1.1.2: The Metric System of Measurement

6.1.2 Learning Objectives

• Describe some of the advantages of the base ten number system
• Define prefixes of the metric measures
• Convert from one unit of measure in the metric system to another unit of measure

The Advantages of the Base Ten Number System

The metric system of measurement takes advantage of our base ten number sys­tem. The advantage of the metric system over the United States system is that in the metric system it is possible to convert from one unit of measure to another simply by multiplying or dividing the given number by a power of 10. This means we can make a conversion simply by moving the decimal point to the right or the left.

Prefixes

Common units of measure in the metric system are the meter (for length), the liter (for capacity), and the gram (for mass). The units are abbreviated m for meter, L for liter, and g for gram. To each of the units a prefix can be attached. The metric prefixes along with their meaning are listed below.

Metric Prefixes and Abbreviations

giga (g): billion
mega (m): million
kilo (k): thousand
hecto (h): hundred
deka (dk): ten
deci (d): tenth
centi (c): hundredth
milli (m): thousandth

For example, if length is being measured,

1 kilometer is equivalent to 1000 meters.                        1 km = 1000 m
1 centimeter is equivalent to one hundredth of a meter.    1 cm = 0.01 m
1 millimeter is equivalent to one thousandth of a meter.    1 mm = 0.001 m

Conversion from One Unit to Another Unit

Let's note three characteristics of the metric system that occur in the metric table of measurements.

1. In each category, the prefixes are the same.
2. We can move from a larger to a smaller unit of measure by moving the decimal point to the right. (or multiplying by 10 for each step)
3. We can move from a smaller to a larger unit of measure by moving the decimal point to the left. (or dividing by 10 for each step)

The following table provides a summary of the relationship between the basic unit of measure (meter, gram, liter) and each prefix, and how many places the decimal point is moved and in what direction.

kilo hecto deka unit deci centi milli

Basic Unit to Prefix
unit to deka	1 to 10	Divide by \(10\)
unit to hecto	1 to 100	Divide by \(10^2\)
unit to kilo	1 to 1,000	Divide by \(10^3\)
unit to deci	1 to 0.1	Multiply by \(10\)
unit to centi	1 to 0.01	Multiply by \(10^2\)
unit to milli	1 to 0.001	Multiply by \(10^3\)

Conversion Table

Listed below, in the unit conversion table, are some of the common metric units of measure.

Unit Conversion Table
Length	\(\text{1 kilometer (km) = 1,000 meters } (m)\)	\(1,000 \times 1\text{m}\)
	\(\text{1 hectometer (hm) = 100 meters}\)	\(100 \times 1 \text{m}\)
	\(\text{1 dekameter (dam) = 10 meters}\)	\(10 \times 1 \text{m}\)
	\(\text{1 meter (m)}\)	\(1 \times 1 \text{m}\)
	\(\text{1 decimeter (dm) = } \dfrac{1}{10} \text{ meter}\)	\(.1 \times 1 \text{m}\)
	\(\text{1 centimeter (cm) = } \dfrac{1}{100} \text{ meter}\)	\(.01 \times 1 \text{m}\)
	\(\text{1 millimeter (mm) = } \dfrac{1}{1,000} \text{ meter}\)	\(.001 \times 1 \text{m}\)
Mass	\(\text{1 kilogram (kg) = 1,000 grams } (g)\)	\(1,000 \times 1\text{g}\)
	\(\text{1 hectogram (hg) = 100 grams}\)	\(100 \times 1 \text{g}\)
	\(\text{1 dekagram (dag) = 10 grams}\)	\(10 \times 1 \text{g}\)
	\(\text{1 gram (g)}\)	\(1 \times 1 \text{g}\)
	\(\text{1 decigram (dg) = } \dfrac{1}{10} \text{ gram}\)	\(.1 \times 1 \text{g}\)
	\(\text{1 centigram (cg) = } \dfrac{1}{100} \text{ gram}\)	\(.01 \times 1 \text{g}\)
	\(\text{1 milligram (mg) = } \dfrac{1}{1,000} \text{ gram}\)	\(.001 \times 1 \text{g}\)
Volume	\(\text{1 kiloliter (kL) = 1,000 liters } (L)\)	\(1,000 \times 1\text{L}\)
	\(\text{1 hectoliter (hL) = 100 liters}\)	\(100 \times 1 \text{L}\)
	\(\text{1 dekaliter (daL) = 10 liters}\)	\(10 \times 1 \text{L}\)
	\(\text{1 liter (L)}\)	\(1 \times 1 \text{L}\)
	\(\text{1 deciliter (dL) = } \dfrac{1}{10} \text{ liter}\)	\(.1 \times 1 \text{L}\)
	\(\text{1 centiliter (cL) = } \dfrac{1}{100} \text{ liter}\)	\(.01 \times 1 \text{L}\)
	\(\text{1 milliliter (mL) = } \dfrac{1}{1,000} \text{ liter}\)	\(.001 \times 1 \text{L}\)

