Arithmetic - Average - Previous Year CAT/MBA Questions
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Read the following scenario and answer the THREE questions that follow.
The upper hinge of a dataset is the median of all the values to the right of the median of the dataset in an ascending arrangement, while the lower hinge of the dataset is the median of all the values to the left of the median of the dataset in the same arrangement. For example, consider the dataset 4, 3, 2, 6, 4, 2, 7. When arranged in the ascending order, it becomes 2, 2, 3, 4, 4, 6, 7. The median is 4 (the bold value), and hence the upper hinge is the median of 4, 6, 7, i.e., 6. Similarly, the lower hinge is 2.
A student has surveyed thirteen of her teachers, and recorded their work experience (in integer years). Two of the values recorded by the student got smudged, and she cannot recall those values. All she remembers is that those two values were unequal, so let us write them as A and B, where A < B. The remaining eleven values, as recorded, are: 5, 6, 7, 8, 12, 16, 19, 21, 21, 27, 29. Moreover, the student also remembers the following summary measures, calculated based on all the thirteen values:
Minimum: 2
Lower Hinge: 6.5
Median: 12
Upper Hinge: 21
Maximum: 29
Which of the following is a possible value of B?
- (a)
2
- (b)
6
- (c)
8
- (d)
13
- (e)
29
Answer: Option C
Text Explanation :
The known observations are: 5, 6, 7, 8, 12, 16, 19, 21, 21, 27, 29
Median of all 13 observations is 12, hence there must be 6 values greather than or equal to 12 and 6 values less than or equal to 12.
We see that for known observations, we already have 6 values (16, 19, 21, 21, 27 and 29) which are greather than 12. Hence, both the unknown values i.e., A and B must be less than 12.
Also, the smallest of these observations is 2 (given) and we know A < B, hence A = 2.
The lowest 6 observations are 2, 5, 6, 7, 8 and B.
The lower hinge is 6.5. Hence, B should take a value such than 6 and 7 are the middle two numbers when arranged in ascending order.
This is possible when B is greater than or equal to 7 and less than or equal to 12.
∴ Possible values of B are 7, 8, 9, 10, 11 or 12.
Hence, option (c).
Workspace:
Based on the information recorded, which of the following can be the average work experience of the thirteen teachers?
- (a)
12
- (b)
12.5
- (c)
13
- (d)
13.5
- (e)
14
Answer: Option E
Text Explanation :
Workspace:
While rechecking her original notes to re-enter the smudged values of A and B in the records, the student found that one of the eleven recorded work experience values that did not get smudged was recorded wrongly as half of its correct value. After re-entering the values of A and B, and correcting the wrongly recorded value, she recalculated all the summary measures. The recalculated average value was 15.
- (a)
7
- (b)
9
- (c)
10
- (d)
12
- (e)
Cannot be determined from the given information.
Answer: Option C
Text Explanation :
Workspace:
Read the following scenario and answer the THREE questions that follow.
A T20 cricket match consists of two teams playing twenty overs each, numbered 1 to 20. The runs scored in any over is a non-negative integer. The run rate at the end of any over is the average runs scored up to and including that over, i.e., the run rate at the end of the k-th over is the average number of runs scored in overs numbered 1, 2, …, k, where 1 ≤ k ≤ 20, k a positive integer. The following table indicates the run rate of a team at the end of some of the overs during a T20 cricket match (correct up to 2 decimal places), where 1 ≤ N – 2 < N + 6 ≤ 20, N a positive integer. It is also known that the team did not score less than 6 runs and more than 15 runs in any over.
Over Number | Run Rate |
N - 2 | 8.00 |
N | 7.43 |
N + 2 | 8.11 |
N + 4 | 8.45 |
N + 6 | 8.08 |
What is the value of N?
- (a)
7
- (b)
13
- (c)
14
- (d)
9
- (e)
12
Answer: Option A
Text Explanation :
Average runs scored till Nth over is 7.43.
∴ Total runs scored in N overs = N × 7.43.
It would be best to check the options for the value of N such that N × 7.43 should be an integer.
This is possible for option (a) i.e., 7 and option (c) i.e., 14.
