Having just written about issues of wording with regard to percentages, we should look at another wording issue that touches on percentages and several other matters of wording. What does “three times larger” mean? How about “300% more”? We’ll focus on one discussion that involved several of us, and referred back to other answers we’ve given.
Percent increase vs. factor
The question, from 2006, started with the idea of a percent increase:
Percent Increase and "Increase by a Factor of ..." A math doctor here recently explained percent increase this way: If we start with 1 apple today and tomorrow have 2 apples, then because 2 - 1 = 1 and 1/1 = 1, we have a 100 percent increase. But can't I also say there was an increase by a factor of 2? Two divided by 1 equals 2, an increase by a factor of 2 -- and also an increase by 200 percent? This is what is confusing me! I'd never been confused about saying "increased by a factor of" and "increased by percent of" until I saw the Dr. Math conversation about finding percentages ... which is a good thing, I guess, because now I know what I didn't know! Thank you for any help.
Unfortunately, Joseph didn’t directly quote from the page he had in mind, and we have never said exactly what he said; so we couldn’t be sure which page it was. Everything he said, however, was correct.
Doctor Rick was the first to answer:
Hi, Joseph. Yes, this can be very confusing, because some statements about increases are ambiguous. When we say "increased by a factor of 2," the word "factor" makes it clear that we mean "multiplied by 2." When we say "increased by 10%," there is only one reasonable interpretation: the amount of the increase is 10% of the original amount. If we meant multiplication by 10%, that would be a decrease -- not an increase! Even when we say "increased by 100%," there is only one reasonable interpretation, since multiplication by 100% is the same as multiplication by 1, and that's still not an increase. When we want to speak of an increase that is greater than the original amount, then ambiguity can arise. In that situation, I much prefer "increase by a factor of 3" or "by a factor of 2.5," etc. I don't know what page you saw -- but have you seen this one? Percent Greater Than vs. Increased http://mathforum.org/library/drmath/view/61774.html See also the page linked there, about the even more confusing phrase "___ times more than" and the like. I am on the side of avoiding the confusing phrases, as a basic principle of communication. If you saw another page and you are still confused by it, please tell me the URL of that page so I can review it with you.
Ambiguity in percent increase
Before we get back to this conversation, we should take a look at the page he referred to, which is a good starting point:
Percent Greater Than vs. Increased What is the difference between the following statements: My profits are 200% bigger than they were last year. and My profits from last year have increased 200%. This is one of the questions we have to answer in my Middle school methods course and I have looked everywhere for the answer. I hope you can help.
I answered this one:
As far as I can see, they mean the same thing; in fact, both are similarly ambiguous. Taken literally, "200% bigger" (or, more formally, larger or greater) and "increased 200%" (or, more completely, increased _by_ 200%) both mean that the increase from one year to the next is 200% of the first year's value, so that the second year's profit is 3 times the first. But both statements are more likely to have been made with the intention of saying that this year's profit is twice last years. English is not very clear in cases like this.
So there is a literal meaning (which mathematicians tend to see as best), and an idiomatic meaning (which ordinary people are more likely to have intended to say). I referred to a page we’ll be looking at below, and then quoted a favorite book of mine that gives a lexicographer’s perspective:
Since writing that, I found a good reference on "two times greater," although it doesn't mention your "200% greater." It is in Merriam Webster's _Dictionary of English Usage_, which under "times" writes The argument in this case is that _times more_ (or _times larger_, _times stronger_, _times brighter_, etc.) is ambiguous, so that "He has five times more money than you" can be misunderstood as meaning "He has six times as much money as you." It is, in fact, possible to misunderstand _times more_ in this way, but it takes a good deal of effort. If you have $100, five times that is $500, which means that "five times more than $100" can mean (the commentators claim) "$500 more than $100," which equals "$600," which equals "six times as much as $100." The commentators regard this as a serious ambiguity, and they advise you to avoid it by always saying "times as much" instead of "times more." Here again, it seems that they are paying homage to mathematics at the expense of language. The fact is that "five times more" and "five times as much" are idiomatic phrases which have - and are understood to have - exactly the same meaning. The "ambiguity" of _times more_ is imaginary: in the world of actual speech and writing, the meaning of _times more_ is clear and unequivocal. It is an idiom that has existed in our language for more than four centuries, and there is no real reason to avoid its use. I think the same applies to "X percent bigger" and "increased [by] X%." There is just enough ambiguity in a technical context that I would want to ask what was intended before assuming anything, but there is no reason to say that they definitely mean different things, or mean something different than "X percent of" or "increased to X percent." I myself would avoid saying these things, just because there are enough people who have heard that they are ambiguous, and would therefore take them the wrong way (whichever that is!).
