Previous Year CAT Questions for Arithmetic

1. CAT 2022 QA Slot 1 | Arithmetic - Average CAT Question

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 number of original students to the number of new students is

(a)
3 : 1
(b)
4 : 1
(c)
1 : 2
(d)
1 : 4

2. CAT 2022 QA Slot 1 | Arithmetic - Average CAT Question

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

(a)
5
(b)
1
(c)
3
(d)
4

Answer: Option A

Explanation :

Sum of the original 3 numbers = 3 × 13 = 39.

Now, $\frac{39 + n}{4}$ = odd = 2k - 1

⇒ 39 + n = 8k – 4

⇒ 43 + n = 8k

For k to be an integer least possible natural value of n = 5

Hence, option (a).

3. CAT 2022 QA Slot 2 | Arithmetic - Average CAT Question

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

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Answer: 14

Explanation :

Given, $\frac{a_{1} + a_{2} + . . . + a_{N}}{N}$ = 300

⇒ a₁ + a₂ + ... + a_N = 300N ...(1)

Also, $\frac{6 a_{1} + a_{2} + . . . + a_{N}}{N}$ = 400

⇒ 6a₁ + a₂ + ... + a_N = 400N ...(2)

(2) - (1)

⇒ 5a₁ = 100N

⇒ a₁ = 20N

Since, a₁ is the least of the given numbers it cannot be more than the average, hence a₁ ≤ 300.

⇒ N ≤ 15

If N = 1, a₁ = 20 and average cannot be equal to 300.

Hence, N can take all values from 2 till 15, i.e., 14 values.

Hence, 14.

4. CAT 2022 QA Slot 3 | Arithmetic - Average CAT Question

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

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Answer: 63

Explanation :

Let the number of students in section B is x and that in A is (x – 10).
⇒ $\frac{(x - 10) \times 32 + x \times 60}{2 x - 10}$ = integer

⇒ $\frac{92 x - 320}{2 x - 10}$ = integer

⇒ $\frac{46 x - 160}{x - 5}$ = integer

⇒ $\frac{46 (x - 5) + 70}{x - 5}$ = integer

⇒ (x - 5) + $\frac{70}{x - 5}$ = integer

⇒ $\frac{70}{x - 5}$ should be an integer

∴ (x – 5) should be a factor of 70 while x > 10

⇒ Highest possible value of x = 75 while lowest possible value is 12

∴ Difference between highest and lowest values of x = 75 – 12 = 63.

Hence, 63.

5. CAT 2022 QA Slot 3 | Arithmetic - Average CAT Question

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)
23
(b)
22.5
(c)
24
(d)
23.5

6. CAT 2021 QA Slot 1 | Arithmetic - Average CAT Question

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

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Answer: 35

Explanation :

Let the number of patients in hospital A = x
∴ The number of patients in hospital B = x + 21

⇒ $\frac{200}{x}$ = $\frac{152}{x + 21}$ + 3

⇒ 200x + 4200 = 152x + 3x² + 63x

⇒ 3x² + 15x - 4200 = 0

⇒ x² + 5x - 1400 = 0

⇒ x = 35 or -40 (not possible)

∴ Number of patients in hospital A = 35

Hence, 35.

7. CAT 2021 QA Slot 1 | Arithmetic - Average CAT Question

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)
20
(b)
16
(c)
18
(d)
26

8. CAT 2021 QA Slot 2 | Arithmetic - Average CAT Question

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

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9. CAT 2021 QA Slot 3 | Arithmetic - Average CAT Question

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

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10. CAT 2020 QA Slot 1 | Arithmetic - Average CAT Question

The mean of all 4-digit even natural numbers of the form ‘aabb’, where a > 0, is

(a)
5050
(b)
4864
(c)
4466
(d)
5544

11. CAT 2020 QA Slot 1 | Arithmetic - Average CAT Question

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)
4
(b)
5
(c)
6
(d)
7

Answer: Option C

Explanation :

Given,

$a + \frac{b + c}{2} = 5$

⇒ 2a + b + c = 10 …(1)

Also, $b + \frac{a + c}{2} = 7$

⇒ 2b + a + c = 14 …(2)

Solving (1) and (2) we get,

b – a = 4

⇒ b = a + 4

Substituting this in the (1)

⇒ 2a + a + 4 + c = 10

⇒ 3a + c = 6

Given all three as positive integers, only possible value for a is 1. (c cannot be 0)

So, when a = 1, c = 3 and b = 5

∴ a + b = 6.

Hence, option (c).

12. CAT 2020 QA Slot 2 | Arithmetic - Average CAT Question

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)
4
(b)
5
(c)
3
(d)
6

Answer: Option A

Explanation :

Let the highest and lowest score be h and l respectively and total score of remaining 8 students be x.

The mean of the lowest 9 scores is 42

⇒ x + l = 9 × 42 = 378 …(1)

The mean of the highest 9 scores is 47

⇒ x + h = 9 × 47 = 423 …(2)

(2) – (1)

⇒ h – l = 423 – 378 = 45

Case 1: Least possible average is when we minimize the highest marks. The least highest marks can be 47.

∴ Lowest score = 47 – 45 = 2

∴ Least average = $\frac{9 \times 47 + 2}{10}$ = 42.5

Case 2: Highest possible average is when we maximize the lowest marks. The highest lowest marks can be 42.

∴ Highest score = 42 + 45 = 87

∴ Highest average = $\frac{9 \times 42 + 87}{10}$ = 46.5

∴ The required difference = 46.5 – 42.5 = 4

Hence, option (a).

