Reverse Plugging In: Basic Technique

One-third of the residents of hall A are juniors. Of the remainder, one-third are seniors, one-third sophomores, and one-third freshmen. Which of the following could be the possible number of total residents in hall A?

Incorrect. [[snippet]] If there are 20 residents in hall A, then 6.33, or $$\frac{1}{3} \times 20$$, are juniors. Since the number of juniors must be an integer, this option is incorrect.

Correct. [[snippet]] Start with the middle option. The number 16 is not divisible by 3, so it is eliminated (the number of juniors must be an integer). The first and last answer choices can be eliminated for the same reason, and you are left with B (12) and D (18) as possible answers. If there are 18 residents in hall A, then $$\frac{1}{3} \times 18=6$$ are juniors. This leaves behind 12 residents ($$18- 6$$). These 12 are to be divided into seniors, sophomores, and freshmen at one-third each. Because 12 is divisible by 3 (i.e., $$\frac{12}{3} = 4$$), this is a possible number for the residents of Hall A. Since there can be only one answer choice, you can stop __PI__ here—this is the correct answer.

Incorrect. [[snippet]] If there are 16 residents in hall A, then 5.33, or $$\frac{1}{3} \times 16$$, are juniors. Since the number of juniors must be an integer, this option is incorrect. See if you can eliminate any of the other answer choices for the same reason.

Incorrect. [[snippet]] If there are 12 residents in hall A, then 4, or $$\frac{1}{3} \times 12$$, are juniors. This leaves behind 8 residents ($$12- 4$$). These 8 are to be divided into seniors, sophomores, and freshmen at one-third each. Because 8 is _not_ divisible by 3 (i.e., $$\frac{12}{3} = 4$$), this option is incorrect.

Incorrect. [[snippet]] If there are 10 residents in hall A, then 3.33, or $$\frac{1}{3} \times 10$$, are juniors. Since the number of juniors must be an integer, this option is incorrect.

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