One example is when planning a long distance coastal or ocean route which yields the total duration, and we know the starting date and time, and want to know the date of arrival.
Another example is when figuring the watch error in cel nav for a watch set some time ago. We might know the watch gains 0.335 seconds every 10 days and it was set on April 14. It is now July 22 and we want to know the watch error.
We could of course start out with "Thirty days has September, April, June and November..." to find number of days for each month (if we did not know this already), and then count out the days. But we don't want to make mistakes at the Nav Table, so it is much easier to look up that (in say 2019) April 14 is day 104 and July 22 is day 203. This difference is 99 days and (99 /10) x 0.335 = 3.3 seconds. The watch error at hand is -3s.
If you had a Nautical Almanac onboard, you would find that there is a Day of the Year (DOY) Calendar in every issue, because they know this comes up in ocean navigation. If you do not have an Almanac, you can use our new compact DOY calendar, which we will keep updated and available online.
Note added Nov 29, 2023: we never did follow up on this project, and instead just recommend the DOY calendar in the Nautical Almanac.... In part because you can just google DOY and get one.