Math 40
Prealgebra
Section 6.4 – Unit Conversion: Metric System of Measurement
6.4 Unit Conversion: Metric System of Measurement
While working the problems in Section 6.3, you may have noticed just how random some of the relationships in
the U.S. Customary system are. Many others before you noticed the same thing and around the year 1790, a
group of French scientists developed the metric system of measurement.
The metric system is more uniform than the U.S. Customary system mainly because it is based on multiples of
10 and is set up very much the like the our number and money systems.
Key Characteristics of the Metric System

The base unit of length is the meter.

The base unit of mass (weight) is the gram.

The base unit of volume (capacity) is the liter.

All other lengths are created from these base units. To indicate these lengths, prefixes are written in
front of the words meter, gram, or liter.
10 of the smaller unit make up 1 of the next larger unit.

We recommend for you to become very familiar with the prefixes and the numerical meanings as it will greatly
help you in the problems of this section. Refer to the following table to help you remember!
Prefix
kilohectodekabase unit
Abbreviation
k
h
da
m, g, or L
Meaning
1000
100
10
1
deci-
d
1
10
centi-
c
1
100
milli-
m
1
1000
Notes
meter, gram, or liter
1
of a decade)
10
Our number system is a decimal (base 10) system
1
100 cents in a dollar (1 cent is
of a dollar)
100
1
100 years in a century (1 year is
of a century)
100
1
1000 years in a millenium (1 year is
of a
1000
millenium)
10 years in a decade (1 year is
You will also want to remember the order of these prefixes and that kilo- indicates the largest measure while
milli- indicates the smallest measure.
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2015 Campeau
Math 40
Prealgebra
Section 6.4 – Unit Conversion: Metric System of Measurement
To further help with the order, the following mnemonic device may help.
king henry died by drinking chocolate milk
Notice that the first letter of each word corresponds to the prefix in the metric system (with b meaning the base
unit—meter, gram, or liter).
Caution: Be extremely careful with identifying which d-prefix you are using. (deka = da vs. deci = d)!
We realize that the above mnemonic device may sound silly to you, but surprisingly the sillier it is, the more
likely you are to remember it. If the mnemonic device above is not appealing to you, consider using
king henry drank both diet cokes Monday,
kittens hate dogs but do chase mice,
kids have dreams but dreams cost money, or making up one on your own.
Now that we have a better understanding of the prefixes, let us look at a few benchmarks for the more
commonly used metric units.
Length
Unit
Abbreviation
kilometer
km
hectometer
dekameter
hm
dam
meter
(base unit)
m
decimeter
dm
centimeter
cm
millimeter
mm
Benchmark
often used for “mile-like” lengths
a little more than half a mile (0.62 miles)
about 5-6 city block
often used for “yard-like” lengths
distance from the doorknob to the ground
a little longer than a yardstick
length of a roll of dimes (100 dimes)
width of hand
often used for “inch-like” lengths
a little less than half an inch (0.39 inches)
width of little finger
half the width of a nickel
often used for very small lengths
thickness of a dime
Example 1: Write the most reasonable metric unit in each blank. Choose from km, m, cm, and mm.
a) I commute 60 _______ to work.
b) A postage stamp is 22 _______ wide.
Solution:
a) 60 km because kilometers are used for “mile-like” distances. 60 km is a little over 30 miles.
b) 22 mm because the width of a postage stamp is very small. 22 cm would be a little less than
11 inches, which is too large for a postage stamp.
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2015 Campeau
Math 40
Prealgebra
Section 6.4 – Unit Conversion: Metric System of Measurement
You Try It 1: Write the most reasonable metric unit in each blank. Choose from km, m, cm, and mm.
a) A newborn baby is 50 _______ long.
b) The classroom is about 12 _______ long.
Remember that in the metric system 10 of the smaller unit make up 1 of the next larger unit. This can help us
visualize units we are not familiar with.
10 mm = 1 cm
10 cm =1 dm
10 dm = 1 m
10 m = 1 dam
10 dam = 1 hm
10 hm = 1 km
It can also help us develop direct relationships for the most common units we see in our everyday lives such as
kilometers, meters, centimeters, and millimeters. We usually memorize these relationships
Common Metric System Relationships for Length
1 kilometer (km) = 1000 meters (m)
1 meter (m) = 100 centimeters (cm)
1 meter (m) = 1000 millimeters (mm)
1 centimeter = 10 millimeters (mm)
Once we have some direct relationships, we can use the techniques of Section 6.3 to convert between units in
the metric system.
Example 2: Use unit fractions to convert 4.5 km to m.
Solution:
4.5 km
1
Write down what is given over denominator 1.
There is a direct relationship between km and m: 1 km = 1000 m
We have two choices for the unit fraction,
Choose
4.5 km  1000 m 


