How Do I Change Units on a Number?
Unit Conversions in the Geosciences

Introduction to Unit Conversions

In the geosciences, we think about how Earth works on a variety of scales. For example, the San Andreas Fault that runs nearly the length of California's coastline is more than 1200 km long. But we talk about movement on the fault in terms of mm per year. So, we have to be able to think about thousands of km and a few mm when talking about the same feature. In addition, a geoscientist who lives in the United States needs to be able to think in terms of English and metric units, because the public thinks in terms of English units (miles, gallons, etc.) and the scientific community uses SI units (kilometers, liters, etc.). This may seem difficult at first, but it is crucial for anyone studying geosciences to be able to move easily from one unit to another (with a little calculation, of course). Luckily, there are some simple steps that, if followed, can help you complete unit conversions with relative ease.

How Do I Do a Unit Conversion?

You can do any unit conversion if you follow a few simple steps. Although there is no single "right" way to do unit conversions, these steps provide one way to learn to do unit conversions. DO NOT SKIP ANY STEPS! Although it may seem tedious, working through unit conversions requires that each of these steps be followed so that you can be sure that you end up with what you want, especially when you are just starting out with learning to do unit conversions.

Below, you can download and print some tables for your use when doing unit conversions:

When you do any unit conversion, you should always know what units you started with and what units you want to end up with. This is key to success at unit conversions.

The Steps

The steps to successfully completing a unit conversion are outlined below. To illustrate the steps, let's use a geologic example: In Southern California, slip on the San Andreas Fault is on the order of 25 km/Myr. How many cm does the San Andreas Fault move in one year?

1. Write (copy) out the units that you are given as a fraction.
   
   Show me how to write example units as a fraction
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   In this case, you are given km/Myr.
   
   `\frac{25\text{km}}{1\text{Myr}}`
   
   OR you could write that you have kilometers (km is a length or distance) and Myr (which is Megayears or million years, a period of time).
2. Write out the units that you want at the end of the conversion as a fraction:
   
   Show how to write cm/year as a fraction
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   In the example, you are asked to convert km/Myr to cm/year, so you want to end up with cm/year.
   
   `\frac{text{cm}}{text{yr}}`
   
   OR you could write that you want your units to have centimeters (cm; a length or distance) and years (yr; a time).
3. Determine appropriate conversion factors. Use tables in your textbook or download one of the tables listed above (for this particular example, you probably only need to know the prefixes for metric system (Acrobat (PDF) 7kB Aug31 11)).
   
   Show conversion factors for the example problem
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   For our example of movement on the San Andreas Fault, we need to convert 1 km to cm and Myr (Mega-years or Million years) to years, so with the handy guide to metric system prefixes, we need three conversion factors:

   • 1 km = 1,000 m
   • 1 m = 100 cm
   • 1 Myr = 1,000,000 yr
     
   
4. Evaluate the appropriate arrangement for conversion factors. That is, copy the conversion factor(s) from step 3 into fraction form so that the units end up canceling. Remember that when you multiply fractions (as you will in step 6 below), you can cancel units ONLY when they appear in the numerator of one fraction and the denominator of another.
   Show appropriate arrangement of conversion factors for the example problem
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   We're going to have to do several steps in this conversion and we have to start somewhere. Let's start with converting km/year to cm/year. Because km is on top, we're going to want to arrange the conversion fraction so that km is in the bottom position (denominator) like so:
   
   `\frac{1000text{m}}{1text{km}}` 
   
   or you could start setting up your conversion:
   

   Show me how

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   `\frac{25\text{km}}{1\text{Myr}} \times \frac{1000text{m}}{1text{km}}`
   
   Now we can cancel km 
   
   `\frac{25}{1\text{Myr}} \times \frac{1000text{m}}{1}`

   
   but even when we cancel km, we're still left with m in the numerator so we need another conversion that has m in the denominator like this:
   
   `\frac{100text{cm}}{1text{m}}`
   
   

   Show me how to continue setting up the conversion

   Hide

   We can add our next conversion factor to the string of fractions we're going to eventually multiply: 
   
   `\frac{25\text{km}}{1\text{Myr}} \times \frac{1000text{m}}{1text{km}} \times \frac{100text{cm}}{1text{m}}`
   and we can cancel appropriate units: 
   
   `\frac{25}{1\text{Myr}} \times \frac{1000}{1} \times \frac{100text{cm}}{1}`

    
   Now we have what we want in the numerator, but we still have Myr in the denominator. So, we need a conversion factor that has Myr in the numerator:
   
   `\frac{1text{Myr}}{1","000","000text{yr}}`
   
   

   Show me the next step in the conversion for the example problem

   Hide

   Add another fraction to the string: 
   
   `\frac{25\text{km}}{1\text{Myr}} \times \frac{1000text{m}}{1text{km}} \times \frac{100text{cm}}{1text{m}} \times \frac{1text{Myr}}{1","000","000text{yr}}`
   
   and then cancel units: 
   
   `\frac{25}{1} \times \frac{1000}{1} \times \frac{100text{cm}}{1} \times \frac{1}{1","000","000text{yr}}`
5. Set up the conversion by writing the fractions in a row with multiplication signs in between.
   Show me how to set up the conversion for the example problem
   Hide
   (You may have already done this if you set the conversion up as you wrote out the conversion factors.) 
   
