Learning goals

  • Recall the PEMDAS (or BODMAS) priority order and why it exists
  • Evaluate multi-step expressions without skipping equal-rank left-to-right rules
  • Spot and fix the most common order-of-operations mistakes

Why order matters

Without agreed rules, the same expression could have different answers. PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction) is the standard for arithmetic and algebra.

BODMAS / BIDMAS are the same idea with different letters. The key insight: multiplication and division share one priority level, and addition and subtraction share another — at each level, work left to right.

Grouping symbols (parentheses, brackets, fraction bars) always come first because they redefine which operations belong together.

PEMDAS sequence

P → E → MD (L→R) → AS (L→R)

Parentheses first, then exponents, then multiplication and division left to right, then addition and subtraction left to right. Do not always multiply before divide — whichever appears first left-to-right wins.

Example 1 — Multiplication before addition

Given

Evaluate: 3 + 4 × 2

Steps

  1. There are no parentheses or exponents.
  2. Multiplication comes before addition: 4 × 2 = 8.
  3. Then add: 3 + 8 = 11.

Answer

3 + 4 × 2 = 11.

Tip: If someone adds first (3 + 4 = 7, then × 2 = 14), they broke PEMDAS.

Example 2 — Parentheses change the result

Given

Evaluate: (3 + 4) × 2

Steps

  1. Parentheses first: 3 + 4 = 7.
  2. Then multiply: 7 × 2 = 14.

Answer

(3 + 4) × 2 = 14.

Example 3 — Equal rank, left to right

Given

Evaluate: 24 ÷ 6 × 2

Steps

  1. Division and multiplication are the same priority.
  2. Work left to right: 24 ÷ 6 = 4.
  3. Then 4 × 2 = 8.

Answer

24 ÷ 6 × 2 = 8 (not 24 ÷ 12 = 2).

Tip: A common mistake is “always multiply before divide.” Equal-rank ops go left to right.

Example 4 — Exponents before multiply

Given

Evaluate: 2 + 3² × 4

Steps

  1. Exponents before multiplication: 3² = 9.
  2. Multiply: 9 × 4 = 36.
  3. Add: 2 + 36 = 38.

Answer

2 + 3² × 4 = 38.

Treating multiplication as always-before division

Common mistake

24 ÷ 6 × 2 → multiply first → 24 ÷ 12 = 2.

Better approach

Same priority: left to right → (24 ÷ 6) × 2 = 8.

Ignoring the fraction bar as grouping

Common mistake

Reading (8 + 2) / 2 + 3 as only dividing 2, then adding freely without structure.

Better approach

A written fraction bar groups the numerator and denominator. Compute each group fully before dividing.

Check your understanding

  1. 1.Evaluate: 10 − 3 × 2

    Answer: 10 − 6 = 4.

  2. 2.Evaluate: 18 ÷ 3 × 3

    Answer: (18 ÷ 3) × 3 = 18.

  3. 3.Evaluate: 5 + 2³

    Answer: 5 + 8 = 13.

  4. 4.Evaluate: 4 × (6 − 1)

    Answer: 4 × 5 = 20.

Key takeaways

  • Parentheses (and other grouping symbols) come first.
  • Exponents are evaluated before multiplication and division.
  • Multiplication and division share the same priority — work left to right.
  • Addition and subtraction come last, also left to right.