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Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring, $I$ an$\mathfrak{m}$-primary ideal of $R$ and $J$ its minimal reduction. We study thedepths of $F(I)$ under certain depth assumptions on $G(I)$ and length conditionon quotients of powers of $I$ and $J$, namely$\sum_{n\geq0}\lambda(\mathfrak{m}I^{n+1}/\mathfrak{m}JI^n)$ and$\sum_{n\geq0}\lambda(\mathfrak{m}I^{n+1} \cap J/\mathfrak{m}JI^n)$.

On the lengths of quotients of ideals and depths of fiber cones