25 Coins in a Dollar
Date: 05/07/2001 at 20:05:37 From: Jessica Johnson Subject: Money Carlos has 25 U.S. coins whose total is $1.00. What are the coins? Money isn't my best math topic. I've tried a lot of different coins, but I'm either too high or too low. I just can't get the right amount.
Date: 05/08/2001 at 15:03:11 From: Doctor Rick Subject: Re: Money Hi, Jessica. It can be done with guess-and-check, if we pick our guesses wisely. For instance, let's start by picking a number of pennies. What if there is just one penny? If you try it a few times, you'll see that whatever the numbers of other coins (nickels, dimes, quarters or half dollars), the total value ends in a 1 or a 6. (It doesn't matter how many coins you have.) For instance, 1 quarter + 3 dimes + 4 nickels = 1*25 + 3*10 + 4*5 = 75 cents 75 cents + 1 penny = 76 cents There's a reason for this: all coins except pennies are multiples of 5 cents. Any amount you make by adding multiples of 5 is also a multiple of 5 (as 75 is in my example). Multiples of 5 end in either 0 or 5, so when you add 1 penny, you get a number that ends in 1 or 6. We can't make 100 cents if there is only one penny, because 100 doesn't end in 1 or 6. What numbers of pennies could we have? Start by supposing you have the largest number of pennies possible. How many non-pennies must there be, so that the total number of coins is 25? How much are the pennies worth? How much must the non-pennies be worth? Can you find this many non-pennies that are worth exactly this much? If not, try the next smaller number of pennies. You can keep going this way if you need to. You'll be trying a lot fewer possibilities than if we hadn't made these observations. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/
Date: 05/08/2001 at 15:11:15 From: Doctor Twe Subject: Re: money Hi - thanks for writing to Dr. Math. Actually, there are 5 different ways to make $1.00 using 25 coins. We know that the solution will involve pennies, because 25 coins that are worth $0.05 (a nickel) or more would have to be worth at least 25 * $0.05 = $1.25 which is more than a dollar. We also know that the number of pennies has to be a multiple of 5 (i.e. 5, 10, 15, ...) because otherwise we'd have a few cents left over. We know it can't be 25 pennies (that would be 25 coins right there, and only totals $0.25) or more. So that leaves four options for the pennies: 5 pennies, 10 pennies, 15 pennies, or 20 pennies. Let's consider each case: 5 PENNIES: That's 5 coins totaling $0.05. That means we need the other 20 coins to total $0.95. Since 20 nickels are worth $1.00 (and, of course, using larger coins would make the total even larger), we can't get $0.95 using 20 coins that aren't pennies. 10 PENNIES: That's 10 coins totaling $0.10. That means we need the other 15 coins to total $0.90. On the average, those coins must be worth: $0.90 / 15 = $0.06 Since this is more than a nickel, it might be possible. Can you come up with a combination of 15 coins (nickels, dimes, quarters or half- dollars only) worth $0.90? 15 PENNIES: That's 15 coins totaling $0.15. That means we need the other 10 coins to total $0.85. On the average, those coins must be worth: $0.85 / 10 = $0.085 (8.5 cents) Again, since this is more than a nickel, it might be possible. Can you come up with a combination of 10 coins (nickels, dimes, quarters or half-dollars only) worth $0.85? 20 PENNIES: That's 20 coins totaling $0.20. That means we need the other 5 coins to total $0.80. On the average, those coins must be worth: $0.80 / 5 = $0.16 Once more, it might be possible. Can you come up with a combination of 5 coins (nickels, dimes, quarters or half-dollars only) worth $0.80? A couple of hints: 1. There are two ways each to do it with 20 pennies and 15 pennies, and there is one way to do it with 10 pennies. 2. Try solving the 20-pennies combinations first. Since there are fewer coins left (only 5 instead of 10 or 15), they should be easier to find. You may find a system that you can then use to find the other solutions. 3. You can use similar reasoning to break these cases down into sub-cases. For example, for the 10-pennies solutions, we are looking for 5 coins worth $0.80. There could be either 0 or 1 half-dollar... There could be either 0, 1, 2, or 3 quarters... and so on. I hope this helps. If you have any more questions, write back. - Doctor TWE, The Math Forum http://mathforum.org/dr.math/
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