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Study Guides > Mathematics for the Liberal Arts

Metric System Basics

Learning Outcomes

  • 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

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

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).

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. 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:

[latex] \displaystyle kg\quad hg\quad dag\quad g\quad d\underbrace{g\quad c}_{1}\underbrace{g\quad m}_{2}g[/latex]

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]. The same method works when you are converting from a smaller to a larger unit, as in the problem: Convert 1 centimeter to kilometers.

[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]

Note that instead of moving to the right, you are now moving to the left—so the decimal point must do the same:

[latex] \displaystyle 1\ cm=0.\underbrace{0}_{5}\underbrace{0}_{4}\underbrace{0}_{3}\underbrace{0}_{2}\underbrace{1}_{1}\ km[/latex].

Try It

How many milliliters are in 1 liter?

Answer: 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.

Now that you have seen how to convert among metric measurements in multiple ways, let’'s revisit the problem posed earlier.

Example

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?

Answer: Convert mg to g.

[latex]5,000\text{ mg}=\text{___ g}[/latex]

[latex] \displaystyle \frac{5,000\text{ mg}}{1}\cdot \frac{1\text{ g}}{1,000\text{ mg}}=\text{ g}[/latex]

[latex] \displaystyle \frac{5,000\cancel{\text{mg}}}{1}\cdot \frac{1\text{ g}}{1,000\ \cancel{\text{mg}}}=\text{ g}[/latex]

[latex] \displaystyle \frac{5,000\cdot 1\text{ g}}{1\cdot 1,000}=\frac{5,000\text{ g}}{1,000}[/latex]

[latex] \displaystyle \frac{5,000\text{ g}}{1,000}=5\text{ g}[/latex]

[latex]5\text{ g}=5,000\text{ mg}[/latex], so the pharmacist did not make a mistake.

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  • Question ID 126793, 126794. 126795. Provided by: Lumen Learning License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.
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