{"id":637,"date":"2016-11-30T22:02:24","date_gmt":"2016-11-30T22:02:24","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/?post_type=chapter&#038;p=637"},"modified":"2019-05-30T16:31:09","modified_gmt":"2019-05-30T16:31:09","slug":"the-metric-system","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/chapter\/the-metric-system\/","title":{"raw":"Metric System Basics","rendered":"Metric System Basics"},"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\n<h2>What Is Metric?<\/h2>\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<h2>Length, Mass, and Volume<\/h2>\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<h2>Prefixes in the Metric System<\/h2>\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<h2>Converting Units Up and Down the Metric Scale<\/h2>\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<h2>Factor Label Method<\/h2>\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>","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<h2>What Is Metric?<\/h2>\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<h2>Length, Mass, and Volume<\/h2>\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<h2>Prefixes in the Metric System<\/h2>\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<h2>Converting Units Up and Down the Metric Scale<\/h2>\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<h2>Factor Label Method<\/h2>\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\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-637\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div 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