Is it correct to say "old adage" or simply the word "adage" itself is enough because most of the adages are old
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An adage is essentially a proverb; a well-known expression regarded to be general truth.
Of course you need not put the word "old" in front of "adage" – adage can stand on its own.
This Ngram shows that the phrase "old adage" is relatively common; about half the time "adage" is used, it appears to be prefixed by "old."
But commonality doesn't imply correctness, especially in this case. "Old adage" may be overused, to the point where it reads trite. (When writing, remember the adage, avoid trite expressions).
Here are three literary quotes with the word adage (the first omits the old; the second leaves it in, and the third sheds light on the answer to your question):