Explaining Division of Fractions

Date: 08/11/99 at 16:26:25
From: Lee Werling
Subject: Tutoring division of fractions

My wife and I are helping some neighbor kids prepare for 6th grade. 
Since they will be entering an advanced program in science and math, 
those are the areas we're concentrating on. In a review of fractions, 
I stumbled when I could not come up with a practical explanation of 
how division of fractions works. Other than stating the rule about 
inverting the second fraction and multiplying to get the correct 
answer, I could not give a concrete example to help them understand 
the concept. For example, in the simple problem of 1/2 divided by 1/2, 
the correct answer is 1. However, I was unable to explain why division 
using whole numbers results in a smaller amount, but dividing by 
fractions produces an answer greater than what you started with. I 
hope I am clear. The kids left a few moments ago and my brain still 
feels like mush.

Thanks for any assistance you can provide.

Sincerely,
Lee Werling

Date: 08/11/99 at 19:42:54
From: Doctor Twe
Subject: Re: Tutoring division of fractions

Hi Lee!

You're right, dividing fractions is confusing and seems 
counterintuitive. The logic, of course, is that division is the 
inverse function of multiplication. So if 1 * 1/2 = 1/2, then 1/2 / 
1/2 = 1. But that is distinctly unsatisfying. The "real world" doesn't 
provide much help, either. There simply aren't many examples in the 
real world of dividing by a fraction. When we visualize division, we 
picture splitting something into more than one part - not less than 
one part.

We can find a few examples of "undoing multiplication" if we search 
hard enough. For example, if Tony Gwynn has a batting average of .400 
(2/5 in fractional form), and he wants to get 80 more hits this 
season, how many at-bats will he have to get? To solve this, we divide 
the hits needed by the rate of getting hits (batting average):

      80     80   5
     ---  =  -- * -  =  200 AB needed.
     2/5      1   2

Similar examples can come from the world of finance. What principle 
must I have in the bank if a 5% (1/20) interest rate earns me $.47 
(47/100)? To solve, we divide the interest amount by the interest 
rate:

     47/100      47   20     47
     ------  =  --- * --  =  --  =  $9.40.
      1/20      100    1      5

If I come up with any better or more satisfying examples, I'll let you 
know.

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

Date: 08/21/99 at 16:27:07
From: Doctor Twe
Subject: Re: Tutoring division of fractions

Hi Lee!

I came up with a few more (and I hope better) examples. The first one 
is nice because it correlates directly with what we do with integers. 
Here they are:

Integer example:

I went to a dairy farm and bought a 10-gallon canister of milk. The 
canister won't fit in my refrigerator, so I want to pour it into 
several 2-gallon jugs. How many jugs do I need?

Solution:

     10
     --  =  5 jugs needed.
      2

Fraction example:

I went to the store and bought 1/2 gallon of milk. The container won't 
fit in my refrigerator (I have a "mini-fridge"), so I want to pour it 
into several 1/8-gallon (one pint) containers. How many containers do 
I need?

Solution:

     1/2
     ---  =  4 containers needed.
     1/8

In both cases, we divide the total quantity of milk by the capacity of 
the containers. This shows that dividing a fraction by a smaller 
fraction produces a value larger than one (you need more than one of 
the smaller containers).

This also demonstrates an alternative way to solve dividing fractions. 
The "real world" problem can be solved using integers by converting 
the quantities to pints. 1/2 gallon = 4 pints, 1/8 gallon = 1 pint. 
Then 4/1 = 4 containers needed. The equivalent mathematical operation 
is called "eliminating the fraction," and is accomplished by 
multiplying both the dividend and divisor by a number that will 
eliminate the denominators. The most efficient value to use is the 
Least Common Multiple (LCM) of the denominators of the two fractions - 
in this example 8.

     8   1/2     8*1/2     4 pints
     - * ---  =  -----  =  -------  =  4 containers.
     8   1/8     8*1/8     1 pint


A second example:
(A) How many 1/2-hour 'Simpsons' episodes can you watch in 1/2 hour?  
(B) How many 1/4-hour 'Rugrats' episodes can you watch in 1/2 hour?

Solution:
(A)
     1/2     1   2
     ---  =  - * -  =  1 'Simpsons' episode.
     1/2     2   1

and

(B)
     1/2     1   4
     ---  =  - * -  =  2 'Rugrats' episodes.
     1/4     2   1

Here again, we can solve the problem by converting to a smaller unit 
(minutes). 1/2 hour = 30 minutes, 1/4 hour = 15 minutes. So 30/30 = 1 
'Simpsons' episode, and 30/15 = 2 'Rugrats' episodes. Eliminating the 
fraction can be accomplished by multiplying both the dividend and 
divisor by 30. (Note that 30 is _not_ the LCM, but it works because it 
is a common multiple of 2 and 4).

(A)
     60   1/2     60*1/2     30 min.
     -- * ---  =  ------  =  -------  =  1 'Simpsons' episode.
     60   1/2     60*1/2     30 min.

(B)
     60   1/2     60*1/2     30 min.
     -- * ---  =  ------  =  -------  =  2 'Rugrats' episodes.
     60   1/4     60*1/4     15 min.


A final thought:

We eliminate the fraction in the "real world" situations by converting 
to a smaller unit (pints instead of gallons, or minutes instead of 
hours). We can think of the mathematical method as converting to a 
smaller base unit. We are counting in 1/8ths (or /60ths, etc.) instead 
of 1's.

You might also like to read the Dr. Math FAQ on dividing fractions:

 http://mathforum.org//dr.math/faq/faq.divide.fractions.html   

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