Now, the runrate at the end of N + 6 overs is 8.08. Checking for N = 7 or 14 such that (N + 6) × 8.08 should be an integer.
N = 7 is the only option satisfying.
⇒ N = 7
Runs scored after:
5th over: 5 × 8 = 40
7th over: 7 × 7.43 = 52
9th over: 9 × 8.11 = 73
11th over: 11 × 8.45 = 93
13th over: 13 × 8.08 = 105
Hence, option (a).
Workspace:
In which of these pairs of over numbers, the team could have scored 22 runs in total?
- (a)
6 and 7
- (b)
7 and 8
- (c)
8 and 9
- (d)
9 and 10
- (e)
10 and 11
Answer: Option D
Text Explanation :
Consider the solution for first question of this set.
Runs scored after:
5th over: 5 × 8 = 40
7th over: 7 × 7.43 = 52
9th over: 9 × 8.11 = 73
11th over: 11 × 8.45 = 93
13th over: 13 × 8.08 = 105
Option (a): In 6th and 7th overs total 52 - 40 = 12 runs are scored. Hence, option (a) is rejected.
Option (b): Minimum runs scored till 6th over 40 + 6 = 46
Maximum runs scored before 9th over 73 - 6 = 67
∴ Maximum runs scored in 7th and 8th over = 67 - 46 = 21
Hence, option (b) is rejected.
Option (c): In 8th and 9th overs total 73 - 52 = 21 runs are scored. Hence, option (c) is rejected.
Option (d): Minimum runs scored till 8th over 52 + 6 = 58
Maximum runs scored before 11th over 93 - 6 = 87
∴ Maximum runs scored in 9th and 10th over = 87 - 58 = 29
Hence, option (d) is the right answer.
Option (e): In 10th and 11th overs total 93 - 73 = 20 runs are scored. Hence, option (e) is rejected.
Hence, option (d).
Workspace:
In which of the following over numbers, the team MUST have scored the least number of runs?
- (a)
7
- (b)
8
- (c)
9
- (d)
10
- (e)
11
Answer: Option A
Text Explanation :
Consider the solution for first question of this set.
Runs scored after:
5th over: 5 × 8 = 40
7th over: 7 × 7.43 = 52
9th over: 9 × 8.11 = 73
11th over: 11 × 8.45 = 93
13th over: 13 × 8.08 = 105
In 6th and 7th overs total 52 - 40 = 12 runs are scored.
Since minimum runs scored in any over is 6, hence in both 6th and 7th overs, 6 runs MUST have been scored.
Hence, option (a).
Workspace:
Amit has forgotten his 4-digit locker key. He remembers that all the digits are positive integers and are different from each other. Moreover, the fourth digit is the smallest and the maximum value of the first digit is 3. Also, he recalls that if he divides the second digit by the third digit, he gets the first digit. How many different combinations does Amit have to try for unlocking the locker?
- (a)
2
- (b)
1
- (c)
4
- (d)
5
- (e)
3
Answer: Option E
Text Explanation :
Let the number be 'abcd' where all digits are positive single-digit distinct integers.
Now, d is the smallest of all the 4 digits while a = b/c.
Also, a ≤ 3.
Case 1: a = 1
Not possible since d is the smallest of all the digits.
Case 2: a = 2 ⇒ b = 2c
Also, since d is the smallest of the 4 digits, d = 1
Now, possible values of (b, c) are (2, 1), (4, 2), (6, 3) and (8, 4)
But c cannot be the least of the 4 digits and all digits must be distinct.
∴ Accepted values of (b, c) are (6, 3) and (8, 4)
⇒ (a, b, c, d) can be (2, 6, 3, 1) or (2, 8, 4, 1)
Case 3: a = 3 ⇒ b = 3c
Also, since d is the smallest of the 4 digits, d = 1 or 2
Now, possible values of (b, c) are (3, 1), (6, 2), (9, 3)
But c cannot be the least of the 4 digits and all digits must be distinct.
∴ Accepted values of (b, c) are (6, 2)
⇒ (a, b, c, d) can be (3, 6, 2, 1)
∴ Total acceptable values of (a, b, c, d) are (2, 6, 3, 1) or (2, 8, 4, 1) or (3, 6, 2, 1) i.e., 3 values.
