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### Marble Puzzle

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Date: 3/10/96 at 0:32:10
From: Cristian Liu
Subject: math

Here is my question: Jason and Bob have 193 marbles altogether.
Bob has 47 marbles less than Jason.  If Jason gives Bob 15
marbles, how many more marbles does Jason have more than Bob?

Thank you.

From Cristian Liu
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```
Date: 3/10/96 at 23:6:27
From: Doctor Jodi
Subject: Re: math

Hi Cristian.  Thanks for your question.

There are two ways I can think of to solve it.

One is that if Jason gives Bob some marbles, he loses 15 and Bob
gains 15 at the same time. This means that by giving away 15
marbles, Jason makes a total difference of 30 marbles.  So, if
before Jason had 47 marbles more than Bob, now he only has 17
marbles more than Bob.

Does that make sense?

The other way to figure out this puzzle is to figure out how many
marbles Jason and Bob each have before and after the exchange.
First, let's write some number sentences.

If Jason and Bob have 193 marbles altogether, we can write that as

j + b = 193

and if Bob has 47 marbles less than Jason, we can write that as

b + 47 = j ( or j - 47 = b)

Now since j and b + 47 are equal, we can substitute one for the
other.

So in our first equation,

j + b = 193

let's substitute b + 47 for j in the first equation to find out
how many marbles Bob has.

(b + 47) + b = 193

We can rewrite this as 2b + 47 = 193

In order to find b, we need to isolate it - to get it alone on one
side of the equation.  In order to keep this as an equality, we
need to do the SAME thing to each side.

So if we want to subtract 47 from the lefthand side to find out
what 2b is equal to, we need to subtract 47 from the righthand
side, too.

2b + 47    =    193
- 47        - 47
____________________
2b         =    146

Now to get b alone, we need to divide by two, right?

2b           = 146
---	       ----
2         	2

which means that

b = 73

Does this make sense so far?

Now, we know that Bob has 73 marbles.  We also know that Jason has
47 MORE than Bob has.

73 + 47 = 120 marbles

So Bob has 73 marbles and Jason has 120.

Let's check to see if the total number of marbles still works.

Yes, 120 + 73 = 193, just what we were told at the beginning of
the problem.

But we still haven't solved the puzzle.  What we want to figure
out is what will be the difference in the number of marbles each
boy has if Jason gives Bob 15 marbles.

That means that Jason, who has 120 marbles, will give away 15,
leaving 105 marbles.  Meanwhile, Bob, who has 73 marbles, will
gain another 15, making a total of 88 marbles.

The difference between the number of marbles Jason has and the
number of marbles Bob has is again 17.

105
- 88
____
17

Does this make sense?  Write us back if you need a better
explanation of some of the steps or if you have any other
questions!

-Doctor Jodi,  The Math Forum
```

```
Date: 3/12/96 at 0:6:57
From: Eric Liu
Subject: math

On the question I sent you, I don't understand why it would be 30
Jason makes in the changes because it only said 15 marbles.

From Cristian Liu
```

```
Date: 3/16/96 at 1:29:14
From: Doctor Jodi
Subject: Re: math

Hi Cristian!

Let me try to explain this again.  Maybe it would help to try with
smaller numbers.  I can understand why this problem is confusing!
Let me write you a few questions.

Ok, say I have 3 marbles and you have 9.  You have 6 more than I
do.

If you give me 2 marbles, then how many do we each have?  How many
more do you have now?

9              vs. 3

***             ***       difference: 6
***
***

give 2

7           vs.  5
***           ***       difference: 2
***            **
*

difference in differences : 6-2 = 4

But only 2 marbles switched hands!  Any idea what is going on
here?

Write back with more questions...

-Doctor Jodi,  The Math Forum

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