If a and b are the numbers, a+b=218 and ab is maximum.
We can write b=218-a, so the product is a(218-a)=218a-a².
This can be written -(a²-218a)=-(a²-218a+109²)+109²=109²-(a-109)².
The expression is maximum 109² when (a-109)² has its minimum value of zero, that is, when a=109. (As a moves away from 109, 109² is reduced by the square of the difference between the value of a and 109, and the square is always positive.)
The maximum product is a=b=109, which can be written as an improper fraction 218/2, but reduced to its lowest terms is 109/1 or the integer 109.