,000\\text{ g}}{1,000}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\frac{5,000\\text{ g}}{1,000}=5\\text{ g}[\/latex]<\/p>\r\n[latex]5\\text{ g}=5,000\\text{ mg}[\/latex], so the pharmacist did not make a mistake.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Applications of Metric Conversions<\/h2>\r\nLearning how to solve real-world problems using metric conversions is as important as learning how to do the conversions themselves. Mathematicians, scientists, nurses, and even athletes are often confronted with situations where they are presented with information using metric measurements, and must then make informed decisions based on that data.\r\n<div class=\"textbox\"><strong>TIP:<\/strong> To solve these problems effectively, you need to understand the context of a problem, perform conversions, and then check the reasonableness of your answer. Do all three of these steps and you will succeed in whatever measurement system you find yourself using.<\/div>\r\n<h3>Understanding Context and Performing Conversions<\/h3>\r\nThe first step in solving any real-world problem is to understand its context. This will help you figure out what kinds of solutions are reasonable (and the problem itself may give you clues about what types of conversions are necessary). Here is an example.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nMarcus bought at 2 meter board, and cut off a piece 1 meter and 35 cm long. How much board is left?\r\n\r\n[reveal-answer q=\"701860\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"701860\"]\r\n\r\nTo answer this question, we will need to subtract.\r\n\r\nFirst convert all measurements to one unit. Here we will convert to centimeters.\r\n\r\n[latex]2\\text{ meters}-1\\text{ meter and }35\\text{ cm}[\/latex]\r\n\r\nUse the factor label method and unit fractions to convert from meters to centimeters.\r\n\r\n[latex] \\displaystyle \\frac{2\\text{ m}}{1}\\cdot \\frac{100\\text{ cm}}{1\\text{ m}}=\\text{ cm}[\/latex]\r\n\r\nCancel, multiply, and solve.\r\n\r\nConvert the 1 meter to centimeters, and combine with the additional 35 centimeters.\r\n\r\nSubtract the cut length from the original board length.\r\n\r\n[latex] \\displaystyle \\frac{2\\ \\cancel{\\text{m}}}{1}\\cdot \\frac{100\\text{ cm}}{1\\ \\cancel{\\text{ m}}}=\\text{ cm}[\/latex]\r\n\r\n[latex] \\displaystyle \\frac{200\\text{ cm}}{1}=200\\text{ cm}[\/latex]\r\n\r\n[latex]1\\text{ meter}+35\\text{ cm}[\/latex]\r\n\r\n[latex]100\\text{ cm}+35\\text{ cm}[\/latex]\r\n\r\n[latex]135\\text{ cm}[\/latex]\r\n\r\n[latex]200\\text{ cm}-135\\text{ cm}[\/latex]\r\n\r\n[latex]65\\text{ cm}[\/latex]\r\n\r\nThere is 65 cm of board left.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nAn example with a different context, but still requiring conversions, is shown below.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nA faucet drips 10 ml every minute. How much water will be wasted in a week?\r\n\r\n[reveal-answer q=\"642392\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"642392\"]\r\n\r\nStart by calculating how much water will be used in a week using the factor label method to convert the time units.\r\n\r\n[latex] \\displaystyle \\frac{10\\ ml}{1\\text{ minute}}\\cdot \\frac{60\\text{ minute}}{1\\text{ hour}}\\cdot \\frac{24\\text{ hours}}{1\\text{ day}}\\cdot \\frac{7\\text{ days}}{1\\text{ week}}[\/latex]\r\n\r\nCancel, multiply and solve.\r\n\r\n[latex] \\displaystyle \\frac{10\\ ml}{1\\text{ }\\cancel{\\text{minute}}}\\cdot \\frac{60\\text{ }\\cancel{\\text{minute}}}{1\\text{ }\\cancel{\\text{hour}}}\\cdot \\frac{24\\text{ }\\cancel{\\text{hours}}}{1\\text{ }\\cancel{\\text{day}}}\\cdot \\frac{7\\text{ }\\cancel{\\text{days}}}{1\\text{ week}}[\/latex]\r\n\r\n[latex] \\displaystyle \\frac{10\\centerdot 60\\centerdot 24\\centerdot 7\\ ml}{1\\centerdot 1\\centerdot 1\\centerdot 1\\text{ week}}[\/latex]\r\n\r\nTo give a more useable answer, convert this into liters.\r\n\r\n[latex] \\displaystyle \\frac{100800\\ ml}{1\\text{ week}}[\/latex]\r\n\r\nCancel, multiply and solve.\r\n\r\n[latex] \\displaystyle \\frac{100800\\text{ ml}}{1\\text{ week}}\\centerdot \\frac{1\\text{ L}}{1000\\text{ ml}}[\/latex]\r\n\r\n[latex] \\displaystyle \\frac{100800\\text{ }\\cancel{\\text{ml}}}{1\\text{ week}}\\centerdot \\frac{1\\text{ L}}{1000\\text{ }\\cancel{\\text{ml}}}[\/latex]\r\n\r\n[latex] \\displaystyle \\frac{100800\\text{ L}}{1000\\text{ week}}[\/latex]=[latex] \\displaystyle 100.8\\frac{\\text{L}}{\\text{week}}[\/latex]\r\n\r\nThe faucet wastes about 100.8 liters each week.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThis problem asked for the difference between two quantities. The easiest way to find this is to convert one quantity so that both quantities are measured in the same unit, and then subtract one from the other.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nA bread recipe calls for 600 g of flour. How many kilograms of flour would you need to make 5 loaves?\r\n\r\n[reveal-answer q=\"838013\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"838013\"]\r\n\r\nMultiplying 600 g per loaf by the 5 loaves,\r\n\r\n[latex]600\\text{g}\\cdot5=3000\\text{g}[\/latex]\r\n\r\nUsing factor labels or the \u0093move the decimal\u0094 method, convert this to 3 kilograms.\r\n\r\nYou will need 3 kg of flour to make 5 loaves.\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=1002&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"200\"><\/iframe>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=117516&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"200\"><\/iframe>\r\n\r\n<\/div>\r\n<h3>Checking your Conversions<\/h3>\r\nSometimes it is a good idea to check your conversions using a second method. This usually helps you catch any errors that you may make, such as using the wrong unit fractions or moving the decimal point the wrong way.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nA bottle contains 1.5 liters of a beverage. How many 250 mL servings can be made from that bottle?\r\n\r\n[reveal-answer q=\"451287\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"451287\"]\r\n\r\nTo answer the question, you will need to divide 1.5 liters by 250 milliliters. To do this, convert both to the same unit. You could convert either measurement.\r\n\r\n[latex]1.5\\text{ L}\\div250\\text{ mL}[\/latex]\r\n\r\nConvert 250 mL to liters\r\n\r\n[latex]250\\text{ mL}=\\text{ ___ L}[\/latex]\r\n\r\n[latex] \\displaystyle \\frac{250\\text{ mL}}{1}\\cdot \\frac{1\\text{ L}}{1000\\text{ mL}}=\\text{___ L}[\/latex]\r\n\r\n[latex] \\displaystyle \\frac{250\\text{ L}}{1000=0.25\\text{ L}}[\/latex]\r\n\r\nNow we can divide using the converted measurement\r\n\r\n[latex]1.5\\text{ L}\\div250\\text{ mL}=\\frac{1.5\\text{ L}}{250\\text{ mL }}=\\frac{1.5\\text{ L}}{0.25\\text{ L}}[\/latex]\r\n\r\n[latex] \\displaystyle \\frac{1.5\\text{ L}}{0.25\\text{ L}}=6[\/latex]\r\n\r\nThe bottle holds 6 servings.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nUnderstanding the context of real-life application problems is important. Look for words within the problem that help you identify what operations are needed, and then apply the correct unit conversions. Checking your final answer by using another conversion method (such as the \u0093move the decimal\u0094 method, if you have used the factor label method to solve the problem) can cut down on errors in your calculations.\r\n<h3 class=\"Subsectiontitleunderline\">Summary<\/h3>\r\nThe metric system is an alternative system of measurement used in most countries, as well as in the United States. The metric system is based on joining one of a series of prefixes, including kilo-, hecto-, deka-, deci-, centi-, and milli-, with a base unit of measurement, such as meter, liter, or gram. Units in the metric system are all related by a power of 10, which means that each successive unit is 10 times larger than the previous one.