2002 AMC 8 Problems/Problem 17
Contents
Problem
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?
Solution 1
We can try to guess and check to find the answer. If she got five right, her score would be . If she got six right her score would be . That's close, but it's still not right! If she got 7 right, her score would be . Thus, our answer is . ~avamarora
Solution 2
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 will represent the number incorrect by .
Solution 3
Suppose she got questions right. Then she got questions wrong. Since she gains 5 points for a correct answer and loses 2 for an incorrect one, we can solve to get that .
Solution 4
We see that Olivia's score is odd. Since subtracting multiples of (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 greater than to see which one works. The smallest odd multiple of greater than is , meaning answers were correct and the remaining answers were wrong. We see that the final score is which is the score Olivia got. Thus, the answer is .
Video Solution
https://youtu.be/8YXPMTjOyvM Soo, DRMS, NM
https://www.youtube.com/watch?v=aTeyOXo6-Uo ~David
Video Solution by OmegaLearn
https://youtu.be/rQUwNC0gqdg?t=1560
~pi_is_3.14
See Also
2002 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
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All AJHSME/AMC 8 Problems and Solutions |
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