Dimensional Analysis
(aka Factor-Label)
This technique involves the use of conversion
factors and writing all measurements with
both numerical values and the unit of
measurement
A conversion factor is where you have the same
amount (entity) represented by two different units of
measurement with their corresponding numerical
values
Conversion Factors
•
•
•
•
•
•
•
Here are some examples
1 foot = 12
__ inches
1 kilometer = ____
1000 meters
1 inch = 2.54 centimeters
4 quarts
1 gallon = __
1 acre = 4840 square yards
24 hours
1 day = ___
Conversion factors…cont.
Conversion factors for 1 ft = 12 in
1 foot
12 inches
or
12 inches
1 foot
There are almost an infinite number of
conversion factors that include meters:
1000 m
1m
1m
,
,
1 km
100 cm 1000 mm
1m
1m
0.9144 yards
,
,
3.28 feet
39.37 inches
1m
Conversion Factors….cont.
• One member of a dinner party orders a
16 ounce steak and another orders a one
pound steak- Compare the two steaks
• They are the same since
16 oz dry wt. = 1 pound
Conversion Factors….cont.
• In grade school we learned that 1
gallon contained 4 quarts or stating
that relationship as an equality:
• 1 gallon = 4 quarts
• Since 1 gallon and 4 quarts represent
the same amount, we have a
Conversion Factor
Conversion Factors….cont.
• Start with 1 gallon = 4 quarts
• Dividing each side by 1 gallon
we get this equation
• 1 gallon =
4 quarts
1 gallon
1 gallon
Since 1 gallon divided by 1 gallon equals 1
• Our equality becomes: 1 =
4 quarts
1 gallon
Conversion Factors….cont.
• Again start with 1 gallon = 4 quarts
• But this time we’ll divide each side of
the equality by 4 quarts
• The resulting equation is
• 1 gallon =
4 quarts
4 quarts
4quarts
Conversion Factors…. Cont.
• The right side of our equation becomes one
because 4 quarts divided by 4 quarts is 1
• 1 gallon =
4 quarts
1
• Rearranging this becomes 1 = 1 gallon
4 quarts
Conversion Factors….cont.
•
•
•
•
A mid-presentation summary
We know that 1 gallon = 4 quarts
Using a little mathematical magic
1 gallon = 1 and
4 quarts = 1
4 quarts
1 gallon
• Why is this an important concept?
Conversion Factors….cont.
• Now a little math review…………….
• What is 5 x 1?
• What is 5 x
2 ?
2
• Both expressions give you the same answerwhy?
• Because 2/2 equals 1 and therefore the
second equation is just like the first and
we did not change the initial value of 5.
Putting It Together
Here’s An Example
• How many quarts are in 15 gallons ?
• Remember we do NOT want to change
the amount represented by 15 gallons,
only the units in quarts
• So we’ll use the conversion factor
between gallons and quarts; that is
1 gallon = 4 quarts
Our Example continued…….
• We set it up like this:
15 gallons x 4 quarts
1 gallon
• Cancel units
• Do the math to complete the problem
• 15 x 4 quarts = 60 quarts
Every measurement must have a unit.
60 quarts
What do I need to do?
• From the problem determine
the following:
– Known quantity (number and units)
which is called the Given
– Identify what the Desired units are
– Conversion factor(s) needed (both
universal and question specific)
Factor label example
Q - How many kilometers are in 47 miles?
(note: 1 km = 0.621 miles)
# km
First write down the
desired quantity
Factor label example
Q - How many kilometers are in 47 miles?
(note: 1 km = 0.621 miles)
# km = 47 mi
Next, equate desired
quantity to the given
quantity
Factor label example
Q - How many kilometers are in 47 miles?
(note: 1 km = 0.621 miles)
# km = 47 mi
Now we have to
choose a conversion
factor
Factor label example
Q - How many kilometers are in 47 miles?
(note: 1 km = 0.621 miles)
# km = 47 mi
1 km
0.621 mi
0.621 mi
1 km
Pick the one that will
allow you to cancel
out miles
Factor label example
Q - How many kilometers are in 47 miles?
(note: 1 km = 0.621 miles)
# km = 47 mi
1 km
0.621 mi
0.621 mi
1 km
Multiply given
quantity by chosen
conversion factor
Factor label example
Q - How many kilometers are in 47 miles?
(note: 1 km = 0.621 miles)
# km = 47 mi
x 1 km
0.621 mi
Cross out common
factors
Factor label example
Q - How many kilometers are in 47 miles?
