Orders of Magnitude - Practice Problems
Solving Earth science problems with Orders of Magnitude
This module is undergoing classroom implementation with the Math Your Earth Science Majors Need project. The module is available for public use, but it will likely be revised after classroom testing.
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 order of magnitude is -14
Mudflow beneath Craig Pwllfa.
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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 ~100,000 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.
~100,000 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.
100,000 km per year = 105 km per year
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 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. That value is your order of magnitude.
The exponent for ~8.9 x 103 m is 3. This means that the order of magnitude is 3
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 stated that the average strain rate in a certain mountain range is on the order of 1 x 10
-15 seconds
-1. How much bigger 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 2 orders of magnitude (Big - 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 researcher's strain rate is an estimated 10 times larger than the global average -OR_ the global average is an estimated 10 times smaller compared to the researcher's findings.
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: 4.66217 x 1012 poise; student #2: 3.8145 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 = 4.66217 x 1012 poise. The exponent on this value is 12; the estimated order of magnitude on this measurement is 12.
Student #2 = 3.8145 x 1012 poise. The exponent on this value is 12; the estimated order of magnitude on this measurement is 12.
Step 3: Subtract the 2 orders of magnitude (Big - Small).
Student #2 - Student #1 = 12 - (12) = 0. This means that the two calculations have the same order of magnitude.
Step 4: To determine an estimated "magnification" use the answer from step three as a power of 10.
10(12 - (12)) = 100 = 1. Because the two student's calculations have the same estimated order of magnitude, there is no estimated magnfication between them. This suggests that neither student's calculation is wildly different compared to the other's.
Calculating Orders of Magnitude
These problems help you practice determining the order of magnitude of values.
Problem 6: Practice calculating the order of magnitude for a span of time that is 598 years
Step 1: Take the base-10 logarithm (log10) 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
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 7: Practice calculating the order of magnitude for an estimate of the diameter of the Milky Way, roughly 9.5x10
17 km
Step 1: Take the base-10 logarithm (log10) of the number.
Step 2: Round the resulting number to the nearest whole number.
17.98 rounds to 18, so the order of magnitude is 18.
Problem 8: 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 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 years
Step 2: Round the resulting number to the nearest whole number.
6.2 rounds to 6. This means that the order of magnitude is 6
Branchiopod fossils in limestone.
Provenance: James St. John, CC BY 2.0, via Wikimedia Commons. https://commons.wikimedia.org/wiki/File:Brachiopods_in_limestone_(Waynesville_Formation,_Upper_Ordovician;_Roaring_Run,_Warren_County,_Ohio,_USA)_3.jpg
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Problem 9: The width of a brachiopod is measured to be 2.2 inches. Calculate the order of magnitude for this measurement.
Step 1: Take the base-10 logarithm (log10) of the number.
Step 2: Round the resulting number to the nearest whole number.
0.3 rounds to 0, so the order of magnitude is 0.
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
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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 10: 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 a 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 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 2a: Take the base-10 logarithm (log10) of the numbers.
White crystals: log10(1 mm2) = 0; Black groundmass: log10(99 mm2) = 1.996
Step 2b: Round the resulting numbers to the nearest whole numbers.
White crystals: = 0 (no rounding needed); Black groundmass: 1.996 rounds to 2
Step 3: Subtract the 2 orders of magnitude (Big - Small).
Black groundmass - White crystals = 2 - 0 = 2: This means that the amount of white crystals versus the surrounding groundmass is different by 2 orders of magnitude.
Step 4: To determine "magnification" use the answer from step three as a power of 10.
10(2-0) = 102 = 100. This means that there are 100 times fewer white crystals than black groundmass -OR_ there is 100 times more black groundmass compared to the white crystals.
Problem 11: A grain of medium sand has a diameter of 0.25 mm. The diameter of the Earth is ~6370 km. The Earth's diameter is how many orders of magnitude larger than a grain of medium sand?
Step 1: Convert units so they are the same units.
Convert both to meters: 0.25 mm = 0.00025 m and 6370 km = 6.37x106 m
Step 2: Compute order of magnitude for diameter of sand grain and Earth 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 2b: Round these to the nearest whole number: sand rounded to -4, Earth rounded to 7.
Step 3: Find difference between the orders of magnitude
7-(-4) = 11. So Earth diameter is 11 orders of magnitude larger than a grain of medium sand.
Problem 12: This graph shows data for the discharge of a river (in units of cubic feet per second) compared to the return period (in units of years). Return periods represent estimates of the time between floods of certain sizes. For instance, the largest floods will have the highest discharge and they happen less frequently (i.e., they have a longer return period). 
Q: The discharge in a flood with a return period of ~1 year is ~2000 cfs. How much more discharge occurs in floods with an order of magnitude longer return period?
Step 1: Make sure that the two values have the same units.
The numbers we get from the x-axis of the graph are in the same units of years so no unit conversion is necessary.
Step 2a: Take the base-10 logarithm (log10) of the numbers.
1 year: log10(1 year) = 0
To get a calculated order magnitude equal to 1, we need a log10(return period) = 0.5 because 0.5 will round to 1. This happens at ~3 years: log10(3 years) = 0.5
Step 2b: Round the results for each number to get the order of magnitude for each value in the comparison.
Return period of 1 year: log10(1) = 0 no rounding needed
Return period of 3 years: log10(3) = 0.5. This rounds to 1.
Step 3. Subtract the order of magnitude of the smaller number from the larger order of magnitude.
Question specific step: Read data from the graph and calculate the difference in discharge.
We need to use the graph to determine the discharge associated with a 3 year return period. This value is 5000 cfs. The question has asked us to determine the difference in discharge between these two return periods.
Discharge (3 year return period) - Discharge (1 year return period) = 5000 cfs - 2000 cfs = 3000 cfs
The answer is 3000 cfs
Next steps
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