\r\n\r\nThis makes converting one metric measurement to another a straightforward process, and is often as simple as moving a decimal point. It is always important, though, to consider the direction of the conversion. If you are converting a smaller unit to a larger unit, then the decimal point has to move to the left (making your number smaller); if you are converting a larger unit to a smaller unit, then the decimal point has to move to the right (making your number larger).\r\n\r\nThe factor label method can also be applied to conversions within the metric system. To use the factor label method, you multiply the original measurement by unit fractions; this allows you to represent the original measurement in a different measurement unit.\r\n<h2>Temperature Scales<\/h2>\r\nTurn on the television any morning and you will see meteorologists talking about the day\u0092s weather forecast. In addition to telling you what the weather conditions will be like (sunny, cloudy, rainy, muggy), they also tell you the day\u0092s forecast for high and low temperatures. A hot summer day may reach 100\u00b0 in Philadelphia, while a cool spring day may have a low of 40\u00b0 in Seattle.\r\n\r\nIf you have been to other countries, though, you may notice that meteorologists measure heat and cold differently outside of the United States. For example, a TV weatherman in San Diego may forecast a high of 89\u00b0, but a similar forecaster in Tijuana, Mexico\u0097, which is only 20 miles south\u0097, may look at the same weather pattern and say that the day\u0092's high temperature is going to be 32\u00b0. What\u0092s going on here? The difference is that the two countries use different temperature scales.\r\n<h3>Measuring Temperature on Two Scales<\/h3>\r\nFahrenheit and Celsius are two different scales for measuring temperature.\r\n<table border=\"0\" cellspacing=\"0\" cellpadding=\"0\">\r\n<tbody>\r\n<tr>\r\n<td>A thermometer measuring a temperature of 22\u00b0 Celsius is shown here.\r\n\r\nOn the Celsius scale, water freezes at 0\u00b0 and boils at 100\u00b0.\r\n\r\nIf the United States were to adopt the Celsius scale, forecast temperatures would rarely go below -30\u00b0 or above 45\u00b0. (A temperature of\r\n\r\n-18\u00b0 may be forecast for a cold winter day in Michigan, while a temperature of 43\u00b0 may be predicted for a hot summer day in Arizona.)\r\n\r\nMost office buildings maintain an indoor temperature between 18\u00b0C and 24\u00b0C to keep employees comfortable.<\/td>\r\n<td colspan=\"2\">\r\n<p style=\"text-align: center;\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/30201023\/image109.jpg\" width=\"224\" height=\"569\" \/><\/p>\r\n<\/td>\r\n<td>A thermometer measuring a temperature of 72\u00b0 Fahrenheit is shown here.\r\n\r\nOn the Fahrenheit scale, water freezes at 32\u00b0 and boils at 212\u00b0.\r\n\r\nIn the United States, forecast temperatures measured in Fahrenheit rarely go below -20\u00b0 or above 120\u00b0. (A temperature of 0\u00b0 may be forecast for a cold winter day in Michigan, while a temperature of 110\u00b0 may be predicted for a hot summer day in Arizona.)\r\n\r\nMost office buildings maintain an indoor temperature between 65\u00b0F and 75\u00b0F to keep employees comfortable.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td><b>Celsius<\/b><\/td>\r\n<td>\r\n<p align=\"right\"><b>Fahrenheit<\/b><\/p>\r\n<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nA cook puts a thermometer into a pot of water to see how hot it is. The thermometer reads 132\u00b0, but the water is not boiling yet. Which temperature scale is the thermometer measuring?\r\n\r\n<\/div>\r\n<h3>Converting Between the Scales<\/h3>\r\nBy looking at the two thermometers shown, you can make some general comparisons between the scales. For example, many people tend to be comfortable in outdoor temperatures between 50\u00b0F and 80\u00b0F (or between 10\u00b0C and 25\u00b0C). If a meteorologist predicts an average temperature of 0\u00b0C (or 32\u00b0F), then it is a safe bet that you will need a winter jacket.\r\n\r\nSometimes, it is necessary to convert a Celsius measurement to its exact Fahrenheit measurement or vice versa. For example, what if you want to know the temperature of your child in Fahrenheit, and the only thermometer you have measures temperature in Celsius measurement? Converting temperature between the systems is a straightforward process as long as you use the formulas provided below.\r\n<div class=\"textbox\">\r\n<h3>Temperature Conversion Formulas<\/h3>\r\nTo convert a Fahrenheit measurement to a Celsius measurement, use this formula.\r\n<p style=\"text-align: center;\">[latex] C=\\frac{5}{9}(F-32)[\/latex]<\/p>\r\nTo convert a Celsius measurement to a Fahrenheit measurement, use this formula.\r\n<p style=\"text-align: center;\">[latex] F=\\frac{9}{5}C+32[\/latex]<\/p>\r\n\r\n<\/div>\r\nHow were these formulas developed? They came from comparing the two scales. Since the freezing point is 0\u00b0 in the Celsius scale and 32\u00b0 on the Fahrenheit scale, we subtract 32 when converting from Fahrenheit to Celsius, and add 32 when converting from Celsius to Fahrenheit.\r\n\r\nThere is a reason for the fractions [latex] \\frac{5}{9}[\/latex] and [latex] \\frac{9}{5}[\/latex], also. There are 100 degrees between the freezing (0\u00b0) and boiling points (100\u00b0) of water on the Celsius scale and 180 degrees between the similar points (32\u00b0 and 212\u00b0) on the Fahrenheit scale. Writing these two scales as a ratio, [latex] \\frac{F{}^\\circ }{C{}^\\circ }[\/latex], gives [latex] \\frac{180{}^\\circ }{100{}^\\circ }=\\frac{180{}^\\circ \\div 20}{100{}^\\circ \\div 20}=\\frac{9}{5}[\/latex]. If you flip the ratio to be [latex] \\frac{\\text{C}{}^\\circ }{\\text{F}{}^\\circ }[\/latex], you get [latex] \\frac{100{}^\\circ }{180{}^\\circ }=\\frac{100{}^\\circ \\div 20}{180{}^\\circ \\div 20}=\\frac{5}{9}[\/latex]. Notice how these fractions are used in the conversion formulas.\r\n\r\nThe example below illustrates the conversion of Celsius temperature to Fahrenheit temperature, using the boiling point of water, which is 100\u00b0 C.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nThe boiling point of water is 100\u00b0C. What temperature does water boil at in the Fahrenheit scale?\r\n\r\nA Celsius temperature is given. To convert it to the Fahrenheit scale, use the formula at the left.\r\n<p style=\"text-align: center;\">[latex]F=\\frac{9}{5}C+32[\/latex]<\/p>\r\nSubstitute 100 for <i>C<\/i> and multiply.\r\n<p style=\"text-align: center;\">[latex]F=\\frac{9}{5}(100)+32[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]F=\\frac{900}{5}+32[\/latex]<\/p>\r\nSimplify [latex]\\frac{900}{5}[\/latex] by dividing numerator and denominator by 5.\r\n<p style=\"text-align: center;\">[latex]F=\\frac{900\\div 5}{5\\div 5}+32[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]F=\\frac{180}{1}+32[\/latex]<\/p>\r\nAdd [latex]180+32[\/latex].\r\n<p style=\"text-align: center;\" align=\"right\">[latex]F=212[\/latex]<\/p>\r\nThe boiling point of water is 212\u00b0F.\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY IT NOW<\/h3>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=1011&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nWater freezes at 32\u00b0F. On the Celsius scale, what temperature is this?\r\n\r\n[reveal-answer q=\"825354\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"825354\"]\r\n\r\nA Fahrenheit temperature is given. To convert it to the Celsius scale, use the formula at the left.\r\n<p style=\"text-align: center;\" align=\"right\">[latex] C=\\frac{5}{9}(F-32)[\/latex]<\/p>\r\nSubstitute 32 for <i>F<\/i> and subtract.\r\n<p style=\"text-align: center;\" align=\"right\">[latex] C=\\frac{5}{9}(32-32)[\/latex]<\/p>\r\nAny number multiplied by 0 is 0.\r\n<p style=\"text-align: center;\" align=\"right\">[latex] C=\\frac{5}{9}(0)[\/latex]<\/p>\r\n<p style=\"text-align: center;\" align=\"right\">[latex] C=0[\/latex]<\/p>\r\nThe freezing point of water is [latex]0^{\\circ}\\text{C}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY IT NOW<\/h3>\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=1010&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\nThe two previous problems used the conversion formulas to verify some temperature conversions that were discussed earlier\u0097the boiling and freezing points of water. The next example shows how these formulas can be used to solve a real-world problem using different temperature scales.