Order of Operations

Order of Operations

When an expression has several operations together — multiplication, addition, brackets — we don't just calculate from left to right. There are agreed-upon rules so that everyone arrives at the same answer. On this page we'll learn the correct order and practise it.

Background and Basic Definitions

Order of operations (what to do first):

  1. Brackets \((\;)\) — everything inside brackets is calculated first.
  2. Multiplication and Division \(\times, \div\) — before addition and subtraction (left to right between them).
  3. Addition and Subtraction \(+, -\) — last (left to right between them).

A handy memory aid: BODMAS — Brackets, Order (powers), Division/Multiplication, Addition/Subtraction.

Why do we need this rule? Without it, the same expression can give different answers. For example \(2 + 3 \times 4\): if we add first \((2+3)\times 4 = 20\), but if we multiply first \(2 + 12 = 14\). The rule says 14 is correct.

Solution Steps

  1. Step 1 — Scan the expression. Are there brackets? If so, calculate what's inside them first.
  2. Step 2 — After brackets, scan left to right for multiplication and division. Carry them out in order.
  3. Step 3 — After multiplication and division, carry out addition and subtraction from left to right.
  4. Step 4 — If there are nested brackets (brackets inside brackets), start with the innermost ones.
  5. Step 5 — Check: write the expression step by step, replacing only one part at a time, to avoid confusion.

Worked Examples

Example 1: Multiplication Before Addition

Problem: Calculate: \( 5 + 3 \times 4 \)

Solution:

  1. No brackets. Multiplication before addition.
  2. \( 3 \times 4 = 12 \).
  3. \( 5 + 12 = 17 \).

Answer: \( 5 + 3 \times 4 = 17 \)

Example 2: Brackets Change Everything

Problem: Calculate: \( (5 + 3) \times 4 \)

Solution:

  1. Brackets — calculate inside first.
  2. \( 5 + 3 = 8 \).
  3. \( 8 \times 4 = 32 \).

Answer: \( (5 + 3) \times 4 = 32 \)

Example 3: Several Operations Together

Problem: Calculate: \( 20 - 2 \times 6 + 4 \)

Solution:

  1. No brackets. Multiplication first.
  2. \( 2 \times 6 = 12 \).
  3. Expression becomes: \( 20 - 12 + 4 \).
  4. Left to right: \( 20 - 12 = 8 \), then \( 8 + 4 = 12 \).

Answer: \( 20 - 2 \times 6 + 4 = 12 \)

Example 4: Division Before Subtraction

Problem: Calculate: \( 30 - 24 \div 6 \)

Solution:

  1. Division first: \( 24 \div 6 = 4 \).
  2. \( 30 - 4 = 26 \).

Answer: \( 30 - 24 \div 6 = 26 \)

Example 5: Expression with Brackets and Multiple Operations

Problem: Calculate: \( 3 \times (8 - 5) + 10 \div 2 \)

Solution:

  1. Brackets first: \( 8 - 5 = 3 \).
  2. Expression: \( 3 \times 3 + 10 \div 2 \).
  3. Multiplication and division (left to right): \( 3 \times 3 = 9 \), \( 10 \div 2 = 5 \).
  4. \( 9 + 5 = 14 \).

Answer: \( 3 \times (8 - 5) + 10 \div 2 = 14 \)

Common Mistakes

✗ Common mistake: Calculating the expression left to right without following the order: \( 5 + 3 \times 4 = 8 \times 4 = 32 \).

✓ The correct way: Multiplication and division always come before addition and subtraction! \( 5 + 3 \times 4 = 5 + 12 = 17 \). Always ask: "Is there a multiplication or division outside brackets?"

✗ Common mistake: Ignoring brackets: calculating \( 2 \times 3 + 4 \) even when \( 2 \times (3 + 4) \) is written.

✓ The correct way: Brackets are a special signal that says: "Calculate me first!" Whatever is inside brackets always comes first.

✗ Common mistake: When addition and subtraction remain after the multiplication is done, calculating subtraction before addition because "subtraction is more important".

✓ The correct way: Addition and subtraction are equal in priority. When only addition and subtraction remain — simply work left to right.

Practice Tips

  • Tip — Remember: BODMAS — Brackets, (Orders), Division/Multiplication, Addition/Subtraction. That's the order!
  • Tip — Want to change the order of calculation? Simply add brackets! \( 2 + 3 \times 4 = 14 \) but \( (2+3) \times 4 = 20 \).
  • Tip — Write each step separately. Don't try to calculate everything in your head at once — that leads to mistakes.
  • Tip — Check: go back over the original expression and mark each operation in the correct order before calculating.

Summary and Key Formulas

  1. Brackets — first.
  2. Multiplication and Division — left to right.
  3. Addition and Subtraction — left to right.

Memory aid: BODMAS.