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### How Many Coins Did Each Originally Have?

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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?

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

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