2020 AMC Question 24

 A light rail network has 21 drivers, but not all of them are required at the same time:
• 15 drivers are required for the Friday night shift.
• 12 drivers are required for the Saturday morning shift.
• 9 drivers are required for the Sunday morning shift.
Given that every driver must work on at least one of these shifts, what is the maximum number of drivers that can work on all three shifts?
 A light rail network has 21 drivers, but not all of them are required at the same time:

@gentlegorilla The way I would approach this problem is to set a variable. Let there be n drivers that work on all 3 shifts we're trying to maximize n. Now, we can separate the drivers into 2 categories: there are n drivers that work on all 3 shifts, and 21n drivers that do not.
Notice that there are 15 + 12 + 9 = 36 shifts total. How many of those shifts are taken up by the drivers that work all 3? How many shifts are remaining for the other 21n drivers? Can you write an equation or inequality now using the restriction that every driver needs to work on at least one shift?
Hope this helped, and let me know if you have questions!