Academy of Health Professions Mathematics Activity
The Metric System
Materials Needed:
Any high school algebra textbook
Calculators
Prerequisite:
AHP Mathematics Activity “Ratio and Proportion”
Metric System Roots
Metric units of measurement have two components: a root and a prefix. The root indicates the
type of measurement you are making.
Meter – length
Liter – volume
Gram – weight
In English measurements, cubic units like “cubic feet” are sometimes used for volume. In metric
units, units like “cubic meters” are also used for volume. This type of volume unit will be
covered later in this module.
The roots are used by themselves, as in “5 meters”, “4.8 liters” or “120 grams”.
To distinguish the abbreviation for liter from the number 1 or the capital
letter I, liter will be abbreviated using a capital L in this module.
From above, we would have 5m, 4.8 L, or 120 g.
A Comparison of Metric Roots and English Units
1 meter = 39.37 inches (rounded)
1 liter = 1.09 quarts (rounded)
453.6 grams (rounded) = 1 pound
NOTE: In “Ratio and Proportion”, you used dimensional analysis with fixed ratios like those
above to convert between the metric system and English measurements.
The metric system roots form the basis of the metric system and so the metric system has a major
advantage over English units of measurement. In English measurement, you had to learn that
inch, foot, yard, furlong, and mile were all units of length. These words don’t look at all alike. In
the metric system, if you see the root “meter”, you know you are measuring length no matter
what the prefix is.
Metric System Prefixes
The metric system derives its power from the fact that roots above are combined with prefixes to
make differently sized units. In English measurement, you needed different words for differently
sized units.
Each prefix itself denotes a power of 10 and placing the prefix before the root means to take that
power of 10 times the root unit. Some prefixes and associated powers of 10 are given below:
Prefix
kilo(k)
hecto(h)
deka(da)
Power
1000
100
10
deci(d)
centi(c)
1
10
1
100
milli(m)
1
1,000
micro( )
1
1,000,000
The symbol  is the Greek letter “mu”. Also, the spelling “deca” is sometimes given for
“deka” but the abbreviation “da” is the same.
So a kilogram is 1000 grams and a centimeter is
1
100
of a meter. For the fractional units (deci,
centi, milli, micro), it may be easier to think about how many make up one of the root unit.
Example 1: How many deciliters are in one liter?
Solution: Using the table, we can see that 1 deciliter = 1 liter. Multiply by 10 on both sides of
10
the equation to get 10 dL = 1L . So there are 10 deciliters in one liter. ■
Exercise 1: How many millimeters are in one meter?
Similar conversions for all the “fractional” prefixes are below:
deci10 dm = 1 m
10 dL = 1 L
10 dg = 1 g
centi100 cm = 1 m
100 cL = 1 L
100 cg = 1 g
milli1000 mm = 1 m
1000 mL = 1 L
1000 mg = 1 g
micro1,000,000  m = 1 m
1,000,000  L = 1 L
1,000,000  g = 1 g
The conversion depends only on the prefix. kilo- means 1000 no matter what root we are using!
Converting Units Within the Metric System
Example 2: How many kilometers are in 2,540 meters?
Solution: We can use dimensional analysis to solve this problem all at once:
 2540 m  1 km  2540 km
2540 m  
 2.540 km


1000
 1  1000 m 
cancel the m unit
So there are 2.54 km in 2,540 m. ■
Exercise 2: How many dekaliters are in 275 centiliters?
Sometimes, we want to convert from one unit to another but not to the root unit. It makes sense
to examine the relationships between the prefixes themselves rather than just comparing them to
the root unit.
Example 3: You have 3,500  g of a substance. How many milligrams do you have?
Solution: We already know:
1,000,000  g = 1 g
and
1000 mg = 1 g
So we must have 1,000,000  g = 1000 mg. Dividing both sides of this equation by 1000, we
find that 1000  g = 1 mg. We can use this fixed ratio to finish the problem:
 3500  g   1 mg  3500 mg
3500  g = 
 3.500 mg


