I have 10 beans. Jim has four times more.
Is this a valid sentence? And, if so, does it mean Jim has 40 or 50 beans?
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I would recommend against using four times more in this context, although it would generally be understood to mean 40. What I would say is four times as many.
Googling four times more and four times as many, it seems that four times more is generally used with uncountable nouns, and four times as many with countable nouns.
four times more cancer-causing tar.
four times more carbon emissions.
four times as many journalists
four times as many males.
four times as many films.
I won't do more than mention the confusing construction four times as many more, which I would recommend against using, as I think it's ambiguous.
The grammatical rule is: use X times as many with countable nouns, and X times as much with uncountable nouns. The construction X times more is grammatical with both, but as many and as much are clearer in this context. Modifying adjectives, however, you should use four times more (or four times as):
four times more likely.
four times more prevalent.
four times more dangerous.
four times more effective.
I think the sentence makes sense to the question how many beans do you have and how many does Jim have.
You have 10 and Jim has four times more.
From a "scientific" perspective, the more than/ as many part of the expression indicates "direction"— think vector, rather than scalar. "More" indicates who has the greater quantity. The "four times" indicates the degree of the comparison — you have four times fewer than Jim. The more/as many and few/less than are analagous to "multiplied by" or "divided by" in mathematics.
Then, as the comments above point out, the correct form to use depends on whether the noun is countable or not.
Using times with a comparative (more, faster, better, etc.) doesn't make mathematical sense. You can't identify the second value in the equation: 4 × (more than you have). How much or many is more than you have? This is unknown for both countable and mass nouns.
What is meant is four times as many (or as much) as you have. You can also say four times what you have. If you always multiply by the known value, you'll be clear and precise. Instead of times more, faster, or better, say times as much, many, fast, or good. Bill Walsh (in Lapsing Into a Comma) and Bryan Garner (in Garner's Modern American Usage, 3rd edition) explain this point well.
To scientists and mathematicians, 4 times more has a precise meaning. 4 times the base (10) is 40 so four times more than 10 is 50.
One could say that 40 is four times ten, but 50 is four times more than 10.
Consider, what is 50% more than 10? 50% is another word for half. I would say that 50% more than 10 is 15 because you multiply 10 by a half and add that to 100. If you take the approach that you just multiply, then you would say that 5 is 50% more than 10?!
What if you made $10 per hour and I told you I made 1 times more than that per hour.
This is very similar to Percent - look that up on wikipedia and it has examples that show how something that is 100% more than something else is twice as large
I was taught that "X times more" is (X+1) times as many, it should be the "more correct" one but like many other terms that are frequently misused to the extent that they become the norm, it is not a surprise if it meant otherwise to some people.
Just try avoid using such words if you are the one writing it.