Add one before you're done

[RachelR514]

Hi! I don't understand the "add one before you're done" rule for counting integers. For example, one of the questions I answered asks you to number raffle tickets from 101 - 350. The Manhattan Prep explanation says to do 350 - 101 + 1 = 250. I thought the answer would be 249. Why do you have to add that extra 1?

[RonPurewal]

just do what you should do with these kinds of rules anyway: try it out on smaller / easier examples.

for instance, consider the whole numbers from 1 to 10.
ok, there are pretty clearly 10 of those, right?
you can see why subtraction DOESN'T work, here -- if you just do 10 minus 1, you'll get 9, and that's wrong.

[RonPurewal]

in any case, i would never be able to remember a rule like this (i wouldn't ever be able to remember whether the rule was "add 1" or "subtract 1").
so, what i'd do instead is just turn the list into "1 through N", through addition or subtraction.

here, for instance, you have 101 through 350.
if you just subtract 100 from all of these, you have 1 through 250. that's very clearly 250 integers, so, there must be 250 integers in the original list.