In an examination, the Data Insights section has 20 questions. Each correct answer awards 3 points, while each incorrect answer results in a 1-point deduction, and there is no penalty for unanswered questions. During his only attempt, Paul scored 35 points on this section. Based on the above information, select for Unattempted the maximum number of questions that Paul could have left unanswered and for Incorrect the maximum number of questions that Paul could have answered incorrectly and still scored 35 points. Make only two selections, one in each column.

Official Solution: To maximize the number of questions that Paul could have left unanswered, we should minimize the number of correct and incorrect answers. The minimum number of questions Paul could have answered correctly to score 35 is 12. In this case, he would have earned 3 * 12 = 36 points, so he must have answered one question incorrectly to reduce the score to 35. This results in a total of 12 + 1 = 13 correct and incorrect answers, making the maximum number of unanswered questions equal to 7. To get the maximum number of questions Paul could have answered incorrectly, let's check the options: 8 incorrect would result in -8 points, meaning 43 points would need to be gained with correct answers, which is not possible (43 is not a multiple of 3). 7 incorrect would result in -7 points, meaning 42 points would need to be gained with correct answers, requiring 14 correct answers. This is not possible because 7 + 14 > 20. 6 incorrect would result in -6 points, meaning 41 points would need to be gained with correct answers, which is not possible (41 is not a multiple of 3). 5 incorrect would result in -5 points, meaning 40 points would need to be gained with correct answers, which is not possible (40 is not a multiple of 3). 4 incorrect would result in -4 points, meaning 39 points would need to be gained with correct answers, requiring 13 correct answers. This is possible with the addition of 3 unanswered questions: 4 + 13 + 3 = 20. Therefore, the maximum number of questions Paul could have answered incorrectly is 4. Correct answer: Unattempted "7" Incorrect "4"

I have revised the question, solution, and formatting by adding more details to enhance clarity

Just wanted to say thanks Bunuel for dropping such quality questions! I just missed out "maximum" keeping in mind like most of the regular TPA problems, I arrived at 3 and 4 thinking 13*3 + 4*-1 = 35 but NO, question was to be answered separately for max values :/

I like the solution - it’s helpful.

I like the solution - it’s helpful. Scored 35 points could be misinterpreted as scored 35 or more points. This interpretation would rely on read 36 points as having scored 35 points in addition to having scored 1 additional point as well (viewing as an accumulation of points). A less ambiguous phrasing may be "received a score of 35", which removes the issue of disregarding "excess points".

I like the solution - it’s helpful.

Bunuel Can I understand why is the question interpretated as two separate cases i.e. the maximum number of unanswered and maximum number of incorrect questions are independent of each other? The word "and" makes me thinks that the answers to both should be solved as a whole. I created two equations where x is correct, y is incorrect, z is unanswered: (1) x + y + z = 20 (2) 3x - y = 35 Adding these two equations together give me 4y + 3z = 25 and the only possible answers is y = 4 and z = 3