A customer wants to mail a package.
The postal clerk determines the cost of the package to be $18.95, but only
6¢ and 9¢ stamps are available. Can the available stamps be used for the exact amount of postage for the package?
Why or why not?
Thank you
Well, if I were a package, I would definitely want the full postage amount for my journey! Unfortunately, in this case, the available stamps won't cut it. You see, the cost of the package is $18.95, and the only stamps available are 6¢ and 9¢. No matter how many of these stamps you use, you won't be able to reach $18.95 exactly. It's like trying to buy a Lamborghini with only a piggy bank full of pennies - it just won't work! So, it looks like we'll need some other denominations of stamps to make up the difference. Keep those pennies for your piggy bank though - they'll come in handy someday!
To determine if the available stamps can be used for the exact amount of postage for the package, we need to find out if it is possible to obtain $18.95 using only 6¢ and 9¢ stamps.
One way to approach this problem is to use a method called the "coin change problem". In this case, we can consider the 6¢ stamp as a coin with a value of 6, and the 9¢ stamp as a coin with a value of 9. We want to find out if it is possible to obtain a total of 1895 (since $18.95 equals 1895 cents) using only these two coin values.
To solve this problem, we can use a dynamic programming approach. We start by creating a table where each cell represents the minimum number of coins needed to obtain a certain value. We initialize the table with infinity values, except for the first cell which is set to 0.
Next, we iterate through each coin one by one and update the table values based on two conditions:
1. If the current coin value is less than or equal to the target value, we consider two possibilities:
a) We include the coin and subtract its value from the target value.
b) We exclude the coin and continue to the next coin.
2. If the current coin value is greater than the target value, we exclude the coin.
By the end of the iteration, we will have the minimum number of coins needed to obtain the target value in the last cell of the table. If this value is less than infinity, it means it is possible to obtain the target value using the available coin values.
In this case, after performing the dynamic programming algorithm, if the last cell of the table contains a finite number, then it is possible to obtain $18.95 using only 6¢ and 9¢ stamps. If the last cell contains infinity, then it is not possible to obtain the exact amount of postage for the package using these stamps.
Note that this algorithm finds the minimum number of coins needed to obtain the target value, but does not provide the exact combination of coins. However, it can be modified to also provide the coin combination.
So, to determine if the available stamps can be used for the exact amount of postage for the package, we need to implement the dynamic programming algorithm described above and check if it is possible to obtain 1895 cents using only 6¢ and 9¢ stamps.