Ben was in charge of ordering 3636 pizzas for the office party. He ordered three types of pizza: cheese, pepperoni, and supreme. The cheese pizzas cost $ 8$8 each, the pepperoni pizzas cost $ 11$11 each, and the supreme pizzas cost $ 14$14 each. He spent exactly twice as much on the pepperoni pizzas as he did on the cheese pizzas. If Ben spent a total of $ 390$390 on pizza, how many pizzas of each type did he buy?

Ben bought 11 cheese pizzas, 16 pepperoni pizzas and 9 supreme pizzas

Step-by-step explanation:

We can solve this problem with a equation system. I will abbreviate the amount of cheese pizzas he buyed as C, the amount of pepperoni ones as P, and the amount of supreme ones as S. I will also remove the $ notation

The system is given as follows

C+P+S = 36 8C + 11P + 14S = 390 11P = 2 * (8C) = 16C

We can easily replace 11P with 16C in equation 2, obtaining

8C + 16C + 14S = 390

24C + 14S = 390

S = (390 - 24C) / 14 = 195/7 - 12/7 C

Note that, also from equation 3, we have that

11P = 16 C

P = 16/11 C

Now, we can obtain the value of C using equation 1 replacing S and P with the founded values

C+P+S = 36

C + 16/11 C + 195/7 - 12/7 C = 36

C + 16/11 C - 12/7 C = 36-195/7 = 57/7

Taking C as common factor we obtain

C (1+16/11-12/7) = 57/7

C * 57/77 = 57/7

C = (57/7) / (57/77) = 11

Now, we replace this value in the formula we obtain for P and S

P = 16/11 * C = 16/11 * 11 = 16

S = 195/7 - 12/7 * 11 = 63/7 = 9

Now, we verify:

C+P+S = 11+16+9 = 36 8C + 11P + 14S = 8*11 + 11*16 + 14*9 = 88 + 176 + 126 = 390 2 (8C) = 2*88 = 176 = 11P

Thus, Ben bought 11 cheese pizzas, 16 pepperoni pizzas and 9 supreme pizzas.