Complicated Age Problem

Date: 02/04/2002 at 20:04:36
From: Ann Butler
Subject: Mathematics trivia question

Eight years ago Mary was half as old as Jane will be when Jane is one 
year older than Tim will be at that time when Mary will be five times 
as old as Tim will be two years from now. Ten years from now Tim will 
be twice as old as Jane was when Mary was nine times as old as Tim.  
When Tim was 1 year old, Mary was 3 years older than Tim will be when 
Jane is three times as old as Mary was 6 years before the time when 
Jane was 1/2 as old as Tim will be when Mary will be 12 years older 
than Mary was when Jane was 1/3 as old as Tim will be when Mary will 
be 3 times as old as she was when Jane was born.

Date: 02/05/2002 at 19:32:10
From: Doctor Jeremiah
Subject: Re: Insanely complicated word problem

Hi Ann,

Yes, it's insanely complicated, but it is still just a word problem.  
That means we can solve it using the same techniques that we use on 
any word problem.

Let's start by making some shorthand names for things:

Let m = Mary's current age
Let t = Tim's current age
Let j = Jane's current age

Now let's look at the first sentence and see if we can break it down 
into something that looks like math.

Eight years ago Mary was half as old as Jane will be when Jane is one 
year older than Tim will be at that time when Mary will be five times 
as old as Tim will be two years from now.  

Well, "as Tim will be two years from now" means t+2 and "five times as 
old as as Tim will be two years from now" must mean 5(t+2) so we can 
replace that part of the sentence with 5(t+2):

Eight years ago Mary was half as old as Jane will be when Jane is one  
year older than Tim will be at that time when Mary will be 5(t+2)

The end of the sentence says "when Mary will be 5(t+2)." So that is 
some time in the future. But how far in the future? Well since 5(t+2) 
is Mary's future age, and m is Mary's current age, 5(t+2) - m is how
many years from now this will happen. So the phrase currently says 
"than Tim will be at that time when ,Mary will be 5(t+2)"  This can be 
rewritten as "than Tim will be 5(t+2) - m years from now."

Eight years ago Mary was half as old as Jane will be when Jane is one 
year older than Tim will be 5(t+2) - m years from now.

So how old will Tim be 5(t+2) - m years from now? He will be his 
current age plus 5(t+2) - m years or t + (5(t+2) - m) years old, so we 
can replace this phrase with the words "when Tim is t + 5(t+2) - m 
years old."

Eight years ago Mary was half as old as Jane will be when Jane is one 
year older than when Tim is t + 5(t+2) - m years old.

So when Jane is 1 year older than Tim's future age, she will be: 
1 + t + 5(t+2) - m years old, so we make ,that change to get:

Eight years ago Mary was half as old as Jane will be when Jane is 
1 + t + 5(t+2) - m years old.

"as Jane will be when Jane is 1 + t + 5(t+2) - m years old" is 
redundant and really means "1 + t + 5(t+2) - m years old," so let's 
make that change:

Eight years ago Mary was half as old as 1 + t + 5(t+2) - m years old.

Now it's pretty simple. "Eight years ago Mary" is just m-8, and "half 
as old as 1 + t + 5(t+2) - m" is "(1 + t + 5(t+2) - m)/2" - which 
gives us:

                  m-8 = (1 + t + 5(t+2) - m)/2

So let's simplify this a bit:

                  m-8 = (1 + t + 5(t+2) - m)/2
                  m-8 = (1 + t + 5t + 10 - m)/2
                  m-8 = (6t + 11 - m)/2
               2(m-8) = 2(6t + 11 - m)/2
              2m - 16 = 6t + 11 - m
          2m - 16 + m = 6t + 11 - m + m
              3m - 16 = 6t + 11
         3m - 16 + 16 = 6t + 11 + 16
                   3m = 6t + 27
               3m / 3 = (6t + 27) / 3
                    m = 6t/3 + 27/3
                    m = 2t + 9

So Mary is currently twice Tim's current age plus nine years. Let's 
test that just to be sure. Let's pick a number for Tim's age. It 
doesn't matter what we pick because it's just for testing the result.
It won't be the right age but for this test it doesn't matter. Let's 
say Tim is currently 1 year old. That means Mary would be 2(1) + 9 = 
11 years old. Does that work out in the original sentence?

Eight years ago Mary was half as old as Jane will be when Jane is one  
year older than Tim will be at that time when Mary will be five times 
as old as Tim will be two years from now.  

"Tim will be two years from now" is 3, because we are testing Tim = 1.

Eight years ago Mary was half as old as Jane will be when Jane is one  
year older than Tim will be at that time when Mary will be five times 
as old as 3 years old.  

"five times as old as 3 years old" means 15 years old.

Eight years ago Mary was half as old as Jane will be when Jane is one 
year older than Tim will be at that time when Mary will be 15.

We are testing with Mary currently 11, so when she is 25 that will be 
4 years from now.

Eight years ago Mary was half as old as Jane will be when Jane is one 
year older than Tim will be 4 years from now.

Since we are testing that Tim is 1, 4 years from now he will be 5.

Eight years ago Mary was half as old as Jane will be when Jane is one 
year older than when Tim is 5.

"one year older than when Tim is 5" means 6, so:

Eight years ago Mary was half as old as Jane will be when Jane is 6.

"as Jane will be when Jane is 6" is redundant and
means "as 6":

Eight years ago Mary was half as old as 6 years old.

"half as old as 6 years old" is 3 years old, so:

Eight years ago Mary was 3 years old.

Since we are testing with Mary being 11, this is a true statement and 
that means our equation m = 2t + 9 is correct.

Now we can go on to the other sentences and make equations out of 
them. After that we just do the algebra to solve the equations and we 
are done.

Here's the next sentence:

Ten years from now Tim will be twice as old as Jane was when Mary was 
nine times as old as Tim.

"nine times as old as Tim." means 9t, so if we do that replacement:

Ten years from now Tim will be twice as old as Jane was when Mary was 
9t.

Mary is currently m, so how many years ago was it when Mary was 9t? It 
was the difference between the current age and the past age: m - 9t. 
So "when Mary was 9t" really means "m - 9t years ago."

Ten years from now Tim will be twice as old as Jane was m - 9t years 
ago.

"as Jane was m - 9t years ago" is simply j - (m - 9t).

Ten years from now Tim will be twice j - (m - 9t).

Again, this is pretty simple. "twice as j - (m - 9t)" is just 
2(j - (m-9t)) and "Ten years from now Tim" is t+10, so we end up with:

               t + 10 = 2(j - (m-9t))

Which we can simplify:

               t + 10 = 2(j - (m-9t))
               t + 10 = 2j - 2(m-9t)
               t + 10 = 2j - 2m + 18t
           t + 10 - t = 2j - 2m + 18t - t
                   10 = 2j - 2m + 17t

But remember that m = 2t + 9 so we can change the m:

                   10 = 2j - 2(2t + 9) + 17t
                   10 = 2j - 4t - 18 + 17t
                   10 = 2j - 18 + 13t
              10 + 18 = 2j - 18 + 13t + 18
                   28 = 2j + 13t

In the last test we pretended that Tim was 1 and that Mary was 11, so 
let's continue to pretend that. Then because of 28 = 2j + 13t Jane 
would be 7.5

If you test these numbers they do actually work.

I know it seems strange that one of the ages would be a fraction, but 
remember these aren't the real ages. They are just ages we are using 
to test our results.

If you do the same thing to the last sentence, you should be able to 
nail down the age of one of the people. Then you can use all the 
equations to figure out everybody's age. But let me know if you get 
stuck.

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