Piece of cake (Help me please)

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absoluzation

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Ruprecht has prepared a huge cake and invited 100 elves to a cake eating party. The elves are numbered from 1 to 100 and visit Ruprecht one by one in the afternoon.
  • The elf with number 1 receives 1 % of the cake.
  • The elf with number 2 receives 2 % of the rest of the cake.
  • The elf with number 3 receives 3 % of the rest of the cake.
  • The elf with number 4 receives 4 % of the rest of the cake.
  • And so on.
  • The elf with number k receives k % of the rest of the cake.
  • And so on.
  • The elf with number 99 receives 99 % of the rest of the cake.
  • The elf with number 100, finally, receives 100 % of the rest of the cake and hence all the remaining cake.
The elf with number N has received the largest piece of cake. Find N.

So I thought N was 10 but if someone has another answer please tell me so I will send my calculations so that I can be corrected.
 
absoluzation said:
Ruprecht has prepared a huge cake and invited 100 elves to a cake eating party. The elves are numbered from 1 to 100 and visit Ruprecht one by one in the afternoon.
  • The elf with number 1 receives 1 % of the cake.
  • The elf with number 2 receives 2 % of the rest of the cake.
  • The elf with number 3 receives 3 % of the rest of the cake.
  • The elf with number 4 receives 4 % of the rest of the cake.
  • And so on.
  • The elf with number k receives k % of the rest of the cake.
  • And so on.
  • The elf with number 99 receives 99 % of the rest of the cake.
  • The elf with number 100, finally, receives 100 % of the rest of the cake and hence all the remaining cake.
The elf with number N has received the largest piece of cake. Find N.

So I thought N was 10 but if someone has another answer please tell me so I will send my calculations so that I can be corrected.
Click to expand...
Please show how you got 10.
 
Using a spreadsheet, the tenth elf gets the most cake. I am very interested in how you got 10. Please post your solution.

ELFShare of remaining cakeCake eaten by elfCake left
100​
1​
0.01​
1​
99​
2​
0.02​
1.98​
97.02​
3​
0.03​
2.9106​
94.1094​
4​
0.04​
3.764376​
90.345024​
5​
0.05​
4.5172512​
85.8277728​
6​
0.06​
5.149666368​
80.67810643​
7​
0.07​
5.64746745​
75.03063898​
8​
0.08​
6.002451119​
69.02818786​
9​
0.09​
6.212536908​
62.81565096​
10​
0.1​
6.281565096​
56.53408586​
11​
0.11​
6.218749445​
50.31533642​
12​
0.12​
6.03784037​
44.27749605​
13​
0.13​
5.756074486​
38.52142156​
14​
0.14​
5.392999018​
33.12842254​
15​
0.15​
4.969263381​
28.15915916​
16​
0.16​
4.505465466​
23.65369369​
 
After seeing HTC's post, I too am very interested in how you got 10!
 
I, too, made a spreadsheet and found the answer.

I also found an algebraic method of finding the answer, by looking for when the ratio of successive amounts taken falls below 1. I'm sure there are nicer ways, so I too want to see the OP's method.
 
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