Order of Operations (PEMDAS/BODMAS) — Rules and Examples

Grade: 6-7 | Topic: Arithmetic

What You Will Learn

By the end of this guide you will know exactly which calculation to do first in any math expression, no matter how many operations it contains. You will understand the PEMDAS and BODMAS mnemonics, work through step-by-step examples ranging from simple to challenging, and avoid the most common mistakes students make when evaluating expressions.

Theory

Why Order of Operations Matters

Consider the expression $3 + 4 \times 2$. If you go left to right you get $14$. If you multiply first you get $11$. Without a universal set of rules, the same expression would give different answers. The order of operations is the agreed-upon convention that ensures every person evaluating an expression arrives at the same result.

The PEMDAS Rule

PEMDAS is the most widely used mnemonic in North America. Each letter stands for one level of priority:

Step	Letter	Meaning	Examples
1	P	Parentheses (and other grouping symbols)	$(\ ),\ [\ ],\ \{\ \}$
2	E	Exponents (powers and roots)	$2^{3},\ \sqrt{9}$
3	M / D	Multiplication and Division (left to right)	$\times,\ \div$
4	A / S	Addition and Subtraction (left to right)	$+,\ -$

The full evaluation sequence:

$\text{Parentheses} \;\rightarrow\; \text{Exponents} \;\rightarrow\; \text{Multiplication / Division} \;\rightarrow\; \text{Addition / Subtraction}$

Critical detail: Multiplication and Division share the same priority. You do not always multiply before you divide. Instead, evaluate whichever comes first when reading left to right. The same rule applies to Addition and Subtraction.

BODMAS — The Same Rules, Different Name

In the UK, Australia, India, and many other countries the mnemonic is BODMAS:

• B — Brackets (same as Parentheses)
• O — Orders (same as Exponents)
• D / M — Division and Multiplication (left to right)
• A / S — Addition and Subtraction (left to right)

PEMDAS and BODMAS produce identical results for every expression. The only difference is the name. Throughout this guide we use PEMDAS, but every rule applies equally to BODMAS.

Nested Parentheses

When an expression has parentheses inside parentheses, work from the innermost set outward:

$5 \times [3 + (2 + 1)^{2}]$

1. Innermost parentheses: $(2 + 1) = 3$
2. Exponent: $3^{2} = 9$
3. Brackets: $3 + 9 = 12$
4. Multiplication: $5 \times 12 = 60$

Worked Examples

Example 1: Basic PEMDAS (Easy)

Problem: Evaluate $8 + 2 \times 5$

Step 1 — Multiplication first (M comes before A): $2 \times 5 = 10$

Step 2 — Addition: $8 + 10 = 18$

Answer: $\boxed{18}$

Example 2: Parentheses Change Everything (Easy)

Problem: Evaluate $(8 + 2) \times 5$

Step 1 — Parentheses first: $8 + 2 = 10$

Step 2 — Multiplication: $10 \times 5 = 50$

Answer: $\boxed{50}$

Notice how the parentheses changed the result from $18$ to $50$ compared to Example 1.

Example 3: All Four Steps (Medium)

Problem: Evaluate $3 + 2^{3} \times (10 - 6) \div 4$

Step 1 — Parentheses: $10 - 6 = 4$

The expression becomes $3 + 2^{3} \times 4 \div 4$.

Step 2 — Exponents: $2^{3} = 8$

The expression becomes $3 + 8 \times 4 \div 4$.

Step 3 — Multiplication and Division left to right: $8 \times 4 = 32$ $32 \div 4 = 8$

The expression becomes $3 + 8$.

Step 4 — Addition: $3 + 8 = 11$

Answer: $\boxed{11}$

Example 4: Left-to-Right Trap (Medium)

Problem: Evaluate $24 \div 6 \times 2$

Students often get this wrong because they think multiplication comes before division. Remember: M and D have equal priority — go left to right.

Step 1 — Division first (it is on the left): $24 \div 6 = 4$

Step 2 — Multiplication: $4 \times 2 = 8$

Answer: $\boxed{8}$

If you had multiplied first ($6 \times 2 = 12$, then $24 \div 12 = 2$), you would get the wrong answer.

Example 5: Nested Grouping Symbols (Challenging)

Problem: Evaluate $2 \times [3 + (4^{2} - 10)] - 5$

Step 1 — Innermost parentheses — exponent first inside them: $4^{2} = 16$

Step 2 — Continue inside parentheses: $16 - 10 = 6$

The expression becomes $2 \times [3 + 6] - 5$.

Step 3 — Brackets: $3 + 6 = 9$

The expression becomes $2 \times 9 - 5$.

Step 4 — Multiplication before subtraction: $2 \times 9 = 18$

Step 5 — Subtraction: $18 - 5 = 13$

Answer: $\boxed{13}$

Common Mistakes

Mistake 1: Multiplying before dividing regardless of position

Consider $12 \div 3 \times 2$:

❌ $12 \div 6 = 2$ (multiplied $3 \times 2$ first)

✅ $4 \times 2 = 8$ (divided $12 \div 3$ first, because it is further left)

Why this matters: Multiplication and division have equal priority. Always read left to right.

Mistake 2: Ignoring parentheses around negative numbers

Consider $(-3)^{2}$ vs. $-3^{2}$:

❌ Assuming $-3^{2} = 9$

✅ $-3^{2} = -(3^{2}) = -9$, while $(-3)^{2} = 9$

Why this matters: Without parentheses, the exponent applies only to $3$, and the negative sign is treated as subtraction.

Mistake 3: Forgetting to apply PEMDAS inside parentheses

Consider $2 \times (3 + 4 \times 5)$:

❌ $(3 + 4) \times 5 = 35$, then $2 \times 35 = 70$

✅ Inside the parentheses, multiply first: $4 \times 5 = 20$, then $3 + 20 = 23$, then $2 \times 23 = 46$

Why this matters: PEMDAS rules still apply inside grouping symbols — parentheses only tell you to evaluate that group first, not to go left to right within it.

Practice Problems

Try these on your own before checking the answers:

1. $7 + 3 \times 4 - 2$
2. $(7 + 3) \times (4 - 2)$
3. $5^{2} - 3 \times 4 + 8 \div 2$
4. $48 \div 8 \times 3 - 2^{2}$
5. $[2 + 3 \times (6 - 2)^{2}] \div 5 - 1$

Click to see answers

1. $7 + 12 - 2 = \mathbf{17}$ — multiply first, then left to right.
2. $10 \times 2 = \mathbf{20}$ — parentheses first, then multiply.
3. $25 - 12 + 4 = \mathbf{17}$ — exponent, then multiply and divide, then left to right.
4. $6 \times 3 - 4 = 18 - 4 = \mathbf{14}$ — divide first (left to right), multiply, exponent, then subtract.
5. Inside parentheses: $(6-2)^{2} = 16$, then $3 \times 16 = 48$, then $2 + 48 = 50$. Finally $50 \div 5 - 1 = 10 - 1 = \mathbf{9}$.

Summary

• The order of operations ensures every person gets the same answer: P-E-M/D-A/S (or B-O-D/M-A/S).
• Evaluate Parentheses (innermost first), then Exponents, then Multiplication and Division left to right, then Addition and Subtraction left to right.
• Multiplication and Division share equal priority — do not always multiply first.
• The same PEMDAS rules apply inside parentheses, not just outside them.

Related Topics

• PEMDAS vs BODMAS — What's the Difference and How to Use Them
• Order of Operations with Fractions and Decimals
• Order of Operations Practice Problems with Answers
• Exponents and Powers — Rules, Examples, and Practice
• Fractions — Complete Guide

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