Initial Publication Date: August 16, 2024 | Revision: October 22, 2026
- First Publication: August 16, 2024
- Revision: October 22, 2026 -- Fixed mistakes in practice problems
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Cite thisOrders of Magnitude - Practice Problems
Solving Earth science problems with Orders of Magnitude
Estimating Orders of Magnitude
These problems will help you practice estimating orders of magnitude.
Problem 1: An estimate for the average global strain rate in mountains is 1 x 10-14 sec-1. What is the estimated order of magnitude of this value?
Step 1: Start with an estimate that is a power of 10 or is written in scientific notation.
The question gives us a value that is written appropriately for this step: 1 x 10-14 sec-1.
Step 2: If your value is not in scientific notation, write your estimate in an exponent form that shows the power of 10.
This step is skipped because we have a value written in scientific notation.
Step 3: Identify the exponent used for your power of ten or for your scientific notation. That value is your order of magnitude.
The exponent for 1 x 10-14 sec-1 is -14. This means that the estimated order of magnitude is -14.
Mudflow beneath Craig Pwllfa.
![[creative commons]](/images/creativecommons_16.png)
Provenance: Alan Bowring via Wikipedia Commons, https://commons.wikimedia.org/wiki/File:Mudflow_beneath_Craig_Pwllfa_-_geograph.org.uk_-_1405853.jpg
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Problem 2: A fast mudflow moves ~200,000 km per year. What is the
estimated order of magnitude of this value?
Step 1: Start with an estimate that is a power of 10 or is written in scientific notation.
~200,000 km per year is an estimate that is a power of 10.
Step 2: Write your estimate in an exponent form that shows the power of 10.
200,000 km per year = 2 x 105 km per year. The coefficient "2" rounds to 1, so for estimation purposes you can consider this as 1 x 105 or 105.
Step 3: Identify the exponent used for your power of ten or for your scientific notation. That value is your order of magnitude.
The exponent for 105 is 5. This means that the estimated order of magnitude is 5.
Problem 3: Mount Everest is the tallest mountain in the world and is ~8,900 m above sea level. What is the estimated order of magnitude of this value?
Step 1: Start with an estimate that is a power of 10 or is written in scientific notation.
~8,900 m is not a power of 10, so we need to write it in scientific notation: ~8,900 m = ~8.9 x 103 m.
Step 2: If your value is not in scientific notation, write your estimate in an exponent form that shows the power of 10.
This step is skipped because our elevation value is in scientific notation.
Step 3: Identify the exponent used for your power of ten or for your scientific notation and round the coefficient to the nearest order of magnitude.
The exponent for ~8.9 x 103 m is "3", but you need to round the coefficient to the nearest order of magnitude. The coefficient of "8.9" rounds to 10. This means your estimation is 10 x 103 m, which is simplified to 1 x 104 m. The value "4" is your estimate order of magnitude.
Comparing Estimated Orders of Magnitude
The questions below provide some practice comparing estimated orders of magnitude and applying that knowledge to determine how much larger or smaller a value is.
Faulting within a mountain range.
Provenance: Raymond M. Coveney, …, CC BY-SA 3.0 via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Glarus_thrust_fault_in_Switzerland%27s_Sardona_Tectonic_Arena,_a_UNESCO_world_heritage_site_-_panoramio.jpg
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Problem 4: In structural geology, a researcher has determined that the average strain rate in a certain mountain range is on the order of 1 x 10
-15 seconds
-1. How much larger or smaller is the
estimated order of magnitude for this rate compared to the estimated order of magnitude associated with the global average strain rate of 1 x 10
-14 seconds
-1?
Step 1: Make sure that the two values have the same units.
Yes, both strain rates are in units of seconds-1.
Step 2: Determine the estimated order of magnitude for both measurements.
Researcher's measurement = 1 x 10-15 seconds-1. The exponent on this value is -15; the estimated order of magnitude on this measurement is -15.
Global estimate = 1 x 10-14 seconds-1. The exponent on this value is -14; the estimated order of magnitude on this measurement is -14.
Step 3: Subtract the two orders of magnitude (large - small).
Global value - Researcher's value = -14 - (-15) = -14 + 15 = 1. This means that the two strain rates differ by 1 order of magnitude.
Step 4: To determine an estimated "magnification," use the answer from step three as a power of 10.
10(-14 - (-15)) = 101 = 10. This means that the global average is 10 times larger than the researcher's measured strain rate.
