Order of Operations (BIDMAS/BODMAS)
The Order of Operations is a set of rules that tells you in which order you should carry out mathematical operations when evaluating an expression. In many curricula worldwide, this is often referred to as BODMAS, BIDMAS, or PEMDAS. While each acronym might slightly differ in its letters, they all convey the same principle: some operations must be done before others.
In this guide, we’ll focus on the BIDMAS/BODMAS approach, common in the UK and other parts of the world. We’ll break down each step and show you how to avoid common pitfalls. By the end, you’ll be confident in evaluating even the trickiest expressions correctly.
1. What is BIDMAS/BODMAS?
The acronym BIDMAS (or BODMAS) stands for:
- Brackets
- Indices (or Orders in BODMAS) — exponents/powers
- Division
- Multiplication
- Addition
- Subtraction
BODMAS is often interpreted as:
- Brackets
- Orders (i.e., powers, square roots, etc.)
- Division
- Multiplication
- Addition
- Subtraction
While the second letter differs (I vs. O), the essence is the same: it refers to exponents, powers, or more generally “orders” such as square/cube operations or roots.
A key point to remember: Division and Multiplication are of the same rank — you perform them in the order they appear from left to right. The same applies for Addition and Subtraction.
2. Detailed Breakdown of the Steps
2.1 Brackets
Solve anything inside brackets (parentheses) first. If you have nested brackets, start from the innermost brackets and move outward.
Example: In (2 + 3) × 4, you must compute (2 + 3) = 5, and then multiply by 4 to get 20.
2.2 Indices (or Orders)
Next, handle exponents and roots. Indices refer to powers and roots, such as squaring (x²), cubing (x³), square roots (√ ), cube roots, etc.
Example: If you have 3² + 2, deal with 3² first = 9, then add 2 to get 11.
2.3 Division and Multiplication
After clearing brackets and indices, move on to Division and Multiplication. Importantly, they share the same priority, so you must go from left to right in the expression.
Example: In 20 ÷ 5 × 3, you do the division first (left to right):
- 20 ÷ 5 = 4
- Then 4 × 3 = 12
It is not 20 ÷ (5 × 3) unless brackets specify otherwise.
2.4 Addition and Subtraction
Finally, perform Addition and Subtraction, again from left to right.
Example: In 10 − 3 + 5:
- First do 10 − 3 = 7
- Then do 7 + 5 = 12
Note that if you did 3 + 5 first, you would incorrectly get 10, and then 10 − 10 = 0, which is wrong for the given expression.
3. Example Problems (Easy to Advanced)
3.1 Easy Examples
Example 1: 5 + 3 × 2
Solution:
- Multiplication before addition: 3 × 2 = 6
- Then add 5 + 6 = 11
Answer: 11
Example 2: (4 + 6) − 2
Solution:
- Solve inside brackets: (4 + 6) = 10
- Subtract 2: 10 − 2 = 8
Answer: 8
3.2 Intermediate Examples
Example 3: 2 + (3² × 4)
Solution Steps:
- Inside the brackets, do indices first: 3² = 9
- Then multiply by 4: 9 × 4 = 36
- Now add 2 + 36 = 38
Answer: 38
Example 4: 20 ÷ (5 − 3)²
Solution Steps:
- Solve inside the brackets: 5 − 3 = 2
- Square that result: (2)² = 4
- Now do the division: 20 ÷ 4 = 5
Answer: 5
Example 5: 3 × 4² − 10
Solution Steps:
- Indices first: 4² = 16
- Multiply by 3: 3 × 16 = 48
- Subtract 10: 48 − 10 = 38
Answer: 38
3.3 Advanced Examples
Example 6: 36 ÷ 6 × (5 − 2²) + 7
Solution Steps:
- Inside brackets, do the exponent: 2² = 4
- Then (5 − 4) = 1
- Now we have 36 ÷ 6 × 1 + 7
- Division and multiplication from left to right:
- 36 ÷ 6 = 6
- Then 6 × 1 = 6
- Finally add 7: 6 + 7 = 13
Answer: 13
Example 7: (2 + 2)² × 3 − 4 ÷ 2
Solution Steps:
- Brackets first: 2 + 2 = 4
- Square it: (4)² = 16
- Multiply by 3: 16 × 3 = 48
- Deal with the subtraction and division from left to right:
- Expression is now: 48 − 4 ÷ 2
- Division first (left to right after multiplication): 4 ÷ 2 = 2
- So we have 48 − 2 = 46
Answer: 46
4. Common Mistakes and Pitfalls
- Forgetting left-to-right rule for operations of the same rank: Division and multiplication must be done in sequence from left to right, not always multiplication first.
- Confusing bracket content: Sometimes expressions inside brackets themselves contain exponents or further brackets. Always solve from the innermost bracket outward.
- Wrongly distributing exponents: Remember that (3 + 2)² is 5² = 25, not 3² + 2². That would be incorrect expansion.
- Combining addition/subtraction incorrectly: Keep the left-to-right flow. For example, 10 − 3 + 2 must be handled as (10 − 3) + 2, not 10 − (3 + 2).
5. Practice Exercises
Try these on your own before checking any answers (if provided) to master the concept.
5.1 Easy Exercises
- Evaluate: 4 + 3 × 2.
- Compute: (2 + 6) ÷ 2.
- Calculate: 10 − 2 × 3.
- Find the result of: 6 + 4².
5.2 Intermediate Exercises
- Compute: (5 − 1)² × 2.
- Simplify: 20 ÷ 5 + 3 × 2.
- Evaluate: 2 × 3² − (4 + 6).
- Solve: (4² + 3) − (8 ÷ 4).
5.3 Advanced Exercises
- Simplify: 3² + 4 ÷ 2 × (3 − 1).
- Compute: (6 − 2)² + 10 ÷ 5.
- Evaluate: 16 ÷ 4 ÷ 2 + 3².
- Find the value of: ((2 + 2)² × 5) − (9 − 3).
6. Summary
Remember the BIDMAS/BODMAS order:
- Brackets
- Indices (or Orders)
- Division
- Multiplication
- Addition
- Subtraction
The crucial points are:
- Always do what’s inside Brackets first, then exponents (Indices/Orders).
- Perform Division and Multiplication in the order they appear from left to right.
- Finally, do Addition and Subtraction in the order they appear from left to right.
Mastering these rules will help you avoid calculation errors in algebra, arithmetic, and beyond. Always take your time to apply the steps carefully. With consistent practice, you’ll find it becomes second nature to follow the correct order of operations.
Order of Operations Quiz
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