Unit Conversions

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One of the things that many students have problems with is unit conversions.  Unfortunately, unit conversions are really important to know how to do.  For example, let's say that somebody wants to do a calculation of the number of moles of water in your body.  This seems like it should be easy, but you probably know your weight in pounds (if you live in the US) rather than kilograms - since mole conversions are in grams, you need to be able to convert between pounds and kilograms to do this.

Fortunately, there's a way around this.  It's called the "factor-label method", or the "t-chart method".  Whatever you call it, the idea is the same.

The idea behind this method is simple.  In every problem, they give you a number you need to convert.  We'll refer to this as "what you know", since it's the given in the problem.  In every problem you're also given something you need to find.  We'll refer to this as "what you want to know".  When doing conversions between what you know and what you want to know, you'll need to have some unit conversion factors.  These factors are along the lines of "there are 12 inches in one foot" - nothing too disturbing.

Let's do an example:

Example:  Convert 10 meters to inches.  There are 2.54 centimeters in 1 inch.

How to solve:

Go through these steps to make your life much easier:

1)    Draw a great big T.  It should look like this:

2)    Put the number the problem gives you to convert in the top left of this T.  Since the problem told you that you've got "10 meters", that's what you should write up there.  Check it out:

3)    Put the unit of that number in the bottom right part of the T.  In this case, the unit is "meters", so you just write that in the bottom right.  Have a look:

4)    Write the unit of what you want in the top right.  Now, in this question we may have a problem.  Here it is:  Do you know what the conversion factor is between meters and inches?  Come on, off the top of your head!  Quick, what is it?  What do you mean you don't know?

Well, we have a problem then.  If you can't convert between meters and inches, I guess you can't do the problem.  It's unsolvable!

Or is it?

Maybe, just maybe we can convert meters to something that can be converted to inches.  If we're really lucky, the question might give you a hint as to what you can try.  Let's take a look at the question again:

Example:  Convert 10 meters to inches.  There are 2.54 centimeters in 1 inch.

I don't know about you, but I just had an idea!  Maybe instead of going directly from meters to inches, we could go from meters to centimeters, and then from centimeters to inches!  I'm a genius!

As a result, we'll put the unit "centimeters" in the top right.

5)    Write the unit conversion factor in front of the units from steps 3 and 4.

There are three ways to do this:

• Sometimes the unit conversion factor is so simple that it doesn't need to be told to you - it's assumed that you know it off the top of your head.  Examples of this include the factors "12 inches in 1 foot", "3 feet in 1 yard", "60 minutes in 1 hour", "24 hours in 1 day", and so on.  If this is the case for a problem, then all you need to do is look back in your memory to figure it out.
• When the unit conversion factor isn't simple, the problem will give it to you.  For example, in this problem, the problem tells you that there are 2.54 centimeters in 1 inch.  That'll probably come in handy, and is obscure enough that you don't need to memorize it.
• When converting from metric units to other metric units, you put a "1" in front of the unit with the prefix, and write the "multiplier" in front of the unit without the prefix.  The common multipliers for metric conversions are as follows:
 Prefix Multiplier micro 0.000001 (10-6) milli 0.001 (10-3) centi 0.01 (10-2) deci 0.1 (10-1) kilo 1,000 (103) mega 1,000,000 (106) giga 1,000,000,000 (109)

In this problem, that's what we need to do because we're converting between meters and centimeters.  Since "centimeters" has a prefix (centi), we write a 1 in front of it.  Since "meters" has no prefix, we put the multiplier for centimeters in front of it.  From the chart above, we can see that the multiplier is 0.01.  Let's do it!

6)    If you're not done with the conversion in one step, draw another vertical line in the t-chart after the first step and start over at step 3.  I'll assume that you can follow along with what we're doing at this point, so I'll just show you the steps you need to follow in a bunch of different diagrams.  If you have problems following what we're doing, refer to the steps above.

Here's what it looks like when you draw the vertical line and stick the unit in the bottom right (from step 3):

Since we want to convert to inches and we have a unit conversion factor between centimeters and inches, write inches in the top right:

Since we know that the unit conversion factor between inches and centimeters is "2.54 centimeters in 1 inch", write "2.54" in front of centimeters and "1" in front of inches:

7)    Since you've got the t-chart filled out, all you have left to do is multiply all the numbers on the top together and divide them by the product of all the numbers on the bottom.  The unit of this answer will be "inches", since "meters" and "centimeters" cancel out.  As a result, we get: