  In this paper, we study the $F$-rationality of the Rees algebra and the
extended Rees algebra of $\mathfrak{m}$-primary ideals in excellent local rings
$(R, \mathfrak{m})$ of prime characteristic. We partially answer some
conjectures and questions raised by N. Hara, K.-i. Watanabe and K.-i. Yoshida
(J. Algebra, pp.153--190, vol 247, 2002).
