One of these things is Not
like the other…
This guide will explain briefly the
concept of units, and the use of a
simple technique with a fancy name—
"dimensional analysis”.
Whatever You Measure,
You Have to Use Units
• A measurement is a way to describe the
world using numbers. We use
measurements to answer questions like,
how much? How long? How far?
• Suppose the label on a ball of string
indicates that the length of the string is
150.
• Is the length 150 feet, 150 m, or 150 cm?
• For a measurement to make sense, it
must include a number and a unit.
Units and Standards
Number
150 feet
Unit
Rule: No naked numbers. They must have units
• Now suppose you and a friend want to
make some measurements to find out
whether a desk will fit through a doorway.
• You have no ruler, so you decide to use
your hands as measuring tools.
Units and Standards
• Even though you
both used hands
to measure, you
didn’t check to
see whether your
hands were the
same width as
your friend’s.
Units and Standards
• In other words, you
didn’t use a
measurement standard,
so you can’t compare
the measurements.
• Hands are a
convenient measuring
tool, but using them
can lead to
misunderstanding.
Units and Standards
• So in order to avoid confusion we use
measurement standards.
• A standard is an exact quantity that
people agree to use to compare
measurements.
Units and Standards
• In the United States, we commonly use
units such as inches, feet, yards, miles,
gallons, and pounds. This is known as
the English system of measurement.
• Most other nations and the scientific
community use the metric system
- a system of measurement based on multiples of ten.
International System of Units
• In 1960, an improvement was made to
the metric system. This improvement is
known as the International System of
Units.
• This system is abbreviated SI from the
French Le Systeme Internationale
d’Unites.
International System of Units
• The standard kilogram is kept in Sèvres,
France.
• All kilograms used throughout the world
must be exactly the same as the kilogram
kept in France because it is the standard.
International System of Units
• Each type of SI
measurement
has a base unit.
• The meter is the
base unit of
length.
International System of Units
• Every type of
quantity measured
in SI has a symbol
for that unit.
• All other SI units are
obtained from these
seven units.
Review
• When we measure something, we
always specify what units we are
measuring in.
• All kinds of units are possible, but in
science we use the SI system.
• Problem! What if I measure something
in inches but I am supposed to give
you the answer using SI units?
Sometimes You Have to Convert
Between Different Units
• How many seconds are in a day?
• How many inches are in a centimeter?
• If you are going 50 miles per hour,
how many meters per second are you
traveling?
• To answer these questions you need
to change (convert) from one unit to
another.
How do you change units?
• Whenever you have to convert a physical
measurement from one dimensional unit to
another, dimensional analysis is the
method used.
(It is also known as the unit-factor method or the factor-label method)
• So what is dimensional analysis?
The converting from one unit system to another.
If this is all that it is, why make such a fuss about it? Very
simple. Wrong units lead to wrong answers. Scientists have
thus evolved an entire system of unit conversion.
Dimensional Analysis
• How does dimensional analysis work?
• It will involve some easy math (Multiplication & Division)
• In order to perform any conversion, you need a
conversion factor.
• Conversion factors are made from any two terms
that describe the same or equivalent “amounts”
of what we are interested in.
For example, we know that:
1 inch = 2.54 centimeters
1 dozen = 12
Conversion Factors
• So, conversion factors are nothing more than
equalities or ratios that equal to each other. In
“math-talk” they are equal to one.
• In mathematics, the expression to the left of the
equal sign is equal to the expression to the right.
They are equal expressions.
• For Example
12 inches = 1 foot
Written as an “equality” or “ratio” it looks like
=1
or
=1
Conversion Factors
or
Hey!
These
look like
fractions!
• Conversion Factors look a lot like fractions, but
they are not!
• The critical thing to note is that the units
behave like numbers do when you multiply
fractions. That is, the inches (or foot) on top and
the inches (or foot) on the bottom can cancel
out. Just like in algebra, Yippee!!
Example Problem #1
• How many feet are in 60 inches?
Solve using dimensional analysis.
• All dimensional analysis problems are set
up the same way. They follow this same
pattern:
What units you have x What units you want
What units you have
The number & units
you start with
The conversion factor
(The equality that looks like a fraction)
= What units you want
The units you
want to end with
Example Problem #1 (cont)
• You need a conversion factor. Something
that will change inches into feet.
