Why Do We Need an Order of Operations?

Many mathematical expressions contain more than one operation.

For example:

$6+4\times5$6+4×5

Should we:

• Add first?
• Multiply first?

If everyone chose a different order, they would get different answers.

To avoid confusion, mathematicians around the world use a standard order of operations.

This ensures that everyone obtains the same answer.

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PEMDAS / BEDMAS

The order of operations is commonly remembered using:

PEMDAS

• P – Parentheses
• E – Exponents
• M – Multiplication
• D – Division
• A – Addition
• S – Subtraction

or

BEDMAS

• B – Brackets
• E – Exponents
• D – Division
• M – Multiplication
• A – Addition
• S – Subtraction

Both acronyms describe the same mathematical rules.

Important: Multiplication and division have equal priority. Work from left to right.

Addition and subtraction also have equal priority. Work from left to right.

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Step 1: Parentheses (Brackets)

Always evaluate expressions inside parentheses first.

Example

$8+(6-2)$8+(6−2)

Step 1:

$6-2=4$6−2=4

Step 2:

$8+4=12$8+4=12

Answer:

$\boxed{12}$12​

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Step 2: Exponents

Evaluate powers after parentheses.

Example

$4+3^2$4+32

Calculate the exponent first.

$3^2=9$32=9

Then:

$4+9=13$4+9=13

Answer:

$\boxed{13}$13​

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Step 3: Multiplication and Division

Perform multiplication and division from left to right.

Example

$24\div6\times2$24÷6×2

First:

$24\div6=4$24÷6=4

Then:

$4\times2=8$4×2=8

Answer:

$\boxed{8}$8​

Notice that multiplication is not always done before division.

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Step 4: Addition and Subtraction

Finally, perform addition and subtraction from left to right.

Example

$18-5+2$18−5+2

First:

$18-5=13$18−5=13

Then:

$13+2=15$13+2=15

Answer:

$\boxed{15}$15​

Addition is not always done before subtraction.

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Worked Examples

Example 1

Evaluate:

$7+3\times4$7+3×4

Multiply first:

$3\times4=12$3×4=12

Then:

$7+12=19$7+12=19

Answer:

$\boxed{19}$19​

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Example 2

Evaluate:

$(9-4)\times6$(9−4)×6

Parentheses first:

$9-4=5$9−4=5

Multiply:

$5\times6=30$5×6=30

Answer:

$\boxed{30}$30​

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Example 3

Evaluate:

$18\div3+7$18÷3+7

Division first:

$18\div3=6$18÷3=6

Then:

$6+7=13$6+7=13

Answer:

$\boxed{13}$13​

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Example 4

Evaluate:

$2^3+5$23+5

Exponent first:

$2^3=8$23=8

Then:

$8+5=13$8+5=13

Answer:

$\boxed{13}$13​

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Evaluating Algebraic Expressions

When an expression contains variables, first substitute the given values.

Example

Evaluate:

$3x+4$3x+4

when:

$x=5$x=5

Step 1:

Substitute:

$3(5)+4$3(5)+4

Step 2:

Multiply:

$15+4$15+4

Step 3:

Add:

$19$19

Answer:

$\boxed{19}$19​

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Another Example

Evaluate:

$2a^2-3$2a2−3

when:

$a=4$a=4

Step 1:

Substitute:

$2(4)^2-3$2(4)2−3

Step 2:

Exponent:

$4^2=16$42=16

Step 3:

Multiply:

$2\times16=32$2×16=32

Step 4:

Subtract:

$32-3=29$32−3=29

Answer:

$\boxed{29}$29​

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Multi-Step Calculations

Many expressions require several steps.

Example

$4+2(7-3)^2$4+2(7−3)2

Step 1:

Parentheses:

$7-3=4$7−3=4

Step 2:

Exponent:

$4^2=16$42=16

Step 3:

Multiplication:

$2\times16=32$2×16=32

Step 4:

Addition:

$4+32=36$4+32=36

Answer:

$\boxed{36}$36​

Working carefully through each step reduces mistakes.

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Common Mistakes

Mistake 1

Ignoring multiplication.

Incorrect:

$5+4\times2=(5+4)\times2=18$5+4×2=(5+4)×2=18

Correct:

$5+8=13$5+8=13

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Mistake 2

Ignoring parentheses.

Incorrect:

$10-3\times2=14$10−3×2=14

Correct:

$10-6=4$10−6=4

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Mistake 3

Forgetting exponents.

Incorrect:

$3+2^3=(3+2)^3=125$3+23=(3+2)3=125

Correct:

$3+8=11$3+8=11

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Mistake 4

Doing multiplication before division (or addition before subtraction) regardless of order.

Incorrect:

$24\div6\times2=2$24÷6×2=2

Correct:

$24\div6=4,\qquad4\times2=8$24÷6=4,4×2=8

Always work from left to right when operations have equal priority.

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Why the Order of Operations Matters

The order of operations allows mathematicians, scientists, engineers, and computer programmers to communicate calculations clearly.

It ensures that:

• Everyone follows the same rules.
• Expressions have only one correct value.
• Mathematical formulas work consistently.

Without these rules, mathematics would be unreliable.

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Real-World Applications

The order of operations is used in many fields.

Science

Calculating:

• Speed.
• Force.
• Energy.
• Chemical formulas.

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Engineering

Design calculations require accurate multi-step mathematics.

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Finance

Calculating:

• Interest.
• Taxes.
• Discounts.
• Investment returns.

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Computing

Computer programs follow the order of operations exactly when evaluating expressions.

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Worked Example

Evaluate:

$5+2(9-5)^2\div4$5+2(9−5)2÷4

Solution

Step 1:

Parentheses:

$9-5=4$9−5=4

Step 2:

Exponent:

$4^2=16$42=16

Step 3:

Multiply and divide from left to right:

$2\times16=32$2×16=32$32\div4=8$32÷4=8

Step 4:

Addition:

$5+8=13$5+8=13

Answer:

$\boxed{13}$13​

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Key Terms

Order of Operations — The agreed sequence for evaluating mathematical expressions.

Parentheses (Brackets) — Symbols used to group parts of an expression that should be evaluated first.

Exponent — A number indicating repeated multiplication.

Substitution — Replacing a variable with a known value.

Expression — A mathematical phrase containing numbers, variables, and operations.

Evaluate — To calculate the value of an expression.

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Key Takeaways

• The order of operations ensures that mathematical expressions are evaluated consistently.
• Follow PEMDAS or BEDMAS:
  1. Parentheses (Brackets)
  2. Exponents
  3. Multiplication and Division (left to right)
  4. Addition and Subtraction (left to right)
• Multiplication and division have equal priority, as do addition and subtraction.
• When evaluating algebraic expressions, substitute the given values before applying the order of operations.
• Solving multi-step calculations carefully helps avoid common mistakes and ensures accurate results.

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Suggested Images

Suggested placement:

1. After "PEMDAS / BEDMAS" – Colourful infographic illustrating the sequence of operations, emphasizing that multiplication/division and addition/subtraction are performed left to right.
2. After "Worked Examples" – Step-by-step visual solution showing each stage of evaluating a multi-operation numerical expression.
3. After "Evaluating Algebraic Expressions" – Diagram demonstrating how to substitute a value for a variable before applying the order of operations.
4. After "Common Mistakes" – Educational poster highlighting frequent errors (such as ignoring brackets or performing multiplication before division regardless of order) alongside the correct solutions.
5. After "Real-World Applications" – Images of a calculator, engineering calculations, and scientific formulas illustrating where the order of operations is used in everyday life and STEM fields.