How many days was Noah's ark on the water after the 40 days and nights of flooding before they actually found land?

150 days. 'And the waters prevailed upon the earth an hundred and fifty days (Genesis 7:24) - ...and after the end of the hundred and fifty days, the waters were abated. (Genesis 8:3). And the ark rested in the seventh month, on the seventeenth day of the month, upon the mountains of Ararat. (Genesis 8:5).

Acccording to Genesis:

'In the six hundredth year of Noah's life, in the second month, the seventeenth day of the month, the same day were all the fountains of the great deep broken up, and the windows of heaven were opened. And the rain was upon the earth forty days and forty nights.' (7: 11 - 12)

'And the waters prevailed upon the earth a hundred an hundred and fifty days.' (7: 24)

'After the end of the hundred and fifty days the waters were abated. And the ark rested in the seventh month, on the seventeenth day of the month, upon the mountains of Ararat. And the waters decreased continually until the tenth month: in the tenth month, on the first day of the month, were the tops of the mountains seen.' (8: 3 - 5)

From the dates given in 7:11 and 8:4, it seems that the 40 days of actual rain are included in the 150 days during which 'the waters prevailed'. So it must have been 110 days after the rain stopped that the Ark came to rest on a (presumably still submerged) Ararat and a further two and a half months (roughly) until the mountain could actually be seen.

The Jewish calendar does not seem to have been regularized in those days: in the modern Jewish calendar there are 147 days from 17 Cheshvan to 17 Nisan (148 if you count both days), and not 150.