Practice!

Sample Problem

190 students take midterms. 178 of them pass the Math exam, 181 pass the Chinese exam and 174 pass the English exam. Then at least students pass all the 3 exams.

Solution

Since we want the least amount, we need to avoid repetitions of counting those who fail the 3 exams. That is, when the numbers do not contains each other, the number of students pass the exams is the least.

190 − 178 = 12 students fail the Math exam.

190 − 181 = 9 students fail the Chinese exam.

190 − 174 = 16 students fail the English exam.

Then at most 12 + 9 + 16 = 37 coins fail the exams. So at least 190 − 37 = 153 students pass all the exams.