Solving Arithmetic Expressions Step-by-Step Guide

Arithmetic expressions form the bedrock of mathematical understanding. To successfully navigate the world of algebra, calculus, and beyond, it’s crucial to master the order of operations and confidently tackle any expression that comes your way. This guide will provide a detailed walkthrough of how to solve various arithmetic expressions, ensuring clarity and accuracy every step of the way. We'll break down the process, making it easy to understand and apply to any arithmetic problem you encounter. This is especially important because mathematical skills are not only essential for academic success but also for everyday life, such as managing finances, calculating expenses, and even cooking. Understanding the fundamentals of arithmetic ensures that you can tackle complex problems with confidence and precision, leading to better decision-making in various aspects of your life.

Understanding the Order of Operations (PEMDAS/BODMAS)

The order of operations is the golden rule in solving arithmetic expressions. Often remembered by the acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction), this principle dictates the sequence in which operations must be performed to arrive at the correct answer. Failing to adhere to this order can lead to incorrect results and a misunderstanding of the underlying mathematical concepts. This is why it's crucial to internalize this principle and apply it consistently in every arithmetic problem you solve. Mastering the order of operations not only enhances your mathematical skills but also develops your logical thinking and problem-solving abilities, valuable assets in any field.

• Parentheses/Brackets: Always start by simplifying expressions inside parentheses or brackets. These grouping symbols indicate that the enclosed operations should be performed before any others. Parentheses and brackets act as containers, holding specific operations that need to be resolved before anything else. They provide a structure for the expression, guiding you through the steps required to find the solution. For instance, if you have an expression like 2 × (3 + 4), you must first solve the addition within the parentheses before proceeding with the multiplication. This ensures the correct sequence of calculations and prevents errors.
• Exponents/Orders: Next, evaluate any exponents or orders (powers and roots). Exponents represent repeated multiplication, and they need to be simplified before multiplication, division, addition, or subtraction. Understanding exponents is critical in various areas of mathematics, including algebra and calculus. For example, in the expression 5^2 + 3 × 2, you would first calculate 5^2, which equals 25, before performing any other operations. This ensures that you handle the powers correctly and move towards the correct solution. Orders, on the other hand, involve roots and radicals, which are the inverse operation of exponents. Solving these correctly is equally crucial for arriving at the right answer.
• Multiplication and Division: Perform multiplication and division from left to right. These operations have equal priority, so the order in which they appear in the expression determines the sequence of calculation. Multiplication and division are inverse operations, and they must be addressed in the order they appear to avoid any discrepancies. For instance, if an expression reads 10 ÷ 2 × 3, you would first divide 10 by 2, which equals 5, and then multiply the result by 3. This step-by-step approach ensures that you're following the mathematical convention and arriving at the correct result. Ignoring this rule can lead to incorrect answers, highlighting the importance of understanding and applying the order of operations.
• Addition and Subtraction: Finally, carry out addition and subtraction from left to right. Similar to multiplication and division, these operations have equal priority and should be performed in the order they appear. Addition and subtraction are fundamental arithmetic operations, and their proper execution is essential for accurate calculations. If you have an expression like 8 + 5 - 3, you would first add 8 and 5, resulting in 13, and then subtract 3 from 13. This ensures that you're following the correct sequence of calculations and arriving at the accurate final answer. This meticulous approach is crucial for mastering arithmetic expressions and building a strong foundation in mathematics.

Problem 1: 25 - 5 × (16 - 12) ÷ 2

To solve the arithmetic expression 25 - 5 × (16 - 12) ÷ 2, we must diligently follow the order of operations, commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This methodical approach ensures that we arrive at the correct solution by addressing each operation in the proper sequence. Let's break down the steps:

1. Parentheses: First, we tackle the expression within the parentheses: (16 - 12). Subtracting 12 from 16 gives us 4. This simplifies the expression to 25 - 5 × 4 ÷ 2. The parentheses act as a container, prioritizing the operations inside them before anything else. This step is crucial as it sets the stage for the subsequent calculations.
2. Multiplication: Next, we perform the multiplication operation. In the expression 25 - 5 × 4 ÷ 2, we multiply 5 by 4, which equals 20. The expression now becomes 25 - 20 ÷ 2. Prioritizing multiplication over division and addition or subtraction is a key aspect of the order of operations. This ensures that we handle the multiplication correctly and move towards the correct final solution.
3. Division: Now, we address the division operation. We divide 20 by 2, which equals 10. The expression is further simplified to 25 - 10. Division, being the inverse of multiplication, is performed next according to PEMDAS. This step ensures that we correctly handle the division aspect of the problem and continue moving towards the correct result.
4. Subtraction: Finally, we perform the subtraction operation. We subtract 10 from 25, which gives us 15. Thus, the final answer to the expression is 15. Subtraction, being the final operation in this case, leads us to the ultimate solution. This step confirms that we have diligently followed the order of operations and arrived at the correct answer.

Therefore, 25 - 5 × (16 - 12) ÷ 2 = 15. This result demonstrates the importance of adhering to PEMDAS to solve arithmetic expressions accurately. Each step is interconnected, and following the correct order ensures a reliable and correct solution.

