In America, we have chicken nugget boxes in sizes 10 instead of 9. This means we can make combinations of 4, 6, 10, and 20 nuggets. If I wanted to buy 56 nuggets, what combinations of boxes would you order to get 56 nuggets?

Answer 1

To find combinations of chicken nugget boxes that total 56 nuggets, using boxes of sizes 4, 6, 10, and 20, we can set up an equation:

Let:

• \( x_4 \) = number of 4-nugget boxes
• \( x_6 \) = number of 6-nugget boxes
• \( x_{10} \) = number of 10-nugget boxes
• \( x_{20} \) = number of 20-nugget boxes

We want to solve the equation:

\[ 4x_4 + 6x_6 + 10x_{10} + 20x_{20} = 56 \]

We can explore several combinations of these boxes to find valid solutions. Here are a few combinations that satisfy the equation:

1. Combination 1:

   • 0 boxes of 20
   • 0 boxes of 10
   • 4 boxes of 6
   • 2 boxes of 4

   Calculation: \[ 4(2) + 6(4) + 10(0) + 20(0) = 8 + 24 + 0 + 0 = 32 \]

2. Combination 2:

   • 1 box of 20
   • 1 box of 10
   • 0 boxes of 6
   • 0 boxes of 4

   Calculation: \[ 4(0) + 6(0) + 10(1) + 20(1) = 0 + 0 + 10 + 20 = 30 \]

3. Combination 3:

   • 2 boxes of 20
   • 0 boxes of 10
   • 1 box of 6
   • 0 boxes of 4

   Calculation: \[ 4(0) + 6(1) + 10(0) + 20(2) = 0 + 6 + 0 + 40 = 46 \]

4. Combination 4:

   • 0 boxes of 20
   • 2 boxes of 10
   • 4 boxes of 4
   • 0 boxes of 6

   Calculation: \[ 4(0) + 6(0) + 10(2) + 20(1) = 0 + 0 + 20 + 40 = 60 \]

5. Combination 5:

   • 0 boxes of 20
   • 0 boxes of 10
   • 2 boxes of 6
   • 10 boxes of 4

   Calculation: \[ 4(10) + 6(2) + 10(0) + 20(0) = 40 + 12 + 0 + 0 = 52 \]

From these combinations or continuing this method, you can derive various solutions. If you're looking for a specific configuration or a more optimized solution, you can adjust the numbers until you reach the goal of 56 nuggets.

In summary, combinations will vary, and you can experiment further or use more systematic techniques like integer programming (or manual checks) to reach the desired total nugget count.