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Converting Units across Measurements

Date: 08/31/2010 at 19:40:33
From: Bally
Subject: All About Conversions 

How do you convert units to a different measurement?

For example, the metric system has kilograms and centimeters; and I know
that "kilo-" means 1000 and "centi-" means 100. But when and why do you
move the decimal spot?



Date: 09/01/2010 at 10:01:16
From: Doctor Ian
Subject: Re: All About Conversions

Hi Bally,

This is going to seem like a long explanation, but if you work through it,
you'll find that you never have to worry about conversions again.

I find the easiest way to think about conversions is to make the
multiplication of units explicit. That is, instead of working with
something like '3 cm,' I always work with '3 * (1 cm).'

How does this help? Well, you can always multiply a quantity by 1, because
it doesn't change the value of the quantity. We use this to find
equivalent fractions, e.g.,

   3   5   15
   - * - = --
   4   5   20

Going in the opposite direction, if we start with something like 15/20 and
identify common factors ...

   15   3 * 5
   -- = -----
   20   4 * 5

... we can cancel out pairs that appear in both the numerator and the
denominator. That is, if we multiply by 5, and then divide by 5, those
operations cancel, so we can get rid of them:

   15   3 * 5   3
   -- = ----- = -
   20   4 * 5   4

Now, suppose we have something like 

   3 * (1 cm)

I can multiply that by 1, but a special form of 1. That is, I can 'make 1'
with a ratio of any two quantities that I know are equal to each other.
For example, 

                   1 * (1 in)
   3 * (1 cm) * -------------
                2.54 * (1 cm)

That is, I can do this because I know that 1 inch and 2.54 centimeters are
defined to be the same distance.

Now, since I'm multiplying by (1 cm) in one place, and dividing by (1 cm)
in another place, those operations cancel out, leaving me with the
equivalent distance in inches:

         1 * (1 in)
   3 * -------------- = (3/2.54) * (1 in)
            2.54 

That is, the unit (1 cm) is a common factor, just like 5 in my earlier
fractions. And I can treat it the same way.

Does this make sense? It's very, very general, and will work for any kinds
of units whatever -- even units you've never heard of. If I tell you that
something has a weight of 3 bizzles, and there are 12 bizzles in a dozzle,
then you can do this:

                     1 * (1 dozzle)
   3 * (1 bizzle) * --------------- = (3/12) * (1 dozzle)
                    12 * (1 bizzle)

It also works for compound units, like speeds:

   30 * (1 mile)   5280 * (1 foot)    1 * (1 hour)    1 * (1 min)
   ------------- * --------------- * ------------- * ------------
    1 * (1 hour)      1 * (1 mile)   60 * (1 min)    60 * (1 sec)
       
   Start with 
   miles per
   hour
                   Now I have feet
                   per hour
                                     Now I have feet
                                     per minute
                                                     And now feet
                                                     per second

After the units cancel, and I do the remaining operations, I've converted
30 miles per hour to an equivalent number of feet per second.

(I happen to know that 60 miles per hour is the same as 88 feet per
second, so this should work out to 44 feet per second. Try it, and see if
that's what you get, just to check your ability to do this kind of
calculation.)

So far, so good? Now, what about metric units? Well, suppose I want to
change centimeters into kilometers. I can change centimeters into
meters, ...

                  1 * (1 m)
   3 * (1 cm) * ------------
                100 * (1 cm)

... and then change meters into kilometers, 

                  1 * (1 m)       1 * (1 km)
   3 * (1 cm) * ------------ * ------------- 
                100 * (1 cm)   1000 * (1 m)

... and the result will be in kilometers. I could, in theory, keep
stringing these together as much as I want: centimeters to inches, inches
to feet, feet to yards, yards to furlongs, furlongs to miles, miles to
light years, and so on. I'm just multiplying by 1 and canceling the units
I don't want, until I get the ones I do want.

Now, what makes metric units easy to deal with is that we're always
working with powers of 10, as opposed to quantities like 2.54 (centimeters
in an inch) or 12 (inches in a foot) or 4 (quarts in a gallon). So my
digits don't change, and I just move the decimal point around. That
eliminates a lot of work -- but it doesn't change the principles of what's
happening.

So what I can do now is cancel my units, 

        1      1     
  3 * --- * ---- * (1 km) 
      100   1000

and I can see that I'm dividing by 100,000. That tells me how far I have
to move the decimal place, right? 

Is it making more sense now?

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
Middle School Measurement
Middle School Ratio and Proportion
Middle School Terms/Units of Measurement

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