DimensionalAnalysis
Converting Units using Conversion Factors
Mr. Gilliland's Physical Science Classes @ SHS

What is Dimensional Analysis?

The 5 Steps of Dimensional Analysis:

Here are the five steps for using DA to convert from one unit to another:

  1. Change your given into a fraction by placing it over 1. This makes it a fraction but does not change it's value.
  2. Write a conversion factor that has the unit you wish to get rid of in the denominator and the unit you want to end up with in the numerator. After you fill in your units, add the numbers. One of the numbers is always a 1 but it can be in either the denominator or the numerator.
  3. Multiply your given times the conversion factor. (A given is what you started with.) When you do this the unit in your given will cancel out leaving you with the unit you wish to convert to.
  4. Multiply the numerators together. Then multiply the denominators together to where you have a fraction. (Do not cross-multiply!)
  5. Change your fraction back to a decimal by dividing the numerator by the denominator and adding the unit. Note
    Note: The unit you end up with will always be in the numerator of the last conversion factor and no units will be left in the denominators since they all cancel out. When you divide a measurement by a number the unit carries over. For example: 5.0 mL/2 = 2.5 mL.

 

Examples of Dimensional Analysis with one conversion factor:

Much of the time, converting from one unit to another involves only the use of one conversion factor.
Here is an example of using Dimensional Analysis to convert 568 mL to liters:

There are 1.6 km in 1 mile. If you were asked to convert the distance from Sarasota to Tampa (55 miles) to kilometers, here is how you would do it:

Examples of Dimensional Analysis with more than one conversion factor:

Sometimes it takes more than one conversion factor to convert one unit to another. This can occur when you are converting a very large unit to a very small one, or vice-versa. In these cases, always convert to your unit to a base unit (grams, meters or liters), then convert to the base unit to the unit you wish to end up with.

For example, if you were asked to convert the length of a 15 cm pencil to kilometers, you would first convert your cm to meters, then meters to kilometers:

Instead of writing individual division lines and multiplication signs, scientist use one long line for the entire conversion and a vertical line for multiplication. So the work above would look like this:

How many ounces in 45 kilograms? You would first need to know that 2.2 lbs = 1 kg. Here is how to convert it:

 

Suppose you want to know the number of seconds in the average person's life (77.2 years). You would begin with 77.2 yrs. over 1 then convert years to days, days to hours, hours to minutes and minutes to seconds. Here is how it would look:

Sig Figs and Dimensional Analysis

When you are converting units in DA, you are either multiplying or dividing. Therefore you use the Sig Fig rule:

In multiplication and division, the result should be rounded off so as to have the same number of significant figures
as in the measurement with the
least number of significant figures.

Some measurements are exact - they are known with complete certainty. For example, there is exactly 100 cm in 1 meter, exactly 60 minutes in 1 hours and exactly 12 inches in 1 foot. For this reason these measurements have an infinate number of significant figures and should NOT be used to round your answer to the correct number of sig figs. Any conversion factor that has both units in metric (examples: 1000 mL/1 L, 1 m/100 cm, 1000 mg/ 1 g) or both units in the English System (examples: 5280 ft/ 1 mi., 1 gallon/ 4 quarts, 3 feet/1 yd.) should not be used in determining sig figs in your answer.

If you have conversion factors that have units in both metric and English, they generally are used to determine sig figs. Examples: 3.8 L / 1.0 gallon, 1.0 km/ 0.6 mi, 39.37 inches/ 1.0 meter. This is because there is not exactly 3.8 liters in 1.0 gallon, not exactly 1.0 km in 106 mi...

As a general rule, it is usually (but not always) your given (what you start out with) that determines the number of sig figs in your answer. If you look at the example above where we converted 77.2 years to seconds, we rounded the answer to 3 sig figs because our given (77.2 years) has 3 sig figs.

For more info on DA, visit Keynotes and download Mr. G's Keynote presentation on Dimensional Analysis (in PDF form).

Mr. Gilliland, 2006