2005 CEMC Gauss (Grade 7) Problems/Problem 25
Contents
Problem
How many different combinations of pennies, nickels, dimes and quarters use coins to total dollar?
Solution 1
We want to combine coins to get cents. Since the combined value of the coins is a multiple of , as is the value of a combination of nickels, dimes and quarters, then the value of the pennies must also be a multiple of . Therefore, the possible numbers of pennies are . We can also see that because there are coins in total, it is not possible to have anything other than , , or pennies. (For example, if we had pennies, we would have other coins which are worth at least cents each, so we would have at least cents in total, which is not possible. We can make a similar argument for , , , , and pennies.) It is also not possible to have or quarters. If we did have or quarters, then the remaining or coins would give us a total value of at least cents, so the total value would be greater than cents. Therefore, we only need to consider , , or quarters.
Possibility : quarters
If we have quarters, this means we have coins with a value of cents. The only possibility for these coins is pennies and nickel.
Possibility : quarter
If we have quarter, this means we have coins with a value of cents. The only possibility for these coins is pennies and nickels.
Possibility : quarters
If we have quarters, this means we have coins with a value of cents. If we had pennies, we would have to have nickels. If we had pennies, we would have to have dimes and nickels. It is not possible to have pennies.
Therefore, there are 4 possible combinations. Thus, the answer is .
Solution 2
We want to use coins to total cents. Let us focus on the number of pennies. Since any combination of nickels, dimes and quarters always is worth a number of cents which is divisible by , then the number of pennies in each combination must be divisible by , since the total value of each combination is cents, which is divisible by . Could there be pennies? If so, then the remaining coins are worth cents. But each of the remaining coins is worth at least cents, so these coins are worth at least cents, which is impossible. So there cannot be pennies. Could there be pennies? If so, then the remaining coins are worth cents. But each of the remaining coins is worth at least cents, so these coins are worth at least cents, which is impossible. So there cannot be pennies. We can continue in this way to show that there cannot be , , , or pennies. Therefore, there could only be , , or pennies. If there are pennies, then the remaining coins are worth cents. Since each of the remaining coins is worth at least cents, this is possible only if each of the coins is a nickel. So one combination that works is pennies and nickels. If there are pennies, then the remaining coins are worth cents. We now look at the number of quarters in this combination. If there are quarters, then we must have nickels and dimes totaling cents. If all of the coins were nickels, they would be worth cents, so we need to change nickels to dimes to increase our total by cents to cents. Therefore, pennies, quarters, nickels and dimes works. If there is quarter, then we must have nickels and dimes totaling cents. Since each remaining coin is worth at least cents, then all of the remaining coins must be nickels. Therefore, pennies, quarter, nickels, and dimes works. If there are quarters, then we must have nickels and dimes totaling cents. This is impossible. If there were more than quarters, the quarters would be worth more than cents, so this is not possible. If there are pennies, then the remaining coins are worth cents in total. In order for this to be possible, there must be quarters (otherwise the maximum value of the coins would be with quarter and dimes, or cents). This means that the remaining coin is worth cents, and so is a nickel. Therefore, pennies, quarters, nickel, and dimes is a combination that works. Therefore, there are combinations that work. The answer is .
See Also
2005 CEMC Gauss (Grade 7) (Problems • Answer Key • Resources) | ||
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CEMC Gauss (Grade 7) |