As a result of my side interest in linguistics, I recognize that human speech is not as logical as we might wish; what a word means is a matter of actual usage in a culture, rather than pure logic. So rather than state that either understanding of “times bigger” is “correct”, I just recognize that people take it in two ways, so you have to ask, or use contextual cues, in order to decide on what is meant.
Confusion about “three times larger”
Back to the original discussion: Joseph responded with specific references, the first of which was that link of mine that Doctor Rick said to “see also”:
I'm sorry, I should have specified the site. In fact, there were two -- and I still don't see the difference between them. Here is the first example, from Larger Than and As Large As http://mathforum.org/library/drmath/view/52338.html 1) "Three times as large as N" means "3 * N." 2) "Three times larger than N" means "4 * N" -- but only if you stop to think about it, as many people do not. Here, I don't understand how something can be 3 times larger and be 4 times N. That sounds really weird to me. If you asked "What is something that is three times as large as N?" then I would say 3N ... but apparently I'd be wrong! I just don't see where my thinking is wrong.
His thinking isn’t wrong on this point: 3N is three times as large as N! He seems just to be letting what he’s read sow doubt about everything.
Here is the second example, from Percentage of Increase http://mathforum.org/library/drmath/view/58131.html You can choose two ways to express your answer now. One is to say: there will be a 550% increase by the year 2000. Or you can say: in the year 2000 the (new value) -- you didn't say what the numbers represented, so I'm a little confused right here -- will be about five and a half times greater than what it was in 1995. Many people don't quite grasp those phrases, especially the latter one. Instead you might wish to say it this way: in 2000 the (new value) will be 6 and a half times what it was in 1995. The difference in the wording is subtle, of course, but important. The number 6 1/2 comes from 325,000 --------- = 6.5 or 6 1/2 50,000 which is NOT a percent increase situation. In this problem, I don't understand the difference between the way the doctor explains the two different ways you can talk about the increase, and the implications of each. The doctor says that 6.5 times is not a percent increase; but can you still say it's 650 percent OF the original? I'm sorry -- this is all very confusing at this point!
So his specific question is this:
- Why wouldn’t “three times as large” and “three times larger” mean the same thing?
- How can “5 1/2 times greater” and “6 1/2 times what it was” mean the same thing?
The two pages quoted are by me (1999) and Doctor Terrel (1997). The first is particularly worth reading in its entirety, as there is a lot more there.
The case for a literal interpretation
Doctor Greenie responded, arguing against laxness on the matter, and making the case for the literalistic interpretation:
I'm going to jump in here, because this is one of my pet peeves. Mathematics is commonly called the exact science. Mathematics must be exact; if it is not, it all falls apart. We can't use ambiguous language in mathematics. I agree that the use of the phrase "x times larger than" is best avoided. However, as a mathematician who believes in using unambiguous language, I cannot accept the proposition that we should be able to interpret "5 times larger than 10" as either 50 or 60. It HAS TO BE ONE OR THE OTHER. And grammatically, "5 times larger than" means the "new" number is 5 times larger than the "old" number; this in turn means the difference between the new and old numbers is 5 times the old number, making the new number 6 times the old number. So the number which is 5 times larger than 10 is 10 + 5(10) = 10 + 50 = 60 (The phrase "... larger than ..." implies comparison by subtraction; the phrase "... as large as ..." implies comparison by division. Sixty is 6 times as large as 10, because 60/10 = 6. But 60 is 5 times larger than 10, because [60 - 10]/10 = 50/10 = 5.) Yes, we hear it all the time in everyday life. Sometimes, we even hear it in the supposedly rigorous world of science -- "an earthquake of magnitude 5 is 10 times greater than one of magnitude 4," and such. But the common idiom of using "10 times greater than" -- when the actual meaning is "10 times as great as" -- has no place in mathematics.
He concluded with an accidental overstatement of what the “other side” says:
I disagree with many of the concessions that other math doctors here have made in interpreting the phrase "x times larger than" as being the same as "x times as large as." On one of the pages I saw, a fellow doctor said that "50% larger than" and "50% as large" mean the same thing. But if my weekly salary last year was $1000 and it is 50% LARGER this year, then it is now $1000 + 50%($1000) = $1000 + $500 = $1500 While if it was $1000 last year and it is 50% AS LARGE this year, then it is now 50%($1000) = $500 If something is 50% larger, then it is larger; if it is 50% as large, then it is smaller. They can't be the same; that is nonsense.
In fact, as he admitted in a subsequent private discussion, he had misremembered what others had said; none of us have claimed that “50% larger than” and “50% as large” mean the same thing. What we say is that, when the percentage or multiplier is greater than 100%, we recognize ambiguity in the likely intent. I think we agree that the phrase should not be used in a mathematical context, and that we both grudgingly interpret it as intended elsewhere.
