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Solution to Puzzle 5:

The following solution is sent by Umesh

Let I got x dollars and y cents from the bank. I gave a check for y dollars and x cents. So the equation is
100x + y - 5 = 2 * (100y + x), or 98x - 199y = 5 ... (A)
First, let us solve the equation
98x - 199y = 1 ... (B)
Using the method described in the party problem in this mathematics corner, let us express (98/199) as a continued fraction with even number of terms including the integer part.
So, (98/199) = 0+(1/(2+1/(32+1/(1+1/(1+1/1))))) or [0;2,32,1,1,1]
Discarding the last term, [0;2,32,1,1] = 0+1/(2+1/(32+1/(1+1/1))) = (65/132).
So (132, 65) is the smallest solution in positive integers to (B).
So, (132*5, 65*5) = (660,325) is a solution to (A).
Now (660+199k, 325+98k), where k is an integer, is the general solution to (B). Putting k = -3, we get the solution as per our requirements x = 63, y = 31.

So, I gave a check for $31.63, I got $63.31, I gave 5c to charity, so the remaining $63.26 is the double of $31.63.

Questions ? Ask me Umesh

Correct Answer was sent by :

Jigar Shah, CA, USA

Correct Answer was given by :
Jigar Shah, CA, USA

Solution to Puzzle 6:

The following solution is sent by Jigar Shah

First Man and goat goes in the boat. Man leaves the goat there, and comes back in the boat. Then the man takes the fodder and leaves it there, and picks up the goat. The man and the goat comes back, then the man takes the lion and leaves the goat back. Man goes across and drops the lion where the fodder is. NOW for the final time, the man comes back to pick up the goat. This will do it.