Order of Operations

When a math expression has more than one operation, you cannot simply work left to right. There is a specific order of operations that everyone follows so that every person gets the same answer from the same expression. In the United States, this order is remembered with the acronym PEMDAS.

The PEMDAS Rules

Critical detail: Multiplication and division have the same priority — you do whichever comes first as you read left to right. The same is true for addition and subtraction. A common mistake is thinking multiplication always comes before division; it does not.

Some people use the phrase “Please Excuse My Dear Aunt Sally” to remember the order.

Outside the U.S., the same rules go by BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction) or BEDMAS. The rules themselves are identical.

Step-by-Step Worked Examples

Example 1: $8 + 2 \times 5$

There are no parentheses or exponents, so start with multiplication:

$8 + 2 \times 5 = 8 + 10 = 18$

Common mistake: Adding $8 + 2$ first to get $10 \times 5 = 50$. That is incorrect because multiplication must be done before addition.

Answer: 18

Example 2: $(6 + 4)^2 \div 5 - 3$

Step 1 — Parentheses:

$(6 + 4)^2 \div 5 - 3 = (10)^2 \div 5 - 3$

Step 2 — Exponents:

$10^2 \div 5 - 3 = 100 \div 5 - 3$

Step 3 — Division (left to right):

$100 \div 5 - 3 = 20 - 3$

Step 4 — Subtraction:

$20 - 3 = 17$

Answer: 17

Example 3: $3 + 6 \times (5 + 4) \div 3 - 7$

Step 1 — Parentheses:

$3 + 6 \times 9 \div 3 - 7$

Step 2 — Multiplication and division, left to right:

$3 + 54 \div 3 - 7 = 3 + 18 - 7$

Step 3 — Addition and subtraction, left to right:

$3 + 18 - 7 = 21 - 7 = 14$

Answer: 14

Nested Parentheses

When parentheses are nested inside other parentheses, work from the innermost set outward.

Example 4: $2 \times [3 + (8 - 5)^2]$

Step 1 — Innermost parentheses:

$2 \times [3 + (3)^2]$

Step 2 — Exponent inside the brackets:

$2 \times [3 + 9]$

Step 3 — Brackets:

$2 \times 12$

Step 4 — Multiplication:

$24$

Answer: 24

Common Mistakes to Avoid

1. Multiplying before dividing regardless of position. Multiplication and division are equal in priority — always work left to right.

   $24 \div 6 \times 2 = 4 \times 2 = 8 \quad \checkmark$ $24 \div 6 \times 2 \neq 24 \div 12 = 2 \quad \text{(wrong)}$

2. Ignoring parentheses. Parentheses override all other rules. Always resolve them first.

3. Applying exponents to a sum without parentheses. Note the difference:

   $3 + 4^2 = 3 + 16 = 19$ $(3 + 4)^2 = 7^2 = 49$

   Without parentheses, only the 4 is squared.

4. Distributing a negative sign incorrectly. When a subtraction precedes parentheses, be careful:

   $10 - (3 + 2) = 10 - 5 = 5$

Real-World Application: Electrician — Total Resistance in a Circuit

An electrician is calculating the total resistance in a circuit that has a series section and a parallel section. The formula for total resistance when a 10 ohm resistor is in series with two parallel resistors of 6 ohms and 12 ohms is:

$R_{\text{total}} = R_{\text{series}} + \frac{R_1 \times R_2}{R_1 + R_2}$

Substituting the values:

$R_{\text{total}} = 10 + \frac{6 \times 12}{6 + 12}$

Step 1 — Parentheses (denominator):

$10 + \frac{6 \times 12}{18}$

Step 2 — Multiplication (numerator):

$10 + \frac{72}{18}$

Step 3 — Division:

$10 + 4$

Step 4 — Addition:

$14$

Answer: The total circuit resistance is 14 ohms. Getting the order wrong here could mean selecting the wrong wire gauge or breaker, which is a serious safety issue.

Quick Reference: PEMDAS at a Glance

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• PEMDAS defines the universal order: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right)
• Multiplication and division share the same priority — always work left to right, not multiplication first
• The same is true for addition and subtraction — left to right
• Parentheses override everything — when in doubt, use them to make your intent clear
• In trades like electrical work, applying the correct order of operations is not just a math exercise — it affects safety and code compliance

Return to Arithmetic for more foundational topics.