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### The Missing Million

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Date: 06/11/2001 at 11:50:24

If I have six million dollars and my friend has three million dollars,
I am wealthier than my friend by three million dollars. If I give my
friend one million dollars, the latter's fortune jumps up to four
million and mine comes down to five million. This means that I am now
wealthier than my friend by only one million.

Since I was wealthier than my friend by three million dollars before
the transaction, shouldn't I be still wealthier than him by two
million after giving him only one million dollars? What happened to
the million dollars that is missing from me? Many people told me that
my one million should be counted twice, i.e. once as a credit for my
friend, and secondly as a debit to my account. I am not convinced by

Thanks.
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Date: 06/11/2001 at 13:14:43
From: Doctor Peterson

Your friends are right. Try picturing it in terms of water rather than
money: I have 6 ounces of water in a glass, and by son has 3 ounces. I
pour 1 ounce from my glass to his. Now I have 5 and he has 4; the
difference has been reduced from 3 to 1, because _I_ have one less and
_he_ has one more. The difference has been reduced on BOTH ENDS, so it
decreased by twice the amount I poured.

|<-------3------->|
him               me
+-----+-----+-----+-----+-----+-----+-----+
+---->|     |<----+
him    me
|<-1->|

Does that help?

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
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Date: 06/12/2001 at 02:45:18

Dr Math,

simpler, although an unanswered question still looms in my son's mind,
i.e. "why should you count his gain to me?"

Again, many thanks for your help.
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Date: 06/12/2001 at 08:37:06
From: Doctor Peterson

Hi again!

It sounds as if he is forgetting the question, which is about the
DIFFERENCE between two people's wealth (or water glasses), not the
INDIVIDUAL amounts. The difference is affected by BOTH amounts; it is
increased by "my" wealth, and decreased by "his" wealth. If my wealth
decreases and his increases, then both changes decrease the
difference, regardless of whether his gain comes from me or from
elsewhere. We are not counting his gain toward me, only toward the
difference between us.

As my picture illustrated, we can model this not only in terms of
money or water, but of position. If I move a centimeter to the left
and he moves a centimeter to the right, then the distance between us
decreases by two centimeters. Here, I can't say I'm giving him a
centimeter, but the effect is the same.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
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Date: 06/13/2001 at 07:08:14

Dr. Math,

Thank you.

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