How can you have different answers to the same expressions with or without the application of the PEMDAS rule?
PEMDAS is a guide to attempt to help reduce ambiguity, but it can result in unintentional interpretations.
If the order of operations is not clearly indicated then without the guidance of PEMDAS they are ambiguous.
Consider a popular example:
If we follow the rules of PEMDAS strictly without embellishment then this expression is the same as:
According to PEMDAS we evaluate the parentheses first, then the division and multiplication from left to right:
#9/3xx(1+2) = 9/3xx3 = 3xx3 = 9#
Note however that historically the
#9-:3(1+2) = 9/(3(1+2)) = 9/(3xx3) = 9/9 = 1#
The purpose of rules like PEMDAS is to try to reduce the ambiguity, but if you want your expressions to be unambiguous it is often helpful to add some parentheses.