Solving Mathematical Expressions Using Order Of Operations

Understanding and applying the correct order of operations is fundamental to solving mathematical expressions accurately. Often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), this set of rules dictates the sequence in which operations must be performed to arrive at the correct answer. This article delves into several mathematical expressions, demonstrating the application of PEMDAS to simplify and solve them effectively. By mastering the order of operations, you'll be able to tackle a wide range of mathematical problems with confidence and precision.

1. 6 + (9 ÷ 3 × 4)

The first expression we will tackle is 6 + (9 ÷ 3 × 4). To solve this correctly, we need to strictly adhere to the order of operations, commonly remembered by the acronym PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). This mnemonic helps us to remember the correct sequence in which to perform mathematical operations, ensuring accurate results.

Firstly, we address the operations within the parentheses. Inside the parentheses, we have both division and multiplication. According to PEMDAS, multiplication and division are performed from left to right. Thus, we start with the division: 9 ÷ 3 = 3. This simplifies the expression inside the parentheses to 3 × 4.

Next, we complete the multiplication within the parentheses: 3 × 4 = 12. Now, the entire expression inside the parentheses has been simplified to a single value, 12. Our original expression now looks like this: 6 + 12.

Finally, we perform the addition: 6 + 12 = 18. Therefore, the solution to the expression 6 + (9 ÷ 3 × 4) is 18. This step-by-step breakdown highlights how crucial it is to follow the correct order of operations. If we were to add before dividing or multiplying, we would arrive at an incorrect answer. PEMDAS provides a clear roadmap, ensuring we navigate through the expression in the proper sequence.

In summary, by systematically working through the expression—first addressing the division within the parentheses, then the multiplication, and finally the addition—we arrive at the accurate solution. This methodical approach not only provides the correct answer but also helps to minimize errors and fosters a deeper understanding of mathematical principles. Whether dealing with simple or complex expressions, the order of operations is the cornerstone of mathematical accuracy, enabling us to solve problems with confidence and precision. By consistently applying this method, we can efficiently and effectively handle any mathematical challenge.

2. 3 × [(9 + 15)] ÷ 8

Let's consider the expression 3 × [(9 + 15)] ÷ 8. This problem involves parentheses (or brackets), multiplication, and division. According to PEMDAS, we must first resolve the operations inside the parentheses before moving on to multiplication and division.

Inside the brackets, we have 9 + 15. Performing this addition, we get 24. The expression now simplifies to 3 × 24 ÷ 8. This intermediate step is crucial because it reduces the complexity of the problem, making it easier to manage and solve accurately. By focusing on one operation at a time, we minimize the chances of making errors.

Now that we have removed the parentheses, we are left with multiplication and division. As per PEMDAS, multiplication and division have equal priority and are performed from left to right. Therefore, we start with the multiplication: 3 × 24. This yields 72. Our expression now looks like this: 72 ÷ 8.

Finally, we perform the division: 72 ÷ 8. This gives us 9. Thus, the solution to the expression 3 × [(9 + 15)] ÷ 8 is 9. This step-by-step approach underscores the significance of adhering to the order of operations. Each operation is performed in its correct sequence, ensuring that we arrive at the accurate answer.

Breaking down the expression into manageable steps—first addressing the parentheses, then multiplication, and finally division—not only simplifies the problem but also provides a clear path to the solution. This method is particularly useful when dealing with more complex expressions, where the order of operations can become crucial in avoiding mistakes. By consistently following PEMDAS, we can confidently solve a wide range of mathematical problems, regardless of their complexity. This methodical approach enhances both our accuracy and our understanding of mathematical principles, fostering a deeper appreciation for the logic and structure inherent in mathematics.

3. 4 × [18 ÷ 2 × (10 - 8)]

Now, let’s analyze the expression 4 × [18 ÷ 2 × (10 - 8)]. This problem presents a slightly more complex scenario with nested parentheses and a mix of multiplication, division, and subtraction. Once again, we rely on PEMDAS to guide us through the correct sequence of operations.

First, we address the innermost parentheses: (10 - 8). Performing this subtraction, we get 2. The expression now becomes 4 × [18 ÷ 2 × 2]. Dealing with the innermost parentheses first simplifies the problem and allows us to proceed systematically.

Next, we focus on the operations within the brackets. Inside the brackets, we have division and multiplication. According to PEMDAS, multiplication and division are performed from left to right. Thus, we start with the division: 18 ÷ 2 = 9. The expression within the brackets now simplifies to 9 × 2.

Continuing within the brackets, we perform the multiplication: 9 × 2 = 18. Now, the entire expression inside the brackets has been simplified to a single value, 18. Our original expression now looks like this: 4 × 18.

Finally, we perform the remaining multiplication: 4 × 18 = 72. Therefore, the solution to the expression 4 × [18 ÷ 2 × (10 - 8)] is 72. This detailed breakdown illustrates how critical it is to tackle expressions step by step, following the order of operations rigorously.

