Find a 4-digit number so that if you put a decimal point in the middle the number you get is the average of the two numbers on either side??
Since the final number with a decimal point is the average of the two 2-digit numbers, the one on the right must be either 50 or 00.
If it is 50, then the one on the left must be 49.
If it is 00, the number on the left must be 00 also. Since 0000 is not considered a 4 digit number, this case should be discarded.
This can also be solved by algebra. Left the number on the left be a, the one on the right be b, then
a+b/100 = (a+b)/2
Solve to get a=49b/50, or 50a=49b, which comes to the same conclusion as above.