1  1 km 
1 km
1000 m
.
or
1000 m
1 km
1000 m
since we want km to divide out.
1 km
Write the unit fraction to the right of what is given and divide out units.
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Math 40
Prealgebra
Section 6.4 – Unit Conversion: Metric System of Measurement
4.5  1000 m 


1  1 
4.5 1000 m

1 1
4500

m
1
 4500 m

Notice we are left with meters, which is what the problem is asking for.
Multiply the numerators. Multiply the denominators.
So 4.5 km = 4500 m
You Try It 2: Use unit fractions to convert 6.42 cm to mm.
Example 3: Use unit fractions to convert 249 cm to m.
Solution:
249 cm
1
Write down what is given over denominator 1.
There is a direct relationship between cm and m: 1 m = 100 cm
We have two choices for the unit fraction,
Choose
 1m 


 100 cm 
249  1 m 



1  100 
249 1 m

100
249 cm
1

1m
100 cm
or
.
100 cm
1m
1m
since we want cm to divide out.
100 cm
Write the unit fraction to the right of what is given and divide out units.
Notice we are left with meters, which is what the problem is asking for.
Multiply the numerators. Multiply the denominators.
249
m
100
 2.49 m
So 249 cm = 2.49 m
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2015 Campeau
Math 40
Prealgebra
Section 6.4 – Unit Conversion: Metric System of Measurement
You Try It 3: Use unit fractions to convert 457 mm to m.
Now that we have some experience with metric units involving length, let us extend our understanding to mass
(weight) and volume (capacity).
Mass (Weight)
Unit
Abbreviation
kilogram
kg
hectogram
dekagram
gram
(base unit)
decigram
centigram
hg
dag
milligram
mg
g
Benchmark
often used for “pound-like” weights
a little more than 2 pounds (2. 2 pounds)
basketball
large cast-iron frying pan
about half of a plain hamburger
2 nickels
dollar bill
stick of gum
dg
cg
often used for extremely small weights
1 grain of salt
medicine and vitamin doses
Example 4: Write the most reasonable metric unit in each blank. Choose from kg, g, and mg.
a) A 12-year old child weighs 40 _______.
b) You mailed a letter that weighed 30 _______.
c) You take 500 _______ of vitamin C every day.
Solution:
a) 40 kg because kilometers are used for “pound-like” weights. 40 km is a little over 80 lbs.
b) 30 g because 30 kg would be much too heavy and 30 mg weighs less than a dollar bill.
c) 500 mg because vitamin doses are given in mg. 500 grams would be around the weight of 2
and a half hamburgers (way too much vitamin C) and 500 kg would be around the weight of
500 basketballs (an impossible dosage to consume).
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2015 Campeau
Math 40
Prealgebra
Section 6.4 – Unit Conversion: Metric System of Measurement
You Try It 4: Write the most reasonable metric unit in each blank. Choose from kg, g, and mg.
a) A banana weighs 150 _______.
b) The suitcase weighed 20 _______.
Volume (Capacity)
Unit
Abbreviation
kiloliter
kL
hectoliter
dekaliter
hL
daL
liter
(base unit)
L
deciliter
centiliter
dL
cL
milliliter
mL
Benchmark
often used for very large capacities
4 bathtubs of water
500-2 liter bottles of soda
5-2 liter bottles of soda
often used for “quart-like” capacities
slightly more than 1 quart (1.06 quarts)
half of a 2 liter bottle of soda
2 teaspoons of water
often used for extremely small capacities
4 drops of water from an eyedropper
medicine doses (1 mL = 1 cubic cm or cc)
Example 5: Write the most reasonable metric unit in each blank. Choose from kL, L, and mL.
a) The nurse gave me 10 _______ of cough syrup.
b) This is a 100 _______ garbage can.
c) The lake holds 50 _______ of water.
Solution:
a) 10 mL because medicine doses are given in mL. 10 L or 10 kL would be too much.
b) 100 L because a garbage can has a “quart-like” capacity. 100 kL would hold way too much
and 100 mL would hold way too little.
c) 50 kL because a lakes generally hold very large capacities.
You Try It 5: Write the most reasonable metric unit in each blank. Choose from kL, L, and mL.
a) A car’s gas tank holds 50 _______.
b) You add 15 _______ of oil to the pancake mix.
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2015 Campeau
Math 40
Prealgebra
Section 6.4 – Unit Conversion: Metric System of Measurement
Just as with length, 10 of the smaller unit make up 1 of the next larger unit. This can help us visualize mass and
volume units in the metric system that we are not familiar with.
10 mg = 1 cg
10 cg =1 dg
10 dg = 1 g
10 g = 1 dag
10 dag = 1 hg
10 hg = 1 kg
10 mL = 1 cL
10 cL = 1 dL
10 dL = 1L
10 L = 1 daL
10 daL = 1 hL
10 hL = 1 kL
Just as with length, we can obtain direct relationships for the most common mass and volume units we see in
our everyday lives such as kilograms, grams, milligrams, kiloliters, liters, and milliliters. We also memorize
these relationships
Common Metric System Relationships for Mass (Weight)
1 kilogram (kg) = 1000 grams (g)
1 gram (m) = 1000 milligrams (mg)
Common Metric System Relationships for Volume (Capacity)
1 kiloliter (kL) = 1000 liters (L)
1 liter (m) = 1000 milliliters (mL)
We similarly use the techniques of Section 6.3 to convert between units in the metric system.
Example 6: Use unit fractions to convert 750 mg to g.
Solution:
750 mg
1
Write down what is given over denominator 1.
There is a direct relationship between mg and g: 1 g = 1000 mg
1g
Choose
since we want mg to divide out.
1000 mg
750 mg  1 g

