AMC 8 2017 Problem 14
AMC8 2017 年第 14 题
百分比★★★☆☆
Chloe and Zoe are both students in Ms. Demeanor's math class. Last night, they each solved half of the problems in their homework assignment alone and then solved the other half together. Chloe had correct answers to only 80% of the problems she solved alone, but overall 88% of her answers were correct. Zoe had correct answers to 90% of the problems she solved alone. What was Zoe's overall percentage of correct answers?
Chloe 和 Zoe 都是 Demeanor 女士数学课上的学生。昨晚她俩各自解决了家庭作业上一半的题目，接着共同解决了剩下的一半。Chloe 自己单独解决的题目里，正确率是 80％，但算上共同完成的题目，总的来说，她的正确率是 88％．Zoe 自已单独解决的题目里，正确率是 90％，问总的来说，Zoe 的正确率是多少？
- A.89
- B.92
- C.93
- D.96
- E.98
正确答案 · Correct Answer
C
解析 · Solution
Solution 1
Let the number of questions that they solved alone be . Let the percentage of problems they correctly solve together be %. As given, Hence, a = 96. Zoe got problems right out of . Therefore, Zoe got percent of the problems correct.
Assume the total amount of problems is 100 per half homework assignment since we are dealing with percentages, not values. Then, we know that Chloe got 80 problems correct by herself and got 176 problems correct overall. We also know that Zoe had 90 problems she did correctly alone. We can see that the total amount of correct problems Chloe and Zoe did together was 176-80=96. Therefore, Zoe did 96+90=186 problems out of 200 problems correctly. This is 93 percent.
In the problem, we can see that Chloe solved 80% of the problems she solved alone, but 88% of her answers are correct. If 80 and another number's average is 88, the other number must be 96. Then, Zoe solved 90% of the problems she did alone, but 96% of her answers were correct. Then, the average of 90 and 96 is 93.
(Slightly different Solution) Suppose we said that there were 100 problems in their assignment. Then, Chloe had 40 correct and 10 incorrect on her portion, and 48 correct and 2 incorrect on the portion she and Zoe solved. Zoe has 45 correct and 5 incorrect on her portion, and 48 correct and 2 incorrect on the portion that she and Chloe solved. Then, Zoe has 48 + 45 = 93 correct answers out of 100, so the answer is 93. -HW73
Let the total number of problems be . Let the percentage of the number of problems that Chloe and Zoe did together and got right be . Chloe got 80% of of the total problems right, hence . We also know Chloe got 88% of problems right altogether, making it total problems right. If we add to the problems Chloe solved alone, that should equal the total Chloe got right, giving the equation . Solving, we get x = 0.48t. We also know Zoe got 90% of of the total problems right, making it . Adding the problems Chloe and Zoe got right together gives . Solving, we get $0.93t$, which is 93%; hence, 93.
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