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### Flipping and Switching Fractions

```
Date: 01/18/2002 at 11:39:19
From: Russell Marks
Subject: Solving n in a fraction

I need the equations for solving the following problems:

n/2 = 5/10

1/n = 5/10

1/2 = n/10

1/2 = 5/n

```

```
Date: 01/18/2002 at 12:47:17
From: Doctor Ian
Subject: Re: Solving n in a fraction

Hi Russell,

The first thing to notice is that there are really only TWO cases that

?   b
1)  - = -           We don't know the numerator on the left.
a   c

a   b
2)  - = -           We don't know the denominator on the left.
?   c

Why?  Because if you have the variable on the other side of the
equation, you can always just switch sides.  That is, if

something = something_else

then it's also true that

something_else = something

The second thing to notice is that there is really only ONE case that
you have to worry about, because if

something = something_else

then it is also true that

1/something = 1/something_else

(so long as neither of them is zero).  So if

a/? = b/c

is true, then so is

1/(a/?) = 1/(b/c)

1*(?/a) = 1*(c/b)                Remember, to divide by a fraction,
you multiply by its inverse.
?/a = c/b

Let's see how we can apply that to your equations:

1)  n/2 = 5/10

We already like this one. We don't need to change it.

2) 1/n = 5/10

Flip both sides to get n/1 = 10/5.

3) 1/2 = n/10

Switch sides to get n/10 = 1/2.

4) 1/2 = 5/n

Switch sides to get 5/n = 1/2.  Then flip to get n/5 = 2/1.

Okay, so now we know that we can always end up with something like

?/a = b/c

What can we do with that?  Well, we can think back to when we first
learned about division. The way division is _defined_ is that

whenever K = MN,  it's also true that   K/M = N   and K/N = M

(again, so long as neither M nor N is zero). Don't just take my word
for this - check out a few examples:

12 = 3 * 4       so 12/3 = 4     and 12/4 = 3

20 = 4 * 5       so 20/4 = 5     and 20/5 = 4

15 = 2 * 7.5     so 15/2 = 7.5   and 15/7.5 = 2

and so on.

Well, we have something that looks like

K/M = N

right?  Let's match up the pieces:

K
|
?/a = b/c
|   \_/
|    |
M    N

This means that, by definition,

K   M
|   |
? = a(b/c)
\_/
|
N

So once you have something like

n/3 = 10/6

you can use the definition of division to write

n = 3*(10/6)

Let's look at an example of the toughest case:

12/4 = 18/?

18/? = 12/4            Switch sides.

?/18 = 4/12            Flip.

Does this help?

any other questions.

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

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