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### What Does 'n' Mean?

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Date: 09/25/2001 at 16:19:54
From: Cristina
Subject: Mathsuperstars

Dear Dr. Math,

I need to know what "n" means in a word problem. Example:

Think about the first number in these pairs.

Three is the first number and 8 is the second.
If 50 is the first number, what is the second?  53
If a number n is the first number, what is the second?

Can you help me?

Cristina Palmer
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Date: 09/25/2001 at 17:48:00
From: Doctor Ian
Subject: Re: Mathsuperstars

Hi Cristina,

In a word problem, 'n' is the name of a number whose value we don't
know yet.

So let's say that I have some pairs of numbers:

(1,2)
(2,4)
(3,6)
(5,10)

You might notice that in each pair, the second number is twice as
large as the first. So if I tell you that I'm thinking of a new pair,
but I'm not going to tell you what the first number is, you can write

(1,2)          2 is 2*1
(2,4)          4 is 2*2
(3,6)          6 is 2*3
(5,10)        10 is 2*5
(n, 2 * n)    The second number is 2 times the first number.

which is a way of writing down a way to _find_ the second number, as
soon as you know the first one.

Here is another set of pairs:

(1,3)
(2,4)
(3,5)
(n, ?)

Can you find a pattern that tells you how to get from the first number
in each pair to the second number?  If so, then you can write the
pattern using 'n' instead of a particular number. For example, if you
think that to get the second number, you add 4, you would write

(n, n+4)

Of course, that's not correct. If you think that to get the second
number, you add 1 and then divide by 2, you would write

(n, (n+1)/2)

That's not correct either. But it shows how you go about using 'n' to
represent numbers that you haven't yet chosen.

This might make more sense if you connect it with a more realistic
situation. Suppose I want to have some people over for dinner, and I
need to know how much food to buy. I know that everyone will have two
pieces of chicken, and I always like to have three pieces left for the
dogs. So let me ask you: How much chicken should I buy?

You can't tell me until you know how many people are coming. The most
you can do is say something like

when you know how many people will be coming, take that
number and multiply by two; then add three. The result will
be the number of pieces of chicken to buy.

This is all we're doing when we write something like

pieces = 2 * n + 3

except it's a lot easier to write 'n' instead of 'the number of people
coming to dinner'.

Does this help?

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
Middle School Algebra
Middle School Definitions