The Missing Million

Date: 06/11/2001 at 11:50:24
From: Aden Ali
Subject: Addition and subtraction

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 
this answer and my children keep on asking me. Please help.

Thanks.

Date: 06/11/2001 at 13:14:43
From: Doctor Peterson
Subject: Re: Addition and subtraction

Hi, Aden.

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/   

Date: 06/12/2001 at 02:45:18
From: aden m. ali
Subject: Re: Addition and subtraction

Dr Math,

I thank you for your kind answer to my question.  The water made it 
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.  
Aden

Date: 06/12/2001 at 08:37:06
From: Doctor Peterson
Subject: Re: Addition and subtraction

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/   

Date: 06/13/2001 at 07:08:14
From: aden m. ali
Subject: Re: Addition and subtraction

Dr. Math,

I am very grateful for your quick answer and your kind attention.  
Thank you. 

Aden