CAT — Dynamic Ranking
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At InnovateX, six employees, Asha, Bunty, Chintu, Dolly, Eklavya, and Falguni, were split into two groups of three each: Elite led by Manager Kuku, and Novice led by Manager Lalu.
At the end of each quarter, Kuku and Lalu handed out ratings to all members in their respective groups. In each group, each employee received a distinct integer rating from 1 to 3.
The score for an employee at the end of a quarter is defined as their cumulative rating from the beginning of the year. At the end of each quarter the employee in Novice with the highest score was promoted to Elite, and the employee in Elite with the minimum score was demoted to Novice. If there was a tie in scores, the employee with a higher rating in the latest quarter was ranked higher.
1. Asha, Bunty, and Chintu were in Elite at the beginning of Quarter 1. All of them were in Novice at the beginning of Quarter 4.
2. Dolly and Falguni were the only employees who got the same rating across all the quarters.
3. The following is known about ratings given by Lalu:
• Bunty received a rating of 1 in Quarter 2.
• Asha and Dolly received ratings of 1 and 2, respectively, in Quarter 3.
What was Eklavya’s score at the end of Quarter 2?
Step 1:
Using Condition (1), we know that Asha, Bunty, and Chintu belonged to the Elite group at the beginning of Quarter 1. Consequently, Dolly, Eklavya, and Falguni must have been in the Novice group at the same time.
Since Asha, Bunty, and Chintu were all in the Novice group at the beginning of Quarter 4, it follows that Dolly, Eklavya, and Falguni had moved to the Elite group by the beginning of Quarter 4.
From Condition (3) and the ratings assigned by Lalu, Bunty was shifted to the Novice group in Quarter 2. This implies that his Quarter 1 rating was 1. The same condition also tells us that Asha and Dolly received ratings of 1 and 2, respectively, in Quarter 3.
According to Condition (2), Dolly consistently received a rating of 2, while Falguni consistently received a rating of 3 across all three quarters. Therefore, Falguni must have been promoted to the Elite group in Quarter 2.
Since Asha was demoted to the Novice group in Quarter 3, her cumulative score at the end of Quarter 2 could not have exceeded Chintu's cumulative score. Hence, their Quarter 1 ratings must have been 2 and 3 in some order. Also, in Quarter 2, Asha and Chintu received ratings of 1 and 2, respectively.
The deductions obtained so far are summarized in the table below.
Step 2:
By the end of Quarter 2, both Dolly and Eklavya had cumulative scores of 4. However, since Eklavya's rating in Quarter 2 was higher than Dolly's, Eklavya was promoted to the Elite group in Quarter 3.
As Chintu was demoted to the Novice group at the beginning of Quarter 4, his cumulative score after Quarter 3 must have been less than or equal to Eklavya's cumulative score. This leads to the conclusion that Chintu and Eklavya received ratings of 1 and 2, respectively, in Quarter 3.
With these deductions, the entire arrangement is now determined and is represented in the table below.
Eklavya's cumulative score at the end of Quarter 2 was 4.
How many employees changed groups more than once up to the beginning of Quarter 4?
Step 1:
Using Condition (1), we know that Asha, Bunty, and Chintu belonged to the Elite group at the beginning of Quarter 1. Consequently, Dolly, Eklavya, and Falguni must have been in the Novice group at the same time.
Since Asha, Bunty, and Chintu were all in the Novice group at the beginning of Quarter 4, it follows that Dolly, Eklavya, and Falguni had moved to the Elite group by the beginning of Quarter 4.
From Condition (3) and the ratings assigned by Lalu, Bunty was shifted to the Novice group in Quarter 2. This implies that his Quarter 1 rating was 1. The same condition also tells us that Asha and Dolly received ratings of 1 and 2, respectively, in Quarter 3.
According to Condition (2), Dolly consistently received a rating of 2, while Falguni consistently received a rating of 3 across all three quarters. Therefore, Falguni must have been promoted to the Elite group in Quarter 2.
