The following study guides are linked to Anne McKenna's website (http://employees.csbsju.edu/amckenna/CH123), with permission by the author.

Dimensional Analysis

            Problem solving inevitably involves changing one set of units into another set of units.  Problem solving does not generate any new information.  Problem solving simply transforms information from one form into another (frequently more useful) form.  There are many methods of solving problems.  The most common and easiest to use is known as the "unit-factor-method" or "factor-label-method" or is sometimes just called "dimensional analysis."

            Dimensional analysis is a technique of calculating with physical quantities in which units are included and treated in the same way as numbers--it sounds a lot harder than it really is.  If you have ever planned a pizza party, you have used dimensional analysis.  Clearly, the amount of pizza and pop you will need will depend on the number of people you expect.  For example, if you are planning a Friday night pizza party and expect 30 people you might begin by estimating how much pizza and pop each person will eat.  Let's assume that each person at the party will drink 4 cans of pop and eat 5 slices of pizza.  Given these assumptions, how much pizza and pop will you need?  The solution is arrived at by dimensional analysis.  First let's consider how much pizza you will need:

Normally, you don't buy this quantity of pizza by the slice; but rather by the whole pizza.  In order to determine the number of pizzas you will need, you must know how many slices there are per pizza.  For a large pizza, there are typically eight slices. Thus, for 150 slices you would need:

This would properly be rounded off to 19 pizzas because you can't buy a partial pizza. 

            This calculation could have also been performed in one step rather than two steps:

Please note that either procedure gives the same answer.  However, there is less chance of a round off error in performing the calculation in a single step.

            How much pop will be needed for the party?

             The pop for the party can be purchased by buying individual cans from a vending machine.  However, this is probably the most expensive way of buying pop.  More conveniently, we would purchase the pop in six packs, twelve packs or twenty-four packs.  In each case we would need:

Note that in each of these calculations, the conversion factor is the number of cans per pack.  In general, the larger quantity per package is more cost effective.  However, sales or coupons may make the price of a smaller package price competitive.

            Consider the following relationship:

If you divide both sides of the equality by the right hand quantity, you get:

Observe that you treat units in the same way as algebraic quantities.  Note too that the right-hand side now equals 1 and no units are associated with it.  Because it is always possible to multiply any quantity by 1 without changing that quantity, we can multiply any length by the factor 1 ft/12 in without changing the actual length.  Essentially, multiplication by this factor only changes the way we express the length (i.e., the units).

            Let's consider how many feet are represented in a length of 31.6 in.  This problem can be solved as follows:

The ratio 1 ft/12 in is called a unit conversion factor (ucf) because it is a factor equal to one that converts a quantity expressed in one unit to a quantity expressed within a given system of measurement.

            Similarly, we could have divided both sides of the relationship 1 ft/12 in by the left side to get:

Thus the relationships:

are both ucfs because the numerator and denominator describe the same distance.  Multiplication of a quantity by these ucfs changes the units in which the quantity is expressed but not its value.  Remember, we can always multiply something by one without changing its value.

NOTE:  Since ucfs are conversion factors within one system of units, all the relationships are defined and hence have an infinite number of significant figures.  Thus use of ucfs do not affect the number of significant figures in an arithmetic result.  In the previous calculation, the quantity 31.6 in determines or limits the number of significant figures in the result.

Consider the following alternative way to set up the above example:

The mathematics of this problem are indeed correct since 1 ft/12 in is equivalent to 12 in/1 ft because any number divided by itself is 1.  However, the rather bizarre units of in2/ft indicate that the equation has been set up incorrectly.  This example shows:

The value of dimensional analysis:

    1.     The units for the answer will come out of the calculations automatically.

     2.     If you make an error in arranging conversion factors in the calculation, the units will be nonsensical.

HINT:  In addition to using dimensional analysis to solve problems, when you finish a calculation, look at the magnitude of your answer and ask yourself whether or not your answer makes any sense; often the units of the answer are correct but the numerical value does not.