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There is a group of 100 students. They study one or more of the 3 subjects among Geography, History and English. The number of students studying English is more than the number of students studying Geography, which, in turn, is more than the number of students who study History, which in turn is more than the number of students who study exactly 2 out of the 3 subjects, which in turn, is more than the number of students who study all 3 subjects. It is known that at least one student studies all 3 subjects.
Q. If it is known that exactly half the students study English, then what is the maximum number of students who study all three?
- a)16
- b)18
- c)15
- d)17
- e)19
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
There is a group of 100 students. They study one or more of the 3 subj...
Solution: If exactly 50 students study English, the maximum values for Geography and History are 49 and 48. A maximum of 147 instances from 100 students. v The students who study exactly 3 is less than the number who study exactly 2, by trial and error we can say the maximum value of exactly 3 would be around 17. (17 x 3) + (18 x 2) + 19 + 20 + 21 = 147 instances from 17 + 18 + 19 + 20 + 21 =95 students v Number of students only add upto 95, maximum value of students who study exactly 3 is at most 16. (16 x 3) + (17 x 2) + 20 + 22 + 23 = 147 instances from 16 + 17 + 20 + 22 + 23 = 98 students We can see that 16 is also not possible and the solution would be 15: (15 x 3) + (17 x 2) + 21 + 23 + 24 = 147 instances from 15 + 17 + 21 + 23 + 24 = 100 students. Hence, option 3.
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Most Upvoted Answer
There is a group of 100 students. They study one or more of the 3 subj...
Understanding the Problem
In this scenario, we have 100 students studying three subjects: Geography, History, and English. The relationships between the number of students studying each subject and the constraints provided are critical to solving the problem.
Key Relationships
- Let \( E \) = number of students studying English
- Let \( G \) = number of students studying Geography
- Let \( H \) = number of students studying History
- Let \( x \) = number of students studying exactly 2 subjects
- Let \( y \) = number of students studying all 3 subjects
Given conditions:
- \( E > G > H > x > y \)
- \( E = 50 \) (exactly half of 100)
Constraints Analysis
Since \( E = 50 \), we know:
- \( G \) must be less than 50, let's denote \( G = 49 \) as a starting point.
- \( H \) must be less than \( G \), so \( H \) can be 48 or lower.
- The number of students studying exactly 2 subjects \( x \) must be less than \( H \).
Calculating the Maximum for \( y \)
Let's represent the maximum feasible numbers for the other variables:
- Start with \( G = 49 \)
- Set \( H = 48 \)
- Then \( x \) could be 47 (just below \( H \))
- Given the inequalities, the number of students studying all three subjects \( y \) must be less than \( x \), so \( y \) could be at maximum 46.
- However, this would violate the condition that at least one student studies all three subjects, and we need the maximum value for \( y \).
To satisfy all conditions:
1. Let \( y = 15 \)
2. Then \( x \) can be \( 16 \) (which is less than \( y \)), meeting \( E > G > H > x > y \).
Conclusion
Thus, the maximum number of students who study all three subjects is 15, making option 'C' the correct answer.
In this scenario, we have 100 students studying three subjects: Geography, History, and English. The relationships between the number of students studying each subject and the constraints provided are critical to solving the problem.
Key Relationships
- Let \( E \) = number of students studying English
- Let \( G \) = number of students studying Geography
- Let \( H \) = number of students studying History
- Let \( x \) = number of students studying exactly 2 subjects
- Let \( y \) = number of students studying all 3 subjects
Given conditions:
- \( E > G > H > x > y \)
- \( E = 50 \) (exactly half of 100)
Constraints Analysis
Since \( E = 50 \), we know:
- \( G \) must be less than 50, let's denote \( G = 49 \) as a starting point.
- \( H \) must be less than \( G \), so \( H \) can be 48 or lower.
- The number of students studying exactly 2 subjects \( x \) must be less than \( H \).
Calculating the Maximum for \( y \)
Let's represent the maximum feasible numbers for the other variables:
- Start with \( G = 49 \)
- Set \( H = 48 \)
- Then \( x \) could be 47 (just below \( H \))
- Given the inequalities, the number of students studying all three subjects \( y \) must be less than \( x \), so \( y \) could be at maximum 46.
- However, this would violate the condition that at least one student studies all three subjects, and we need the maximum value for \( y \).
To satisfy all conditions:
1. Let \( y = 15 \)
2. Then \( x \) can be \( 16 \) (which is less than \( y \)), meeting \( E > G > H > x > y \).
Conclusion
Thus, the maximum number of students who study all three subjects is 15, making option 'C' the correct answer.
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There is a group of 100 students. They study one or more of the 3 subjects among Geography, History and English. The number of students studying English is more than the number of students studying Geography, which, in turn, is more than the number of students who study History, which in turn is more than the number of students who study exactly 2 out of the 3 subjects, which in turn,is more than the number of students who study all 3 subjects. It is known that at least one student studies all 3 subjects.Q. If it is known that exactly half the students study English, then what is the maximum number of students who study all three?a)16b)18c)15d)17e)19Correct answer is option 'C'. Can you explain this answer?
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