Percent Increase and "Increase by a Factor of ..."

Date: 08/01/2006 at 17:53:57
From: joseph
Subject: diff between factor of 2 increase vs 100 percent increase

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.

Date: 08/01/2006 at 21:12:56
From: Doctor Rick
Subject: Re: diff between factor of 2 increase vs 100 percent increase

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.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 08/02/2006 at 13:41:31
From: joseph
Subject: diff between factor of 2 increase vs 100 percent increase

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 dont see where my thinking is 
wrong.

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!

Thanks for your help so far.

Date: 08/02/2006 at 16:14:18
From: Doctor Greenie
Subject: Re: diff between factor of 2 increase vs 100 percent increase

Hello, Joseph --

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.

About 20 years ago, I started to notice that the old phrase "I 
couldn't care less" was giving way to "I could care less."  Thanks to 
intonation and context, we know that the new idiomatic phrase "I 
could care less" is supposed to mean the same as the old phrase; but 
if we interpret it literally, we see that it means exactly the 
opposite.  In mathematics, we can't use language like that, since it 
would leave our readers to guess what we really mean.

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.

- Doctor Greenie, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 08/03/2006 at 11:09:57
From: Doctor Peterson
Subject: Re: diff between factor of 2 increase vs 100 percent increase

Hi, Joseph.

As the author of one of the pages you're asking about, I want to make 
sure we've answered your question!

I'll make some comments below:

As joseph wrote to Dr. Math on 08/02/2006 at 13:41:31 (Eastern Time),
>Here is the first site and 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 dont see where my
>thinking is wrong.

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."

Again, I'm not convinced that this is really the only logically 
correct conclusion, and as I quoted in ...

  Percent Greater Than vs. Increased
    http://mathforum.org/library/drmath/view/61774.html 

... 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.

>Here is the second site and 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? 

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!


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 08/03/2006 at 13:28:27
From: joseph
Subject: Thank you (diff between factor of 2 increase vs 100 percent 
increase)

Thanks for all the help.  The more I get help from you guys, the more 
I'm realizing the issue is language!!  I think I have a good grasp on 
the ambiguities now.  Thanks again.

Date: 08/03/2006 at 13:39:44
From: joseph
Subject: diff between factor of 2 increase vs 100 percent increase

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? 

Thanks,
Joseph.

Date: 08/03/2006 at 15:47:45
From: Doctor Peterson
Subject: Re: diff between factor of 2 increase vs 100 percent increase

Hi, Joseph.

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!


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 
  

Date: 08/04/2006 at 00:13:37
From: joseph
Subject: Thank you (diff between factor of 2 increase vs 100 percent 
increase)

It seems english is a bit TOO flexible in these cases!  Thanks for 
all your help.