Problem 5: Two students have calculated values for viscosity based on field measurements they gathered from the same locality. They come up with two different values; student #1 calculates 6.66217 x 1012 poise; student #2 calculates 2.8145 x 1012 poise. What is the difference in the estimated orders of magnitude for their calculations?
Step 1: Make sure that the two values have the same units.
Yes, both viscosity values are in units of poise.
Step 2: Determine the estimated order of magnitude for both measurements.
Student #1 = 6.66217 x 1012 poise. The exponent on this value is 12, but the coefficient "6.66217" rounds to 10. 10 x 1012 simplifies to 1 x 1013, so the estimated order of magnitude on this measurement is 13.
Student #2 = 2.8145 x 1012 poise. The exponent on this value is 12 and the coefficient "2.8145" rounds to 1, or 1 x 1012. The estimated order of magnitude on this measurement is 12.
Step 3: Subtract the two orders of magnitude (large - small).
Student #1 - Student #2 = 13 - 12 = 1. This means that the two calculations are one order of magnitude different.
Step 4: To determine an estimated "magnification," use the answer from step three as a power of 10.
10(13 - 12) = 101 = 10. The calculation by student #1 is 10 times larger than the calculation by student #2. This practice problem shows how orders of magnitude may require a bit more attention than just a quick look at the exponents.
Provenance: Rory McFadden, Carleton College
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Problem 6: According to the flow velocity vs grain size graph to the right, a 1.0 mm sand grain will be deposited when it is in a river with a flow velocity of 1.0 cm/sec. Notice that the intersection of the two coordinates is within the deposition field. What is the minimum change in the
estimated order of magnitude needed for the flow velocity to transport the 1.0 mm sand grain? How many times faster is this velocity?
Step 1: Make sure the two values are in the same units.
Both values are shown on the graph and both the x- and y-axis are logarithmic, meaning each tick marks represents a one order of magnitude increase. Instead of directly comparing the values, you can use the graph to determine the change in orders of magnitude.
Step 2: Determine the estimated orders of magnitude for both measurements. For this problem, use the graph to determine the flow velocity needed to transport the 1.0 mm sand grains.
The measurements are on tick marks of the graph axes and both represent 0 orders of magnitude.
Shown in the image to the right, make a straight line from 1.0 mm sand grain tick mark until you cross into the transport field to determine the required flow velocity. The intersection of the transport field and 1.0 mm sand grain is around 11 cm/sec. The 11 cm/sec value is the flow velocity needed to transport the sand grains and it represents 1 order of magnitude.
Step 3: Subtract the two orders of magnitude.
The transport flow velocity minus the deposition flow velocity = 1 - 0 = 1. This means the change in flow velocity needed is one order of magnitude.
Step 4: To determine how many times faster this flow velocity, use the answer from step three as a power of 10.
101 = 10. Showing that the change in the minimum flow velocity needed to transport a sand grain is 10 times faster than the flow velocity of 1.0 cm/sec at which it was deposited.
Provenance: Rory McFadden, Carleton College
Reuse: This item is offered under a Creative Commons Attribution-NonCommercial-ShareAlike license http://creativecommons.org/licenses/by-nc-sa/3.0/ You may reuse this item for non-commercial purposes as long as you provide attribution and offer any derivative works under a similar license.
Problem 7: The figure to the right shows a dataset of earthquakes that occurred in a one year period. The y-axis shows the number of earthquakes with magnitude greater than the magnitude on the x-axis. Notice the y-axis is logarithmic with major tick marks increasing by one order of magnitude. The y-axis shows linear change in earthquake magnitude. The physical change in energy release increases exponentially with each step up in magnitude, but you don't need to consider that for this type of problem. Roughly 10 earthquakes with M > 7 occurred during this year, how many estimated orders of magnitude more M > 5 events occurred during that year?
Step 1: Make sure the two values are in the same units.
Both values needed to answer this question are in number of events and can be determined directly from the graph.
Step 2: Determine the estimated orders of magnitudes. For this problem, use the graph to determine the number of M>5 events.
Starting at the "7" on the x-axis, you should see that M>7 events has a value at 10 on the y-axis. The graph displays that there were 10 M>7 events. Starting at "5" on the x-axis, you should see that M>5 events has a value at 1000 on the y-axis. This means there were 1000 M>5 events. The estimated orders of magnitude for 10 and 1000 are 1 and 3, respectively.
Step 3: Subtract the two orders of magnitude.
