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

Question 1117906: 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 needed both brake and exhaust system repairs?
Answer by ikleyn(52832) (Show Source): You can put this solution on YOUR website!
.

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.

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To get more acquaintance with this circle of problems, look into the lesson
- Counting elements in sub-sets of a given finite set
in this site.

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