Solving for Any Variable...
Rearranging Equations: Practice Problems

Practice rearranging equations below using the "rules" that you have just learned. Answers are provided (but try doing them on your own before peeking!) You can download a sheet with all the questions (Acrobat (PDF) 127kB Jul25 09) so you can print them out to try on your own.

Velocity, Distance, and Time

Many problems in the geosciences deal with the concept of velocity, from stream flow to plate motion. Being able to manipulate the equation for velocity (v = d/t) gives you a powerful tool for understanding the world around you. And there are two other equations that you can get by rearranging the equation that you probably already know for velocity (also rate)!

Problem 1: Generally, we know the equation for velocity (a rate) to be:

`text{v}=\frac{text{d}}{text{t}}`

Where v = velocity, d = distance and t = time.

This equation can be rearranged so that you have an equation for distance (d) and time (t).

1. Rearrange the velocity equation to create an equation for distance (d).
   Show me how
   Hide
   Remember to first ask yourself: Which variable do I want to isolate? In this case, we want to isolate d. So we begin by working on moving all other variables to the other side:
   
   

   1. Multiply both sides by t:
      
      `\frac{text{d}}{text{t}} times text{t}=text{v}\times text{t}`
   2. Cancel t's where appropriate:
      
      `text{v}\times text{t}=text{d}`
      
      
      Show me an alternate way to do steps 1 and 2
      Hide
      The two steps above are exactly what you do when you cross-multiply. If you are more familiar (or comfortable) with doing it this way, please feel free to do so.
      
   3. And you end with an equation for d: If you are more comfortable with the unknown being on the left, you can also rearrange it to read:
      
      `text{d}=text{v}\times text{t}`
2. Rearrange the velocity equation to create an equation for time (t).
   Show me how
   Hide
   Remember to first ask yourself: Which variable do I want to isolate? In this case, you want to isolate t. But t is on the bottom of the right side (in the denominator of the fraction). We want to get it out from under that d. The first three steps of the process are exactly the same as the question above, if you start with the original equation.

   1. Multiply both sides by t:
      
      `\frac{text{d}}{text{t}} times text{t}=text{v}\times text{t}`
       
   2. Cancel t's where appropriate: 
      
      `text{v}\times text{t}=text{d}`
      
      The two steps above are exactly what you do when you cross-multiply. If you are more familiar (or comfortable) with doing it this way, please feel free to do so.
   3. And you end up with an equation for d:
      
      `text{v}\times text{t}=text{d}`
       
   4. Now we need to isolate t by dividing both sides of the equation by v :
      
      `\frac{text{v}\times text{t}}{text{v}}=\frac{text{d}}{text{v}}`
       
   5. Cancel v's where appropriate:
      
      `text{t}=\frac{text{d}}{text{v}}`
   6. And we end up with an equation for t:
      
      `text{t}=\frac{text{d}}{text{v}}`

 

Use the equations that you manipulated above to solve the following problems:

Problem 2: A wave traveling downward from the surface of the ocean at 1.5 km/sec takes six seconds to reflect off the ocean floor. How deep is the ocean at that site?
Problem 3: Imagine that you are working with Ms. Homeowner to understand the groundwater flow in her area. She is particularly interested in an underground tank that is located 2.6 km from her home. You have measured the velocity of the groundwater to be 0.033 km/day. About how long will it take any contaminants leaking from the tank to reach Ms. Homeowner's well?  

Density

Density plays an important role in our understanding of the physical properties of Earth materials. The equation for density is similar to that for velocity and, as such, it can be manipulated so that you can solve for any of the variables involved. The next few problems utilize the equation for density:

`\rho text{(density)}=\frac{text{mass}}{text{volme}}`

which can be shortened to:

`\rho=\frac{text{m}}{text{v}}`

Problem 1: Manipulate (rearrange) the density equation (above) to create an equation for mass.

 
Problem 2: Manipulate the density equation to create an equation for volume.  
Use the equations that you manipulated above to solve the following problems: 

Problem 3: Continental crust has a density of about 2.75 g/cm³. What volume would 1000 g (1 kg) of continental crust occupy?  
Problem 4: The outer core has a density of about 10.5 g/cm³. What volume would 1000 g (1 kg) of the outer core occupy?  

Isostasy

Isostasy is an important concept in the geosciences that is related to density. The concept of isostasy explains why continental crust sits so much higher than oceanic crust. It involves properties of an object like height and density. It also explains why only a portion of an iceberg is visible above water. The following problems involve an equation that is important to understanding the way that continental crust/oceanic crust behave on the Earth. If you need help visualizing the problem, click on the illustration below to see a larger version.

The equation for calculating the height of an object above the "fluid" in which it is floating is:

`H_text{total}\times \left(1-\frac{\rho_text{object}}{\rho_text{fluid}}\right)=H_text{above}`

where:

H_total = total height of the object 
ρ_object = density of the object of interest (e.g., iceberg, continental crust, etc.) 
ρ_fluid = density of the fluid in which the object is floating (e.g., seawater, mantle, etc.) 
H_above = height of the object visible above the fluid.

Problem 1: Rearrange the equation above to solve for H_total.

Problem 2: Use the equation that you generated in Problem 1 to calculate the thickness (H_total) of continental crust if:

• the density of continental crust (ρ_object) is 2.79 g/cm³,
• the density of the mantle (ρ_fluid) is 3.3 g/cm³,
• and, the average height of the continents above the mantle is 4.5 km (H_above).

 

Problem 3: Use the information from Problem 2 to calculate the thickness of crust that lies below the mantle (e.g., H_below).

Next Steps

TAKE THE QUIZ!!  

I think I'm competent with manipulating equations and I am ready to take the quiz! This link takes you to WAMAP. If your instructor has not given you instructions about WAMAP, you may not have to take the quiz.

Still Need More Practice?

• PurpleMath has some pages to help you solve for a given variable.
• Massey University in New Zealand has some pages on rearranging equations with many worked examples.
• MathMax.com's introductory algebra book has a rearranging equations worksheet (Acrobat (PDF) 4kB Oct5 07) and answers (but not worked solutions) (Acrobat (PDF) 9kB Oct5 07) that you can download to work through rearranging equations.
• S. Peters Collegiate School (in the UK) Maths Department) has two worksheets (and their solutions) that you can download below:
  • Rearranging Equations 1 (Microsoft Word 21kB Oct5 07) | Answers to Rearranging Equations (Microsoft Word 20kB Oct5 07)
  • Rearranging Equations 2 (Microsoft Word 22kB Oct5 07) | Answers to Rearranging Equations 2 (Microsoft Word 28kB Oct5 07)