Making $5 Using 50 CoinsDate: 12/02/2005 at 15:41:47 From: Haley Subject: 50 coins = $5 How many ways can you make $5 with 50 coins, without dimes? My dad and I found 26, but my teacher says 37 is the most. My dad made an Excel page to help me but I'm stuck. It was for extra credit, but I would like to know the ones I missed. Date: 12/03/2005 at 01:58:24 From: Doctor Greenie Subject: Re: 50 coins = $5 Hi, Haley -- My first comment about this extra credit problem is that it is far too complex for a 12-year-old. Very few talented high school students would be able to solve this problem. And perhaps your teacher can't solve it correctly either--because I found 38 ways instead of the 37 he says is the most. Here is a table copied from an Excel spreadsheet showing my solutions. The first column is the solution number. The next five columns are, respectively, the numbers of pennies, nickels, dollar coins, half dollars, and quarters. The last two columns are the Excel checks to show that, for each of the solutions, the number of coins is 50 and the total value of the coins is $5.00 (500 cents). 1 40 2 3 1 4 50 500 2 40 2 2 4 2 50 500 3 40 2 1 7 0 50 500 4 35 3 2 0 10 50 500 5 35 3 1 3 8 50 500 6 35 3 0 6 6 50 500 7 30 4 0 2 14 50 500 8 35 8 2 4 1 50 500 9 30 9 2 0 9 50 500 10 30 9 1 3 7 50 500 11 30 9 0 6 5 50 500 12 25 10 0 2 13 50 500 13 30 14 3 1 2 50 500 14 30 14 2 4 0 50 500 15 25 15 2 0 8 50 500 16 25 15 1 3 6 50 500 17 25 15 0 6 4 50 500 18 20 16 0 2 12 50 500 19 25 20 3 1 1 50 500 20 20 21 2 0 7 50 500 21 20 21 1 3 5 50 500 22 20 21 0 6 3 50 500 23 15 22 0 2 11 50 500 24 20 26 3 1 0 50 500 25 15 27 2 0 6 50 500 26 15 27 1 3 4 50 500 27 15 27 0 6 2 50 500 28 10 28 0 2 10 50 500 29 10 33 2 0 5 50 500 30 10 33 1 3 3 50 500 31 10 33 0 6 1 50 500 32 5 34 0 2 9 50 500 33 5 39 2 0 4 50 500 34 5 39 1 3 2 50 500 35 5 39 0 6 0 50 500 36 0 40 0 2 8 50 500 37 0 45 2 0 3 50 500 38 0 45 1 3 1 50 500 I didn't solve the problem using an Excel spreadsheet; I used the spreadsheet to verify the solutions I found. The organization of the spreadsheet does, however, demonstrate the process I used to find the solutions. I will try to describe that process below. To me, the key to finding a method of solution to this problem is the fact that the dollar coins, half dollars, and quarters together can only make totals which are multiples of 25 cents. Since the desired total is a multiple of 25 cents, this means the total value of the pennies and nickels must also be a multiple of 25 cents. I decided, somewhat arbitrarily, to start my investigation with the largest possible number of pennies. We obviously couldn't use 50 pennies, so my first try was with 45 pennies. 45 pennies together with 1 nickel makes 50 cents. That means we have 4 coins left to make the remaining $4.50. But the largest coin we have is a dollar--so we can't make $4.50 with 4 coins. So next we try 40 pennies; and we need 2 nickels to make a total of 50 cents. In this case, we need to make the remaining $4.50 using 8 coins. If 4 of those other 8 coins are dollars, then we have 4 coins left to make 50 cents; we can't do that with just quarters and half dollars. If 3 of those other 8 coins are dollars, then we have 5 coins left to make $1.50. This we can do--with 1 half dollar and 4 quarters (solution #1 in the list above). If 2 of those other 8 coins are dollars, then we have 6 coins left to make $2.50. This too we can do--with 4 half dollars and 2 quarters (solution #2 in the list above). If 1 of those other 8 coins are dollars, then we have 7 coins left to make $3.50. And this we can do -- with 7 half dollars (solution #3 in the list above). If none of the other 8 coins are dollars, then we have to make $4.50 with 8 coins, the largest of which is a half dollar. 8 half dollars does not make $4.50, so we don't have a solution here. Now let's look at the combinations of dollars, half dollars, and quarters we found using 40 pennies and 2 nickels: dollars halves quarters ------------------------- 3 1 4 2 4 2 1 7 0 To get from one solution to the next, we do the following: use one fewer dollar coin, 3 more half dollars, and two fewer quarters. We add three coins and subtract three coins, and the total value stays the same. So in the rest of our investigation, whenever we find one solution, we can find others by using one less dollar, three more half dollars, and two less quarters. (Or we might be able to find other solutions by doing the opposite--using one more dollar, three less half dollars, and two more quarters.) I won't go through the details any further; I will just outline a bit more of the process. We have found all the solutions using 40 pennies and 2 nickels; the next thing we should try is finding combinations using 35 pennies and 3 nickels. It turns out we can't use 4 or 3 dollar coins with this combination; but with 2 dollar coins we can complete the $5 using 0 half dollars and 10 quarters (solution #4 in the list above). From there, using our method of trading coins, we can find solutions #5 and 6 in the list above. When we next try combinations using 30 pennies and 4 nickels, we find solution #7 in the list above. There are no other solutions in which the total value of the pennies and nickels is 50 cents. So next we look for combinations in which the total value of the pennies and nickels is 75 cents. Continuing in this fashion, a great deal of work finds the 38 solutions shown in the list.... I hope all this helps. Please write back if you have any further questions about any of this. - Doctor Greenie, The Math Forum http://mathforum.org/dr.math/ Date: 12/03/2005 at 11:06:52 From: Haley Subject: Thank you (50 coins = $5) Thank you for the quick responce. My dad also tried to show me the same way, but after a while I could not think of any more. I'll let my teacher know there are 38 answers. Thanks again. Haley |
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