Suppose that today is your last of work. You are now saying thanks to one of your colleagues.
(1) Thank you for taking time out of your busy days to run errands for me.
(2) Thank you for taking the time out of your busy days to run errands for me.
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Sign up to join this communitySuppose that today is your last of work. You are now saying thanks to one of your colleagues.
(1) Thank you for taking time out of your busy days to run errands for me.
(2) Thank you for taking the time out of your busy days to run errands for me.
They both sound a bit unnatural, but the first choice is preferable.
"The" is a definite article, and should refer to something definite and specific. Therefore, if you wanted to say "the", you might say:
Thank you for taking the time out of your day (today), to run errands for me.
That is one day, and one block of time, so the word "the" fits.
On the other hand, if you are talking about "busy days", which are many days, and many different times, the uncountable version of the word "time" (without an article) sounds better.
And, which usages would be the most natural? In addition to the above mentioned example, you could also construct a sentence without "out of your days":
Thank you for taking time to run errands for me.
Thank you for taking the time to assist me.
and, you could insert the phrase "on many occasions" there.