The Pi (π) Club at Math High School was planning a fundraiser to sell pepperoni pizza. A pizza company donated enough ingredients to make the equivalent of 300 extra large pepperoni pizzas (16” diameter). In trying to determine the number of pizzas of each size to make, they surveyed students in the school. Based on the survey, the club decided that more than half of the pizzas they make should be extra large; there would be more large (14” diameter) than medium (12” diameter) and between 40 and 50 small (10” diameter). The club members plan to sell the pizzas for $11(small), $15 (medium), $18 (large), and $20 (extra-large).
How many pizzas does the club make for each size?
If there are w,x,y,z pizzas of the sizes, then we have
100w+144x+196y+256z = 300*256
z >= 150
y > x
40 <= w <= 50
You have said nothing about how much they need to make from the sales, so the price information is just noise.
To determine the number of pizzas of each size to make, let's break down the problem step by step.
Step 1: Calculate the total number of pizzas to be made.
The problem states that the club received enough ingredients to make the equivalent of 300 extra-large pizzas. Since the extra-large pizzas are the largest size, we can assume that the total number of pizzas will not exceed 300.
Step 2: Determine the proportion of extra large pizzas.
The problem mentions that more than half of the pizzas made should be extra large. Let's assume that exactly half of the pizzas are extra large. Therefore, the proportion of extra large pizzas would be 1/2.
Step 3: Calculate the number of extra large pizzas.
To find the number of extra-large pizzas, we multiply the total number of pizzas (300) by the proportion of extra large pizzas (1/2).
Number of Extra Large Pizzas = 300 * (1/2) = 150
Step 4: Determine the proportion of large pizzas.
Since the problem didn't specify a specific proportion for large pizzas, we can assume it to be anything greater than zero. Let's set it to 1/4, which means one-fourth of the total pizzas would be large.
Step 5: Calculate the number of large pizzas.
We multiply the total number of pizzas (300) by the proportion of large pizzas (1/4).
Number of Large Pizzas = 300 * (1/4) = 75
Step 6: Determine the proportion of medium pizzas.
Since the large pizzas have been determined, we know that the medium pizzas must be less than the number of large pizzas. Again, no specific proportion was given, so let's set it to 1/6, meaning one-sixth of the total pizzas would be medium.
Step 7: Calculate the number of medium pizzas.
We multiply the total number of pizzas (300) by the proportion of medium pizzas (1/6).
Number of Medium Pizzas = 300 * (1/6) = 50
Step 8: Calculate the number of small pizzas.
The problem states that there should be between 40 and 50 small pizzas. Since there are no other constraints, we can choose any number within that range. Let's choose 45.
Number of Small Pizzas = 45
So, the club should make the following number of pizzas for each size:
Extra Large: 150
Large: 75
Medium: 50
Small: 45