Systems and Scales of Measurement
In the United States, both the U.S. customary measurement system and the metric system 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 kilometer, liter, and milligram to measure the length, volume, and weight of different objects. These measurement units are part of the metric system. 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 10 is not present in the U.S. customary systemthere are 12 inches in a foot, and 3 feet in a yard and 5,280 feet in a mile! 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. 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. In the United States, temperatures are usually measured using the Fahrenheit scale, while most countries that use the metric system use the Celsius scale to record temperatures. Learning about the different scales, including how to convert between them will help you figure out what the weather is going to be like, no matter which country you find yourself in.Learning Objectives
In this lesson you will learn how to do the following:- Describe the general relationship between the U.S. customary units and metric units of length, weight/mass, and volume
- Define the metric prefixes and use them to perform basic conversions among metric units
- Solve application problems using metric units
- State the freezing and boiling points of water on the Celsius and Fahrenheit temperature scales.
- Convert from one temperature scale to the other, using conversion formulas
Metric System Basics
What Is Metric?
The metric system uses units such as meter, liter, and gram to measure length, liquid volume, and mass, just as the U.S. customary system uses feet, quarts, and ounces to measure these. 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 meter is part of all of these units. 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 getting bigger or smaller by 10 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.Length, Mass, and Volume
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.Length | Mass | Volume |
basic units | ||
meter | gram | liter |
other units you may see | ||
kilometer | kilogram | dekaliter |
centimeter | centigram | centiliter |
millimeter | milligram | milliliter |
The handle of a shovel is about 1 meter. | A paperclip weighs about 1 gram. | A medium-sized container of milk is about 1 liter. |
Common Measurements in Customary and Metric Systems | |
Length | 1 centimeter is a little less than half an inch. |
1.6 kilometers is about 1 mile. | |
1 meter is about 3 inches longer than 1 yard. | |
Mass | 1 kilogram is a little more than 2 pounds. |
28 grams is about the same as 1 ounce. | |
Volume | 1 liter is a little more than 1 quart. |
4 liters is a little more than 1 gallon. |
Prefixes in the Metric System
The metric system is a base 10 system. This means that each successive unit is 10 times larger than the previous one. 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 prefix. To tell whether the unit is measuring length, mass, or volume, you look at the base.Prefixes in the Metric System | ||||||
kilo- | hecto- | deka- | meter gram liter | deci- | centi- | milli- |
1,000 times larger than base unit | 100 times larger than base unit | 10 times larger than base unit | base units | 10 times smaller than base unit | 100 times smaller than base unit | 1,000 times smaller than base unit |
- A kilogram is 1,000 times larger than one gram (so 1 kilogram = 1,000 grams).
- A centimeter is 100 times smaller than one meter (so 1 meter = 100 centimeters).
- A dekaliter is 10 times larger than one liter (so 1 dekaliter = 10 liters).
Measuring Mass in the Metric System | ||||||
kilogram (kg) | hectogram (hg) | dekagram (dag) | gram (g) | decigram (dg) | centigram (cg) | milligram (mg) |
1,000 grams | 100 grams | 10 grams | gram | 0.1 gram | 0.01 gram | 0.001 gram |
Try It Now
Which of the following sets of three units are all metric measurements of length? A) inch, foot, yard B) kilometer, centimeter, millimeter C) kilogram, gram, centigram D) kilometer, foot, decimeterAnswer: B) kilometer, centimeter, millimeter All of these measurements are from the metric system. You can tell they are measurements of length because they all contain the word meter.
Example
Convert 1 centimeter to kilometers.Answer: Identify locations of kilometers and centimeters.
