Order of Operations (PEMDAS)

The order of operations is a set of rules that tells you which calculations to perform first in a math expression. Without these rules, the same expression could produce different answers depending on the order you choose. The standard acronym is PEMDAS (in the US) or BODMAS (in the UK), and every math, science, and trade formula depends on following it correctly.

The PEMDAS Rules

Critical detail: Multiplication and division have equal priority — you work left to right. The same applies to addition and subtraction. PEMDAS does not mean multiplication always comes before division.

Step-by-Step Examples

Example 1: Basic PEMDAS

Evaluate $3 + 4 \times 2$.

Step 1: No parentheses. No exponents.

Step 2: Multiplication before addition: $4 \times 2 = 8$.

Step 3: Addition: $3 + 8 = 11$.

Answer: $11$

Common wrong answer: $14$ (if you add first: $(3 + 4) \times 2 = 14$). Without parentheses, multiplication comes first.

Example 2: Parentheses Change Everything

Evaluate $(3 + 4) \times 2$.

Step 1: Parentheses first: $3 + 4 = 7$.

Step 2: Multiply: $7 \times 2 = 14$.

Answer: $14$

Example 3: Exponents in the Mix

Evaluate $5 + 2^3 \times 3$.

Step 1: No parentheses.

Step 2: Exponents: $2^3 = 8$.

Step 3: Multiplication: $8 \times 3 = 24$.

Step 4: Addition: $5 + 24 = 29$.

Answer: $29$

Example 4: Left-to-Right Rule for Division and Multiplication

Evaluate $24 \div 6 \times 2$.

Both division and multiplication have equal priority, so go left to right.

Step 1: $24 \div 6 = 4$.

Step 2: $4 \times 2 = 8$.

Answer: $8$

Common wrong answer: $2$ (if you multiply first: $24 \div 12 = 2$). Always go left to right when operations have equal priority.

Example 5: Left-to-Right Rule for Addition and Subtraction

Evaluate $15 - 7 + 3$.

Step 1: $15 - 7 = 8$.

Step 2: $8 + 3 = 11$.

Answer: $11$

Example 6: Complex Expression

Evaluate $2 \times (6 + 3)^2 - 10 \div 5$.

Step 1 — Parentheses: $6 + 3 = 9$.

The expression becomes $2 \times 9^2 - 10 \div 5$.

Step 2 — Exponents: $9^2 = 81$.

The expression becomes $2 \times 81 - 10 \div 5$.

Step 3 — Multiplication and Division (left to right):

$2 \times 81 = 162$

$10 \div 5 = 2$

The expression becomes $162 - 2$.

Step 4 — Subtraction: $162 - 2 = 160$.

Answer: $160$

Nested Grouping Symbols

Sometimes parentheses contain other parentheses, or expressions use brackets $[\,]$ and braces $\{\,\}$ for clarity. Always work from the innermost grouping outward.

Example 7: Nested Parentheses

Evaluate $3 \times [2 + (4 + 1)^2]$.

Step 1 — Inner parentheses: $4 + 1 = 5$.

The expression becomes $3 \times [2 + 5^2]$.

Step 2 — Exponent inside brackets: $5^2 = 25$.

The expression becomes $3 \times [2 + 25]$.

Step 3 — Brackets: $2 + 25 = 27$.

The expression becomes $3 \times 27$.

Step 4 — Multiply: $3 \times 27 = 81$.

Answer: $81$

Fraction Bars as Grouping Symbols

A fraction bar acts like a giant set of parentheses — it groups the entire numerator and the entire denominator.

$\frac{8 + 4}{3} \quad \text{means} \quad (8 + 4) \div 3$

Example 8: Fraction Bar

Evaluate $\frac{10 + 6}{2^2}$.

Step 1 — Numerator: $10 + 6 = 16$.

Step 2 — Denominator: $2^2 = 4$.

Step 3 — Divide: $\frac{16}{4} = 4$.

Answer: $4$

Example 9: More Complex Fraction Bar

Evaluate $\frac{3^2 + 7}{2 \times (1 + 3)}$.

Numerator: $3^2 + 7 = 9 + 7 = 16$.

Denominator: $2 \times (1 + 3) = 2 \times 4 = 8$.

$\frac{16}{8} = 2$

Answer: $2$

Real-World Application: Electrician — Total Resistance

An electrician calculates the total resistance in a circuit with resistors in both series and parallel. The formula might look like:

$R_{\text{total}} = 10 + \frac{1}{\frac{1}{20} + \frac{1}{30}}$

Step 1 — Inner fractions: $\frac{1}{20} = 0.05$ and $\frac{1}{30} \approx 0.0333$.

Step 2 — Add the fractions: $0.05 + 0.0333 = 0.0833$.

Step 3 — Divide: $\frac{1}{0.0833} \approx 12$.

Step 4 — Add the series resistance: $10 + 12 = 22$ ohms.

Answer: The total resistance is 22 ohms. Without following the order of operations, the electrician could calculate the wrong resistance and size the circuit breaker incorrectly.

Real-World Application: Nursing — Dosage Calculation

A nurse uses this formula to calculate an IV flow rate in drops per minute:

$\text{Rate} = \frac{\text{Volume (mL)} \times \text{Drop factor (drops/mL)}}{\text{Time (minutes)}}$

For 500 mL of fluid with a 15 drops/mL set over 240 minutes:

$\text{Rate} = \frac{500 \times 15}{240} = \frac{7{,}500}{240} = 31.25 \text{ drops/min}$

In practice, a nurse would round to 31 drops per minute since partial drops cannot be delivered.

The fraction bar groups the numerator and denominator separately — you must evaluate the entire numerator and denominator before dividing.

Common Mistakes to Avoid

1. Thinking M always comes before D. Multiplication and division are equal — go left to right. $8 \div 2 \times 4 = 16$, not $1$.

2. Thinking A always comes before S. Same rule: left to right. $10 - 3 + 2 = 9$, not $5$.

3. Ignoring the fraction bar as a grouping symbol. $\frac{6 + 2}{4}$ is $\frac{8}{4} = 2$, not $6 + \frac{2}{4} = 6.5$.

4. Forgetting exponents before multiplication. $2 \times 3^2$ is $2 \times 9 = 18$, not $6^2 = 36$.

5. Not working inside-out with nested grouping. Always start with the innermost parentheses.

Practice Problems

Key Takeaways

• PEMDAS dictates the order: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).
• Multiplication and division have equal priority — always work left to right.
• Addition and subtraction have equal priority — always work left to right.
• Fraction bars are grouping symbols — compute numerator and denominator separately before dividing.
• For nested grouping, work from the innermost set outward.
• Correctly applying the order of operations is essential in every trade formula.

Return to Pre-Algebra for more topics in this section.