13. CAT 2020 QA Slot 3 | Arithmetic - Average CAT Question

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)
3
(b)
4
(c)
1
(d)
2

14. CAT 2020 QA Slot 3 | Arithmetic - Average CAT Question

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

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15. CAT 2019 QA Slot 1 | Arithmetic - Average CAT Question

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)
53
(b)
51
(c)
49
(d)
48

16. CAT 2019 QA Slot 2 | Arithmetic - Average CAT Question

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
(b)
5
(c)
4.5
(d)
3.5

17. CAT 2018 QA Slot 1 | Arithmetic - Average CAT Question

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

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Answer: 60

Explanation :

Let there be ‘n’ tests and their average be ‘x’.

By the given conditions,

When first 10 tests are not considered:
⇒ $\frac{n x - 200}{n - 10}$ = x + 1

⇒ x = $\frac{n x - 200}{n - 10}$ - 1 ...(1)

When last 10 tests are not considered:
⇒ $\frac{n x - 300}{n - 10}$ = x - 1

⇒ x = $\frac{n x - 300}{n - 10}$ + 1 ...(2)

Equating (1) and (2)
∴ $\frac{n x - 200}{n - 10}$ - 1 = $\frac{n x - 300}{n - 10}$ + 1

∴ $\frac{n x - 200}{n - 10}$ - $\frac{n x - 300}{n - 10}$ = 2

∴ $\frac{100}{n - 10}$ = 2

∴ n = 60

Hence, 60.

18. CAT 2018 QA Slot 1 | Arithmetic - Average CAT Question

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)
27
(b)
28
(c)
26
(d)
25

19. CAT 2018 QA Slot 2 | Arithmetic - Average CAT Question

Let a₁, a₂, ... , a₅₂ be positive integers such that a₁ < a₂ < ... < a₅₂. Suppose, their arithmetic mean is one less than the arithmetic mean of a₂, a₃, ..., a₅₂. If a₅₂ = 100, then the largest possible value of a₁ is

(a)
45
(b)
23
(c)
48
(d)
20

20. CAT 2017 QA Slot 1 | Arithmetic - Average CAT Question

An elevator has a weight limit of 630 kg. It is carrying a group of people of whom the heaviest weighs 57 kg and the lightest weighs 53 kg. What is the maximum possible number of people in the group?

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21. CAT 2017 QA Slot 2 | Arithmetic - Average CAT Question

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)
28
(c)
32
(d)
26

Answer: Option C

Explanation :

Let the average height of 22 toddlers be ‘x’ inches.

Now as the average height of the 2 toddlers who leave the group is one-third the average height of the entire group 22, their average = x/3 inches.

Further, the average height of the balance 20 toddlers in terms of x will be ‘x + 2’ inches. So the average height of all 22 toddlers can be expressed as

x = $\frac{2 (\frac{x}{3}) + 20 (x + 2)}{22}$

Hence, 22x = 2x/3 + 20x + 40

⇒ x = 30

So the average height of the balance

20 toddlers will be 30 + 2 or 32 inches.

Hence, option (c).

22. CAT 2007 QA | Arithmetic - Average CAT Question

Each question is followed by two statements A and B. Answer each question using the following instructions.
Mark (1) if the question can be answered by using statement A alone but not by using statement B alone.
Mark (2) if the question can be answered by using statement B alone but not by using statement A alone.
Mark (3) if the question can be answered by using both the statements together but not by using either of the statements alone.
Mark (4) if the question cannot be answered on the basis of the two statements.

The average weight of a class of 100 students is 45 kg. The class consists of two sections, I and II, each with 50 students. The average weight, W_I, of Section I is smaller than the average weight, W_II, of Section II. If the heaviest student, say Deepak, of Section II is moved to Section I, and the lightest student, say Poonam, of Section I is moved to Section II, then the average weights of the two sections are switched, i.e., the average weight of Section I becomes W_II and that of Section II becomes W_I. What is the weight of Poonam?

A. W_II – W_I = 1.0
B. Moving Deepak from Section II to I (without any move from I to II) makes the average weights of the two sections equal.

(a)
1
(b)
2
(c)
3
(d)
4

23. CAT 2007 QA | Arithmetic - Average CAT Question

Consider the set S = {2, 3, 4, ..., 2n + 1}, where n is a positive integer larger than 2007. Define X as the average of the odd integers in S and Y as the average of the even integers in S. What is the value of X – Y?

(a)
0
(b)
1
(c)
n/2
(d)
n + 1/2n
(e)
2008

24. CAT 2007 QA | Arithmetic - Average CAT Question

Ten years ago, the ages of the members of a joint family of eight people added up to 231 years. Three years later, one member died at the age of 60 years and a child was born during the same year. After another three years, one more member died, again at 60, and a child was born during the same year. The current average age of this eight member joint family is nearest to:

(a)
23 years
(b)
22 years
(c)
21 years
(d)
25 years
(e)
24 years

25. CAT 2002 QA | Arithmetic - Average CAT Question

A boy finds the average of 10 positive integers. Each integer contains two digits. By mistake, the boy interchanges the digits of one number say ba for ab. Due to this, the average becomes 1.8 less than the previous one. What was the difference of the two digits a and b?

(a)
4
(b)
2
(c)
6
(d)
8

Answer: Option B

Explanation :

Let the Arithmetic Mean of the 10 numbers be x and s be the sum of the remaining 9 numbers.

$\frac{x + 10 a + b}{10} = x$ ...(i)

Interchanging the number ab with ba,

$\frac{x + 10 b + a}{10} =$ x + 1.8 ... (ii)

Subtracting (i) from (ii), we get,

10b + a − (10a + b) = 9(b − a) = 18

b − a = 2

Hence, option (b).

Arithmetic - Average - Previous Year CAT/MBA Questions

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