I am interested in a (reasonably) quantified explanation of why the
work week in most of the world is 40 hrs and not for example 20? How
did we arrive to the number 40.
Both current 40 hour workweek and its predecessor, 48 hours workweek with 6 days of work are a result of 8 hours workday. 8 hours comes from the division of 24 hours into three equal parts: 8h sleep, 8h rest and 8h work.
JoaoBotelho and Brian Romanchuk are right - it was direct result of the XIX and early XX century organized labour’s struggles, which was based on employees observations of their work fatigue rather than a result of a scientific research. Perhaps the first, who tried to measure the effect of a 8h workday from the productivity standpoint was Henry Ford, who introduced 8 hour-workday in 1914.
[...] Ford said, “We have settled on the eight-hour day ... because it
so happens that this is the length of time which we find gives the
best service for men, day in and day out.” This notion that the
institution of the eight-hour day was a matter of “proven” efficiency,
was based on no solid calculation showing the specific superiority of
eight hours. Still, Ford could point to several ways in which the new
schedule served the company. The switch from two nine-hours to three
eight-hour shifts provided for an extra six hours of production. [...]
Ford and his advisors also saw efficiency as served by the eight-hour,
five-dollar day in that the new labor policy decreased turnover and
absenteeism and increased labor discipline. [...] Labor turnover
dropped by 90 percent, and absenteeism was at least halved. According
to some figures it plummeted from 10 percent to .3 percent per day.
(source: Our Own Time: A History of American Labor and the Working Day, David R. Roediger, Philip Sheldon Foner - please refer to this source for broader context and more information on the 40 hours workweek in the U.S.)
Around the time of the I Word War, legislators started introducing 8 hour-workday (both in 40 and 48 hours-a-week model) pressed by workers movements, but there weren’t any hard calculations on this issue.
Later studies confirmed employees and Ford’s observations: there is a workhour-health limit of around 40 hours on average. The last research I know of, Hour-glass ceilings: Work-hour thresholds, gendered health inequities (Social Science & Medicine, 2017 vol. 176), calculated this limit to 43.5 h per week for men and 38 h per week for woman.
As to your second question, please make it into a different post and we'll answer it.