 1000 mg
1

750 1 g
11000
750

g
1000
 0.75 g





Write the unit fraction to the right of what is given and divide out units.
Notice we are left with meters, which is what the problem is asking for.
Multiply the numerators. Multiply the denominators.
So 750 mg = 0.75 g
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2015 Campeau
Math 40
Prealgebra
Section 6.4 – Unit Conversion: Metric System of Measurement
You Try It 6: Use unit fractions to convert 3420.5 g to kg.
Example 7: Use unit fractions to convert 56,000 mL to kL.
Solution:
Write down what is given over denominator 1.
56000 mL
1
There is no direct relationship between milliliters and kiloliters!
Write down any relationships involving mL and kL: 1 L = 1000 mL and 1 kL = 1000 L
Figure out a path to get from mL to kL: mL  L  kL
Choose the appropriate unit fractions and write them in order to the right of what is given.
1L
since we want to cancel out mL and be left with L.
1000 mL
1 kL
Then choose
since we want to cancel out L and be left with kL.
1000 L
First choose
56000 mL  1 L   1 kL 



1
 1000 mL   1000 L 
Now divide out units to make sure you will be left with kL in the final answer.
  1 kL 
56000 mL  1 L



1
 1000 mL   1000 L 
Multiply the numerators. Multiply the denominators.
56000 11 kL

11000 1000

56000
kL
1000000
Divide the numerator by the denominator.
 0.056 kL
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2015 Campeau
Math 40
Prealgebra
Section 6.4 – Unit Conversion: Metric System of Measurement
You Try It 7: Use unit fractions to convert 423,500 mL to kL.
After some practice of using unit fractions to covert metric measurements, you may have noticed that in all
cases you are either multiplying or dividing by a power of 10.
In chapter 5, we discussed a shortcut to use when we multiply or divide by a power of 10. It had to do with
moving the decimal point to the left (when dividing) or right (when multiplying) based on how many zeros were
in the power of 10.
We combine this shortcut with conversions among metric units and call this procedure Using Metric Stairs to
Perform Conversions
We start off with sketching 7 steps. On each step, we fill in the metric prefixes and whatever base the problem
is asking you to convert between.
We remember the order by using king henry died by drinking chocolate milk,
where b, the base, is either meter, gram, or liter.
mm
cm
dm
m
dam
hm
km
kg
kL
2015 Campeau
hg
dag
g
dg
cg
mg
mL
cL
dL
L
daL
hL
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Math 40
Prealgebra
Section 6.4 – Unit Conversion: Metric System of Measurement
How to Use Metric Stairs to Perform Conversions
1) Make a quick sketch of the seven steps. Label each step with the ordered prefixes
along with the base given in the problem.
2) Identify the unit that is given and mark this on the corresponding step.
3) Identify the unit you must convert to and mark this on the corresponding step.
3) Now decide whether to move the decimal point to the left or right. We do this by
seeing where we start on the stairs and seeing where we end on the stairs.

