PEMDAS: Order Of Operations Rules, Examples And Tricks

What Does PEMDAS Stand For?

PEMDAS is an acronym where each letter stands for one type of math operation, listed in the order you do them.

The most popular memory phrase in US classrooms is "Please Excuse My Dear Aunt Sally." Each word starts with the same letter as a step in PEMDAS.

The Order Of Operations Rules, Step By Step

The order of operations has four levels, not six. Multiplication and division share one level, and addition and subtraction share another. Within each shared level, you work from left to right.

Step 1: Parentheses And Grouping Symbols

Anything inside parentheses gets worked out first, no matter what operation it is. Parentheses are how you tell the reader, "do this part before anything else."

• 2 × (3 + 4) = 2 × 7 = 14
• Without parentheses: 2 × 3 + 4 = 6 + 4 = 10

Parentheses are not the only grouping symbols. All of these count as "P":

• Brackets [ ] and braces { }, used when parentheses are nested
• A fraction bar, which groups everything above it and everything below it
• Absolute value bars | |
• A square root sign, which groups everything underneath it

With nested symbols, start with the innermost pair and work outward:

2 × [3 + (5 – 1) × 2] = 2 × [3 + 4 × 2] = 2 × [3 + 8] = 2 × 11 = 22

Step 2: Exponents And Roots

After grouping symbols, work out exponents and roots. An exponent tells you how many times to multiply a number by itself, so 4² = 4 × 4 = 16.

Stacked exponents go from the top down. In 2³², the 3² part comes first:

• 2³² = 2⁹ = 512
• Not (2³)² = 8² = 64

Watch the negative sign. An exponent applies only to what sits directly under it:

• -3² = -(3 × 3) = -9
• (-3)² = (-3) × (-3) = 9

If you need to check a tricky power, an exponent calculator shows the result instantly.

Step 3: Multiplication And Division, Left To Right

Multiplication and division are equal partners. Do whichever comes first as you read from left to right.

Example: 12 ÷ 6 × 3 ÷ 2

1. 12 ÷ 6 = 2
2. 2 × 3 = 6
3. 6 ÷ 2 = 3

Doing the multiplication first would give a wrong answer.

Step 4: Addition And Subtraction, Left To Right

Addition and subtraction work the same way as multiplication and division. They share a level, and you go left to right.

Example: 20 – 4 + 3

• Left to right: 20 – 4 = 16, then 16 + 3 = 19 ✓
• Addition first (wrong): 4 + 3 = 7, then 20 – 7 = 13 ✗

A and S appear in order in the word, but they are equal in rank.

Why &Quot;Please Excuse My Dear Aunt Sally&Quot; Can Mislead You

The phrase helps you remember the letters, but it hides the fact that M and D, and A and S, are pairs. Many students read it as six separate steps.

Some teachers write it as P E (MD) (AS) or draw a box around MD and AS to show they belong together. Another way to think about it: division is multiplying by a fraction (÷ 4 is the same as × 1/4), and subtraction is adding a negative (- 3 is + (-3)). Seen that way, there are really only two operations at those levels.

PEMDAS Examples, From Easy To Hard

These eight examples show the order of operations in action. Try each one yourself before reading the steps.

Example 1: Multiplication Before Addition

3 + 6 × 2

1. Multiply: 6 × 2 = 12
2. Add: 3 + 12 = 15

Example 2: Parentheses Change Everything

(3 + 6) × 2

1. Parentheses: 3 + 6 = 9
2. Multiply: 9 × 2 = 18

Same numbers as Example 1, but the parentheses change the answer from 15 to 18.

Example 3: An Exponent And Division

4 + 8 ÷ 2² – 1

1. Exponent: 2² = 4
2. Divide: 8 ÷ 4 = 2
3. Add and subtract left to right: 4 + 2 – 1 = 5

Example 4: Every Level At Once

10 – 2 × 3 + 4² ÷ 8

1. Exponent: 4² = 16
2. Multiply and divide left to right: 2 × 3 = 6, then 16 ÷ 8 = 2
3. Now you have 10 – 6 + 2
4. Left to right: 10 – 6 = 4, then 4 + 2 = 6

Example 5: Parentheses With An Exponent Outside

3 × (2 + 4)² – 10 ÷ 5

1. Parentheses: 2 + 4 = 6
2. Exponent: 6² = 36
3. Multiply and divide: 3 × 36 = 108, and 10 ÷ 5 = 2
4. Subtract: 108 – 2 = 106

Example 6: Division Before Multiplication

50 – (12 + 18) ÷ 6 × 2

1. Parentheses: 12 + 18 = 30
2. Left to right: 30 ÷ 6 = 5, then 5 × 2 = 10
3. Subtract: 50 – 10 = 40

Example 7: A Fraction Bar As A Grouping Symbol

(8 + 4) / (2 × 3), written as a fraction with 8 + 4 on top and 2 × 3 on the bottom

1. Top: 8 + 4 = 12
2. Bottom: 2 × 3 = 6
3. Divide: 12 ÷ 6 = 2

When you type this into a calculator, you must add the parentheses yourself. Typing 8 + 4 ÷ 2 × 3 gives 14, not 2.