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nTwo scientists are doing an experiment designed to identify the boiling point of an unknown liquid. One scientist gets a result of 120\u00b0C; the other gets a result of 250\u00b0F. Which temperature is higher and by how much?\r\n\r\n[reveal-answer q=\"607680\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"607680\"]\r\n\r\nOne temperature is given in \u00b0C, and the other is given in \u00b0F. To find the difference between them, we need to measure them on the same scale.\r\n\r\nWhat is the difference between 120\u00b0C and 250\u00b0F?\r\n\r\nUse the conversion formula to convert 120\u00b0C to \u00b0F.\r\n\r\n(You could convert 250\u00b0F to \u00b0C instead; this is explained in the text after this example.)\r\n<p style=\"text-align: center;\">[latex] F=\\frac{9}{5}C+32[\/latex]<\/p>\r\nSubstitute 120 for <i>C<\/i>.\r\n<p style=\"text-align: center;\">[latex] F=\\frac{9}{5}(120)+32[\/latex]<\/p>\r\nMultiply.\r\n<p style=\"text-align: center;\">[latex] F=\\frac{1080}{5}+32[\/latex]<\/p>\r\nSimplify [latex] \\frac{1080}{5}[\/latex] by dividing numerator and denominator by 5.\r\n<p style=\"text-align: center;\">[latex] F=\\frac{1080\\div 5}{5\\div 5}+32[\/latex]<\/p>\r\nAdd [latex]216+32[\/latex].\r\n<p style=\"text-align: center;\">[latex] F=\\frac{216}{1}+32[\/latex]<\/p>\r\nYou have found that [latex]120^{\\circ}\\text{C}=248^{\\circ}\\text{F}[\/latex].\r\n<p style=\"text-align: center;\">[latex] F=248[\/latex]<\/p>\r\nTo find the difference between 248\u00b0<i>F<\/i> and 250\u00b0F, subtract.\r\n<p style=\"text-align: center;\">[latex]250^{\\circ}\\text{F}-248^{\\circ}\\text{F}=2^{\\circ}\\text{F}[\/latex]<\/p>\r\n250\u00b0F is the higher temperature by 2\u00b0F.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nYou could have converted 250\u00b0F to \u00b0C instead, and then found the difference in the two measurements. (Had you done it this way, you would have found that [latex]250^{\\circ}\\text{F}=121.1^{\\circ}\\text{C}[\/latex], and that 121.1\u00b0C is 1.1\u00b0C higher than 120\u00b0C.) Whichever way you choose, it is important to compare the temperature measurements within the same scale, and to apply the conversion formulas accurately.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nTatiana is researching vacation destinations, and she sees that the average summer temperature in Barcelona, Spain is around 26\u00b0C. What is the average temperature in degrees Fahrenheit?\r\n\r\n<\/div>\r\n<h3>Summary<\/h3>\r\nTemperature is often measured in one of two scales: the Celsius scale and the Fahrenheit scale. A Celsius thermometer will measure the boiling point of water at 100\u00b0 and its freezing point at 0\u00b0; a Fahrenheit thermometer will measure the same events at 212\u00b0 for the boiling point of water and 32\u00b0 as its freezing point. You can use conversion formulas to convert a measurement made in one scale to the other scale.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Describe the general relationship between the U.S. customary units and metric units of length, weight\/mass, and volume<\/li>\n<li>Define the metric prefixes and use them to perform basic conversions among metric units<\/li>\n<li>Solve application problems using metric units<\/li>\n<\/ul>\n<ul>\n<li>State the freezing and boiling points of water on the Celsius and Fahrenheit temperature scales.<\/li>\n<li>Convert from one temperature scale to the other, using conversion formulas<\/li>\n<\/ul>\n<\/div>\n<p>In the United States, both the <b>U.S. customary measurement system<\/b> and the <b>metric system<\/b> are used, especially in medical, scientific, and technical fields. In most other countries, the metric system is the primary system of measurement. If you travel to other countries, you will see that road signs list distances in kilometers and milk is sold in liters. People in many countries use words like \u0093kilometer,\u0094 \u0093liter,\u0094 and \u0093milligram\u0094 to measure the length, volume, and weight of different objects. These measurement units are part of the metric system.<\/p>\n<p>Unlike the U.S. customary system of measurement, the metric system is based on 10s. For example, a liter is 10 times larger than a deciliter, and a centigram is 10 times larger than a milligram. This idea of \u009310\u0094 is not present in the U.S. customary system\u0097there are 12 inches in a foot, and 3 feet in a yard and 5,280 feet in a mile!<\/p>\n<p>So, what if you have to find out how many milligrams are in a decigram? Or, what if you want to convert meters to kilometers? Understanding how the metric system works is a good start.<\/p>\n<p>In this section we will discover the basic units used in the metric system, and show how to convert between them. We will also explore temperature scales.\u00a0In the United States, temperatures are usually measured using the <b>Fahrenheit<\/b> scale, while most countries that use the metric system use the <b>Celsius<\/b> scale to record temperatures. Learning about the different scales\u0097, including how to convert between them\u0097 will help you figure out what the weather is going to be like, no matter which country you find yourself in.<\/p>\n<p>&nbsp;<\/p>\n<h2>Metric System Basics<\/h2>\n<h3>What Is Metric?<\/h3>\n<p>The metric system uses units such as <b>meter<\/b>, <b>liter<\/b>, and <b>gram<\/b> to measure length, liquid volume, and mass, just as the U.S. customary system uses feet, quarts, and ounces to measure these.<\/p>\n<p>In addition to the difference in the basic units, the metric system is based on 10s, and different measures for length include kilometer, meter, decimeter, centimeter, and millimeter. Notice that the word \u0093meter\u0094 is part of all of these units.<\/p>\n<p>The metric system also applies the idea that units within the system get larger or smaller by a power of 10. This means that a meter is 100 times larger than a centimeter, and a kilogram is 1,000 times heavier than a gram. You will explore this idea a bit later. For now, notice how this idea of \u0093getting bigger or smaller by 10\u0094 is very different than the relationship between units in the U.S. customary system, where 3 feet equals 1 yard, and 16 ounces equals 1 pound.<\/p>\n<h3>Length, Mass, and Volume<\/h3>\n<p>The table below shows the basic units of the metric system. Note that the names of all metric units follow from these three basic units.<\/p>\n<table cellpadding=\"0\" style=\"border-spacing: 0px;\">\n<tbody>\n<tr>\n<td><b>Length<\/b><\/td>\n<td><b>Mass<\/b><\/td>\n<td><b>Volume<\/b><\/td>\n<\/tr>\n<tr>\n<td colspan=\"3\"><i>basic units<\/i><\/td>\n<\/tr>\n<tr>\n<td>meter<\/td>\n<td>gram<\/td>\n<td>liter<\/td>\n<\/tr>\n<tr>\n<td colspan=\"3\"><i>other units you may see<\/i><\/td>\n<\/tr>\n<tr>\n<td>kilometer<\/td>\n<td>kilogram<\/td>\n<td>dekaliter<\/td>\n<\/tr>\n<tr>\n<td>centimeter<\/td>\n<td>centigram<\/td>\n<td>centiliter<\/td>\n<\/tr>\n<tr>\n<td>millimeter<\/td>\n<td>milligram<\/td>\n<td>milliliter<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>In the metric system, the basic unit of length is the meter. A meter is slightly larger than a yardstick, or just over three feet.<\/p>\n<p>The basic metric unit of mass is the gram. A regular-sized paperclip has a mass of about 1 gram.<\/p>\n<p>Among scientists, one gram is defined as the mass of water that would fill a 1-centimeter cube. You may notice that the word \u0093mass\u0094 is used here instead of \u0093weight.