Associated Topics || Dr. Math Home || Search Dr. Math

### 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

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.
http://mathforum.org/dr.math/