• Remember
12 inches = 1 foot
Written as an “equality” or “ratio” it looks like
60 inches x
=
5 feet
(Mathematically all you do is: 60 x 1  12 = 5)
What units you have x What units you want
What units you have
= What units you want
Example Problem #1 (cont)
• The previous problem can also be written
to look like this:
• 60 inches
1 foot
= 5 feet
12 inches
• This format is more visually integrated,
more bridge like, and is more appropriate
for working with factors. In this format, the
horizontal bar means “divide,” and the
vertical bars mean “multiply”.
Dimensional Analysis
• The hardest part about dimensional
analysis is knowing which conversion
factors to use.
• Some are obvious, like 12 inches = 1 foot,
while others are not. Like how many feet
are in a mile.
Example Problem #2
• You need to put gas in the car. Let's
assume that gasoline costs $3.35 per
gallon and you've got a twenty dollar bill.
How many gallons of gas can you get with
that twenty? Try it!
• $ 20.00
1 gallon
= 5.97 gallons
$ 3.35
(Mathematically all you do is: 20 x 1  3.35 = 5.97)
Example Problem #3
• What if you had wanted to know not how many
gallons you could get, but how many miles you
could drive assuming your car gets 24 miles a
gallon? Let's try building from the previous
problem. You know you have 5.97 gallons in the
tank. Try it!
• 5.97 gallons 24 miles
= 143.28 miles
1 gallon
(Mathematically all you do is: 5.97 x 24  1 = 143.28)
Example Problem #3
• There's another way to do the previous
two problems. Instead of chopping it up
into separate pieces, build it as one
problem. Not all problems lend
themselves to working them this way but
many of them do. It's a nice, elegant way
to minimize the number of calculations
you have to do. Let's reintroduce the
problem.
Example Problem #3 (cont)
• You have a twenty dollar bill and you need
to get gas for your car. If gas is $3.35 a
gallon and your car gets 24 miles per
gallon, how many miles will you be able to
drive your car on twenty dollars? Try it!
• $ 20.00
1 gallon
24 miles
$ 3.35
1 gallon
= 143.28 miles
(Mathematically all you do is: 20 x 1  3.35 x 24  1 = 143.28 )
Example Problem #4
• Try this expanded version of the previous
problem.
• You have a twenty dollar bill and you need
to get gas for your car. Gas currently costs
$3.35 a gallon and your car averages 24
miles a gallon. If you drive, on average,
7.1 miles a day, how many weeks will you
be able to drive on a twenty dollar fill-up?
Example Problem #4 (cont)
• $ 20.00 1 gallon 24 miles 1 day
$ 3.35
1 week
1 gallon 7.1 miles 7 days
= 2.88 weeks
(Mathematically : 20 x 1  3.35 x 24  1 x 1  7.1 x 1  7 = 2.88 )
Dimensional Analysis
• So you can have a simple 1 step problem
or a more complex multiple step problem.
Either way, the set-up of the problem
never changes.
• You can even do problems where you
don’t even understand what the units are
or what they mean. Try the next problem.
Example Problem #5
• If Peter Piper picked 83 pecks of pickled
peppers, how many barrels is this?
• Peter Piper picked a peck of pickled peppers... Or so the rhyme
goes. (What in the world is a peck?)
• You need help for this one. As long as you
have information (conversion factors) you
can solve this ridiculous problem.
Example Problem #5 (cont)
• Use this info: A peck is 8 dry quarts: a bushel is 4
pecks or 32 dry quarts; a barrel is 105 dry quarts.
•
WHAT?! Rewrite them as conversion factors if the info
is not given to you that way.
8 dry quarts
1 peck
or
1 peck .
8 dry quarts
32 dry quarts
1 bushel
105 dry quarts
1 barrel
4 pecks
1 bushel
or
or
or
1 bushel .
32 dry quarts
1 barrel .
105 dry quarts
1 bushel
4 pecks
Example Problem #5
• Pick the conversion factors that will help
get to the answer. 83 pecks
1 barrel.
Hint: Look for units that will cancel each other.
• 83 pecks 8 dry quarts
1 peck
1 barrel
105 dry quarts
= 6.3 barrels
(Mathematically : 83 x 8  1 x 1  105 = 6.3 )
Review
• Dimensional Analysis (DA) is a method
used to convert from one unit system to
another. In other words a math problem.
• Dimensional Analysis uses Conversion
factors . Two terms that describe the same or equivalent
“amounts” of what we are interested in.
• All DA problems are set the same way.
Which makes it nice because you can do
problems where you don’t even understand
what the units are or what they mean.