Hence, option (e).
Workspace:
Five students appeared for an examination. The average mark obtained by these five students is 40. The maximum mark of the examination is 100, and each of the five students scored more than 10 marks. However, none of them scored exactly 40 marks. Based on the information given, which of the following MUST BE true?
- (a)
At least, three of them scored a maximum of 40 marks
- (b)
At least, three of them scored more than 40 marks
- (c)
At least, one of them scored exactly 41 marks
- (d)
At most, two of them scored more than 40 marks
- (e)
At least, one of them scored less than 40 marks
Answer: Option E
Text Explanation :
Average of 5 studens is 40, hence their total marks = 200
If the average is 40, there are two possibilities:
Case 1: All score exactly 40, but that is not possible.
Case 2: At least one scores more than 40 and at least one scores less than 40.
Hence, option (e).
Workspace:
A firm pays its five clerks Rs. 15,000 each, three assistants Rs. 40,000 each and its accountant Rs. 66,000. Then the mean salary in the firm comprising of these nine employees exceeds its median salary by rupees
- (a)
14600
- (b)
14000
- (c)
15480
- (d)
15200
- (e)
14720
Answer: Option B
Text Explanation :
To get the median salary, we arrange the salaries of the 9 employees in numerical ascending order.
The salaries when arranged in ascending numerical order will be as follows
15000, 15000, 15000, 15000, 15000, 40000, 40000, 40000, 66000
Now the median of these 9 values will be the middle value i.e., 5th value which is 15,000.
∴ Median = 15000
Mean Salary of these 9 employees = (15000×5 + 40000×3 + 66000)/(5 + 3 + 1) = 29,000
∴ Mean salary exceeds Median Salary by 29000 -15000 = 14000.
Hence, option (b).
Workspace:
The median of 11 different positive integers is 15 and seven of those 11 integers are 8, 12, 20, 6, 14, 22, and 13.
Statement I: The difference between the averages of four largest integers and four smallest integers is 13.25.
Statement II: The average of all the 11 integers is 16.
Which of the following statements would be sufficient to find the largest possible integer of these numbers?
- (a)
Statement I only.
- (b)
Statement II only.
- (c)
Both Statement I and Statement II are required.
- (d)
Neither Statement I nor Statement II is sufficient.
- (e)
Either Statement I or Statement II is sufficient.
Answer: Option E
Text Explanation :
Three integers are not known.
Using Statement I:
Average of four smallest integers
= (6 + 8 + 12 + 13)/4 = 39/4
∴ Average of four largest integers
In order to get the largest possible integer, two of the three unknown integers must be lowest possible i.e., 16 and 17.
So, the largest possible integer
= 92 – 22 – 20 – 17 = 33
Statement I can answer the question independently.
Using Statement II:
Sum of 11 integers = 11 × 16 = 176
Sum of the given integers = 110
∴ Sum of three unknown integers = 66
In order to get the largest possible integer, two of the three unknown integers must be lowest possible i.e., 16 and 17.
So, the largest possible integer
= 66 – 16 – 17 = 33
Statement II also answers the question independently.
Hence, option (e).
Workspace:
Ramesh analysed the monthly salary figures of five vice presidents of his company. All the salary figures are integers. The mean and the median salary figures are Rs. 5 lakh, and the only mode is Rs. 8 lakh. Which of the options below is the sum (in Rs. lakh) of the highest and the lowest salaries?
- (a)
9
- (b)
10
- (c)
11
- (d)
12
- (e)
None of the above.
Answer: Option A
Text Explanation :
Mean of the salaries of the five vice presidents is Rs. 5 lakhs. Hence, sum of the salaries of the five vice presidents = 25 lakhs.
Now, median of the salaries is Rs. 5 lakhs and 8 is the only mode.
Hence, the highest salary and second highest salaries are both 8 lakhs.
Hence, sum of two lowest salaries = 25 – (5 + 8 + 8) = 4 Lakhs.
As 8 is the only mode hence, the only combination of the lowest salaries is 1 lakhs and 3 lakhs.
Hence, lowest salary = Rs. 1 lakh.
Hence, the required sum = 8 + 1 = 9 lakhs.
Hence, option (a).
Workspace:
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