Out of 35 students, 18 have at least one brother and 19 have at least one sister. 3 students have both brothers and sisters. How many students do not have either a brother or a sister?

Answer 1

Hi April D.,

Again, an ambiguously written problem. The issue I have with it is the "three students have both brothers and sisters". If it had said, "each at least one brother and one sister", that would be unambiguous. Do you see the difference?

Let's assume your questioner really means the latter. The way to conceptalize this is as a line-up. First, put the 18 students having at least one brother (each?!) on the left of the line, followed by 19 students who have at least one sister (each?!) on the right. Count them up, that's 37 people. Since you only have 35 students, there's obviously overlap! In fact, you know exactly how much overlap: it's 3 students (who have each a brother and a sister). So move your line segment on the right over 3 spaces to the left. Those 3 who overlap are in fact just one student each, not 2! You now have in line 15 with only >= 1 brother, 3 with >=1 bother and 1 sister, and 16 with only >= 1 sister. But that's only 34 students in the line now! Therefore, that 35th student spot must be someone not initially lined up, who has neither a brother nor a sister. It's OK that they suddenly "popped into view", since you only took subsets of the class when you did your initial lineup and shift operations. So just 1 student. Seems statistically low, by the way -- you might model assuming siblings ave=2, standard deviation=1, (and other data sets), randomly "seeded into classes"....

-- Cheers, --Mr. d.