Yeah, the math doesn’t work out really all that well. I think it’s mostly because the joke is intentionally confusing the 8-hour day of work with the 24-hour calendar day.

Here’s the numbers I get.

Days of the year: 365

Nonweekend days: 365-23*52 = 261. So far so good.

16 hours per day not spent at work implies that 8 hours per day are at work, leaving 261*8 = 2,088 hours of work per year, which is actually 2,088/24 = 87 days, less than the 91 the joke mentions.

Lunch and coffee break equal 1.5 hours per day of the 261 days, which basically just means you’re at work for 6.5 hours a day instead of 8, meaning we should have here 6.5 * 261 = 1,696.5 hours or about 70.7 days, which is way more than the 22 the joke mentions. And I have no idea where the joke is getting its numbers at this point.

(Although, just for the coffee break part, 23 days of 0.5-hour coffee breaks means that there is 15 times more work time than coffee break time, and multiplying that out gives 368 days, which means that the boss is giving you a coffee break on your day off. So maybe that’s how it’s doing it. Terrible rounding, here, too.)

Anyway, taking the math further, 2 sick days, 5 holidays, and 14 vacation days is another subtraction of 21*6.5 hours, which gives 1,696.5 - 136.5 = 1,560 hours or 65 days, and here it’s clear the boss is conflating the 6.5-hour work days with actual 24-hour days.