\u0094 In the sciences and technical fields, a distinction is made between weight and mass. Weight is a measure of the pull of gravity on an object. For this reason, an object\u0092s weight would be different if it was weighed on Earth or on the moon because of the difference in the gravitational forces. However, the object\u0092s mass would remain the same in both places because mass measures the amount of substance in an object. As long as you are planning on only measuring objects on Earth, you can use mass\/weight fairly interchangeably\u0097but it is worth noting that there is a difference!<\/p>\n<p>Finally, the basic metric unit of volume is the liter. A liter is slightly larger than a quart.<\/p>\n<table cellpadding=\"0\" style=\"width: 602px; border-spacing: 0px;\">\n<tbody>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/30201020\/image106.jpg\" width=\"162\" height=\"109\" alt=\"image\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/30201021\/image107.jpg\" width=\"148\" height=\"104\" alt=\"image\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/30201022\/image108.jpg\" width=\"94\" height=\"117\" alt=\"image\" \/><\/td>\n<\/tr>\n<tr>\n<td>The handle of a shovel is about 1 meter.<\/td>\n<td>A paperclip weighs about 1 gram.<\/td>\n<td>A medium-sized container of milk is about 1 liter.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Though it is rarely necessary to convert between the customary and metric systems, sometimes it helps to have a mental image of how large or small some units are. The table below shows the relationship between some common units in both systems.<\/p>\n<table cellpadding=\"0\" style=\"width: 507px; border-spacing: 0px;\">\n<tbody>\n<tr>\n<td><\/td>\n<td><b>Common Measurements in Customary and Metric Systems<\/b><\/td>\n<\/tr>\n<tr>\n<td><i>Length<\/i><\/td>\n<td>1 centimeter is a little less than half an inch.<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>1.6 kilometers is about 1 mile.<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>1 meter is about 3 inches longer than 1 yard.<\/td>\n<\/tr>\n<tr>\n<td><i>Mass<\/i><\/td>\n<td>1 kilogram is a little more than 2 pounds.<\/td>\n<\/tr>\n<tr>\n<td><i>\u00a0<\/i><\/td>\n<td>28 grams is about the same as 1 ounce.<\/td>\n<\/tr>\n<tr>\n<td><i>Volume<\/i><\/td>\n<td>1 liter is a little more than 1 quart.<\/td>\n<\/tr>\n<tr>\n<td><i>\u00a0<\/i><\/td>\n<td>4 liters is a little more than 1 gallon.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>Prefixes in the Metric System<\/h3>\n<p>The metric system is a base 10 system. This means that each successive unit is 10 times larger than the previous one.<\/p>\n<p>The names of metric units are formed by adding a prefix to the basic unit of measurement. To tell how large or small a unit is, you look at the <b>prefix<\/b>. To tell whether the unit is measuring length, mass, or volume, you look at the base.<\/p>\n<table cellpadding=\"0\" style=\"border-spacing: 0px;\">\n<tbody>\n<tr>\n<td colspan=\"7\"><b>Prefixes in the Metric System<\/b><\/td>\n<\/tr>\n<tr>\n<td><i>kilo-<\/i><\/td>\n<td><i>hecto-<\/i><\/td>\n<td><i>deka-<\/i><\/td>\n<td>meter<\/p>\n<p>gram<\/p>\n<p>liter<\/td>\n<td><i>deci-<\/i><\/td>\n<td><i>centi-<\/i><\/td>\n<td><i>milli-<\/i><\/td>\n<\/tr>\n<tr>\n<td>1,000 times <b>larger<\/b> than base unit<\/td>\n<td>100 times <b>larger<\/b> than base unit<\/td>\n<td>10 times <b>larger<\/b> than base unit<\/td>\n<td>base units<\/td>\n<td>10 times <b>smaller<\/b> than base unit<\/td>\n<td>100 times <b>smaller<\/b> than base unit<\/td>\n<td>1,000 times <b>smaller<\/b> than base unit<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Using this table as a reference, you can see the following:<\/p>\n<ul>\n<li>A kilogram is 1,000 times larger than one gram (so 1 kilogram = 1,000 grams).<\/li>\n<li>A centimeter is 100 times smaller than one meter (so 1 meter = 100 centimeters).<\/li>\n<li>A dekaliter is 10 times larger than one liter (so 1 dekaliter = 10 liters).<\/li>\n<\/ul>\n<p>Here is a similar table that just shows the metric units of measurement for mass, along with their size relative to 1 gram (the base unit). The common abbreviations for these metric units have been included as well.<\/p>\n<table cellpadding=\"0\" style=\"border-spacing: 0px;\">\n<tbody>\n<tr>\n<td colspan=\"7\"><b>Measuring Mass in the Metric System<\/b><\/td>\n<\/tr>\n<tr>\n<td>kilogram<\/p>\n<p>(kg)<\/td>\n<td>hectogram<\/p>\n<p>(hg)<\/td>\n<td>dekagram<\/p>\n<p>(dag)<\/td>\n<td>gram<\/p>\n<p>(g)<\/td>\n<td>decigram<\/p>\n<p>(dg)<\/td>\n<td>centigram<\/p>\n<p>(cg)<\/td>\n<td>milligram<\/p>\n<p>(mg)<\/td>\n<\/tr>\n<tr>\n<td>1,000 grams<\/td>\n<td>100 grams<\/td>\n<td>10 grams<\/td>\n<td>gram<\/td>\n<td>0.1 gram<\/td>\n<td>0.01 gram<\/td>\n<td>0.001 gram<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Since the prefixes remain constant through the metric system, you could create similar charts for length and volume. The prefixes have the same meanings whether they are attached to the units of length (meter), mass (gram), or volume (liter).<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Which of the following sets of three units are all metric measurements of <strong>length<\/strong>?<\/p>\n<p>A) inch, foot, yard<\/p>\n<p>B) kilometer, centimeter, millimeter<\/p>\n<p>C) kilogram, gram, centigram<\/p>\n<p>D) kilometer, foot, decimeter<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q728320\">Show Solution<\/span><\/p>\n<div id=\"q728320\" class=\"hidden-answer\" style=\"display: none\">\n<p>B) kilometer, centimeter, millimeter<\/p>\n<p>All of these measurements are from the metric system. You can tell they are measurements of length because they all contain the word \u0093meter.\u0094<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom5\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=126793&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"200\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=126794&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"200\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=126795&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"200\"><\/iframe><\/p>\n<\/div>\n<h3>Converting Units Up and Down the Metric Scale<\/h3>\n<p>Converting between metric units of measure requires knowledge of the metric prefixes and an understanding of the decimal system\u0097that\u0092s about it.<\/p>\n<p>For instance, you can figure out how many centigrams are in one dekagram by using the table above. One dekagram is larger than one centigram, so you expect that one dekagram will equal many centigrams.<\/p>\n<p>In the table, each unit is 10 times larger than the one to its immediate right. This means that 1 dekagram = 10 grams; 10 grams = 100 decigrams; and 100 decigrams = 1,000 centigrams. So, 1 dekagram = 1,000 centigrams.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>How many milligrams are in one decigram?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q363102\">Show Solution<\/span><\/p>\n<div id=\"q363102\" class=\"hidden-answer\" style=\"display: none\">\n<p>Identify locations of milligrams and decigrams.<\/p>\n<table>\n<tbody>\n<tr>\n<td>kg<\/td>\n<td>hg<\/td>\n<td>dag<\/td>\n<td>g<\/td>\n<td>dg<\/td>\n<td>cg<\/td>\n<td>mg<\/td>\n<\/tr>\n<tr>\n<td>^<\/td>\n<td>^<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Decigrams (dg) are larger than milligrams (mg), so you expect there to be many mg in one dg.<\/p>\n<p>Dg is 10 times larger than a cg, and a cg is 10 times larger than a mg.<\/p>\n<table>\n<tbody>\n<tr>\n<td>[latex]\\times10[\/latex]<\/td>\n<td>[latex]\\times10[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>kg<\/td>\n<td>hg<\/td>\n<td>dag<\/td>\n<td>g<\/td>\n<td>dg<\/td>\n<td>cg<\/td>\n<td>mg<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\downarrow[\/latex]<\/td>\n<td>[latex]\\uparrow[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">[latex]\\rightarrow[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Since you are going from a larger unit to a smaller unit, multiply.<\/p>\n<p><i>Multiply: 1 \u00b7 10 \u00b7 10, to find the number of milligrams in one decigram.\u00a0<\/i><\/p>\n<p style=\"text-align: center;\">[latex]1\\text{ dg}\\cdot10\\cdot10=100\\text{ mg}[\/latex]<\/p>\n<p>There are 100 milligrams (mg) in 1 decigram (dg).