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Different Combinations of Coins

Date: 2/18/96 at 17:32:11
From: "Susan L. Riglin"
Subject: Help with Problem

Hello Dr. Math:

Here is a problem my son brought home from 6th grade.  Usually 
the problems are much harder and my husband (a college graduate) 
would get together with another father (also a college graduate) 
to try to solve them.  They spend a lot of time solving the 
problems and then try to explain them to the kids.  We thought 
we would give your service a try for help.  
Francisco had some change in his pocket.  He gave his friends, 
Mike and Kevin, these three clues to see if they could guess how 
much he had:

* The coins equaled exactly one dollar.
* He had no more than 100 coins.
* He had at least one coin.

What combination of coins could Francisco have had in his 
We started doing the tedious task by doing the following, but 
decided that there must be a formula or something to help 
instead of doing what we started below.

Example of what we started:
100 pennies
95 pennies, 5 nickel
90 pennies, 1 dime
90 pennies, 2 nickels
85 pennies, 1 dime, 2 nickel
80 pennies, 4 nickels

We appreciate any help you can give.


Date: 3/17/96 at 3:34:45
From: Doctor Jodi
Subject: Re: Help with Problem

Hi Susan!
Thanks for your question. 

There are two formulae that must be satisfied:

1 < number of coins < 100

100 cents = 1*silver_dollars + 50 * half_dollars + 25 * quarters 
+ 10 * dimes + 5 * nickels + 1 * pennies.

There are a LOT of solutions to this problem.  I would first 
make a list of the maximum number of each sort of coin:

1 silver dollar
2 half dollars
4 quarters
10 dimes
20 nickels 
100 pennies 

Here we have 6 different solutions, using the maximum number of 
each coin.  Now we can go through other solutions using these 

silver dollar - this is the only solution with this coin, since 
all by itself this makes a dollar.

half dollar - could use one coin; how many ways can you make up 
the other 50 cents:  2 quarters, 5 dimes, 10 nickels, 50 
pennies, 1 quarter and 2 dimes and 1 nickel, 1 quarter and 1 
dime and two nickels, 1 quarter and 25 pennies, etc.

With the pennies, you should find that any multiple of 5 will 
work, since you can add nickels, dimes, etc. (since you can't 
have 97 pennies - this doesn't make a dollar, and you can't make 
a dollar out of just pennies unless you have 100)

so that means that you can use
pennies with combinations of other coins.

This process goes on.... I can't think of any way to shorten it, 
but please let us know if you find one.  

-Doctor Jodi,  The Math Forum

Associated Topics:
Middle School Word Problems

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