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

### Explaining Fractions

```
Date: 11/06/2001 at 10:10:11
From: Ellen Roddy
Subject: Explaining fractions

Dr. Math,

I am a practicum student in a 8th grade math class. The teacher and
I don't know how to explain to the kids why you can do this manuever.

c = d/g

c/d = g

d/c = g

```

```
Date: 11/06/2001 at 11:11:20
From: Doctor Greenie
Subject: Re: Explaining fractions

Hello, Ellen -

You can't do that maneuver, as you have shown it. Your first and last
equations are equivalent; the middle one is different.

You can get from the first form to the last form with the following
middle step:

c = d/g

c/d = 1/g  [divide both sides of previous equation by d]

d/c = g  [take reciprocals of both sides of previous equation]

Compare this middle equation with the middle equation as you showed
it....

I hope this helps.  Write back if you have any further questions on
this.

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

```
Date: 11/08/2001 at 15:33:41
From: Doctor Ian
Subject: Re: Explaining fractions

Hi Ellen,

It may be easier to understand what's going on here by going back to
basics.  The following are three different ways of expressing the very
same fact:

1)  cg = d

2)   g = d/c

3)   c = d/g

For example,

1) 3*4 = 12

2)   4 = 12/3

3)   3 = 12/4

This is, in fact, how we _define_ 'division':

If ab = c, then c/a = b and c/b = a.

So this is _why_ you can transform equations (2) and (3) into each
other.

As for _how_ you go about transforming them, Dr.Greenie showed you one
way.  I like to get rid of denominators whenever I can, so I would
have done it somewhat differently:

c = d/g

cg = d       [Multiply both sides by g]

g = d/c     [Divide both sides by c]

In each case, we followed the basic rule of algebra, which is that you
can do anything (except divide by zero) to both sides of an equation
without changing the truth of the equation.  So that's another way of
justifying _why_ the transformation works.

To sum up, you can do this transformation because:

1) It follows directly from the definition of division.

2) There exists a sequence of multiplications and divisions
that leads from either equation to the other.

Students may differ on which explanation they find more convincing.

more, or if you have any other questions.

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Middle School Algebra
Middle School Division
Middle School 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