SOLUTION: A person was in charge of ordering 26 pizzas for the office party. He ordered three types of​ pizza: cheese,​ pepperoni, and supreme. The cheese pizzas cost $6 ​each, the pep

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Question 1138848: A person was in charge of ordering 26 pizzas for the office party. He ordered three types of​ pizza: cheese,​ pepperoni, and supreme. The cheese pizzas cost $6 ​each, the pepperoni pizzas cost $9 ​each, and the supreme pizzas cost $12 each. He spent exactly twice as much on the pepperoni pizzas as he did on the cheese pizzas. If the person spent a total of $222 on​ pizza, how many pizzas of each type did he​ buy?
Answer by greenestamps(13367) About Me  (Show Source):
You can put this solution on YOUR website!


c = # of cheese pizzas
p = # of pepperoni pizzas
s = # of supreme pizzas

c%2Bp%2Bs+=+26 the total number of pizzas was 26
6c%2B9p%2B12s+=+222 the total cost was $222
9p+=+2%286c%29 the cost of the pepperoni pizzas was twice the cost of the cheese pizzas

There is an endless number of different paths for solving that system of 3 equations in 3 variables. The path I chose involved many twists and turns and was not particularly pleasant....

So instead of finishing the problem that way, let's spend a little effort to set up the problem using a single equation in a single variable and see if the resulting path to the solution is easier.

He spent twice as much on the $9 pepperoni pizzas as he spent on the $6 cheese pizzas:

9p+=+2%286c%29
9p+=+12c
3p+=+4c

Using this, we can let p=4x and c=3x; then 3p=4c=12x.

And then with p=4x and c=3x, and with 26 pizzas in all, the number of supreme pizzas is 26-7x.

Then...

6%283x%29%2B9%284x%29%2B12%2826-7x%29+=+222
18x%2B36x%2B312-84x+=+222
-30x%2B312+=+222
90+=+30x
x+=+3

ANSWER:
cheese: 3x = 9
pepperoni: 4x = 12
supreme: 26-7x = 5

With the two ways I chose to solve the problem, the little extra effort required to set up an equation using a single variable resulted in an equation that required far less effort to solve than the system of 3 equations in 3 variables.

While it is of course important to understand how to set up a problem like this with 3 variables directly from the given information, it is also important to know that very often the overall effort required to solve a problem will be greatly reduced if you can set up the problem using a single variable.