Since Asha was demoted to the Novice group in Quarter 3, her cumulative score at the end of Quarter 2 could not have exceeded Chintu's cumulative score. Hence, their Quarter 1 ratings must have been 2 and 3 in some order. Also, in Quarter 2, Asha and Chintu received ratings of 1 and 2, respectively.
The deductions obtained so far are summarized in the table below.
Step 2:
By the end of Quarter 2, both Dolly and Eklavya had cumulative scores of 4. However, since Eklavya's rating in Quarter 2 was higher than Dolly's, Eklavya was promoted to the Elite group in Quarter 3.
As Chintu was demoted to the Novice group at the beginning of Quarter 4, his cumulative score after Quarter 3 must have been less than or equal to Eklavya's cumulative score. This leads to the conclusion that Chintu and Eklavya received ratings of 1 and 2, respectively, in Quarter 3.
With these deductions, the entire arrangement is now determined and is represented in the table below.
No employee changed groups more than once up to the beginning of Quarter 4.
What was Bunty’s score at the end of Quarter 3?
Step 1:
Using Condition (1), we know that Asha, Bunty, and Chintu belonged to the Elite group at the beginning of Quarter 1. Consequently, Dolly, Eklavya, and Falguni must have been in the Novice group at the same time.
Since Asha, Bunty, and Chintu were all in the Novice group at the beginning of Quarter 4, it follows that Dolly, Eklavya, and Falguni had moved to the Elite group by the beginning of Quarter 4.
From Condition (3) and the ratings assigned by Lalu, Bunty was shifted to the Novice group in Quarter 2. This implies that his Quarter 1 rating was 1. The same condition also tells us that Asha and Dolly received ratings of 1 and 2, respectively, in Quarter 3.
According to Condition (2), Dolly consistently received a rating of 2, while Falguni consistently received a rating of 3 across all three quarters. Therefore, Falguni must have been promoted to the Elite group in Quarter 2.
Since Asha was demoted to the Novice group in Quarter 3, her cumulative score at the end of Quarter 2 could not have exceeded Chintu's cumulative score. Hence, their Quarter 1 ratings must have been 2 and 3 in some order. Also, in Quarter 2, Asha and Chintu received ratings of 1 and 2, respectively.
The deductions obtained so far are summarized in the table below.
Step 2:
By the end of Quarter 2, both Dolly and Eklavya had cumulative scores of 4. However, since Eklavya's rating in Quarter 2 was higher than Dolly's, Eklavya was promoted to the Elite group in Quarter 3.
As Chintu was demoted to the Novice group at the beginning of Quarter 4, his cumulative score after Quarter 3 must have been less than or equal to Eklavya's cumulative score. This leads to the conclusion that Chintu and Eklavya received ratings of 1 and 2, respectively, in Quarter 3.
With these deductions, the entire arrangement is now determined and is represented in the table below.
Bunty’s score at the end of Quarter 3 was 5.
For how many employees can the scores at the end of Quarter 3 be determined with certainty?
Step 1:
Using Condition (1), we know that Asha, Bunty, and Chintu belonged to the Elite group at the beginning of Quarter 1. Consequently, Dolly, Eklavya, and Falguni must have been in the Novice group at the same time.
Since Asha, Bunty, and Chintu were all in the Novice group at the beginning of Quarter 4, it follows that Dolly, Eklavya, and Falguni had moved to the Elite group by the beginning of Quarter 4.
From Condition (3) and the ratings assigned by Lalu, Bunty was shifted to the Novice group in Quarter 2. This implies that his Quarter 1 rating was 1. The same condition also tells us that Asha and Dolly received ratings of 1 and 2, respectively, in Quarter 3.
According to Condition (2), Dolly consistently received a rating of 2, while Falguni consistently received a rating of 3 across all three quarters. Therefore, Falguni must have been promoted to the Elite group in Quarter 2.
Since Asha was demoted to the Novice group in Quarter 3, her cumulative score at the end of Quarter 2 could not have exceeded Chintu's cumulative score. Hence, their Quarter 1 ratings must have been 2 and 3 in some order. Also, in Quarter 2, Asha and Chintu received ratings of 1 and 2, respectively.
The deductions obtained so far are summarized in the table below.
