Math Age Word Problem

Date: 03/20/2001 at 20:45:11
From: Rosalynd Nicholl
Subject: Math Age Word Problem

My mom and I searched your archives and even looked at your many 
examples.  We are still stuck! Mom spent all night but had no success.  
My fourth grade teacher gave us this problem.

Bob is as old as John will be when Bob is twice as old as John was 
when Bob's age was half the sum of their present ages. John is as old 
as Bob was when John was half the age he will be ten years from now.  
How old are John and Bob?

My teacher gave us the answer, but she asked us to figure out how 
they came up with it.  I cannot break down the sentences.

Bob is 40
John is 30

How can this BE?

Date: 03/21/2001 at 17:01:21
From: Doctor Peterson
Subject: Re: Math Age Word Problem

Hi, Rosalynd.

This is a challenging problem even for adults, and I can't imagine 
doing it without at least a basic knowledge of algebra. I'll interpret 
it for you and show you the algebraic approach, and maybe I'll think 
of a simpler way while I work.

We have two unknown ages, and also three unknown times ("when ...") in 
addition to the present. I'll call their current ages B and J, and the 
times +X, -Y, and -Z. Time +X, the first one mentioned, is in the 
future, X years from now, when everyone is X years older than they are 
now; times -Y and -Z are in the past, when everyone was Y or Z years 
younger, respectively. You'll see where those fit in in a moment.

Now we can clarify the problem:

  Bob is [now] as old as John will be [at time +X] when Bob is 
  twice as old as John was [at time -Y] when Bob's age was half 
  the sum of their present ages [now]. John is [now] as old as 
  Bob was [at time -Z] when John was half the age he will be ten 
  years from now [at time +10].

We can break it down into separate sentences to make it even clearer:

  Bob is now as old as John will be at time +X.
  At time +X, Bob will be twice as old as John was at time -Y.
  At time -Y, Bob's age was half the sum of their present ages [now].
  John is now as old as Bob was at time -Z.
  At time -Z, John was half the age he will be at time +10.

Finally, we can translate each new sentence into an equation:

  Bob is now as old as John will be at time +X.
    B = J+X
  At time +X, Bob will be twice as old as John was at time -Y.
    B+X = 2(J-Y)
  At time -Y, Bob's age was half the sum of their present ages [now].
    B-Y = (B+J)/2
  John is now as old as Bob was at time -Z.
    J = B-Z
  At time -Z, John was half the age he will be at time +10.
    J-Z = (J+10)/2

Now here comes the hard work. Don't try to follow everything I say; I 
promise to come back down to earth at the end, so skim until you see 
something that says I'm back with you. Your mom may or may not want to 
try to wade through it all; my main point is to show that it really is 
far beyond you!

All we've done so far is to figure out what everything means, and put 
it into algebraic terms. Now we have to do actual algebra, eliminating 
the unknowns we don't care about, until we can find J and B. First 
I'll simplify the equations:

           B = J + X
       B + X = 2J - 2Y
    2(B - Y) = B + J --> B - 2Y = J
           J = B - Z
    2(J - Z) = J + 10 --> J - 2Z = 10

The last equation gives me Z, which I can put into the fourth 
equation:

       Z = J/2 - 5

       J = B - (J/2 - 5)

    3J/2 = B + 5

      3J = 2B + 10

Now I can set that aside, and put Y from the third equation into the 
second:

       Y = (B - J)/2

   B + X = 2J - (B - J)

  2B + X = 3J

I can put X from this into the first equation:

       X = 3J - 2B

       B = J + (3J - 2B)

      3B = 4J

Finally, I can put J from this into the equation I set aside:

            J = 3B/4

       3*3B/4 = 2B + 10

    9B/4 - 2B = 10

          B/4 = 10

            B = 40

And

   J = 3*40/4 = 30

Whew! We're done, but you surely couldn't follow all that algebra. Is 
there any solution you could follow? Not that I can see. But what you 
can do - and perhaps what you were really expected to do - is just to 
check that the answer is right, and see that it really makes sense. 
Even that is hard if you haven't solved the problem so as to know the 
values of my X, Y, and Z. I'll just start that process, by looking at 
the (easier) second statement.

  John is [now] as old as Bob was [at time -Z] when
  John was half the age he will be ten years from now [at time +10].

How old will John be ten years from now? 30 + 10 = 40. What is half of 
that? 20. When was John that age? 30 - 20 = 10 years ago. (That's my 
Z.) How old was Bob then? 40 - 10 = 30. That's John's age now, so it 
works.

That's still not easy, but it's much more reasonable to expect you to 
do this than to actually solve the problem.

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