Cancelling (or Converting) Units

Quarts are smaller than gallons; every gallon has four quarts. Since I'm converting from a larger unit (gallons) to a smaller unit (quarts), my answer needs to be a bigger number. So I multiply:

Then:

Miles are bigger than yards; there are 1760 yards in every mile. Since I'm converting from a smaller unit (yards) to a bigger unit (miles), my answer needs to be a smaller number. So I divide:

Then:

One of these measurements is in terms of "miles per hour" and the other is in terms of "meters per second". In order to compare the two speeds, I'll need to convert one of the measurements into the units for the other measures. That way, I'll be comparing "apple to apples".

So, in some way or another, I need to convert between "miles" and "meters" and between "hours" and "seconds". Looking in the back of my textbook (which is frequently a handy resource, along with the endpages of many dictionaries), I find a table of common conversion factors, from which I pull out these as being probably helpful:

I could have chosen other conversion factors, if I'd felt like it. But these factors provide connections, one way or another, between "seconds" and "hours" and between "miles" and "meters", so they'll get the job done.

To compare these two rates of speed, I need them to be in the same units. Flipping a coin, I decide that I'll convert the "80 miles per hour" to "meters per second". I need to set things up so the units will cancel:

Why did I put "1 hour" on top and "60 mins" underneath? Because I started with "80 miles per hour", so "hours" started out underneath. I want "hours" to cancel off (in favor of "seconds", eventually), so the conversion factor for hours and minutes needed to have "hours" on top. That meant that "60 mins" had to be underneath.

And that dictated the orientation of the next factor: Since "60 mins" was underneath and since I'd need "minutes" to cancel at some point, then the "1 min" (from the conversion factor for minutes and seconds) had to be on top. This in turn meant that "60 secs" had to be underneath. And since I'm wanting a final answer of "per seconds", I want the seconds underneath, so this works out just right.

In the same way, what they gave me has "miles" on top so, in the "1 mile to 5280 feet" conversion factor, I need the "miles" on the bottom, so it cancels off. This will put the "feet" on top. So, in the "1 foot to 12 inches" conversion factor, I'll need the "foot" on the bottom; this will put the "inches" on top. Then the "1 inch to 2.54 centimeters" conversion factor will need the "inch" on the bottom, leaving the "centimeters" on top.

The last conversion factor is "100 centimeters to 1 meter"; since "centimeters" in the previous factor were on top, then I'll need "centimeters" on the bottom, leaving the "meters" on top. Since I'm wanting a final result that is in terms of "meters per second" (that is, meters divided by seconds), I want the meters on top, so this part worked out right, too.

Putting it all together, I get the following long string:

(If you're not sure of my multiplication set-up above, then go grab a sheet of scratch-paper and do the described steps yourself.)

Now I cancel off the units:

Since the units cancel, leaving me with the "meters per second" that I need (circled above), I know the numbers must be in the right places. To get my answer, all I have to do is grab a calculator and simplify the fraction multiplication:

This says that 80 miles per hour is equivalent to just under 36 meters per second. Forty is more than thirty-six, so: