Lesson 2-4

The Factor-Label Method or Dimensional Analysis

     During the course of this year you will be required to solve many types of problems that involve units.  Often, you will be required to change from one unit to another.  This is not actually very hard, because it is something that you do in your day-to-day life.  For example, the running time of a movie may be listed as 90 minutes.  You could convert that to hours using a conversion factor.  A conversion factor is an expression for the relationship between units.  In this case you would be using the relationship 1 hour = 60 minutes.    You can set up a factor-label problem as shown below:

Step 1.   Show what you are given on the left, and what units you want on the right.

90 min                                                                       = ? hrs

Step 2.   Insert the required conversion factors to change between units.  In this case we need only one conversion factor, and we show it as the fraction, 1hr/60min.  We put the units of minutes on the bottom so that they will cancel out with the minutes on the top of the given.

90 min x   1 hr                                                             = ? hrs
               ---------
               60 min

Step 3.   Cancel units where you can, and solve the math.

          90 min x    1 hr                                                             = 1.5 hrs
               ---------
               60 min

     Of course, most of us can do the above calculation in our heads.  This is because we are very familiar with the units and the conversion factors involved.  Not all conversions will be that easy, but if you follow the steps correctly, there should be little chance for mistake. Follow the example below.

     Example 1.  A student determines that the density of a certain material is 4.46 g/cm³. What would be the density of this material in g/L? 

     Well, in order to solve this problem you must remember that 1000 cm³ = 1L.  Then follow the same steps as the previous problem.

Step 1.   Show what you are given on the left, and what units you want on the right.

                   4.46  g/cm³                                                                         = ? g/L 

Step 2.   Insert the required conversion factors to change between units.  Note that I have changed the "look" of the fractions to show the cancellation of units more clearly.

                    4.46    g            1000 cm³                                                         ? g
                             ---      x     -----------                                                                =      ^---
                                       cm^{3               1 L                                                                                          L}

Step 3.   Cancel units where you can, and solve the math.

           4.46  g                1000 cm³                                                                4460 g
                    ---       x    -----------                                                                      =     ^-----------
                            cm^{3               1 L                                                                                                        L}

^{Answer - 4460 g/L  (note that we are showing the correct number of significant digits.)}

Example 2.  Imagine that water is leaking from a container, at a rate of 1.2 ml/hour.  If this rate does not change, how many liters of water will be lost in a week? 

We can make a list of the conversion factors that we will need.

1 L = 1000 ml                24 h = 1 day                7 day = 1 week

Step 1.   Show what you are given on the left, and what units you want on the right.

                   1.2ml                                                                                  ? L
                   -------                                                                             =   ------
                        h                                                                                    week     

Step 2.   Insert the required conversion factors to change between units. 

                    1.2ml           1 L             24 h          7 day                                        ? L
                   -------    x  ------------ x ----------- x -----------                                   =   ------
                        h           1000 ml         1 day       1 week                                       week     

Step 3.   Cancel units where you can, and solve the math.

                    1.2ml           1 L             24 h          7 day                                     0.2016 L
                   -------    x  ------------ x ----------- x -----------                            =   ---------------
                        h           1000 ml         1 day        1 week                                       week

We must round to two significant digits, as shown in the original problem.

Answer - 0.20 L/week