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From Representing Two Bags of Marbles, a Way to Solve All Kinds of Problems

Date: 04/03/2011 at 22:11:39
From: Sarah
Subject: Trying to find original values of algebraic expressions

David had 2 bags of marbles. After David sold 44 marbles from Bag X to his
friend, the number of marbles in Bag Y was 5/7 of the number of marbles in
Bag X.

Given that there were 2/5 as many marbles in Bag Y as in Bag X originally,
find the number of marbles in Bag X at first.

It is confusing to understand the parts of the equation. I get as far as
representing the first step as x - 44, and the second step as y = (5/7)x,
but I find it REALLY difficult to set up one expression and solve it. 



Date: 04/05/2011 at 10:30:26
From: Doctor Ian
Subject: Re: Trying to find original values of algebraic expressions

Hi Sarah,

It can often be helpful to make small changes, rather than trying to write
down equations in a single step. 

Here's the problem again:

   David had 2 bags of marbles. 
   After David sold 44 marbles from Bag X to his friend, 
   the number of marbles in Bag Y was 5/7 of the number 
   of marbles in Bag X. 

   Given that there were 2/5 as many marbles in Bag Y as in 
   Bag X originally, find the number of marbles in Bag X at 
   first. 

We're asked to find the original number of marbles in bag X. So we might
start by giving that quantity a name, like 'x.' And because we'll want to
talk about the original number of marbles in bag Y, we can give that a
name as well: 'y.'

Now we can try rewriting the problem using that notation:

   David had 2 bags of marbles.

   Bag X had x marbles originally.

   Bag Y had y marbles originally. 

   After David sold 44 marbles from Bag X to his friend, 
   the number of marbles in Bag Y was 5/7 of the number 
   of marbles in Bag X. 

   Given that there were 2/5 as many marbles in Bag Y as in 
   Bag X originally, find x.

Let's look at that next-to-last paragraph. David is removing 44 marbles
from bag X. That means the number of marbles will be reduced by 44, right?
So the number of marbles in bag x will be (x - 44). Now we have

   David had 2 bags of marbles. 

   Bag X had x marbles originally.

   Bag Y had y marbles originally. 

   After David sold some marbles, y was 5/7 of (x - 44).

   Given that there were 2/5 as many marbles in Bag Y as in 
   Bag X originally, find x.

And that looks like an equation:

   David had 2 bags of marbles. 

   Bag X had x marbles originally.

   Bag Y had y marbles originally. 

   After David sold some marbles, y = (5/7)(x - 44).

   Given that there were 2/5 as many marbles in Bag Y as in 
   Bag X originally, find x.

And that last paragraph describes an original relationship between x and
y, doesn't it? But it's kind of confusing, so I'd try some concrete
numbers to make sure I'm interpreting it correctly.

For example, what if there are 5 marbles in bag X? In symbols,

   if   x = 5,
   then y = ?

Now it's clear that there would have to be 2 marbles in bag Y, since 2 is
2/5 of 5. So the relationship between x and y is

   y = (2/5)x

I can put that back in the problem:

   David had 2 bags of marbles. 

   Bag X had x marbles originally.

   Bag Y had y marbles originally. 

   After David sold some marbles, y = (5/7)(x - 44).

   Originally, y = (2/5)x.

   What is x? 

So now I have two equations, in two variables, and the details of the
problem are irrelevant for purposes of solving the system. That is, I just
have

   y = (5/7)(x - 44)

   y = (2/5)x

Does this make sense?

You'll probably never have to solve a problem like this in real life, but
there's a valuable lesson to be learned from working through problems like
this in such a small-step-by-small-step way, because that method applies
to all kinds of problems that you WILL encounter. It's not so much a 'math
technique' as a way of thinking about problems in general.

What I did here was make small changes to what I had, to get something I
preferred, or at least understood better. In spirit, it's the same sort of
thing you do when you solve an equation like

   12x + 9 = 4x + -1

I'd like that equation better if I had all the x terms on one side, so I can make a 
small change:

   -4x + 12x + 9 = -4x + 4x + -1

I can look at those two equations, and see that they mean the same thing. So when I 
simplify the new equation, to get ...

   8x + 9 = -1

... I know that it has the same solution set as the one I started with.
And I just keep doing that, until I get something that is so simple that I
can deal with it easily.

What I did before was the same sort of thing, but using words and
sentences instead of equations.

If there is a secret to solving problems, I think this is it: When you're
faced with a problem, instead of looking directly for a solution, look for
a way to turn it into an easier problem.

Does this help? 

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



Date: 04/06/2011 at 08:57:56
From: Sarah
Subject: Thank you (Trying to find original values of algebraic expressions)

Thanks!
Associated Topics:
Middle School Algebra
Middle School Word Problems

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