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Question: The average score in an examination of 10 students of a class is 60. If the scores of the top five students are not considered, the average score of the remaining students falls by 5. The pass mark was 40 and the maximum mark was 100. It is also known that none of the students failed. If each of the top five scorers had distinct integral scores, the maximum possible score of the topper is:
- 87
- 95
- 99
- 100
- 103
Approach Solution (1)
The given problem can be solved using the concepts of averages.
1. Let the 10 students be s1, s2, s3, s4, s5, s6, s7, s8, s9, s10. The average score of the 10 students is 60.
(s1 + s2 + s3 + s4 + s5 + s6 + s7 + s8 + s9 + s10)/10 = 60,
s1 + s2 + s3 + s4 + s5 + s6 + s7 + s8 + s9 + s10 = 600 ……..(1)
2. Let the top 5 scorer students be s6, s7, s8, s9, s10. The average score of the remaining students except the top scorers is 55,
(s1 + s2 + s3 + s4 + s5)/5 = 55,
s1 + s2 + s3 + s4 + s5 = 275……...(2)
3. Subtract equation 2 from equation 1,
s6 + s7 + s8 + s9 + s10 = 325.
Assume s10 has scored the highest among all the students.
The least possible scores of s6, s7, s8, s9 are 55, 56, 57, and 57 as the scores are slightly greater than or equal to the average scores.
Hence, the maximum possible score by the topper is,
s10 = 325 - 55 - 56 - 57 - 58,
Marks scored by the topper = 325 - (226),
Highest possible marks scored by the topper = 99.
Correct option: C
Approach Solution (2)
The average score in an examination of 10 students of a class is 60
The total score = 10 x 60 = 600
The 5 smallest scores have an average of 55
The total score of the 5 smallest scores = 275
From above, the total score of the 5 largest scores = 600 - 275 = 325
Say the 5 largest scores are a, b, c, d, and e (where a < b < c < d < e, since each of the top five scorers had distinct scores). We want to maximize e. To maximize e, we need to minimize a, b, c, and d. The least value of a, is 55 (The least score of the top 5 , a, should be equal to the highest of the bottom 5 and to minimize the highest of the bottom 5, all scores from the bottom 5 should be equal). In this case the least values of b, c, and d are 56, 57, and 58 respectively:
a + b + c + d + e = 55 + 56 + 57 + 58 + e = 325;
e = 99
Correct option: C
Approach Solution (3)
If the value of all others except the largest is 55
Largest will be 60 x 10 - 55 x 9 = 600 - 495 = 105
However, the top 5 are distinct and the smallest of these can be 55, so remaining THREE will be 1, 2, & 3 more so the total combined value will be 1 + 2 + 3 or 6 more
Therefore the largest will be 6 less than max possible
= 105 - 6 = 99
Correct option: C
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