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### Math Age Word Problem

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

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