Set A has 20 readings whose smallest is 5 and whose range is 12. Set B has 30 readings whose smallest is 20 and whose range is 9. What is the range of the 50 readings taken together?

A range is decided by the two extreme readings and nothing else, so recover them first. Set A runs from 5 to 5 + 12 = 17 and Set B runs from 20 to 20 + 9 = 29. Together the readings run from the smallest of the two minima, 5, to the largest of the two maxima, 29, so the combined range is 29 − 5 = 24. The value 21 adds the two ranges, 12 + 9, which would only be right if Set B began exactly where Set A ended; here there is a gap between 17 and 20, and that gap is part of the combined spread. The value 12 is simply the larger of the two ranges, and 3 is the gap 20 − 17 between the sets. Note that the sizes 20 and 30 never enter the working — unlike a mean, a range takes no account of how many readings there are.