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### Girls and Coins

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Date: 01/29/2001 at 21:17:25
From: shana
Subject: Coin combinations

Five girls stopped for a soda on the way home from school. The girls
were amazed to find that each of them had exactly six coins. Each of
them had at least one of four types of coins (penny, nickel, dime, and
quarter).

1. Betty had one more nickel than Alice, but Alice had 5 cents more
in all than Betty.

2. Donna had one more nickel than Betty, but Betty had 5 cents more
in all than Donna.

3. Eve had one more nickel than Carol, but Carol had 5 cents more
in all than Eve.

4. Carol had one more penny than Donna, she had 1 cent more than
Donna, and she also had 1 cent more in all than Donna.

How much money did each girl have? Make a chart of the possible
combinations.
```

```
Date: 01/30/2001 at 13:11:34
From: Doctor Keith
Subject: Re: Coin combinations

Hi Shana,

When you have a problem like this, with lots of information, a chart
is the only way to go.  Luckily our chart does not have to include
four of the six coins because we already know what they are.  We only
need to worry about the two we don't know. Let's make a chart:

| Coin 1 | Coin 2 | Notes
------+--------+--------+-------
Alice |        |        |
------+--------+--------+-------
Betty |        |        |
------+--------+--------+-------
Carol |        |        |
------+--------+--------+-------
Donna |        |        |
------+--------+--------+-------
Eve   |        |        |

Now we have to go through the clues step by step.  We will start with
clue 1.

> 1. Betty had one more nickel than Alice, but Alice had 5 cents more
in all than Betty.

We can tell that Betty has at least one nickle (she may have more but
she at least has one). We also see that Alice has 5 cents more than
Betty, so we hold onto this information for later. We can mark this in
our chart:

| Coin 1 | Coin 2 | Notes
------+--------+--------+-------
Alice |        |        | 5 cents more than Betty
------+--------+--------+-------
Betty | Nickle |        |
------+--------+--------+-------
Carol |        |        |
------+--------+--------+-------
Donna |        |        |
------+--------+--------+-------
Eve   |        |        |

We now move on to hint 2:

> 2. Donna had one more nickel than Betty, but Betty had 5 cents more
in all than Donna.

This one gives us a lot of information. For Donna to have a nickle
more than Betty, Donna must have two and Betty must have one.  Donna
has a total of 10 centsand Betty has 5 cents more, so Betty has 15
cents.  Thus Betty must have a dime also, since there is no other coin
you can add to a nickle to get 15 cents.

| Coin 1 | Coin 2 | Notes
------+--------+--------+-------
Alice |        |        | 5 cents more than Betty
------+--------+--------+-------
Betty | Nickle |  Dime  |
------+--------+--------+-------
Carol |        |        |
------+--------+--------+-------
Donna | Nickle | Nickle |
------+--------+--------+-------
Eve   |        |        |

At this point I am going to leave filling the chart in to you.  I will
ask you some questions to think about which should lead you to the

Now that we know how much Betty has, can you tell me how much Alice
has?  To get that amount what are the only two coins she could have?

Now look at hint 3:

> 3. Eve had one more nickel than Carol, but Carol had 5 cents more
in all than Eve.

and compare with hint 1:

> 1. Betty had one more nickel than Alice, but Alice had 5 cents more
in all than Betty.

How are these similar?  How should you fill in the chart?

Finally you need to look at hint 4:

> 4. Carol had one more penny than Donna, she had 1 cent more than
Donna, and she also had 1 cent more in all than Donna.

What is the total of Donna's two coins? Since Carol has 1 cent more,
how much does Carol have? What must the coins be that Carol has to
total this?

Now that you have Carol's total, you can go back to hint 3 and find
out what Eve has, since Carol has 5 cents more than Eve.

Thanks for writing in. If you have any questions on what we did or
other problems in the future, write back and we will be glad to help.

Best wishes,

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