Consider the 50-term sums:

\(S = \dfrac1{1\cdot 2 } +\dfrac1{3\cdot4} + \cdots + \dfrac1{99\cdot100} \).

\(T = \dfrac1{51\cdot 100 } + \dfrac1{52\cdot99} + \cdots + \dfrac1{99\cdot 52} + \dfrac1{100\cdot 51} \).

Find \(\dfrac ST\).

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