AMC 8 2002 Problem 17
AMC8 2002 年第 17 题
方程与不等式★★★☆☆
In a mathematics contest with ten problems, a student gains 5 points for a correct answer and loses 2 points for an incorrect answer. If Olivia answered every problem and her score was 29, how many correct answers did she have?
在一个有十道题的数学竞赛中，一个学生答对一题得 5 分，答错一题扣 2 分。如果 Olivia 回答了所有问题，她的分数是 29 分，那么她答对多少题？
- A.5
- B.6
- C.7
- D.8
- E.9
正确答案 · Correct Answer
C
解析 · Solution
Solution 1
We can try to guess and check to find the answer. If she got five right, her score would be (5*5)-(5*2)=15. If she got six right her score would be (6*5)-(2*4)=22. That's close, but it's still not right! If she got 7 right, her score would be (7*5)-(2*3)=29. Thus, our answer is .
We can start with the full score, 50, and subtract not only 2 points for each incorrect answer but also the 5 points we gave her credit for. This expression is equivalent to her score, 29. Let be the number of questions she answers correctly. Then we represent the number incorrect by 10-x. $$$$$$$$
Suppose she got questions right. Then she got 10 - x questions wrong. Since she gains 5 points for a correct answer and loses 2 for an incorrect one, we can solve 5x - 2(10 - x) = 29 to get that .
We see that Olivia's score is odd. Since subtracting multiples of 2 (even numbers) does not change a number's parity (odd or even), Olivia's score from only her correct answers must be odd. Then, we test odd multiples of 5 greater than 29 to see which one works. The smallest odd multiple of 5 greater than 29 is 35, meaning 7 answers were correct and the remaining 3 answers were wrong. We see that the final score is 7(5)-3(2)=35-6=29, which is the score Olivia got. Thus, the answer is 7.
Suppose represents the number of correctly answered questions and represents the number of incorrectly answered questions. We can model the situation with the following system of equations:
$\begin{cases} x + y = 10 \5x - 2y = 29 \end{cases}
$ Multiply the first equation by 2 to get: $2x + 2y = 20$ Now add this equation to the second equation: $(2x + 2y) + (5x - 2y) = 20 + 29$ $7x = 49$ Solve for $$: $x = 7$ -N.继续练习