Training Whole Numbers Order of Operations (PEMDAS)
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Order of Operations (PEMDAS)

20 min Whole Numbers

Order of Operations

When a mathematical expression contains more than one operation, the order in which you perform them matters enormously. The expression 2 + 3 × 4 equals 14, not 20, because multiplication is performed before addition. The rules that govern this ordering are called the order of operations, commonly remembered by the acronym PEMDAS.

PEMDAS stands for Parentheses, Exponents, Multiplication and Division (left to right), and Addition and Subtraction (left to right). These rules are not arbitrary conventions — they ensure that every person who reads a mathematical expression arrives at the same answer.

In this lesson you will practice applying PEMDAS to increasingly complex expressions, building the precision and discipline that every future math course will demand.

When an expression contains multiple operations, PEMDAS gives the universally-agreed evaluation order.

PEMDAS
LetterOperationNotes
PParentheses (grouping symbols)Innermost first: $( \;)$, $[ \;]$, $\{ \;\}$
EExponentsPowers and roots
M / DMultiplication / DivisionEqual priority — left to right
A / SAddition / SubtractionEqual priority — left to right
Common Mistake

Multiplication does not always come before division. They have equal priority and are performed left to right. Same for addition and subtraction.

Example 1

Evaluate: $3 + 4 \times 2$

Multiply first: $4 \times 2 = 8$. Then add: $3 + 8 = 11$.

$$3 + 4 \times 2 = 11$$

Example 2

Evaluate: $(6 + 2) \times 5 - 4^2$

  1. P: $6 + 2 = 8$
  2. E: $4^2 = 16$
  3. M: $8 \times 5 = 40$
  4. S: $40 - 16 = 24$

$$(6 + 2) \times 5 - 4^2 = 24$$

Example 3

Evaluate: $48 \div 6 \times 2$

Equal priority — left to right: $48 \div 6 = 8$, then $8 \times 2 = 16$.

$$48 \div 6 \times 2 = 16$$

Example 4

Evaluate: $2 + 3 \times (8 - 5)^2 \div 9$

  1. P: $8 - 5 = 3$
  2. E: $3^2 = 9$
  3. M: $3 \times 9 = 27$
  4. D: $27 \div 9 = 3$
  5. A: $2 + 3 = 5$

$$2 + 3 \times (8 - 5)^2 \div 9 = 5$$

Example 5

Evaluate: $5 \times [4 + 3 \times (10 - 8)]$

  1. Innermost: $10 - 8 = 2$
  2. $3 \times 2 = 6$
  3. $4 + 6 = 10$
  4. $5 \times 10 = 50$

$$5 \times [4 + 3 \times (10 - 8)] = 50$$

Example 6

Evaluate: $\dfrac{20 + 4}{6} + 7 \times 2$

The fraction bar acts as a grouping symbol:

  1. Numerator: $20 + 4 = 24$
  2. $24 \div 6 = 4$
  3. $7 \times 2 = 14$
  4. $4 + 14 = 18$

$$\frac{20 + 4}{6} + 7 \times 2 = 18$$

Example 7

Evaluate: $100 - 3 \times 2^3 + 5$

  1. E: $2^3 = 8$
  2. M: $3 \times 8 = 24$
  3. A/S left to right: $100 - 24 + 5 = 81$

$$100 - 3 \times 2^3 + 5 = 81$$

Practice Problems

Evaluate each expression.

1. $8 + 3 \times 5$
2. $(8 + 3) \times 5$
3. $36 \div 4 + 2 \times 3$
4. $5^2 - 3 \times 4 + 1$
5. $2 \times (7 + 3)^2$
6. $100 - 4 \times (5 + 3)$
7. $18 \div 3 \div 2$
8. $15 - 10 + 3$
9. $4 + 6 \times \frac{12}{4}$
10. $3 \times [2 + 4 \times (5 - 2)]$
11. $2^4 + 3^2 - 5 \times 2$
12. $(12 - 4)^2 \div (2 + 2)$
13. $\frac{8 + 4}{3} \times 5 - 2^3$
14. $50 - 2 \times 3^2 + 4 \div 2$
15. $7 \times 8 - 6 \times 9 + 2$
Show Answer Key

1. $23$

2. $55$

3. $9 + 6 = 15$

4. $25 - 12 + 1 = 14$

5. $2 \times 100 = 200$

6. $100 - 32 = 68$

7. $6 \div 2 = 3$ (left to right)

8. $5 + 3 = 8$ (left to right)

9. $4 + 6 \times 3 = 4 + 18 = 22$

10. $3 \times [2 + 12] = 3 \times 14 = 42$

11. $16 + 9 - 10 = 15$

12. $64 \div 4 = 16$

13. $4 \times 5 - 8 = 12$

14. $50 - 18 + 2 = 34$

15. $56 - 54 + 2 = 4$

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