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A vending operator stocked a machine using only two sizes of snack pack: small packs holding 6 snacks each and large packs holding 15 snacks each. The operator used fewer than 12 packs in total. How many large packs did the operator use?
(1) The machine was stocked with exactly 132 snacks.
(2) The operator used twice as many small packs as large packs.
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Correct answer
A
Let s be the number of small packs and L the number of large packs, both non-negative integers, with s + L < 12. Statement (1) gives 6s + 15L = 132, which divides by 3 to 2s + 5L = 44. Since 2s and 44 are both even, 5L must be even, so L must be even.
Listing even L and checking the fewer-than-12 bound: L = 0 needs 22 packs, L = 2 needs 19, L = 4 needs 16, L = 6 needs 13, all too many; L = 8 gives s = 2, which is 10 packs and is allowed; L = 10 gives a negative s. Exactly one value survives, L = 8, so statement (1) is sufficient.
Statement (2) gives s = 2L, so the total is 3L, and any L with 3L < 12 works, namely 0, 1, 2, or 3, leaving L undetermined. So statement (1) alone fixes L and statement (2) alone does not, and the answer is A.
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