Step 2:
By the end of Quarter 2, both Dolly and Eklavya had cumulative scores of 4. However, since Eklavya's rating in Quarter 2 was higher than Dolly's, Eklavya was promoted to the Elite group in Quarter 3.
As Chintu was demoted to the Novice group at the beginning of Quarter 4, his cumulative score after Quarter 3 must have been less than or equal to Eklavya's cumulative score. This leads to the conclusion that Chintu and Eklavya received ratings of 1 and 2, respectively, in Quarter 3.
With these deductions, the entire arrangement is now determined and is represented in the table below.
The scores of four employees—Bunty, Dolly, Eklavya, and Falguni—at the end of Quarter 3 can be determined with certainty.
Which of the following statements is/are NECESSARILY true?
I. Asha received a rating of 2 in Quarter 1.
II. Asha received a rating of 1 in Quarter 2.
Both I and II
Neither I nor II
Only I
Only II
Only II
Step 1:
Using Condition (1), we know that Asha, Bunty, and Chintu belonged to the Elite group at the beginning of Quarter 1. Consequently, Dolly, Eklavya, and Falguni must have been in the Novice group at the same time.
Since Asha, Bunty, and Chintu were all in the Novice group at the beginning of Quarter 4, it follows that Dolly, Eklavya, and Falguni had moved to the Elite group by the beginning of Quarter 4.
From Condition (3) and the ratings assigned by Lalu, Bunty was shifted to the Novice group in Quarter 2. This implies that his Quarter 1 rating was 1. The same condition also tells us that Asha and Dolly received ratings of 1 and 2, respectively, in Quarter 3.
According to Condition (2), Dolly consistently received a rating of 2, while Falguni consistently received a rating of 3 across all three quarters. Therefore, Falguni must have been promoted to the Elite group in Quarter 2.
Since Asha was demoted to the Novice group in Quarter 3, her cumulative score at the end of Quarter 2 could not have exceeded Chintu's cumulative score. Hence, their Quarter 1 ratings must have been 2 and 3 in some order. Also, in Quarter 2, Asha and Chintu received ratings of 1 and 2, respectively.
The deductions obtained so far are summarized in the table below.
Step 2:
By the end of Quarter 2, both Dolly and Eklavya had cumulative scores of 4. However, since Eklavya's rating in Quarter 2 was higher than Dolly's, Eklavya was promoted to the Elite group in Quarter 3.
As Chintu was demoted to the Novice group at the beginning of Quarter 4, his cumulative score after Quarter 3 must have been less than or equal to Eklavya's cumulative score. This leads to the conclusion that Chintu and Eklavya received ratings of 1 and 2, respectively, in Quarter 3.
With these deductions, the entire arrangement is now determined and is represented in the table below.
Only Statement II is necessarily true.
Three participants – Akhil, Bimal and Chatur participate in a random draw competition for five days. Every day, each participant randomly picks up a ball numbered between 1 and 9. The number on the ball determines his score on that day. The total score of a participant is the sum of his scores attained in the five days. The total score of a day is the sum of participants’ scores on that day. The 2-day average on a day, except on Day 1, is the average of the total scores of that day and of the previous day. For example, if the total scores of Day 1 and Day 2 are 25 and 20, then the 2-day average on Day 2 is calculated as 22.5. Table 1 gives the 2-day averages for Days 2 through 5.
Participants are ranked each day, with the person having the maximum score being awarded the minimum rank (1) on that day. If there is a tie, all participants with the tied score are awarded the best available rank. For example, if on a day Akhil, Bimal, and Chatur score 8, 7 and 7 respectively, then their ranks will be 1, 2 and 2 respectively on that day. These ranks are given in Table 2.
The following information is also known.
1. Chatur always scores in multiples of 3. His score on Day 2 is the unique highest score in the competition. His minimum score is observed only on Day 1, and it matches Akhil’s score on Day 4.
2. The total score on Day 3 is the same as the total score on Day 4.
3. Bimal’s scores are the same on Day 1 and Day 3.
What is Akhil's score on Day 1?
