Averages , Ratio and Proportion Question 8 - CAT 2022
Five students, including Amit, appear for an examination in which possible marks are integers between 0 and 50, both inclusive. The average marks for all the students is 38 and exactly three students got more than 32. If no two students got the same marks and Amit got the least marks among the five students, then the difference between the highest and lowest possible marks of Amit is:
Step-by-Step Explanation
Total marks scored by all 5 students = 5 × 38 = 190 units.
Let the individual scores be ranked in descending order:
x₁ > x₂ > x₃ > x₄ > x₅
Since Amit scored the lowest, Amit's score is x₅.
- Find Amit's lowest possible score (x₅):
To make Amit's score as small as possible, we must maximize the scores of the other 4 students.
The problem states that exactly three students scored strictly more than 32 (x₁, x₂, x₃ > 32), which means x₄ cannot be greater than 32 (x₄ ≤ 32).
Maximizing these values using distinct integers under the 50-mark limit:
x₁ = 50, x₂ = 49, x₃ = 48, x₄ = 32
Sum of these maximum scores = 50 + 49 + 48 + 32 = 179
Amit's lowest possible score = 190 − 179 = 11
- Find Amit's highest possible score (x₅):
Amit scored the lowest, so his score is bounded above by x₄.
We know that exactly three students scored more than 32, so x₄ can be at most 32.
Since all scores are distinct integers, Amit's score (x₅) can be at most:
Amit's highest possible score = 32 − 1 = 31
(We check this upper bound: If x₅ = 31 and x₄ = 32, the remaining three students must sum to 190 − 32 − 31 = 127. Choosing distinct integers greater than 32 like 33, 44, and 50 satisfies all conditions perfectly).
Difference = 31 − 11 = 20
Correct Option: D
💡 FundaMakers Tip:
The constraint "exactly three students scored more than 32" creates a sharp threshold line right between x₃ and x₄. This locks x₄ ≤ 32, which serves as the upper bound to minimize Amit's score and the lower bound to maximize it.