Dimensional Analysis Tutorial

Using Two or More Conversion Factors

Problems Using Two or More Conversion Factors

In the last tutorial, you learned to use dimensional analysis to convert directly from one set of units to a different set of units using a single conversion factor. There are many times, however, where there is not a direct relationship between one set of units and another. For example, how would you convert from inches to meters? Most textbooks don't give you a relationship between these two units. So, what do you do???

In a case like this, you have to map out a strategy using relationships between units that you either know or can easily look up in a text. For the conversion between inches and meters, your text gives you the relationship between inches and centimeters and between centimeters and meters. The strategy in such a problem would be to use dimensional analysis to convert from inches to centimeters and then from centimeters to meters. Let's look at an actual example.

Example: A table is 74.2 inches long. What is its length in meters?

  1. First, notice that this problem involves converting from US to metric units. Look in your text for relationships between various units for length. The relationship 2.54 cm = 1 in is given in your text. You can use this relationship to convert from inches to cm. You should also remember all of the metric to metric relationships covered in class as well. In this case you'll need to remember that 1 m = 100 cm. Thus, you can convert from cm to m to obtain your final answer.
  2. Write down the units that you are looking for and an equal sign:

    m

    =
  3. Write down the length and the units you were given in the problem:

    m

    =

    74.2 in
  4. Put a multiplication sign after the number (and units) you were given and draw a line:
    m = 74.2 in x ______
  5. Convert from inches to centimeters by writing a conversion factor above and below the line you've drawn in such a way that the inches cancel out:
    • put "1 in" on the bottom
    • put "2.54 cm" on the top
      m = 74.2 in x
      2.54 cm
      1 in
  6. Cancel out the inches and compare the units you have left (cm) to the units you need at the end (m). Since they aren't the same, put a multiplication sign after the first conversion factor and draw another line.
    m = 74.2 in x
    2.54 cm
    		 x
    ______
    1 in
  7. Convert from centimeters to meters by writing a conversion factor above and below the line you've drawn. Since you want to get rid of (cancel out) the cm, put "100 cm" on the bottom. Since your new units should be meters, put "1 m" on the top.
    m = 74.2 in x
    2.54 cm
    		 x
    1 m
    1 in
    100 cm
  8. Cancel out the centimeters and compare the units you have left (m) to the units you need at the end (m). Since they are the same, put an equals sign after the last conversion factor and do the math. Remember to report your answer using the correct number of significant figures as well as the correct units.
    m = 74.2 in x
    2.54 cm
    		 x
    1 m
    		 = 1.88 m
    1 in
    100 cm
  9. NOTE: In order to avoid rounding errors, do NOT do any of the math until you have used all of the conversion factors needed to convert from your initial units to the ones you want.

Example: Convert 1.59 x 106 cm to miles.

  1. First, map out your strategy by looking at the conversion factors that are available to you. Don't forget to consider the metric-to-metric conversion factors that you must know. In this case, there are several possible strategies that you could use. They will all give the same answer. Two that you might want to consider would be:
    • cm to m to km to mi
    • cm to in to ft to mi
  2. In this example, we will use the first strategy (cm to m to km to mi).
  3. Write down the units that you're looking for and an equal sign.
    mi
    =
  4. Write down the information you were given (including the units). Put a multiplication sign after the units and then draw a line.
    mi
    =
    1.59x 106 cm
    x
    _______
  5. Convert from cm to m by writing the appropriate conversion factor above and below the line:
    mi
    =
    1.59x 106 cm
    x
    1 m
    100 cm
  6. Cancel out the units and compare the one that is left (m) with the one you're looking for (mi). Since they are not the same, multiply by your next conversion factor:
    mi
    =
    1.59x 106 cm
    x
    1 m
    		 x
    1 km
    100 cm
    1000 m
  7. Cancel out the units and compare the one that is left (km) with the one you're looking for (mi). Since they are not the same, multiply by your next conversion factor:
    mi
    =
    1.59x 106 cm
    x
    1 m
    		 x
    1 km
    		 x
    1.609 mi
    100 cm
    1000 m
    1 km
  8. Cancel out the units and compare the one that is left (mi) with the one you're looking for (mi). Since they are the same, put an equal sign and do the math. Report your answer using the correct units and significant figures.
    mi
    =
    1.59x 106 cm
    x
    1 m
    		 x
    1 km
    		 x
    1 mi
    100 cm
    1000 m
    1.609 km
                     
    mi = 9.88 mi
  9. Remember, do not do any of the math until you have used all of the conversion factors that are necessary. This will help avoid unnecessary rounding errors.

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Practice Problems:

  1. 1.430 x 105 mg = _______ kg
  2. 15.25 gal = _______ mL
  3. 0.66 mi/min = _______km/hr

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