How is the Zakat calculated for a sum of 10,000 dinars accumulated over varying periods spanning 10 years, knowing that parts of it have had a full year pass over them, and the exact amounts cannot be determined? And is it considered circulating money upon which Zakat is not due, or is it due on the entire amount?

If the questioner has money in the bank that reaches the nisab, then Zakat is obligatory on the surplus amounts, even if these surpluses do not reach the nisab on their own. This is because they are subsidiary to the existing wealth, and all of it is considered one category. Since the surplus cannot be determined precisely, it should be estimated based on the predominant assumption (ghalabat al-dhann), and one-quarter of one-tenth (2.5%) should be paid for each past year to clear one's liability.

However, if the money was accumulated before the known amount was in the bank, then Zakat is not obligatory on it unless it reaches the nisab and a full year (hawl) has passed over it. Its hawl begins from the date it reached the nisab by itself or by being combined with other wealth.

If the time of reaching the nisab is unknown, then one must investigate and act upon the predominant assumption. The nisab for currencies is equivalent to 85 grams of gold. Zakat does not lapse with the passage of time. Whoever delays it has sinned and must pay it for all the delayed years. One should act upon their best judgment in estimating the wealth and the number of years if there is doubt.