Dimensional Analysis
Solving Scientific Problems
Mathematically
• I CAN solve mathematical conversions using
Dimensional Analysis.
• Dimensional Analysis is an orderly and
systematic approach used to make
MATHEMATICAL CONVERSIONS from one unit
to another easy to accomplish.
• Since measurements require both a NUMBER
and a UNIT, this method is often referred to as
the FACTOR LABEL method because UNITS
MUST BE INCLUDED with each measurement.
Using the Factor Label Method
• Let’s use a simple example to learn this
method:
• If someone were to tell you that they had 275
eggs and asked you how many dozen eggs
that represented, how would you find that
answer?
• Before solving that problem, you would have
to know that 12 eggs make up 1 DOZEN.
• Once you knew that, solving the problem
would be easy.
• You would simple divide 275 by 12 to
determine how many dozen eggs you have.
• Written out mathematically, your calculation
would look something like this:
•
•
275 eggs
1 dozen = 22.9 dozen
12 eggs
• From this set up, do you see any UNITS that
could be canceled DIAGONALLY?
• 275 eggs
•
1 dozen = 22.9 dozen
12 eggs
• Notice the EGGS canceled diagonally, leaving
the final unit of DOZENS as the only unit for
the answer.
• This is called the FACTOR LABEL METHOD:
– Conversions involve NUMBERS with UNITS.
• What was the relationship between 12 eggs
and 1 dozen?
• Since 12 eggs make 1 dozen, this relationship
is called a CONVERSION FACTOR.
– A conversions factor represents a number of
smaller units that make 1 of a larger unit, as in this
case 12 eggs (smaller units) = 1 dozen (larger
unit).
– Dimensional Analysis is written using a CROSSBRACKET like this:
A dimensional analysis always begins with WHAT
YOU ALREADY KNOW and is written in the upper
corner of the frame:
=
X
This is usually what’s GIVEN IN THE PROBLEM.
– In our previous example, we were given 275 eggs.
• Generally NOTHING (put an X in it) is written
in the bottom of the first frame:
275 eggs
•
X
• The SECOND FRAME contains the
CONVERSION FACTOR:
12 EGGS = 1 DOZEN
• How do you know which value GOES ON TOP
and which value GOES ON BOTTOM?
That’s easy….when everything is added to the
frame, ALL UNITS EXCEPT THE FINAL ONE
MUST CANCEL DIAGONALLY!
– If they do not, something is in the wrong place!
EXAMPLE
• Now let’s set up our egg problem using
DIMENSIONAL ANALYSIS:
•
275 EGGS
1 DOZEN
=
X
12 EGGS
275 Dozen
12
Once all the numbers are in place and UNITS ARE
CANCELED…everything ACROSS the TOP LINE is
MULTIPLIED, then everything ACROSS the BOTTOM LINE
is MULTIPLIED, then the TOP is DIVIDED by the BOTTOM
for the FINAL ANSWER.
PRACTICE PROBLEM
• There are 39.37 inches in 1 meter.
Set up a DIMENSIONAL ANALYSIS
problem to convert 8.75 meters
to inches.
•
8.75 meters
39.37 inches
=
X
1 meter
318.601 inches
1
Once all the numbers are in place and UNITS ARE
CANCELED…everything ACROSS the TOP LINE is
MULTIPLIED, then everything ACROSS the BOTTOM LINE
is MULTIPLIED, then the TOP is DIVIDED by the BOTTOM
for the FINAL ANSWER.
= 318.601 INCHES
YOUR TURN
SHOW YOUR WORK as DIMENSIONAL ANALYSIS
• Problem #1 How many inches are there in 45
meters if each meter is 39.37 inches?
• How many ounces are contained in 37.5
pounds assuming 1 pound is 16 ounces?
Practice with CONVERSION FACTORS
•
Below are some common CONVERSION FACTORS. If you do NOT know these, use
Google or another search engine to find them.
•
1. 1 foot = _____ inches
2. 1 meter = ______ centimeter
•
3. 1 pound = ______ ounces
4. 1 pound = ______ grams
•
5. 1 mile = ______ feet
6. 1 mile = _______ kilometers
•
7. 1 liter = ______ ounces
8. 1 gallon = _______ ounces
•
9. 1 ton = _______ pounds
10. 1 meter = ______ millimeters
•
11. 1 gallon = ______ milliliters
12. 1 inch = _____ centimeters