Question

mind workout
date: 8-1-26
a shop sells apple juice in containers of 2 ℓ, 3 ℓ and 5 ℓ.
kate wants to buy 12 ℓ of apple juice.
what are the different ways she can buy the apple juice?
(images of 2ℓ, 3ℓ, 5ℓ apple juice containers)
make a list to help you.

Explanation:

Step1: Start with 5 ℓ containers

Let the number of 5 ℓ containers be \( x \), 3 ℓ be \( y \), and 2 ℓ be \( z \). We have \( 5x + 3y + 2z = 12 \), \( x,y,z \geq 0 \) integers.

• \( x = 0 \): Then \( 3y + 2z = 12 \).
• \( y = 0 \): \( 2z = 12 \Rightarrow z = 6 \) (6 of 2 ℓ)
• \( y = 2 \): \( 3\times2 + 2z = 12 \Rightarrow 2z = 6 \Rightarrow z = 3 \) (2 of 3 ℓ, 3 of 2 ℓ)
• \( y = 4 \): \( 3\times4 + 2z = 12 \Rightarrow 2z = 0 \Rightarrow z = 0 \) (4 of 3 ℓ)
• \( x = 1 \): Then \( 5 + 3y + 2z = 12 \Rightarrow 3y + 2z = 7 \).
• \( y = 1 \): \( 3\times1 + 2z = 7 \Rightarrow 2z = 4 \Rightarrow z = 2 \) (1 of 5 ℓ, 1 of 3 ℓ, 2 of 2 ℓ)
• \( y = 3 \): \( 3\times3 + 2z = 7 \Rightarrow 2z = -2 \) (invalid, \( z < 0 \))
• \( x = 2 \): Then \( 10 + 3y + 2z = 12 \Rightarrow 3y + 2z = 2 \).
• \( y = 0 \): \( 2z = 2 \Rightarrow z = 1 \) (2 of 5 ℓ, 1 of 2 ℓ)
• \( y \geq 1 \): \( 3y \geq 3 > 2 \) (invalid)

Step2: List all valid combinations

From above, the combinations are:

1. 6 containers of 2 ℓ (\( 6\times2 = 12 \))
2. 4 containers of 3 ℓ (\( 4\times3 = 12 \))
3. 2 containers of 3 ℓ and 3 containers of 2 ℓ (\( 2\times3 + 3\times2 = 6 + 6 = 12 \))
4. 1 container of 5 ℓ, 1 container of 3 ℓ, and 2 containers of 2 ℓ (\( 5 + 3 + 2\times2 = 5 + 3 + 4 = 12 \))
5. 2 containers of 5 ℓ and 1 container of 2 ℓ (\( 2\times5 + 2 = 10 + 2 = 12 \))

Answer:

The different ways are:

• 6 containers of 2 ℓ.
• 4 containers of 3 ℓ.
• 2 containers of 3 ℓ and 3 containers of 2 ℓ.
• 1 container of 5 ℓ, 1 container of 3 ℓ, and 2 containers of 2 ℓ.
• 2 containers of 5 ℓ and 1 container of 2 ℓ.