How Old are Ben and Kris?

Date: 10/17/1999 at 17:44:56
From: Emily
Subject: Age word problems

I have been given several word problems on age like: A man is 13 times 
as old as his son is. In 10 years he will be 3 times as old as his son 
will be. How old are they now?

The book does show how to solve that particular problem, but gives a 
new twist on the problems they solve!

Here's one. Please give me a step-by-step on how to solve:

Five years ago Ben was 2/3 as old as Kris. Ten years from now he will 
be 5/6 as old as Kris. How old are they now?

Thanks.

Date: 11/28/1999 at 10:21:29
From: Doctor Aileen
Subject: Re: Age word problems

Hi Emily,
    
One way to solve this problem is to write a system of equations with 
two variables. Let x be Kris' age now and y be Ben's age. Then x - 5 
is how old Kris was 5 years ago. Since Ben was two thirds of Kris' age 
5 years ago, then 

  2/3*(Kris' age five years ago) = Ben's age five years ago.

Therefore

     2/3*(x - 5) = y - 5

The problem also tells you that in 10 years from the present time 
Ben's age will be 5/6 Kris' age. In other words 

  5/6 * (Kris' age in 10 years) = Ben's age in 10 years or:

     5/6 * (x + 10) = y + 10

You need to find an x and y that will be true for both equations. To 
do this you need to solve for the variables one at a time. By 
subtracting the second equation from the first you can formulate a 
third equation that has only one variable.

                  2/3*(x-5)  = y - 5
              -   5/6*(x+10) = y + 10
     --------------------------------
     (2/3*(x-5) - 5/6*(x+10) = -15

The equations can be either subtracted or added, whatever works to 
eliminate one variable.

Now you can find the value of this variable. After doing so you can 
substitute it back into one of the original equations. Setting up the 
equations is the most important aspect of the problem. Once you have 
the proper equations it is simply a matter of solving. Also, if you 
are uncomfortable with the method of eliminating a variable, you can 
try graphing both equations. The intersection point represents the x 
and y values that satisfy both equations.

I hope this helps.

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

Date: 1/20/2002 at 13:01:58
From: Judy Ann Brown
Subject: Re: Age word problems

Here's the same problem done in two more ways:

Five years ago Ben was 2/3 as old as Kris. Ten years from now he will
be 5/6 as old as Kris. How old are they now?

We can set up a chart:

        5 years ago   now          in ten years
Ben     (2/3)x        (2/3)x + 5   (2/3)x + 15
Kris    x             x + 5        x + 15

From the problem we know that

   Ben's age in 10 years = 5/6 Kris' age in 10 years
   
   (2/3)x + 15 = 5/6(x + 15)

       4x + 90 = 5x + 75  (we multiplied both sides by 6 to get rid 
                           of the fractions)
            15 = x

If we know that Kris was 15 five years ago, he is now 20.
Ben was 2/3(15) or 10 five years ago, so he is now 15.
In ten years Ben will be 25 and Kris will be 30.


There is also an elementary guess-and-check approach to solving this 
problem that does not use variables.

Five years ago Kris had to be an age that was a multiple of 3. The 2/3 
for Ben's age tells me so. Let's make a chart and guess:

Let's start with 3, because that is the youngest Kris could have been 
5 years ago.

GUESS 1: Kris is 3

      5 years ago   now   in ten years
Ben   2             7     17  (more than 5/6 of 18, so 3 is too little)
Kris  3             8     18

GUESS 2: Kris is 30

      5 years ago   now   in ten years
Ben   20            25    35  (less than 5/6 of 45, so 30 is too much)
Kris  30            35    45

I can also tell that Kris' age in ten years must be a multiple of 6. 
But that's only going to happen if I start out with an odd number for 
his age, because I will have to add 15, and the only way to get an 
even number when one of the addends is odd is to have the second addend 
also odd.

GUESS 3: Kris is 21

      5 years ago   now   in ten years
Ben   14            19    29  (less than 5/6 of 36, so 21 is too much)
Kris  21            26    36

I know that I'm getting close in my guesswork, but I need a smaller 
number for Kris' age, and it must be an odd multiple of 3. I choose 15.

GUESS 4: Kris is 15

      5 years ago   now   in ten years
Ben   10            15    25  (this is exactly 5/6 of 30)
Kris  15            20    30

I know I have a correct solution. But what was the question? How old 
are they now...?  Ben is 15, and Kris is 20.

- Judy Ann Brown