Master the Rules for Solving Mathematical Expressions
The Order of Operations is a set of rules that tells us which calculations to perform first when solving mathematical expressions with multiple operations. Without these rules, different people would get different answers to the same problem!
(Used in USA)
"Please Excuse My Dear Aunt Sally"
(Used in UK, India, Australia)
Both methods give the same result!
Multiplication and Division have the same priority — work from
LEFT to RIGHT.
Addition and Subtraction have the same priority — work from
LEFT to RIGHT.
❌ Common Mistake: Don't always do multiplication before division!
❌ Common Mistake: Don't always do addition before subtraction!
Solve everything inside ( ), [ ], and { } from innermost to outermost.
Example: \( (3 + 5) \times 2 = 8 \times 2 = 16 \)
Calculate powers, square roots, and other exponents.
Example: \( 3^2 + 4 = 9 + 4 = 13 \)
Perform whichever comes first as you read from left to right.
Example: \( 12 \div 4 \times 3 = 3 \times 3 = 9 \)
Perform whichever comes first as you read from left to right.
Example: \( 10 - 3 + 5 = 7 + 5 = 12 \)
Solve: \( 6 \times 4 + 8 \div 2 \)
Step 1: Multiply and divide first (left to right)
\( 6 \times 4 = 24 \)
Now we have: \( 24 + 8 \div 2 \)
\( 8 \div 2 = 4 \)
Now we have: \( 24 + 4 \)
Step 2: Add
\( 24 + 4 = 28 \)
✓ Answer: 28
Solve: \( (8 + 5) - 3 \times 2^2 \)
Step 1: Parentheses first
\( (8 + 5) = 13 \)
Now we have: \( 13 - 3 \times 2^2 \)
Step 2: Exponents
\( 2^2 = 4 \)
Now we have: \( 13 - 3 \times 4 \)
Step 3: Multiplication
\( 3 \times 4 = 12 \)
Now we have: \( 13 - 12 \)
Step 4: Subtraction
\( 13 - 12 = 1 \)
✓ Answer: 1
Solve: \( 18 \div 9 \times 2 \)
⚠️ Many students get this wrong!
Step 1: Work left to right (division comes first)
\( 18 \div 9 = 2 \)
Now we have: \( 2 \times 2 \)
Step 2: Multiply
\( 2 \times 2 = 4 \)
✓ Answer: 4 (NOT 1!)
❌ If you did multiplication first: \( 18 \div (9 \times 2) = 18 \div 18 = 1 \) — This is WRONG!
Solve: \( 3 + 6 \times (5 + 4) \div 3 - 7 \)
Step 1: Parentheses
\( (5 + 4) = 9 \)
Now: \( 3 + 6 \times 9 \div 3 - 7 \)
Step 2: Multiplication and Division (left to right)
\( 6 \times 9 = 54 \)
Now: \( 3 + 54 \div 3 - 7 \)
\( 54 \div 3 = 18 \)
Now: \( 3 + 18 - 7 \)
Step 3: Addition and Subtraction (left to right)
\( 3 + 18 = 21 \)
\( 21 - 7 = 14 \)
✓ Answer: 14
Wrong Thinking: "PEMDAS says M comes before D, so multiply first."
Correct Rule: Multiplication and Division have equal priority. Work left to right!
Example: \( 8 \div 2 \times 4 \)
❌ Wrong: \( 8 \div (2 \times 4) = 8 \div 8 = 1 \)
✓ Correct: \( (8 \div 2) \times 4 = 4 \times 4 = 16 \)
Wrong Thinking: "A comes before S in PEMDAS."
Correct Rule: Addition and Subtraction have equal priority. Work left to right!
Example: \( 10 - 3 + 5 \)
❌ Wrong: \( 10 - (3 + 5) = 10 - 8 = 2 \)
✓ Correct: \( (10 - 3) + 5 = 7 + 5 = 12 \)
Correct Rule: Always work from the innermost parentheses outward.
Example: \( 5 + [3 \times (2 + 4)] \)
✓ Step 1: \( (2 + 4) = 6 \) → \( 5 + [3 \times 6] \)
✓ Step 2: \( [3 \times 6] = 18 \) → \( 5 + 18 \)
✓ Step 3: \( 5 + 18 = 23 \)
Correct Rule: PEMDAS/BODMAS applies inside parentheses too!
Example: \( (3 + 2 \times 4) \)
❌ Wrong: \( (5 \times 4) = 20 \)
✓ Correct: \( (3 + 8) = 11 \) (multiply before adding, even inside parentheses!)
PEMDAS: "Please Excuse My Dear Aunt Sally"
BODMAS: "Big Oranges Don't Make Any Sense"
GEMS: "Grouping, Exponents, Multiply/Divide, Add/Subtract"
If you're unsure about the order, use parentheses to show what should be calculated first. This makes your work clearer and helps avoid mistakes.
Example: \( 2 + 3 \times 4 \) can be written as \( 2 + (3 \times 4) \) to clarify.
As you work through a problem, underline or highlight the operation you're about to perform. This helps you track your progress and catch errors.
Remember that MD and AS are pairs that work together left to right:
Parentheses → Exponents → (MD) Left to Right →
(AS) Left to Right
Most scientific calculators follow PEMDAS/BODMAS. Use them to verify your answer, but always work through the problem by hand first to understand the process.
