Summing Ratio of Integers Over Their Digital Sums

Algebra Level 5

\[ \dfrac{ 100} { 1 + 0 + 0 } + \dfrac{ 101 } {1 + 0 + 1 } + \cdots + \dfrac{ \overline{abc} } { a + b + c } + \cdots + \dfrac{ 998 } { 9 + 9 + 8 } + \dfrac{ 999 } { 9 + 9 + 9 } = \, ? \]

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