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### Dividing by a Fraction

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

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
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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/
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Associated Topics:
Elementary Division
Elementary Fractions

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