How Much Money is Left?

Date: 1 May 1995 01:13:19 -0400
From: Anonymous
Subject: Math problem

This problem was given to a sixth grader. I tried to work it using what I
remembered of high school algebra but I couldn't arrive at an answer that
worked. Any help in solving would be appreciated. Thanks in advance.

Work Backwords - Lesson A
A lady has a certain number of dollar bills in her purse. She has no other
money. She spends half the money on a hat and gives $1 to a beggar outside
the store. She spends half the remaining dollars for lunch and tips the
waiter $2. She then spends half the remaining dollars for a book, and just
before she goes home she spends $3 on a hot fudge sundae. She now has $1
left. How many dollars did she begin with?

Thanks again.

Bob Marvin
RMARVIN@aol.com

Date: 1 May 1995 10:30:56 -0400
From: Daniel Eisenbud
Subject: Re: Math problem

Before the woman spends three dollars on her sundae, she has four left
(since she has one left afterwards.)  This is half of what she had
before she bought the book, so she had eight dollars then.  Before she
spent the two dollars for a tip, she had eight dollars.  You can
probably see the pattern from this; if you need more help finishing
the problem, please write back.

-Dan "Dr. Math" Eisenbud

Date: 1 May 1995 11:57:50 -0400
From: Stephen Weimar
Subject: Re: Math problem

Hi Bob,

We'd be interested to see what you tried.  Please tell us if you're willing.

I see that Dan has offered one response based on the strategy in the
problem's title where it says "Work Backwords."  Did you receive his
answer?

Now, as for doing this with algebra, I find that it helps to go one step at
a time.  Perhaps the most important part is deciding what to focus on.
Since we are told at the end what she has left, I'm going to focus all the
way through on what she has left.

1. She starts with some amount we'll call x.
2. She spends half so she has 1/2 of x left (I'll write that here as x/2).
3. She gives away a dollar so she has x/2 - $1 left.
4. Spends half so she has x/4 - $.50 left.
5. Makes a $2 tip so she has x/4 - $2.50 left.
6. Spends half so she has x/8 - $1.25 left.
7. Spends $3 on sundae so she has x/8 - $4.25 left.
8. She has $1 left.

Can you solve it from here?

Step 4 or 6 is probably the place where most errors will be made, not
taking half of everything.  In step three $1 was taken away from the half.
There are two fourths in that half so each fourth is missing $.50. A
typical algebra error is to not treat x/2 -1 as a unit so that everything
you do to one part of it must be done to the rest.  This is why we use
parentheses: 1/2[x/2 -1] means take half of x/2 and subtract half of 1.

To make this concrete, imagine a batch of brownies not yet cut into chunks.
Cut it in half and give half the batch to me. Thank you. Still without
cutting it up into little chunks, cut out of your half one brownie for you
to eat now.  After you finish eating the brownie the sixth grader walks in,
sees the half of the batch that you have, and asks for half of it. Where do
you make the cut?  How much of the missing brownie is missing from each
quarter?  We are all watching to see if you do math the way you share
brownies.  : )

-- steve