What the literal interpretation means
Then it was my turn to respond, as the author of the first page Joseph had asked about, wanting to make sure he understood both why people take it literally as they do, and how we should think about it.
First, on “Larger Than and As Large As”, I said this:
Joseph, your thinking is RIGHT: if M is three times AS LARGE AS N, then M = 3N. That's what statement (1) above says. But if we break statement (2) apart carefully (some would say TOO carefully :-)), then it means something different from what people usually mean by it. If I said "M is 50 larger than N," I would mean that if you ADD 50 to N, you get M: M = N + 50. And if I said, "M is 50% larger than N," I would mean that if you add 50% OF N to N, you get M; that is, I mean that M is 50% of N added to N: M = N + (0.50)N. Now, though I'm not entirely sure I agree with this, technically minded people often apply the same thinking to (2), for the sake of consistency. The "larger than" means we add something to N. And what do we add? Three times N. So by this thinking, M = N + 3N = 4N. So "three times LARGER THAN N" means the same as "four times AS LARGE AS N."
I then referred to the usage book quoted above, adding:
English usage experts think it is nonsense. My feeling is that this thinking puts a little too much weight on consistency, and is just too weird for the general public to follow. English is not known for consistency! So we need to recognize that in everyday usage, (1) and (2) really mean the same thing. When we accept that, though, we set ourselves up for the opposite confusion: Cases like "50% larger" and "3 times larger" no longer follow the same pattern, and our language becomes inconsistent, which really bothers mathematicians! As Dr. Rick pointed out, this means that there are gray areas where it's hard to be sure what someone means, so it may be best just to avoid using these phrases in mathematical contexts.
Finally, I commented on Joseph’s other quote, from Doctor Terrel:
Joseph, here the doctor was saying that a 550% INCREASE means adding 550% of the original value to the original value, which means 650% OF the original value. In the other terminology, "5 1/2 times greater" (there again, "greater" is taken to refer to the increase) is the same as "6 1/2 times as much." When he says that the 6 1/2 is not a percent increase, he doesn't mean that it hasn't increased, just that he is talking about multiplying by 650% rather than adding 650%. When you think in terms of increase (adding), it is a 550% increase. Now, the "percent increase" case is pretty standard, because it IS technical terminology (though ordinary people reading it can get confused, so it's still risky). The "times greater" case is more disputable, since that sounds less technical. Most people don't demand absolute consistency from language; they are happy to understand "times greater" idiomatically. I hope that clears up some of the confusion. It isn't all cleared up yet at our end. You will definitely get different opinions as to what it all REALLY means!
Increase by a factor
Joseph replied, returning to his initial question:
That really cleared things up for me and I appreciate your time in driving home the differences! The last question I would like to ask is, how do you deal with factors? If someone says something has changed by a factor of ... or is less/greater than by a factor of ..., do we use the same rules that you've discussed above? Or when using the word "factor," are things a bit different?
As Dr. Rick pointed out in the first response, "factor" is used to make it clear that multiplication, rather than addition, is the cause of an increase. Just as "increased by a factor of 2" means "twice as large," so does "greater by a factor of 2." And "decreased by a factor of 2" and "less by a factor of 2" both mean "half as much" (divided by 2). I can't think of a context in which that would not be true -- but English is flexible enough that I probably shouldn't guarantee anything!
Isn’t English fun?
In closing, here is a more recent question (2015) where I summarized this complicated issue:
As Big As vs. Bigger Than I'm having difficulty convincing my 5th graders that "as big as" and "bigger than" do not mean the same thing. For example, when asked, "How many times larger is 10,000 than than 100?" they answer "100." Their tests and homework are full of this misunderstanding! How would you suggest telling them they are wrong?
I referred to most of the pages we’ve seen above, and added:
To be honest, my feeling (basically unchanged since the first of those) is that although your understanding is common among thoughtful people, it is a case of excessive consistency. Mathematical people want a certain word structure to always have the same meaning, so we relate "x times bigger" to "x percent bigger," and that to "x bigger," and want to take all in an incremental (additive) sense. But taken on its own, it is perfectly logical to interpret "x times bigger" as "bigger, as a result of multiplication by x." And human language is not completely consistent; we have idioms all over the place that we interpret with no trouble. Having said that, I think that math books should avoid that form, because there is just enough truth to your thinking, and enough extra expectation of careful use of words in a math book, that it can be confusing. What I would do is to make a brief mention of the fact that many people take it as you do, but then point out that your book is using the phrase in the way it is usually intended in the real world. If I were writing the textbook, I would reverse this: almost always use "x times as big," but mention somewhere the fact that many people use "x times bigger" to mean the same thing, and briefly discuss the controversy before moving on to other things.