They absolutely are not talking about percentages. They're talking about factors. A factor is a term in a multiplication. In standard US English, increasing something by a factor means to multiply it by that number. Even the GMAT uses that phrase in that way. You absolutely can split hairs on this and the phrasing can be confusing but it is absolutely understood that "increased by a factor of 4" means to multiply by 4. If one was talking in terms of percentages, they would say something like "increased by 300%".
>* increasing something by a factor means to multiply it by that number*
That's absolutely wrong... When you multiply something you scale it by a factor. When you add to something you increase it.
1) 4A is 400% of A.
2) 4A in an increase of 300%.
3) 4A is an increase of 3A.
4) 4A is an increase of a factor of 3.
In the most charitable interpretation you are confusing the terms of exponential growth with linear scaling and misapplying from one to the other. A growth factor is not the same as an increase in a factor.
Go ahead and tell that to everyone who has been using that phrase to mean what I describe. I seriously searched for a single case of someone using the phrase in the manner you're insisting is correct and came up empty. Instead, I found a large number of resources backing up what I and OP have been stating. Even many professional mathematicians acknowledge the ambiguity but agree that the phrase means to multiply. You may disagree with how the phrase came to be but it clearly has been taken to mean to multiply.
In addition to the short explanation in the last link, some of the comments on that page may help shed light, such as:
I'm a college student, and for the past few years I've been an SAT
tutor to pay the bills. 95% of my students do not know how to
calculate percentage change. I spend hours teaching kids this.
They usually start to get it after working with the formula several
times, and then I show them this:
for any number "n"
100% increase = n*2 = double
200% increase = n*3 = triple
300% increase = n*4 = quadruple
...
x*100% increase = n*(x+1)
When math is being discussed, even simple math, I'm going to go with the mathematical explanation not the arguments of populism. That's especially true for this case, where people are corrected in school and out of school constantly for the mistake you are promulgating.
You seem to be stuck on the translation here. These are two different phrases with different meanings. Yes, they both refer to increases. Yes, they both can be interpreted as multiplications on the original term. No, they are not the same. You found exactly one case which does not use the same verbage and specifically points to percentages. I don't know how else to say that different phrases in English carry different meanings. You can't just say "a factor of X" is equivalent to "X00%". This just is not understood to be true. Even the GMAT and GRE present questions in this way with the understanding that "a factor of X" means to multiply by X.
I'm not going to continue to explain this to you. There's really nothing more to be said. If you attempt to solve problems presented in the format of "increased by a factor of X" such that X is not presented as a percentage, you will be marked as incorrect more often than not.
I'm not even explicitly arguing for a populism argument. I'm stating that the problem is presented in a different format than you are claiming it to be.
