Order of Operations (BODMAS/PEMDAS)

Why we need a rule at all

Look at this simple expression:

3 + 4 × 2

What is the answer? If you work strictly left to right, you get 3 + 4 = 7, then 7 × 2 = 14. But if you do the multiplication first, you get 4 × 2 = 8, then 3 + 8 = 11. Two different answers from the same expression!

This is exactly the problem the order of operations solves. Mathematicians agreed on one fixed order so that everyone in the world gets the same answer from the same expression. Without it, calculators, textbooks and engineers would all disagree, and maths would fall apart. The correct answer above is 11, because multiplication is done before addition.

Two memory words describe this agreed order: BODMAS (common in the UK) and PEMDAS (common in the US). They are the same rule wearing different clothes.

BODMAS and PEMDAS side by side

BODMAS	PEMDAS	Meaning
Brackets	Parentheses	( ) — do these first
Orders	Exponents	Indices/powers like 3² and roots
Division	Multiplication	Equal level — left to right
Multiplication	Division	Equal level — left to right
Addition	Addition	Equal level — left to right
Subtraction	Subtraction	Equal level — left to right

"Orders" is the old British word for indices or powers — numbers raised to an exponent, like 5² or 2³. If you would like a refresher, see Exponents and Powers.

The single most important thing to understand is in the next section.

The two "equal pairs"

The letters make it look as if division always beats multiplication, and addition always beats subtraction. That is not true. These operations sit in two equal pairs:

• Division and multiplication are equal in rank. Do whichever comes first when reading left to right.
• Addition and subtraction are equal in rank. Again, work left to right.

This is the trap that catches most students. Let's see why it matters.

Example — 12 ÷ 4 × 3.

• Wrong way: "M before D", so 4 × 3 = 12, then 12 ÷ 12 = 1. ✗
• Right way: left to right. 12 ÷ 4 = 3, then 3 × 3 = 9. ✓

Because division appears first, you do it first. The answer is 9.

Working through the order step by step

Here is the full method, applied to a longer expression.

Example 1 — Evaluate 5 + 2 × (8 − 3)².

1. Brackets first. Inside the brackets: 8 − 3 = 5. The expression becomes 5 + 2 × 5².
2. Indices next. 5² = 25. Now we have 5 + 2 × 25.
3. Division and multiplication. 2 × 25 = 50. Now we have 5 + 50.
4. Addition and subtraction. 5 + 50 = 55.

So 5 + 2 × (8 − 3)² = 55.

Example 2 — Evaluate 36 ÷ 6 + 4 × 2 − 1.

1. No brackets, no indices.
2. Division and multiplication, left to right: 36 ÷ 6 = 6, and 4 × 2 = 8. Now we have 6 + 8 − 1.
3. Addition and subtraction, left to right: 6 + 8 = 14, then 14 − 1 = 13.

Example 3 — A tricky subtraction. Evaluate 20 − 8 + 3.

1. Addition and subtraction are equal, so go left to right.
2. 20 − 8 = 12, then 12 + 3 = 15.
3. Common mistake: doing 8 + 3 = 11 first gives 20 − 11 = 9, which is wrong. Always honour the left-to-right rule.

How brackets change everything

Brackets are your power tool. They let you force an operation to happen first, overriding the usual order. Compare these two carefully:

Expression	Steps	Answer
6 + 2 × 5	2 × 5 = 10, then 6 + 10	16
(6 + 2) × 5	6 + 2 = 8, then 8 × 5	40
18 − 6 ÷ 3	6 ÷ 3 = 2, then 18 − 2	16
(18 − 6) ÷ 3	18 − 6 = 12, then 12 ÷ 3	4

The numbers are identical, yet the answers differ wildly. If you ever want a calculation to happen in a particular order, add brackets and remove all doubt.

Nested brackets

Sometimes brackets sit inside brackets. Always work from the innermost pair outward.

Example 4 — Evaluate 2 × [3 + (10 − 4)].

1. Innermost brackets: 10 − 4 = 6. Now 2 × [3 + 6].
2. Outer brackets: 3 + 6 = 9. Now 2 × 9.
3. Multiply: 2 × 9 = 18.

Indices and a common trap

Indices come right after brackets. Be careful which number the power applies to.

Example 5 — Evaluate 2 + 3². Only the 3 is squared, not the 2 + 3. So 3² = 9, then 2 + 9 = 11. The answer is not 5² = 25.

Example 6 — Evaluate (2 + 3)². Now the brackets group the sum first: 2 + 3 = 5, then 5² = 25. The brackets tell you to square the whole sum.

A quick checklist

When you face any expression, run down this list in order:

1. B / P — Are there brackets? Solve inside them first (innermost first).
2. O / E — Any indices or roots? Do them next.
3. D / M — Division and multiplication, left to right.
4. A / S — Addition and subtraction, left to right.

A practice activity

Try these without a calculator, then check using the steps above:

• (a) 7 + 6 × 2
• (b) (7 + 6) × 2
• (c) 48 ÷ 8 ÷ 2
• (d) 5 × 4 − 6 ÷ 2
• (e) 3 + 2 × 4²
• (f) 100 − (4 + 6)² ÷ 5

Answers: (a) 19; (b) 26; (c) 48 ÷ 8 = 6, then 6 ÷ 2 = 3; (d) 20 − 3 = 17; (e) 4² = 16, 2 × 16 = 32, then 3 + 32 = 35; (f) (10)² = 100, 100 ÷ 5 = 20, then 100 − 20 = 80.

Challenge: add brackets to 4 + 2 × 3 − 1 so that the answer becomes 17. (Solution: (4 + 2) × 3 − 1 = 18 − 1 = 17.)

Where this leads

The order of operations is the grammar of mathematics — it tells you how to "read" an expression correctly. You will rely on it constantly in Algebra Basics, where expressions mix brackets, powers and variables, and in Solving Linear Equations, where you must simplify each side carefully. Practise reading expressions slowly, one priority level at a time, and the rules will soon feel automatic.