The M>5 events minus the M>7 events = 3 - 1 = 2. This means there were 2 orders of magnitude more M>5 events than M>7 events.
Step 4: To determine how many times more events, use the answer from step 3 as a power of 10.
102 = 100. Showing there were 100 times more M>5 events than M>7 events during the year.
Calculating Orders of Magnitude
These problems help you practice determining the order of magnitude of values.
Problem 8: What is the calculated order of magnitude for a span of time that is 598 years.
Step 1: Take the base-10 logarithm (log
10) of the number.
Step 2: Round the resulting number to the nearest whole number.
2.8 rounds to 3. This means that the order of magnitude is 3.
Notice that you would get the same value for the estimated order of magnitude. The 598 rounds up to 1,000, which can be written as to 1 x 103. Using the exponent, you can see the estimated order of magnitude is also 3.
Milky Way galaxy
Provenance: Pablo Carlos Budassi, CC BY-SA 4.0 via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Milky_way.png
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Problem 9: What is
calculating the order of magnitude for the diameter of the Milky Way, which is roughly 9.5x10
17 km?
Step 1: Take the base-10 logarithm (log
10) of the number.
Step 2: Round the resulting number to the nearest whole number.
17.98 rounds to 18, so the calculated order of magnitude is 18.
Notice that you would get the same value for the estimated order of magnitude. The coefficient "9.5" rounds up to 10. 9.5 x 1017 rounds to 10 x 1017, which simplifies to 1 x 1018. This shows 18 is also the estimated order of magnitude.
Problem 10: Some estimates suggest that it takes ~1.5 million years for organic material to reach the conditions needed to convert it to a hydrocarbon. What is the calculated order of magnitude for this amount of time?
Step 1: Take the base-10 logarithm (log10) of the number.
log10(1,500,000 years) = 6.2.
Step 2: Round the resulting number to the nearest whole number.
6.2 rounds to 6. This means that the calculated order of magnitude is 6.
Comparing Calculated Orders of Magnitude
The questions below provide some practice comparing orders of magnitude and applying that knowledge to determine how much larger or smaller a value is.
White crystals in black groundmass
![[reuse info]](/images/information_16.png)
Provenance: This image has been adjusted by Farthing from American Society for Testing and Materials, ASTM D-2488, “Standard Practice for Description and Identification of soils (Visual-Manual Procedure),” ASTM Annual Book of Standards, Volume 04.08 on Soil and Rock, Section 4 - Construction, West Con- shohocken, PA, 1996.
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Problem 11: Earth scientists regularly make visual estimates for modal abundances: E.g., How much of a rock is made of quartz? How many of the sand particles are shells? How much of the soil is root material? In this image, there are white crystals surrounded by black groundmass. If ~99 mm
2 of the circle is black and ~1 mm
2 is white, what is the difference in the
calculated order of their magnitudes?
Step 1: Make sure that the two values have the same units
Yes, both the amount of black groundmass and white crystals are provided in mm2.
Step 2: Take the base-10 logarithm (log10) of the numbers.
White crystals: log10(1 mm2) = 0; Black groundmass: log10(99 mm2) = 1.996.
Step 3: Subtract the 2 orders of magnitude (large- small).
Black groundmass - White crystals = 1.996 - 0 = 1.996.
Step 3b: Round the value to the nearest whole number.
The value 1.996 rounds to 2, which means the area of the black groundmass is 2 orders of magnitude larger than the area of the white crystals
Step 4: To determine "magnification," use the answer from step 3b as a power of 10.
102 = 100. This means that the area of black groundmass is 100 times larger than the area of white crystals.
Problem 12: A grain of medium sand has a diameter of 0.25 mm. The diameter of the Earth is ~6,370 km. The Earth's diameter is how many calculated orders of magnitude larger than a grain of medium sand?
Step 1: Convert units so they are the same.
Convert both to meters: 0.25 mm = 0.00025 m and 6,370 km = 6.37x106 m.
Step 2: Compute the order of magnitude for the diameter of a sand grain and Earth's diameter.
Step 2a: Compute the log10 of the diameters: For sand, log(0.00025) = -3.6 and for Earth, log(6.37x106) = 6.8.
Step 3: Find the difference between the orders of magnitude
6.8-(-3.6) = 10.4.
Step 3b: Round this value to the nearest whole number. The number 10.4 rounds to 10. So Earth's diameter is 10 orders of magnitude larger than a grain of medium sand.
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