After playing with my calculator for a while I tried doing $\frac{9876543210}{0123456789}$ and it came out as $80.000000729$ which came really close to a whole number so I tried it for the first 16 number bases and I got this:
Base 2 - 2 ->In base 2:($\frac{10}{01}$)->In base 10: ($\frac{3}{1}$)
Base 3 - 4.2 ->In base 3:( $\frac{210}{012}$) ->In base 10:($\frac{21}{5}$)
Base 4 - 8.444444444444445 ->In base 4:($\frac{3210}{0123}$) ->In base 10:($\frac{228}{27}$)
Base 5 - 15.103092783505154 ->In base 5:($\frac{43210}{01234}$) ->In base 10:($\frac{2930}{194}$)
Base 6 - 24.01608579088472 ->In base 6:($\frac{543210}{012345}$) ->In base 10:($\frac{44790}{1865}$)
Base 7 - 35.00183606557377 ->In base 7:($\frac{6543210}{0123456}$) ->In base 10:($\frac{800667}{22875}$)
Base 8 - 48.00016355570094 ->In base 8:($\frac{76543210}{01234567}$) ->In base 10:($\frac{16434824}{342391}$)
Base 9 - 63.00001189405568 ->In base 9:($\frac{876543210}{012345678}$) ->In base 10:($\frac{381367044}{6053444}$)
Base 10 - 80.000000729 ->In base 10:($\frac{9876543210}{0123456789}$) ->In base 10:($\frac{9876543210}{123456789}$)
Base 11 - 99.00000003855433 ->In base 11:($\frac{A9876543210}{0123456789A}$) ->In base 10:($\frac{282458553905}{2853116705}$)
Base 12 - 120.00000000179136 ->In base 12:($\frac{BA9876543210}{0123456789AB}$) ->In base 10:($\frac{8842413667692}{73686780563}$)
Base 13 - 143.00000000007418 ->In base 13:($\frac{CBA9876543210}{0123456789ABC}$) ->In base 10:($\frac{300771807240918}{2103299351334}$)
Base 14 - 168.00000000000276 ->In base 14:($\frac{DCBA9876543210}{0123456789ABCD}$) ->In base 10:($\frac{11046255305880158}{65751519677857}$)
Base 15 - 195.00000000000009 ->In base 15:($\frac{EDCBA9876543210}{0123456789ABCDE}$) ->In base 10:($\frac{435659737878916200}{2234152501943159}$)
Base 16 - 224 ->In base 16:($\frac{FEDCBA9876543210}{0123456789ABCDEF}$) ->In base 10:($\frac{18364758544493064000}{81985529216486900}$)
This took me too long to make... Well anyways this thing seems to approach a whole number.
Another thing I noticed was that the answers were about Base n = $n^2-2n$ if you ignore the exception of base 2...
Can anyone give me some sort of explanation?
The denominator in base $b$ is very close to the first repeat of the fraction $\frac 1{(b-1)^2}$. For example in base $10$ we have $$\frac 1{81}=0.012345679012345679\ldots$$ which differs from your denominator by excluding the $8$. The pattern is general. If you express $\frac 1{25_{10}}$ in base $6$ the repeat is $.01235_6$. The denominator is then very close to $\frac {b^b}{(b-1)^2}$If you add $1$ to your fraction, the numerator becomes $b^b-1$ so your fraction is essentially $\frac {b^b-1}{\frac {b^b}{(b-1)^2}}-1=(b-1)^2-1$