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? Possible answers { 1, 2, 3, 4, 5 }
This is a problem from the second Bergen Academies sample admission test for middle schoolers. It measures methodical thinking. This question reminds me of the old SAT where a there was a penalty for wrong answers to discourage guessing. I’m sure SAT candidates will be happy to know that the new SAT no longer penalizes wrong answers. However, read this question carefully and do not conclude, that since you’ve been subjected to all types of sadistic exams, that you know the grading requirement. For example, do not assume that you’re also penalized for NOT answering questions. I’ve seen students fly over the statement that “blank answers are not counted” and bang their heads trying to solve this question. Also, wrong answers reduce your score, not add to it.
An algebraic approach to solving this problem will take too long. Try it. The strategy for solving this problem should be a trial-and-error approach.
Since right answers are worth 2 points, then right answers can only be even valued. Since wrong answers deduct ½ a point, then the presence of odd scores or half scores indicate wrong answers. Thus the score sets the minimum bound for right answers. In this case, we can safely assume that at minimum 88.5 / 2 = 44 right answers. But it has to be more right answers since points were deducted. So, we’ll start our trial-and-error starting at 45.
Create a table thusly;
|
Right (x2) |
Wrong (x -0.5) |
Score |
|
45 x 2 = 90 |
1 x -0.5 = -0.5 |
89.5 |
|
ditto |
2 x -0.5 = -1.0 |
89.0 |
|
ditto |
3 x -0.5 = -1.5 |
88.5 which is the score! |
The more astute student will have realized that (2 x Right - 88.5) / 0.5 = number of Wrong answers.
We can continue to see if there are more possible answers. If Right = 46, then Wrong = (2 x 46 - 88.5) / 2 = 7. That’s not possible since there are only 50 questions; and 46 right + 7 wrong is greater than 50 questions.
So what’s the right answer? If you chose 3, it’s wrong!
It’s common to choose 3 as the answer since finding the number of wrong answers takes the most effort, and most test takers stop there. If you chose 3, please re-read the question carefully.
The question asks “How many answers did he leave blank?” for which the correct response is there are 2 questions left blank. Don’t get suckered into incomplete answers! If you’re methodical and created that table with a column for Blanks, you’ll likely not step into this type of trap.
|
Right (x2) |
Wrong (x -0.5) |
Score |
Blank |
|
45 x 2 = 90 |
1 x -0.5 = -0.5 |
89.5 |
4 |
|
ditto |
2 x -0.5 = -1.0 |
89.0 |
3 |
|
ditto |
3 x -0.5 = -1.5 |
88.5 the score and so... |
2 is the answer! |
Ironically, it’s often the student that derives that great “number of Wrong answers” formula that forgets to solve for blanks.
Here is a short video with some great test taking tips.
Contact:
Ruby Yao and Benedict Zoe, Mathnasium of Fort Lee
201-969-6284 (WOW-MATH), [email protected]
246 Main St. #A
Fort Lee, NJ 07024
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(201) 461-6284
(201) 969-6284