1
1000

  1000  g 
So you have 3.5 mg of the substance. ■
The “micro-to-milli” conversion is very common in the sciences, so it makes sense to learn the
ratio for it. That is,
1000  m = 1 mm
1000  L = 1 L
1000  g = 1 g
So there are 1000 “micro” in one “milli” regardless of the root used.
Exercise 3: How many centigrams are in 25 kilograms?
If you look at the solution to Example 3, you will see that converting from micrograms to
milligrams resulted in a movement of the decimal point. This is another powerful aspect of the
metric system – conversion between units is accomplished by an appropriate movement of the
decimal point.
Let’s examine the answer from Example 2 more closely, remembering that micrograms are
smaller than milligrams:
3500  g = 3.5 mg
larger
smaller
unit
smaller
larger
unit
So in converting from a smaller unit to a larger unit, the number out in front went from a larger
number to a smaller number. That is, since milligrams are larger than micrograms, it doesn’t
take as many of them to equal the same weight in micrograms.
Compare to English Units
There are 16 ounces in a pound so the ounce is a smaller unit than the pound. A small cat that
weighs 6 pounds would have an equivalent weight of 96 ounces. We needed more ounces for
the same weight since the ounces are a smaller unit than the pound.
The microgram-to-milligram conversion factor we found was 1000  g = 1 mg. See the three
zeroes in the 1000? That tells you how many spaces to move the decimal point. Since mg are
larger than micrograms, we must move the decimal point 3 spaces to result in a smaller number
since fewer milligrams are needed to make the same weight as in micrograms.
If you convert to a unit with a larger prefix, the number in front must get smaller so move the
decimal point to the left. If you convert to a unit with a smaller prefix, the number in front must
get larger, so move the decimal point to the right.
In order to do this correctly, you must know the conversion factor between the prefixes AND
which prefix represents the larger quantity.
Example 4: Find the conversion factor to convert from deciliters to milliliters and use it to
convert 250 mL to dL.
Solution: We already know 10 dL = 1 L and 1000 mL = 1 L so 10 dL = 1000 mL. Dividing
by 10, we see that 1 dL = 100 mL.
The conversion factor has 100 in it (two zeroes) so we need to move the decimal point two
spaces. We want to convert to dL (a larger unit) so we need fewer dL to equal the same number
of mL. So move the decimal point to the left:
250 mL = 250.0 mL = 2.50 dL
every number has
a decimal point
So there are 2.5 dL in 250 mL. ■
move the decimal
point two spaces
to the left
Example 5: How many centigrams are in 5.8 dekagrams?
Solution: We already know that 10 g = 1 dag so divide by 10 to see that 1 g = 1 dag. Also,
10
100 cg = 1 g. Putting these together, we see that 100 cg = 1 dag . Multiplying by 10, we get
10
the conversion factor 1000 cg = 1 dag.
The conversion factor has 1000 in it so we will move the decimal point 3 spaces. But this time,
we are converting to a smaller unit so it will take more centigrams to give the equivalent weight
of 5.8 dekagrams. Therefore, the number must get bigger, so move the decimal point to the
right:
5.8 dag = 5.800 dag = 5800 cg
add as many zeroes
after the decimal
point as necessary
move the decimal
point three spaces
to the right
So there are 5800 cg in 5.8 dag. ■
Once you master the prefixes, here is an easy way to remember which way to move the decimal
point when converting between units with the same root:
L M N O P Q R S
As your units get larger (L), move your decimal point to the left (L is on the left). As your units
get smaller (S), move your decimal point to the right (S is on the right).
Example 6: Convert 475 mg to dg.
Solution: Decigrams are larger than milligrams. So we want to convert to a larger unit (L) so we
need to move the decimal point to the left (L is on the left).
We know 1000 mg = 1 g and 10 dg = 1 g so we have 1000 mg = 10 dg. Dividing by 10, we get
the conversion factor 100 mg = 1 dg. The conversion factor has 100 in it, so we move the
