Faster Than Algebra

Date: 05/14/2002 at 10:30:59
From: Chris Floyd
Subject: algebra

My students are working on a competition problem and I can't for the 
life of me figure it out:  "The sides of a rose garden in the shape of 
a right triangle are in a ratio of 8:15:17.  If the perimeter is 60 
feet, what is the length of the shortest side?

Date: 05/14/2002 at 12:51:59
From: Doctor Peterson
Subject: Re: algebra

Hi, Chris.

If the sides were actually 8, 15, and 17, the perimeter would be 
8+15+17 = 40. You want the perimeter to be 60/40 of that (that is, 
3/2); so you just have to multiply each side by 3/2. The sides will 
be 12, 22.5, and 25.5 feet. But you only need to report the 12.

This approach, taking the simplest guess and seeing what it takes to 
change the answer into what you want, is an ancient one and often 
faster than algebra! The equivalent, using algebra, is to call the 
sides 8x, 15x, and 17x, and write an equation for the perimeter.

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

Date: 05/15/2002 at 09:49:47
From: Chris Floyd
Subject: algebra

Question submitted via WWW:
Thank you so much for your help with what seemed like a trivial 
problem; I just couldn't visualize it. I appreciate your time.