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Cancelling / Converting Units (page 1 of 2) To convert between units, you're usually given one measure and asked to convert to another measure. For instance, you'll be given some volume in "gallons" and be asked to convert to "fluid ounces". You'll have been given (or else you can easily find) conversion units that are suitable to the task. In these simple scenarios, all you have to do to convert is remember a fairly simple rule:
Here's how it works:
Quarts are smaller than gallons; every gallon has four quarts. Since I'm converting from a larger unit (gallons) to a smaller unit (quarts), I need a bigger number. So I multiply: (3)(4) = 12 Answer: 12 quarts
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), I need a smaller number. So I divide: 7920 ÷ 1760 = 4.5 Answer: 4.5 miles The above are examples of one-step conversions: You use one conversion factor (the equivalence between two measures or units) to convert from the one unit to the other. But sometimes conversions are more complicated, or you're not sure which unit is "bigger". This applies especially in the case of conversions between English and metric units. For instance, which is "bigger", decaliters or Imperial gallons? Or consider rates: which is "bigger", 80 miles an hour or 40 meters per second? The term "bigger" doesn't even really apply here. For these sorts of conversion, we use as many conversion factors as we need, setting up a long multiplication so the units we don't want cancel out. Note: I'm not talking here about numbers cancelling out, like when you're multiplying fractions. Instead, I'm talking about treating the units ("feet", "miles", "seconds", etc) as though they were numbers, and cancelling them.
Okay, I need to convert from "miles" to "meters" and from "hours" to "seconds". Looking in the back of my textbook (which is frequently a handy resource), I find the following conversion factors among the many listed: 60 seconds : 1 minute Depending on the source and my predilection, I could have chosen other conversion factors. But these provide connections, one way or another, between "seconds" and "hours" and between "miles" and "meters", so they'll do. 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 start setting 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" was underneath. I want "hours" to cancel off, 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. Continuing:Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved
Cancel off the units:
Since the units cancel, leaving me with the "meters per second" that I need, I know the numbers must then be in the right places. So to get my answer, all I have to do is grab a calculator and simplify:
This says that 80 miles per hour is equivalent to just under 36 meters per second, so: 40 meters per second is faster than 80 miles per hour. This method of converting units can actually be quite useful: it got me through a chemistry class! I didn't have a clue what the instructor was talking about, but on the test questions he gave only the exact information needed, and if I set up everything so the units cancelled, I always got the right answer. While I'm not advocating being ignorant of chemistry, I think you get my point: This is a powerful technique. Cancelling units (also known as "unit analysis" or "dimensional analysis") is based on the principal that multiplying something by "1" doesn't change the value, and that any value divided by the same value equals "1". Top | 1 | 2 | Return to Index Next >>
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Copyright © 2006-2008 Elizabeth Stapel | About | Terms of Use |
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![[ 80 miles / hour ] [ 1 hour / 60 mins ] [ 1 min / 60 secs ]](units/units15.gif){kind=small}
![[80 miles/hour][1 hour/60 min][1 min/60 sec][5280 ft/1 mile][12 in/1 ft][2.54 cm/1 in][1 m/100 cm]](units/units16.gif){kind=small}
![[80/1][1/60][1/60][5280/1][12/1][2.54/1][1/100] = 35.7632 (approx.)](units/units18.gif){kind=small}
