Let M n be a compact, simply connected n ( 3)-dimensional Riemannian manifold without boundary and S n be the unit sphere Euclidean space R n+1 . We derive a differentiable sphere theorem whenever the manifold concerned satisfies that the sectional curvature K M is not larger than 1, while Ric(M ) n+2 4 and the volume V (M ) is not larger than (1 + η)V (S n ) for some positive number η depending only on n. Keywords k-th Ricci curvature, Hausdorff convergence, differentiable sphere theorem, harmonic coordinate, harmonic radius MSC(2000): 53C20, 53C23, 53C24 Citation: Wang P H, Wen Y L. A differentiable sphere theorem with positive Ricci curvature and reverse volume pinching.