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Comparing Sums of Odd and Even Integers

Date: 03/31/2004 at 19:13:20
From: jules
Subject: total of odd & even numbers

Which is greater, the sum of all of the even numbers that are less 
than 100, or the sum of all of the odd numbers that are less than 100?
It takes too long to add them all up!

Date: 04/01/2004 at 13:52:26
From: Doctor Ian
Subject: Re: total of odd & even numbers

Hi Jules,

It would take a long time to add them all up, wouldn't it?  There is a
formula you can use to find such sums, but let's skip that and see if
we can be clever and just think about the question.

Let's look at a smaller version of the same problem, where we go up to
10 instead of 100.  We want to compare

  0 + 2 + 4 + 6 + 8  

  1 + 3 + 5 + 7 + 9

without actually adding them up.  What if we line the numbers up like

  0, 2, 4, 6, 8  
  1, 3, 5, 7, 9

In each case, the number on the bottom is larger than the one on top,
right?  And we have the same number of numbers in each list.  So if we
add the ones on the bottom, we have to come out with something larger
than if we add the ones on top, don't we? 

Here's another way to look at it.  Suppose we start even smaller, with
the numbers less than 2:

   Even: 0
   Odd:  1

The sum of the odds is larger, right?  Now let's add a couple more

   Even: 0, 2
   Odd:  1, 3

Since I added a larger number to the odds, the sum of the odds must
_still_ be larger, right?  And every time I do that, the same thing

   Even: 0, 2, 4
   Odd:  1, 3, 5           5 > 4

   Even: 0, 2, 4, 6
   Odd:  1, 3, 5, 7        7 > 6

   Even: 0, 2, 4, 6, 8
   Odd:  1, 3, 5, 7, 9     9 > 8

Does this help? 

- Doctor Ian, The Math Forum
Associated Topics:
Elementary Addition
Elementary Number Sense/About Numbers
Elementary Puzzles
Middle School Number Sense/About Numbers
Middle School Puzzles

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