The Importance of Defining Variables Clearly in Algebra

Date: 11/01/2007 at 10:40:36
From: Cathy
Subject: writing a formula for a word problem

A sample of dimes and quarters total $18.00.  If there are 111 coins
in all, how many are there of each coin?

I have x + y = $18.00 but I'm not sure where to go from there.

Date: 11/01/2007 at 12:07:48
From: Doctor Peterson
Subject: Re: writing a formula for a word problem

Hi, Cathy.

Before you can write an equation, you have to have a clear idea of 
what the variables MEAN.  Are x and y supposed to be the NUMBER of 
dimes and the NUMBER of quarters, or the VALUE of the dimes and the 
VALUE of the quarters?  The former is what the problem is asking for; 
but your equation treats them as the latter.

Let's start by defining the variables:

  Let x = number of dimes
  Let y = number of quarters

Now, if you knew the number of dimes, how would you calculate the 
VALUE of those dimes (in dollars)?  Write an expression that does 
that calculation.  Then write a similar expression for the value of 
the quarters.

Your equation will say

  value of dimes + value of quarters = 18

Use your expressions to write this equation.

Then you'll have to write another equation that says

  number of dimes + number of quarters = 111

Keeping the ideas of "number" and "value" separate is the key to 
solving this kind of problem.

If you have any further questions, feel free to write back.

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