Problem 2: (6 ÷ 3) × (11 - 4)

To solve the expression (6 ÷ 3) × (11 - 4), we again apply the order of operations, specifically PEMDAS or BODMAS. This systematic approach ensures we handle each operation in the correct sequence, leading us to the accurate final answer. Neglecting the order of operations can result in incorrect solutions, emphasizing the importance of this mathematical principle. Let's break down the solution step-by-step:

1. Parentheses: Our initial focus is on the expressions within the parentheses. We have two sets of parentheses in this expression: (6 ÷ 3) and (11 - 4). First, let's solve (6 ÷ 3). Dividing 6 by 3, we get 2. Next, we solve (11 - 4). Subtracting 4 from 11, we get 7. So, the expression now simplifies to 2 × 7. Parentheses prioritize these enclosed operations, guiding us to resolve them before proceeding with other calculations. This ensures that we handle these parts of the expression first, maintaining the mathematical integrity of the solution process.
2. Multiplication: With the parentheses resolved, we are left with a simple multiplication: 2 × 7. Multiplying 2 by 7, we get 14. Therefore, the final answer to the expression is 14. Multiplication completes the solution, providing the final numerical result after addressing the parentheses. This step reinforces the importance of following the correct order of operations to ensure accuracy.

Hence, (6 ÷ 3) × (11 - 4) = 14. This solution underscores the significance of following the order of operations to accurately solve arithmetic expressions. Each step builds upon the previous one, leading to the correct final result. By consistently applying PEMDAS or BODMAS, we can confidently tackle any arithmetic problem, regardless of its complexity.

Problem 3: 9 × 3 + (20 - 18)

When tackling the expression 9 × 3 + (20 - 18), we once again adhere to the fundamental principle of the order of operations, often abbreviated as PEMDAS or BODMAS. This methodical approach ensures that we address each operation in the correct sequence, preventing errors and leading us to the accurate solution. Understanding and applying this principle is crucial for mastering arithmetic and progressing to more complex mathematical concepts. Let's break down the solution step by step:

1. Parentheses: Our initial focus is on the expression within the parentheses: (20 - 18). Subtracting 18 from 20, we get 2. This simplifies the expression to 9 × 3 + 2. Parentheses serve as containers that prioritize the operations within them. By resolving the expression inside the parentheses first, we ensure that we are following the correct order of operations. This step sets the foundation for the subsequent calculations, ensuring the integrity of the solution process.
2. Multiplication: Next, we address the multiplication operation. In the expression 9 × 3 + 2, we multiply 9 by 3, which gives us 27. The expression now becomes 27 + 2. Prioritizing multiplication before addition is a key aspect of the order of operations. This step ensures that we handle the multiplication aspect of the problem correctly before moving on to addition, which could otherwise lead to an incorrect answer.
3. Addition: Finally, we perform the addition operation. We add 27 and 2, which gives us 29. Therefore, the final answer to the expression is 29. Addition completes the solution process, providing the final numerical result after addressing the parentheses and multiplication. This step confirms that we have diligently followed the order of operations and have arrived at the correct answer.

Thus, 9 × 3 + (20 - 18) = 29. This solution exemplifies the importance of adhering to the order of operations in arithmetic expressions. Each step builds logically upon the previous one, ensuring that we arrive at the correct and accurate final result. By consistently applying PEMDAS or BODMAS, we can approach any arithmetic problem with confidence and precision.

Problem 4: (47 - 17) + 10 × 3

To solve the arithmetic expression (47 - 17) + 10 × 3, we strictly adhere to the order of operations, commonly known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). This principle is the cornerstone of accurate mathematical calculations, ensuring that we perform operations in the correct sequence. Neglecting this order can lead to incorrect solutions and a misunderstanding of the underlying mathematical concepts. Let's break down the solution step-by-step:

1. Parentheses: Our primary focus is on the expression within the parentheses: (47 - 17). Subtracting 17 from 47 gives us 30. The expression now simplifies to 30 + 10 × 3. Parentheses prioritize the operations enclosed within them, making this subtraction the first step in our calculation. Resolving the parentheses first ensures that we set the stage for subsequent calculations in the correct order, maintaining the mathematical integrity of the process.
2. Multiplication: Next, we perform the multiplication operation. In the expression 30 + 10 × 3, we multiply 10 by 3, which equals 30. The expression further simplifies to 30 + 30. Prioritizing multiplication over addition is a crucial aspect of the order of operations. This step ensures that we handle the multiplication correctly before moving on to addition, which could otherwise lead to an inaccurate result.
3. Addition: Finally, we address the addition operation. We add 30 and 30, which gives us 60. Therefore, the final answer to the expression is 60. Addition is the final operation in this case, bringing us to the ultimate solution. This step confirms that we have diligently followed the order of operations, accurately completing each stage of the calculation.

Thus, (47 - 17) + 10 × 3 = 60. This solution highlights the critical importance of following the order of operations to achieve accuracy in arithmetic expressions. Each step logically builds upon the previous one, ensuring that we arrive at the correct final result. By consistently applying PEMDAS or BODMAS, we can confidently tackle any arithmetic problem, regardless of its complexity, and build a strong foundation in mathematical skills.

Conclusion

In conclusion, mastering arithmetic expressions requires a thorough understanding and application of the order of operations (PEMDAS/BODMAS). By consistently following these rules, you can confidently solve any arithmetic problem with accuracy. The examples provided illustrate the importance of each step in the process, from simplifying parentheses to performing multiplication, division, addition, and subtraction in the correct sequence. These skills are fundamental to mathematical proficiency and are essential for success in more advanced topics. Remember, practice makes perfect, so continue to challenge yourself with various arithmetic expressions to strengthen your understanding and problem-solving abilities. This not only enhances your mathematical skills but also develops critical thinking and logical reasoning, valuable assets in both academic and everyday life scenarios.