| Table 1: 2-day averages for Days through 5 | |||
|---|---|---|---|
| Day 2 | Day 3 | Day 4 | Day 5 |
| 15 | 15.5 | 16 | 17 |
| 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 |
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
- 7
- 5
- 8
The Correct Option is B
Solution and Explanation
- The 2-day average for Day 2 is 15. Thus, the equation for Day 1 and Day 2's total score is:
$$ \text{Total Day 1} + \text{Total Day 2} = 2 \times 15 = 30 $$
The 2-day average for Day 3 is 15.5. Thus, the equation for Day 2 and Day 3's total score is:
$$ \text{Total Day 2} + \text{Total Day 3} = 2 \times 15.5 = 31 $$
The 2-day average for Day 4 is 16. Thus, the equation for Day 3 and Day 4's total score is:
$$ \text{Total Day 3} + \text{Total Day 4} = 2 \times 16 = 32 $$
The 2-day average for Day 5 is 17. Thus, the equation for Day 4 and Day 5's total score is:
$$ \text{Total Day 4} + \text{Total Day 5} = 2 \times 17 = 34 $$
Total scores of Days 3 and 4 are equal, so:
$$ \text{Total Day 3} = \text{Total Day 4} = 16 $$ - Now, we have:
$$ \text{Total Day 2} + 16 = 31 \Rightarrow \text{Total Day 2} = 15 $$
Substituting into first equation for Day 1 & Day 2:
$$ \text{Total Day 1} + 15 = 30 \Rightarrow \text{Total Day 1} = 15 $$
So, Total Day 1's score is 15. - Participants' ranks for Day 1 are:
- Akhil: 1
- Bimal: 2
- Chatur: 3
- Akhil's rank is 1, he has the highest score on Day 1:
- The scores can be distributed considering conditions:Thus, Akhil's score on Day 1 is 7.
Participants Score Akhil 7 Bimal 5 Chatur 3
Who attains the maximum total score?
- Cannot be determined
- Akhil
- Bimal
- Chatur
The Correct Option is D
Solution and Explanation
To solve the problem of determining who attains the maximum total score, we must analyze the information and utilize logical reasoning based on the constraints and data provided.
Given:
- 2-day averages from Day 2 to Day 5.
- Ranks for each day.
- Specific constraints about the participants' scores.
Let's calculate the total scores for Days 1 through 5:
| Day | Total Score |
|---|---|
| Day 1 | x |
| Day 2 | y |
| Day 3 | z |
| Day 4 | z |
| Day 5 | a |
From the 2-day averages:
- (x + y) / 2 = 15 ⟹ x + y = 30
- (y + z) / 2 = 15.5 ⟹ y + z = 31
- (z + z) / 2 = 16 ⟹ z = 16
- (z + a) / 2 = 17 ⟹ a = 18
Solving these:
- z = 16
- y + 16 = 31 ⟹ y = 15
- x + 15 = 30 ⟹ x = 15
- z + a = 34 ⟹ a = 18
Total scores per day:
- Day 1: 15
- Day 2: 15
- Day 3: 16
- Day 4: 16
- Day 5: 18
Given Rank Constraints:
- Chatur’s score is always a multiple of 3
- Lowest score on Day 1 (equal to Akhil’s score on Day 4)
- Highest unique score on Day 2
Chatur’s Scores:
- Day 1: 6 (same as Akhil’s Day 4 score)
- Day 2: 9 (unique and highest)
- Day 3: 6
- Day 4: 6
- Day 5: 12
Bimal’s Scores:
- Day 1: 5
- Day 2: 4
- Day 3: 5
- Day 4: 7
- Day 5: 7
- Total: 28
Akhil’s Scores:
- Day 1: 4
- Day 2: 2
- Day 3: 5
- Day 4: 3
- Day 5: 5
- Total: 19
Chatur’s Total Score: 6 + 9 + 6 + 6 + 12 = 39
Conclusion: Chatur attains the maximum total score.
