Times Greater Than, Times As Much As

Date: 05/02/99 at 01:39:30
From: Kenny Ng
Subject: As much as & Greater than

I would like to ask you the following questions.

1) A number is five times greater than x. Will this number be 6x 
   or 5x?

2) A number is five times as much as x. Will this number be 5x or 6x?

I myself think that the answer to question 1 should be 6x, and the 
answer to question 2 should be 5x. But my maths teacher disagrees. 
Can you help me?

Date: 05/03/99 at 17:06:19
From: Doctor Peterson
Subject: Re: As much as & Greater than

Hi, Kenny. Interesting question!

I would say that both are 5x, though I see what you are saying, and 
there are similar phrases where I would definitely say 6x is correct:

3) increased by 500% (not the amount, but the increase, is 500% of the 
   original value.)

4) 500% greater than x (Note that "50% greater than" clearly means 
   "150% of".)

5) 5 times as much again (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 take (1) as only 5x is that I can't picture using "times" 
in this way in an incremental sense; I wouldn't say "1/2 times 
greater."

This is why we restrict our use of words in math more than in everyday 
English (or any other language), 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 "five times x." And I would try to avoid saying "five 
times greater," because it is definitely on the edge, and might be 
taken either way.

Out of curiosity I just did a quick search for the phrases "times 
more" and "times greater," and found things like "2.1 times more 
effective 
than" in an ad; "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 if 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, 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 
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'm 
going to keep looking for 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/