Without a standard set of rules, the same math expression could produce different answers depending on which operation you tackle first. The order of operations eliminates that ambiguity so everyone arrives at the same result every time.
The order of operations is a set of rules that tells you which calculations to perform first when an expression contains more than one operation. Without these rules, the expression 3 + 4 × 2 could be read as either 14 or 11 depending on whether you add or multiply first.
In the United States, the convention is remembered by the acronym PEMDAS. In the UK and other countries you may see BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction) or BIDMAS — they all describe the same rules, just with different names.
Each letter in PEMDAS represents a step. Work through them in order from top to bottom:
A common misconception is that multiplication always comes before division, or that addition always comes before subtraction. In reality, M and D are evaluated together from left to right, and A and S are evaluated together from left to right.
Example 1: 3 + 4 × 2
Step 1 — Multiply first: 4 × 2 = 8
Step 2 — Then add: 3 + 8 = 11
Answer: 11 (not 14). A common mistake is to add 3 + 4 first, which ignores the rule that multiplication is evaluated before addition.
Example 2: (5 + 3) × 2
Step 1 — Parentheses first: 5 + 3 = 8
Step 2 — Then multiply: 8 × 2 = 16
Answer: 16. The parentheses override the normal priority and force the addition to happen before the multiplication.
Example 3: 12 ÷ 4 + 3² × 2
Step 1 — Exponents: 3² = 9
Step 2 — Division (left to right): 12 ÷ 4 = 3
Step 3 — Multiplication: 9 × 2 = 18
Step 4 — Addition: 3 + 18 = 21
Answer: 21. Exponents are handled before multiplication and division, and all three come before addition.
Example 4: 2 × (3 + 4)² ÷ 7 − 1
Step 1 — Parentheses: 3 + 4 = 7
Step 2 — Exponents: 7² = 49
Step 3 — Multiplication (left to right): 2 × 49 = 98
Step 4 — Division (left to right): 98 ÷ 7 = 14
Step 5 — Subtraction: 14 − 1 = 13
Answer: 13. This example combines every level of PEMDAS. Working step by step keeps you from making mistakes.
Even experienced students slip up on order of operations. Watch out for these frequent errors:
3 + 4 × 2, you must multiply first. If you want the addition to happen first, use parentheses: (3 + 4) × 2.8 ÷ 2 × 4, work left to right: 8 ÷ 2 = 4, then 4 × 4 = 16. Do not jump to the multiplication first.2 × 3², calculate the exponent first: 3² = 9, then 2 × 9 = 18. Multiplying 2 × 3 first gives the wrong answer of 36.2 × ((1 + 3) × 5), always work from the innermost parentheses outward. Evaluate 1 + 3 = 4 first, then 4 × 5 = 20, then 2 × 20 = 40.The most popular mnemonic in the US is “Please Excuse My Dear Aunt Sally” — the first letter of each word matches Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.
If you learned math outside the US, you may know it as BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction) or BIDMAS (Brackets, Indices, Division, Multiplication, Addition, Subtraction). The order of operations is exactly the same — only the acronym differs.
Whichever mnemonic you prefer, always remember that M/D and A/S are resolved left to right, not strictly in the letter order of the acronym.
Ready to practice?