By systematically addressing each operation in its correct sequence—starting with the innermost parentheses, then working through division and multiplication from left to right, and finally performing the remaining multiplication—we ensure accuracy and minimize the risk of errors. This methodical approach not only leads to the correct answer but also deepens our understanding of mathematical principles. Whether dealing with simple or intricate expressions, adhering to PEMDAS is essential for mathematical precision and confidence. By consistently applying this method, we can efficiently solve a wide array of mathematical challenges, enhancing our problem-solving skills and fostering a greater appreciation for the logical structure of mathematics.

4. (15 - 6) + (14 - 1) × 8

The fourth expression we will solve is (15 - 6) + (14 - 1) × 8. This expression includes parentheses, subtraction, addition, and multiplication. To solve it correctly, we must adhere to PEMDAS, ensuring each operation is performed in the correct order. This methodical approach is crucial for achieving accurate results in mathematics.

First, we address the operations within the parentheses. We have two sets of parentheses in this expression: (15 - 6) and (14 - 1). Let's solve them one by one. The first set (15 - 6) equals 9. The second set (14 - 1) equals 13. Now, our expression simplifies to 9 + 13 × 8. By addressing the parentheses first, we reduce the complexity of the overall expression and set the stage for the subsequent operations.

Next, according to PEMDAS, we must perform multiplication before addition. So, we multiply 13 by 8: 13 × 8 = 104. The expression now looks like this: 9 + 104. This step highlights the importance of adhering to the order of operations, as performing addition before multiplication would lead to an incorrect answer.

Finally, we perform the addition: 9 + 104 = 113. Therefore, the solution to the expression (15 - 6) + (14 - 1) × 8 is 113. This step-by-step breakdown demonstrates the critical role of PEMDAS in solving mathematical expressions. Each operation is performed in its proper sequence, ensuring accuracy and clarity in the solution process.

By systematically working through the expression—first addressing the parentheses, then performing the multiplication, and finally completing the addition—we arrive at the accurate solution. This methodical approach not only provides the correct answer but also reinforces the importance of following the order of operations in mathematics. Whether dealing with simple or complex expressions, a clear understanding and application of PEMDAS are essential for mathematical proficiency and problem-solving success. This consistent practice enhances our ability to tackle a wide range of mathematical challenges with confidence and precision.

5. 2 × 15 [3 + 2 × (10 - 9)]

Lastly, let’s solve the expression 2 × 15 [3 + 2 × (10 - 9)]. This problem is more intricate, involving nested parentheses, multiplication, and addition. As with previous examples, we will rely on PEMDAS to guide us through the correct order of operations. This methodical approach is crucial for maintaining accuracy and clarity throughout the problem-solving process.

First, we address the innermost parentheses: (10 - 9). Performing this subtraction, we get 1. The expression now becomes 2 × 15 [3 + 2 × 1]. Dealing with the innermost parentheses first simplifies the problem and allows us to proceed systematically through the remaining operations.

Next, we focus on the operations within the brackets. Inside the brackets, we have addition and multiplication. According to PEMDAS, we must perform multiplication before addition. So, we multiply 2 by 1: 2 × 1 = 2. The expression within the brackets now simplifies to 3 + 2.

Continuing within the brackets, we perform the addition: 3 + 2 = 5. Now, the entire expression inside the brackets has been simplified to a single value, 5. Our original expression now looks like this: 2 × 15 × 5. By systematically simplifying the expression within the brackets, we reduce the complexity of the overall problem and set the stage for the final operations.

Finally, we perform the remaining multiplications from left to right. First, 2 × 15 = 30. Then, 30 × 5 = 150. Therefore, the solution to the expression 2 × 15 [3 + 2 × (10 - 9)] is 150. This step-by-step breakdown highlights the importance of addressing each operation in its correct sequence, as dictated by PEMDAS.

By systematically working through the expression—starting with the innermost parentheses, then performing multiplication before addition within the brackets, and finally completing the remaining multiplications—we ensure accuracy and minimize the risk of errors. This methodical approach not only leads to the correct answer but also reinforces the importance of following the order of operations in mathematics. Whether dealing with simple or complex expressions, a clear understanding and application of PEMDAS are essential for mathematical proficiency and problem-solving success. This consistent practice enhances our ability to tackle a wide range of mathematical challenges with confidence and precision.

In conclusion, mastering the order of operations is fundamental to accurately solving mathematical expressions. By consistently applying PEMDAS, we can approach complex problems with confidence and ensure precise results. This skill is not only essential in mathematics but also valuable in various fields that require logical and analytical thinking. The expressions discussed in this article serve as practical examples of how PEMDAS works, reinforcing the importance of a systematic approach to mathematical problem-solving.