Why do we need to have rules when we are multiplying, dividing, adding, and subtracting?

Understanding ARITHMETIC

To have a solid understating of the rules involved in adding subtracting and diving we will begin with an example.  \begin{align*} \{ \frac{20}{13} \cdot [ \frac{7}{5} - ( \frac{16}{11} \cdot \frac{15}{2} \cdot ( \frac{5}{4} + \frac{3}{2} )) ] \} \end{align*}

In order to get the answer for this expression we have to follow the rules of operations. That is, whatever is inside the inner most bracket is computed first, then multiplication or division, and then addition or subtraction. The reason we develop these rules is so that everyone gets the same answer when they solve for the same expression otherwise everyone will get different answers depending on what operation they do first.  For example, Does, $2 \cdot 5+3 = 10+3 = 13$, or does it equal $ 2 \cdot 5+3 = 2 \cdot 8 = 16$? The point is that we do not know unless we have a set of rules that can help us to remove that confusion. For this reason we have an order of operations. In this example the correct answer is 13 because the order of operations tell us that multiplication comes before addition. 

Lets try to solve for our long example above. The rule tells us that brackets comes first, but which one. There are three types of brackets { }, [ ], ( ), but the reason we have these is because we want to make it easy for us to see our expression. We will solve for the inner most bracket first. $\frac{5}{4}+\frac{3}{2} = \frac{(5+2*3)}{4} = \frac{5+6}{4} = \frac{11}{4}$.

The next bracket is $\frac{16}{11} \cdot \frac{15}{2} \cdot \frac{11}{4}$. [to be completed]


BEDMAS

An easy way to remember the order of operation is by using BEDMAS.

  1. B for bracket,
  2. E for exponent, 
  3. D  for division, 
  4. M for multiplication, 
  5. A for addition,
  6. and S for subtraction. 

Lets try another example:

5.17 + [3.2*(4.4/(3.3-1.1))]

= 5.17 + [3.2*(4.4/(2.2))]

= 5.17 + [3.2*(2)]

= 5.17 + [6.4]

= 11.57