Associated Topics || Dr. Math Home || Search Dr. Math

### Is Division Sharing?

```
Date: 09/08/2001 at 18:58:00
From: Emily
Subject: Fraction

Hi, Dr Math,

Why when I do a fraction division (e.g. 1/2 divided by 1/2 = 1) is the
answer bigger than the first fraction in question (i.e. 1/2)?
Shouldn't division mean sharing, so logically the answer is smaller
than the first fraction?

Thanks, Emily
```

```
Date: 09/09/2001 at 00:15:48
From: Doctor Ian
Subject: Re: Fraction

Hi Emily,

It's not strictly true that 'division means sharing'. In fact,
division is just another way of looking at multiplication. For
example, all of the following are just different ways of saying the
same thing:

3 times 4 equals 12.
4 times 3 equals 12.
12 divided by 3 equals 4.
12 divided by 4 equals 3.

It's sort of like looking at this picture,

/\
/  \
/    \
+------+
|      |
|      |
+------+

and knowing that both of the following descriptions

The triangle is above the square.

The square is below the triangle.

are equally true, since they are really just different ways of saying
the same thing.

In other words, whenever it is true that

a * b = c

it must also be true that

c / a = b

and

c / b = a

(unless either a or b is zero, in which case all bets are off).
That's what we _mean_ by division.

Now, let's think about what this means for division by 1/2.  If

c / (1/2) = a

then it must be true that

c = a * (1/2)

which means that a must be _larger_ than c.

Thinking about division as 'sharing' is one way that teachers try to
make the concept simpler for students to understand. But it's
important to realize that these simplifications usually lead you in
the wrong direction once you start looking at things more carefully.

Does this help?  Write back if you'd like to talk about this some
more, or if you have any other questions.

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Elementary Division
Elementary Fractions

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search