km | hm | dam | m | dm | cm | mm |
^ | ^ |
[latex]\div10[/latex] | [latex]\div10[/latex] | [latex]\div10[/latex] | [latex]\div10[/latex] | [latex]\div10[/latex] | ||
km | hm | dam | m | dm | cm | mm |
^ | [latex]\leftarrow[/latex] | [latex]\leftarrow[/latex] | [latex]\leftarrow[/latex] | [latex]\leftarrow[/latex] | ^ |
[latex]1\text{ cm}\div10\div10\div10\div10\div10=0.00001\text{ km}[/latex]
1 centimeter (cm) = 0.00001 kilometers (km).Try It Now
Factor Label Method
There is yet another method that you can use to convert metric measurementsthe factor label method. You used this method when you were converting measurement units within the U.S. customary system. 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 unit equivalents and unit fractions 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.) Also, notice that two new prefixes have been added here: mega- (which is very big) and micro- (which is very small).Unit Equivalents | Conversion Factors | |
1 meter = 1,000,000 micrometers | [latex] \displaystyle \frac{1\ m}{1,000,000\ \mu m}[/latex] | [latex] \displaystyle \frac{1,000,000\ \mu m}{1\ m}[/latex] |
1 meter = 1,000 millimeters | [latex] \displaystyle \frac{1\ m}{1,000\ mm}[/latex] | [latex] \displaystyle \frac{1,000\ mm}{1\ m}[/latex] |
1 meter = 100 centimeters | [latex] \displaystyle \frac{1\ m}{100\ cm}[/latex] | [latex] \displaystyle \frac{100\ cm}{1\ m}[/latex] |
1 meter = 10 decimeters | [latex] \displaystyle \frac{1\ m}{10\ dm}[/latex] | [latex] \displaystyle \frac{10\ dm}{1\ m}[/latex] |
1 dekameter = 10 meters | [latex] \displaystyle \frac{1\ dam}{10\ m}[/latex] | [latex] \displaystyle \frac{10\ m}{1\ dam}[/latex] |
1 hectometer = 100 meters | [latex] \displaystyle \frac{1\ hm}{100\ m}[/latex] | [latex] \displaystyle \frac{100\ m}{1\ hm}[/latex] |
1 kilometer = 1,000 meters | [latex] \displaystyle \frac{1\ km}{1,000\ m}[/latex] | [latex] \displaystyle \frac{1,000\ m}{1\ km}[/latex] |
1 megameter = 1,000,000 meters | [latex] \displaystyle \frac{1\ Mm}{1,000,000\ m}[/latex] | [latex] \displaystyle \frac{1,000,000\ m}{1\ Mm}[/latex] |
Example
Convert 7,225 centimeters to meters.Answer: Meters is larger than centimeters, so you expect your answer to be less than 7,225.
[latex]7,225\text{ cm}=\text{___ m}[/latex]
Using the factor label method, write 7,225 cm as a fraction and use unit fractions to convert it to m.[latex] \displaystyle \frac{7,225\ cm}{1}\cdot \frac{1\ m}{100\ cm}=\_\_\_ m[/latex]
Cancel similar units, multiply, and simplify.[latex] \displaystyle \frac{7,225\ \cancel{cm}}{1}\cdot \frac{1\text{ m}}{100\ \cancel{\text{cm}}}=\_\_\_m[/latex]
[latex] \displaystyle \frac{7,225}{1}\cdot \frac{1\text{ m}}{100}=\frac{7,225}{100}\text{m}[/latex]
[latex] \displaystyle \frac{7,225\text{ m}}{100}=72.25\text{ m}[/latex]
[latex]7,225\text{ centimeters}=72.25\text{ meters}[/latex]
Try It Now
Convert 32.5 kilometers to meters.Answer: 32,500 meters [latex] \displaystyle \frac{32.5\text{ km}}{1}\cdot \frac{1,000\text{ m}}{1\text{ km}}=\frac{32,500\text{ m}}{1}[/latex]. The km units cancel, leaving the answer in m.
Example
Water freezes at 32°F. On the Celsius scale, what temperature is this?Answer: A Fahrenheit temperature is given. To convert it to the Celsius scale, use the formula at the left.
[latex] C=\frac{5}{9}(F-32)[/latex]
Substitute 32 for F and subtract.[latex] C=\frac{5}{9}(32-32)[/latex]
Any number multiplied by 0 is 0.[latex] C=\frac{5}{9}(0)[/latex]
[latex] C=0[/latex]
The freezing point of water is [latex]0^{\circ}\text{C}[/latex].TRY IT NOW
Example
Two scientists are doing an experiment designed to identify the boiling point of an unknown liquid. One scientist gets a result of 120°C; the other gets a result of 250°F. Which temperature is higher and by how much?Answer: One temperature is given in °C, and the other is given in °F. To find the difference between them, we need to measure them on the same scale. What is the difference between 120°C and 250°F? Use the conversion formula to convert 120°C to °F. (You could convert 250°F to °C instead; this is explained in the text after this example.)
[latex] F=\frac{9}{5}C+32[/latex]
Substitute 120 for C.[latex] F=\frac{9}{5}(120)+32[/latex]
Multiply.[latex] F=\frac{1080}{5}+32[/latex]
Simplify [latex] \frac{1080}{5}[/latex] by dividing numerator and denominator by 5.[latex] F=\frac{1080\div 5}{5\div 5}+32[/latex]
Add [latex]216+32[/latex].[latex] F=\frac{216}{1}+32[/latex]
You have found that [latex]120^{\circ}\text{C}=248^{\circ}\text{F}[/latex].[latex] F=248[/latex]
To find the difference between 248°F and 250°F, subtract.[latex]250^{\circ}\text{F}-248^{\circ}\text{F}=2^{\circ}\text{F}[/latex]
250°F is the higher temperature by 2°F.Try It Now
Tatiana is researching vacation destinations, and she sees that the average summer temperature in Barcelona, Spain is around 26°C. What is the average temperature in degrees Fahrenheit?Summary
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° and its freezing point at 0°; a Fahrenheit thermometer will measure the same events at 212° for the boiling point of water and 32° as its freezing point. You can use conversion formulas to convert a measurement made in one scale to the other scale.Licenses & Attributions
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