Order of Operations in Algebra – BODMAS, PEMDAS, BIDMAS

Basic Algebra in Full | Lesson 8

🧠 What You’ll Learn

In this lesson, you’ll learn how the Order of Operations truly works in mathematics and algebra, including the deeper mathematical ideas behind BODMAS, PEMDAS, BIDMAS, operator ownership, operator families, and reciprocal thinking.

You’ll learn how to:
• Understand what the Order of Operations actually means and why it exists
• Understand the purpose of BODMAS, PEMDAS, and BIDMAS
• Recognize why multiplication and division belong to the same operator family
• Recognize why addition and subtraction belong to the same operator family
• Understand the concept of operator ownership and how operators belong to terms
• Understand how operator families make algebraic simplification easier and more flexible
• Convert division into multiplication using reciprocals correctly
• Simplify arithmetic expressions involving multiple operators
• Apply the true hierarchy of operations using brackets, powers, multiplication families, and addition families
• Understand why the left-to-right rule is often unnecessary when operator ownership is understood properly
• Solve expressions involving brackets and nested operations
• Apply order of operations inside brackets and grouped expressions
• Simplify algebraic expressions involving variables and like terms
• Combine powers, fractions, reciprocals, multiplication, and addition within the same expression
• Avoid common BODMAS and PEMDAS mistakes students make
• Understand how mathematical clarity and proper notation prevent ambiguity
• Build the structural thinking needed for more advanced algebra later in the series

By the end of this lesson, you’ll be able to simplify arithmetic and algebraic expressions confidently while understanding the mathematical reasoning behind every step.

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🧩Practice Exercises

These exercises help you build from foundation to advanced understanding.

You can:

  • Start at your level
  • Or work through all levels step by step

✍️ Attempt exercises before checking answers.

🎯 Levels

Each level matches a stage of learning and shows
what you should be able to understand and solve at that point.

🟢 Foundation → You’re learning this topic for the first time
🟠 Developing → You understand the basics and are building on them
⚪ Confident → You’re ready for deeper, multi-step problems
⚫ Advanced → You want to challenge your understanding further

📘 Not sure where you fit? → View Level Guide

💡 Each set of exercises is designed to match what students at that stage are typically expected to handle.

Start with your level, then try the next one when you feel ready.

  1. Simplify $7 + 5 – 9$
  2. Simplify $18 \div 3 \times 2$
  3. Simplify $4 + 6 \times 5$
  4. Simplify $(12 – 7) + 9$
  5. Simplify $20 \div 5 + 3 \times 4$
Show Answers
  1. $3$
  2. $12$
  3. $34$
  4. $14$
  5. $16$

  1. Simplify $15 – 6 + 8 – 4$
  2. Simplify $24 \div 6 \times 9$
  3. Simplify $5 + 4 \times (7 – 3)$
  4. Simplify $(18 – 6) \div 3 + 2 \times 5$
  5. Simplify $8y – 3y + 6 – 4$
Show Answers
  1. $13$
  2. $36$
  3. $21$
  4. $14$
  5. $5y + 2$

  1. Simplify $14 + 8 \times (5 – 2)^2 \div 6$
  2. Simplify $30 \div (4 + 2) \times 8 – 5$
  3. Simplify $7a + 24a \div 6 – 5a$
  4. Simplify $(3^2 + 7) \times 2 – 18 \div 3$
  5. Simplify $12 + 9x \times (6 – 4) \div 3x$
Show Answers
  1. $26$
  2. $35$
  3. $6a$
  4. $26$
  5. $18$

  1. Simplify $48 \div (2 + 2)^2 \times 6 + 5$
  2. Simplify $(5^2 – 18 \div 3) \times 4 + 6$
  3. Simplify $36x \div (3x) + 4(7 – 2)^2 – 9$
  4. Simplify $20 + 18 \div (3^2) \times 6 – (8 – 5)^2$
  5. Simplify $\left(64 \div (4 + 4) \times 3\right) + 15x \div 5x – (2^3 – 1)$
Show Answers
  1. $23$
  2. $82$
  3. $103$
  4. $23$
  5. $20$

🔁 Continue Learning

⏮️ Back to Basic Algebra in Full
⏭️ Next Lesson → Lesson 9: Linear Equations in One Variable

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