Question

cookies are sold singly or in packages of 7 or 21. with this packaging how many ways can you buy 42 cookies?

Answer 1

There are 10 different ways to buy 42 cookies using the packaging options of single cookies, sets of 7, or sets of 21. This is determined by solving the equation x + 7y + 21z = 42 for non-negative integers x, y, and z.

Step-by-step explanation:

To determine the different ways you can buy 42 cookies given the packaging options, we need to consider combinations of the packages. The packages come in single cookies, sets of 7, or sets of 21. Let's use 'x' to represent the number of single cookies, 'y' for the number of 7-cookie packages, and 'z' for the number of 21-cookie packages. The equation to solve is:

x + 7y + 21z = 42

We are looking for non-negative integer solutions to this equation since you can't buy negative packages of cookies. Here are the possible combinations:

1. 42 singles (x=42, y=0, z=0)
2. 6 packages of 7 (x=0, y=6, z=0)
3. 2 packages of 21 (x=0, y=0, z=2)
4. 1 package of 21 and 3 packages of 7 (x=0, y=3, z=1)
5. 1 package of 21 and 21 singles (x=21, y=0, z=1)
6. 5 packages of 7 and 7 singles (x=7, y=5, z=0)
7. 4 packages of 7 and 14 singles (x=14, y=4, z=0)
8. 3 packages of 7 and 21 singles (x=21, y=3, z=0)
9. 2 packages of 7 and 28 singles (x=28, y=2, z=0)
10. 1 package of 7 and 35 singles (x=35, y=1, z=0)

Counting all the possibilities, there are a total of 10 different ways to buy 42 cookies with the given packaging options.

Answer 2

You can buy 42 cookies in 12 ways.

Step-by-step explanation:

The first way of buying 42 cookies is by buying 2 packages of 21 (21x2 = 42).

The second way is by buying 1 package of 21 and 3 packages of 7 (21 + 3x7 = 42).

The third way is by buying 1 package of 21 and 21 single cookies (21 + 21 = 42).

The fourth way is by buying 1 package of 21, 1 package of 7 and 14 single cookies (21 + 7 + 14 = 42).

The fifth way is by buying 1 package of 21, 2 packages of 7 and 7 single cookies (21 + 14 + 7 = 42).

The sixth way is by buying 6 packages of 7 (7x6 = 42).

The seventh way is by buying 5 packages of 7 and 7 single cookies (7x5 + 7 = 42).

The eight way is by buying 4 packages of 7 and 14 single cookies (7x4 + 14 = 42).

The ninth way is by buying 3 packages of 7 and 21 single cookies (7x3 + 21 = 42).

The tenth way is by buying 2 packages of 7 and 28 single cookies (7x2 + 28 = 42).

The eleventh way is by buying 1 package of 7 and 35 single cookies (7 + 35 = 42).

Finally, the twelfth way is by buying 42 single cookies (42 = 42).