If you have to go up the stairs to get from the starting unit to the ending
unit, make the original number larger by moving the decimal point to the
right. Here you have just multiplied by a power of 10.

If you have to go down the stairs to get from the starting unit to the ending
unit, make the original number smaller by moving the decimal point to the
left. Here you have just divided by a power of 10.

The number of places you move the decimal point depends on how
many steps it takes to get from the starting unit to the ending unit.
4) Write your final answer with its appropriate decimal location and units. If
applicable, make sure to fill in any necessary zeros
Example 8: Use the metric stairs to convert 421.3 m to km.
Solution:
Make a quick sketch of the stairs and label them.
mm
Mark the starting unit step (meter).
cm
Mark the ending unit step (kilometer).
dm
m
dam
hm
3 steps down: make the number
smaller by moving the decimal
point 3 places to the left.
So, 421.3 m = 0.421 3 km = 0.4213 km
km
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2015 Campeau
Math 40
Prealgebra
Section 6.4 – Unit Conversion: Metric System of Measurement
You Try It 8: Use the metric stairs to convert 4820.5 m to km.
Example 9: Use the metric stairs to convert 7 hg to cg.
Solution:
Make a quick sketch of the stairs and label them.
mg
Mark the starting unit step (hectogram).
cg
Mark the ending unit step (centigram).
dg
g
dag
hg
4 steps up: make the number
larger by moving the decimal
point 4 places to the right.
So, 7 hg = 7.0 hg = 7 0000. cg = 70, 000 cg
kg
You Try It 9: Use the metric stairs to convert 16 hg to cg.
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2015 Campeau
Math 40
Prealgebra
Section 6.4 – Unit Conversion: Metric System of Measurement
Example 10: Use the metric stairs to convert 0.092 cL to L.
Solution:
Make a quick sketch of the stairs and label them.
mL
Mark the starting unit step (centiliter).
cL
Mark the ending unit step (liter).
dL
L
daL
hL
2 steps down: make the number
smaller by moving the decimal
point 2 places to the left.
So, 0.092 cL = 0.00 092 L = 0.00092 L
kL
You Try It 10: Use the metric stairs to convert 0.003 cL to L.
Example 11: Use the metric stairs to convert 0.3 dag to g.
Solution:
Make a quick sketch of the stairs and label them.
mg
Mark the starting unit step (dekagram).
cg
Mark the ending unit step (gram).
dg
g
dag
hg
kg
2015 Campeau
1 step up: make the number
larger by moving the decimal
point 1 place to the right.
So, 0.3 dag = 0 3. g = 3 g
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Math 40
Prealgebra
Section 6.4 – Unit Conversion: Metric System of Measurement
You Try It 11: Use the metric stairs to convert 0.46 dag to g.
Applications
Many applications involving metric units don’t always compare units consistently. For example, a problem
might give a figure in centimeters and then ask a question about meters. Then you might have to answer further
questions involving your conversion.
Like all applications, you will have to read each carefully, paying close attention to the units, and determine
exactly what the problem is asking for. It is your choice whether you want to use unit fractions, metric stairs, or
a combination of both techniques to solve each application.
Example 12: Connie needs a 315 cm piece of plastic tubing for her garden. The price is $5.50 per meter.
How much will it cost Connie?
Solution:
One Method
The amount Connie needs is given in centimeters but the price is $5.50 per meter.
We can use the unit stairs to convert 315 centimeters to meters.
The meter step is two steps down from the centimeter step so we make the number smaller by
moving the decimal point 2 places to the left.
315 cm = 315.0 cm =3 .15 0 m = 3.15 m
Now we use unit fractions to convert 3.15 meters to price per meter.
Rewrite the “per” statement: $5.50 per meter =
Write 3.15 m with denominator 1:
$5.50
1m
3.15 m
1
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2015 Campeau
Math 40
Prealgebra
Section 6.4 – Unit Conversion: Metric System of Measurement
Attach
$5.50
to the right of 3.15 m.
1m
3.15 m  $5.50 