(note: 1 km = 0.621 miles)
# km = 47
x 1 km
0.621
Cross out common
factors
Factor label example
Q - How many kilometers are in 47 miles?
(note: 1 km = 0.621 miles)
# km = 47
x 1 km
0.621
Are the units now
correct?
Factor label example
Q - How many kilometers are in 47 miles?
(note: 1 km = 0.621 miles)
# km = 47
x 1 km
0.621
Yes. Both sides have
km as units.
Factor label example
Q - How many kilometers are in 47 miles?
(note: 1 km = 0.621 miles)
# km = 47
x 1 km
0.621
Yes. Both sides have
km as units.
Factor label example
Q - How many kilometers are in 47 miles?
(note: 1 km = 0.621 miles)
# km = 47
x 1 km
0.621
= 75.7 km
Now finish the math.
Factor label example
Q - How many kilometers are in 47 miles?
(note: 1 km = 0.621 miles)
# km = 47
x 1 km
0.621
= 75.7 km
The final answer is
76 km (correct sig fig)
Summary
The previous problem was not that hard.
In other words, you probably could have
done it faster using a different method.
However, for harder problems the factor
label method is easiest.
Let’s answer the beginning
questions
• The fastest human is reported to be able to
run at a rate of 27 mph, while the fastest
fish can swim at a rate of 31 m/s.
• Which one is faster?
• Both must be in the same units, so we must
convert one.
• Does it matter which one?
• NO.
Factor label example
Question: 27 mph is equal to how many
m/s? factors needed: 1 mi = 1.609 km
1 hr = 60 min
1000 m = 1 km
1 min = 60 sec
# m/s
First write down the
desired quantity
Factor label example
Q – 27 mph is equal to how many m/s?
factors needed: 1 mi = 1.609 km
1 hr = 60 min
1000 m = 1 km
1 min = 60 sec
# m/s = 27 mi/hr
Next, equate
desired quantity to
the given quantity
Factor label example
Q – 27 mph is equal to how many m/s?
factors needed: 1 mi = 1.609 km
1 hr = 60 min
1000 m = 1 km
1 min = 60 sec
# m = 27 mi
x
s
1 hr
Now we have to
choose conversion
factors
Factor label example
Q – 27 mph is equal to how many m/s? factors
needed: 1 mi = 1.609 km
1 hr = 60 min
1000 m = 1 km
1 min = 60 sec
# m = 27 mi
s
1 hr
1.609 km
1 mi
1 mi
1.609 km
Pick the one that will allow
you to cancel out miles
Factor label example
Q – 27 mph is equal to how many m/s? factors
needed: 1 mi = 1.609 km
1 hr = 60 min
1000 m = 1 km
1 min = 60 sec
# m = 27 mi
s
1 hr
1.609 km
X
1 mi
Multiply given quantity by
chosen conversion factor
Factor label example
Q – 27 mph is equal to how many m/s? factors
needed: 1 mi = 1.609 km
1 hr = 60 min
1000 m = 1 km
1 min = 60 sec
# m = 27 mi
s
1 hr
1.609 km
X
1 mi
Cross out common
factors
Factor label example
Q – 27 mph is equal to how many m/s? factors
needed: 1 mi = 1.609 km
1 hr = 60 min
1000 m = 1 km
1 min = 60 sec
# m = 27
s
1 hr
1.609 km
X
1
NO,
both
sides
Are the
units
now
correct?
aren’t
equal
Factor label example
Q – 27 mph is equal to how many m/s? factors
needed: 1 mi = 1.609 km
1 hr = 60 min
1000 m = 1 km
1 min = 60 sec
# m = 27
s
1 hr
1.609 km X1000 m
X
1
1 km
Must
Cross
choose
out
common
another factors
factor
Factor label example
Q – 27 mph is equal to how many m/s? factors
needed: 1 mi = 1.609 km
1 hr = 60 min
1000 m = 1 km
1 min = 60 sec
# m = 27
s
1 hr
1.609
X
1
X1000
m
1
NO, must choose
Do
units
match?
another factor
Factor label example
Q – 27 mph is equal to how many m/s? factors
needed: 1 mi = 1.609 km
1 hr = 60 min
1000 m = 1 km
1 min = 60 sec
# m = 27
s
1 hr
1.609
X
1
X1000
1
m
X
1 hr
60 min
Cross out common
Do units match
factors
Factor label example
Q – 27 mph is equal to how many m/s? factors
needed: 1 mi = 1.609 km
1 hr = 60 min
1000 m = 1 km
1 min = 60 sec
#m=
s
27
1
1.609
X
1
X1000
1
m
X
1
60 min
NO
Cross
NO,–must
must
out common
choose
choose
Do
units
match?
