Questions: Question #4 Mr. Jordan's class is having a pizza party with the 100 they earned in a competition. A cheese pizza costs 10, and a pepperoni pizza costs 15. They must order at least 2 pepperoni pizzas. They also must order at least twice as many cheese pizzas as pepperoni pizzas. What is the greatest total number of pizzas Mr. Jordan's class can order?

Solution Steps

To solve this problem, we need to maximize the total number of pizzas while adhering to the given constraints. We can set up a system of inequalities based on the cost and the required number of pizzas, and then iterate through possible values to find the maximum number of pizzas.

1. Let $c$ be the number of cheese pizzas and $p$ be the number of pepperoni pizzas.
2. The total cost constraint is $10c + 15p \leq 100$.
3. The constraint for pepperoni pizzas is $p \geq 2$.
4. The constraint for cheese pizzas is $c \geq 2p$.

We will iterate over possible values of $p$ starting from 2 and calculate the corresponding $c$ to maximize the total number of pizzas $c + p$.

Let $c$ be the number of cheese pizzas and $p$ be the number of pepperoni pizzas. The constraints based on the problem are:

1. Cost constraint: $10c + 15p \leq 100$
2. Minimum pepperoni pizzas: $p \geq 2$
3. Cheese pizzas must be at least twice the pepperoni pizzas: $c \geq 2p$

To maximize the total number of pizzas, we need to maximize $c + p$ under the given constraints. We start with the minimum value for $p$ and calculate the corresponding $c$ while ensuring the cost does not exceed $100.

Starting with $p = 2$:

• Minimum $c = 2 \times 2 = 4$
• Check if $10c + 15p \leq 100$:
  • For $p = 2$ and $c = 4$: $10(4) + 15(2) = 40 + 30 = 70$ (valid)
  • Increase $c$ to find maximum combinations while keeping the cost under $100.

Continuing this process for higher values of $p$ (up to the maximum possible based on the cost), we find that the maximum total number of pizzas occurs when $p = 2$ and $c = 7$, yielding a total of $2 + 7 = 9$.

Final Answer