Unit Analysis and Conversion Factors

Date: 02/01/2003 at 16:07:47
From: Melissa
Subject: Conversions

How to convert the following?

1. 7.5 X 10^4 nm to kilometers
2. 3.9 X 10^5 mg to decigrams
3. 2.21 X 10^-4 dL to microliters

Date: 02/02/2003 at 23:11:17
From: Doctor Wigton
Subject: Re: Conversions

Hi Melissa,

Converting metric units is a very useful thing, and it's something 
that will eventually become very easy for you with practice. But I 
will teach you a way to convert pretty much anything to anything.  
Luckily, the metric system is easy to work with, so this will not be 
too difficult. Sigh, if only we (the U.S.) would switch to metric... 
but I digress.

To start, to write out unit analysis, we put our data into a form 
like this:

        |        |
  ------|--------------
        |        |

Now we multiply the numbers within the boxes across the top. Then we 
divide that by the bottom numbers multiplied out. For example:

    a   |    2   |  x
  ------|-------------- = (2*a*x) / (16)
        |    4   |  4

Now you are probably wondering what all these variables and numbers
are. I am getting to that. Just understand the principle of how to
use the boxes once they are filled out.

What happens with these boxes is, we put in what we have, and what 
comes out at the end, (2*a*x) / 16 in our last case, is the final 
answer.

So let's start with a simple example. Convert 3 feet to inches. We 
already know the answer, but it's always best to start simple.

So we put what we know into the first box.

  3 feet|        |
  ------|--------------
        |        |

Don't forget to write the unit as well, as it will be needed to check
your work. Now the next box will contain a number on top, and one on
the bottom. This is our conversion factor. It is in this case:

     1 foot = 12 inches

Put it in the box so that feet cancel out.  In other words, feet will 
be on the bottom.

  3 feet| 12 inch|
  ------|--------------
        | 1 foot |

Now we see that the unit 'feet' cancels out. What are we left with?  
Inches. This was what we were trying to convert to, so we know that 
is as far as we need to go. Now multiply out for the final answer.

  3 feet| 12 inch|
  ------|--------------  = 36 inches
        | 1 foot |

That's all well and good, but what if we have more than just this 
simple example? Don't worry, our box can be extended as long as 
necessary. Let's try a more complicated example.

Convert 1 year to seconds.  Start with our first box.

  1 year|        |
  ------|--------------
        |        |

Now add what we know.

  1 year| 365 days|
  ------|--------------
        | 1 year  |

As you can see, this is the conversion to days, so we are not done.  
We keep going, adding to our boxes...

  1 year| 365 days| 24 hours
  ------|--------------------
        | 1 year  | 1 day

This leads us to hours, so we keep going.  Skipping some steps, we 
get:

  1 year| 365 days| 24 hours  | 60 min | 60 sec
  ------|--------------------------------------
        | 1 year  | 1 day     | 1 hour | 1 min

So:  years, days, hours, and minutes cancel, leaving us with the 
equivalent in seconds.  This is what we want.  Solving:

  1 year| 365 days| 24 hours  | 60 min | 60 sec
  ------|--------------------------------------  = 31536000 seconds
        | 1 year  | 1 day     | 1 hour | 1 min

As long as each unit conversion is valid, we can convert pretty much 
anything to anything. For example:

How many games of Boggle can I play in a year? (Assuming no stopping!)

  1 year| 365 days| 24 hours  | 60 min | 1 Boggle Game
  ------|---------------------------------------------- = 175200 Games
        | 1 year  | 1 day     | 1 hour | 3 min

Thus you can also see, the units on the bottom do not need to be 
equal to 1, and frequently aren't once the conversions become more 
complicated.  

So let's do one of your examples. I won't do them all, but I think 
you'll be able to do the rest on your own after one.

Convert 7.5x10^4 nm to km. I like to write 7.5x10^4 as 7.5E4 instead: 
it is shorter and easier to type, and also a pretty common practice.  
Just keep in mind that they are one and the same.

Start with what we have:

  7.5E4 nm|        |
  --------|--------------
          |        |

Now, how many nanometers are in a meter?  I'm sure in your text you 
will find that 1E9 nm = 1 m.

  7.5E4 nm|  1 m   |
  --------|--------------
          | 1E9 nm |

Now, we want the answer in km, not m.  So we need another conversion 
factor.  The one we need is:  1 km = 1000 m.

  7.5E4 nm|  1 m   | 1 km
  --------|----------------
          | 1E9 nm | 1000 m

Are we done?  Yes, because the nm's cancel, and m's cancel.  This
leaves us with an answer in km.  Now just multiply through to get:

  7.5E4 nm|  1 m   | 1 km
  --------|----------------  = 7.5E-8 km (aka 7.5x10^-8 km)
          | 1E9 nm | 1000 m

After you're done, try to check your work with common sense. We know 
a nanometer is very small (about ten times the size of the hydrogen 
atom). So then we know that the equivalent in km will be very small.  
This is true, so our answer is probably correct.

Overall, the process is pretty simple, don't you think? One more note: 
this works for more than 1-dimensional units as well. You can convert 
miles/hour to meters/second and so on. In the end, just make sure your 
units are what were desired.  

I hope this helps.  If you have any other questions about it, please 
write back.

Sincerely,

- Doctor Wigton, The Math Forum
  http://mathforum.org/dr.math/