Question

Last year Isabella took $7$ math tests and received $7$ different scores, each an integer between $91$ and $100,$ inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was $95.$ What was her score on the sixth test?

$92$

$94$

$96$

$98$

$100$

Answer: E

Concepts:divisibility mean bounding to limit cases

Difficulty rating: 2040

Solution:

Let $S$ be the sum of all seven scores. Then $S$ is a multiple of $7$ with $658 \le S \le 679,$ so $S \in \{658, 665, 672, 679\}.$ Since the average after six tests is an integer, $S - 95$ is a multiple of $6,$ which forces $S = 665.$ Then the first six scores sum to $570,$ a multiple of $5;$ the average after five tests is an integer, so the first five scores also sum to a multiple of $5,$ making the sixth score a multiple of $5.$ Since all scores differ and the seventh is $95,$ the sixth must be $100.$

Thus, the correct answer is E.

2017 AMC 12B Problem 21

Answer: E

Difficulty rating: 2040

Solution:

Problem 21 in Other Years