6
5
7
8
7
Step 1:
Let the scores on Day 1, Day 2, Day 3, Day 4, and Day 5 be D1, D2, D3, D4, and D5, respectively.
From the given information,
- D1 + D2 = 30
- D2 + D3 = 31
- D3 + D4 = 32
- D4 + D5 = 34
Using condition (2),
D3 = D4 = 16
Therefore,
- D1 = 15
- D2 = 15
- D3 = 16
- D4 = 16
- D5 = 18
From conditions (1) and (3), let the unknown scores be represented as a, b, and c.
Since the total score on Day 3 is 16,
2a + c = 16
Also, c > a.
The possible pairs are (4, 8) and (5, 6).
Since all of Chatur's scores are multiples of 3, the only valid choice is:
a = 5, c = 6
This also implies that Chatur's unique highest score occurs on Day 2, so
b = 3.
Step 2:
The known scores are:
- Akhil: Day 1 = 7, Day 2 = x, Day 3 = 5, Day 4 = 3, Day 5 = p
- Bimal: Day 1 = 5, Day 2 = y, Day 3 = 5, Day 4 = 7, Day 5 = q
- Chatur: Day 1 = 3, Day 2 = 9, Day 3 = 6, Day 4 = 6, Day 5 = 6
Since Chatur's score of 9 is the unique highest score,
9 > x > y
and
q > 6 > p.
Using the daily totals,
x + y = 6
and
p + q = 12.
The possible values are:
- (x, y) = (5, 1) or (4, 2)
- (p, q) = (5, 7) or (4, 8)
Who attains the maximum total score?
Akhil
Chatur
Bimal
Cannot be determined
Chatur
Step 1:
Let the scores on Day 1, Day 2, Day 3, Day 4, and Day 5 be D1, D2, D3, D4, and D5, respectively.
From the given information,
- D1 + D2 = 30
- D2 + D3 = 31
- D3 + D4 = 32
- D4 + D5 = 34
Using condition (2),
D3 = D4 = 16
Therefore,
- D1 = 15
- D2 = 15
- D3 = 16
- D4 = 16
- D5 = 18
From conditions (1) and (3), let the unknown scores be represented as a, b, and c.
Since the total score on Day 3 is 16,
2a + c = 16
Also, c > a.
The possible pairs are (4, 8) and (5, 6).
Since all of Chatur's scores are multiples of 3, the only valid choice is:
a = 5, c = 6
This also implies that Chatur's unique highest score occurs on Day 2, so
b = 3.
Step 2:
The known scores are:
- Akhil: Day 1 = 7, Day 2 = x, Day 3 = 5, Day 4 = 3, Day 5 = p
- Bimal: Day 1 = 5, Day 2 = y, Day 3 = 5, Day 4 = 7, Day 5 = q
- Chatur: Day 1 = 3, Day 2 = 9, Day 3 = 6, Day 4 = 6, Day 5 = 6
Since Chatur's score of 9 is the unique highest score,
9 > x > y
and
q > 6 > p.
Using the daily totals,
x + y = 6
and
p + q = 12.
The possible values are:
- (x, y) = (5, 1) or (4, 2)
- (p, q) = (5, 7) or (4, 8)
What is the minimum possible total score of Bimal?
Step 1:
Let the scores on Day 1, Day 2, Day 3, Day 4, and Day 5 be D1, D2, D3, D4, and D5, respectively.
From the given information,
- D1 + D2 = 30
- D2 + D3 = 31
- D3 + D4 = 32
- D4 + D5 = 34
Using condition (2),
D3 = D4 = 16
Therefore,
- D1 = 15
- D2 = 15
- D3 = 16
- D4 = 16
- D5 = 18
From conditions (1) and (3), let the unknown scores be represented as a, b, and c.
Since the total score on Day 3 is 16,
2a + c = 16
Also, c > a.
The possible pairs are (4, 8) and (5, 6).
Since all of Chatur's scores are multiples of 3, the only valid choice is:
a = 5, c = 6
This also implies that Chatur's unique highest score occurs on Day 2, so
b = 3.
