Official Solution: At a school fundraiser, a snack table sold four types of items: fruit cups, granola bars, sandwich boxes, and salad bowls. A fruit cup cost $3, a granola bar cost $5, a sandwich box cost $9, and a salad bowl cost $14. If the snack table sold 21 items in total, at least one of each type, and collected $180, how many granola bars were sold? Let \(F\), \(G\), \(S\), and \(L\) be the numbers of fruit cups, granola bars, sandwich boxes, and salad bowls sold, respectively.
Given:
\(F + G + S + L = 21\) (i)
\(3F + 5G + 9S + 14L = 180\) (ii)
The question asks for \(G\).
(1) The number of granola bars sold was equal to the number of sandwich boxes sold.
So \(G = S\).
Then from (i):
\(F + 2G + L = 21\)
and from (ii):
\(3F + 14G + 14L = 180\)
Multiply the first equation by 3:
\(3F + 6G + 3L = 63\)
Subtract from \(3F + 14G + 14L = 180\):
\(8G + 11L = 117\)
Now since \(G\) and \(L\) must be positive integers, we should check whether this equation gives one or more than one set of solutions for \((G, L)\):
\(11L = 117 - 8G\)
So, \(117 - 8G\) must be a positive multiple of 11: \(110, 99, 88, 77, \ldots, 11\). Only \(G = 5\) and \(L = 7\) work. So \(G = 5\).
Sufficient.
(2) The snack table sold 4 fruit cups and 5 sandwich boxes.
So \(F = 4\) and \(S = 5\).
From the total number of items:
\(4 + G + 5 + L = 21\)
\(G + L = 12\)
From the total revenue:
\(3 * 4 + 5G + 9 * 5 + 14L = 180\)
\(12 + 5G + 45 + 14L = 180\)
\(5G + 14L = 123\)
So, we have two distinct linear equations with two unknowns: \(G + L = 12\) and \(5G + 14L = 123\). We can solve and get the value of \(G\).
Sufficient.
Answer: D