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First, the relevant rule excerpts.

Favored Class Options

Some of these alternate class rewards only add +1/2, +1/3, +1/4, or +1/6 on a roll (rather than +1) each time the reward is selected. When applying this type of result to a die roll, always round down (minimum 0). For example, a half-orc with gunslinger as her favored class adds a +1/4 bonus on attack rolls and a +1/2 bonus on damage rolls when using the pistol whip deed each time she selects that favored class bonus, though this means the net effect is +0 on each after selecting it once (because both +1/4 and +1/2 rounds down to 0). After 20 levels, this benefit gives the half-orc a +5 bonus on attack rolls and a +10 bonus on damage rolls when using the pistol whip deed (in addition to the base value from being a 20th-level gunslinger).

Rounding

Rounding: Occasionally the rules ask you to round a result or value. Unless otherwise stated, always round down. For example, if you are asked to take half of 7, the result would be 3.

Half-Orc Favored Class Options

Inquisitor: Add a +1/2 bonus on Intimidate checks and Knowledge checks to identify creatures.

Stern Gaze

Inquisitors are skilled at sensing deception and intimidating their foes. An inquisitor receives a morale bonus on all Intimidate and Sense Motive checks equal to 1/2 her inquisitor level (minimum +1).

I am creating a Half-Orc Inquisitor. I plan on taking the Half-Orc's alternative FCO at each level-up. The bonus granted by the FCO is untyped. The bonus granted by Stern Gaze is a morale type bonus. Therefore, in the absence of any text stating the contrary, these two bonuses stack because they are of different types and come from different sources. At 1st level, I will have a +1 bonus to Intimidate from Stern Gaze because even though it rounds down, it explicitly states that the minimum bonus is +1. I will also have a +1/2 bonus to Intimidate at 1st level from the FCO.

My question is this: How do these fractional bonuses stack?

Do I add these fractional bonuses together first and then round down the resulting total bonus? Do I round each fractional bonus down first and then add them together? Is there another possibility that I am not considering?

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Favored class options and fractional bonuses

I know it's odd, but let me directly address this part of your own answer:

Dyndrilliac: If fractional bonuses were intended to be rounded down prior to being added together, then logically, the FCO would never provide any bonus because it adds +1/2 each time it is selected. So in order to receive any bonus at all, it must be selected multiple times and those individual +1/2 bonuses need to be added together prior to rounding in order to ever result in a positive integer.

Emphasis mine. I get what you're saying, and I understand your frustration. I fear, however, there's an intermediate step that might've been overlooked. Racial Favored Class Options says

Many... alternate class rewards add only 1/2, 1/3, 1/4, or 1/6 to a roll (rather than 1) each time the reward is selected, or add 1/2, 1/3, 1/4, or 1/6 to a class ability.... When applying such a benefit to a die roll or a class ability, always round down (minimum 0). You might thus need to select such an option several times before the benefit applies.

And Alternate Racial Rules says

When choosing one of these favored class options, the reward is gained on a level-by-level basis. Unless otherwise noted, these benefits always stack with themselves.

Emphasis mine. Hence, a favored class option bonus stacks with itself even if it's a fraction. That is, a favored class option is picked, the favored class option bonus is computed and stacks with itself, and then the rounding happens. For example, a favored class option that's +½ and picked twice isn't two +0s (+½ each time and rounded down each time) but one lone +1. Favored class options are a rare case of specifically stacking fractions, and keeping fractions doesn't happen with, for example, the inquisitor's extraordinary ability stern gaze, which just generates a value that can be a fraction and is rounded down when it does.

A fractional bonus from a favored class option does not get rounded before it's added with itself for the same reason that a creature picking the generic favored class option +1 hp twice means the creature gets from that option +2 hp instead of only +1 nonstacking hp twice: it's the same source so it normally doesn't stack, but the game specifically says it does so it does but only with itself. But the special ability stern gaze has no provision for keeping fractions with itself or another feature, so the special ability stern gaze is always rounded down normally before it's stacked with anything; there's just no option to add +½ of a bonus from the special ability stern gaze to another +½ from somewhere else as stern gaze is separately and independently computed as "1/2 [the inquisitor's] inquisitor level (minimum +1)."

I'm unaware of any other sources that stack fractional bonuses before rounding. Unless the argument is that The game does not say you can't stack then round, there's no way for a half-orc or tiefling inquisitor that takes the inquisitor favored class option to do so. That means, unfortunately, the way it works is the way that folks that disagree with you say it works.

On rounding fractional bonuses

The bonuses from the the inquisitor's extraordinary ability stern gaze and the half-orc or tiefling's inquisitor favored class option are, obviously, from different and discrete sources. The bonuses can't be applied before rounding because each bonus already exists separately as a unique bonus.

That is, the first feature grants a +1½ bonus therefore granting the creature a +1 bonus (if nothing indicates otherwise), and the second feature grants a +1½ bonus grants therefore granting the creature a +1 bonus (if nothing indicates otherwise), and, because those bonuses already exist, it doesn't matter to the game that those bonuses end up applied to the same third thing.

