A test has 50 questions. each right answer is worth 2 points; each wrong answer deducts 0.5 points; blank answers are not counted. a student got a score of 88.5. how many answers did he leave blank?

ANSWER: There are 2 blank answers   EXPLANATION   Let The number of right answers be ‘r’ The number of wrong answers be ‘w’ The number of blank answers be ‘b’   r + w + b = 50 This means r + w ≤ 50   Then we know, Right answers = 2 marks Wrong answers = -0.5 mark Blank Answers = 0 marks   2r – 0.5w = 88.5 2r = 88.5 + 0.5w                               … (Equation I)                      Since the score as .5, we know that there is at least one wrong answer, and the number of wrong answers is an odd number.   Since the score is 88.5, and each right answer gives 2 marks There are at more than 44 (i.e. 88/2) right answers   Since r + w ≤ 50, and possible values of w are odd numbers If r = 45, Possible values of w are 1, 3, and 5 If r = 46, Possible values of w are 1, 3 If r = 47, Possible values of w are 1, 3 If r = 48, The only possible value of w is 1 If r = 49,The only possible value of w is 1     Since 2r = 88.5 + 0.5w  (Equation I) We test for possible values:   If r = 45 2r = 88.5 + 0.5w   2(45) = 88.5 + 0.5w   90 = 88.5 + 0.5w 0.5w = 90 – 88.5 0.5w = 1.5 w = 3   So, If there are 45 right answers There are 3 wrong answers r + w + b = 50 45 + 3 + b = 50 48 + b = 50 b = 50 – 48 b = 2 Then, there are 2 blank answers.   If r = 46 2r = 88.5 + 0.5w 2(46) = 88.5 + 0.5w 92 = 88.5 + 0.5w 0.5w = 92 – 88.5 0.5w = 3.5 w = 7   So, If there are 46 right answers There are 7 wrong answers We know that r + w ≤ 50 46 + 7 = 53 So 46 and higher numbers are not possible solutions.   The only possible solution is: There are 45 right answers There are 3 wrong answers There are 2 blank answers