Final answer:
To find how many ways you can buy 60 cookies using single cookies or packages of 10 or 30, you solve for non-negative integer solutions to the equation x + 10y + 30z = 60 and count the combinations.
Step-by-step explanation:
The question concerns the number of ways to purchase a specific quantity of an item when given different packaging options. Here, we want to find out how many ways you can buy 60 cookies when they are sold singly or in packages of 10 or 30.
To solve this, we can set up a mathematical equation where x, y, and z represent the number of single cookies, 10-cookie packages, and 30-cookie packages, respectively. The equation would be:
x + 10y + 30z = 60
We are looking for all non-negative integer solutions to this equation. By examining the possible combinations, we can systematically list out the ways to achieve the total of 60 cookies using these packaging options.
- If we start by using no 30-cookie packages, we can have 0 to 6 packages of 10, with the remainder being single cookies.
- When using one 30-cookie package, we can have 0 to 3 packages of 10, with the remainder being single cookies.
- Lastly, if we use two 30-cookie packages, we don't need any 10-cookie packages, and we can just have 0 singles.
By counting all the combinations, we can determine how many different ways there are to purchase exactly 60 cookies.