Dimensional Analysis (The Factor Label Method) Most calculations in science involve measured quantities.  In such calculations, the units in which  quantities are measured must be treated mathematically just as the numerical parts of the quantities are.  For example, in multiplying 1.2 cm by 2.0 cm, there are two separate calculations to be carried out.   First, it is necessary to multiply the two numbers: 1.2 X 2.0 = 2.4     Second, it is necessary to multiply the two units: cm X cm = cm2.  The complete answer then, is 1.2 cm X 2.0 cm = 2.4 cm2. This concept can be applied in the solution of many problems.  The application depends on the use of a "conversion factor".  A conversion factor is a fraction in which the numberator adn the denominator both represent the same measurement.    For example, the fraction 100 cm    1 m   is a unit factor since both the numerator and denominator represent the same length (one meter).  The solved examples illustrate the use of unit factors in solving problems by dimensional analysis. Table of Conversion Factors Example_1: Convert 45.3 cm to its equivalent measurement in mm. Solution: Select a conversion factor which will convert the unit "cm" to the unit "mm". The appropriate conversion factor is:  10 mm / 1 cm.  Arrange the problem so that the given measurement, when multiplied by the correct unit factor, will yield an answer with the proper label: 45.3 cm  X  10 mm  =  453 mm                     1 cm Example_2: Change a speed of 72.4 miles per hour to its equvalent in meters per second. In this example, several conversion factors are needed. One to change the miles into meters and the other to change hours into seconds? 72.4 mi  X  1760 yd  X  36 inches  X    1 meter  X     1 hr       =  32.7 m    1 hr              1 mi           1 yd             39 inches      3600 s                  s Example_3: The density of mercury is 13.6 g/mL.  What is the mass in kilograms of a 2 L commercial flask of mercury? Set up the problem so that the calculation will yield a result with a mass in grams. 13.6 g  X  1000 mL X  2 L X   1 kg    =  27.2 kg   1 mL           1 L                     1000 g Dimensional Analysis Practice Problems Level 1 Dimensional Analysis Practice Problems Level 2 Dimensional Analysis Practice Problems Level 3