Order of Operations

When evaluating a math expression with multiple operations (like addition, subtraction, multiplication, and division), we need a standard set of rules so everyone gets the same answer. This set of rules is called the Order of Operations.

The Rules to Follow

To solve a numerical expression correctly, always follow these steps in order:

1. Parentheses and Grouping Symbols: First, perform the operations inside parentheses $()$, brackets $[]$, and braces $\{\}$. If there are grouping symbols inside other grouping symbols (nested), always start from the inside and work your way out.
2. Multiplication and Division: Next, perform all multiplication and division from left to right.
3. Addition and Subtraction: Finally, perform all addition and subtraction from left to right.

Example Problems

Let's walk through some examples to see the order of operations in action.

Example 1: Calculate $(3 + 4) \times 5 - 2$

• Step 1: Solve inside the parentheses first. $3 + 4 = 7$ Now substitute that back in. The expression becomes: $7 \times 5 - 2$
• Step 2: Multiply before subtracting. $7 \times 5 = 35$ The expression becomes: $35 - 2$
• Step 3: Subtract. $35 - 2 = 33$

Answer: $33$

Example 2: Evaluate $2 \times \{(6 + 4) \div 2\}$

• Step 1: Start with the innermost grouping symbols, which are the parentheses $()$. $6 + 4 = 10$ The expression becomes: $2 \times \{10 \div 2\}$
• Step 2: Next, solve the operation inside the braces $\{\}$. $10 \div 2 = 5$ The expression becomes: $2 \times 5$
• Step 3: Multiply. $2 \times 5 = 10$

Answer: $10$

Example 3: Calculate $48 \div (2 \times 4) + 7$

• Step 1: Solve inside the parentheses first. $2 \times 4 = 8$ The expression becomes: $48 \div 8 + 7$
• Step 2: Divide before you add. $48 \div 8 = 6$ The expression becomes: $6 + 7$
• Step 3: Add. $6 + 7 = 13$

Answer: $13$