AMC10 2017 B

AMC10 2017 B · Q25

AMC10 2017 B · Q25. It mainly tests Averages (mean), Logic puzzles.

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?

去年Isabella参加了7次数学测验，得到7个不同的分数，每个分数是91到100之间的整数。每次测验后她注意到其测验平均分是整数。第七次测验的分数是95。她的第六次测验分数是多少？

(A) 92 92

(B) 94 94

(D) 98 98

(E) 100 100

Answer

Correct choice: (E)

正确答案：（E）

Solution

Let S be the sum of Isabella’s 7 scores. Then S is a multiple of 7, and 658 = 91 + 92 + 93 + · · · + 97 ≤ S ≤ 94 + 95 + 96 + · · · + 100 = 679, so S is one of 658, 665, 672, or 679. Because S − 95 is a multiple of 6, it follows that S = 665. Thus the sum of Isabella’s first 6 scores was 665 − 95 = 570, which is a multiple of 5, and the sum of her first 5 scores was also a multiple of 5. Therefore her sixth score must have been a multiple of 5. Because her seventh score was 95 and her scores were all different, her sixth score was 100.

设S为Isabella的7个分数之和。那么S是7的倍数，且658=91+92+93+⋯+97≤S≤94+95+96+⋯+100=679，因此S是658、665、672或679之一。因为S-95是6的倍数，故S=665。于是前6次分数之和是665-95=570，是5的倍数，前5次分数之和也是5的倍数。因此第六次分数必须是5的倍数。因为第七次是95且分数均不同，故第六次分数是100。

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