SOLUTION: Of 75 cars that were inspected, 10 needed brake repairs and 18 needed exhaust system repairs. If the brakes or exhaust systems on 53 of the cars did not need repair, how many cars

You are given a "universal" set A of 75 cars, and two subsets A of 10 cars and B of 18 cars.


You also are given that the complement to the union of the sets A and B has 53 elements; hence, the union of A and B has  75 - 53 = 22 elements.


Thus we know that  n(AUB) = 23  (the number of elements in the union),  and  n(A) = 10,  n(B) = 18.


Then the number of elements in the intersection  n(AnB) = n(A) + n(B) - n(AUB) = 10 + 18 - 23 = 5.


Answer.  5 cars needed both brake and exhaust system repairs.