Practice Maths

Order of Operations

Key Terms

BODMAS / BIDMAS
The agreed order of operations: Brackets, Orders (Indices), Division & Multiplication (left to right), Addition & Subtraction (left to right).
brackets
Grouping symbols evaluated first; work from the innermost pair outward.
order (index)
Powers and roots — evaluated after brackets but before multiplication, division, addition, or subtraction.
equal priority
Division and multiplication share the same priority (resolve left to right); so do addition and subtraction.
  1. Brackets (innermost first)
  2. Orders / Indices (powers and roots)
  3. Division and Multiplication (left to right)
  4. Addition and Subtraction (left to right)
Common error: Addition does NOT come before subtraction — they are equal priority and must be done left to right. Same for division and multiplication.

Worked Examples

Example 1: 3 + 4 × 2 = 3 + 8 = 11 (not 14!)

Example 2: (3 + 4) × 2 = 7 × 2 = 14

Example 3: 20 − 4 × (2 + 3)2 ÷ 5

= 20 − 4 × 52 ÷ 5   (brackets)

= 20 − 4 × 25 ÷ 5   (indices)

= 20 − 100 ÷ 5   (multiply left to right)

= 20 − 20 = 0

Why Order of Operations Exists

Consider 2 + 3 × 4. If you calculate left to right, you get 5 × 4 = 20. But if you do multiplication first, you get 2 + 12 = 14. Both methods give different answers from the same expression — which is right? Mathematics needs a universal agreement so that everyone gets the same answer. The agreed convention is BODMAS: Brackets, Orders (indices/powers), Division and Multiplication (left to right), Addition and Subtraction (left to right). So 2 + 3 × 4 = 2 + 12 = 14.

This isn't arbitrary. Multiplication represents repeated addition, so 3 × 4 = 12 must be evaluated as a single unit before being added to 2. The convention reflects the mathematical structure of expressions, not just a human agreement.

Brackets: Always First

Brackets override every other operation. Work from the innermost brackets outward. For (3 + 42) × 2: inside the bracket, indices come first: 42 = 16. Then: (3 + 16) × 2 = 19 × 2 = 38. For nested brackets [(2 + 3) × 4] − 1: innermost (2 + 3) = 5, then 5 × 4 = 20, then 20 − 1 = 19. Always resolve the innermost pair of brackets first, then work outward.

Fraction bars act like brackets. In (2 + 6)/(3 − 1): calculate numerator 2 + 6 = 8, denominator 3 − 1 = 2, then divide: 8 ÷ 2 = 4. Think of the fraction bar as saying "do the top completely and the bottom completely before dividing."

Key Tip: Multiplication and division are equal priority — do them left to right, not multiplication first. Same for addition and subtraction. So 20 ÷ 4 × 2: left to right gives 5 × 2 = 10. If you did multiplication first: 4 × 2 = 8, then 20 ÷ 8 = 2.5 — wrong!

Indices (Orders) in BODMAS

Indices are evaluated before multiplication and division, but after brackets. In 3 + 23 × 4: first the index 23 = 8, then multiplication 8 × 4 = 32, then addition 3 + 32 = 35. In 4 × (2 + 1)2: first the bracket (2 + 1) = 3, then the index 32 = 9, then 4 × 9 = 36.

The worked example from Key Ideas: 20 − 4 × (2 + 3)2 ÷ 5. Step 1 brackets: (2+3) = 5. Step 2 index: 52 = 25. Step 3 multiplication: 4 × 25 = 100. Step 4 division: 100 ÷ 5 = 20. Step 5 subtraction: 20 − 20 = 0. Careful step-by-step working prevents errors.

Key Tip: Write one step per line when evaluating complex expressions. Don't try to collapse multiple steps into one line — it leads to errors and makes checking impossible. Show each BODMAS stage clearly. Examiners reward correct working even if the final answer has a small arithmetic slip.

BODMAS with Negative Numbers

Combine BODMAS with integer sign rules carefully. For −22 + 3 × (−4): Step 1 indices: −22 = −4 (note: −22 means −(22), not (−2)2). Step 2 multiplication: 3 × (−4) = −12. Step 3 addition: −4 + (−12) = −16. As a real-world application: thermostats, bank calculations, physics problems, and computer programs all rely on consistent order of operations. Getting it right is essential for any technical field.

