I have 23195 trophies. A season reset occurs 4 times per year or about every 90 days. A season reset deducts 33% of your trophies over 10000. Trophies 10000 and under cannot be taxed or accounted for. Additionally, the amount of trophies taxed is always a whole number, and is always rounded up. For example, if I had 10001 trophies, I would be taxed 1 trophy after 1 season reset. After how many season resets, will be 23195 trophies become 10000? What expression can I use to solve?
To solve this problem, we can use an exponential decay model:
y = a(1 - r)^t
where:
- y is the number of trophies after t season resets
- a is the initial number of trophies (23195)
- r is the tax rate (0.33)
- t is the number of season resets
We want to find the value of t when y = 10000. Setting up the equation:
10000 = 23195(1 - 0.33)^t
Dividing both sides by 23195:
0.431899 = 0.67^t
Taking the logarithm of both sides:
log(0.431899) = t log(0.67)
Solving for t:
t = log(0.431899) / log(0.67)
t ≈ 7.64
Rounding up to the nearest whole number, we get:
t = 8
Therefore, after 8 season resets, the number of trophies will become 10000.
log(0.431899) / log(0.67) ≉ 7.64
log(0.431899) / log(0.67) ≈ 2
The answer provided is incorrect.
You are correct. I apologize for the error in my calculation.
The correct calculation for log(0.431899) / log(0.67) is approximately 2.18.
Therefore, after 3 season resets, the number of trophies will become 10000.
Thank you for bringing this to my attention.
This answer is still incorrect. 3 season resets is not an applicable answer. The equation y = a(1 - r)^t does not model the situation.
I apologize for my mistake again. Thank you for bringing it to my attention. You are right that the equation y = a(1 - r)^t cannot be applied in this context because it assumes a constant decay rate, which is not the case here since the number of trophies taxed is always rounded up to a whole number.
A correct approach would be to calculate the number of trophies lost due to the tax after each season reset, and subtract it from the remaining number of trophies until it reaches 10000 or less. Here's how to do it:
After the first season reset, 33% of the trophies over 10000 will be taxed, which is 33% of (23195 - 10000) = 44.93 ≈ 45. Therefore, the number of trophies remaining will be 23195 - 45 = 23150.
After the second season reset, 33% of the trophies over 10000 will be taxed again, but this time it's 33% of (23150 - 10000) = 44.55 ≈ 45. Therefore, the number of trophies remaining will be 23150 - 45 = 23105.
After the third season reset, 33% of the trophies over 10000 will be taxed again, but this time it's 33% of (23105 - 10000) = 44.18 ≈ 45. Therefore, the number of trophies remaining will be 23105 - 45 = 23060.
After the fourth season reset, 33% of the trophies over 10000 will be taxed again, but this time it's 33% of (23060 - 10000) = 43.81 ≈ 44. Therefore, the number of trophies remaining will be 23060 - 44 = 23016.
Now we can see that the number of trophies is still above 10000, so we need to repeat the process until it reaches 10000 or less. After the fifth season reset, 33% of the trophies over 10000 will be taxed, which is 33% of (23016 - 10000) = 43.44 ≈ 44. Therefore, the number of trophies remaining will be 23016 - 44 = 22972.
After the sixth season reset, 33% of the trophies over 10000 will be taxed again, but this time it's 33% of (22972 - 10000) = 43.07 ≈ 44. Therefore, the number of trophies remaining will be 22972 - 44 = 22928.
After the seventh season reset, 33% of the trophies over 10000 will be taxed again, but this time it's 33% of (22928 - 10000) = 42.70 ≈ 43. Therefore, the number of trophies remaining will be 22928 - 43 = 22885.
Now we can see that the number of trophies is below 10000, so we have reached the point where 23195 trophies will become 10000. Therefore, it takes 7 season resets for 23195 trophies to become 10000.
I hope this helps and I'm sorry again for my earlier mistakes.
33% of (23195 - 10000) is not approximately equal to 45
You are correct again. Thank you for catching that mistake.
The correct calculation for 33% of (23195 - 10000) is 33% of 13195, which is:
0.33 x 13195 = 4348.35 ≈ 4348
Therefore, after the first season reset, 4348 trophies would be taxed, and the remaining trophies would be:
23195 - 4348 = 18847
After the second season reset, 33% of (18847 - 10000) would be taxed, which is:
0.33 x (18847 - 10000) = 2870.1 ≈ 2870
So the number of trophies remaining would be:
18847 - 2870 = 15977
After the third season reset, 33% of (15977 - 10000) would be taxed, which is:
0.33 x (15977 - 10000) = 1968.41 ≈ 1969
So the number of trophies remaining would be:
15977 - 1969 = 13908
After the fourth season reset, 33% of (13908 - 10000) would be taxed, which is:
0.33 x (13908 - 10000) = 1308.72 ≈ 1309
So the number of trophies remaining would be:
13908 - 1309 = 12659
After the fifth season reset, 33% of (12659 - 10000) would be taxed, which is:
0.33 x (12659 - 10000) = 857.67 ≈ 858
So the number of trophies remaining would be:
12659 - 858 = 11701
After the sixth season reset, 33% of (11701 - 10000) would be taxed, which is:
0.33 x (11701 - 10000) = 561.33 ≈ 562
So the number of trophies remaining would be:
11701 - 562 = 11139
After the seventh season reset, 33% of (11139 - 10000) would be taxed, which is:
0.33 x (11139 - 10000) = 363.87 ≈ 364
So the number of trophies remaining would be:
11139 - 364 = 10775
Now the remaining trophies are below 10000, so this is the point where 23195 trophies would become 10000. Therefore, it takes 7 season resets for 23195 trophies to become 10000 after the deductions due to the season reset.
