MathBench > Measurement

Follow the Units

Following slowly

the press roomBen Bradlee (editor in chief): Where's the goddamn story?

Bob Woodward: The money's the key to whatever this is.

Ben Bradlee: Says who?

Howard Simons (local editor): Deep Throat.

Ben Bradlee: Who?

Howard Simons: Oh, that's Woodward's garage freak...

Ben Bradlee: Garage Freak? Jesus, what kind of a crazy f*****g story is this? Who did you say?

Howard Simons: He's on deep background, I call him deep... throat.

Here's what we did so far:

Let's take that a little more slowly.

How the heck is multiplying by 5280 feet ⁄ 1 mile the same as multiplying by 1?

It's easy to see why fractions like 2 ⁄ 2 work, but not so easy with 5280 feet ⁄ 1 mile.

In fact, if you didn't have the units there, it WOULDN'T work. Clearly, 5280 ⁄ 1 does not equal 1!!

But, with the units, we know that the top and the bottom actually are the same. That's because 5280 feet actually are the same as 1 mile. So this fraction is the same as 1.

OK, but that means that multiplying by 1 mile ⁄ 5280 feet would also be the same as multiplying by 1, right?

Right! Even if you flip the fraction, it will still be equal to 1. But (as we'll see in a minute) some fractions-that-equal-1 are a lot more useful than others.

But still, why would I want to do this?

Because it allows you to convert units.

For example, when you're in busy parking garage, 4.5 miles doesn't sound like much. It sounds a bit better in feet though:

Wait, how did you make "miles" go away?

Here's the really cool thing: you can cancel out units in fractions EXACTLY LIKE you cancel out numbers. So, the miles cancel out, and you are left with kilometers only.

So, since 5280 feet ⁄ 1 mile and 1 mile ⁄ 5280 feet are both equal to 1, how do I know which one to choose?

Remember, you need to cancel out the miles, so pick the one that does the trick. Follow the units!! But, no worries: if you pick the wrong one, you'll notice when you try to get the final answer:

4 miles * 1 mile ⁄ 5280 feet = ??? miles2/ft.

Gaack! miles2/ft is definitely not the units you were going for!! If this happens, it's probably because you need to flip a fraction to make the units work out. Just make sure you flip BOTH the numbers AND the units!

Hey, wait a minute, I already know how to convert miles and feet. Why are you trying to make me learn a new method???!

Yup, so do I:

miles * 5280 -> feet, or

feet / 5280 -> miles

The problem is, sometimes I forget which one I'm supposed to multiply, and which one I divide. The fraction method is much more forgiving. All I need to remember is that 1 mile = 5280 feet. From there, I can build the fraction I need, and if I goof up, it's obvious because the units don't work out.

So are you telling me that a scientist with a labcoat and a supercomputer would look up conversion factors and write equations? Wouldn't they just use a computer for that??

No, they wouldn't "just use a computer," because there's no easy way to do that - in fact, I look up conversion factors all the time (and I have a few memorized). Using these equations is second nature for me. (Now, using pencil and paper rather than a calculator or a spreadsheet - that is truly crazy! Use those electronics, people! It's not like we're back in the 1960's.)

So you use a big book of conversion tables? Where do you keep it, under your pillow?

No, usually I just use Google, like this: