Duelling Percentages

Date: 06/11/2002 at 06:05:03
From: Pete
Subject: Percentage question driving me mad! 

Hiya!

Can you tell me why, for instance, when you add 15% to 100, you get
115, but if you subtract 15%, you get 86.95...instead of 85?

Date: 06/11/2002 at 08:20:33
From: Doctor Ian
Subject: Re: Percentage question driving me mad! 

Hi Pete, 

Are you sure about that?  

  100 + (15% of 100) = 100 + (0.15 * 100) 

                     = 100 + 15

                     = 115


  100 - (15% of 100) = 100 - (0.15 * 100) 

                     = 100 - 15

                     = 85

I think what you're asking is why, when you reduce something by 
15%, increasing the reduced figure by the same percentage doesn't 
get you back to where you started, e.g., 

        85 + (15% of 85) < 100

  86.95 + (15% of 86.95) = 100

The clearest way to show why it works this way is to use symbols 
instead of concrete values.  Suppose we start with some amount, 
A, and we reduce it by some percentage, P.  Then the new value is

  A' = A(1 - P)

Now if we increase the reduced amount, by the same percentage, we 
get 

  A'' = A'(1 + P)

      = A(1 - P)(1 + P)

      = A(1 - P^2)

which is smaller than what we started with!

It can also be easier to grasp if you use extreme examples.  If 
you start with $100, and I take 99% of it, that leaves you with 
$1.  If I increase your dollar by 99 percent, will that get you 
anywhere near $100?  

And here is a third way to look at it.  Suppose we start with 
some amount, 

  +---+---+---+---+---+
  |   |   |   |   |   |
  +---+---+---+---+---+
  |   |   |   |   |   |
  +---+---+---+---+---+
  |   |   |   |   |   |
  +---+---+---+---+---+
  |   |   |   |   |   |
  +---+---+---+---+---+
  |   |   |   |   |   |
  +---+---+---+---+---+


and reduce it by some percentage (say, removing one item out of 
every 5):

  +---+---+---+---+---+
  |   |   |   |   | - |
  +---+---+---+---+---+
  |   |   |   |   | - |
  +---+---+---+---+---+
  |   |   |   |   | - |
  +---+---+---+---+---+
  |   |   |   |   | - |
  +---+---+---+---+---+
  |   |   |   |   | - |
  +---+---+---+---+---+

Now we want to increase it by the same percentage (i.e., adding 
one item for every 5):

  +---+---+---+---+---+
  |   |   |   |   | - |
  +---+---+---+---+---+
  |   |   |   |   | - |
  +---+---+---+---+---+
  |   |   |   |   | - |
  +---+---+---+---+---+
  |   |   |   |   | - |
  +---+---+---+---+---+
  |   |   |   |   | - |
  +---+---+---+---+---+
  | + | + | + | + | 
  +---+---+---+---+

The increase, being the SAME fraction of a SMALLER number, will 
be smaller than the reduction, so we end up replacing only SOME 
of the amount that was removed:

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

The difference between the increase and the reduction is the 
square of the percentage. 

Does this help? 

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

Date: 06/12/2002 at 05:23:02
From: Pete
Subject: Percentage question driving me mad! 

Hello Ian.

Thanks very much for replying yesterday to my question.

The problem lay in the fact that I was using a desktop calculator, 
and the -TAX button for some unknown reason does not give minus 10% 
of 100 as 90.

Any ideas why this is so, as it happens on all models?

Thanks....

Date: 06/12/2002 at 07:34:11
From: Doctor Ian
Subject: Re: Percentage question driving me mad! 

Hi Pete,

I've never seen a -TAX button on a calculator, but it sounds to me 
as though it might be assuming that you know the price WITH tax, 
and the tax rate, and you want to figure out the price WITHOUT 
tax. 

For example, if the price with tax is $100, and the tax is 10%, 
and the price without tax is P, then

  P(1 + 0.10) = $100

            P = $100 / 1.10

              = 90.91

In other words, a retail price of $90.91, with 10% tax added to 
it, comes to $100. 

Does that make sense? 

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