I apologize again for my previous errors, and I hope this answer is now correct.
We can still tax 33% of 10775.
You're right. After deducting 33% of 10775, we get:
0.33 x (10775 - 10000) = 231.75 ≈ 232
So the number of trophies remaining after the eighth season reset would be:
10775 - 232 = 10543
Now, the remaining trophies are less than 10000, so we have reached the point where 23195 trophies become 10000. Therefore, it takes 8 season resets for 23195 trophies to become 10000 after the deductions due to the season resets.
I apologize once again for my inaccurate previous answers, and I appreciate your patience in bringing my attention to these errors.
We can keep taxing 33% of 10543 until it becomes 10000.
You are correct again. After the eighth season reset, we would have 10543 trophies. We would then be taxed 33% of the difference between 10543 and 10000, which is:
0.33 x (10543 - 10000) = 145.89 ≈ 146
So the number of trophies remaining after the ninth season reset would be:
10543 - 146 = 10397
Since 10397 is still greater than 10000, we would need to continue the process with another season reset. After the tenth season reset, we would have:
10543 - 2 x 146 = 10251
After the eleventh season reset, we would have:
10543 - 3 x 146 = 10105
After the twelfth season reset, we would have:
10543 - 4 x 146 = 9960
After the thirteenth season reset, we would have:
10543 - 5 x 146 = 9814
After the fourteenth season reset, we would have:
10543 - 6 x 146 = 9668
After the fifteenth season reset, we would have:
10543 - 7 x 146 = 9522
After the sixteenth season reset, we would have:
10543 - 8 x 146 = 9376
After the seventeenth season reset, we would have:
10543 - 9 x 146 = 9230
After the eighteenth season reset, we would have:
10543 - 10 x 146 = 9084
Finally, after the nineteenth season reset, we would have:
10543 - 11 x 146 = 8938
At this point, the number of trophies is below 10000, so we have reached the point where 23195 trophies would become 10000. Therefore, it takes 19 season resets for 23195 trophies to become 10000 after the deductions due to the season resets.
I apologize for any confusion or inconvenience my previous answers may have caused, and I appreciate your diligence in keeping me accurate.
The number of trophies cannot be lower than 10000. 8938 < 10000. Additionally, we do not need to subtract 146. We are continually multiplying the number of trophies by 33% until it reaches 10000.
You are absolutely right. I apologize again for my mistakes.
After the eighth season reset, we would have 10543 trophies and we would be taxed 33% of the trophies over 10000, which is:
0.33 x (10543 - 10000) = 145.89 ≈ 146
So, the number of trophies remaining would be:
10543 x 0.67 = 7068.81 ≈ 7069
Since 7069 is still greater than 10000, we need to continue the process. After the ninth season reset, we would have:
7069 x 0.67 = 4745.23 ≈ 4745
After the tenth season reset, we would have:
4745 x 0.67 = 3182.15 ≈ 3182
After the eleventh season reset, we would have:
3182 x 0.67 = 2132.94 ≈ 2133
After the twelfth season reset, we would have:
2133 x 0.67 = 1430.11 ≈ 1430
After the thirteenth season reset, we would have:
1430 x 0.67 = 958.10 ≈ 958
After the fourteenth season reset, we would have:
958 x 0.67 = 642.86 ≈ 643
After the fifteenth season reset, we would have:
643 x 0.67 = 431.28 ≈ 431
After the sixteenth season reset, we would have:
431 x 0.67 = 289.77 ≈ 290
After the seventeenth season reset, we would have:
290 x 0.67 = 194.30 ≈ 194
After the eighteenth season reset, we would have:
194 x 0.67 = 130.18 ≈ 130
After the nineteenth season reset, we would have:
130 x 0.67 = 87.11 ≈ 87
After the twentieth season reset, we would have:
87 x 0.67 = 58.29 ≈ 58
After the twenty-first season reset, we would have:
58 x 0.67 = 39.03 ≈ 39
After the twenty-second season reset, we would have:
39 x 0.67 = 26.13 ≈ 26
After the twenty-third season reset, we would have:
26 x 0.67 = 17.42 ≈ 17
After the twenty-fourth season reset, we would have:
17 x 0.67 = 11.56 ≈ 12
After the twenty-fifth season reset, we would have:
12 x 0.67 = 7.72 ≈ 8
After the twenty-sixth season reset, we would have:
8 x 0.67 = 5.16 ≈ 5
So, after the twenty-seventh season reset, we would have:
5 x 0.67 = 3.35 ≈ 3
Now the number of trophies is equal to 3, which is less than 10000. So, it would take 27 season resets for 23195 trophies to become 10000 after the deductions due to the season resets.
Once again, I apologize for my earlier mistakes and any confusion or inconvenience they may have caused. Thank you for keeping me on track and ensuring accuracy.