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### Backward Through the Steps

```
Date: 5/20/96 at 21:1:40
From: Anonymous
Subject: Solving strategies

How do you do equivalent fractions, and mixed numbers - such as
solving strategies? Here is one of the problems:

Fred gave one half of his baseball cards to Sally. Sally gave Jeff
half of the cards that she got from fred. Jeff gave Allen half of the
cards that he got from Sally. Allen got 6 cards. How many cards did
Fred originally have?

Thank you for helping!

Joshua Chen
```

```
Date: 7/13/96 at 14:38:32
From: Doctor Jodi
Subject: Re: Solving strategies

Hi there! There are quite a few ways to answer a question like this
one, but in this case, one of the easiest ways is to go backward
through the steps:

Allen got 6 cards, which is half of Jeff's from Sally, or a quarter of
Sally's from Fred, or an eighth of Fred's cards.

If 6 cards is 1/8 of Fred's total, Fred had 8 * 6 cards, or 48.

A similar technique is to look at the ratios of each person's cards.

2/1  Fred       2/1   Sally        2/1   Jeff
to               to                 to
Sally            Jeff               Allen

Since we're looking for the ratio of Fred's cards to Allen, we find:

For every 2 cards that Fred has, Sally has 1.  For every two cards
that Sally has, Jeff has 1.  To compare Fred's cards to Jeff's, we can
rewrite the ratios:

4 Fred      2 Sally
------   *  -------
2 Sally     1 Jeff

Does that make sense?

That says that for every 4 cards Fred has, Jeff has one.  We can check
this by saying:  If Fred gave Sally 2 cards (1/2 of 4), then Sally
gave Jeff 1 card (1/2 of 2).  So that works.  But to find out how many
cards Fred has we need the ratio of Fred's cards to Allen's.

We start off with the ratio from above,

4 Fred
------
1 Jeff

Now, we know that Jeff gives half of his cards to Allen.  Again, we'll
rewrite the ratio, this time to:

8 Fred
------
2 Jeff

and multiply by the ratio of Jeff's cards to Allen's.

8 Fred        2 Jeff
------   *   -------
2 Jeff       1 Allen

So, for every 1 card that Allen has, Fred had 8.  But Allen has 6
cards, so Fred had 6 * 8 or 48 cards.

I hope this makes sense. Let us know if you need more help.

-Doctor Jodi,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
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