In a class of 25 pupils, 19 have scientific calculators and 14 have graphic calculators. If xpupils have both and y pupils have neither, what are the largest and smallest possiblevalues of x and y?

Answer 1

The largest value of X is, obviously, 14. There are only 14 graphic calculators, so if every student has who has a graphic calculator also has a scientific calculator, then there are 14.

The smallest possible value of X is more complex. 19+14=33, which is how many students we would need for there to be no overlap. Since there's only 25 students, there must be some overlap.

33-25=8, the difference between how many students that we need for no overlap and the number that we have. So let's do some quick math to verify. The number of students who have only scientific, plus the number of students who have only graphic, plus the number who have both should be 25. (19-8) + (14-8) + 8 = 25. So 8 is the smallest possible overlap. Any less than that would require more than 25 students.

We already know, at this point, what the smallest possible value of Y is: zero. We already demonstrated that there aren't enough students to spread these calculators out with zero overlap, so we already know that it's possible for every student to have at least one calculator, leaving 0 who have none.

The most students who will have neither is the same scenario as the most students who have both: 14 people have both. 19-14 =5. So there's 14 people with both, leaving 5 with only a scientific calculator. 25-5-14 = 6 people who have neither in this case. 6 is the highest possible value of Y.

In total:

X is between 8 and 14. Y is between 0 and 6.