   `\frac{25\text{km}}{1\text{Myr}} \times \frac{1000text{m}}{1text{km}} \times \frac{100text{cm}}{1text{m}} \times \frac{1text{Myr}}{1","000","000text{yr}}`
   
6. Evaluate. Do the original units cancel so that you end up with what the question is asking for? If not, repeat steps 2 and 3 until they do!
   Show units canceled for the example problem
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   `\frac{25}{1} \times \frac{1000}{1} \times \frac{100text{cm}}{1} \times \frac{1}{1,000,000text{yr}}` 
   (like units are canceled with the same color in the example above) Note that the only units that are not canceled are cm (in the numerator) and yr (in the denominator) so the units on your answer will be cm/yr!
7. Now, let's do some arithmetic: To complete the conversion, we multiply all the numerators and then all the denominators (or multiply across the top and bottom).
   Show me how to multiply fractions for the example problem
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   `\frac{25}{1} \times \frac{1000}{1} \times \frac{100text{cm}}{1} \times \frac{1}{1","000","000text{yr}}=\frac{2","500","000text{cm}}{1","000","000text{yr}}` 
   
   If you are unsure about how I got the numbers here, you can review what is meant by "multiply across top and bottom" below:

   Show me more complete step-by-step instructions

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   When fractions are multiplied, you need to keep track of numbers on the top (also called the numerator) and bottom (also called the denominator).

   1. It's easier to keep track of these numbers if you write the fractions like this: `\frac{25\text{km}}{1\text{Myr}}`. In this case, the numerator is 25 km and the denominator is 1 Myr.
   2. To multiply fractions, treat top and bottom numbers separately (multiply top numbers and then multiply bottom numbers). 
      
      `\frac{25}{1} \times \frac{1000}{1} \times \frac{100text{cm}}{1} \times \frac{1}{1","000","000text{yr}}=\frac{25 \times 1000 \times 100 text{cm} \times 1}{1 \times 1 \times 1 \times 1","000","000text{yr}}`
   3. Write out your new numbers as a fraction (the top product as the numerator; the bottom product as the denominator) 
      
      `\frac{25}{1} \times \frac{1000}{1} \times \frac{100text{cm}}{1} \times \frac{1}{1","000","000text{yr}}=\frac{25 \times 1000 \times 100 text{cm} \times 1}{1 \times 1 \times 1 \times 1","000","000text{yr}}=\frac{2","500","000text{cm}}{1","000","000text{yr}}`
8. Reduce the fraction by dividing numerator by denominator once you have the top and bottom multiplied.
   Show me how to reduce fractions
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   `\frac{25}{1} \times \frac{1000}{1} \times \frac{100text{cm}}{1} \times \frac{1}{1","000","000text{yr}}=\frac{2","500","000text{cm}}{1","000","000text{yr}}=\frac{2.5text{cm}}{text{yr}}`
   
   This can also be written as 2.5 cm/yr (said "centimeters per year"). 
   
9. EVALUATE (again). Is this a reasonable number?
   Show reasonable plate motions
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   Considering that plate motions are in the range of 1–10 cm/yr, 2.5 cm/yr is a reasonable answer!

You can download and print a sheet with the unit conversion steps (Acrobat (PDF) 44kB Apr11 08)to use while completing the practice problems.

Where Are Unit Conversions Used in the Geosciences?

• Plate tectonics: converting rates of plate motion, etc.
• Topographic maps: converting scales
• Rivers and streams: converting rates of flow, slope, etc.
• Groundwater: converting rates of flow
• Glaciers: converting rates of flow (or retreat), etc.
• Geologic time: converting time, rates of deposition, etc.

And many other topics . . .

Next Steps

I am ready to PRACTICE!

If you think you have a handle on the steps above, click on this bar to try practice problems with worked answers.
Or, if you want even more practice, see 'More help' below.

More Help with Unit Conversions

Some tables of conversion factors

• NIST (National Institute of Standards and Technology) has an extensive list of weights and measures. There are specific tables for metric units of measure, U.S. units of measure and U.S. and metric equivalents.
• SOSMath has several tables of conversion factors for linear measure (length/distance), square (area) measure and http://www.sosmath.com/algebra/unitconv/unit4/unit4.html 'cubic (volume) measure'.
• The geology department at Rutgers has a list of important conversion factors in geology.
• The engineering department at Auburn University has a long list of conversion factors.

Help with unit conversions (dimensional analysis)

• Alabama Learning Exchange has a handout about basic mathematical skills needed for the workplace.
• Alysion.org has a page with step-by-step instructions for dimensional analysis and some fun with dimensional analysis.
• Oak Road Systems (Stan Brown) has a discussion of converting units and how to go about it.

Help with multiplying fractions

• SOS Math has an entire unit on fractions. These pages include a list of rules for the mathematics of fractions, reducing fractions, multiplying fractions.

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This page was written and compiled by Dr. Jennifer M. Wenner, Geology Department, University of Wisconsin, Oshkosh, and Dr. Eric M. Baer, Geology Program, Highline Community College