AMC 8 2019 Problem 7
AMC8 2019 年第 7 题
统计与数据图表★★☆☆☆
Shauna takes five tests, each worth a maximum of 100 points. Her scores on the first three tests are 76 , 94 , and 87 . In order to average 81 for all five tests, what is the lowest score she could earn on one of the other two tests?
Shauna 一共考了 5 场考试，每场的满分都是 100 分。前三场考试的得分是 76，94 和 87，为了使得所有 5 场考试的平均分是 81 分，那么剩下 2 场考试中其中 1 场考试的最低得分可以是多少分？
- A.48
- B.52
- C.66
- D.70
- E.74
正确答案 · Correct Answer
A
解析 · Solution
Solution 1
We should notice that we can turn the information we are given into a linear equation and just solve for our set variables. I'll use the variables and for the scores on the last two tests. We can now cross multiply to get rid of the denominator. 257+x+y = 405, x+y = 148. Now that we have this equation, we will assign as the lowest score of the two other tests, so x = 100, y=48. Now we know that the lowest score on the two other tests is 48.
Right now, she scored 76, 94, and 87 points, for a total of 257 points. She wants her average to be 81 for her 5 tests, so she needs to score 405 points in total. This means she needs to score a total of 405-257= 148 points in her next 2 tests. Since the maximum score she can get on one of her 2 tests is 100, the least possible score she can get is 48.
We can compare each of the scores with the average of 81: , , , . So the last one has to be $-33$ (since all the differences have to sum to 0 ), which corresponds to 81-33 = 48.
We know that she scored 76, 94, and 87 points on her first 3 tests for a total of 257 points and that she wants her average to be 81 for her 5 total tests. Therefore, she needs to score a total of 405 points. In addition, one of the final 2 tests needs to be the maximum of 100 points, to make the final test score—the one that we are looking for—the lowest score possible for her to earn. We can now see here that the sum of 76+94+100 has a units digit of 0 and that the final test score must have a units digit ending with a 5. Now, 87 needs to be added to a number that makes the sum divisible by 5. Among the answer choices of 48, 52, 66, 70, and 74, only 48 has a units digit that works. ( 48+87=135, giving us a units digit of 5.)
~ saxstreak For Shauna to get an average of 81, she needs to score a total of $81×5 = 405$. One of the final two tests must earn a score of 100 to ensure the final test score is the lowest possible. $76+94+87+100 = 357$, and $405-357 = 48$. 81 times 5 = 405. Adding the units digits of 76, 94, 87, and 100 gives 7. To end with five the fifth score ends with 8 so the answer is 48.
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