Finding Ages: Tot and Teen

Date: 12/01/2002 at 16:30:53
From: Joanna Sumber
Subject: Word problem

Question: Tot is now half as old as Teenager was when Tot was a third 
as old as Teenager is now.

Assume:
1) The age is a whole number.
2) Teenager is a teenager.

I do not understand exactly what the question is asking. I have 
discussed it with many of my peers and they are also unsure of the 
intent of the question. Please help.

Date: 12/02/2002 at 16:41:26
From: Doctor Ian
Subject: Re: Word problem

Hi Joanna,

What makes this tricky is that you're dealing with ages at two
different times, which means that in addition to representing their
ages now, you have to be able to represent their ages at an earlier
time as well. To do that, you need a third variable. 

Let's say that Teenager is now T years old, and Tot is now t years
old. Some number of years ago - call it y - Tot was 1/3 of Teenager's
current age:

     t - y = (1/3)T

And right now, Tot is half as old as Teenager was then:

         t = (1/2)(T - y)

So now we have two equations with three unknowns, which means that
there are infinitely many real solutions, which seems like bad news.

However, the problem specifies that all three values (t, T, and y)
must be integers. So that makes a unique solution possible. 

One way to find the solution would be to try each possible value for
T: 13, 14, 15, 16, 17, 18, and 19. Choose one of the values, 
substitute it for T, and you'll end up with two equations in two
unknowns, which you can solve for t and y. If you get integers for
both, you've found a solution. 

That seems like a lot of work, doesn't it? But note that if t and y 
are integers, then (t - y) must be an integer, which means that (1/3)T
must be an integer. So T must be divisible by 3. That eliminates most 
of the possibilities. 

Can you take it from here? 

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