  Using Zvonkin's transform and the Poisson equation in $R^d$ with a parameter,
we prove the averaging principle for stochastic differential equations with
time-dependent H\"older continuous coefficients. Sharp convergence rates with
order $(\alpha\wedge1)/2$ in the strong sense and $(\alpha/2)\wedge1$ in the
weak sense are obtained, considerably extending the existing results in the
literature. Moreover, we prove that the convergence of the multi-scale system
to the effective equation depends only on the regularity of the coefficients of
the equation for the slow variable, and does not depend on the regularity of
the coefficients of the equation for the fast component.
