Order of Operations

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Skill: order-of-operations

Order of Operations

Without agreed-upon rules, 3 + 4 × 2 could be 11 or 14. PEMDAS is the contract that makes every expression mean exactly one thing — learn the precedence ladder and the left-to-right pairs.

1 Understand

The core idea in plain language.

PEMDAS (or GEMDAS) is a precedence ladder: Parentheses (and other grouping), Exponents, Multiplication and Division, Addition and Subtraction. Higher rungs always go first.

M/D are one rung, not two: multiply and divide left to right as they appear. In 12 ÷ 3 × 2, divide first (leftmost): 4 × 2 = 8. Doing the × first gives 2 — wrong. The same pairing rule holds for A/S.

Grouping symbols — parentheses (), brackets [], braces {} — outrank everything. Nested groups work inside-out: in 2 × [3 + (4 − 1)], compute (4 − 1) = 3 first, then [3 + 3] = 6, then 2 × 6 = 12.

A fraction bar is a grouping symbol in disguise: in (8 + 4)/3, the entire top is computed before dividing. So are the bars of absolute value and the top of a square root.

2 See it

Diagrams that make the idea visual.

The precedence ladder

Read top to bottom — do the highest unfinished rung first:

  1. Grouping ( ) [ ] { } — inside out
  2. Exponents — right to left for stacks
  3. × and ÷ — left to right, as a pair
  4. + and − — left to right, as a pair
1 · Grouping ( ) [ ]2 · Exponents3 · × and ÷ (left to right)4 · + and − (left to right)
PEMDAS: higher rungs first; ×/÷ and +/− are pairs, worked left to right.

Left-to-right pairs: 12 ÷ 3 × 2

× and ÷ share a rung, so scan left to right: the ÷ comes first.

12 ÷ 3 × 2 → 4 × 2 → 8.

Multiplying first gives 12 ÷ 6 = 2 — the classic pair trap.

12 ÷ 3 × 2÷ first (leftmost) → 4 × 2= 8
Same rung → leftmost operation wins: 12 ÷ 3 × 2 = 8.

3 The key rule

The PEMDAS contract
Grouping → Exponents → ×/÷ (left to right) → +/− (left to right)

Two operations on the same rung are done left to right — never “multiplication before division” as a rule. When in doubt, add parentheses to say what you mean.

4 Worked examples

Follow each step. The pattern is always the same.

Example 1 Multiplication before addition
  1. ×/÷ outrank +/−: do 4 × 2 = 8 first.
  2. Then 3 + 8 = 11.
3 + 4 × 2 = 11

Check: Left-to-right would give (3 + 4) × 2 = 14 — wrong. 11 is correct.

Example 2 Parentheses change everything
  1. Grouping first: (2 + 3) = 5.
  2. Then 5 × 4 = 20.
(2 + 3) × 4 = 20

Check: Compare with 2 + 3 × 4 = 14. Parentheses moved the answer from 14 to 20.

Example 3 Exponents plus a pair
  1. Exponents first: 3² = 9 → 2 + 9 × 2.
  2. ×/÷ next: 9 × 2 = 18 → 2 + 18.
  3. Add: 20.
2 + 3² × 2 = 20

Check: Rungs in order: exponent, then ×, then +. 20 is correct.

Example 4 Fraction bar as grouping
  1. The bar groups the top: compute 8 + 4 = 12 first.
  2. Then divide: 12 ÷ 3 = 4.
(8 + 4)/3 = 4

Check: Dividing only the 4 (8 + 4/3) would be wrong — the bar groups the whole top.

5 Common mistakes

These errors show up on almost every quiz. Spot them now and they will never cost you points.

1. Computing left to right, ignoring order
Wrong
3 + 4 × 2 = 14 — “I just went left to right.”
Right
Multiply first: 3 + 8 = 11.
Left-to-right applies only within a rung. ×/÷ always outrank +/−.
2. Doing multiplication before division always
Wrong
12 ÷ 3 × 2 = 2 — “M comes before D in PEMDAS.”
Right
They share a rung: go left to right → 4 × 2 = 8.
PEMDAS is four rungs, not six. M/D are one rung; A/S are one rung.
3. Forgetting nested grouping works inside-out
Wrong
2 × [3 + (4 − 1)] = 2 × 7 + … — adding before finishing the inner group.
Right
Innermost first: (4 − 1) = 3, then [3 + 3] = 6, then 12.
Parentheses inside brackets resolve first. Work from the inside out.

6 Key vocabulary

Say these words like you mean them.

PEMDAS / GEMDAS
The precedence order: Parentheses/Grouping, Exponents, ×/÷, +/−.
Precedence
Which operation goes first when several compete.
Grouping symbols
Parentheses ( ), brackets [ ], braces { } — and fraction bars.
Nested
Groups inside groups; resolve from the inside out.
Left-to-right pair
×/÷ (and +/−) share a rung and are done in left-to-right order.
Evaluate
Compute the single value an expression represents.

7 Quick check

Try each one on paper first, then reveal the answer.

1. Evaluate: 10 − 2 × 3.
Answer
Multiply first: 10 − 6 = 4.
2. Evaluate: (10 − 2) × 3.
Answer
Group first: 8 × 3 = 24.
3. Evaluate: 20 ÷ 4 + 2².
Answer
Exponent: 4; division: 5; then 5 + 4 = 9.
4. Evaluate: 3 × (2 + 5²).
Answer
Inside: 5² = 25, 2 + 25 = 27; then 3 × 27 = 81.

Key points to remember

  • Grouping → Exponents → ×/÷ → +/− — the four rungs, in order.
  • × and ÷ are one rung: work them left to right. Same for + and −.
  • Nested groups resolve inside out.
  • A fraction bar groups its top (and bottom) before dividing.
  • Exponent stacks go right to left: 2^3^2 = 2^(3^2) = 512.
  • Parentheses are free — add them any time they make your meaning clearer.
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