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### Dividing Fractions: Why Invert?

```
Date: 01/24/2001 at 10:15:44
From: MOHD. ZULASYRAF B. MOHD. ZAKI
Subject: Dividing fractions

Why do we have to invert the fraction in dividing fractions?
```

```
Date: 01/24/2001 at 11:41:09
From: Doctor Greenie
Subject: Re: Dividing fractions

Hi, Mohd -

Let's first think about the division operation.  When I see the
division problem

24
---- = ?
4

I can think of it as asking "how many 4's are there in 24?"

So if I have the division problem

6
-------
1
---
2

I can think of it as asking "how many halves are there in 6?"  The way
I answer that question is to say that it takes 2 halves to make a
whole, so in 6 wholes there must be 6*2 = 12 halves. In this case I
have multiplied 6 by the reciprocal of (1/2), which is (2/1) = 2.

The preceding example was easy to understand, because 1/2 is a very
simple fraction. When the fractions are more complicated, like

3
---
4
-------
2
---
7

it is very hard to see what is going on. So let's look at the
mathematical reason for inverting when dividing fractions.

In any fraction

a
---
b

where a and b are both non-zero, it is true that

a     b
--- * --- = 1
b     a

So if I need to perform the division problem

3
---
4
-------
2
---
7

I simplify the expression by multiplying both numerator and
denominator by 7/2, so that the denominator becomes 1:

3         7          3     7         3     7
---       ---        --- * ---       --- * ---
4         2          4     2         4     2
------- * -------  =  -----------  =
2         7             1
---       ---
7         2

So you can see that the original problem, 3/4 divided by 2/7, is the
same as the multiplication problem 3/4 times 7/2.

- Doctor Greenie, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Elementary Fractions
Middle School Fractions

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