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

Thanks for your help.

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 
you have to worry about:

      ?   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.

       ? = 18(4/12)        The answer.  

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/