Sometimes it helps to rewrite \( a \div b \) as \( a \times \frac{1}{b} \). This can make complex expressions easier to understand.
Example: \( 8 \div 2 \times 4 = 8 \times \frac{1}{2} \times 4 = 16 \)
Try these problems on your own, then click to reveal the solutions!
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| Order | Operation | Symbol/Example | Direction |
|---|---|---|---|
| 1st | Parentheses/Brackets | \( ( ) [ ] \{ \} \) | Innermost first |
| 2nd | Exponents/Orders | \( x^2, \sqrt{x}, x^n \) | Top to bottom |
| 3rd | Multiplication & Division | \( \times \div \) | Left to Right |
| 4th | Addition & Subtraction | \( + - \) | Left to Right |
Parentheses → Exponents → Multiply/Divide (Left to Right) → Add/Subtract (Left to Right)
The order of operations ensures everyone solves math problems the same way and gets the same answer. Master this, and you'll avoid 90% of common calculation errors!
PEMDAS is an acronym used in the United States to remember the order of operations in mathematics. It stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. The mnemonic "Please Excuse My Dear Aunt Sally" helps students remember this sequence. PEMDAS ensures that mathematical expressions are solved consistently by everyone.
BODMAS stands for Brackets, Orders, Division, Multiplication, Addition, and Subtraction. It is used in the UK, India, Australia, and other countries. While PEMDAS and BODMAS use different terminology (Parentheses vs. Brackets, Exponents vs. Orders), they represent the same mathematical rules and produce identical results. The order of Multiplication/Division and Addition/Subtraction is interchangeable in both systems.
No! This is one of the most common misconceptions. Multiplication and division have equal priority. You perform whichever operation comes first when reading the expression from left to right. For example, in \(12 \div 4 \times 3\), you first divide (12 ÷ 4 = 3), then multiply (3 × 3 = 9), not the other way around.
No! Similar to multiplication and division, addition and subtraction have equal priority. You perform these operations from left to right in the order they appear. For example, in \(10 - 5 + 3\), you first subtract (10 - 5 = 5), then add (5 + 3 = 8).
When you have nested parentheses like \([5 + (3 \times 2)]\), always start with the innermost brackets first and work your way outward. First solve \((3 \times 2) = 6\), then solve \([5 + 6] = 11\). Different bracket types [ ], { }, ( ) all serve the same purpose and should be handled from inside out.
"Orders" in BODMAS refers to powers (exponents), square roots, and other indices. It includes any operation where a number is raised to a power, such as \(2^3\), \(5^2\), or \(\sqrt{16}\). This is equivalent to the "Exponents" in PEMDAS. Orders are calculated after brackets but before multiplication and division.
The order of operations is essential in many real-world applications: programming and coding (computers follow PEMDAS), financial calculations (interest rates, loan payments), engineering and science (formulas and equations), cooking and recipes (scaling ingredients), and everyday problem-solving. Without standard rules, the same calculation would give different results to different people.
Scientific calculators and most modern calculators follow PEMDAS/BODMAS rules automatically. However, basic 4-function calculators may not—they often calculate in the order you enter the numbers. When in doubt, use parentheses to ensure your calculator performs operations in the correct order, or verify by calculating manually.
This is a famous viral math problem! Following strict PEMDAS: First, solve parentheses: \((2 + 2) = 4\). Then work left to right for multiplication/division: \(8 \div 2 = 4\), then \(4 \times 4 = 16\). So the answer is 16. Some argue for 1, interpreting 2(4) as a single term, but standard mathematical notation gives 16.
Yes! The order of operations applies inside parentheses as well. For example, in \((3 + 2 \times 4)\), you must multiply before adding inside the parentheses: \(2 \times 4 = 8\), then \(3 + 8 = 11\). Don't make the mistake of thinking parentheses means "just add everything inside."
GEMDAS stands for Grouping, Exponents, Multiplication, Division, Addition, and Subtraction. The "G" for Grouping is sometimes preferred because it includes all grouping symbols (parentheses, brackets, braces, and even the fraction bar or radical sign), not just parentheses. GEMDAS follows the same rules as PEMDAS but uses more inclusive terminology.
Be careful with negative bases! \((-3)^2 = 9\) (the negative is inside the parentheses, so the whole base is squared), but \(-3^2 = -9\) (only 3 is squared, then negated). The placement of parentheses is crucial. When in doubt, use parentheses to clearly show what you mean.
In most educational systems, students begin learning the order of operations in 4th or 5th grade (ages 9-11). Basic concepts like "multiply before add" may be introduced earlier, while more complex applications with exponents and nested parentheses are typically covered in 6th through 8th grade. The concept is reinforced throughout high school mathematics.
A fraction bar acts as a grouping symbol (like parentheses). Solve the numerator completely, solve the denominator completely, then divide. For example, in \(\frac{6 + 2}{4 - 2}\), first calculate the top (6 + 2 = 8), then the bottom (4 - 2 = 2), then divide (8 ÷ 2 = 4). The fraction bar has the highest priority.
Yes! Think of PEMDAS as having only 4 levels, not 6: P → E → (M/D) → (A/S). Multiplication and Division share one level (go left to right). Addition and Subtraction share another level (go left to right). Another tip: think of subtraction as "adding a negative" and division as "multiplying by a fraction" to avoid confusion.
