The answer is 0.05.

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$Let\quad the\quad bat\quad cost\quad x\$\quad and\quad the\quad ball\quad y\$\quad$

$so,\quad x+y=1.10$

$and\quad x=1+y$

$putting\quad x=1+y\quad in\quad first\quad equation,\quad we\quad get$

$1+1+y=1.10$

$\Rightarrow 1+2y=1.10$

$\Rightarrow 2y=0.1$

$\Rightarrow y=\boxed{ 0.05 }$