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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

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/
```
Associated Topics:
Middle School Algebra
Middle School Word Problems

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