Subtracting with Mixed Units

Date: 02/10/2009 at 22:45:12
From: David
Subject: Subtracting with mixed units

What is 4 T minus 1 T 12 lb 6 oz?

I'm not sure how to subtract to get smallest unit.  I think the answer
is 2 T 1,999 lb 10 oz.

Date: 02/11/2009 at 10:10:59
From: Doctor Ian
Subject: Re: Subtracting with mixed units

Hi David,

Suppose I have 4 one-ton weights, and I'd like to give you a one-ton
weight, 12 one-lb weights, and 6 one-oz weights.  How might I do that?
 And what would I have left? 

I could trade a one-ton weight for 2000 one-lb weights:

    3 T   2000 lb   0 oz
  - 1 T     12 lb   6 oz
  ----------------------

And I could trade a one-lb weight for 16 one-oz weights:

    3 T   1999 lb  16 oz
  - 1 T     12 lb   6 oz
  ----------------------

And now I'm good to go:


    3 T   1999 lb  16 oz
  - 1 T     12 lb   6 oz
  ----------------------
    2 T   1987 lb  10 oz

It's the same idea as breaking bills into smaller bills, or into
coins, or breaking coins into smaller coins to make change.  And you
can use this for any situation where units are grouped into larger
units, e.g., times (hours, minutes, seconds), distances (miles, feet,
inches), volumes (gallons, quarts, pints, cups), and so on. 

For that matter, it's really what we're doing when we subtract
something like

   213
  - 87
  ----

We can think of that as

    2*100  1*10   3*1
  -        8*10   7*1
  -------------------

We can trade a hundred for 10 tens,

    1*100 11*10   3*1
  -        8*10   7*1
  -------------------

and a ten for 10 ones,

    1*100 10*10  13*1
  -        8*10   7*1
  -------------------

and now the subtraction is straightforward:

    1*100 10*10  13*1
  -        8*10   7*1
  -------------------
    1*100  2*10   6*1  =  126

The method taught in schools involves trying to do this in place, by
crossing things out and writing little 1's over the columns to track
the exchanges:

                     0  1             1  10  1
   2 1 3          2  /   3            /   /   3
  -  8 7   ->   -    8   7    ->   -      8   7
  ------        ----------         ------------

                Now we have        Now we have 1
                13 ones, and       hundred, 10 tens,
                no tens.           and 13 ones,
                                   so we can do the
                                   subtraction.

That works because each group is always 10 times as large as the
previous group--something that doesn't hold when we're dealing with
measurements instead of pure numbers.  

(For example: there are 60 seconds to a minute, 60 minutes to an hour,
and 24 hours to a day; there are 8 ounces to a cup, 2 cups to a pint,
2 pints to a quart, and 4 quarts to a gallon; there are 12 inches to a
foot, 3 feet to a yard, and 1760 yards to a mile; and so on. 
Americans in particular seem to be fond of using group sizes that are
multiples of 2, and 3, rather than multiples of 10.)  

The school method saves some writing, but at the cost of 
transparency.  So there is a temptation to try to memorize the steps,
without really understanding why they work, or how the method can be
generalized. 

Anyway, is this looking familiar?  Does it make sense? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 

The Math Forum @ Drexel is a research and educational enterprise
of the Drexel School of Education: http://www.drexel.edu/soe/