Arithmetic

Mean, Median & Mode

Medium

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

$21$

$20$

$24$

$22$

✅ Correct Option: 2

We need to find the range of Amit's possible marks given several constraints.

We have:

$5$ students total (including Amit)

Marks range from $0$ to $50$ (integers only)

Average marks $= 38$ , so total marks $= 38 \times 5 = 190$

Exactly $3$ students scored more than $32$

No two students have the same marks

Amit has the lowest marks among all five

To minimize Amit's marks, we need to maximize the marks of the other four students.

Since exactly $3$ students scored more than $32$ :

$3$ students scored $33$ or higher

$2$ students (including Amit) scored $32$ or lower

To maximize the others' marks:

Give the highest possible marks to the $3$ students who scored above $32$ : $50$ , $49$ , $48$

The fourth student gets the maximum possible while staying $\leq 32$ : $32$

Amit gets: $190 - (50 + 49 + 48 + 32) = 190 - 179 = 11$

To maximize Amit's marks, we need to minimize the marks of the other four students while respecting all constraints.

Since Amit has the lowest marks:

Amit can score at most $31$ (anything higher would violate the "lowest marks" condition)

One other student must score $32$ (to satisfy the constraint that exactly $3$ scored above $32$ )

Now we need to distribute the remaining marks among the $3$ students who scored above $32$ :

Remaining total $= 190 - 31 - 32 = 127$

These $3$ students must score more than $32$ and have different marks

The minimum they could theoretically score is $33$ , $34$ , $35$ (total $= 102$ ), but we need $127$ .

So we have an extra $127 - 102 = 25$ marks to distribute.

One valid distribution: $33$ , $45$ , $49$ (check: $33 + 45 + 49 = 127$ ✓)

This gives us the maximum possible marks for Amit: $31$

Difference $=$ Maximum $-$ Minimum $= 31 - 11 = 20$

The constraint "exactly $3$ students scored more than $32$ " is crucial - it means exactly $2$ students scored $32$ or below. Since Amit has the lowest marks, this severely limits his possible range while keeping the total at $190$ .

CAT 2017 Slot 1

A class consists of $20$ boys and $30$ girls. In the mid-semester examination, the average score of the girls was $5$ higher than that of the boys. In the final exam, however, the average score of the girls dropped by $3$ while the average score of the entire class increased by $2$ . The increase in the average score of the boys is

CAT 2020 Slot 3

Dick is thrice as old as Tom and Harry is twice as old as Dick. If Dick's age is $1$ year less than the average age of all three, then Harry's age, in years, is

CAT 2024 Slot 2

A company has 40 employees whose names are listed in a certain order. In the year 2022, the average bonus of the first 30 employees was Rs. 40000, of the last 30 employees was Rs. 60000, and of the first 10 and last 10 employees together was Rs. 50000. Next year, the average bonus of the first 10 employees increased by 100%, of the last 10 employees increased by 200% and of the remaining employees was unchanged. Then, the average bonus, in rupees, of all the 40 employees together in the year 2023 was

2025 Slot 2

The average number of copies of a book sold per day by a shopkeeper is 60 in the initial seven days and 63 in the initial eight days, after the book launch. On the ninth day, she sells 11 copies less than the eighth day, and the average number of copies sold per day from second day to ninth day becomes 66. The number of copies sold on the first day of the book launch is

CAT 2022 Slot 3

In an examination, the average marks of students in sections $A$ and $B$ are $32$ and $60$ , respectively. The number of students in section $A$ is $10$ less than that in section $B$ . If the average marks of all the students across both the sections combined is an integer, then the difference between the maximum and minimum possible number of students in section $A$ is

CAT 2020 Slot 2

In a group of $10$ students, the mean of the lowest $9$ scores is $42$ while the mean of the highest $9$ scores is $47$ . For the entire group of $10$ students, the maximum possible mean exceeds the minimum possible mean by

Solution

Related questions:

Keyboard Shortcuts

CAT Past Year Questions (Topic-Wise)