<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Convert 3,085 milligrams to grams.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q353889\">Show Solution<\/span><\/p>\n<div id=\"q353889\" class=\"hidden-answer\" style=\"display: none\">\n<p>One gram is 1,000 times larger than a milligram, so you can move the decimal point in 3,085 three places to the left.<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom15\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=1000&amp;theme=oea&amp;iframe_resize_id=mom15\" width=\"100%\" height=\"250\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom100\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=1001&amp;theme=oea&amp;iframe_resize_id=mom100\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"mom13\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=1005&amp;theme=oea&amp;iframe_resize_id=mom13\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Convert 1 centimeter to kilometers.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q4330\">Show Solution<\/span><\/p>\n<div id=\"q4330\" class=\"hidden-answer\" style=\"display: none\">\n<p>Identify locations of kilometers and centimeters.<\/p>\n<table>\n<tbody>\n<tr>\n<td>km<\/td>\n<td>hm<\/td>\n<td>dam<\/td>\n<td>m<\/td>\n<td>dm<\/td>\n<td>cm<\/td>\n<td>mm<\/td>\n<\/tr>\n<tr>\n<td>^<\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td><\/td>\n<td>^<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Kilometers (km) are larger than centimeters (cm), so you expect there to be less than one km in a cm.<\/p>\n<p>Cm is 10 times smaller than a dm; a dm is 10 times smaller than a m, etc.<\/p>\n<p>Since you are going from a smaller unit to a larger unit, divide.<\/p>\n<table>\n<tbody>\n<tr>\n<td>[latex]\\div10[\/latex]<\/td>\n<td>[latex]\\div10[\/latex]<\/td>\n<td>[latex]\\div10[\/latex]<\/td>\n<td>[latex]\\div10[\/latex]<\/td>\n<td>[latex]\\div10[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>km<\/td>\n<td>hm<\/td>\n<td>dam<\/td>\n<td>m<\/td>\n<td>dm<\/td>\n<td>cm<\/td>\n<td>mm<\/td>\n<\/tr>\n<tr>\n<td>^<\/td>\n<td>[latex]\\leftarrow[\/latex]<\/td>\n<td>[latex]\\leftarrow[\/latex]<\/td>\n<td>[latex]\\leftarrow[\/latex]<\/td>\n<td>[latex]\\leftarrow[\/latex]<\/td>\n<td>^<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Divide: [latex]1\\div10\\div10\\div10\\div10\\div10[\/latex], to find the number of kilometers in one centimeter.<em>\u00a0<\/em><\/p>\n<p style=\"text-align: center;\">[latex]1\\text{ cm}\\div10\\div10\\div10\\div10\\div10=0.00001\\text{ km}[\/latex]<\/p>\n<p>1 centimeter (cm) = 0.00001 kilometers (km).<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom200\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=998&amp;theme=oea&amp;iframe_resize_id=mom200\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<p>Once you begin to understand the metric system, you can use a shortcut to convert among different metric units. The size of metric units increases tenfold as you go up the metric scale. The decimal system works the same way: a tenth is 10 times larger than a hundredth; a hundredth is 10 times larger than a thousandth, etc. By applying what you know about decimals to the metric system, converting among units is as simple as moving decimal points.<\/p>\n<p>Here is the first problem from above: How many milligrams are in one decigram? You can recreate the order of the metric units as shown below:<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle kg\\quad hg\\quad dag\\quad g\\quad d\\underbrace{g\\quad c}_{1}\\underbrace{g\\quad m}_{2}g[\/latex]<\/p>\n<p>This question asks you to start with 1 decigram and convert that to milligrams. As shown above, milligrams is two places to the right of decigrams. You can just move the decimal point two places to the right to convert decigrams to milligrams: [latex]\\displaystyle 1\\ dg=1\\underbrace{0}_{1}\\underbrace{0}_{2}.\\ mg[\/latex].<\/p>\n<p>The same method works when you are converting from a smaller to a larger unit, as in the problem: Convert 1 centimeter to kilometers.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle k\\underbrace{m\\quad h}_{5}\\underbrace{m\\quad d}_{4}\\underbrace{am\\quad }_{3}\\underbrace{m\\quad d}_{2}\\underbrace{m\\quad c}_{1}m\\quad mm[\/latex]<\/p>\n<p>Note that instead of moving to the right, you are now moving to the left\u0097so the decimal point must do the same:<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle 1\\ cm=0.\\underbrace{0}_{5}\\underbrace{0}_{4}\\underbrace{0}_{3}\\underbrace{0}_{2}\\underbrace{1}_{1}\\ km[\/latex].<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>How many milliliters are in 1 liter?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q95548\">Show Solution<\/span><\/p>\n<div id=\"q95548\" class=\"hidden-answer\" style=\"display: none\">\n<p>There are 10 milliliters in a centiliter, 10 centiliters in a deciliter, and 10 deciliters in a liter. Multiply: [latex]10\\cdot10\\cdot10[\/latex], to find the number of milliliters in a liter, 1,000.<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom10\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=999&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<h3>Factor Label Method<\/h3>\n<p>There is yet another method that you can use to convert metric measurements\u0097the <b>factor label method<\/b>. You used this method when you were converting measurement units within the U.S. customary system.<\/p>\n<p>The factor label method works the same in the metric system; it relies on the use of unit fractions and the cancelling of intermediate units. The table below shows some of the <b>unit equivalents<\/b> and <b>unit fractions<\/b> for length in the metric system. (You should notice that all of the unit fractions contain a factor of 10. Remember that the metric system is based on the notion that each unit is 10 times larger than the one that came before it.)<\/p>\n<p>Also, notice that two new prefixes have been added here: mega- (which is very big) and micro- (which is very small).<\/p>\n<table cellpadding=\"0\" style=\"border-spacing: 0px;\">\n<tbody>\n<tr>\n<td><b>Unit Equivalents<\/b><\/td>\n<td colspan=\"2\"><b>Conversion Factors<\/b><\/td>\n<\/tr>\n<tr>\n<td>1 meter = 1,000,000 micrometers<\/td>\n<td>[latex]\\displaystyle \\frac{1\\ m}{1,000,000\\ \\mu m}[\/latex]<\/td>\n<td>[latex]\\displaystyle \\frac{1,000,000\\ \\mu m}{1\\ m}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>1 meter = 1,000 millimeters<\/td>\n<td>[latex]\\displaystyle \\frac{1\\ m}{1,000\\ mm}[\/latex]<\/td>\n<td>[latex]\\displaystyle \\frac{1,000\\ mm}{1\\ m}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>1 meter = 100 centimeters<\/td>\n<td>[latex]\\displaystyle \\frac{1\\ m}{100\\ cm}[\/latex]<\/td>\n<td>[latex]\\displaystyle \\frac{100\\ cm}{1\\ m}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>1 meter = 10 decimeters<\/td>\n<td>[latex]\\displaystyle \\frac{1\\ m}{10\\ dm}[\/latex]<\/td>\n<td>[latex]\\displaystyle \\frac{10\\ dm}{1\\ m}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>1 dekameter = 10 meters<\/td>\n<td>[latex]\\displaystyle \\frac{1\\ dam}{10\\ m}[\/latex]<\/td>\n<td>[latex]\\displaystyle \\frac{10\\ m}{1\\ dam}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>1 hectometer = 100 meters<\/td>\n<td>[latex]\\displaystyle \\frac{1\\ hm}{100\\ m}[\/latex]<\/td>\n<td>[latex]\\displaystyle \\frac{100\\ m}{1\\ hm}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>1 kilometer = 1,000 meters<\/td>\n<td>[latex]\\displaystyle \\frac{1\\ km}{1,000\\ m}[\/latex]<\/td>\n<td>[latex]\\displaystyle \\frac{1,000\\ m}{1\\ km}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>1 megameter = 1,000,000 meters<\/td>\n<td>[latex]\\displaystyle \\frac{1\\ Mm}{1,000,000\\ m}[\/latex]<\/td>\n<td>[latex]\\displaystyle \\frac{1,000,000\\ m}{1\\ Mm}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>When applying the factor label method in the metric system, be sure to check that you are not skipping over any intermediate units of measurement!