What is the minimum possible total score of Bimal?
Solution and Explanation
Let's consider the following table for minimum possible total score of Bimal:
| Day 1 | Day 2 | Day 3 | Day 4 | Day 5 | Total Score | |
|---|---|---|---|---|---|---|
| Akhil | 7 | \(\frac{4}{5}\) | 5 | 3 | \(\frac{5}{4}\) | 23 / 24 / 25 |
| Bimal | 5 | \(\frac{2}{1}\) | 5 | 7 | \(\frac{7}{8}\) | 27 / 26 / 25 |
| Chatur | 3 | 9 | 6 | 6 | 6 | 30 |
| Total score | 15 | 15 | 16 | 16 | 18 | 80 |
From the table, we can see that the minimum score obtained by Bimal is 25.
So, the correct answer is: 25.
If the total score of Bimal is a multiple of 3, what is the score of Akhil on Day 2?
- 4
- 5
- 6
- Cannot be determined
The Correct Option is A
Solution and Explanation
Let's see the final table of total scores of Bimal:
| Day 1 | Day 2 | Day 3 | Day 4 | Day 5 | Total Score | |
| Akhil | 7 | \(\frac{4}{5}\) | 5 | 3 | \(\frac{5}{4}\) | 23 / 24 / 25 |
| Bimal | 5 | \(\frac{2}{1}\) | 5 | 7 | \(\frac{7}{8}\) | 27 / 26 / 25 |
| Chatur | 3 | 9 | 6 | 6 | 6 | 30 |
| Total Score | 15 | 15 | 16 | 16 | 18 | 80 |
The question says that Bimal's total score is a multiple of 3, which means his total score is 27.
This means Akhil's total score is 23.
Akhil scores 23 when his scores on Days 1, 2, 3, 4, and 5 are 7, 4, 5, 3, and 4.
So, Akhil's score on Day 2 is 4.
If Akhil attains a total score of 24, then what is the total score of Bimal?
Solution and Explanation
Bimal's Total Score
To determine Bimal's total score given Akhil's total score of 24, let's analyze the data step-by-step:
2-Day Averages:
- Day 2: \(15\)
- Day 3: \(15.5\)
- Day 4: \(16\)
- Day 5: \(17\)
Let the total scores for each day be:
- Day 1: \(x\)
- Day 2: \(y\)
- Day 3 and Day 4: \(z\)
- Day 5: \(w\)
Using the average equations:
- \(\frac{x + y}{2} = 15 \Rightarrow x + y = 30\)
- \(\frac{y + z}{2} = 15.5 \Rightarrow y + z = 31\)
- \(\frac{z + z}{2} = 16 \Rightarrow z = 16\)
- \(\frac{z + w}{2} = 17 \Rightarrow w = 18\)
- \(y = 15\), \(x = 15\)
So, total scores per day:
- Day 1: \(15\)
- Day 2: \(15\)
- Day 3: \(16\)
- Day 4: \(16\)
- Day 5: \(18\)
Participants’ Scores
- Chatur: Always multiples of 3
- Chatur's highest score is on Day 2 (unique highest)
- Lowest score on Day 1 equals Akhil's score on Day 4 (\(= 3\))
Assigning scores:
- Day 1: Chatur = 3, Akhil = 7 (rank 1), Bimal = 5
- Day 2: Chatur = 9, Akhil = 4 (rank 2), Bimal = 2 (rank 3)
- Day 3: Chatur = 6, Akhil = 5, Bimal = 5
- Day 4: Chatur = 6, Akhil = 3, Bimal = 7
- Day 5: Chatur = 9, Akhil = 4, Bimal = 5
Total Scores:
- Akhil: \(7 + 4 + 5 + 3 + 5 = 24\)
- Bimal: \(5 + 2 + 5 + 7 + 7 = 26\)
- Chatur: \(3 + 9 + 6 + 6 + 9 = 33\)
Therefore, Bimal's total score is: 26
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