How Do I Calculate Rates?
Calculating Changes through Time in the Geosciences

Introduction to Rates

Change and time are two of the main themes in the geosciences. For example, geomorphologists study changes in landscape such as beach erosion, geochemists study chemical changes in rocks as they are weathered, climatologists study long-term changes in global and regional weather and climate. Because geologists are interested in how rapidly changes can occur, we also look at the time over which those changes take place. Combined, this creates a rate, the rapidity in which change occurs.

What Is a Rate?

Any change with respect to time is called a rate.

This is represented mathematically by

`text{R}=\frac{Deltatext{X}}{Deltatext{t}}`

Where R is the rate, ΔX is the change in whatever you are looking at (it could be temperature, pressure, distance, or anything else) and Δt is the change in time. In mathematics and many science fields, Δ means "change."

There are many rates in the geosciences: plate tectonic velocities (distance over time), crystal growth rates (change in crystal size over time), river and groundwater discharge (volume change over time), decay rates (change in number of parent isotopes over time), or any variable that is divided by time.

How Do I Calculate a Rate?

Here are some steps for calculating a rate. Under each step you can find an example in which we determine a velocity of a glacier's retreat.

In 1892, the terminus of Nisqually Glacier, on the southern side of Mt. Rainier in Washington, was located 40 meters above the current position of the Nisqually river bridge. In 1951, the terminus was located 790 m above the bridge. Calculate Nisqually Glacier's rate of retreat from the bridge (in m per year) between 1892 and 1951.
The steps to answer this question are:

1. Determine which changing variable is ΔX and which is Δt.
   Show me what variables are changing
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   When calculating a rate, one variable is always time (Δt), and it is most often easiest to pick this out. In this problem, it is the time that has elapsed, which is Δt. The other property that is changing is the position (the location of the terminus), so this is ΔX.
2. Calculate ΔX and Δt to determine the change in the variable(s).
   Show me how to calculate changes in the variables
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   In the glacier problem, there are two variables that change: position (ΔX) and time (Δt). 
   Let's begin by calculating the change in position (ΔX). We need to know the difference between the 1892 and the 1951 positions, so we should subtract:
   790 m – 40 m = 750 m = ΔX
   Next we need to calculate the amount of time that elapsed. In other words, how many years are there between 1892 and 1951? 
   1951 – 1892 = 59 years = Δt
3. Calculate the rate using ΔX and Δt.
   Show me how to calculate the rate using ΔX and Δt
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   Remember,
    
   `text{R}=\frac{Deltatext{X}}{Deltatext{t}}`
   
   so to calculate the rate, you need to divide ΔX by Δt.

   Show me how to divide ΔX by Δt

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   In step 2, you found that ΔX = 750m and Δt=59 years, so
    
   `text{R}=\frac{750\ m}{59\ yr}=12.7\ \frac{m}{yr}`
   
   This is your answer, that the rate of retreat is 12.7 meters per year.
4. Check to see what the units on your final number should be.
   Show me how to check units
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   In this case, we are asked to calculate a rate in meters per year. Do we have units of meters and years? Yes!

   Show me what to do if my units are not what the question is asking for

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   If your units are not the same as what is needed at the end of the problem (for example, if we had km and years), you can visit the unit conversions module to learn how to convert units. Just don't forget to come back and finish this page (just hit the "back" button on your browser)!!
5. Evaluate your answer. Does that number seem reasonable?
   Show me how to evaluate the number
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   This is one of the most important steps; if the number you had gotten was 12,700 meters per year then the glacier would have retreated more than the height of the mountain in a single year (not likely). If you had gotten .0078 meters per year, that is less than a centimeter (a quarter of an inch or so) and that would not be measurable. As it turns out, in North America glaciers typically have been retreating at rates of 3–25 m per year. Since our rate falls somewhere in the middle, it's probably reasonable.
     

Determining Rates from a Graph

Geoscientists often present data in a graph. Any graph that has time as the horizontal axis can be used to determine a rate. In these cases, the rate is the slope of the line on the graph (many of you will know this as "rise over run") or the change in the vertical axis variable divided by the change in time (on the horizontal axis).

For practice, let's take a look at the plot at the right (you can click on it to make it bigger). In 1950, the population was approximately 2.5 billion people. In 2000, world population had grown to a little more than 6 billion people. So, let's calculate the rate of population growth from 1950 to 2000.
Slope is the rise over the run; this is the same as our equation for a rate since the change in time on the horizontal axis and is thus the "run."

One slightly confusing thing is that our change in population, ΔX, is on the vertical axis (sometimes called the y-axis,) while Δt is on the horizontal axis (often called the x-axis.) To avoid confusion, use the rate equation above or "slope is rise over run."

 

`text{Rate}=\frac{text{rise}}{text{run}}=\frac{Delta text{X}}{Delta text{t}}=\frac{6-2.5\ text{billion people}}{2000-1950\ text{years}}=\frac{3.5\ text{billion people}}{50\ text{years}}=0.07\ frac{text{billion people}}{text{year}}`

So the rate of population growth between 1950 and 2000 is 0.07 billion people per year (that is 70 million people added to the planet every year on average!).

Where Are Rates Used in Geology?

Rates are used in a range of topics in introductory geoscience courses: 

• plate tectonics
• rocks (igneous, metamorphic, and sedimentary)
• Earth history
• streams and groundwater
• glaciers
• mountain building and erosion
• climate change
• population growth
• and almost any other chapter in your textbook.

You will probably revisit the concept of rates many times. You can always come back to review how to calculate a rate!

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, continue below to get more help.

I Know the Rate, but This Question Is Asking for Something Different!

Sometimes we have information about the rate but are asked to solve for another variable (time, distance, volume, etc.). The basic equation is still the same, but you will need to rearrange it to solve for a different variable. The Math You Need, When You Need It has an entire module dedicated to rearranging equations (if you click that link, don't forget to come back and practice your newly learned steps!). The Rearranging Equations sample problems have several problems (right at the top of the page) that help you solve for distance and/or time if you know the velocity!

Resources for Rates and Calculating Velocity

Reaction rates in chemistry helps students calculate reaction rates for chemical reactions. 

Calculating rates of change and slope for a graph describes and explains the process of finding the slope of a function, and thus the rate. This is from M. Casco Learning Center and has some nice mathematical applets.

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