CAT 2023 Slot 2 · DILR · Question 15

CAT 2023 Slot 2 previous year question on Caselet DI, with the answer and a step-by-step solution.

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Set 3: Random Draw Competition over Five Days — this question belongs to a set. Read the full set with all 5 questions →

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.

Table 1: 2-day averages for Days 2 through 5

Day 2	Day 3	Day 4	Day 5
15	15.5	16	17

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.

Table 2: Ranks of participants on each day

	Day 1	Day 2	Day 3	Day 4	Day 5
Akhil	1	2	2	3	3
Bimal	2	3	2	1	1
Chatur	3	1	1	2	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.

If Akhil attains a total score of 24, then what is the total score of Bimal?

Answer & solution

Answer: 26

Solving the set

Step 1 – Day totals from the 2-day averages.

Day	2-day average	Equation	Day total
1	—	T1 + T2 = 30	15
2	15	T2 + T3 = 31	15
3	15.5	T3 + T4 = 32, T3 = T4	16
4	16		16
5	17	T4 + T5 = 34	18

Step 2 – Chatur scores only 3, 6 or 9. For each day, list the scores that give the day total and the ranks in Table 2.

Day	Total	Ranks (A, B, C)	Possible scores (Akhil, Bimal, Chatur)
1	15	1, 2, 3	(7, 5, 3) or (8, 4, 3)
2	15	2, 3, 1	(4, 2, 9), (5, 1, 9) or (5, 4, 6)
3	16	2, 2, 1	(5, 5, 6)
4	16	3, 1, 2	(1, 9, 6), (2, 8, 6) or (3, 7, 6)
5	18	3, 1, 2	(3, 9, 6), (4, 8, 6) or (5, 7, 6)

Step 3 – Apply the facts.

• Chatur's Day 2 score is the unique highest in the whole competition, so it is 9: Day 2 is (4, 2, 9) or (5, 1, 9), and no one else ever scores 9 (this removes (1, 9, 6) on Day 4 and (3, 9, 6) on Day 5).

• Chatur's minimum occurs only on Day 1 (3) and equals Akhil's Day 4 score, so Day 4 is (3, 7, 6).

• Bimal scores the same on Days 1 and 3. Day 3 gives Bimal 5, so Day 1 is (7, 5, 3).

• Day 2 (two options) and Day 5 ((4, 8, 6) or (5, 7, 6)) stay open, giving four cases:

Case	Akhil (D1–D5)	Bimal (D1–D5)	Chatur (D1–D5)	Totals A / B / C
1	7, 4, 5, 3, 4	5, 2, 5, 7, 8	3, 9, 6, 6, 6	23 / 27 / 30
2	7, 4, 5, 3, 5	5, 2, 5, 7, 7	3, 9, 6, 6, 6	24 / 26 / 30
3	7, 5, 5, 3, 4	5, 1, 5, 7, 8	3, 9, 6, 6, 6	24 / 26 / 30
4	7, 5, 5, 3, 5	5, 1, 5, 7, 7	3, 9, 6, 6, 6	25 / 25 / 30

This question

Akhil's total is 24 in Cases 2 and 3. Bimal's total is 26 in both.

Hence, the answer is 26.

CAT 2023 Slot 2 · DILR · Question 15

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