1  1m 
Now divide out units to make sure you will be left with just the price in the final answer.
3.15 m  $5.50 


1  1m 

3.15  $5.50 


1  1 
Multiply the numerators. Multiply the denominators.
Round to the nearest cent.

3.15  $5.50
1 1

$17.325
1
 $17.33
State the answer. It will cost Connie $17.33 for the plastic tubing.
Another Method
Since we know the direct relationship between centimeters and meters, we could also solve this
problem strictly using unit fractions.
315 cm  1 m   $5.50 



1  100 cm   1 m 

315 1  $5.50
1 100 1

$1732.5
100
 $17.325
 $17.33
Which is the same answer we obtained above!
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2015 Campeau
Math 40
Prealgebra
Section 6.4 – Unit Conversion: Metric System of Measurement
You Try It 12: Ground turkey is on sale at $8.99 per kilogram. How much will 350 g cost?
Example 13: A floor tile that measures 30 cm by 30 cm weighs 185 g. How many kilograms would a stack of
24 tiles weigh?
Solution:
One Method
The weight of one tile is given but the problem is asking for the weight of 24 tiles.
Multiply 185 g by 24: 185 g  24   4440 g .
The weight of 24 tiles is in grams but the problem is asking for the weight in kilograms.
We can use the unit stairs to convert 4440 grams to kilograms.
The kilogram step is three steps down from the gram step so we make the number smaller by
moving the decimal point 3 places to the left.
4440 g = 4440.0 g = 4.440 0 kg = 4.44 kg
State the answer. 24 tiles weigh 4.44 kg.
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2015 Campeau
Math 40
Prealgebra
Section 6.4 – Unit Conversion: Metric System of Measurement
Another Method
If you prefer to solve this problem using strictly unit fractions, then
24 tiles  185 g   1 kg 



1  1 tile   1000 g 

24  185   1 kg 



1  1   1000 

24 185 1 kg
1 1 1000

4440
kg
1000
 4.44 kg
Which is the same answer we obtained earlier!
You Try It 13: One 8 ounce soda contains 45 mg of caffeine. If Dominic usually drinks four 8 ounce sodas
each day, how many grams of caffeine will he consume in one week?
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2015 Campeau
Math 40
Prealgebra
Section 6.4 – Unit Conversion: Metric System of Measurement
Example 14: A pool caretaker puts 750 mL of chlorine into the swimming pool each day. How many liters
should be ordered to have a 30-day supply on hand? If chlorine is sold in containers that hold 4
L, how many containers should be ordered for this 30-day supply?
Solution:
The daily of amount chlorine is given but the problem is asking for a 30-day supply of chlorine.
Multiply 750 mL by 30: 750 mL  30   22,500 mL .
The 30-day supply of chorine is in milliliters but the problem is asking for the 30-day supply in
liters.
We can use the unit stairs to convert 22,500 milliliters to liters.
The liter step is three steps down from the milliliter step so we make the number smaller by
moving the decimal point 3 places to the left.
22,500 mL = 22,500.0 mL = 22.500 0 L = 22.5 L
To answer the second question, we want to figure out how many 4 L are in 22.5 L.
This language indicates that we must divide 22.5 L by 4 L: 22.5 L  4 L  5.625 containers.
It is impossible to purchase 0.625 container. This must be rounded up to 1 full container.
State the answer to both questions.
A 30-day supply is 22.5 L. The pool caretaker must order 6-4 liter containers of chlorine for
this 30-day supply.
You Try It 14: A pool caretaker puts 340 mL of chlorine into the spa each day. How many liters
should be ordered to have a 60-day supply on hand? If chlorine is sold in containers that hold 4
L, how many containers should be ordered for this 60-day supply?
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