Now, the scores in the table are the cumulative scores. We can easily see which recruits did not participate in which round from the fact that the cumulative scores do not change for a particular person in a particular round. For example, D did not participate in the first round, A did not participate in round 3, B did not participate in round 4, C did not participate in round 2 and E did not participate in round 5.
Let us assume that every round lasts for ‘x’ minutes and by the end of 5th round, each recruit shall have shot for 4x minutes. Now, the average cumulative score for A after round 5 is 60.4 meaning that the total score is = 60.4 * 4x = 241.6x and so on for other teams also.
Now, it can be deduced from observation that the maximum score by any recruit shall be in a round where the cumulative score has seen the maximum jump. So, the possible candidates are:
- A in round 5
- B in round 2
- C in round 3
- E in round 3
D is not valid because its score is decreasing in most of the rounds.
Let’s evaluate these cases:
- Now, at the end of round 5, the score of A is 241.6 and his time was 3x at the end of round 4 so, the score at the end of round 4 = 54.4 * 3x making the score for round 5 for A as = 78.4x
- At the end of round B, the score of B = 146x and the score = 67.2x at the end of round one. So, the score in round 2 = 78.8x
- Similarly, for C, the score in round 3 = 78.2x
- Score for E in round 3 = 69.6x
Hence, it is clear that the maximum score is for B in round 2 = 78.8x and the nearest integral score is if x is a multiple of 5 giving the maximum score as = 394
This is the answer to the first question.
Now, we have to estimate the score for C in round 4. So, the cumulative score for C at the end of round 4 = 61.2667 and the time spent = 3x
The score at the end of round 3 = 69.7 and the time spent = 2x
So, the score in round 4 = 44.4x
X should be 5 to make it integral so, the score can be 222 or any multiple.
Now, to calculate the minimum points scored by any recruit in any round, the process is mostly similar to the first question. The cases where the points decrease by the possible cases are:
- A in round 2
- B in round 3
- C in round 4
- E in round 2
Evaluating these options now,
- Score of A in round 2 = 48.8x
- Score of B in round 3 = 47.8x
- Score of C in round 4 = 44.4x
- Score of E in round 2 = 43.8x
So, the minimum score is for E in round 2 and it can be 219 or any multiple.
The score of D in round 5 = 61.6x and x = 10 minutes for it to be 616 minutes.
Now, the score of B in round 3 = 47.8x = 478