Dimensional analysis is a tool to help you solve problems. Dimensional analysis can tell you if a problem is set up improperly. Dimensional analysis can tell you if a problem is set up correctly.
So, what is dimensional analysis? Dimensional analysis is, essentially, writing the units down for every number and then canceling out units as if they were algebraic terms.
Let us do a short and easy example. How many inches are there in one yard?
First: what do we know? one yard
Second: where will we be when we get to the end of the problem? inches
Third: what relationship do we know between the two?
36 inches = 1 yard
To solve this problem we would take:
FOR CLARITY, I AM GOING TO WRITE EVERYTHING IN THIS CHAPTER AS A FRACTION. IT IS EASIER TO SEE HOW THE UNITS CANCEL OUT WHEN WRITTEN THIS WAY.
There are two important factors in this problem which most people get without thinking about them. The first is that we need to get rid of the unit of "yards" and introduce the unit of "inches." The second is the use of the 36 inches/yard conversion factor. Where does this conversion come from?
36 inches = 1 yard.
If we divide both sides of the equation by "1 yard " we will get:
The right side of the equality is = 1, therefore, 
.
So, when we multiplied by this conversion factor, we were, mathematically speaking, multiplying by "1".
The conversion factors are the relationships between any TWO things. If we want to relate more than two things, we need to use more than one relationship. If the relationships are done properly (and are all equal to "1" ) then the problem will be correct.
Things to remember about dimensional analysis:
1) If the units come out right, then the problem is almost always set up right.
2) If the units are not wrong, then the problem is still OK. You may not be doing a problem the shortest possible way, but it will still work out OK.
Let's do a more complicated example:
Convert 
to 
First: we are given
Second: we want 
There are two unit conversions we need to do here: miles to feet
and hours to seconds.
Let us do the two steps and then combine the answers. (You can do this as one step if you want to, it is OK)
What information do we know?
1760 yards = 1 mile
5280 feet = 1 mile
12 inches = 1 foot
36 inches = 1 yard
The second piece seems most useful:
Notice that the unit of miles cancels out and we now have the unit of feet in the denominator which is what we want to have at the end of the problem.
The second step is to convert hours to seconds:
What information do we know:
60 seconds = 1 minute
60 minutes = 1 hour
(Notice the unit of hours cancels out.)
(notice the unit of minutes cancels out)
Now we have completed the two steps. We are ready to get the final answer. We want feet in the numerator and seconds in the denominator. We need to divide 316800 feet by 3600 seconds:
Let's do another example. how many tablespoons in 1 gallon:
What do we know: 1 gallon
What do we want: tablespoons
What reasonable relationships do we know:
4 tablespoons = quarter cup
4 quarter cups = 1 cup
2 cups = 1 pint
2 pints = 1 quart
2 quarts = half gallon
2 half gallons = 1 gallon
(there are shorter ways, but what the heck)
We need to keep track of our units and make sure that the units seem reasonable at each step. We will do this problem two times. The first time will be to do each step individually. The second time we will do all of the steps in sequence in one giant problem. Either way is OK, you choose which one you like best. Remember, watch your units.
(gallons
cancel)
(half gallons
cancel)
(quarts
cancel)
(pints
cancel)
(cups cancel)
and now we know that 256 tablespoons = 1 gallon. (aren't you excited)
If you do it in one step, it will look something like this:
Here are some useful relationships (in the English system) which you may need to solve these problems.
12 inches = 1 foot
3 feet = 1 yard
1760 yards = 1 mile
5280 feet = 1 mile
16 ounces = 1 pound
2000 pounds = 1 ton
7000 grains = 1 pound
12 thingys = 1 dozen
20 thingys = 1 score
144 thingys = 1 gross
SAMPLE PROBLEMS
The answer to the SAMPLE PROBLEMS are given in APPENDIX I.
1) a) Convert 
to 
. (use
365 days = 1 year)
b) Convert 180 pounds into ounces.
c) Convert $2.14 into pennies.
2) a) Convert 6.6 gallons into quarts.
b) Convert 
into 
(rate
of sea floor spreading)
c) Convert 20 years into seconds.
HOMEWORK
1) Convert 2.55 tons into ounces.
2) Which weighs more: 1000 grains of flour or 5X10-5 tons of flour?
3) 2.5 gross hamburger patties weighs 0.045 tons. What size is each hamburger patty in pounds.
4) Convert 1 ton into grains.
5) A recipe calls for four cups of liquid. How many tablespoons would you need to get four cups?
E-mail answers to Jim Tarter