Mastering the order of operations becomes significantly easier once you learn to recognize the common traps that lead to incorrect answers. Avoiding these frequent mistakes will instantly improve your accuracy in algebra and arithmetic.
1. Calculating Everything Strictly From Left to Right
One of the most widespread errors is ignoring operational precedence and solving expressions purely from left to right.
Consider the expression: 7 + 3 × 4
- Incorrect: Adding 7 + 3 = 10 first, then multiplying 10 × 4 = 40.
- Correct: Multiplication holds higher priority. Calculate 3 × 4 = 12 first, then add 7 + 12 = 19.
2. Thinking Multiplication Always Comes Before Division
A frequent misconception with the PEMDAS rule is assuming multiplication must strictly precede division.
As detailed in our guide on the left-to-right rule, multiplication and division hold equal priority. You must evaluate whichever operation appears first on the left.
Consider: 30 ÷ 5 × 2
- Work from left: 30 ÷ 5 = 6
- Then multiply: 6 × 2 = 12
3. Ignoring or Misinterpreting Parentheses
Grouping symbols explicitly define operations that must be evaluated first. As covered in our article on solving math expressions with parentheses, always resolve everything inside brackets before attempting outside operations.
For example: (6 + 4) × 3
Do not multiply 4 × 3 first. Resolve the parentheses (6 + 4 = 10), then multiply 10 × 3 = 30.
4. Forgetting the Left-to-Right Rule for Addition and Subtraction
Just like multiplication and division, addition and subtraction share equal priority rank.
Consider: 20 − 7 + 3
- Work from left: 20 − 7 = 13
- Then add: 13 + 3 = 16
Performing 7 + 3 first to get 20 − 10 = 10 is a classic calculation trap.
5. Attempting to Solve Long Expressions Mentally
Trying to perform multiple steps in your head increases cognitive load and causes avoidable errors. Always write down intermediate results line by line.
A Reliable Step-by-Step Method
- Identify and resolve all grouping symbols.
- Evaluate exponents and roots.
- Perform multiplication and division from left to right.
- Perform addition and subtraction from left to right.
- Review each line to verify your arithmetic.