What Does 'n' Mean?

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

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/