Step 2:
The known scores are:
- Akhil: Day 1 = 7, Day 2 = x, Day 3 = 5, Day 4 = 3, Day 5 = p
- Bimal: Day 1 = 5, Day 2 = y, Day 3 = 5, Day 4 = 7, Day 5 = q
- Chatur: Day 1 = 3, Day 2 = 9, Day 3 = 6, Day 4 = 6, Day 5 = 6
Since Chatur's score of 9 is the unique highest score,
9 > x > y
and
q > 6 > p.
Using the daily totals,
x + y = 6
and
p + q = 12.
The possible values are:
- (x, y) = (5, 1) or (4, 2)
- (p, q) = (5, 7) or (4, 8)
25 is the minimum possible total score of Bimal when y = 1 and q = 7.
If the total score of Bimal is a multiple of 3, what is the score of Akhil on Day 2?
5
4
6
Cannot be determined
4
Step 1:
Let the scores on Day 1, Day 2, Day 3, Day 4, and Day 5 be D1, D2, D3, D4, and D5, respectively.
From the given information,
- D1 + D2 = 30
- D2 + D3 = 31
- D3 + D4 = 32
- D4 + D5 = 34
Using condition (2),
D3 = D4 = 16
Therefore,
- D1 = 15
- D2 = 15
- D3 = 16
- D4 = 16
- D5 = 18
From conditions (1) and (3), let the unknown scores be represented as a, b, and c.
Since the total score on Day 3 is 16,
2a + c = 16
Also, c > a.
The possible pairs are (4, 8) and (5, 6).
Since all of Chatur's scores are multiples of 3, the only valid choice is:
a = 5, c = 6
This also implies that Chatur's unique highest score occurs on Day 2, so
b = 3.
Step 2:
The known scores are:
- Akhil: Day 1 = 7, Day 2 = x, Day 3 = 5, Day 4 = 3, Day 5 = p
- Bimal: Day 1 = 5, Day 2 = y, Day 3 = 5, Day 4 = 7, Day 5 = q
- Chatur: Day 1 = 3, Day 2 = 9, Day 3 = 6, Day 4 = 6, Day 5 = 6
Since Chatur's score of 9 is the unique highest score,
9 > x > y
and
q > 6 > p.
Using the daily totals,
x + y = 6
and
p + q = 12.
The possible values are:
- (x, y) = (5, 1) or (4, 2)
- (p, q) = (5, 7) or (4, 8)
Total score of Bimal is a multiple of 3 when y = 2 and q = 8.
Then, score of Akhil on Day 2 will be 4.
If Akhil attains a total score of 24, then what is the total score of Bimal?
Step 1:
Let the scores on Day 1, Day 2, Day 3, Day 4, and Day 5 be D1, D2, D3, D4, and D5, respectively.
From the given information,
- D1 + D2 = 30
- D2 + D3 = 31
- D3 + D4 = 32
- D4 + D5 = 34
Using condition (2),
D3 = D4 = 16
Therefore,
- D1 = 15
- D2 = 15
- D3 = 16
- D4 = 16
- D5 = 18
From conditions (1) and (3), let the unknown scores be represented as a, b, and c.
Since the total score on Day 3 is 16,
2a + c = 16
Also, c > a.
The possible pairs are (4, 8) and (5, 6).
Since all of Chatur's scores are multiples of 3, the only valid choice is:
a = 5, c = 6
This also implies that Chatur's unique highest score occurs on Day 2, so
b = 3.
Step 2:
The known scores are:
- Akhil: Day 1 = 7, Day 2 = x, Day 3 = 5, Day 4 = 3, Day 5 = p
- Bimal: Day 1 = 5, Day 2 = y, Day 3 = 5, Day 4 = 7, Day 5 = q
- Chatur: Day 1 = 3, Day 2 = 9, Day 3 = 6, Day 4 = 6, Day 5 = 6
Since Chatur's score of 9 is the unique highest score,
9 > x > y
and
q > 6 > p.
Using the daily totals,
x + y = 6
and
p + q = 12.
The possible values are:
- (x, y) = (5, 1) or (4, 2)
- (p, q) = (5, 7) or (4, 8)
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