Distinction Between Mass and Weight
There is a distinction between mass and weight. The weight of a body is related to gravity whereas the mass of a body is not. For example, your weight on the earth is different than it is on the moon, but your mass is the same in both places. Mass is a measure of a body's resistance to motion. The more massive a body, the more resistant it is to motion. Also, more massive bodies weigh more than less massive bodies.

Converting Metric Units
To convert from one metric unit to another metric unit:

1. Determine the location of the original number on the metric scale (pictured in each of the following examples).
2. Multiply or divide the original number by 10 raised to the same number of places as is necessary to move to the metric unit you wish to go to. (If changing from a larger to a smaller unit, multiply by 10. If changing from a smaller to a larger unit, divide by 10)

We can also convert from one metric unit to another using unit fractions. Both methods are shown in the following example.

Example 1

Convert 3 kilograms to grams.

Solution

a. 3 kg can be written as 3.0 kg. Then,

Since we are moving from a larger unit to a smaller unit. Multiply the original number by 10 three times.

\(3 \times 10^3=3000\) grams

Thus, \(\text{3 kg = 3,000 g}\).

b. We can also use unit fractions to make this conversion.

Since we are converting to grams, and \(\text{1,000 g = 1 kg}\). we choose the unit fraction \(\dfrac{\text{1,000 g}}{\text{1 kg}}\) since grams is in the numerator.

\(\begin{array} {rcl} {\text{3 kg}} & = & {\text{3 kg} \cdot \dfrac{\text{1,000 g}}{\text{1 kg}}} \\ {} & = & {3 \cancel{\text{ kg}} \cdot \dfrac{\text{1,000 g}}{1 \cancel{\text{ kg}}}} \\ {} & = & {3 \cdot 1,000 \text{ g}} \\ {} & = & {3,000 \text{ g}} \end{array}\)

Example 2

Convert 67.2 hectoliters to milliliters.

Solution

Since we are moving from a larger unit to a smaller unit. Multiply the original number by 10 five times.

\(67.2 \times 10^5=6,720,000\) milliliters

Thus, \(\text{67.2 hL = 6,720,000 mL}\).

Example 3

Convert 100.07 centimeters to meters.

Solution

Since we are moving from a smaller unit to a larger unit. Divide the original number by 10 two times.

\(100.07 \div 10^2=1.0007\) meters

Thus, \(\text{100.07 cm = 1.0007m}\).

Example 4

Convert 0.16 milligrams to grams.

Solution

Since we are moving from a smaller unit to a larger unit. Divide the original number by 10 three times.

\(0.16 \div 10^3=0.00016\) grams

Thus, \(\text{0.16 mg = 0.00016}\).

Example 5

A medication is sold in 5 milligram size capsules. The pharmacist receives the medication that that is used to fill the capsules in 3.5 kilogram bags. How many capsules can be produced?

Solution

First we need to convert from kilograms to milligrams. Since we are moving from a larger to a smaller unit. Multiply the original number by 10 six times.

\(3.5 \times 10^6=3,500,000\) milligrams

Each capuse holds 5 milligrams, so we divide the number we just found by 5 to determine the number of capsules that can be filled.

\(3,500,000\div 5=700,000\) capsules

Thus, 3.5 kilograms of the medication produce 700,000 capsules.

 

Try It Now 1

Make the following conversions. If a fraction occurs, convert it to a decimal rounded to two decimal places.

1. 411 kilograms to grams
2. 5.626 liters to centiliters
3. 80 milliliters to kiloliters
4. 150 milligrams to centigrams
5. 2.5 centimeters to meters

Answer

1. 411,000 g
2. 562.6 cL
3. 0.00008 kL
4. 15 cg
5. 0.025 m