<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Convert 7,225 centimeters to meters.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q461145\">Show Solution<\/span><\/p>\n<div id=\"q461145\" class=\"hidden-answer\" style=\"display: none\">\n<p>Meters is larger than centimeters, so you expect your answer to be less than 7,225.<\/p>\n<p style=\"text-align: center;\">[latex]7,225\\text{ cm}=\\text{___ m}[\/latex]<\/p>\n<p>Using the factor label method, write 7,225 cm as a fraction and use unit fractions to convert it to m.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\frac{7,225\\ cm}{1}\\cdot \\frac{1\\ m}{100\\ cm}=\\_\\_\\_ m[\/latex]<\/p>\n<p>Cancel similar units, multiply, and simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\frac{7,225\\ \\cancel{cm}}{1}\\cdot \\frac{1\\text{ m}}{100\\ \\cancel{\\text{cm}}}=\\_\\_\\_m[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\frac{7,225}{1}\\cdot \\frac{1\\text{ m}}{100}=\\frac{7,225}{100}\\text{m}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\frac{7,225\\text{ m}}{100}=72.25\\text{ m}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]7,225\\text{ centimeters}=72.25\\text{ meters}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Convert 32.5 kilometers to meters.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q574914\">Show Solution<\/span><\/p>\n<div id=\"q574914\" class=\"hidden-answer\" style=\"display: none\">\n<p>32,500 meters<\/p>\n<p>[latex]\\displaystyle \\frac{32.5\\text{ km}}{1}\\cdot \\frac{1,000\\text{ m}}{1\\text{ km}}=\\frac{32,500\\text{ m}}{1}[\/latex].<\/p>\n<p>The km units cancel, leaving the answer in m.<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom500\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=18877&amp;theme=oea&amp;iframe_resize_id=mom500\" width=\"100%\" height=\"200\"><\/iframe><\/p>\n<\/div>\n<p>Now that you have seen how to convert among metric measurements in multiple ways, let\u0092&#8217;s revisit the problem posed earlier.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>If you have a prescription for 5,000 mg of medicine, and upon getting it filled, the dosage reads 5g of medicine, did the pharmacist make a mistake?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q600572\">Show Solution<\/span><\/p>\n<div id=\"q600572\" class=\"hidden-answer\" style=\"display: none\">\n<p>Convert mg to g.<\/p>\n<p style=\"text-align: center;\">[latex]5,000\\text{ mg}=\\text{___ g}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\frac{5,000\\text{ mg}}{1}\\cdot \\frac{1\\text{ g}}{1,000\\text{ mg}}=\\text{ g}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\frac{5,000\\cancel{\\text{mg}}}{1}\\cdot \\frac{1\\text{ g}}{1,000\\ \\cancel{\\text{mg}}}=\\text{ g}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\frac{5,000\\cdot 1\\text{ g}}{1\\cdot 1,000}=\\frac{5,000\\text{ g}}{1,000}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\frac{5,000\\text{ g}}{1,000}=5\\text{ g}[\/latex]<\/p>\n<p>[latex]5\\text{ g}=5,000\\text{ mg}[\/latex], so the pharmacist did not make a mistake.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Applications of Metric Conversions<\/h2>\n<p>Learning how to solve real-world problems using metric conversions is as important as learning how to do the conversions themselves. Mathematicians, scientists, nurses, and even athletes are often confronted with situations where they are presented with information using metric measurements, and must then make informed decisions based on that data.<\/p>\n<div class=\"textbox\"><strong>TIP:<\/strong> To solve these problems effectively, you need to understand the context of a problem, perform conversions, and then check the reasonableness of your answer. Do all three of these steps and you will succeed in whatever measurement system you find yourself using.<\/div>\n<h3>Understanding Context and Performing Conversions<\/h3>\n<p>The first step in solving any real-world problem is to understand its context. This will help you figure out what kinds of solutions are reasonable (and the problem itself may give you clues about what types of conversions are necessary). Here is an example.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Marcus bought at 2 meter board, and cut off a piece 1 meter and 35 cm long. How much board is left?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q701860\">Show Solution<\/span><\/p>\n<div id=\"q701860\" class=\"hidden-answer\" style=\"display: none\">\n<p>To answer this question, we will need to subtract.<\/p>\n<p>First convert all measurements to one unit. Here we will convert to centimeters.<\/p>\n<p>[latex]2\\text{ meters}-1\\text{ meter and }35\\text{ cm}[\/latex]<\/p>\n<p>Use the factor label method and unit fractions to convert from meters to centimeters.<\/p>\n<p>[latex]\\displaystyle \\frac{2\\text{ m}}{1}\\cdot \\frac{100\\text{ cm}}{1\\text{ m}}=\\text{ cm}[\/latex]<\/p>\n<p>Cancel, multiply, and solve.<\/p>\n<p>Convert the 1 meter to centimeters, and combine with the additional 35 centimeters.<\/p>\n<p>Subtract the cut length from the original board length.<\/p>\n<p>[latex]\\displaystyle \\frac{2\\ \\cancel{\\text{m}}}{1}\\cdot \\frac{100\\text{ cm}}{1\\ \\cancel{\\text{ m}}}=\\text{ cm}[\/latex]<\/p>\n<p>[latex]\\displaystyle \\frac{200\\text{ cm}}{1}=200\\text{ cm}[\/latex]<\/p>\n<p>[latex]1\\text{ meter}+35\\text{ cm}[\/latex]<\/p>\n<p>[latex]100\\text{ cm}+35\\text{ cm}[\/latex]<\/p>\n<p>[latex]135\\text{ cm}[\/latex]<\/p>\n<p>[latex]200\\text{ cm}-135\\text{ cm}[\/latex]<\/p>\n<p>[latex]65\\text{ cm}[\/latex]<\/p>\n<p>There is 65 cm of board left.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>An example with a different context, but still requiring conversions, is shown below.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>A faucet drips 10 ml every minute. How much water will be wasted in a week?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q642392\">Show Solution<\/span><\/p>\n<div id=\"q642392\" class=\"hidden-answer\" style=\"display: none\">\n<p>Start by calculating how much water will be used in a week using the factor label method to convert the time units.<\/p>\n<p>[latex]\\displaystyle \\frac{10\\ ml}{1\\text{ minute}}\\cdot \\frac{60\\text{ minute}}{1\\text{ hour}}\\cdot \\frac{24\\text{ hours}}{1\\text{ day}}\\cdot \\frac{7\\text{ days}}{1\\text{ week}}[\/latex]<\/p>\n<p>Cancel, multiply and solve.<\/p>\n<p>[latex]\\displaystyle \\frac{10\\ ml}{1\\text{ }\\cancel{\\text{minute}}}\\cdot \\frac{60\\text{ }\\cancel{\\text{minute}}}{1\\text{ }\\cancel{\\text{hour}}}\\cdot \\frac{24\\text{ }\\cancel{\\text{hours}}}{1\\text{ }\\cancel{\\text{day}}}\\cdot \\frac{7\\text{ }\\cancel{\\text{days}}}{1\\text{ week}}[\/latex]<\/p>\n<p>[latex]\\displaystyle \\frac{10\\centerdot 60\\centerdot 24\\centerdot 7\\ ml}{1\\centerdot 1\\centerdot 1\\centerdot 1\\text{ week}}[\/latex]<\/p>\n<p>To give a more useable answer, convert this into liters.<\/p>\n<p>[latex]\\displaystyle \\frac{100800\\ ml}{1\\text{ week}}[\/latex]<\/p>\n<p>Cancel, multiply and solve.<\/p>\n<p>[latex]\\displaystyle \\frac{100800\\text{ ml}}{1\\text{ week}}\\centerdot \\frac{1\\text{ L}}{1000\\text{ ml}}[\/latex]<\/p>\n<p>[latex]\\displaystyle \\frac{100800\\text{ }\\cancel{\\text{ml}}}{1\\text{ week}}\\centerdot \\frac{1\\text{ L}}{1000\\text{ }\\cancel{\\text{ml}}}[\/latex]<\/p>\n<p>[latex]\\displaystyle \\frac{100800\\text{ L}}{1000\\text{ week}}[\/latex]=[latex]\\displaystyle 100.8\\frac{\\text{L}}{\\text{week}}[\/latex]<\/p>\n<p>The faucet wastes about 100.8 liters each week.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>This problem asked for the difference between two quantities. The easiest way to find this is to convert one quantity so that both quantities are measured in the same unit, and then subtract one from the other.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>A bread recipe calls for 600 g of flour. How many kilograms of flour would you need to make 5 loaves?