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