Cancelling / Converting Units

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 the volume to “fluid ounces”. They will have given you (or else you can easily find) the 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:

• going to smaller units mean going to bigger numbers, so multiply
• going to bigger units mean going to smaller numbers, so divide

Here’s how it works:

• Convert 3 gallons to quarts.
  (3)(4) = 12

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:

Answer: 12 quarts

• Convert 7920 yards to miles.
  7920 ÷ 1760 = 4.5

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:

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? It’s hard to see how the term “bigger” would 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.

• Which is faster, going 80 miles an hour or going 40 meters per second?
  60 seconds : 1 minute
  60 minutes : 1 hour
  1 mile : 5280 feet
  1 foot : 12 inches
  2.54 centimeters : 1 inch
  100 centimeters : 1 meter
  

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, along with the endpages of many dictionaries), I find the following conversion factors among the many listed:

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 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, 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. Copyright © Elizabeth Stapel 1999-2009 All Rights Reserved

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

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. So to get my answer, all I have to do is grab a calculator and simplify the fraction multiplication:

• 40 meters per second is faster than 80 miles per hour.

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

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 powerfultechnique. 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“.

• Suppose an object is moving at 66 ft/sec. How fast would you have to drive a car to keep pace with this object?

• 
  
  • 

  

A car’s speedometer doesn’t measure feet per second, so you’ll have to convert to some other measurement. You choose miles per hour. You know the following conversions: 1 minute = 60seconds, 60 minutes = 1 hour, and 5280 feet = 1 mile. If 1 minute equals 60 seconds (and it does), then

The fact that the conversion can be stated in terms of “1“, and that the conversion ratio equals “1” no matter which value is on top, is crucial to the process of cancelling units.

We have a measurment in terms of feet per second; we need a measurement in terms of miles per hour. To convert, we start with the given value with its units (in this case, “feet over seconds”) and set up our conversion ratios so that all undesired units are cancelled out, leaving us in the end with only the units we want. Here’s what it looks like:

Why did we set it up like this? Because, just like we can cancel duplicated factors when we multiply fractions, we can also cancel duplicated units:

I would have to drive at 45 miles per hour.

How did I know which way to put the ratios? How did I know which units went on top and which went underneath? I didn’t. Instead, I started with the given measurement, wrote it down complete with its units, and then put one conversion ratio after another in line, so that whichever units I didn’t want were eventually canceled out. If the units cancel correctly, then the numbers will take care of themselves.

If, on the other hand, I had done something like, say, the following:

…then nothing would have cancelled, and I would not have gotten the correct answer. By making sure that the units cancelled correctly, I made sure that the numbers were set up correctly too, and I got the right answer. This “setting up so the units cancel” is a crucial aspect of this process.

• You are mixing some concrete for a home project, and you’ve calculated according to the directions that you need six gallons of water for your mix. But your bucket isn’t calibrated, so you don’t know how much it holds. On the other hand, you just finished a two-liter bottle of soda. If you use the bottle to measure your water, how many times will you need to fill it?

• • = (6 × 3.785) liters = 22.71 L

  

  

For this, I take the conversion factor of 1 gallon = 3.785 liters. This gives me:

Since my bottle holds two liters, then:

I should fill my bottle completely eleven times, and then once more to about one-third capacity.

On the other hand, I might notice that the bottle also says “67.6 fl.oz.”, right below where it says “2.0L“. Since there are 128 fluid ounces in one (US) gallon, I might do the calculations like this:

• = 11.3609467456… bottles Copyright © Elizabeth Stapel 1999-2009 All Rights Reserved

…which, considering the round-off errors in the conversion factors, compares favorably with the answer I got previously.

• You find out that the average household in Mesa, Arizona, uses about 0.86 acre-feet of water every year. You get your drinking water home-delivered in those big five-gallon bottles for the water dispenser. How many of these water bottles would have to be stacked in your driveway to equal 0.86 acre-feet of water?

• 
  • = 56,050.04592…. bottles

  

The conversion ratios are 1 acre = 43,560 ft², 1ft³ = 7.481 gallons, and five gallons = 1 water bottle. First I have to figure out the volume in one acre-foot. An acre-foot is the amount that it would take to cover one acre of land to a depth of one foot. How big is 0.86 acres, in terms of square feet?

If I then cover this 37,461.6 ft² area to a depth of one foot, this would give me 0.86 acre-feet of water, or (37,461.6 ft²)(1 ft deep) = 37,461.6 ft³ volume of water. But how many bottles does this equal?

…or about 56,000 bottles every year.

This works out to about 150 bottles a day. Can you imagine “living close to nature” and having to lug all that water in a bucket? Thank heaven for modern plumbing!

• You’ve been watching a highway construction project that you pass on the way home from work. They’ve been moving an incredible amount of dirt. You call up the information line, and find out that, when all eighty trucks are running with full crews, the project moves about nine thousand cubic yards of dirt each day. You think back to the allegedly “good old days” when work was all done manually, and wonder how many wheelbarrowsful of dirt would be equivalent to nine thousand cubic yards of dirt. You go to your garage, and see that your wheelbarrow is labeled on its side as holding six cubic feet. Since people wouldn’t want to overfill their barrows, spill their load, and then have to start over, you assume that this stated capacity is a good measurement. How many wheelbarrow loads would it take to move the same amount of dirt as those eighty trucks?

• • = 40,500 wheelbarrows

  

The conversion ratios are 1 wheelbarrow = 6 ft³ and 1 yd³ = 27 ft³. Then I get:

Wow; 40,500 wheelbarrow loads!

Even ignoring the fact the trucks drive faster than people can walk, it would require an amazing number of people just to move the loads those trucks carry. No wonder there weren’t many of these big projects back in “the good old days”!

When I was looking for conversion-factor tables, I found mostly Javascript “cheetz” that do the conversion for you, which isn’t much help in learning how to do the conversions yourself. But along with finding the above tables of conversion factors, I also found a table of currencies, a table of months in different calendars, the dots and dashes of Morse Code, how to tell time using ships’ bells, and theBeaufort scale for wind speed.

http://www.purplemath.com/modules/units2.htm