Dimensional Analysis

Dimensional analysis is a powerful tool for converting units within and between measuring systems. Keeping track of the 'units' in an equation assures some confidence in the numeric outcome. Conversion factors expresses a basic relationships between units:

English units:

12 inches   1 foot     1 yard     3 feet   5280 ft 1 mile
1 foot 12 inches 3 feet 1 yard 1 mile 5280 ft

SI units:

1000 m     1 km        1 m     100 cm 1000 mm   1 cm     10 mm  
1 km 1000 m 100 cm 1 m 1 m 10 mm 1 cm

English to SI units:

   1 inch       1 pound     1 quart  
2.54 cm 454 g .95 liters

Process

  1. Place desired units on the right of the equals sign.
  2. Place the given units to the left of the equals.
  3. Select the denominator of the conversion factor so that it has the same units as the given unit.
  4. Select the numerator of the conversion factor so that it has the same units as the desired unit.
  5. Insert the actual numbers appropriate for the conversion factor.
  6. The given unit and the conversion factor unit will "cancel", leaving just the desired unit.
  7. Complete the indicated multiplication.

Example 1

Example of an English system single conversion: 108 inches = ?? feet

  1. Place desired units on the right of the equals sign.
      x     =   ______ feet
  2. Place the given units to the left of the equals.
    108 inches   x       =   ______ feet
  3. Select the denominator of the conversion factor so that it has the same units as the given unit.
    108 inches   x               =   ______ feet
        inches    
  4. Select the numerator of the conversion factor so that it has the same units as the desired unit.
    108 inches   x   foot   =   ______ foot
        inches    
  5. Insert the actual numbers appropriate for the conversion factor.
    108 inches   x   1 foot   =   ______ feet
        12 inches    
  6. The given unit and the conversion factor unit will "cancel", leaving just the desired unit.
    108 inches   x   1 foot   =   ______ feet
        12 inches    
  7. Complete the indicated multiplication
    108   x   1 foot   =   9 feet
        12    
 

Example 2

Example of an English to SI system single conversion: 3 pounds = ?? grams

  1. Place desired units on the right of the equals sign.
      x     =   ______ grams
  2. Place the given units to the left of the equals.
    3 pounds   x       =   ______ grams
  3. Select the denominator of the conversion factor so that it has the same units as the given unit.
    3 pounds   x               =   ______ grams
        pounds    
  4. Select the numerator of the conversion factor so that it has the same units as the desired unit.
    3 pounds   x   grams   =   ______ grams
        pounds    
  5. Insert the actual numbers appropriate for the conversion factor.
    3 pounds   x   454 grams   =   ______ grams
        1 pound    
  6. The given unit and the conversion factor unit will "cancel", leaving just the desired unit.
    3 pounds   x   454 grams   =   ______ grams
        1 pounds    
  7. Complete the indicated multiplication
    3   x   454 grams   =   1362 grams
        1    
 

Example 3

Example of an English system double conversion: 1.2 miles = ?? inches

  1. Place desired units on the right of the equals sign.
      x     =   ______ inches
  2. Place the given units to the left of the equals.
    1.2 miles   x       =   ______ inches
  3. Select the denominator of the conversion factor so that it has the same units as the given unit.
    1.2 miles   x               =   ______ inches
        miles    
  4. Select the numerator of the conversion factor so that it has the same units as the desired unit.
    1.2 miles   x   inches   =   ______ inches
        miles    
  5. Insert the actual numbers appropriate for the conversion factor.
    1. OOPS, we don't know a conversion from miles to inches.
      1.2 miles   x   ? inches   =   ______ inches
          ? miles    
    2. We will have to add another conversion factor we do know.
      1.2 miles   x   feet   x               =   ___ inches
          miles        
    3. For the second conversion factor we must convert from feet to inches.
      1.2 miles   x   feet   x   inches   =   ___ inches
          miles   feet    
    4. Now we can add the approproaite numbers.
      1.2 miles   x   5280 feet   x   12 inches   =   ___ inches
          1 mile   1 foot    
  6. The given unit and the conversion factor unit will "cancel", leaving just the desired unit.
    1.2 miles   x   5280 feet   x   12 inches   =   ___ inches
        1 mile   1 foot    
  7. Complete the indicated multiplication
    1.2   x   5280   x   12 inches   =   76,032 inches
        1   1