Dividing by a Fraction

Date: 06/29/99 at 14:22:29
From: Dorothy Carletta
Subject: Division of fractions producing a larger number

When you divide, your answer should be smaller. Twenty-four divided by 
six is four. Four is lower than twenty-four. 

When you divide fractions, like 1/4 divided by 1/4, the answer is a 
larger number than 1/4. Why is this? I try telling myself that 
fractions are less than 1 and therefore the math may be different. One 
strategy we use in teaching children is that their answer must make 
sense. Is your answer smaller when you divide? I can't answer this.

Date: 06/29/99 at 16:57:28
From: Doctor Peterson
Subject: Re: Division of fractions producing a larger number

Hi, Dorothy.

Yes, it's important for our math to make sense. If our number sense 
disagrees with the answer, either the answer or our sense has to be 
fixed. If we leave them both as they are, we'll be in trouble. In this 
case, what seems natural and obvious when we're used to working with 
whole numbers has to be modified when we introduce fractions.

Let's think about why dividing by a whole number produces a smaller 
result. Dividing 24 by 4 means I want to find out how many 4's it 
takes to make 24. If I have 24 sticks

     ||||||||||||||||||||||||
                24

I can divide 24 by 4 by counting out 6 groups of 4 sticks:

     ||||  ||||  ||||  ||||  ||||  ||||
       1     2     3     4     5     6

Since each group contains more than one stick, there are fewer groups 
than sticks.


Now let's divide 6 by 1/4. That means I want to find out how many 
1/4's it takes to make 6. If I have 6 sticks

     ||||||
        6

and I break each of them into 4 pieces:

     ||||  ||||  ||||  ||||  ||||  ||||  
       1     2     3     4     5     6

I find that there are 24 quarter-sticks. Since each stick has been 
turned into 4 pieces, there are more pieces than sticks - the result 
of the division is greater than the original number of sticks. (By the 
way, this also helps to explain why dividing by a fraction is the same 
as multiplying by the reciprocal.)

You're exactly right that when you divide by a number less than one, 
things turn upside down, and the quotient is now larger than the 
dividend. In fact, you're multiplying by the reciprocal, which is a 
number greater than one, and you therefore increase the number. 
Similarly, multiplying by a number greater than one increases a 
number, and multiplying by a number less than one decreases it:

     24 / 4 = 6   smaller       6 * 4 = 24  bigger

    6 / 1/4 = 24  bigger     24 * 1/4 = 6   smaller

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