Inches, Feet, Yards, Miles, and Unit Fractions

Date: 01/29/2002 at 22:43:25
From: Lyann
Subject: Changing customary units

I need to know if you multiply or divide in changing feet to inches, 
feet to yards, miles to yards, feet to miles, yards to miles, yards to 
feet.

Date: 01/29/2002 at 23:32:03
From: Doctor Peterson
Subject: Re: Changing customary units

Hi, Lyann.

Here are two good ways to keep track of this.

First, just picture replacing one unit with several of the other. To 
change yards to feet, remember that a yard is 3 feet; so if you have 
5 yards, say, you can replace each yard with 3 feet. That makes it 
5 times 3 feet:

       1 yard      1 yard      1 yard      1 yard      1 yard
    +-----------+-----------+-----------+-----------+-----------+
    +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
     1ft 1ft 1ft 1ft 1ft 1ft 1ft 1ft 1ft 1ft 1ft 1ft 1ft 1ft 1ft

If you were changing feet to yards, you would replace in the other 
direction: given 15 feet, you would make groups of 3 feet and replace 
each group with one yard. How many groups are there? 15 divided by 3 
is 5 groups, so 15 feet is 5 yards.

If the units are not lengths, you can still pretend they are. Every 
minute is 60 seconds; every pound is 16 ounces; and so on. The same 
method still works.

Second, you can use units like fractions. Since 1 yard is the same as 
3 feet, the "unit fraction"

    3 feet
    ------
    1 yard

is 1, since the numerator and denominator represent the same amount. 
So if you multiply by this fraction, you won't change the meaning of 
the amount you started with. If we start with 5 yards, and multiply, 
we get

              3 feet   5 yards * 3 feet   5 * 3 feet
    5 yards * ------ = ---------------- = ---------- = 15 feet
              1 yard   1 yard                 1

Notice that I "canceled" the "yard" in the numerator and the "yard" in 
the denominator just as if I were simplifying a fraction by canceling 
common factors.

This may be a little hard for you to follow now, but eventually it 
will be the best method for handling units in complicated problems. By 
choosing the unit fraction that will cancel out the units you want to 
get rid of, you can convert all sorts of units.

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