An approximate answer is one which is close to the correct answer. Likewise, we can talk of an approximate model, or approximate methods in mathematics.
The etymology is from the Latin ad, "to" and proximus, "closest", see e.g. here. I don't think this can be disputed. Occasionally proximate is used as a synonym, though that word has a different meaning in legal English.
So if an approximate answer is close, a very approximate answer should be very close.
In practice, "very approximate" or "very approximately" is overwhelmingly used with the opposite meaning of "roughly", indicating that an answer or estimate is not necessarily accurate. For examples, search Google.
Is there any justification for this use? Is there any justification for my suggested use? Should we avoid the phrase altogether?
Best Answer
I follow your reasoning, but the point of an approximation is that the exact answer is not available. So the salient feature is that it is not exact. Therefore, very approximate is used to mean very inexact.
An approximation which is especially accurate is a very close approximation. When this near-exactness is expressed in adjective form, I think the word approximate is generally avoided, in favor of phrases like almost exact or very nearly exact.