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q838013\">Show Solution<\/span><\/p>\n<div id=\"q838013\" class=\"hidden-answer\" style=\"display: none\">\n<p>Multiplying 600 g per loaf by the 5 loaves,<\/p>\n<p>[latex]600\\text{g}\\cdot5=3000\\text{g}[\/latex]<\/p>\n<p>Using factor labels or the \u0093move the decimal\u0094 method, convert this to 3 kilograms.<\/p>\n<p>You will need 3 kg of flour to make 5 loaves.<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom5\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=1002&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"200\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=117516&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"200\"><\/iframe><\/p>\n<\/div>\n<h3>Checking your Conversions<\/h3>\n<p>Sometimes it is a good idea to check your conversions using a second method. This usually helps you catch any errors that you may make, such as using the wrong unit fractions or moving the decimal point the wrong way.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>A bottle contains 1.5 liters of a beverage. How many 250 mL servings can be made from that bottle?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q451287\">Show Solution<\/span><\/p>\n<div id=\"q451287\" class=\"hidden-answer\" style=\"display: none\">\n<p>To answer the question, you will need to divide 1.5 liters by 250 milliliters. To do this, convert both to the same unit. You could convert either measurement.<\/p>\n<p>[latex]1.5\\text{ L}\\div250\\text{ mL}[\/latex]<\/p>\n<p>Convert 250 mL to liters<\/p>\n<p>[latex]250\\text{ mL}=\\text{ ___ L}[\/latex]<\/p>\n<p>[latex]\\displaystyle \\frac{250\\text{ mL}}{1}\\cdot \\frac{1\\text{ L}}{1000\\text{ mL}}=\\text{___ L}[\/latex]<\/p>\n<p>[latex]\\displaystyle \\frac{250\\text{ L}}{1000=0.25\\text{ L}}[\/latex]<\/p>\n<p>Now we can divide using the converted measurement<\/p>\n<p>[latex]1.5\\text{ L}\\div250\\text{ mL}=\\frac{1.5\\text{ L}}{250\\text{ mL }}=\\frac{1.5\\text{ L}}{0.25\\text{ L}}[\/latex]<\/p>\n<p>[latex]\\displaystyle \\frac{1.5\\text{ L}}{0.25\\text{ L}}=6[\/latex]<\/p>\n<p>The bottle holds 6 servings.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Understanding the context of real-life application problems is important. Look for words within the problem that help you identify what operations are needed, and then apply the correct unit conversions. Checking your final answer by using another conversion method (such as the \u0093move the decimal\u0094 method, if you have used the factor label method to solve the problem) can cut down on errors in your calculations.<\/p>\n<h3 class=\"Subsectiontitleunderline\">Summary<\/h3>\n<p>The metric system is an alternative system of measurement used in most countries, as well as in the United States. The metric system is based on joining one of a series of prefixes, including kilo-, hecto-, deka-, deci-, centi-, and milli-, with a base unit of measurement, such as meter, liter, or gram. Units in the metric system are all related by a power of 10, which means that each successive unit is 10 times larger than the previous one.<\/p>\n<p>This makes converting one metric measurement to another a straightforward process, and is often as simple as moving a decimal point. It is always important, though, to consider the direction of the conversion. If you are converting a smaller unit to a larger unit, then the decimal point has to move to the left (making your number smaller); if you are converting a larger unit to a smaller unit, then the decimal point has to move to the right (making your number larger).<\/p>\n<p>The factor label method can also be applied to conversions within the metric system. To use the factor label method, you multiply the original measurement by unit fractions; this allows you to represent the original measurement in a different measurement unit.<\/p>\n<h2>Temperature Scales<\/h2>\n<p>Turn on the television any morning and you will see meteorologists talking about the day\u0092s weather forecast. In addition to telling you what the weather conditions will be like (sunny, cloudy, rainy, muggy), they also tell you the day\u0092s forecast for high and low temperatures. A hot summer day may reach 100\u00b0 in Philadelphia, while a cool spring day may have a low of 40\u00b0 in Seattle.<\/p>\n<p>If you have been to other countries, though, you may notice that meteorologists measure heat and cold differently outside of the United States. For example, a TV weatherman in San Diego may forecast a high of 89\u00b0, but a similar forecaster in Tijuana, Mexico\u0097, which is only 20 miles south\u0097, may look at the same weather pattern and say that the day\u0092&#8217;s high temperature is going to be 32\u00b0. What\u0092s going on here? The difference is that the two countries use different temperature scales.<\/p>\n<h3>Measuring Temperature on Two Scales<\/h3>\n<p>Fahrenheit and Celsius are two different scales for measuring temperature.<\/p>\n<table cellpadding=\"0\" style=\"border-spacing: 0px;\">\n<tbody>\n<tr>\n<td>A thermometer measuring a temperature of 22\u00b0 Celsius is shown here.<\/p>\n<p>On the Celsius scale, water freezes at 0\u00b0 and boils at 100\u00b0.<\/p>\n<p>If the United States were to adopt the Celsius scale, forecast temperatures would rarely go below -30\u00b0 or above 45\u00b0. (A temperature of<\/p>\n<p>-18\u00b0 may be forecast for a cold winter day in Michigan, while a temperature of 43\u00b0 may be predicted for a hot summer day in Arizona.)<\/p>\n<p>Most office buildings maintain an indoor temperature between 18\u00b0C and 24\u00b0C to keep employees comfortable.<\/td>\n<td colspan=\"2\">\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1141\/2016\/11\/30201023\/image109.jpg\" width=\"224\" height=\"569\" alt=\"image\" \/><\/p>\n<\/td>\n<td>A thermometer measuring a temperature of 72\u00b0 Fahrenheit is shown here.<\/p>\n<p>On the Fahrenheit scale, water freezes at 32\u00b0 and boils at 212\u00b0.<\/p>\n<p>In the United States, forecast temperatures measured in Fahrenheit rarely go below -20\u00b0 or above 120\u00b0. (A temperature of 0\u00b0 may be forecast for a cold winter day in Michigan, while a temperature of 110\u00b0 may be predicted for a hot summer day in Arizona.)<\/p>\n<p>Most office buildings maintain an indoor temperature between 65\u00b0F and 75\u00b0F to keep employees comfortable.<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><b>Celsius<\/b><\/td>\n<td>\n<p style=\"text-align: right;\"><b>Fahrenheit<\/b><\/p>\n<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>A cook puts a thermometer into a pot of water to see how hot it is. The thermometer reads 132\u00b0, but the water is not boiling yet. Which temperature scale is the thermometer measuring?<\/p>\n<\/div>\n<h3>Converting Between the Scales<\/h3>\n<p>By looking at the two thermometers shown, you can make some general comparisons between the scales. For example, many people tend to be comfortable in outdoor temperatures between 50\u00b0F and 80\u00b0F (or between 10\u00b0C and 25\u00b0C). If a meteorologist predicts an average temperature of 0\u00b0C (or 32\u00b0F), then it is a safe bet that you will need a winter jacket.<\/p>\n<p>Sometimes, it is necessary to convert a Celsius measurement to its exact Fahrenheit measurement or vice versa. For example, what if you want to know the temperature of your child in Fahrenheit, and the only thermometer you have measures temperature in Celsius measurement? Converting temperature between the systems is a straightforward process as long as you use the formulas provided below.<\/p>\n<div class=\"textbox\">\n<h3>Temperature Conversion Formulas<\/h3>\n<p>To convert a Fahrenheit measurement to a Celsius measurement, use this formula.<\/p>\n<p style=\"text-align: center;\">[latex]C=\\frac{5}{9}(F-32)[\/latex]<\/p>\n<p>To convert a Celsius measurement to a Fahrenheit measurement, use this formula.<\/p>\n<p style=\"text-align: center;\">[latex]F=\\frac{9}{5}C+32[\/latex]<\/p>\n<\/div>\n<p>How were these formulas developed? They came from comparing the two scales. Since the freezing point is 0\u00b0 in the Celsius scale and 32\u00b0 on the Fahrenheit scale, we subtract 32 when converting from Fahrenheit to Celsius, and add 32 when converting from Celsius to Fahrenheit.