Larger Than and As Large As

Date: 08/01/99 at 11:56:33
From: Tracy M.
Subject: "Larger than" and "As large as"

Hi, Dr. Math:

I have another question to ask you, but this question is about English 
as related to math. We had a word problem that stated:

The school band has 6 members in the percussion section. The brass 
section is three times larger than the percussion section, and the 
rest of the band members make up the woodwind section. The band has 39 
members altogether. How many members make up the woodwind section?

My answer to the question was:

     39 - 6 - (6 + 6 * 3) = 9

but it was marked wrong by the TA. He said that it should be:

     39 - 6 - 6 * 3 = 15

The point of contention is the difference between the English 
expressions "three times larger than" and "three times as large as."
I think these two expressions are different. In this word problem, the 
brass section should have 6 + 6 * 3 = 24 members, since the brass 
section is "three times larger than" the percussion section. The TA's 
39 - 6 - 6 * 3 = 15 means that the brass section is "three times as 
large as" the percussion section.

I started to get confused because a teacher in the general office who 
manages the TAs also said the TA was right. I asked some other people, 
they also said the TA was right.

Dr. Math, could you please tell me if these two English expressions 
are different? If they are not, how do you say it in English the way I 
understood the question, i.e., the brass section has (6 + 6 * 3) 
members? I could not find an expression for it by myself if "three 
times larger than" also means (6 * 3) in this word problem.

Thank you.
Tracy

Date: 08/02/99 at 17:11:30
From: Doctor Peterson
Subject: Re: "Larger than" and "As large as"

Hi, Tracy.

Technically, you are right, though I'm ambivalent about this 
particular phrase. We have two different phrases:

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.

There are similar phrases where I would definitely say the equivalent 
interpretation is correct:

3) "N increased by 300%" means "4 * N." (Not the new amount, but the 
increase, is 300% of the original value.)

4) "300% greater than N" means "4 * N." (Note that "50% greater than" 
clearly means "150% of" and is not the same as "50% of.")

5) "3 times as much again as N" means "4 * N." (Likewise, "half as 
much again" means to add half to what you already have.)

These could be called "incremental multiplication," where the 
multiplication gives not the final number but the amount to be added 
to the original number.

The reason I'm ambivalent about case (2) is that I can't picture using 
"times" in this way in an incremental sense; I wouldn't say "1/2 times 
greater." It's just not a natural way to say what you're taking it to 
say, so I naturally tend to assume the speaker really meant to say 
"3 times as large."

This is why we restrict our use of words in math more than in everyday 
English (or any other language), in order to avoid ambiguity, so 
mathematicians are sometimes seen as being too picky about words. 
These phrases all really belong to the everyday world, and before we 
can really do math on them they have to be translated into more proper 
and careful phrases like "three times N." And I would try to avoid 
saying "three times larger," because it is definitely on the edge, and 
might be taken either way.

Out of curiosity I did a quick Internet search for the phrases "times 
more" and "times greater," and found things like "100 times more 
efficient," "10 times more detail," and "60 times more objects" 
describing telescopes, "100 times faster than" describing a 
super-computer, and "10 times greater" in an explanation of the 
Richter scale - the latter clearly is meant to mean "10 times as 
much," and I think all are intended that way. So although the phrase 
is correctly interpreted your way, it's certainly so common to misuse 
it that I would always ask "do you mean 10 times as much?" before 
doing the math.

I also searched the Dr. Math archives to see how my colleagues use the 
phrase, and found one case where a question asks "how many times more 
likely is it," and the Doctor quietly rephrases it in his answer, 
saying "58 times as likely." But in the following answer, the phrase 
"five and a half times greater" is explicitly taken as equivalent to 
"6 and a half times what it was," agreeing with you (with a warning 
that "many people don't quite grasp those phrases"):

  http://mathforum.org/dr.math/problems/zone7.23.97.html   

So here's my answer: "N times more than X" technically should mean 
(N+1)X, but is so commonly used to mean NX that it would be dangerous 
to follow the former interpretation without asking questions. I 
haven't yet found a dictionary or other authoritative source to 
support one view or the other (or both, most likely).

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