A women walks into a bank to cash out her check.
By mistake the bank teller gives her dollar amount in change, and her cent amount in dollars.
On the way home she spends 5 cents, and then suddenly she notices that she has twice the amount of her check.
How much was her check amount?

The check was for dollars 31.63.
The bank teller gave her dollars 63.31
She spent .05, and then she had dollars 63.26, which is twice the check.
Let x be the dolars of the check, and y be the cent.
The check was for 100x + y cent
He was given 100y + x cent
Also
100y + x - 5 = 2(100x + y)
Expanding this out and rearranging, we find:
98y = 199x + 5
Which doesn't look like enough information to solve the problem except that x and y must be whole numbers, so:
199x ≡ -5 (mod 98)
98*2*x + 3x ≡ -5 (mod 98)
3x ≡ -5 ≡ 93 (mod 98)
This quickly leads to x = 31 and then y = 63
Alternative solution by substitution:
98y = 199x + 5
y = (199x + 5)/98 = 2x + (3x + 5)/98
Since x and y are whole numbers, so must be (3x + 5)/98.
Call it z = (3x+5)/98
so 98z = 3x + 5, or 3x = 98z - 5 or x = (98z - 5)/3
or x = 32z-1 + (2z-2)/3.
Since everything is a whole number, so must be (2z-2)/3.
Call it w = (2z-2)/3, so 3w = 2z-2 so z = (3w+2)/2 or
z = w + 1 + w/2. So w/2 must be whole, or w must be even.
So try w = 2. Then z = 4. Then x = 129. Then y = 262.
if you decrease y by 199 and x by 98, the answer is the same:
y = 63 and x = 31.

Sally and her younger brother were fighting. Their mother was tired of the fighting, and decided to punish them by making them stand on the same piece of newspaper in such a way that they couldn't touch each other. How did she accomplish this?

Sally's mother slid a newspaper under a door and made Sally stand on one side of the door and her brother on the other.

When Manish was three years old he carved a nail into his favorite tree to mark his height. Six years later at age nine, Manish returned to see how much higher the nail was. If the tree grew by five centimeters each year, how much higher would the nail be.

The nail would be at the same height since trees grow at their tops.

A man is sitting in a pub feeling rather poor. He sees the man next to him pull a wad of £50 notes out of his wallet.
He turns to the rich man and says to him, 'I have an amazing talent; I know almost every song that has ever existed.'
The rich man laughs.
The poor man says, 'I am willing to bet you all the money you have in your wallet that I can sing a genuine song with a lady's name of your choice in it.'
The rich man laughs again and says, 'OK, how about my daughter's name, Joanna Armstrong-Miller?'
The rich man goes home poor. The poor man goes home rich.
What song did he sing?

Even though the odds are always in favor of the gambling house, why does the establishment insist on a house limit on stakes?

Every casino in the world would go bankrupt without a house limit on stakes. Without it, gamblers would keep doubling their stakes until they won. No matter how bad a losing streak they were on, they would eventually win. For more information, search: Martingale