AMC 8 2018 Problem 13
AMC8 2018 年第 13 题
方程与不等式★★★☆☆
Lalia took five math tests, each worth a maximum of 100 points. Laila's score on each test was an integer between 0 and 100, inclusive. Laila received the same score on the first four tests, and she received a higher score on the last test. Her average score on the five tests was 82. How many values are possible for Laila's score on the last test?
参加了 5 场数学考试，每场考试的满分都是 100 分。Laila 每场考试的得分都是 0 到 100 之间(包括 0 和 100)的整数。Laila 前四场考试得分相同，最后一场考试得分比前四场每场得分都高。已知五场考试的平均分是 82 分，那么最后一场考试的得分可能有多少个取值？
- A.4
- B.5
- C.9
- D.10
- E.18
正确答案 · Correct Answer
A
解析 · Solution
Solution 1
Say Laila gets a value of on her first 4 tests, and a value of on her last test. Thus, . Because and are different, must be less than 82 and must be greater than 82. When decreases by 1, must increase by 4 to keep the total constant. The greatest value for is 98 (as y=100 would make non-integer). In the range , only 4 values for result in integer values for : 86, 90, 94 and 98. Thus, the answer is 4.
The average score is 82 which leads us to suppose that Laila got all 82 points for the tests. We know that Laila got the same points in the first four tests and they are all lower than the last test. Let the first four tests be 81 points, then the last test must be 86 points to keep the average fixed. Continue to decrement the first four tests to identify other possible combinations. The possible points for the fifth test are 86, 90, 94, 98. The answer is 4.
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