Dividing a Fraction by a Fraction

Date: 08/11/97 at 23:18:44
From: Chris Tafoya
Subject: Dividing a fraction by a fraction

Dr. Math,

The cookie scenario below is an excellent example to visualize why
dividing a whole number by a fraction causes the answer to be
larger than the original whole number. But what about dividing a 
fraction by a fraction?  The scenario becomes incomprehensible when 
the "5 cookies" become a half a cookie.

Do you have another example/scenario that can help students visualize
a problem such as:

  1/3 / 1/2 = 2/3

The abstract concepts have been explained tremendously. Is there a
concrete way? If I have a third of a pie, and I want to divide that 
third of a pie by 1/2, why does the answer become 2/3 of the pie??

Thanks,
Chris "from the old school" Tafoya

Date: 03/21/97 at 04:19:54
From: Doctor Mitteldorf
Subject: Re:  Divide by a Fraction

Dear Tim,

Lots of people find this confusing.  If you divide 5 by 2, the 
answer is 2.5.  If you divide 5 by 1/2, do you expect the same thing 
as dividing by 2?

If you divide by a number bigger than 1, it always reduces the number. 
If you divide by 1, it doesn't change anything. Does that make you 
think that dividing by a fraction less than 1 should INCREASE the 
number?

How many kids can you serve with 5 cookies if each kid gets 2 cookies? 
You can serve 2 kids (with enough left over for 1/2 a kid).  

How many kids can you serve with 5 cookies if each kid gets 1/2 a 
cookie?  That's 5 divided by 1/2.

-Doctor Mitteldorf,  The Math Forum
 Check out our web site!

Date: 08/12/97 at 04:27:50
From: Doctor Mike
Subject: Re: Dividing a fraction by a fraction

>1/3 / 1/2 = 2/3
   
   Maybe you could think about it this way.  For the 5/2 = 2.5 you
   could think of how many 2-cookie servings you can make out of
   5 cookies. You get two full 2-cookie servings plus half of a
   2-cookie serving. For the 5/half = 10 you could think of how 
   many half-cookie servings you could get out of 5 cookies.  
    
   For the (1/3)/(1/2) = 2/3 it's probably clearer to write it as
   (2/6)/(3/6) = 2/3 and ask how times you could get a (3/6)-cookie
   serving out of 2/6 of a cookie.  You can't!  You get **zero**
   (3/6)-cookie servings. But you can get PART OF A (3/6)-cookie
   serving. In fact you get exactly "two thirds of a (3/6)-cookie
   serving. I'll leave it up to you to decide whether what I just
   said is incomprehensible.   
   
   It may be clearer to keep it (1/3)/(1/2) = 2/3. Then say, 
   "how many cookie-halves can you get out of a third of
   a cookie?" The answer would then be, "You can't get ANY cookie-
   halves out of a third of a cookie, BUT you CAN get two thirds of
   a cookie-half from a third of a cookie. WE NEED MORE COOKIES! 

-Doctor Mike,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/