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?
Because it's a practice problem I already have the answer. I just need help with how to get to the answer.

A: 127 caramel corn boxes, 91 buttered microwave popcorn boxes, 182 lightly buttered microwave popcorn boxes

Answer 1

$\bf \begin{cases} c=\textit{caramel corn}\\ b=\textit{buttered microwave popcorn}\\ p=\textit{lightly buttered popcorn} \end{cases}$

so... he sold 400 items, that means, whatever those quantities of c,b,p are, they add up to 400, thus

c+b+p = 400

now, the "c" sold for $10
the "b" sold for $8
and the "p" sold for $7

so, the amount of revenue on the "c" is 10 * c, or 10c
the revenue on "b" is 8*b or 8b
and the revenue on "p" is 7*p or 7p

whatever those amounts are, they ended up in $3,272
thus
10c + 8b + 7p = 3,272

now, he sold more lightly buttered, or "p", than just buttered corn or "b"
he actually sold twice as much, so whatever "b" is, "p" is twice that much
or 
p = 2b

thus      $\bf \begin{cases} c+b+\boxed{p}=400\\\\ \boxed{p}=2b\qquad thus\\\\ c+b+\boxed{2b}=400\qquad or\\\\ c+3b=400\implies c=\underline{400-3b}\\ --------------\\ 10c+8b+7p=3272\qquad or\\\\ 10(\underline{400-3b})+8b+7(\boxed{2b})=3272 \end{cases}$

solve for "b", to see how many buttered microwave popcorn Mike sold

how many lightly buttered?  well, p = 2b
how many caramel ones?   well  c = 400 - 3b