## whodunnit puzzle

### 494 / 858 = ·575757575757...

It is the only possible correct answer from a vast choice of fractions.
Algebra rather than a brute force approach solves it.

Replace the alphabets in the fraction EVE/DID = .IT
Let .ITITIT be X....(1)
Multiplying by 100, we get 100X = IT.ITIT ....(2)
(2) - (1) gives 99X = IT
So, EVE / DID = IT/99
99 * EVE = IT * DID

Clearly the LHS is divisible by 11, therefore either IT or DID must be a multiple of 11
As it cannot be IT, it must be DID, which implies 2D = I or 2D - I = 11
but IT * DID is divisible by 99
so if IT is divisible by 9
then DID may be (121, 242, 363, 484, 616, 737, 858, 979)
IT may be ( 18, 27, 36, 45, 54, 63, 72, 81)

But DID*IT /99 should result in EVE
but 121*27, 242*45, 363*63, 484*81, 616*18, 737*36, 858*54, 979*72
do not result in the pattern EVE when divided by 99

or DID and IT are both divisible by 3
Either
DID ={363} and IT={60, 63, 69}
or
DID ={858} and IT={51, 54, 57}

Of these possibilities, only 858*57/99 gives EVE, 494.
Finally we have 494/858 = 0.5757575757

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