Example 8: The Viral Puzzle

56 ÷ 7 + 9 × 3 – 12

1. Multiply and divide left to right: 56 ÷ 7 = 8, then 9 × 3 = 27
2. Now you have 8 + 27 – 12
3. Left to right: 8 + 27 = 35, then 35 – 12 = 23

Why Order Of Operations Matters

The order of operations exists so that everyone gets the same answer from the same expression. Without a shared rule, 3 + 6 × 2 could mean 15 or 18, depending on who reads it.

It shows up in everyday math more than people expect:

• Shopping: 3 shirts at $12 plus $5 shipping is 3 × 12 + 5 = $41. Adding first would give 3 × 17 = $51.
• Recipes: doubling a recipe that needs 2 cups plus 1/2 cup is 2 × (2 + 1/2) = 5 cups. Without the parentheses, 2 × 2 + 1/2 = 4 1/2 cups.
• Formulas: the area of a circle, A = πr², means square the radius first, then multiply by π.

Every algebra and science formula assumes you know these rules.

PEMDAS Vs BODMAS, BIDMAS, BEDMAS And GEMDAS

PEMDAS, BODMAS, BIDMAS and BEDMAS all describe the same order of operations. They just use different words for the same steps.

Notice that BODMAS puts D before M, while PEMDAS puts M before D. That does not mean the rules differ. In both systems, multiplication and division share one level and go left to right. The order of the letters is just the order of the words.

"Orders" and "indices" are British terms for exponents. GEMDAS swaps "parentheses" for "grouping" to remind students that brackets, fraction bars and root signs count too.

The 8 ÷ 2(2 + 2) Debate: Why PEMDAS Goes Viral

Puzzles like 8 ÷ 2(2 + 2) spread online because people get two different answers: 16 and 1. The problem is not PEMDAS itself. It is that the expression is written in an unclear way.

The strict PEMDAS reading gives 16:

1. Parentheses: 2 + 2 = 4
2. Now it is 8 ÷ 2 × 4
3. Left to right: 8 ÷ 2 = 4, then 4 × 4 = 16

The other reading gives 1:

Some people, and some older textbooks, treat 2(4) as one tightly joined unit. This is called implied multiplication (or multiplication by juxtaposition). Under that view, 2(4) = 8 happens first, so 8 ÷ 8 = 1.

Under the standard rules taught in US schools, the answer is 16. A clear writer would avoid the problem and use 8 ÷ (2(2 + 2)) if they meant 1, or (8 ÷ 2)(2 + 2) if they meant 16.

The lesson is simple. If an expression could be read two ways, add parentheses.

Order Of Operations In Calculators, Excel And Code

Most calculators and programs follow PEMDAS, but a few have quirks that matter when you type in formulas.

The Excel rule surprises many people. In math class, -3² = -9. In Excel, the minus sign is applied first, so =-3^2 returns 9. Microsoft lists this in its Excel operator precedence guide. To get -9 in Excel, type =-(3^2).

For homework, use a scientific calculator that respects the order of operations, and add parentheses whenever you are unsure.

When Students Learn The Order Of Operations

In schools that follow the Common Core State Standards, students start using grouping symbols in grade 5 and learn the full order of operations with exponents in grade 6.

Algebra depends on these rules, which is why an algebra calculator shows each step in PEMDAS order.

Common PEMDAS Mistakes To Avoid

Most PEMDAS mistakes come from treating the letters as six separate steps or from ignoring hidden grouping. This table shows the ones that cost the most points.

A good habit: rewrite the whole expression after each step. One line per step makes mistakes easy to spot.

Practice Problems With Answers

Try these six problems, then check below.

1. 7 + 3 × 4
2. 18 ÷ 3 × 2 + 4² – (6 + 1)
3. 5 + 2 × 3² – (8 – 4) ÷ 2
4. (15 – 3) ÷ 4 + 2³ × 2
5. 100 ÷ (5 × 2²) + 3 × 7
6. 9 – 3 ÷ (1/3) + 1

Answers

1. 19. 3 × 4 = 12, then 7 + 12.
2. 21. Parentheses: 7. Exponent: 16. Left to right: 18 ÷ 3 = 6, 6 × 2 = 12. Then 12 + 16 – 7 = 21.
3. 21. Exponent: 9. Parentheses: 4. Then 2 × 9 = 18 and 4 ÷ 2 = 2. Finally 5 + 18 – 2 = 21.
4. 19. 12 ÷ 4 = 3, and 2³ × 2 = 8 × 2 = 16. Then 3 + 16 = 19.
5. 26. 2² = 4, 5 × 4 = 20, 100 ÷ 20 = 5, 3 × 7 = 21. Then 5 + 21 = 26.
6. 1. Dividing by 1/3 is the same as multiplying by 3, so 3 ÷ (1/3) = 9. Then 9 – 9 + 1 = 1. A fraction calculator can confirm fraction steps like this one.

Summary

PEMDAS is the US memory aid for the order of operations: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. The key rule is that multiplication and division share a level, as do addition and subtraction, and each pair is worked from left to right.

Grouping symbols include brackets, fraction bars and root signs. Exponents apply only to what sits directly under them, so -3² = -9. BODMAS, BIDMAS and BEDMAS follow the same rules with different names.

Next step: work through the six practice problems above, writing one line per step. Then try writing the viral puzzle 8 ÷ 2(2 + 2) in two clear ways, one that equals 16 and one that equals 1.

Frequently Asked Questions