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What should be the minimum no of students in a class room in which 25% student score 75 marks,10% student score 85 marks,15 % student score less than 50 marks, 25% score 60 marks and remaining student score more than 90 marks..

My Approach:

75 MARKS------------------------------->25%

80 MARKS------------------------------->10%

50 MARKS(LESS THAN)-------------->15%

60 MARKS------------------------------->25%

90 MARKS(REMAINING)------------>25%

How to solve this problem after this as I am unable to do after it?

2)What would be the answer if it would be atleast and almost number of students?

Community
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justin takro
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2 Answers2

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If you take the greatest common divisor of your percentages (which is easy to see is 5), then the minimum number of students is $100 / 5 = 20$ (Why?)

user2566092
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  • Why 100/5? I have not understood – justin takro Sep 22 '15 at 14:50
  • Why I should H.C.F also? – justin takro Sep 22 '15 at 14:58
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    Its obvious that with 100 students in the class, all the %age figures will yield whole numbers. To minimize the no. of students in the class means to maximize the divisor of 100 such that the %age figures still remain whole numbers. Note that the %age figures will be proportionately reduced as the total no. of students will get reduced. Let's assume this divisor is x. Hence, 100/x, 25/x, 15/x, 10/x, etc. must also be whole numbers. Which means, x must be a divisor of all of the %age figures just like 100. Maximum value of x, by definition, must be the HCF of the %age figures :) – Deepak Gupta Sep 22 '15 at 15:02
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    The other answer by @tomi gives the same reasoning using some least common multiple reasoning instead of greatest common divisor. However the logic is exactly the same for this particular problem, so if you understand that answer better, then go with that one. – user2566092 Sep 22 '15 at 15:03
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The key to this is understanding that you can't have a fraction of a student. So if 25% (one quarter) of students score 75 marks, that means that the number of students must be a multiple of 4.

Because 10% (one tenth) of students score 80 marks, the number of students must be a multiple of 10.

The 15% sounds trickier. Three twentieths of the number of students must be an integer - how do we accommodate that? But in fact it is not a problem at all. We already know that the number of students must be a multiple of 4 and a multiple of 10. The Lowest Common Multiple of 4 and 10 is 20, so the number of students must be a multiple of 20. If so, then three twentieths of that number will be an integer, so there is no further constraint.

tomi
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  • What would be atleast and almost number of students? – justin takro Sep 22 '15 at 14:57
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    @justintakro If you mean "what is the maximum number of students", there is no maximum: Any number of students which is $20n$ for some positive integer $n$ will work (and these are the only values that work). The minimum, as stated in both answers, is when $n = 1$ which gives 20 students. – user2566092 Sep 22 '15 at 15:16