Numerical answer
In a group of 100 people, 70 read newspaper A, 60 read newspaper B and 50 read newspaper C. Every person reads at least one of the three newspapers. What is the maximum possible number of people who read all three newspapers?
Show answer and solution
Answer: 40
Let a, b, c be the numbers reading exactly one, exactly two and all three newspapers.
People: a + b + c = 100. Readings: a + 2b + 3c = 70 + 60 + 50 = 180.
Subtracting: b + 2c = 80, so c = (80 − b)/2 ≤ 40, with equality when b = 0.
Feasibility of c = 40, b = 0: only A = 70 − 40 = 30, only B = 20, only C = 10, all three = 40; total 100 ✓, all regions non-negative.
Maximum = 40.
Tip: For max/min of the triple overlap, write the two equations (people and total readings) and push the other variable to its extreme, then check feasibility.
People: a + b + c = 100. Readings: a + 2b + 3c = 70 + 60 + 50 = 180.
Subtracting: b + 2c = 80, so c = (80 − b)/2 ≤ 40, with equality when b = 0.
Feasibility of c = 40, b = 0: only A = 70 − 40 = 30, only B = 20, only C = 10, all three = 40; total 100 ✓, all regions non-negative.
Maximum = 40.
Tip: For max/min of the triple overlap, write the two equations (people and total readings) and push the other variable to its extreme, then check feasibility.
Saved on this device. Sign in or create a free account to keep it on every device.