  Let $\psi: \mathbb{N} \to [0,1/2]$ be given. The Duffin-Schaeffer conjecture,
recently resolved by Koukoulopoulos and Maynard, asserts that for almost all
reals $\alpha$ there are infinitely many coprime solutions $(p,q)$ to the
inequality $|\alpha - p/q| < \psi(q)/q$, provided that the series
$\sum_{q=1}^\infty \varphi(q) \psi(q) / q$ is divergent. In the present paper,
we establish a quantitative version of this result, by showing that for almost
all $\alpha$ the number of coprime solutions $(p,q)$, subject to $q \leq Q$, is
of asymptotic order $\sum_{q=1}^Q 2 \varphi(q) \psi(q) / q$. The proof relies
on the method of GCD graphs as invented by Koukoulopoulos and Maynard, together
with a refined overlap estimate coming from sieve theory, and number-theoretic
input on the "anatomy of integers". The key phenomenon is that the system of
approximation sets exhibits "asymptotic independence on average" as the total
mass of the set system increases.