Mastery Practice

  1. Simple BODMAS (No Negatives) Fluency

    1. 3 + 4 × 2
    2. 10 − 8 ÷ 4
    3. (5 + 3) × 4
    4. 62 − 10
    5. 3 × (4 + 2) − 5
    6. 20 ÷ (2 + 3) + 1
    7. 4 + 32 × 2
    8. (7 − 3)2 + 5

  2. BODMAS with Integers (Negatives Included) Fluency

    1. (−3) + 4 × 2
    2. (−2)2 − 5
    3. −3 × (4 − 7)
    4. 12 ÷ (−4) + 6
    5. (−5)2 − 3 × 4
    6. 2 × (−3)2 − 10
    7. (−4 + 10) ÷ (−2)
    8. −2 × (3 + (−7))

  3. BODMAS with Fractions and Decimals Fluency

    1. ½ + ¼ × 4
    2. (0.5 + 1.5) × 3
    3. 3⁄4 ÷ ½ − ¼
    4. 0.3 × (2 + 0.7)
    5. (½)2 + ¾
    6. (1.2 + 0.8) ÷ (0.5 × 4)
    7. ¾ × (1 + 1⁄3)
    8. 2.52 − 3 × 1.5

  4. Insert Brackets to Make the Statement True Understanding

    Add one pair of brackets to make each equation correct:

    1. 3 + 4 × 2 = 14
    2. 5 × 3 + 2 − 4 = 5
    3. 20 ÷ 4 + 1 = 4
    4. 3 × 4 + 22 = 24
    5. 7 + 3 × 2 − 1 = 19
    6. 8 − 2 × 3 + 1 = 24
    7. 10 ÷ 2 + 3 = 2
    8. 6 + 4 ÷ 2 × 3 = 15

  5. Find and Correct the Error Understanding

    Each working below contains an order-of-operations mistake. Identify the error and write the correct answer:

    1. Problem: 5 + 3 × 4. Student wrote: 5 + 3 = 8, then 8 × 4 = 32.
    2. Problem: 18 ÷ 3 + 3. Student wrote: 18 ÷ 6 = 3.
    3. Problem: 4 × 23. Student wrote: (4 × 2)3 = 83 = 512.
    4. Problem: 6 − 2 × 3 + 1. Student wrote: (6 − 2) = 4, then 4 × (3 + 1) = 16.
    5. Problem: (−2)2 + 3. Student wrote: −22 + 3 = −4 + 3 = −1.
    6. Problem: 12 ÷ 4 ÷ 3. Student wrote: 12 ÷ (4 ÷ 3) = 12 ÷ 4⁄3 = 9.

  6. Multi-Operation Word Problems Problem Solving

    1. A cinema charges $12 per adult ticket and $8 per child ticket. A family of 2 adults and 3 children attend. Write a single mathematical expression and evaluate it to find the total cost.
    2. James earns $15 per hour. He works 6 hours per day for 5 days, then spends $35 on a gift. Write an expression and calculate how much he has left.
    3. A square garden has side length (3 + 2) metres. A path of width 1 m surrounds it on all sides. Write an expression for the total area of the garden including the path, then evaluate it.
    4. Evaluate: 5 × (32 − 4) ÷ (2 + 3) − 1
    5. Evaluate: 3 × [4 − (22 ÷ 4)] + 5 × 2
    6. A school tuckshop sells pies for $3.50 each and drinks for $2.00 each. On Monday, 45 pies and 60 drinks were sold. Write a single expression with correct order of operations and calculate the total revenue.

  7. Nested Brackets Understanding

    Evaluate each expression, working from the innermost brackets outward:

    1. 3 × [(4 + 2) − (8 ÷ 4)]
    2. [(5 + 3) × 2 − 6] ÷ 5
    3. 22 × [(3 + 1)2 − 10]
    4. 10 − [3 × (2 + 1) − 42]
    5. [(12 ÷ 3) + (2 × 5)] ÷ (7 − 3)
    6. 5 × [2 − (3 − (1 + 4))]

  8. Write the Expression Understanding

    Write a mathematical expression (using correct order of operations) for each description, then evaluate it:

    1. Double the sum of 7 and 3, then subtract 4.
    2. The square of (9 minus 4), divided by 5.
    3. Subtract 3 from 10, then multiply the result by 2 squared.
    4. Add 6 to the product of 3 and 4, then divide by the sum of 1 and 8.

  9. BODMAS with Integers, Fractions and Brackets Problem Solving

    1. (−3)2 × [4 − (2 + 3)2] ÷ (−21)
    2. [(−4 + 10) ÷ 2]2 − 5 × (−3)
    3. (½ + ¼) ÷ [(¾)2 × 4] + 1
    4. A shop has a flat-screen TV originally priced at $1800. The shop applies a 20% discount, then adds 10% GST to the discounted price. Using a single expression with correct order of operations, find the final price.
    5. Mia earns $18 per hour. In one week she works 5 shifts of 6 hours and 2 shifts of 4 hours. She spends $35 on lunch and $120 on petrol that week. Write a single expression and find her remaining earnings.

  10. True or False? Justify Each Problem Solving

    State whether each statement is true or false. Show working to justify your answer:

    1. 3 + 42 = 49
    2. (3 + 4)2 = 32 + 42
    3. 18 ÷ 3 ÷ 2 = 18 ÷ (3 ÷ 2)
    4. 2 × 5 − 3 × 2 = (2 × 5) − (3 × 2)
    5. (−2)4 = 24
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