Introduction to Dimensional Analysis
Dimensional analysis is a math tool used to change units. With this method, it is easy to
change very complex units provided you know the necessary conversion definitions.
This method is also referred to as unit conversion, factor analysis, unit analysis, DA,
factor-label method, unit multipliers or unit fractions.
Here are some basic concepts that are used in dimensional analysis:
 Any whole number can be written as a fraction.
A number can be changed to a fraction and still keep its value by placing 1 as the
denominator.
Ex. 4 
4
1
 The value of a quantity is not changed when it is multiplied by 1.
When a number is multiplied by 1, the value of the number remains unchanged.
Ex.
12(1) = 12
 One (1) can be written as a fraction in many ways.
As long as the numerator of the fraction equals the denominator of the fraction,
the fraction is equal to one.
1
1 2 7 45
  
1 2 7 45
 Unit fractions are also fractions that equal 1.
The difference is these fractions are in units. Any definition involving units can be
made into a unit fraction.
12 inches = 1 foot
1 
12 inches
1 foot

1 foot
12 inches
The unit labels are crucial to forming unit fractions. If the labels are removed, the
statement is no longer true.
Since unit fractions equal 1, it is possible to multiply a quantity by a unit fraction
and not change the value.
 Unit labels cancel in the same way that common factors reduce to 1 in
fraction multiplication.
4
5
4


5
7
7
1 yard
1 foot
1 yard


3 feet 12 inches
36 inches
Unit fractions are used to convert the units of a given number while retaining the number’s
value. Remember, the unit fraction equals one. When a number and a unit fraction are
multiplied, the number retains its value because any number that is multiplied by one stays the
same. The only change will be in the units assigned to the value. When a measurement is
changed using unit fractions, the appearance of the measurement changes but not its size.
Units of Capacity
Units of Length
1 ft = 12 in
1 in = 2.54 cm
1 yd = 3 ft
= 36 in
1 yd = .9144 m
1 mile =5280 ft
1 km = .621 mi
1 cm = 10 mm
1 dm = 10 cm
1 m = 10 dm
= 100 cm
= 1000 mm
1 km = 1000 m
1 gal = 4 qt
= 8 pt
1 qt = 2 pt
1 pt = 2 cu
1 cup = 8 oz
= 16 Tbs
1 fl oz = 2 Tbs
1 kL = 1 daL
1 qt = .9463 L
1 fl oz =
29.57 mL
1 hL = 10 daL
1 daL = 10 L
1 L = 1000 mL
= 10 dL
1 dL = 10 cL
Example #1: Change 4 miles into yards.
1.
Write 4 miles as a fraction.
4 miles
1
2.
Since there is not a definition that relates miles to yards, unit fractions are used to change
miles into feet and then into yards. The unit fractions for miles and feet are
1 mile
5280 feet
or
. Because we do not want miles in the final answer, we
5280 feet
1 mile
multiply by the unit fraction that will allow us to cancel the mile units.
4 miles 5280 feet

1
1 mile
We do not want feet in the final answer so we find the unit fractions that will allow us to
1 yard
3 feet
cancel feet:
or
. We use the first one.
3 feet
1 yard
4 miles =
4 miles =
3.
4 miles 5280 feet 1 yard


1
1 mile
3 feet
Now that we have the desired units (yard), we multiply all of the numerators together and
all of the denominators together.
4 miles 5280 feet 1 yard 21120 yards
4 miles =



1
1 mile
3 feet
3
To get the final result, divide the numerator by the denominator.
4 miles =
4 miles 5280 feet 1 yard 21120 yards



 7040 yards
1
1 mile
3 feet
3
Example #2: Change 640 fluid ounces into pints.
640 ounces 
640 ounces
1 cup
1 pint
640 pints



 40 pints
1
8 ounces 2 cups
16
The above examples involve changing one unit to another. Some problems are more
complicated than that. For these we use as many unit fractions as we need, setting up a long
multiplication, so the units we don’t want cancel out.
Units of Time
Units of Weight
1 millennium = 1000 yrs
1 cent = 100 yr
1 yr = 12 mo
= 365 days
1 day = 24 hrs
1 hr = 60 min
1 min = 60 sec
1 T = 2000 lb
1 lb = 16 oz
1 oz = 16 dram
1 dram =
27.34 grain
1 tonne =
=2205 lb
1 lb = .4536 kg
1 oz = 28.35 g
1 dr = 1.772 g
Units of Area
2
1 ft = 144 in2
1 in2 = 6.452 cm2
2
2
1 yd = 9 ft
1 yd2 = .8361 m2
1 acre = 160 rd2 1 a =.4047 ha
= 4840 yd2
1 mi2 = 640
acres
Example #3: Which is faster: 80 miles an hour or 40 meters per second?
To compare two values, they must have the same units. Using the definitions given in the
preceding tables, it is possible to change miles to feet to inches to centimeters to meters and
hours to minutes to seconds. If the tables gave different definitions, we could choose different
unit fractions. We use whatever definitions we have available.
1.
We begin by changing one set of units. Let’s start with the miles.
80 miles 5280 feet 12 inches 2.54 cm 1 meter




hour
1 miles
1 foot
1 inch 100 cm
The units for this problem are now meters per hour.
2.
To finish the unit change, we multiply by the unit fractions that will change hours to seconds.
80 miles 5280 feet 12 inches 2.54 cm 1 meter
1 hour
1 minute






hour
1 miles
1 foot
1 inch 100 cm 60 minutes 60 seconds
The time units have now changed from hours to seconds and the units for the problem are
meters/second.
3.
Now that we have the desired units, we multiply all of the numerators together and all of the
denominators together.
80 miles 12,874,752 meters

hour
360,000 seconds
To get the final result, divide the numerator by the denominator.
80 miles
12,874,752 meters 35.76 meters


hour
360,000 seconds
second
This tells us that 80 miles per hour is equivalent to a little less than 36 meters per second, so 40
meters per second is faster than 80 miles per hour.
Example #4: You are planning a trip to Mexico and, in the process, you find out the cost of gasoline
there is 7 peso per liter. How does that compare to the cost of gas in the US? ($1 = 12 peso)
7 peso 1 dollar .9463 liter 4 quarts
26.50 dollars
2.21 dollars





1 liter 12 peso 1 quart
1 gallon
12 gallons
gallon
7 peso per liter is about the same as $2.21 per gallon.
Example #5: You are installing a fence and you need to mix some concrete to set the posts. The
directions say you need five gallons of water for your mix. Your bucket isn’t marked so you
don’t know how much it holds. However, you just finished a two-liter bottle of soda. If you use
the bottle to measure your water, how many times will you need to fill it?
5 gallons
4 quarts .9463 liters 1 bottle
18.93 bottles




1 bag of mix 1 gallon
1 quart
2 liters
2 bag of mix

9.46 bottles
1 bag of mix
You need to use about 9 1/2 bottles of water.
Suggestions for using Dimensional Analysis
1.
Determine what you want to know. (How many miles in an hour?)
Rephrase it using “per”. (miles per hour)
2.
Determine what you already know.
3.
Determine the unit fractions that may be needed. You will need enough to form
a “bridge” from what you know to what you want to know.
4.
From what you know, pick a starting factor.
5.
Select a unit fraction that cancels out units in the starting factor that you don’t
want.
6.
Continue picking unit fractions until only the desired units remain.
7.
Do the math. Multiply the numerators together. Multiply the denominators
together. Divide the numerator by the denominator. Be sure to write the labels
that go with the answer.
8.
Take a few seconds and ask yourself if the answer makes sense. If it doesn’t,
check your work.