Order of Operations (PEMDAS/BODMAS): Complete Guide with Examples

If you've ever got a different answer to a maths problem than someone else — and you were both sure you were right — the culprit is almost certainly order of operations.

The order of operations is a set of rules that tells you which part of a mathematical expression to calculate first. Without these rules, the same expression could produce different answers depending on who's solving it.

What Is PEMDAS / BODMAS?

PEMDAS (used in the USA) and BODMAS (used in the UK, India, and Australia) are acronyms for the same set of rules — just with slightly different wording.

The order is: Brackets → Powers → Division/Multiplication → Addition/Subtraction

Note: Division and multiplication have equal priority (left to right). Addition and subtraction have equal priority (left to right).

Why Do We Need These Rules?

Without an agreed order, the expression 2 + 3 × 4 would be ambiguous:

• If you add first: (2 + 3) × 4 = 5 × 4 = 20
• If you multiply first: 2 + (3 × 4) = 2 + 12 = 14

The agreed rules say multiplication comes before addition, so the correct answer is 14.

The Rules Explained

1. Brackets / Parentheses First

Always solve whatever is inside brackets before anything else.

(3 + 4) × 2 = 7 × 2 = 14

Nested brackets: work from the innermost outward.

2 × (3 + (4 − 1)) = 2 × (3 + 3) = 2 × 6 = 12

2. Exponents / Orders (Powers and Roots)

After brackets, calculate any powers or square roots.

2 + 3² = 2 + 9 = 11

4 × √16 = 4 × 4 = 16

3. Multiplication and Division (Left to Right)

These two operations have equal priority. When they appear together, work from left to right.

12 ÷ 4 × 3 = 3 × 3 = 9    ✓ (left to right)
12 ÷ 4 × 3 ≠ 12 ÷ 12 = 1  ✗ (doing × before ÷ is wrong)

4. Addition and Subtraction (Left to Right)

Same principle — equal priority, work left to right.

10 − 3 + 2 = 7 + 2 = 9    ✓
10 − 3 + 2 ≠ 10 − 5 = 5   ✗

Worked Examples

Example 1: Basic

8 + 2 × 5 − 3
= 8 + 10 − 3        (multiplication first)
= 18 − 3            (left to right)
= 15

Example 2: With Brackets

(8 + 2) × (5 − 3)
= 10 × 2            (brackets first)
= 20

Example 3: With Exponents

3 + 4² ÷ 2
= 3 + 16 ÷ 2        (exponent first)
= 3 + 8             (division before addition)
= 11

Example 4: Complex

5 × (2 + 3)² − 10 ÷ 2
= 5 × 5² − 10 ÷ 2   (brackets first)
= 5 × 25 − 10 ÷ 2   (exponent)
= 125 − 5           (× and ÷ left to right)
= 120

Example 5: The Classic Viral Problem

6 ÷ 2(1 + 2) — this expression goes viral regularly because people disagree on the answer.

Step 1: Bracket → 1 + 2 = 3
Step 2: Expression becomes 6 ÷ 2 × 3
Step 3: Left to right → 6 ÷ 2 = 3, then 3 × 3 = 9

The answer is 9. The confusion arises because some people treat 2(3) as a single term. In standard mathematical convention, division and multiplication have equal priority and are evaluated left to right.

Practice Problems

Try these before checking the answers:

1. 3 + 4 × 2
2. (3 + 4) × 2
3. 2³ + 4 × 3 − 5
4. 20 ÷ (2 + 3) × 4
5. 6 + 2 × 3² − 4 ÷ 2

Answers:

1. 3 + 8 = 11
2. 7 × 2 = 14
3. 8 + 12 − 5 = 15
4. 20 ÷ 5 × 4 = 4 × 4 = 16
5. 6 + 2 × 9 − 2 = 6 + 18 − 2 = 22

Common Mistakes

Treating multiplication before division as a strict rule — Multiplication and division have equal priority. Always work left to right when both appear together.

Forgetting to work through nested brackets inside-out — Solve the innermost brackets first.

Applying exponents to the wrong part — In -3², the exponent applies only to 3, giving you -(9) = -9, not (-3)² = 9. Use brackets: (-3)² if you want to square the negative number.

Ignoring implied multiplication — 2(3) means 2 × 3. It follows the same rules as explicit multiplication.

Why BODMAS and PEMDAS Give the Same Answer

Despite the different names, both acronyms describe the same priority. In BODMAS, "DM" represents division and multiplication together (equal priority). In PEMDAS, "MD" similarly represents multiplication and division together. The acronym order doesn't mean multiplication comes before division — they're equal.

Quick Reference Card

Read Next

• Percentage Calculations: Every Formula
• How to Calculate Area
• Fractions, Decimals and Percentages