Two celebrated extensions of Helly's theorem are the fractional Helly theorem of Katchalski and Liu (1979) and the quantitative volume theorem of Bárány, Katchalski, and Pach (1982). Improving on several recent works, we prove an optimal combination of these two results. We show that given a family F of n convex sets in ℝd such that at least α(nd+1) of the (d + 1)-tuples of F have an intersection of volume at least 1, then one can select Ωd,α(n) members of F whose intersection has volume at least Ωd(1). Furthermore, with the help of this theorem, we establish a quantitative version of the (p, q) theorem of Alon and Kleitman. Let p ≥ q ≥ d + 1 and let F be a finite family of convex sets in ℝd such that among any p elements of F, there are q that have an intersection of volume at least 1. Then, we prove that there exists a family T of Op,q(1) ellipsoids of volume Ωd(1) such that every member of F contains at least one element of T. Finally, we present extensions about the diameter version of the quantitative Helly theoerm.