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Averages

Q.No: 1
Test Name : CAT 2017 Actual Paper 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

A
9.5
B
10
C
4.5
D
6
Answer key/Solution
Solution:
Q.No: 2
Test Name : CAT 2017 Actual Paper Slot 1

Suppose, log3 x = log12 y = a, where x, y are positive numbers. If G is the geometric mean of x and y, and log6 G is equal to

A
√a
B
2a
C
a/2
D
a
Answer key/Solution
Solution:
Q.No: 3
Test Name : CAT 2017 Actual Paper Slot 2

The average height of 22 toddlers increases by 2 inches when two of them leave this group. If the average height of these two toddlers is one-third the average height of the original 22, then the average height, in inches, of the remaining 20 toddlers is

A
30
B
23
C
32
D
26
Answer key/Solution
Solution:
Q.No: 4
Test Name : CAT 2018 Actual Paper Slot 1

In an apartment complex, the number of people aged 51 years and above is 30 and there are at most 39 people whose ages are below 51 years. The average age of all the people in the apartment complex is 38 years. What is the largest possible average age, in years, of the people whose ages are below 51 years?

A
26
B
25
C
28
D
27
Answer key/Solution
Solution:
Q.No: 5
Test Name : CAT 2018 Actual Paper Slot 1

A CAT aspirant appears for a certain number of tests. His average score increases by 1 if the first 10 tests are not considered, and decreases by 1 if the last 10 tests are not considered. If his average scores for the first 10 and the last 10 tests are 20 and 30, respectively, then the total number of tests taken by him is

Answer key/Solution
Solution:
Q.No: 6
Test Name : CAT 2018 Actual Paper Slot 2

The arithmetic mean of x, y and z is 80, and that of x, y, z, u and v is 75, where u=(x+y)/2 and v=(y+z)/2. If x ≥ z, then the minimum possible value of x is

Answer key/Solution
Solution:
Q.No: 7
Test Name : CAT 2019 Actual Paper Slot 1

Ramesh and Gautam are among 22 students who write an examination. Ramesh scores 82.5. The average score of the 21 students other than Gautam is 62. The average score of all the 22 students is one more than the average score of the 21 students other than Ramesh. The score of Gautam is

A
51
B
53
C
48
D
49
Answer key/Solution
Solution:
Q.No: 8
Test Name : CAT 2019 Actual Paper Slot 2

The average of 30 integers is 5. Among these 30 integers, there are exactly 20 which do not exceed 5. What is the highest possible value of the average of these 20 integers?

A
4.5
B
4
C
5
D
3.5
Answer key/Solution
Solution:
Q.No: 9
Test Name : CAT Actual Paper 2020 Slot-1

Let A, B and C be three positive integers such that the sum of A and the mean of B and C is 5. In addition, the sum of B and the mean of A and C is 7. Then the sum of A and B is

A
6
B
7
C
5
D
4
Answer key/Solution
Solution:
Q.No: 10
Test Name : CAT Actual Paper 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

A
3
B
6
C
5
D
4
Answer key/Solution
Solution:
Q.No: 11
Test Name : CAT Actual Paper 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

Answer key/Solution
Solution:
Q.No: 12
Test Name : CAT Actual Paper 2020 Slot-3

A batsman played n + 2 innings and got out on all occasions. His average score in these n + 2 innings was 29 runs and he scored 38 and 15 runs in the last two innings. The batsman scored less than 38 runs in each of the first n innings. In these n innings, his average score was 30 runs and lowest score was x runs. The smallest possible value of x is

A
1
B
2
C
4
D
3
Answer key/Solution
Solution:
Q.No: 13
Test Name : CAT Actual Paper 2021 Slot-1

Suppose hospital A admitted 21 less Covid infected patients than hospital B, and all eventually recovered. The sum of recovery days for patients in hospitals A and B were 200 and 152, respectively. If the average recovery days for patients admitted in hospital A was 3 more than the average in hospital B then the number admitted in hospital A was

Answer key/Solution
Solution:
Q.No: 14
Test Name : CAT Actual Paper 2021 Slot-1

Onion is sold for 5 consecutive months at the rate of Rs 10, 20, 25, 25, and 50 per kg, respectively. A family spends a fixed amount of money on onion for each of the first three months, and then spends half that amount on onion for each of the next two months. The average expense for onion, in rupees per kg, for the family over these 5 months is closest to

A
16
B
26
C
20
D
18
Answer key/Solution
Solution:
Q.No: 15
Test Name : CAT Actual Paper 2021 Slot-2

In a football tournament, a player has played a certain number of matches and 10more matches are to be played. If he scores a total of one goal over the next 10matches, his overall average will be 0.15 goals per match. On the other hand, if hescores a total of two goals over the next 10 matches, his overall average will be 0.2goals per match. The number of matches he has played is

Answer key/Solution
Solution:
Q.No: 16
Test Name : CAT Actual Paper 2021 Slot-3

The arithmetic mean of scores of 25 students in an examination is 50. Five of these students top the examination with the same score. If the scores of the other students are distinct integers with the lowest being 30, then the maximum possible score of the toppers is

Answer key/Solution
Solution:
Q.No: 17
Test Name : CAT Actual Paper 2022 Slot-1

The average of three integers is 13. When a natural number n is included, the average of these four integers remains and odd integer. The minimum possible value of n is

A
3
B
1
C
5
D
4
Answer key/Solution
Solution:
Q.No: 18
Test Name : CAT Actual Paper 2022 Slot-1

For any natural number n, suppose the sum of the first n terms of an arithmetic progression is (n + 2n2). If the nth term of the progression is divisible by 9, then the smallest possible value of n is