decimal point two spaces and we already know to move it to the left.
475 mg = 475.0 mg = 4.75 dg
locate the
decimal
point
So 475 mg is the same as 4.75 dg. ■
Exercise 4: Convert 2300 mL to L.
Exercise 5: Convert 12 km to dm.
move the decimal
point two spaces
to the left
A Nice Pattern for Prefixes
For each root, we list the units from largest to smallest (omit the micro-units right now):
km, hm, dam, m, dm, cm, mm
kL, hL, daL, L, dL, cL, mL
kg, hg, dag, g, dg, cg, mg
Converting from one unit to one next to it results in a conversion factor of 10. So these “onestep” conversions result in moving the decimal point one space (in the appropriate direction, of
course!)
So a two-step conversion results in moving the decimal point two spaces. A three-step
conversion results in moving the decimal point three spaces. And so on.
Combine this with our LMNOPQRS trick and we see the following:
larger units
smaller units
L M N O P Q R S
km hm dam m dm cm mm
kL hL daL L dL cL mL
kg hg dag g dg cg mg
Example 7: Convert 26,000 mg to hg.
Solution: The units mg to hg is a 5-step jump:
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
mg to cg
cg to dg
dg to g
g to dag
dag to hg
So we need to move the decimal point 5 spaces. Since we are converting to a larger unit, we
move the decimal point to the left.
26000 mg = 26000.0 mg = 0.26000 hg
locate the
decimal
point
So there are 0.26 hg in 26,000 mg. ■
Exercise 6: Convert 1500 cL to daL.
Exercise 7: Convert 250 km to mm.
move the decimal
point five spaces
to the left
NOTE: Units with the micro- prefix were not included in the table above since micro- is a threestep jump from milli-. There actually are prefixes between milli- and micro-, they just aren’t
commonly used. In fact, the deka- and hecto- prefixes are rarely used themselves but were
included above for completeness of the one-step table. So just remember that:
milli- to micro- is a three-step jump
and
micro – is smaller than milli-
Example 8: Convert 7500  g to dg.
Solution: Count the steps:
micro- to milli- (three steps)
milli- to centi- (one step)
centi- to deci- (one step)
So there are a total of 5 steps from micro- to deci-. Since we are converting to a larger unit, we
move the decimal point to the left.
7500  g = 7500.0  g = 0.07500 dg
So 7500  g = 0.75 dg. ■
Exercise 8: Convert 58 cL to  L.
Exercise 9: Convert 150 cm to hm.
insert extra 0’s as
needed to finish
moving decimal point
Volume in Cubic Units
The cubic meter is the basic volume unit in the metric system. Think of a box with length, width
and height all equal to one meter. That is one cubic meter.
Cubic meters can be converted to cubic decimeters, cubic centimeters, cubic millimeters and
cubic micrometers by cubing the conversion factors. For example:
Example 9: Convert 1 cubic meter to cubic centimeters.
Solution: We can cube the conversion factor:
1 m = 100 cm
1 m3 = (1 m)3 = 100 cm   1, 000, 000 cm3
3
So there are 1,000,000 cubic centimeters in one cubic meter. ■
The abbreviation for cubic units usually uses an exponent of 3. However, cubic centimeters arise
so frequently they have their own special notation: cc.
It is important to note that conversions between cubic units and liters are quite common. In fact,
1 cubic centimeter = 1 milliliter
or
1 cc = 1 mL
Putting all our metric system and dimensional analysis knowledge together, we can perform
computations like this:
Example 10: How many liters are in 8 cubic meters?
Solution: We will convert as follows:
cubic meters
cubic centimeters
milliliters
We have the following conversion factors:
1 cubic meter = 1,000,000 cubic centimeters
1 cubic centimeter = 1 milliliter
1000 milliliters = 1 liter
liters
The computation is done all at once below:
 8 m3   1, 000, 000 cc   1 mL  1 L 
8 m3  

  1 cc  1000 mL   8000 L
3
 1  


1
m



So there are 8,000 L in 8 cubic meters. ■
Exercise 10: Find the relevant conversion factors and use them to convert 2 L to cc’s.
Exercise 11: A chart indicates that a patient had a urine output of 750 cc for the day. A fellow
student exclaims, “That’s more than a gallon of urine!” Is it? (Hint: A gallon is the same as 4
quarts.)