Order Of Operations With Parentheses Worksheet

Understanding the order of operations is crucial in mathematics. It ensures that mathematical expressions are solved correctly and consistently. One of the key components of the order of operations is parentheses which help determine which calculations should be performed first.

This topic provides a detailed explanation of order of operations how parentheses affect calculations and a worksheet with practice problems to help improve your math skills.

What is the Order of Operations?

The order of operations is a set of rules that dictate the sequence in which mathematical operations should be performed. The standard rule is often remembered using the acronym PEMDAS:

• P – Parentheses
• E – Exponents
• MD – Multiplication and Division (from left to right)
• AS – Addition and Subtraction (from left to right)

Without following this order different people could get different answers to the same problem.

Why Are Parentheses Important?

Parentheses group certain numbers and operations together ensuring they are calculated first. Ignoring parentheses can lead to incorrect results.

Example Without Parentheses:

Solve: 5 + 3 × 2

Following PEMDAS multiplication comes before addition:
5 + (3 × 2) = 5 + 6 = 11

Example With Parentheses:

Solve: (5 + 3) × 2

Since parentheses come first we solve (5 + 3) = 8 then multiply:
8 × 2 = 16

As seen in these examples parentheses can completely change the outcome of a mathematical expression.

Rules for Solving Expressions with Parentheses

1. Always solve the expressions inside parentheses first.
2. If there are multiple parentheses work from the innermost set outward.
3. Follow PEMDAS rules inside parentheses.
4. After solving the parentheses proceed with exponents multiplication division addition and subtraction.

Step-by-Step Examples

Example 1: Simple Parentheses

Solve: (4 + 6) ÷ 2

Step 1: Solve inside parentheses → 4 + 6 = 10
Step 2: Divide → 10 ÷ 2 = 5

Example 2: Multiple Parentheses

Solve: (3 + 2) × (8 ÷ 4)

Step 1: Solve inside parentheses → (3 + 2) = 5 and (8 ÷ 4) = 2
Step 2: Multiply → 5 × 2 = 10

Example 3: Nested Parentheses

Solve: ((6 + 4) ÷ 2) + 3

Step 1: Solve inside parentheses → (6 + 4) = 10
Step 2: Divide → 10 ÷ 2 = 5
Step 3: Add → 5 + 3 = 8

Common Mistakes to Avoid

1. Ignoring Parentheses – Always solve what’s inside parentheses first.
2. Skipping Steps – Follow PEMDAS carefully step by step.
3. Misordering Multiplication and Addition – Remember that multiplication/division comes before addition/subtraction.
4. Forgetting Nested Parentheses – Solve from the innermost parentheses outward.

Worksheet: Practice Problems

Try solving the following problems using the correct order of operations with parentheses.

Basic Problems

1. (7 + 3) × 2
2. (12 ÷ 4) + 6
3. 8 × (5 – 3)
4. (9 + 6) ÷ 3
5. (10 – 2) × (3 + 1)

Intermediate Problems

6. (15 – 5) × (8 ÷ 2)
7. (4 + 6) ÷ (2 × 2)
8. (18 ÷ 3) + (5 × 2)
9. (7 × 3) – (12 ÷ 4)
10. (20 ÷ (5 + 5)) × 4

Advanced Problems

11. ((8 + 2) ÷ 2) × 5
12. (6 × (9 – 3)) ÷ 3
13. ((12 ÷ 4) + (3 × 2)) × 2
14. (5 × (4 + 2)) – (9 ÷ 3)
15. ((30 ÷ 5) + 2) × (8 – 6)

Answers to Worksheet

Basic Problems

1. 20
2. 9
3. 16
4. 5
5. 32

Intermediate Problems

6. 40
7. 2.5
8. 16
9. 17
10. 8

Advanced Problems

11. 25
12. 12
13. 16
14. 26
15. 12

Understanding the order of operations with parentheses is essential for solving mathematical expressions correctly. By following PEMDAS rules students can avoid common mistakes and ensure they arrive at the correct answer.

Use the worksheet provided to practice and reinforce these concepts. With regular practice mastering the order of operations becomes much easier!