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Study Guides > Math for Liberal Arts: Co-requisite Course

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 system—there 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
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. The basic metric unit of mass is the gram. A regular-sized paperclip has a mass of about 1 gram. Among scientists, one gram is defined as the mass of water that would fill a 1-centimeter cube. You may notice that the word “mass” is used here instead of “weight.” 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’s 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’s 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—but it is worth noting that there is a difference! Finally, the basic metric unit of volume is the liter. A liter is slightly larger than a quart.
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.
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.
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
Using this table as a reference, you can see the following:
  • 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).
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.
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
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).

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, decimeter

Answer: 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
^ ^
Kilometers (km) are larger than centimeters (cm), so you expect there to be less than one km in a cm. Cm is 10 times smaller than a dm; a dm is 10 times smaller than a m, etc. Since you are going from a smaller unit to a larger unit, divide.
[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] ^
Divide: [latex]1\div10\div10\div10\div10\div10[/latex], to find the number of kilometers in one centimeter. 

[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 measurements—the 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]
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!

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

The two previous problems used the conversion formulas to verify some temperature conversions that were discussed earlier—the 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.

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.

You could have converted 250°F to °C 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°C is 1.1°C higher than 120°C.) Whichever way you choose, it is important to compare the temperature measurements within the same scale, and to apply the conversion formulas accurately.

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.

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