A
8
B
7
C
9
D
4
Answer key/Solution
Solution:
Q.No: 19
Test Name : CAT Actual Paper 2022 Slot-1

The average weight of students in a class increases by 600 gm when some new students join the class. If the average weight of the new students is 3 kg more than the average weight of the original students, then the ratio of the number of original students to the number of new students is

A
1 : 2
B
1 : 4
C
3 : 1
D
4 : 1
Answer key/Solution
Solution:
Q.No: 20
Test Name : CAT Actual Paper 2022 Slot-2

The average of a non-decreasing sequence of N numbers a1, a2, …, aN is 300. If a1 is replaced by 6a1, the new average becomes 400. Then the number of possible values of a1 is

Answer key/Solution
Solution:
Q.No: 21
Test Name : CAT Actual Paper 2022 Slot-2

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
20
B
21
C
22
D
24
Answer key/Solution
Solution:
Q.No: 22
Test Name : CAT Actual Paper 2022 Slot-2

If a and b are non-negative real numbers such that a + 2b = 6, then the average of the maximum and minimum possible values of (a + b) is

A
3
B
4.5
C
4
D
3.5
Answer key/Solution
Solution:
Q.No: 23
Test Name : CAT Actual Paper 2022 Slot-2

Manu earns Rs. 4000 per month and wants to save an average of Rs. 550 per month in a year. In the first nine months, his monthly expense was Rs. 3500, and he foresees that, tenth month onward, his monthly expense will increase to Rs. 3700. In order to meet his yearly savings target, his monthly earnings, in rupees, from the tenth month onward should be

A
4300
B
4200
C
4350
D
4400
Answer key/Solution
Solution:
Q.No: 24
Test Name : CAT Actual Paper 2022 Slot-3

The average of all 3-digit terms in the arithmetic progression 38, 55, 72, ..., is

Answer key/Solution
Solution:
The common difference of the AP 38, 55, 72, … is 17.
The smallest 3-digit number in the AP is 106 and the largest 3-digit number in it is 990.
So 106 + 123 + … + 973 + 990
= 106 + (106 + 17) + (106 + 2 × 17) + … +
(106 + 52 × 17)
= 53 × 106 + 17(1 + 2 + 3 + … + 52)
= 53 × 106 + 17 × 52 × 53/2
= 53 × 106 + 17 × 26 × 53
Hence, average of all 3-digit numbers
= (53 × 106 + 17 × 26 × 53)/53 = 106 + 442 = 548.
Q.No: 25
Test Name : CAT Actual Paper 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

Answer key/Solution
Solution:
Q.No: 26
Test Name : CAT Actual Paper 2022 Slot-3

Consider six distinct natural numbers such that the average of the two smallest numbers is 14, and the average of the two largest numbers is 28. Then, the maximum possible value of the average of these six numbers is

A
24
B
23.5
C
22.5
D
23
Answer key/Solution
Solution:
The sum of the two smallest numbers = 14 × 2 = 28
The sum of the two largest numbers = 28 × 2 = 56 To maximize the average of all the 6 numbers, we must try to maximize the two numbers in between.
This is possible when the two largest numbers are 27 and 29.
The maximum average case is
a, b, 25, 26, 27, 29; where a + b = 28v Hence, the average of these 6 numbers
= (a + b + 25 + 26 + 27 + 29)/6
= (28 + 25 + 26 + 27 + 29)/6 = 22.5.
Q.No: 27
Test Name : CAT Actual Paper 2023 Slot 1

In an examination, the average marks of 4 girls and 6 boys is 24. Each of the girls has the same marks while each of the boys has the same marks. If the marks of any girl is at most double the marks of any boy, but not less than the marks of any boy, then the number of possible distinct integer values of the total marks of 2 girls and 6 boys is

A
22
B
19
C
21
D
20
Answer key/Solution
Solution:
Q.No: 28
Test Name : CAT Actual Paper 2023 Slot 2

In a company, 20% of the employees work in the manufacturing department. If the total salary obtained by all the manufacturing employees is one-sixth of the total salary obtained by all the employees in the company, then the ratio of the average salary obtained by the manufacturing employees to the average salary obtained by the non-manufacturing employees is

A
6 : 5
B
4 : 5
C
5 : 4
D
5 : 6
Answer key/Solution
Solution:
Q.No: 29
Test Name : CAT Actual Paper 2023 Slot 3

There are three persons A, B and C in a room. If a person D joins the room, the average weight of the persons in the room reduces by x kg. Instead of D, if person E joins the room, the average weight of the persons in the room increases by 2x kg. If the weight of E is 12 kg more than that of D, then the value of x is

A
1.5
B
0.5
C
2
D
1
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The common difference of the AP 38, 55, 72, … is 17.
The smallest 3-digit number in the AP is 106 and the largest 3-digit number in it is 990.
So 106 + 123 + … + 973 + 990
= 106 + (106 + 17) + (106 + 2 × 17) + … +
(106 + 52 × 17)
= 53 × 106 + 17(1 + 2 + 3 + … + 52)
= 53 × 106 + 17 × 52 × 53/2
= 53 × 106 + 17 × 26 × 53
Hence, average of all 3-digit numbers
= (53 × 106 + 17 × 26 × 53)/53 = 106 + 442 = 548.

Solution:

Solution:
The sum of the two smallest numbers = 14 × 2 = 28
The sum of the two largest numbers = 28 × 2 = 56 To maximize the average of all the 6 numbers, we must try to maximize the two numbers in between.
This is possible when the two largest numbers are 27 and 29.
The maximum average case is
a, b, 25, 26, 27, 29; where a + b = 28v Hence, the average of these 6 numbers
= (a + b + 25 + 26 + 27 + 29)/6
= (28 + 25 + 26 + 27 + 29)/6 = 22.5.

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