TL : DR version : if your opponent has over 143.5 armor and your allies do not have armor reduction debuffs, get last whisper over the black cleaver, otherwise black cleaver is a better armor reduction item. if your allies do have armor reduction debuffs, you'll need to read further for how to calculate the break even point.

ok I just noticed quite a few people out there putting last whisper as a "core" item in their build but I know for fact that last whisper is a conditional item. why is it conditional? because it provides a lot less compared to black cleaver can.

now for those of you who still dont know, flat is applied before percentage, and reduction is applied before penetration, and penetration can not bring armor values down to 0

so heres the formula for those who are interested

TA = Target's armor, FAR = flat armor reduction, PAR = Percentage armor reduction, FAP = flat armor pen, PAP = percent armor pen

all the values of each sub catagory are added together, so if you had last whisper (40% armor pen) and weapon expertise (10 armor pen) you had a total of 50% armor pen

((TA-FAR)*(1-PAR))-FAP)*(1-PAP) = final armor, though like I said earlier, penetration can not bring the value below 0 so if the FAR and PAR brings it down to 0 or lower, the rest of the calculations is ignored, if FAP and PAP brings the value below 0 then the value remains 0

now that we know how to calculate this value, we can use it to determine at what armor value the enemy must have in order for last whisper to reduce more armor than black cleaver. we know that black cleaver reduces target's armor by 45 at max stacks, so at what armor value does 45 = 40%? it is 112.5, meaning that ((TA-FAR)*(1-PAR))-FAP) = 112.5, not TA = 112.5

most people who buy last whisper will have armor pen runes and sunder mastery which add about 31 armor pen (not including yellow and blue armor pen runes) so you would add this value in order to make sure armor pen from last whisper is the same as black cleaver which means that (TA-FAR)*(1-PAR) must = to 143.5 armor.

but wait there are still many other champions out there that also have armor reduction abilities as well, that apply a debuff thus must be counted in, and theres way too many to calculate, but you now have the base of how to calculate how much armor your opponent must have before last whisper becomes a better choice in terms of armor reduction.

or you can just pick up a black cleaver and skip the math since black cleaver is a constant value. or if you have no allies who apply an armor reduction debuff on your opponents, then you know that your opponent must have at least 143.5 armor before last whisper becomes a viable option over black cleaver. but then again black cleaver is a debuff and benefits everyone as well and that cant be ignored, dont forget this is a team game, and thus you fight in a team.