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The Highest Grade (Posted on 2003-12-11) Difficulty: 3 of 5
I had someone tell me, after seeing what grade they got on several quizzes, that he got the highest grade (after rounding to the nearest percent) for that letter, for all 5 quizzes. Each quiz he took, he got a different letter grade on it.

When the grades are calculated for these quizzes, it is the number of questions you got right divided by the number of questions total on the quiz. Then the score is taken as a percent (the decimal is multiplied by 100) and then rounded to the nearest percent (.5 and above rounds up, below .5 rounds down)

The grading scale works so that:

100-90 A
89-80 B
79-70 C
69-60 D
0-59 F

What are the fewest number of questions possible on each quiz?

For example, if someone got 6 questions right out of 7 questions total, it would be 6/7 or about 85.7%, which rounds to 86, which isn't the highest B possible, (86 is not equal to 89) Since no number over 7 can be in the 88.5 up to 89.5 range, there couldn't have been exactly 7 questions on the quiz. There also couldn't have been 8 questions on the B quiz. 7/8 or 88 percent isn't the highest B possible; it's too low, and 8/8 or 100 percent is too high.

No Solution Yet Submitted by Gamer    
Rating: 3.4000 (5 votes)

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Solution Continued fractions Comment 15 of 15 |

We could start with big quizzes, say of 100000 questions each.

Then the scores for the quizzes could be, say, 100000/100000, 89499/100000, etc. I assume Gamer's idea was to ask for convergents to these large numbers. 

We want to have the least number of questions that provides the closest convergent to each of these numbers. Since Gamer asks 'What are the fewest number of questions possible on each quiz?' I don't believe it's necessary for the number of questions to be the same in each quiz (and there is a neat answer if they aren't).

To achieve this we need the continued fraction expansion of each the scores in the big quizzes. 

89499/100000 [0; 1, 8, 1, 1, 10, 2, 2, 2, 7, 1, 1, 2], which we can truncate to [0; 1, 8, 1, 1]  = 17/19, or  0.894736842.

79499/100000 [0; 1, 3, 1, 7, 5, 2, 3, 3, 1, 1, 2, 3], which we can truncate to [0; 1, 3, 1, 7] = 31/39  or 0.794871795 

69499/100000 [0; 1, 2, 3, 1, 1, 2, 3, 2, 4, 1, 7, 5]  which we could truncate to [0; 1, 2, 3, 1] = 9/13 or 0.692307692; but if we allow another couple of terms, we obtain [0; 1, 2, 3, 1, 1, 2] = 41/59 or 0.694915254, a much better fit.

59499/100000 [0; 1, 1, 2, 7, 1, 1, 2, 2, 9, 1, 1, 10]  which we could truncate to [0; 1, 1, 2, 7] = 22/37 or 0.594594595, but again we can get a much better fit from [0; 1, 1, 2, 7, 1, 1] = 47/79 or 0.594936709.

If so we have a nice sequence of fractions with the closed form: (20n-1-2n^2)/(20n-1), which confirms that there was indeed just one question in the test for which an A was scored.



Edited on January 4, 2017, 2:59 am
  Posted by broll on 2017-01-04 02:44:59

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