a Math Olympiad quiz, 5 points are awarded for a correct answer and 2 points are ducted for a wrong answer or if a question is left unanswered. Shirley attempted all 30 estions and her total score for the quiz was not more than 66 . Find the maximum mber of correct answers she obtained.

To solve this problem, let's denote the number of correct answers Shirley gave as \( c \) and the number of wrong answers as \( w \). Since she attempted all 30 questions, we have: 1. \( c + w = 30 \) For scoring, penalties are applied to incorrect answers and unanswered questions. Since she didn't leave any questions unanswered but received penalties for incorrect answers, her total score can be expressed as: 2. \( \text{Total Score} = 5c - 2w \) Given that her total score was not more than 66, we can set up an inequality: 3. \( 5c - 2w \leq 66 \) From equation (1), we can substitute \( w = 30 - c \) into equation (3): \[ 5c - 2(30 - c) \leq 66 \] This simplifies to: \[ 5c - 60 + 2c \leq 66 \] Combining like terms gives: \[ 7c - 60 \leq 66 \] Adding 60 to both sides results in: \[ 7c \leq 126 \] Dividing by 7 leads to: \[ c \leq 18 \] Now we can conclude that the maximum number of correct answers Shirley could have obtained is 18. In this case, she would have 12 wrong answers (since \( w = 30 - c = 30 - 18 \)), and we can check the score: \[ \text{Score} = 5(18) - 2(12) = 90 - 24 = 66 \] Therefore, Shirley’s maximum number of correct answers is **18**!