How Many Coins Did Each Originally Have?

Date: 10/22/2001 at 21:46:37
From: kayla
Subject: I need help!

I am totally stuck - I have no idea how to solve this problem:

Natalie has some nickels, Dirk has some dimes, and Quincy has some 
quarters.  Dirk has five more dimes than Quincy has quarters.  If 
Natalie gives Dirk a nickel, Dirk gives Quincy a dime, and Quincy 
gives Natalie a quarter, they will all have the same amount of 
money.  How many coins did each originally have?

Please help me

Date: 10/23/2001 at 00:41:58
From: Doctor Ian
Subject: Re: I need help!

Hi Kayla,
  
We know that Quincy has some number of quarters, but we don't know how 
many, so let's say he has Q quarters. Similarly, let's say that Dirk 
has D dimes, and Natalie has N nickels. 

In terms of cents, the amount of money that each person has is

  Quincy:  25 * Q

  Dirk:    10 * D

  Natalie:  5 * N

What else do we know?   We know that D is 5 more than Q, right?  So

  D = Q + 5

and in fact, this means that we can forget about D altogether:

  Quincy:  25 * Q

  Dirk:    10 * (Q + 5)

  Natalie:  5 * N

Now, the exchanges begin. Let's keep track of what happens. First, 
Natalie gives Dirk a nickel:

  Quincy:  25 * Q

  Dirk:    10 * (Q + 5) + 5

  Natalie:  5 * N - 5

Next, Dirk gives Quincy a dime:

  Quincy:  25 * Q + 10

  Dirk:    10 * (Q + 5) + 5 - 10 

  Natalie:  5 * N - 5

Finally, Quincy gives Natalie a quarter:

  Quincy:  25 * Q + 10 - 25

  Dirk:    10 * (Q + 5) + 5 - 10 

  Natalie:  5 * N - 5 + 25

Now they all have the same amount of money. Well, if Dirk and Quincy 
have the same amount of money, then

  Quincy has:            Dirk has:

  25 * Q + 10 - 25   =   10 * (Q + 5) + 5 - 10

If you solve this for Q, you'll know how many quarters Quincy started 
out with, which you can use to find out everything else. 

Does this help? 

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