Question

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

• A. 21
• B. 24
• C. 20
• D. 22

Answer 1

The Correct Option is C

Let's determine the difference between the highest and lowest possible marks of Amit. Given:

• Average marks for all five students is 38, so the total marks = 38 × 5 = 190.
• Exactly three students scored more than 32.
• No two students have the same marks.
• Amit has the least marks.

Let's denote the marks of the five students as a, b, c, d, and e, with a being Amit's marks and a < b < c < d < e.

Since exactly three students scored more than 32, then:

• c, d, e > 32

Let’s assign the minimum possible values satisfying the above conditions:

• c = 33, d = 34, e = 35

Then:

• b > a, and a, b ≤ 32 for conditions to meet.

Let's calculate:

• a + b + 33 + 34 + 35 = 190
• a + b = 190 - 102 = 88
• The possible maximum value of a is when a < b and b is maximized<= 32, hence:
• b = 32, a = 88 - 32 = 56 (Not possible as a ≤ 32)
• Therefore, adjust: b = 31, a = 88 - 31 = 57 (Still not valid)
• To ensure a ≤ 32, it requires:
• b = 32, a = 31

For the least marks for Amit satisfying no same marks & conditions:

• Assign different higher possible to c, d, e:
• Example: c = 40, d = 41, e = 42
• a + b + 40 + 41 + 42 = 190
• a + b = 190 - 123 = 67
• Minimum a = 12, achieved when b = 55 (as the highest possible under 62)

The difference in highest and lowest possible for Amit thus is 31 - 11 = 20.

Highest	31
Lowest	11
Difference	20

Answer 2

Given:
The average marks of all 5 students = \(38\) 

Step 1: Calculate total marks of all students
\(\text{Total marks} = 5 \times 38 = 190\)

Step 2: To find the minimum marks of Amit
This occurs when the other 4 students score the maximum possible marks.
Suppose the 4 highest scores are: \(50, 49, 48, 32\)
\(\text{Sum of these scores} = 50 + 49 + 48 + 32 = 179\)
So, minimum marks of Amit = \(190 - 179 = 11\)

Step 3: To find the maximum marks of Amit
Since Amit scored the least, the maximum possible value of his marks is just below the next lowest student’s marks.
From the group of marks: \(50, 49, 48, 32\), the least of these is 32,
So, \(\text{Maximum possible marks of Amit} = 31\)

Step 4: Find the difference between highest and lowest possible marks of Amit
\(31 - 11 = 20\)

Final Answer: \(20\)
Correct Option: (C)