 Proof: I don't know if this was posted elsewhere, but I couldn't find it anywhere so I just did it myself.

The Conclusion is: Barrels stack multiplicatively

For +45 AD, I bought 3 executioner's callings to mimic an LDR or a Mortal Reminder without any armor penetration passive.

With level 13(Level 5 Barrel, Level 5 Q) and 153 AD, Gangplank does 272 damage with a Q and 452 damage with a Q+Barrel against an enemy with 0 armor.

Using this to our advantage, we can get the values for an enemy with 100 armor by cutting those values in half. Of course, I didn't go the lazy way and do that, I tested the values using practice dummies.

Our goal now is to see if Gangplank Barrels have armor penetration that stacks additively, or multiplicatively with the armor penetration from items. However, the physical reduction equation is a very annoying calculation to do, so we will use a convenient values method instead.

Since we know that 100 armor is 50% physical damage reduction and 0 armor is 0% physical damage reduction, we will try to reach 100 effective armor using math. This is shown with the 170 armor test. We are looking for a number x, that when reduced by 40%, equals 100. This word problem ends up being 0.6x=100, which x is approximately 170. Because the practice tool doesn't allow me to increment in values of 1, we will have to round it to 170 instead of getting the most precise value.

So now we have the value 170, we can proceed. The barrel does 452 damage against an enemy with no armor, so it should do half of that for an enemy with 100 effective armor. Since barrels ignore 40% armor, we can see 170*(100%-40%) = approximately 100 effective armor(True value is actually 102, but it's close enough to 100 that we can just ignore the 2 armor). So now, we use Barrel+Q against an enemy with 170 armor. The value should be approximately cut in half, and success, it is. We get 224 damage, which is approximately half of 452 (224*2=448). The 4 missing damage discrepancy is a result of being unable to put precisely 2 armor on a practice dummy.

Now we have a way to calculate the barrel damage with the armor, we can proceed to apply this method to more complex values. We know 2 possibilities exist. The additive stacking and multiplicative stacking. So let's create test values to account for both. With an LDR, if it stacks additively, it's 40%+35% armor ignore, which is 75% ignore armor. For multiplicatively, it's reduced by 40% then that number is reduced by 35%, or the other way around. To calculate the armor ignore, we use a multiplier method. To reduce by 40% is the same as multiplying by 0.6, and to reduce by 35% is the same as multiplying by 0.65. So, if we reduce an armor amount, A, by 40% and 35%, we are effectively multiplying it by 0.6 then 0.65. So A*0.6*0.65=0.39. Since we are effectively multiplying A by 0.39, it's actually a 61% reduction of Armor, not a 39% reduction.

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So let's create two convenient armor values using these methods. Let's consider 75% armor ignore, the additive path. *0.25 = 100. Dividing both sides gives us that the value of that hypothetical armor amount is 400. Using the same method for 61% armor ignore, the multiplicative path, we are able to achieve the equation *0.39=100. Dividing both sides by 0.39, we get that is approximately 256.4, which we will just round up to 260. So now, we have 2 values; we have 400 representing the additive path, and 260 representing the multiplicative path. If we use our barrel+Q on one of these values of armor, one will be exactly half of 452 since that one will have an effective 100 armor. This is because we set up our equations in a way such that after the barrel and armor penetration apply, the armor should be 100, which we know is a perfect, exact, and convenient 50% damage reduction.

After I tested both armor values on two separate dummies, looking at the charts, it is clear to see that 260 armor with LDR yielded the same value as 170 armor without LDR, and 400 armor with LDR did not. Now, it has been proven beyond a doubt that barrels stack multiplicative with armor penetration. I repeated this experiment with Mortal Reminder, keeping in mind it has 25% armor penetration instead of 35% so as not to make a foolish error that would give false results, and I reached the same conclusion. Furthermore, we can see a phenomenon of repeating damage values in my results, which further proves they stack multiplicatively. For example, we can see that 220 armor Barrels without mortal reminder have the same damage as 290 armor Barrels with mortal reminder. Using the fact that 220 armor is approximately a 25% armor reduction of 290 armor, we can deduce that the values are the same because Mortal Reminder ignores the 25% FIRST, then the barrel damage applies to the amount of armor post Mortal Reminder reduction.

Now we know since it is multiplicative, we now know that with LDR, your barrels will ignore 61% total armor, and with Mortal Reminder, your barrels ignore 55% total armor. This means that although LDR has 10% more armor ignore than mortal reminder, the math shows that it's actually only a 6% difference.

Conclusion/TL;DR:

LDR is not that much better than Mortal Reminder for Gangplank's Barrels since the armor penetration stacks multiplicatively. It only actually ignores 6% more armor than Mortal Reminder does, not the full 10%. This means it's probably better to buy Mortal Reminder over LDR in almost all cases, since Grievous Wounds is better than 6% armor penetration in most cases. Most teams will have some sort of healing through support, life steal, or other abilities.