Engineering Insights
Unit Conversion - An Easy Way To Do It
Unit Conversion is a good skill to have, whether you’re solving a science problem, planning a trip or just need to compare numbers of two different measurement systems. Converting numbers can be as simple as using a single conversion factor or might get trickier when having to use multiple ones.
In this month’s article, we’ll start with a simple conversion factor example and move onto more difficult ones.
One thing to keep in mind. Anytime you multiply your number by a conversion factor, you’re actually just multiplying it by one.
Here’s the first example: If you have something 2 inches long and want to convert it to millimeters, the standard method is to multiply it by 25.4.
Like this: 2 in x 25.4mm/in = 50.8 in.mm/in = 50.8 mm That’s simple enough, but to help keep the units clearer, it can be rewritten like this: 
It’s not a bad idea to drop the multiplications signs because they can get in the way if a conversion gets long. Just remember that you have to multiply everything on the top together and everything on the bottom too.
The same units that are on top and bottom cancel each other out, so you end up with this: 
It looks a bit cumbersome performing the conversion like this, and it should. When only a simple conversion factor is needed, you don’t need to go to this much effort. But it’s a very useful method if you’ve got a lot of units to consider.
Looking at a different example, let’s say you want to convert 72 inches to yards, but you can’t remember how many inches are in a yard, but you do know the number of feet in a yard and inches in a foot.
The conversion would look like this:
or simply:
You check yourself by looking at each column and making sure they equal 1. Does 1 foot = 12 inches? Yes. Does 1 yard = 3 feet. Yes. We’re good to go. When operating this way, the cancellation of units above and below the horizontal line is critical. First, we see that there’s an inch unit above and below the horizontal line, so they get canceled. 
Next, we see that there’s a foot unit above and below the line so they get canceled too.
You can see here that the only unit left is yards. So now multiply the top value and the bottom values, combine their units and divide them. This is a really simple example so there’s no combining of units.
So 72 inches = 2 yards. We knew that going into this, but it’s good to see how the method works.
Let’s get a bit more complicated.
We want to convert 1200 feet / sec to miles per hour. Imagine we can’t remember how many seconds are in an hour, so we do it like this with intermediate units that cancel themselves out. 
Besides the first one, every column there has a value of one.
Canceling units and multiplying gives us this: 
Remember, make sure that every column equals one and you’ll be good to go. What happens if you get one of the columns upside down like this?
Though the third column is equal to one, it’s upside down because the units are in reversed position. You would try to cancel the units and end up with this mess. 
resulting in:
Oh boy! Something went wrong, very wrong. That tells you it’s time to go back and check your work. Something (at least one thing) must be upside down! It can get a bit trickier with units that are squared or cubed:
Converting a density of 500 lbs/ft³ to g/cm³ might appear as a challenge but just write everything out in single units and you’ll be OK. Do this first, before you begin: 
That is the way of breaking apart that cubic foot unit into individual feet because it’s much easier to convert feet to inches and then to centimeters instead of cubic feet to cubic inches and then to cubic centimeters. Who on earth remembers that there are 1728 cubic inches to a cubic foot? Now, since I don’t know how many centimeters are in a foot, this is going to take quite a few steps. 
Section A is the number that needs to be converted, section ‘B’ converts pounds into kilograms, section ‘C’ converts kilograms into grams, section ‘D’ converts feet into inches, and section ‘E’ converts inches into centimeters.
Canceling units looks like this: 
And multiplying it all together results in:
Sure, you could write that as 8.03 g / cm.cm.cm but it’s customary to write them with an exponent that matches the number of instances, i.e. cm.cm = cm2 and cm.cm.cm = cm3.
That means that if you had a 1 cubic foot of stainless steel weighing 500 lbs and cut a cubic centimeter from one of the corners you would know, with confidence, that it weighs 8 grams. Why you would ever want to do that, I have no idea.
To summarize:
The basic steps to modifying units associated with a number are as follows:
- If you have a simple conversion factor to get from one unit to another, use it
- If you have squared or cubed units associated with your starting number (or even 4th or 5th powers), split them apart as shown above
- Multiply by as many conversions (values of one) as you need to, keeping in mind which way you place them so that units cancel properly
- Cancel units as necessary
- Multiply your top numbers together and bottom numbers together, dividing the results
- Transfer over the units that are left from your calculation (canceling) to the new number, and there’s your answer!
Hopefully this has helped you navigate the world of unit conversion. With this method you can tackle numbers and their units from the simplest to the most complex.
