Order of Operations

The order of operations is the set of rules that tells you which part of a math problem to do first. Work inside parentheses first. Then do exponents. Next, multiply and divide from left to right. Last, add and subtract from left to right. Many students remember it as PEMDAS.

Before this lesson: You should know how to multiply and divide integers, add and subtract integers, and work with exponents.

Why We Need an Order of Operations

Look at 7 + 3 \times 4. If you add first, you get 10 \times 4 = 40. If you multiply first, you get 7 + 12 = 19. Both can’t be right.

The order of operations is an agreement. Everyone does the steps in the same order, so everyone gets the same answer. The correct answer here is 19.

The Order of Operations Rule (PEMDAS)

1. Parentheses: do the work inside grouping symbols first. This includes ( ), [ ] and the fraction bar.
2. Exponents: find any powers, like 3^2.
3. Multiplication and Division: do these in order from left to right.
4. Addition and Subtraction: do these in order from left to right.

Multiplication does not always come before division. They share a step, so you go left to right. The same is true for addition and subtraction.

A fraction bar works like parentheses. Simplify the whole top and the whole bottom before you divide.

Worked Examples

Example 1: Multiply before you add

Evaluate 7 + 3 \times 4.

Step 1. There are no parentheses or exponents.

Step 2. Multiply: 3 \times 4 = 12. Now you have 7 + 12.

Step 3. Add: 7 + 12 = 19.

Answer: 19

Example 2: Divide and multiply from left to right

Evaluate 20 \div 4 \times 5.

Step 1. Division and multiplication share a step. Go left to right, so divide first: 20 \div 4 = 5.

Step 2. Multiply: 5 \times 5 = 25.

Answer: 25. If you multiplied first, you would get 1, which is wrong.

Example 3: Parentheses and exponents

Evaluate 2 \times (3 + 5)^2 - 10.

Step 1. Parentheses: 3 + 5 = 8. Now you have 2 \times 8^2 - 10.

Step 2. Exponents: 8^2 = 64. Now you have 2 \times 64 - 10.

Step 3. Multiply: 2 \times 64 = 128.

Step 4. Subtract: 128 - 10 = 118.

Answer: 118

Example 4: Grouping symbols inside grouping symbols

Evaluate 18 - 3 \times [4 + (5 - 3)^2] \div 6.

Step 1. Start with the innermost parentheses: 5 - 3 = 2.

Step 2. Inside the brackets, do the exponent, then add: 4 + 2^2 = 4 + 4 = 8. Now you have 18 - 3 \times 8 \div 6.

Step 3. Multiply and divide from left to right: 3 \times 8 = 24, then 24 \div 6 = 4.

Step 4. Subtract: 18 - 4 = 14.

Answer: 14

Example 5: Word problem with movie tickets

Maya buys 3 adult tickets for $12 each and 2 child tickets for $8 each. She uses a $5 coupon. Write one expression for the total, and find it.

Step 1. Write the expression: 3 \times 12 + 2 \times 8 - 5.

Step 2. Multiply first: 3 \times 12 = 36 and 2 \times 8 = 16. Now you have 36 + 16 - 5.

Step 3. Add and subtract from left to right: 36 + 16 = 52, then 52 - 5 = 47.

Answer: Maya pays $47.

Common Mistakes with the Order of Operations

Mistake	Example of the error	Fix
Adding before multiplying	7 + 3 \times 4 = 10 \times 4 = 40	Multiply first: 7 + 12 = 19.
Always multiplying before dividing	20 \div 4 \times 5 = 20 \div 20 = 1	Go left to right: 5 \times 5 = 25.
Multiplying before the exponent	2 \times 3^2 = 6^2 = 36	Do the exponent first: 2 \times 9 = 18.
Always adding before subtracting	10 - 3 + 2 = 10 - 5 = 5	Go left to right: 7 + 2 = 9.
Ignoring the fraction bar	\frac{12 + 6}{5 - 2} treated as 12 + 6 \div 5 - 2	Simplify the top and bottom first: \frac{18}{3} = 6.

PEMDAS, BODMAS and GEMS

Different countries use different memory tricks. They all describe the same rules.

Trick	Stands for	Where it is common
PEMDAS	Parentheses, Exponents, Multiplication and Division, Addition and Subtraction	United States
BODMAS	Brackets, Orders, Division and Multiplication, Addition and Subtraction	UK, India, Australia
BEDMAS	Brackets, Exponents, Division and Multiplication, Addition and Subtraction	Canada
GEMS	Grouping symbols, Exponents, Multiplication and Division, Subtraction and Addition	Some US schools

GEMS is helpful because it reminds you that brackets and fraction bars count as grouping symbols, not just parentheses.

Practice Problems

1. Evaluate 6 + 4 \times 2.
2. Evaluate 18 - 12 \div 3.
3. Evaluate (6 + 4) \times 2.
4. Evaluate 24 \div 6 \times 2.
5. Evaluate 5 + 3^2.
6. Evaluate 2 \times (7 - 4)^2.
7. Evaluate \frac{15 + 9}{2 \times 3}.
8. Evaluate 10 - 2 \times [3 + (8 - 6)].
9. Evaluate -8 + 6 \div (-2).
10. Jon buys 4 notebooks for $3 each and one pack of pens for $5. Write an expression for the total cost and evaluate it.

Show answers

1. 6 + 8 = 14
2. 18 - 4 = 14
3. 10 \times 2 = 20
4. Left to right: 4 \times 2 = 8
5. 5 + 9 = 14
6. 2 \times 3^2 = 2 \times 9 = 18
7. Top: 24. Bottom: 6. \frac{24}{6} = 4
8. 10 - 2 \times [3 + 2] = 10 - 2 \times 5 = 10 - 10 = 0
9. Divide first: 6 \div (-2) = -3. Then -8 + (-3) = -11.
10. 4 \times 3 + 5 = 12 + 5 = 17. The total is $17.

Order of Operations FAQ

What does PEMDAS stand for?

PEMDAS stands for Parentheses, Exponents, Multiplication, Division, Addition and Subtraction. A common way to remember it is “Please Excuse My Dear Aunt Sally.”

Do you multiply or divide first?

Neither one always comes first. Multiplication and division share a step, so you do whichever comes first from left to right.

Is BODMAS the same as PEMDAS?

Yes. BODMAS uses different words, like “brackets” for parentheses and “orders” for exponents. Both give the same answer every time.

What is 8 ÷ 2(2 + 2)?

By the usual rules, the answer is 16: 8 \div 2 \times 4 = 4 \times 4 = 16. Some people read it as 1, so it is better to write clear parentheses and avoid this kind of problem.

Summary

• The order of operations tells you which step to do first.
• Do grouping symbols first, then exponents.
• Multiply and divide from left to right, then add and subtract from left to right.
• A fraction bar groups the top and the bottom.
• PEMDAS, BODMAS, BEDMAS and GEMS all describe the same rules.

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