5) A friend brought small bags of cookies to sell at a fairly large Holiday Bowl

Question

5) A friend brought small bags of cookies to sell at a fairly large Holiday Bowl Game Watch Party. Three kinds of cookies were sold: Stars (sold for $1 per bag), Circles (sold for $0.75 per bag) and Stars and Stripes (sold for $1.50 per bag).

He brought the cookies to the Watch Party in three large boxes that were totally filled to capacity. By volume, it is a known fact that each bag of Stars fills up 1% of a box (1/100th), each bag of Circles fills up 1/120th of a box while each bag of Stars and Stripes fills up 1.25% of a box (1/80th). HINT: Don’t concern yourself with what each box held; view this as an aggregate limit of numbers of cookies..

All cookies brought were sold, a total of 300 bags. The total amount of money raised was $312. All bags of cookies brought in the boxes were sold.

Can you determine how many of each of the three cookie types were brought/sold?

Explanation / Answer

Let x1, x2, and x3 be the number of bags of Stars, Circles, and Stars & Stripes cookies respectively

1x1 + 0.75x2 + 1.5x3 = 312 (Revenue) ----(1)
1x1 + 1x2 + 1x3 = 300 (total bags) ----(2)
(x1/100) + (x2/120) + (x3/80) = 3 (three large boxes)
or, 12x1 + 10x2 + 15x3 = 3600 ----(3)

Subtracting (2) from (1),

0.5x3 - 0.25x2 = 12 or, 2x3 - x2 = 48 or, 10x2 = 20x3 - 480 ----(4)

Replacing (4) into (3), we get,

12x1 + 20x3 - 480 + 15x3 = 3600 or, 12x1 + 35x3 = 4080 -- (5)

Again perform, (1) - 0.75*(2) to get

0.25x1 + 0.75x3 = 312 - 0.75*300 = 87 --- (6)

Solving (5) and (6), we will get x3 = 96; x1 = 60

From (4), x2 = 2x3 - 48 = 192 - 48 = 144

So, final answer is x1= 60 bags; x2 = 144 bags; x3 = 96 bags

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