Meaning of Multiplying Fractions

Date: 12/09/1999 at 20:10:51
From: Mary
Subject: What does it mean to multiply a fraction by a fraction?

I am doing an essay assignment, and I got stuck on this question: What 
does it mean to multiply a fraction by a fraction? My sister and I 
think multiplying a fraction by a fraction just means to multiply a 
fraction by a fraction!

Please help.

Date: 12/09/1999 at 23:28:20
From: Doctor Peterson
Subject: Re: What does it mean to multiply a fraction by a fraction?

Hi, Mary.

Well, what does it mean to multiply a whole number by a whole number?

Two times three means the total of two groups of three things; here 
are two groups of three squares:

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

So 2 times 3 is 6.

Now what does it mean to multiply a fraction times a whole number?

One half times six means half a group of six; here's half a group of 
three squares:

     +-------+-------+-------+
     |       |       |       |
     +-------+-------+-------+
     :       :       :       :
     :.......:.......:.......:

1/2 times 3 is 3/2, because I've cut the squares into six halves, of 
which I'm keeping three.

We do the same thing to multiply a fraction times a fraction; here's 
half a group of 2/3 of a square:

     +-----+..:
     |  :  |  :
     +-----+..:
     :  :  :  :
     :..:..:..:

1/2 times 2/3 is 2/6, because by cutting the square into two slices 
one way and three the other way, I have six pieces, of which I'm 
keeping only two.

So a fraction times anything is just that fraction OF that amount; 
break it into as many pieces as the denominator says, and keep as many 
of them as the numerator says.

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