How Old Are John and Julia?

Date: 05/17/2001 at 22:36:16
From: Deepa
Subject: Age word problem

Julia is as old as John will be when Julia is twice as old as John was 
when Julia's age was half the sum of their present ages.

John is as old as Julia was when John was half the age he will be 10 
years from now.

How old are John and Julia?

Date: 05/18/2001 at 15:41:44
From: Doctor Jaffee
Subject: Re: Age word problem

Hi Deepa,

I had a lot of fun working on this problem. Here is what I did.

First, let x = John's present age and y = Julia's present age.

     x + 10 = John's age in 10 years

     x + 10
     ------ = 1/2 of John's age in 10 years
       2

Now, it would be helpful to know when John was (x + 10)/2. Well, I am 
56 years old and if I want to know when was I 20, I just subtract 
56 - 20 and find that I was 20, 36 years ago.
        x + 10
So, x - ------ will tell us how many years ago John was (x + 10)/2.
          2

How old was Julie then? Well, if I want to know how old my 42-year-old 
brother was 36 years ago, I just subtract 42 - 36 and learn that he 
was 6.
                                        x + 10
So, Julie's age must have been y - (x - -------). But, if that is what
                                          2
John's age is right now, we have:

                     x + 10
     x =   y - ( x - ------ )  
                       2

I went through this same reasoning process with the rest of the 
information in the problem and came up with another equation in two 
variables. Then I solved the system of equations.

Give it a try and if you want to check your answer, write back. If you 
are having difficulties, let me know what you have done so far and 
I'll help you out.

Good luck.

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

Date: 05/18/2001 at 16:01:05
From: Doctor Rick
Subject: Re: Age word problem

Hi, Deepa. I see that Dr. Jaffee answered you before I could, but I 
have a different approach, and you never know which might work better 
for you.

Here is how I tackle age problems like this, which involve more than 
one point in time. First, I define variables for the ages of each 
person NOW:

     x = Julia's age in years now
     y = John's age in years now

There are other things we don't know at the outset: we don't know how 
far in the future or past the other times are. I therefore define a 
variable for each time in the problem other than the present (the word 
"when" is usually a signal that you need another variable):

     r = number of years in the future when Julia is twice as old
     s = number of years in the past when Julia's age was half the sum
     t = number of years in the past when John was half the age

Now you can write equations, breaking the problem down clause by 
clause:

   Julia is as old as John will be ["r" years from now]:
   x = y + r

   ["r" years from now,] Julia is twice as old as John was ["s" years 
   ago]:
   x + r = 2(y - s)

   ["s" years ago,] Julia's age was half the sum of their present 
ages:
   x - s = (1/2)(x + y)

That takes care of the first sentence of the problem. The second 
sentence is easier, because it only involves two times: now, and "t" 
years ago. (There are really three times mentioned, but one is a known 
time in the future, 10 years from now, so we don't need to introduce a 
variable for that time.) I'll let you write two equations for that 
sentence.

Now you have 5 unknowns and 5 equations, so all that's left is to 
solve them!

It's possible, as Dr. Jaffee shows, to write just two equations in the 
two unknowns x and y right off the bat. I'm not embarrassed, though, 
to introduce as many variables as I need at the beginning. Those extra 
variables can be eliminated quickly enough once you start solving, and 
they give me confidence that I really understand what those convoluted 
sentences are saying! (We'd never talk like that in real life, would 
we?)

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

Date: 05/19/2001 at 00:28:51
From: Deepa Thirugnanam
Subject: Re: Age word problem

Hello Dr. Rick,

Thanks for coming down to my level to explain the age word problem 
(John and Julia...) I was really very confused. Now I am clear. I am 
very grateful to you and Dr. Jaffee.

Bye,
Deepa

Date: 05/19/2001 at 00:28:51
From: Deepa Thirugnanam
Subject: Re: Age word problem

Hello Dr. Jaffee,

It's Deepa. Thanks a lot for giving me a clear procedure to solve the 
problem.

I got the solution. Julia's age is 40 and John's age is 30. Is that 
right?

Bye,
Deepa

Date: 05/21/2001 at 16:17:17
From: Doctor Jaffee
Subject: Re: Age word problem

Hi Deepa,

Yes, your solution is completely accurate. I'm glad that I was able to 
help you. I noticed that Dr. Rick also sent you a very good response. 
I think I learned a little from his method, also.

Thanks for writing to Ask Dr. Math.

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