PEMDAS vs BODMAS — What's the Difference and How to Use Them

Grade: 6-7 | Topic: Arithmetic

What You Will Learn

After this lesson you will know exactly what PEMDAS and BODMAS stand for, understand that they are the same rules with different names, master the left-to-right rule for operations with equal priority, and confidently evaluate any arithmetic expression using the correct order.

Theory

The Two Mnemonics Side by Side

PEMDAS and BODMAS are mnemonics — memory aids — for the same set of mathematical rules. Different countries simply use different words:

Priority	PEMDAS (North America)	BODMAS (UK, Australia, India)	What It Means
1st	P — Parentheses	B — Brackets	Evaluate grouped expressions first
2nd	E — Exponents	O — Orders	Powers and roots
3rd	M/D — Multiplication/Division	D/M — Division/Multiplication	Left to right, equal priority
4th	A/S — Addition/Subtraction	A/S — Addition/Subtraction	Left to right, equal priority

Other regional variations include BEDMAS (Brackets, Exponents, Division, Multiplication, Addition, Subtraction — used in Canada) and GEMS (Grouping, Exponents, Multiply/Divide, Subtract/Add).

They all describe the same four levels of priority. No matter which mnemonic you learned, your answers will be identical.

Why the Order of Letters Confuses Students

The biggest source of confusion: PEMDAS lists Multiplication before Division, while BODMAS lists Division before Multiplication. Students often think this means you should always multiply before dividing (PEMDAS) or always divide before multiplying (BODMAS).

Neither is correct. Multiplication and Division have equal priority. You always evaluate them left to right — whichever appears first in the expression gets done first.

The same rule applies to Addition and Subtraction: equal priority, left to right.

The Four Priority Levels in Detail

Level 1 — Parentheses / Brackets

Evaluate everything inside grouping symbols first. If there are nested groups (parentheses inside brackets), work from the innermost group outward.

$\text{Grouping symbols: } ( \;), \; [ \;], \; \{ \;\}$

Level 2 — Exponents / Orders

Calculate powers ($2^3 = 8$) and roots ($\sqrt{16} = 4$) next.

Level 3 — Multiplication and Division (left to right)

Scan from left to right. Perform each multiplication or division as you encounter it:

$12 \div 3 \times 2 = 4 \times 2 = 8 \quad \text{(divide first — it is on the left)}$

Level 4 — Addition and Subtraction (left to right)

Scan from left to right. Perform each addition or subtraction as you encounter it:

$10 - 4 + 3 = 6 + 3 = 9 \quad \text{(subtract first — it is on the left)}$

A Step-by-Step Process

For any expression, follow this checklist:

1. Simplify inside all parentheses/brackets (innermost first)
2. Evaluate all exponents/orders
3. Perform multiplication and division from left to right
4. Perform addition and subtraction from left to right

Worked Examples

Example 1: Simple PEMDAS/BODMAS (Easy)

Problem: Evaluate $6 + 3 \times 4$

Step 1 — Multiplication before addition (Level 3 before Level 4):

$3 \times 4 = 12$

Step 2 — Addition:

$6 + 12 = 18$

Answer: $\boxed{18}$

Example 2: Left-to-Right Rule for Division and Multiplication (Easy)

Problem: Evaluate $20 \div 4 \times 3$

Step 1: Division and multiplication have equal priority. Go left to right — division comes first:

$20 \div 4 = 5$

Step 2: Multiplication:

$5 \times 3 = 15$

Answer: $\boxed{15}$

If you multiplied first ($4 \times 3 = 12$, then $20 \div 12 \approx 1.67$), you would get the wrong answer. The left-to-right rule is critical.

Example 3: PEMDAS with Parentheses and Exponents (Medium)

Problem: Evaluate $(5 + 3)^2 - 4 \times 6$

Step 1 — Parentheses:

$5 + 3 = 8$

Expression becomes $8^2 - 4 \times 6$.

Step 2 — Exponents:

$8^2 = 64$

Expression becomes $64 - 4 \times 6$.

Step 3 — Multiplication:

$4 \times 6 = 24$

Expression becomes $64 - 24$.

