You do not have to look too far to find excellent readers who do not write much or who are not excellent writers. But the reverse is impossible. To excel at writing, one has to read voraciously, eclectically, and thoughtfully.
Truth be told, quality reading and writing are inseparable. The illustration shows how many strong writers approach their craft. First, they read to find inspiration, generate ideas, rewrite notes, seek corroborating details, check facts, or borrow from their previously written material. Next, they draft all their thoughts to get a rough idea of what they're trying to write. Then, they read aloud to catch the stream of logic and to detect ambiguity, repetition, and awkwardness. Finally, they write again for clarity, conciseness, grace, and vigor.
So excellent writers read to write and write to read. They need to answer not one question but two: What have you written lately? and What have you read lately?