A recipe for mac and cheese calls for 6 cups of pasta for every 4 cups of cheese. Suppose a restaurant is making a large batch of mac and cheese using 15 cups of cheese. How much pasta would they need?

Question

Start by labeling the double number line with the cups of cheese and cups of pasta that the recipe calls for.

Answer 1

To solve this problem, we can set up a double number line based on the original recipe ratio of cups of pasta to cups of cheese.

1. Original Recipe Ratios:

   • The recipe calls for 6 cups of pasta for every 4 cups of cheese.

   We can express this in the form of a ratio:

   • Pasta: Cheese = 6:4

2. Simplifying the Ratio:

   • The ratio simplifies to 3:2 (by dividing both numbers by 2). This means for every 2 cups of cheese, we need 3 cups of pasta.

3. Setting Up the Double Number Line:

   • We can use the following increments based on the ratios:
     • 2 cups of cheese ↔ 3 cups of pasta
     • 4 cups of cheese ↔ 6 cups of pasta
     • 6 cups of cheese ↔ 9 cups of pasta
     • 8 cups of cheese ↔ 12 cups of pasta
     • 10 cups of cheese ↔ 15 cups of pasta
     • 12 cups of cheese ↔ 18 cups of pasta
     • 14 cups of cheese ↔ 21 cups of pasta
     • 16 cups of cheese ↔ 24 cups of pasta

4. Finding the Amount of Pasta Needed for 15 Cups of Cheese:

   • For every 2 cups of cheese, we need 3 cups of pasta. Therefore, we can find the amount of pasta needed for 15 cups of cheese by first determining how many batches of 2 cups of cheese are in 15 cups: \[ \text{Number of 2-cup batches} = \frac{15 \text{ cups of cheese}}{2 \text{ cups of cheese}} = 7.5 \]
   • Since each batch of 2 cups of cheese requires 3 cups of pasta, we multiply the number of batches by the corresponding amount of pasta: \[ \text{Pasta needed} = 7.5 \text{ batches} \times 3 \text{ cups of pasta per batch} = 22.5 \text{ cups of pasta} \]

So, the restaurant would need 22.5 cups of pasta for 15 cups of cheese.

In summary:

• For 15 cups of cheese, the restaurant needs 22.5 cups of pasta.