Dividing the Prize Money

Date: 05/24/2000 at 03:57:29
From: Mike Sawbridge
Subject: A problem that my students and I can't solve

Dear Dr. Math:

The problem below was found at the NRICH site:

   http://nrich.maths.org/mathsf/journalf/may00/prob1.html   

We can't find a solution. It seems like a tough one. Help us please.

   Rollerball - Problem

Last weekend Mrs. Shoe won a prize and she gave her winnings to her 
children in order. The first child received 100 pounds and one tenth 
of the remainder. The second child received 200 pounds and one tenth 
of the remainder. The third child received 300 pounds and one tenth of 
the remainder, and so on.

After sharing out her winnings in this way she found that she had 
divided the money equally amongst all her children. How many children 
were there?
 
Mike Sawbridge

Date: 05/24/2000 at 08:16:41
From: Doctor Jerry
Subject: Re: A problem that my students and I can't solve

Hi Mike,

Let c1 be the amount the first child received; from the problem:

     c1 = 100 + (p-100)/10

also,

     c2 = 200 + (p-c1-200)/10

Equations could be written for other children, but this turns out not 
to be necessary. The reason is that we are also told that p/n = c1,
p/n = c2, etc, where n is the number of children.

If you write out p/n = c1 you'll find:

       (10-n)p = 900n

Also, writing out p/n = c2 (and using c1 in the expression for c2) 
you'll find:

     (100-9n)p = 17100n

If you solve these two equations in n and p you'll find n = 9 and
p = 8100.

Nice problem.

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