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Integer restriction

by yellowho » Thu Jan 20, 2011 12:13 am
On a certain test, 9 points are awarded for each correct answer, and 7 points are
deducted for each incorrect or unanswered question. If Sally received a total score of zero points,
how many questions are on the test?

(1) One of Sally's incorrect answers was later deemed correct, which made her new score 16.
(2) There are fewer than 30 questions on the test.

Total has to be a multiple of 16. Only one set of number under 30 that produces the given paramaters. Let's 2) said There are fewer than 33 questions on the test. Then it is E?
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by Rahul@gurome » Thu Jan 20, 2011 1:08 am
yellowho wrote:On a certain test, 9 points are awarded for each correct answer, and 7 points are deducted for each incorrect or unanswered question. If Sally received a total score of zero points, how many questions are on the test?

(1) One of Sally's incorrect answers was later deemed correct, which made her new score 16.
(2) There are fewer than 30 questions on the test.
As Sally got a total of zero points, the scenario must be as follows,
  • 9*(Number of questions answered correctly) = 7*(Number of other questions)
If we assume number of questions answered correctly = x and other questions = y, then (9x - 7y) = 0. Thus, x must be multiple of 7 and y must be the same multiple of 9, i.e. if x is 3*7 then y has to be 3*9 etc. Therefore, we can write x as 7n and y as 9n for some positive integer n.

Now, total number of questions = (x + y) = (7n + 9n) = 16n --> A multiple of 16

Statement 1: It's quiet obvious that if one of Sally's incorrect answers was later deemed correct, 9 points for correct answer as well as 7 points for reduction of an incorrect answer will be added to her score, i.e. her new score will be (0 + 9 + 7) = 16. Hence it is not a new information.

Not sufficient

Statement 2: As there are fewer than 30 questions on the test and total numbers of question is a multiple of 16, total number of question is either 0 or 16. Now if it is a real test it must have at least 1 question, hence the number of questions in the test is 16.

Sufficient

The correct answer is B.
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by Rahul@gurome » Thu Jan 20, 2011 1:10 am
yellowho wrote:Total has to be a multiple of 16. Only one set of number under 30 that produces the given paramaters. Let's 2) said There are fewer than 33 questions on the test. Then it is E?
Yes.
Then there will be two possible number of questions, 16 and 32.
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by Rahul@gurome » Thu Jan 20, 2011 1:11 am
yellowho wrote:Total has to be a multiple of 16. Only one set of number under 30 that produces the given paramaters. Let's 2) said There are fewer than 33 questions on the test. Then it is E?
Yes.
Then there will be two possible number of questions, 16 and 32.
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