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### Times Greater Than, Times As Much As

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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?
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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
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/
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Associated Topics:
Elementary Multiplication
High School Basic Algebra
Middle School Algebra

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