Reason for Order of Operations – Part 1

Elementary Math (K-6th) Tutorial

Reason for Order of Operations - Part 1


In this tutorial, we will use a simple mixed addition and multiplication example to illustrate the reason for order of operations

Sample Problem

I had 3 batteries at home. I bought 2 packs of batteries at a grocery store. Each pack has 4 batteries. How many batteries do I have now?


If we do not use multiplication, the mathematical sentence is

3 + 4 + 4 (A)

= 11 batteries

If we write the mathematical sentence using multiplication, we get

3 + 2 x 4 (B)

Then, we re-write multiplication to addition, we have

3 + 4 + 4 (C)

If we add the last two numbers first, noting that the order of addition does not change the answer, we have

3 + 8 (D)

and the final answer is 11 batteries.

People first created addition and subtraction from counting. Multiplication is derived from addition, or it is a simplified way to write repeated addition. By performing multiplication first, we basically convert the mathematical sentence in (B) to (C) which is the same as (A) without using multiplication. Performing multiplication first is equivalent to converting multiplication back to addition (i.e. from (B) to (C)) but usually, we skip (C) in our calculation.

We need to perform multiplication first because multiplication is a derived operation. This step is equivalent to reverting multiplication to addition. Therefore, order of operations is not simply a rule. There is a reason behind this rule.

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