Step 4 — Subtraction:

$64 - 24 = 40$

Answer: $\boxed{40}$

Example 4: Addition and Subtraction Left to Right (Medium)

Problem: Evaluate $15 - 8 + 3 - 2 + 6$

There are no parentheses, exponents, multiplication, or division. Just work left to right:

Step 1: $15 - 8 = 7$

Step 2: $7 + 3 = 10$

Step 3: $10 - 2 = 8$

Step 4: $8 + 6 = 14$

Answer: $\boxed{14}$

Example 5: Multi-Level Expression (Challenging)

Problem: Evaluate $2 + 3 \times (8 - 2)^2 \div 9 - 1$

Step 1 — Parentheses:

$8 - 2 = 6$

Expression becomes $2 + 3 \times 6^2 \div 9 - 1$.

Step 2 — Exponents:

$6^2 = 36$

Expression becomes $2 + 3 \times 36 \div 9 - 1$.

Step 3 — Multiplication and division left to right:

$3 \times 36 = 108$

$108 \div 9 = 12$

Expression becomes $2 + 12 - 1$.

Step 4 — Addition and subtraction left to right:

$2 + 12 = 14$

$14 - 1 = 13$

Answer: $\boxed{13}$

Common Mistakes

Mistake 1: Always multiplying before dividing because of the M-D order in PEMDAS

Expression: $18 \div 2 \times 3$

❌ Multiply first: $2 \times 3 = 6$, then $18 \div 6 = 3$.

✅ Left to right: $18 \div 2 = 9$, then $9 \times 3 = 27$.

Why this matters: The letters in PEMDAS do not mean multiplication always comes first. M and D share the same level. Always go left to right.

Mistake 2: Always dividing before multiplying because of the D-M order in BODMAS

Expression: $5 \times 4 \div 2$

❌ Divide first: $4 \div 2 = 2$, then $5 \times 2 = 10$.

✅ Left to right: $5 \times 4 = 20$, then $20 \div 2 = 10$.

Interestingly, in this case both approaches give 10 — but this is a coincidence. The correct method is always left to right, and for other expressions the wrong order will give wrong answers.

Why this matters: BODMAS does not mean "always divide before multiplying" — the same left-to-right rule applies.

Mistake 3: Forgetting PEMDAS rules apply inside parentheses too

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

❌ Inside parentheses, adding first: $(3 + 4) \times 5 = 35$, then $2 \times 35 = 70$.

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

Why this matters: Parentheses tell you to evaluate the group first, but within that group all PEMDAS/BODMAS rules still apply.

Practice Problems

Try these on your own before checking the answers:

1. $9 + 6 \times 2 - 3$
2. $30 \div 5 \times 2 + 1$
3. $(4 + 6) \div 2 + 3^2$
4. $100 - 3 \times (2^3 + 4) \div 6$
5. $8 \div 4 \div 2 + 3 \times 2$

Click to see answers

1. $9 + 12 - 3 = \mathbf{18}$ — multiply first, then left to right.
2. $6 \times 2 + 1 = 12 + 1 = \mathbf{13}$ — divide then multiply (left to right), then add.
3. $10 \div 2 + 9 = 5 + 9 = \mathbf{14}$ — parentheses, then exponent, then divide, then add.
4. $2^3 + 4 = 12$, then $3 \times 12 = 36$, then $36 \div 6 = 6$, finally $100 - 6 = \mathbf{94}$.
5. $8 \div 4 = 2$, then $2 \div 2 = 1$, then $3 \times 2 = 6$, finally $1 + 6 = \mathbf{7}$.

Summary

• PEMDAS and BODMAS are different mnemonics for the same four levels of priority.
• Level 1: Parentheses/Brackets. Level 2: Exponents/Orders. Level 3: Multiplication and Division (left to right). Level 4: Addition and Subtraction (left to right).
• Multiplication does NOT always come before division, and division does NOT always come before multiplication. They have equal priority — evaluate left to right.
• The same left-to-right rule applies to addition and subtraction.
• PEMDAS rules apply inside parentheses, not just outside them.

Related Topics

• Order of Operations (PEMDAS/BODMAS) — Rules and Examples
• Order of Operations with Fractions and Decimals
• Order of Operations Practice Problems with Answers

---

Need help with PEMDAS or BODMAS problems?

Take a photo of your math problem and MathPal will solve it step by step.

Open MathPal