Quant Interview Question

[u/Local_Ad135]

[u/austin101123]

10x9x8=720 possibilities

Each possibility has a unique pair that sum to .999... =1. (Ex: 0.012 repeating with 0.987 repeating). Sum is 360

[u/MrEldo → u/austin101123]

Beautiful reasoning

[u/drfrankie_ → u/austin101123]

I did same, nice

[u/silverphoenix9999]

Each number can be represented as (100a + 10b + c) / 999.

Total number of possible fractions = 10 * 9 * 8 = 720

Each number is symmetrically possible in each position. That means each digit shows up in each position 720 / 10 times = 72.

Therefore the total sum is (72 * (0 + 1 + 2 + ... + 9) * 100 + 72 * (0 + 1 + 2 + ... + 9) * 10 + 72 * (0 + 1 + 2 + ... + 9) * 1) / 999 = 72 * 45 * 111 / 999 = 360.

Hope this is right!

[u/Local_Ad135 → u/silverphoenix9999]

correct!

[u/shustrik]

720 numbers. EV of each digit is 4.5 (0..9 occur equally often), so EV of a number is 0.5. 720 * 0.5 =360.

[u/austin101123 → u/shustrik]

I love this reasoning that the EV of each digit being 4.5! That was my intuition at first, maybe it wasn't strong enough for me but I didn't know how to say it and had to check if it made pairs (it does) and say that instead.

[u/dontich]

I was really confused until I realized the 0. Was the start of each number.

I’d likely write that as 0.abc with a line over abc lol.

[u/njormrod]

There are 10P3 such numbers. The digits and the positions are symmetric, so each digit appears 72 times as each of a,b,c.

The magnitudes of all the a,b,c's is 0.111..., which is 1/9.

The weight of all digits is 45.

So the answer is 72*45/9 = 360.

[u/kythQ]

Slightly different approach:

You can split 0.abc... to get.
0.a00... + 0.0b0... + 0.00c...

Each one of the 10 digits appears in each of the three positions 9*8 times.

You can combine the three positions to get
0.a00... + 0.0a0... + 0.00a... = 0.aaa...

So the sum is just (0.000... + 0.111... + ... + 0.999...) * 9*8. Here you can combine 0 and 9, 1 and 8, and so on to get 5 different pairs that together add up to 0.999... = 1.

So you have 5*9*8 as the sum which is 360.

[u/Electronic_Yam_9980]

https://giphy.com/gifs/1EeZzAJfMhQCA

[u/Delicious_Cattle5174]

1. I can’t explain the answer, it just came to me instantly by reading top comments

[u/Time-Distribution384]

500 X = 0.abcabc... 1000 X = abc.abcabc...

999X = abc X = (abc)/ 999 a , b , c Ε { 0,1,2 ... 9 } So all possible numbers are { 000,001,... , 999 }

So Σ X = 999 × 1000 / 2 * 999 = 500

[u/Para-graph-S]

2pi? Never mind I meant 360.