For his troops' popcorn fundraiser sale, Mike sold caramel corn for $10, buttered microwave popcorn for $8, and lightly buttered popcorn for $7. By the end of the fundraiser, he had sold 400 items and made $3,272. If he sold twice as many lightly buttered microwave popcorn boxes as the buttered popcorn, how many boxes of each type of popcorn did he sell?

Question

For his troops' popcorn fundraiser sale, Mike sold caramel corn for $10, buttered microwave popcorn for $8, and lightly buttered popcorn for $7. By the end of the fundraiser, he had sold 400 items and made $3,272. If he sold twice as many lightly buttered microwave popcorn boxes as the buttered popcorn, how many boxes of each type of popcorn did he sell?

Answer 1

x = caramel corn

y = buttered
z = lightly buttered

x + y + z = 400
10x + 8y + 7z = 3272
2z + z + x = 400 <= here since he sold twice as many LB popcorn as buttered, I replaced the LB with 2z.

OK, since we have two of these equal to 400...

2z + z + x = x + y + z
3z = y + z
2z = y
Substitute this in for y.

x + 3z = 400 => 10x + 30y = 4000
10x + 16z + 7z = 3272 => 10z + 23z + 3272

Now subtract these two equations. 7z = 728, z = 104
2(104) = y
y = 208

104 + 208 + x = 400
x = 88

88 caramel popcorn, 208 lightly buttered and 104 buttered.

Answer 2

To solve this problem, let's assign variables to represent the number of each type of popcorn sold.

Let's say:
x = number of buttered microwave popcorn boxes sold
y = number of lightly buttered popcorn boxes sold

We are given the following information:
- Mike sold a total of 400 items, so:
x + y = 400 -- Equation 1

- The cost of selling caramel corn, buttered microwave popcorn, and lightly buttered popcorn was $3,272, so:
10x + 8y + 7(400 - x - y) = 3272 -- Equation 2
(Note: 400 - x - y represents the number of caramel corn boxes sold)

Now, let's solve this system of equations to find the values of x and y.

Let's start by simplifying Equation 2:
10x + 8y + 2800 - 7x - 7y = 3272
3x + y + 2800 = 3272
3x + y = 472 -- Equation 3

Now we have a system of two equations:
x + y = 400 -- Equation 1
3x + y = 472 -- Equation 3

We can solve this system of equations by substitution or elimination. Let's use elimination:

Subtract Equation 1 from Equation 3:
(3x + y) - (x + y) = 472 - 400
2x = 72
x = 36

Now, substitute the value of x in Equation 1 to find y:
36 + y = 400
y = 400 - 36
y = 364

Therefore, Mike sold 36 boxes of buttered microwave popcorn and 364 boxes of lightly buttered popcorn.