K-triple

Date: 9/15/96 at 14:10:46
From: Anonymous
Subject: K-triple

Let a "k-triple" be defined as (k/2, k, 3/2k) for any number k.
Which of the following is a k-triple?

  a. (0, 5, 10)
  b. (4 1/2, 5, 6 1/2)
  c. (25, 50, 75)
  d. (250, 500, 1000)
  e. (450, 500, 650)

I do not understand this question. Could you please explain it?

Date: 9/16/96 at 11:45:18
From: Doctor Mike
Subject: Re: K-triple

Hello,

This looks like an exercise to help you get used to abstract 
symbolic thought processes, which is a step or 2 beyond just 
numerical calculations.  The idea of a "k-triple" is a made-up one 
having no importance, except that it helps to get you to think about 
some important things. 

First of all, the idea of a triple here is just a grouping of three 
things in a particular order.  The fact that a variable "k" is used 
just means that not just one triple is involved, but perhaps many of 
them, all of which follow a particular pattern.

What pattern?  For one thing, the first one in the k-triple must be 
exactly half of the second one.  With this observation alone you can 
eliminate (a.) and (b.) and (e.).  Choices (c.) and (d.) both work SO 
FAR since 25 is half of 50 and 250 is half of 500, respectively.  

The pattern also requires that the last in the k-triple must be 3/2 
times the middle one. Is 3/2*(500) equal to 1000?  No, so (d.) does 
not follow the pattern. Is 3/2*(50) equal to 75?  Yes, so (c.) is the 
only one of the five that fully qualifies.  Clear?

(1,2,3) and (2222,4444,6666) and many more are also k-triples. 
  
  I hope this helps. 

-Doctor Mike,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/