another
factors
factor
Factor label example
Q – 27 mph is equal to how many m/s? factors
needed: 1 mi = 1.609 km
1 hr = 60 min
1000 m = 1 km
1 min = 60 sec
#m=
s
27
1
1.609
X
1
1000
m
1
X
X
X 1 min
1
60 min 60 s
NO
Cross
NO,–must
must
out common
choose
choose
Do
units
match?
another
factors
factor
Factor label example
Q – 27 mph is equal to how many m/s? factors
needed: 1 mi = 1.609 km
1 hr = 60 min
1000 m = 1 km
1 min = 60 sec
#m=
s
27
1
1.609
X
1
1000
m
1
X
X
1
60
X
1
60 s
NO
Cross
NO,–must
must
out common
choose
choose
Dounits
units match?
Do
match?
another
factors
factor
Factor label example
Q – 27 mph is equal to how many m/s? factors
needed: 1 mi = 1.609 km
1 hr = 60 min
1000 m = 1 km
1 min = 60 sec
#m=
s
27
1
1.609
X
1
1000
m
1
X
X
1
60
X
1
60 s
NO
Cross
NO,–must
must
out common
choose
choose
Dounits
units match?
Do
match?
another
factors
factor
YES !
Factor label example
Q – 27 mph is equal to how many m/s? factors
needed: 1 mi = 1.609 km
1 hr = 60 min
1000 m = 1 km
1 min = 60 sec
#m=
s
27
1
1.609
X
1
= 12.0675 m/s
= 12 m/s
(correct sig fig)
1000
m
1
X
X
1
60
X
1
60 s
Do the math
• The fastest human is reported to be able to
run at a rate of 27 mph, while the fastest fish
can swim at a rate of 31 m/s. Which one is
faster? How much faster?
• Human: 27 mph = 12 m/s
• Fish: 31 m/s
• Which one is fastest?
• How much faster?
31m/s – 12 m/s = 19 m/s
Working with metric/SI
quantity
base unit
• length
meter
• mass
gram
• volume
liter
SI Base Units
Base Quantity
Name
Symbol
meter
m
Mass
kilogram
kg
Time
seconds
s
Electric current
ampere
A
Thermodynamic
temperature
Kelvin
K
Amount of
substance
mole
mol
Luminous
intensity
candela
cd
Length
Base Unit
gram
meter
liter
Working with metric/SI
m
Working with metric/SI
When converting within the metric system it is
helpful to remember:
“1 always goes with the prefix”
the value of the prefix goes with
base unit
Conversions with metric/SI
• Example: How many meters are in 12 km?
# m = 12 km x 1000 m = 1.2 x 104 m
1 km
Base Unit
gram
meter
liter
Use chart
to get
conversion
Conversions with metric/SI
• Example: How many cm are in 1.3 m?
# cm = 1.3 m x 1 cm
0.01 m
Base Unit
gram
meter
liter
Use chart
to get
conversion
Conversions with metric/SI
• Example: How many cm are in 1.3 m?
# cm = 1.3 m x 1 cm = 1.3 x 102 cm
0.01 m
Conversions with metric/SI
When given a problem with 2 “prefixes”
always go from prefix to base/base to prefix.
Example: converting cm to km – convert cm
(prefix) to meters (base), then meters (base)
to km (prefix).
These are often referred to as “2 step
problems”
Conversions with metric/SI
• Example: How many km are in 2.7 x 104mm?
• This would be considered a “2-step problem”.
• There are 2 prefixes – km to mm
Conversions with metric/SI
• Example: How many km are in 2.7 x 104mm?
# km = 2.7 x 104mm x 0.001 m
1 mm
Base Unit
gram
meter
liter
Use chart
to get
conversion
Conversions with metric/SI
• Example: How many km are in 2.7 x 104mm?
Use chart
get
# km = 2.7 x 104mm x 0.001 m x 1tokm
1 mm conversion
1000 m
Base Unit
gram
meter
liter
Conversions with metric/SI
• Example: How many km are in 2.7 x 104mm?
# km = 2.7 x 104mm x 0.001 m x 1 km =
1 mm 1000 m
0.027 km
• Take time now to work on the
practice problems
• Ask questions if you need help!!