Finding Areas using Unit Conversions

Date: 3/19/96 at 1:38:33
From: Heikki Jokipii
Subject: Too simple a question

Dear Dr. Math,

   I am not an American and I am not a child in the
   American elementary school - and that is exactly  
   the reason why I write this letter. 

   I am interested in the following problem which is
   presumably very elementary for you, but I can not
   find an answer to it here in Finland where I live
   - I simply can't find your elementary school
   text-books in math here. It is so elementary that
   I could not find anything of it in your archives,
   either.

   As you surely know, we in Europe use the metric system. 
   In the metric system, calculating an area of a rectangle
   is very simple: just one multiplication and there 
   it is. If you then want to express the result, say,
   in centimeters instead of kilometers, you only have
   to move the decimal point.

   But how do you use to count the area, expressed in
   your traditional units. For example: how do you calculate the
   area of a rectangle, where one side is, say,
   2 miles, 3 furlongs, 4 yards, 2 feet and 5 inches and
   the other, say, 1 mile, 7 furlongs, 47 yards, 1 feet 
   and 11 inches? The result in acres and so on?

   I can solve this problem, of course, in my way.
   But there have to be a way that is more systematic
   and I know that there is: the way you Americans (and
   English) use. To be more exact: I mean the way you do the 
   calculation(s) with pencil and paper.

   If you kindly would describe me this systematic
   calculating method I would be very grateful.

   Yours

   Heikki Jokipii

Date: 3/19/96 at 13:14:5
From: Doctor Aaron
Subject: Re: Too simple a question

This type of a calculation is a pain with English units because 
unlike metric units they don't have a natural decimal 
representation.  Eventually we will have to write the area as a 
decimal or use inconvenient units like square inches or absurd 
units like inches*miles.
 
One way to do it is to convert everything to inches, multiply, and 
reconvert into square miles.

2 feet = 24 inches.
4 yards = 12 feet = 144 inches.
3 furlongs = 1980 feet = 23,760 inches.
2 miles = 10,560 feet = 126,720 inches.

So 2 miles + 3 furlongs + 4 yards + 2 feet + 5 inches = 150,653 
inches.

Similarly, 1 mile + 7 furlongs + 47 yards + 1 feet + 11 inches = 
120,515 inches.

To get the area, we just multiply the sides (hopefully with a 
calculator) to get 18,155,846,295 square inches.  Since there are 
4,014,489,600 square in a square mile (5280feet/mile * 12 
inches/foot)^2, we can divide to get 4.522 square miles.

We could've also converted to miles before the multiplication to 
avoid dealing with such big numbers.

Then 150,653 inches = 2.377 miles, and 120,515 inches = 1.902 
miles, so the area of the rectangle spanned by these two sides 
will be 2.377*1.902 = 4.521 square miles.  There is a rounding 
error in the last decimal place, but the two answers basically 
agree with each other.

I think that the tediousness of this seemingly simple calculation 
illustrates the ease of working with the metric system.

-Doctor Aaron,  The Math Forum