This Maths Problem Needs Dixie

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cancel2 2022

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64-equals-65.gif
 
Three men walk into a hotel and ask for a room. The hotel clerk tells them it will be $30 for the night. Each men reach into their wallet and pull out a $10 bill. They paid $10 each for the room, totaling $30. But later, the clerk realizes he made a mistake, the bill was supposed to only be $25. He gives the bellhop $5 and tells him to return it to the men. On the way up to their room, the bellhop begins to contemplate how he can divide $5 between 3 people, being he is a Liberal, he decided the best thing to do would be, give each man $1 each ($3) and he would keep the other $2. So that is what he did. Now, this means, since each man originally paid $10 and each man got $1 back, they paid $9 each for the room. But... $9x3 =$27 Plus the $2 the bellhop kept, makes $29... where is the other dollar???
 
Dixie said:
Three men walk into a hotel and ask for a room. The hotel clerk tells them it will be $30 for the night. Each men reach into their wallet and pull out a $10 bill. They paid $10 each for the room, totaling $30. But later, the clerk realizes he made a mistake, the bill was supposed to only be $25. He gives the bellhop $5 and tells him to return it to the men. On the way up to their room, the bellhop begins to contemplate how he can divide $5 between 3 people, being he is a Liberal, he decided the best thing to do would be, give each man $1 each ($3) and he would keep the other $2. So that is what he did. Now, this means, since each man originally paid $10 and each man got $1 back, they paid $9 each for the room. But... $9x3 =$27 Plus the $2 the bellhop kept, makes $29... where is the other dollar???
Click to expand...
What dollar? The room wasn't $30
 
Dixie said:
Three men walk into a hotel and ask for a room. The hotel clerk tells them it will be $30 for the night. Each men reach into their wallet and pull out a $10 bill. They paid $10 each for the room, totaling $30. But later, the clerk realizes he made a mistake, the bill was supposed to only be $25. He gives the bellhop $5 and tells him to return it to the men. On the way up to their room, the bellhop begins to contemplate how he can divide $5 between 3 people, being he is a Liberal, he decided the best thing to do would be, give each man $1 each ($3) and he would keep the other $2. So that is what he did. Now, this means, since each man originally paid $10 and each man got $1 back, they paid $9 each for the room. But... $9x3 =$27 Plus the $2 the bellhop kept, makes $29... where is the other dollar???
Click to expand...

This is a hoary old chestnut.

[h=2]Solution[/h] The initial payment of $30 is accounted for as the clerk takes $25, the bellhop takes $2, and the guests get a $3 refund. It adds up. After the refund has been applied, we only have to account for a payment of $27. Again, the clerk keeps $25 and the bellhop gets $2. This also adds up.
There is no reason to add the $2 and $27 – the $2 is contained within the $27 already. Thus the addition is meaningless. Instead the $2 should be subtracted from the $27 to get the revised bill of $25.
This becomes clearer when the initial and net payments are written as simple equations. The first equation shows what happened to the initial payment of $30:
$30 (initial payment) = $25 (to clerk) + $2 (to bellhop) + $3 (refund) The second equation shows the net payment after the refund is applied (subtracted from both sides):
$27 (net payment) = $25 (to clerk) + $2 (to bellhop) Both equations make sense, with equal totals on either side of the equal sign. The correct way to get the bellhop's $2 and the guests $27 on the same side of the equal sign ("The bellhop has $2, and the guests paid $27, how does that add up?") is to subtract, not add:
$27 (final payment) - $2 (to bellhop) = $25 (to clerk) This is clearly not a paradox, and involves only the switching of subtraction for addition. Each patron has paid $9 for a total of $27. The storyteller adds the $2 that the bellhop pilfered, but he should have subtracted the $2 to make a total of $25 paid. So 3 X $9 = $27, which accounts for the $25 room and the $2 theft.

http://en.wikipedia.org/wiki/Missing_dollar_riddle
 
Aoxomoxoa said:
64-equals-65.gif
Click to expand...

Solution:

The 64 = 65 paradox arises from the fact that the edges of the four pieces, which lie along the diagonal of the formed rectangle, do not coincide exactly in direction. This diagonal is not a straight segment line but a small lozenge (diamond-shaped figure), whose acute angle is

arctan 2/3 - arctan 3/8 = arctan 1/46

which is less than 1 degree 15' . Only a very precise drawing can enable us to distinguish such a small angle. Using analytic geometry or trigonometry, we can easily prove that the area of the "hidden" lozenge is equal to that of a small square of the chessboard.


...SEE? Isn't "1/3" Great? Thank you "1/3!"
 
Dixie said:
Solution:

The 64 = 65 paradox arises from the fact that the edges of the four pieces, which lie along the diagonal of the formed rectangle, do not coincide exactly in direction. This diagonal is not a straight segment line but a small lozenge (diamond-shaped figure), whose acute angle is

arctan 2/3 - arctan 3/8 = arctan 1/46

which is less than 1 degree 15' . Only a very precise drawing can enable us to distinguish such a small angle. Using analytic geometry or trigonometry, we can easily prove that the area of the "hidden" lozenge is equal to that of a small square of the chessboard.


...SEE? Isn't "1/3" Great? Thank you "1/3!"
Click to expand...

After all the grief you've taken over 1/3, I'm glad you can still laugh about it at times.
 
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