The quickest way to tackle a problem such as this is to look how each restriction reduces the set of possible solutions. Take the most restrictive conditions first, so that you are always dealing with the smallest possible set of answers.
There are between 75 and 125 cars on the lot. { x| 75 < x < 125} (this renders a infinite solution set finite)
AND exactly one ninth of the cars are out of state, so the total number of cars must be evenly divisible by nine. (This eliminates 88.89% of possible solutions)
x ∈ {81, 90, 99, 108, 117}
x ∈ {81, 90, 99, 108, 117}
AND exactly one fourth of the cars are red, so the number of cars must be evenly divisible by 4. (This eliminates 75% of possible solutions)
x = 108
x = 108
Notice that the numbers 88.89% and 75% refer to the elimination of possible answers from the infinite list of counting numbers. In practice, once we reduce the list of possible solutions to a finite number, that percentage is often no longer exact.
By by arranging our restrictions according to how restrictive they are we work with smaller sets. Had we applied the restrictions in the order they were given in the problem, we would have gone:
would you rather work with a set with 5 elements or one with 13?
counting numbers
75 < x < 125
x ∈ {76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124}
x = 108
75 < x < 125
x ∈ {76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124}
x = 108
would you rather work with a set with 5 elements or one with 13?
(You could also have combined the second and third conditions by noticing that between them they require any number in the solution set to be divisible by 36, but as an exercise it makes more sense to go through the answer step by step.)
