(9/2)/(3/4): A Picture

Date: 03/27/2003 at 16:20:15
From: Julie
Subject: Fractions

When dividing fractions, why do you have to invert and multiply?

In my college class we were trying to find a division fraction 
question and everything we came up with turned into a multiplication 
fraction problem. Why is that?

Date: 03/27/2003 at 17:02:05
From: Doctor Peterson
Subject: Re: Fractions

Hi, Julie.

I'd like to know how your division questions "turned into" 
multiplications; it sounds as if you may have come close to seeing 
why we invert and multiply. Division problems naturally turn into 
multiplication problems when you see them the right way.

Let's take a simple, practical division question: I want to cut a 
board that is 4 1/2 feet long (that is, 9/2 foot) into pieces that 
are 3/4 foot long. How many pieces will I get?

Here's the board:

    +-----------+-----------+-----------+-----------+-----+
    |                                                     |
    +-----------+-----------+-----------+-----------+-----+

Here's one piece:

    +--------+
    |        |
    +--------+

I could, of course, just cut until I use up the board:

    +--------+--------+--------+--------+--------+--------+
    |        |        |        |        |        |        |
    +--------+--------+--------+--------+--------+--------+

But I'd like to think first and find the answer mathematically.

One way (believe it or not!) is to use a common denominator. That 
means choosing a different unit to measure in. Since both 4 1/2 feet 
and 3/4 foot can be easily expressed in inches, I'll use that (which 
amounts to using 12 as a common denominator). So I want to divide 4 
1/2 * 12 = 54 inches by 3/4 * 12 = 9 inches; the answer is 6. (In 
fact, that's how I drew my picture; each character is one inch.)

But that's not the way we usually teach division of fractions. We 
want to divide in a way that uses the numerator and denominator of 
our fractions. So let's take those one at a time. Rather than work 
with the 3/4 foot boards, let's think about a 1/4 foot piece, paying 
attention to the denominator alone for now. How many 1/4 foot pieces 
will fit into our 4 1/2 foot board? Well, there will be 4 pieces in 
each foot, so we want to MULTIPLY 4 1/2 feet by 4, which is the 
DENOMINATOR of our divisor. We find that 18 of these smaller pieces 
will fit in our board.

But we don't want 1/4 foot pieces; we want 3/4 foot pieces. So we 
have to take those 1/4 foot pieces and glue each three of them back 
together! How many will we end up with? Well, for every three of the 
18 pieces we have, we will end up with only one piece; so we will 
DIVIDE 18 by 3, the NUMERATOR of the divisor. That gives 6, which is 
our answer.

Here's a picture of what we did:

    +-----------------------------------------------------+
    |                                                     | 4 1/2 ft
    +-----------------------------------------------------+
    times 4 fourths per foot
    +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
    |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  | 18
    +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
    divided by 3 pieces per fourth
    +--------+--------+--------+--------+--------+--------+
    |        |        |        |        |        |        | 6
    +--------+--------+--------+--------+--------+--------+

So we multiplied by the denominator, 4, and then divided by the 
numerator, 3, to get the answer.

If you have any further questions, feel free to write back.

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