Weird Fraction

Date: 06/11/2003 at 02:00:55
From: Charlotte
Subject: Weird fraction

  9 x 11 + 18 x 22 + 27 x 33 + 36 x 44
---------------------------------------   = ?
 22 x 27 + 44 x 54 + 66 x 81 + 88 x 108

Could you tell me the shortest and easiest way of solving this 
problem?

Date: 06/11/2003 at 09:32:23
From: Doctor Ian
Subject: Re: Weird fraction

Hi Charlotte, 

You can't use cancellation directly, because you have sums instead of 
just products. However, you can use the distributive property to 
factor out things that could be canceled. For example, in the 
numerator,

  9*11 + 18*22 + 27*33 + 36*44

each term is divisible by 9, 

  9(1*11 + 2*22 + 3*33 + 4*44)

and each term is divisible by 11, 

  9*11(1*1 + 2*2 + 3*3 + 4*4)

Try doing something like this to the denominator, and see what 
happens. 

I hope this helps.  Write back if you'd like to talk more
about this, or anything else.

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