Why Danes use it?
The force of habit
Using week numbers is a habit. Week numbers were officially implemented in a Danish calendar in 1973, however, people already used it before that. Several generations grew up with this system, taking it for granted, so many use it without a second thought.
The role of the education system
It seems to me that those who have a touchpoint with the education system are more likely to use week numbers. From what I have experienced, both schools and universities make curriculum plans by week, so people who are in the education system or are parents of school-aged children are exposed to week numbers often enough to somewhat remember them and use them in contexts outside education.
Convenience
It's not every Dane who uses the week system, but those who do find it convenient. Looking at time from a week perspective gives an overview that is divided into manageable chunks.
Weeks are also convenient for making plans far into the future. For example, the school break is in the same week every year, even if the dates vary by a few days. It is common for parents to take those weeks off work. So much so that you notice emptier offices, slower business responses, and more crowded entertainment and leisure institutions. Workplaces know not to make big plans, and you know not to get over your head expecting swift answers to your emails in week 7 and week 42.
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