<\/p>\n<p>There is a reason for the fractions [latex]\\frac{5}{9}[\/latex] and [latex]\\frac{9}{5}[\/latex], also. There are 100 degrees between the freezing (0\u00b0) and boiling points (100\u00b0) of water on the Celsius scale and 180 degrees between the similar points (32\u00b0 and 212\u00b0) on the Fahrenheit scale. Writing these two scales as a ratio, [latex]\\frac{F{}^\\circ }{C{}^\\circ }[\/latex], gives [latex]\\frac{180{}^\\circ }{100{}^\\circ }=\\frac{180{}^\\circ \\div 20}{100{}^\\circ \\div 20}=\\frac{9}{5}[\/latex]. If you flip the ratio to be [latex]\\frac{\\text{C}{}^\\circ }{\\text{F}{}^\\circ }[\/latex], you get [latex]\\frac{100{}^\\circ }{180{}^\\circ }=\\frac{100{}^\\circ \\div 20}{180{}^\\circ \\div 20}=\\frac{5}{9}[\/latex]. Notice how these fractions are used in the conversion formulas.<\/p>\n<p>The example below illustrates the conversion of Celsius temperature to Fahrenheit temperature, using the boiling point of water, which is 100\u00b0 C.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>The boiling point of water is 100\u00b0C. What temperature does water boil at in the Fahrenheit scale?<\/p>\n<p>A Celsius temperature is given. To convert it to the Fahrenheit scale, use the formula at the left.<\/p>\n<p style=\"text-align: center;\">[latex]F=\\frac{9}{5}C+32[\/latex]<\/p>\n<p>Substitute 100 for <i>C<\/i> and multiply.<\/p>\n<p style=\"text-align: center;\">[latex]F=\\frac{9}{5}(100)+32[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]F=\\frac{900}{5}+32[\/latex]<\/p>\n<p>Simplify [latex]\\frac{900}{5}[\/latex] by dividing numerator and denominator by 5.<\/p>\n<p style=\"text-align: center;\">[latex]F=\\frac{900\\div 5}{5\\div 5}+32[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]F=\\frac{180}{1}+32[\/latex]<\/p>\n<p>Add [latex]180+32[\/latex].<\/p>\n<p style=\"text-align: center; text-align: right;\">[latex]F=212[\/latex]<\/p>\n<p>The boiling point of water is 212\u00b0F.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>TRY IT NOW<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=1011&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Water freezes at 32\u00b0F. On the Celsius scale, what temperature is this?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q825354\">Show Solution<\/span><\/p>\n<div id=\"q825354\" class=\"hidden-answer\" style=\"display: none\">\n<p>A Fahrenheit temperature is given. To convert it to the Celsius scale, use the formula at the left.<\/p>\n<p style=\"text-align: center; text-align: right;\">[latex]C=\\frac{5}{9}(F-32)[\/latex]<\/p>\n<p>Substitute 32 for <i>F<\/i> and subtract.<\/p>\n<p style=\"text-align: center; text-align: right;\">[latex]C=\\frac{5}{9}(32-32)[\/latex]<\/p>\n<p>Any number multiplied by 0 is 0.<\/p>\n<p style=\"text-align: center; text-align: right;\">[latex]C=\\frac{5}{9}(0)[\/latex]<\/p>\n<p style=\"text-align: center; text-align: right;\">[latex]C=0[\/latex]<\/p>\n<p>The freezing point of water is [latex]0^{\\circ}\\text{C}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>TRY IT NOW<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=1010&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<p>The two previous problems used the conversion formulas to verify some temperature conversions that were discussed earlier\u0097the boiling and freezing points of water. The next example shows how these formulas can be used to solve a real-world problem using different temperature scales.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Two scientists are doing an experiment designed to identify the boiling point of an unknown liquid. One scientist gets a result of 120\u00b0C; the other gets a result of 250\u00b0F. Which temperature is higher and by how much?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q607680\">Show Solution<\/span><\/p>\n<div id=\"q607680\" class=\"hidden-answer\" style=\"display: none\">\n<p>One temperature is given in \u00b0C, and the other is given in \u00b0F. To find the difference between them, we need to measure them on the same scale.<\/p>\n<p>What is the difference between 120\u00b0C and 250\u00b0F?<\/p>\n<p>Use the conversion formula to convert 120\u00b0C to \u00b0F.<\/p>\n<p>(You could convert 250\u00b0F to \u00b0C instead; this is explained in the text after this example.)<\/p>\n<p style=\"text-align: center;\">[latex]F=\\frac{9}{5}C+32[\/latex]<\/p>\n<p>Substitute 120 for <i>C<\/i>.<\/p>\n<p style=\"text-align: center;\">[latex]F=\\frac{9}{5}(120)+32[\/latex]<\/p>\n<p>Multiply.<\/p>\n<p style=\"text-align: center;\">[latex]F=\\frac{1080}{5}+32[\/latex]<\/p>\n<p>Simplify [latex]\\frac{1080}{5}[\/latex] by dividing numerator and denominator by 5.<\/p>\n<p style=\"text-align: center;\">[latex]F=\\frac{1080\\div 5}{5\\div 5}+32[\/latex]<\/p>\n<p>Add [latex]216+32[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]F=\\frac{216}{1}+32[\/latex]<\/p>\n<p>You have found that [latex]120^{\\circ}\\text{C}=248^{\\circ}\\text{F}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]F=248[\/latex]<\/p>\n<p>To find the difference between 248\u00b0<i>F<\/i> and 250\u00b0F, subtract.<\/p>\n<p style=\"text-align: center;\">[latex]250^{\\circ}\\text{F}-248^{\\circ}\\text{F}=2^{\\circ}\\text{F}[\/latex]<\/p>\n<p>250\u00b0F is the higher temperature by 2\u00b0F.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>You could have converted 250\u00b0F to \u00b0C instead, and then found the difference in the two measurements. (Had you done it this way, you would have found that [latex]250^{\\circ}\\text{F}=121.1^{\\circ}\\text{C}[\/latex], and that 121.1\u00b0C is 1.1\u00b0C higher than 120\u00b0C.) Whichever way you choose, it is important to compare the temperature measurements within the same scale, and to apply the conversion formulas accurately.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Tatiana is researching vacation destinations, and she sees that the average summer temperature in Barcelona, Spain is around 26\u00b0C. What is the average temperature in degrees Fahrenheit?<\/p>\n<\/div>\n<h3>Summary<\/h3>\n<p>Temperature is often measured in one of two scales: the Celsius scale and the Fahrenheit scale. A Celsius thermometer will measure the boiling point of water at 100\u00b0 and its freezing point at 0\u00b0; a Fahrenheit thermometer will measure the same events at 212\u00b0 for the boiling point of water and 32\u00b0 as its freezing point. You can use conversion formulas to convert a measurement made in one scale to the other scale.<\/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-1060\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Introduction: Systems and Scales of Measurement. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID 126793, 126794. 126795. <strong>Authored by<\/strong>: Lumen Learning. <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>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Question ID 1002. <strong>Authored by<\/strong>: Brooks, Kelly. <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>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 117516. <strong>Authored by<\/strong>: Volpe, Amy. <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>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 1011, 1010. <strong>Authored by<\/strong>: Lippman, David. <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>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">All rights reserved content<\/div><ul class=\"citation-list\"><li>MAT 081 Unit 12 Problem 18. <strong>Authored by<\/strong>: Volpe, Amy. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/bHWyUBIcQHw\">https:\/\/youtu.be\/bHWyUBIcQHw<\/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":21,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Introduction: Systems and Scales of Measurement\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Question ID 126793, 126794. 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