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Using Guess and Check to Start an Algebra Problem

Date: 02/15/2008 at 12:17:41
From: Shoba
Subject: two by two equations

Laura is three times as old as Maria was when Laura was as old as 
Maria is now.  In two years Laura will be twice as old as Maria was 
two years ago.  Find their present ages.

I didn't get how to write the equation for the first part.  Its 
really a very brain twisting problem.  Please help me to write the 
equations.  I will try the rest of the things.



Date: 02/17/2008 at 08:47:55
From: Doctor Ian
Subject: Re: two by two equations

Hi Shoba,

One way to get an equation or equations for a problem like yours is to
make a guess and check it, without evaluating the arithmetic.  Let me
show you what I mean by that. 

Here's the problem again:

  Laura is three times as old as Maria was when Laura was as 
  old as Maria is now. In two years Laura will be twice as 
  old as Maria was two years ago. Find their present ages.

The first thing I'll do is change things so instead of Laura and
Maria, we're talking about their AGES, since that's what we're
interested in:

  (Laura's age now) is three times (Maria's age when Laura was as 
  old as Maria is now). (Laura's age in two years) will be twice    
  (Maria's age two years ago). Find their present ages.

I'm just going to guess that Laura is 20, and Maria is 12.  Now I
start working through the problem.  First, I'll figure out what all
the ages in the problem are, given my guesses:

  Laura's age now:                20

  Maria's age now:                12

  Laura's age in two years:       20 + 2

  Maria's age when Laura was 12:  12 - (20 - 12)

  Maria's age two years ago:      12 - 2

Second, I'll check to see if everything works out:

  (Laura's age now) is three times 
  (Maria's age when Laura was as old as Maria is now)
     
    20 = 3 * (12 - (20 - 12))

  (Laura's age in two years) will be twice    
  (Maria's age two years ago)

    20 + 2 = 2 * (12 - 2)

Now, if I do the arithmetic, 

    20 = 3 * (12 - (20 - 12))

    20 = 3 * (12 - 8)

    20 = 3 * 4

    20 = 12      

and

    20 + 2 = 2 * (12 - 2)

        22 = 2 * 10

        22 = 20

I see that the equations aren't true, which means my guesses are
wrong.  But here's the good news:  Since I didn't do the arithmetic
when I first wrote the equations, I can see where my guesses are...
and I can replace them with variables to represent the correct values.
If I let L represent Laura's age now, and M represent Maria's age
now, I get

    L = 3 * (M - (L - M))

    L + 2 = 2 * (M - 2)

That's two equations, with two unknowns, so now I can use any of the
methods I've learned for dealing with equations like this.

The important point is this:  By working with guesses, I get to do
simple calculations, which lets me use my common sense to be confident
I'm doing the right thing.  That's much harder if I start with
variables.  Once I know I'm doing the right thing, I can replace my
guesses with variables, giving me equations I can solve... but ONLY if
I defer all the arithmetic.

Does this make sense? 

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



Date: 02/18/2008 at 03:15:58
From: Shoba
Subject: Thank you (two by two equations)

Hi Ian,

I am very grateful to you.  The problem really squeezed my brain and
could not analyse it.  You have analysed it so clearly that it helped
me a lot.  My heartfelt thanks to you.  You guys are doing a good
job!!!  THANKS!!!
Associated Topics:
Middle School Algebra
Middle School Word Problems

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