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 pocket? ***************** 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. Thanks.
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 and 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 coins. 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 95 90 85 . . . 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
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