A house rule can fix this by, for instance, keeping fractions when two or more things add fractional bonuses to the same thing, but that house rule's contrary to—I'm pretty sure—everything else in the game about computing bonuses, requiring confirmation first that another fractional bonus will be added to that third thing before knowing to keep the fractional bonus.

Really, when determining a bonus's value, the game usually cares less about where a bonus goes (seeing after it's computed if it stacks, for example) and much more where it's from (because that usually determines its value), so such a house rule may create some confusion, especially among long-time players.

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The fractional bonuses should be added prior to being rounded.

If fractional bonuses were intended to be rounded down prior to being added together, then logically, the FCO would never provide any bonus because it adds +1/2 each time it is selected. So in order to receive any bonus at all, it must be selected multiple times and those individual +1/2 bonuses need to be added together prior to rounding in order to ever result in a positive integer.

Given a 2nd level character who has selected the FCO twice, you can demonstrate this using the mathematical floor function:

  1. floor(1/2) + floor(1/2) = 0
  2. floor(1/2 + 1/2) = floor(1) = 1

You might argue that the fractional bonuses from the FCO get added together prior to rounding due to being a special case. I would counter-argue that the only special case explicitly indicated by the rules is that FCO bonuses stack despite being from the same source, unless the text says otherwise. If the logic of this line of thought is that two bonuses that stack are added together prior to being rounded down, then in order to be internally consistent that logic must apply to this case as well because it has already been shown that the bonus from Stern Gaze stacks with the bonus from the FCO.

You might also argue that the rules for Racial Favored Class Options have a provision in the rules for keeping fractions, whereas the Stern Gaze class feature does not.

Many... alternate class rewards add only 1/2, 1/3, 1/4, or 1/6 to a roll (rather than 1) each time the reward is selected, or add 1/2, 1/3, 1/4, or 1/6 to a class ability.... When applying such a benefit to a die roll or a class ability, always round down (minimum 0). You might thus need to select such an option several times before the benefit applies.

Emphasis mine. In fact, the text regarding Racial Favored Class Options has no provision for keeping fractions. Fractional values are rounded down as usual. The provision merely states that rounding does not occur until being applied to a die roll or class ability. The Inquisitor's Half-Orc FCO is not applied to a class ability. It is applied directly to the modifier for Intimidate checks, which are D20 die rolls. The default rounding rule states that values which are the results of computations are rounded. I would counter-argue that because the FCO applies to the Intimidate check, and the Stern Gaze class feature applies to the Intimidate check, rounding does not occur until the total resulting modifier for the Intimidate check is computed.

Doing it my way results in the following bonus progression to Intimidate:

  1. +1 after rounding [+1.0 from Stern Gaze and +0.5 from the FCO(s)]
  2. +2 after rounding [+1.0 from Stern Gaze and +1.0 from the FCO(s)]
  3. +3 after rounding [+1.5 from Stern Gaze and +1.5 from the FCO(s)]
  4. +4 after rounding [+2.0 from Stern Gaze and +2.0 from the FCO(s)]
  5. +5 after rounding [+2.5 from Stern Gaze and +2.5 from the FCO(s)]

... and so on such that the total bonus is always a 1:1 representation of the Inquisitor's class level.

Doing it the other way results in the following bonus progression to Intimidate:

  1. +1 after rounding [+1.0 from Stern Gaze and +0.5 from the FCO(s)]
  2. +2 after rounding [+1.0 from Stern Gaze and +1.0 from the FCO(s)]
  3. +2 after rounding [+1.5 from Stern Gaze and +1.5 from the FCO(s)]
  4. +4 after rounding [+2.0 from Stern Gaze and +2.0 from the FCO(s)]
  5. +4 after rounding [+2.5 from Stern Gaze and +2.5 from the FCO(s)]

... and so on such that at 3rd level and every odd level after that, the total bonus will be one less than the Inquisitor's class level.

Either way, at every even Inquisitor level, the bonus is equal to the Inquisitor level. The upper limit on the bonus is the same in both instances: a maximum bonus of +20 for a 20th level Inquisitor. That being the case, with the lack of an explicit line of text from the rules telling us with absolute certainty which way of crunching the numbers is the correct way, I opt to take the path which benefits the character the most for the odd levels in between. This is also consistent with the fact that both the FCO and the class feature benefit the Intimidate skill. Clearly, we can infer that the designer's intended the character to be extremely good at intimidation. I choose to believe that intent extends to the odd character levels, and not just the even ones.

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The answer really comes down to a different question. "Do you round individual bonuses, or the total modifier?"

If you round individual bonuses, then they each have to hit a full integer value before they are of any use. If you round the total modifier, then three +1/6 and one +1/2 modifier are the same as a +1 modifier.

I don't see anything in the rules that definitively says whether the rounding happens as each individual bonus is calculated, or after all the bonuses are added together.

Frankly, as a GM, I'd be inclined to rule in favor of the PCs on this one, and round the total modifier rather than the individual bonuses. As a player, I avoid favored class bonuses like this, simply because it's just another fiddly bunch of numbers to have to deal with, and it feels like poor design.

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