Proportion or Analogy?

Date: 12/12/2001 at 19:01:16
From: John Aguilar
Subject: Not Sure

What does this problem mean?

   9:3::49: 

This is the first time I've seen this problem.

Date: 12/13/2001 at 10:52:40
From: Doctor Ian
Subject: Re: Not Sure

Hi John,

I think you're supposed to fill in the missing value, 

  9:3 :: 49:?

This is a standard form for analogies. Usually they don't involve 
numbers, though. For example, if I say

  Indiana Jones:Harrison Ford :: Dirty Harry:?

there is a relation between the first two items - that is, the 
character Indiana Jones is played in films by Harrison Ford. What you 
want to do is find the item that has the same relation to the third 
item - that is, the character Dirty Harry is played in films by whom?  
In this case, the answer would be 

  Indiana Jones:Harrison Ford :: Dirty Harry:Clint Eastwood

The thing about analogies with numbers is that it is usually possible 
to come up with more than one possible answer, and what you want to do 
is find the 'best' answer.  For example, we might note that in

  9:3 :: 49:?

one relation between the first two items is that 3 is 1/3 of 9.  So 
one possible answer is to replace '?' with 1/3 of 49:

  9:3 :: 49:(49/3)

But that's not a very satisfying answer. Another relation is that 3 is 
6 less than 9. So another possible answer is

  9:3 :: 49:43  

But again, this isn't a very satisfying answer. Can you find another 
relation between 9 and 3? Here are some clues:

  9:3 :: 4:2

  9:3 :: 16:4

  9:3 :: 25:5

Does this help?

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

Date: 12/13/2001 at 12:07:48
From: Doctor Peterson
Subject: Re: Not Sure

Hi, John.

This is an old notation for a "proportion," or an equation involving 
ratios;

    9:3 :: 49:?

means the same as

    9/3 = 49/x

You would solve it traditionally by cross-multiplication:

    9*x = 49*3

      x = 49*3/9 = 49/3 = 16 1/3

But I have to agree with Dr. Ian that the numbers in this case suggest 
that it may have been meant not as a proportion but as an analogy, and 
very likely the one he hinted at in the end. This notation for 
proportions is probably still used in some places or fields, and 
certainly in some older books that still exist, so it can be useful to 
recognize it; but I don't like to see it used in this more general way 
in a mathematical context. I believe the notation for analogies in 
non-mathematical logic was taken from the mathematical notation for 
proportion, because a proportion is an example of an analogy. It 
appears to have come full circle!

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