Mastering Arithmetic Operations A Step-by-Step Guide

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This article will guide you through solving a series of basic arithmetic problems. We will break down each problem step-by-step, explaining the order of operations and the logic behind each calculation. By the end of this guide, you will have a solid understanding of how to tackle similar problems with confidence. Let's dive in!

1. Solving 7 × 2 - (9 + 2)

When approaching arithmetic problems, it's crucial to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order ensures we solve problems consistently and accurately. In this first problem, 7 × 2 - (9 + 2), we have a combination of multiplication, subtraction, and parentheses. Following PEMDAS, we first address the parentheses. Within the parentheses, we have the addition operation (9 + 2). Performing this addition gives us 11. So, the expression now becomes 7 × 2 - 11. Next, we move to multiplication. We have 7 × 2, which equals 14. Our expression is now simplified to 14 - 11. Finally, we perform the subtraction. 14 - 11 equals 3. Therefore, the solution to the problem 7 × 2 - (9 + 2) is 3. This problem highlights the importance of tackling operations within parentheses first, followed by multiplication, and lastly subtraction. By following this order, we avoid errors and arrive at the correct answer. Let's move on to the next problem and further solidify our understanding of these principles. Remember, consistent practice and attention to detail are key to mastering arithmetic operations. Each problem is an opportunity to refine our skills and build confidence in our ability to solve complex expressions.

In summary, the steps for solving 7 × 2 - (9 + 2) are:

  1. Parentheses: (9 + 2) = 11
  2. Multiplication: 7 × 2 = 14
  3. Subtraction: 14 - 11 = 3

Therefore, 7 × 2 - (9 + 2) = 3.

2. Solving (6 ÷ 3) × (11 - 4)

This problem, (6 ÷ 3) × (11 - 4), presents another opportunity to apply the order of operations (PEMDAS). We again begin with parentheses. Here, we have two sets of parentheses: (6 ÷ 3) and (11 - 4). Let's tackle them one by one. First, (6 ÷ 3). Dividing 6 by 3 yields 2. Next, we solve (11 - 4). Subtracting 4 from 11 gives us 7. Now, our expression has been simplified to 2 × 7. With only one operation remaining—multiplication—we complete the calculation. Multiplying 2 by 7 results in 14. Therefore, the answer to (6 ÷ 3) × (11 - 4) is 14. This problem reinforces the significance of addressing parentheses before other operations. It also demonstrates how breaking down a problem into smaller, manageable steps can make complex calculations easier to handle. By first simplifying the expressions within the parentheses, we reduced the problem to a simple multiplication, minimizing the chance of errors. Consistent application of PEMDAS is essential for success in arithmetic and beyond. As we continue to practice, we develop an intuitive understanding of the order in which operations should be performed, allowing us to solve problems more efficiently and accurately. Each problem we solve is a step forward in building our mathematical foundation.

To recap, the steps for solving (6 ÷ 3) × (11 - 4) are:

  1. Parentheses (Division): (6 ÷ 3) = 2
  2. Parentheses (Subtraction): (11 - 4) = 7
  3. Multiplication: 2 × 7 = 14

Thus, (6 ÷ 3) × (11 - 4) = 14.

3. Solving 9 + 3 + (20 - 18)

Our third problem, 9 + 3 + (20 - 18), offers another chance to practice the order of operations. Following PEMDAS, we start with the parentheses. Inside the parentheses, we have (20 - 18). Subtracting 18 from 20 gives us 2. Our expression now becomes 9 + 3 + 2. With only addition operations remaining, we can proceed from left to right. First, we add 9 + 3, which equals 12. Our expression is now 12 + 2. Finally, we add 12 + 2, resulting in 14. Therefore, the solution to 9 + 3 + (20 - 18) is 14. This problem highlights the simplicity of solving expressions once the parentheses are addressed and only one type of operation remains. When faced with multiple additions (or multiple subtractions), we can work through the problem sequentially from left to right, ensuring accuracy. This approach simplifies the calculation process and reduces the likelihood of errors. Mastering these fundamental arithmetic skills is crucial for more advanced mathematics. Each problem we solve reinforces our understanding of the basic principles and builds confidence in our abilities. By consistently applying the order of operations and breaking down problems into manageable steps, we can tackle even complex calculations with ease.

In summary, the steps for solving 9 + 3 + (20 - 18) are:

  1. Parentheses: (20 - 18) = 2
  2. Addition: 9 + 3 = 12
  3. Addition: 12 + 2 = 14

Hence, 9 + 3 + (20 - 18) = 14.

4. Solving (47 - 17) + 10 × 3

In this problem, (47 - 17) + 10 × 3, we again emphasize the importance of following the order of operations (PEMDAS). We start with the parentheses. Inside the parentheses, we have (47 - 17). Subtracting 17 from 47 gives us 30. The expression now becomes 30 + 10 × 3. According to PEMDAS, multiplication comes before addition. Therefore, we perform the multiplication 10 × 3, which equals 30. Our expression is now 30 + 30. Finally, we perform the addition. 30 + 30 equals 60. Thus, the solution to (47 - 17) + 10 × 3 is 60. This problem reinforces the critical distinction between multiplication and addition and the order in which they must be performed. Failing to multiply before adding would lead to an incorrect answer. Understanding and applying the correct order of operations is fundamental to success in arithmetic and algebra. By consistently following PEMDAS, we ensure accuracy and avoid common errors. Each problem we solve solidifies our understanding of these principles and builds our confidence in tackling more complex mathematical challenges. Remember, practice is key to mastering these skills.

To summarize, the steps for solving (47 - 17) + 10 × 3 are:

  1. Parentheses: (47 - 17) = 30
  2. Multiplication: 10 × 3 = 30
  3. Addition: 30 + 30 = 60

Therefore, (47 - 17) + 10 × 3 = 60.

5. Solving 10 ÷ [9 - (2 × 2)]

This final problem, 10 ÷ [9 - (2 × 2)], introduces nested parentheses, adding a layer of complexity. However, the principle of the order of operations (PEMDAS) remains our guiding star. We begin by addressing the innermost parentheses first. Inside the inner parentheses, we have (2 × 2). Multiplying 2 by 2 gives us 4. Now, the expression within the brackets becomes [9 - 4]. Next, we perform the subtraction inside the brackets. 9 - 4 equals 5. Our expression is now simplified to 10 ÷ 5. Finally, we perform the division. Dividing 10 by 5 results in 2. Therefore, the answer to 10 ÷ [9 - (2 × 2)] is 2. This problem emphasizes the importance of working from the innermost parentheses outwards. By systematically simplifying the expression layer by layer, we can avoid confusion and ensure accuracy. Nested parentheses may seem daunting at first, but by breaking the problem down into smaller steps and consistently applying PEMDAS, we can solve them confidently. Attention to detail and a methodical approach are essential skills in mathematics, and this problem provides an excellent opportunity to practice them. Remember, each challenge overcome strengthens our mathematical foundation and prepares us for more advanced concepts.

In conclusion, the steps for solving 10 ÷ [9 - (2 × 2)] are:

  1. Inner Parentheses (Multiplication): (2 × 2) = 4
  2. Outer Parentheses (Subtraction): [9 - 4] = 5
  3. Division: 10 ÷ 5 = 2

Thus, 10 ÷ [9 - (2 × 2)] = 2.

By mastering the order of operations and practicing consistently, you can confidently tackle a wide range of arithmetic problems. Remember to break down complex problems into smaller, manageable steps, and always double-check your work. Happy calculating!