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BODMAS rule

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Understanding the Order of Operations: Why BODMAS Matters

Imagine you are solving a math expression like 5 + 3 x 2. If you add first, you get (5 + 3) x 2 = 8 x 2 = 16. But if you multiply first, you get 5 + (3 x 2) = 5 + 6 = 11. Which answer is correct?

This example shows why the order of operations is essential. Without a standard rule, everyone might solve the same expression differently, leading to confusion and incorrect answers. To avoid this, mathematicians use a universally accepted rule called BODMAS.

The BODMAS rule tells us the order in which to perform operations in any mathematical expression so that everyone arrives at the same, correct answer.

The BODMAS Rule Explained

The acronym BODMAS stands for:

  • B - Brackets: Solve expressions inside brackets first.
  • O - Orders: Calculate powers (exponents) and roots next.
  • D - Division: Perform division operations.
  • M - Multiplication: Perform multiplication operations.
  • A - Addition: Perform addition operations.
  • S - Subtraction: Perform subtraction operations.

Important: Division and multiplication have the same priority and are solved from left to right. The same applies to addition and subtraction.

graph TD    B[Brackets] --> O[Orders (Exponents and Roots)]    O --> DM[Division and Multiplication (Left to Right)]    DM --> AS[Addition and Subtraction (Left to Right)]

This flowchart shows the sequence of operations clearly. Always start with brackets, then orders, followed by division/multiplication, and finally addition/subtraction.

Worked Examples

Example 1: Simplify 5 + (8 x 2) - 6 / 3 Easy
Simplify the expression using the BODMAS rule.

Step 1: Solve inside the brackets first: (8 x 2) = 16.

Expression becomes: 5 + 16 - 6 / 3

Step 2: Next, perform division: 6 / 3 = 2.

Expression becomes: 5 + 16 - 2

Step 3: Now perform addition and subtraction from left to right:

5 + 16 = 21

21 - 2 = 19

Answer: 19

Example 2: Simplify (12 / 4 + 3²) x 2 Medium
Simplify the expression using BODMAS.

Step 1: Solve inside the brackets first.

Inside brackets: 12 / 4 + 3²

Step 2: Division first: 12 / 4 = 3.

Expression inside brackets becomes: 3 + 3²

Step 3: Calculate the exponent: 3² = 9.

Expression inside brackets becomes: 3 + 9 = 12.

Step 4: Now multiply by 2 outside the brackets: 12 x 2 = 24.

Answer: 24

Example 3: Simplify 18 - [6 + (2 x 3)²] / 3 Hard
Simplify the expression step-by-step using BODMAS.

Step 1: Start with the innermost brackets: (2 x 3) = 6.

Expression becomes: 18 - [6 + 6²] / 3

Step 2: Calculate the exponent: 6² = 36.

Expression becomes: 18 - [6 + 36] / 3

Step 3: Add inside the square brackets: 6 + 36 = 42.

Expression becomes: 18 - 42 / 3

Step 4: Perform division: 42 / 3 = 14.

Expression becomes: 18 - 14

Step 5: Perform subtraction: 18 - 14 = 4.

Answer: 4

Example 4: Calculate total cost: If 3 items cost Rs.250 each and a discount of 10% is applied on the total, find the amount payable. Medium
Use BODMAS to calculate the final amount after discount.

Step 1: Calculate total cost before discount: 3 x Rs.250 = Rs.750.

Step 2: Calculate discount amount: 10% of Rs.750.

Convert percentage to decimal: 10% = 10 / 100 = 0.10.

Discount = 0.10 x Rs.750 = Rs.75.

Step 3: Subtract discount from total cost: Rs.750 - Rs.75 = Rs.675.

Answer: The amount payable after discount is Rs.675.

Example 5: Evaluate 5 + 2 x (15 / (3 + 2))² Hard
Simplify the expression using BODMAS.

Step 1: Solve innermost brackets: (3 + 2) = 5.

Expression becomes: 5 + 2 x (15 / 5)²

Step 2: Perform division inside brackets: 15 / 5 = 3.

Expression becomes: 5 + 2 x 3²

Step 3: Calculate exponent: 3² = 9.

Expression becomes: 5 + 2 x 9

Step 4: Perform multiplication: 2 x 9 = 18.

Expression becomes: 5 + 18

Step 5: Perform addition: 5 + 18 = 23.

Answer: 23

BODMAS Rule

\[B \to O \to D \div M \to A \pm S\]

Order in which operations should be performed: Brackets, Orders (powers and roots), Division and Multiplication (left to right), Addition and Subtraction (left to right).

B = Brackets
O = Orders (exponents and roots)
D = Division
M = Multiplication
A = Addition
S = Subtraction

Percentage Calculation

\[\text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100\]

Used to calculate percentage values in problems involving discounts, profit, loss, etc.

Part = portion value
Whole = total value

Formula Bank

BODMAS Rule
\[ B \to O \to D \div M \to A \pm S \]
where: B = Brackets, O = Orders (exponents and roots), D = Division, M = Multiplication, A = Addition, S = Subtraction
Percentage Calculation
\[ \text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100 \]
where: Part = portion value, Whole = total value

Tips & Tricks

Tip: Always solve expressions inside brackets first, no matter how simple they look.

When to use: When simplifying any expression with brackets.

Tip: For division and multiplication, proceed from left to right as they have the same precedence.

When to use: When both division and multiplication appear in the same expression level.

Tip: Similarly, for addition and subtraction, solve from left to right.

When to use: When both addition and subtraction appear together.

Tip: Convert percentages to decimals before performing multiplication or division.

When to use: In problems involving percentage calculations.

Tip: Use estimation to quickly check if your answer is reasonable.

When to use: After solving complex expressions to verify correctness.

Common Mistakes to Avoid

❌ Ignoring the order of operations and solving from left to right without following BODMAS.
✓ Always follow BODMAS strictly to avoid incorrect answers.
Why: Students often rush and assume left-to-right is always correct.
❌ Treating multiplication as having higher precedence than division or vice versa.
✓ Remember division and multiplication have equal precedence; solve left to right.
Why: Misconception that multiplication always comes before division.
❌ Not simplifying expressions inside nested brackets first.
✓ Always simplify innermost brackets first before moving outward.
Why: Students overlook nested brackets complexity.
❌ Forgetting to apply percentage as a decimal in calculations.
✓ Convert percentage to decimal by dividing by 100 before multiplying.
Why: Confusion between percentage values and their decimal equivalents.
❌ Incorrectly handling exponents inside brackets.
✓ Apply exponents after simplifying the expression inside the brackets if applicable.
Why: Misunderstanding the order of operations involving powers.
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