Normalization

Date: 08/01/2001 at 13:12:00
From: michael robinson
Subject: Algebra

How do I figure out this question?

90 + 70 + 88 + 94 + x / 5 = 85

I can get it to 3.42 / 5 = 85

but then I am stuck.

Date: 08/01/2001 at 16:22:11
From: Doctor Ian
Subject: Re: Algebra

Hi Michael,

It looks as if you meant to write

  90 + 70 + 88 + 94 + x
  --------------------- = 85
            5

In other words, you want to find x such that the average is 85.  

If you add up all the numbers on the left, you get

   342 + x
   ------- = 85
      5

which is sort of like what you got, except it looks as if you forgot 
about the x. (Oops!) From here, you can 

  1. multiply both sides of the equation by 5, and then
  2. subtract 342 from both sides of the equation

to get your answer. 

But there is an easier way to do this. Note that if the average is 85, 
then

  90 + 70 + 88 + 94 + x   85 + 85 + 85 + 85 + 85
  --------------------- = ----------------------
            5                       5

Now, if we subtract 85 from each term, we get

  5 + (-15) + 3 + 9 + x   0 + 0 + 0 + 0 + 0
  --------------------- = -----------------
            5                     5

Now we have smaller numbers, which are easier to deal with:

            17 - 15 + x 
            ----------- = 0
                 5


                  2 + x = 0

which tells you that x = -2... which, in our new context, means that x 
is two less than 85, or 83. 

What's going on here?  It's called normalization. We could draw the
original figures as a bar graph:

                                               85
                                               v    
  90 |||||||||||||||||||||||||||||||||||||||||||||
  70 |||||||||||||||||||||||||||||||||||
  88 ||||||||||||||||||||||||||||||||||||||||||||
  94 |||||||||||||||||||||||||||||||||||||||||||||||
   x ?

We want to know what value to choose for x so that the average is 85.  
Now, what if we subtract 20 from everything? 

                                     65
                                     v    
  70 |||||||||||||||||||||||||||||||||||
  50 |||||||||||||||||||||||||
  68 ||||||||||||||||||||||||||||||||||
  74 |||||||||||||||||||||||||||||||||||||
   x ?

We get basically the same picture, but shifted to the right. Do you 
see why the average here will be 20 less than the average in the 
original picture? 

What I did in the 'easy' method was to subtract the average from each 
term, to get the difference from the average:

   5                 0|||||
 -15  |||||||||||||||0  
   3                 0|||
   9                 0|||||||||
   x  ?

Now it's easier to see that the other items add up to slightly more 
than zero, which means that x is going to have to be slightly less 
than zero. 

But when you just use numbers, what it means is that if you add up all 
the other normalized numbers, you know that x has to be whatever it 
would take to get that sum to be zero. Which is to say, it's the sum 
of the other numbers multiplied by -1!

Which way is 'easier' depends on which operations you're more 
comfortable with. I don't mind doing normalizations in my head, but I 
hate multiplying numbers bigger than about 12, because I tend to make 
silly mistakes (even when I use a calculator). Other people find 
normalizations difficult to understand, but they don't mind 
multiplying big numbers.  

There's always more than one way to solve a problem, and it's 
important to learn which methods work best for you. One nice thing 
about solving a problem in two different ways, though, is that if both 
methods give you the same result, you can be pretty confident that 
it's correct.

I hope this helps. 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/