2004 AMC 12B Problem 11

Below is the professionally curated solution for Problem 11 of the 2004 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2004 AMC 12B solutions, or check the answer key.

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Concepts:mean bounding to limit cases

Difficulty rating: 1440

11.

All the students in an algebra class took a $100$ -point test. Five students scored $100,$ each student scored at least $60,$ and the mean score was $76.$ What is the smallest possible number of students in the class?

$10$

$11$

$12$

$13$

$14$

Solution:

Each score of $100$ is $24$ above the mean, so the five contribute $120$ points above $76.$ These must be balanced by points below the mean, and each remaining student is at most $76 - 60 = 16$ below. So at least $\dfrac{120}{16} = 7.5,$ hence $8$ more students are needed, for a total of $13.$ Five $100$ s and eight $61$ s achieve this.

Thus, the correct answer is D.

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