Solving Mathematical Expressions Using Order Of Operations A Detailed Explanation

In mathematics, solving expressions accurately requires a clear understanding of the order of operations. This order, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which mathematical operations should be performed. This article aims to clarify how to apply the order of operations to solve expressions, using the example question: Which of these expressions is equal to $6+(2+3) \times 5$? We will explore each option, step-by-step, to identify the correct answer. By the end of this discussion, you'll have a solid grasp of how to tackle similar problems with confidence.

Decoding the Original Expression: $6+(2+3) \times 5$

To accurately evaluate the expression $6+(2+3) \times 5$, we must adhere strictly to the order of operations, commonly remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Understanding and applying PEMDAS is crucial for correctly solving mathematical expressions and ensuring accurate results. Let's break down the original expression step by step to illustrate this principle. The first step in our calculation is to address the parentheses. Inside the parentheses, we have the operation $(2+3)$. Performing this addition, we get $5$. This simplifies our expression to $6 + 5 \times 5$. Next, according to PEMDAS, we must perform any multiplication or division before addition or subtraction. In our simplified expression, we have the multiplication operation $5 \times 5$. Multiplying these numbers yields $25$. Now, our expression is further simplified to $6 + 25$. Finally, we perform the addition. Adding $6$ and $25$ gives us a final result of $31$. Therefore, $6+(2+3) \times 5 = 31$. This methodical approach, following the order of operations, is essential to avoid errors and arrive at the correct answer. Each step—simplifying within parentheses, performing multiplication, and then addition—plays a critical role in the process. By understanding and applying PEMDAS consistently, one can confidently solve mathematical expressions of varying complexity. This example serves as a foundational exercise in mastering the order of operations, a skill that is invaluable in mathematics and beyond. The importance of adhering to this order cannot be overstated, as it ensures consistency and accuracy in mathematical calculations, making it a cornerstone of mathematical problem-solving.

Evaluating Option A: $5+4 \times(5-6)$

Let's examine option A, which is the expression $5+4 \times(5-6)$. To determine if this expression is equal to the original expression's value of $31$, we again follow the PEMDAS order of operations. This systematic approach is essential for accurately evaluating mathematical expressions. Applying PEMDAS ensures that we perform operations in the correct sequence, avoiding common errors and arriving at the correct result. The first step in evaluating this expression is to address the parentheses. Inside the parentheses, we have the subtraction operation $(5-6)$. Performing this subtraction, we get $-1$. This simplifies our expression to $5 + 4 \times (-1)$. The next operation to perform, according to PEMDAS, is multiplication. We have $4 \times (-1)$, which equals $-4$. Our expression now becomes $5 + (-4)$, which can also be written as $5 - 4$. Finally, we perform the addition or subtraction. Subtracting $4$ from $5$ gives us a final result of $1$. Therefore, $5+4 \times(5-6) = 1$. Comparing this result to the value of the original expression, which is $31$, we can see that option A is not equal to the original expression. The significant difference in the results underscores the importance of carefully following the order of operations and accurately executing each step. This detailed evaluation highlights how each operation contributes to the final outcome and reinforces the necessity of a methodical approach in mathematical problem-solving. By breaking down the expression into manageable steps and applying PEMDAS consistently, we can confidently determine the correct value and compare it with other expressions.

Analyzing Option B: $(4 \times 5)+3$

Next, we evaluate option B, which is the expression $(4 \times 5)+3$. To ascertain whether this expression equals the original expression's value of $31$, we continue to adhere to the order of operations (PEMDAS). This consistent application of PEMDAS is vital for ensuring the accuracy of our calculations. PEMDAS guides us through the steps necessary to correctly solve mathematical expressions, starting with parentheses and moving through exponents, multiplication and division, and finally, addition and subtraction. In this expression, we first address the operation within the parentheses. We have $4 \times 5$ inside the parentheses. Performing this multiplication gives us $20$. The expression now simplifies to $20 + 3$. The next and final step is to perform the addition. Adding $20$ and $3$ yields a result of $23$. Therefore, $(4 \times 5)+3 = 23$. Comparing this result, $23$, to the value of the original expression, which is $31$, we find that option B does not equal the original expression. This discrepancy emphasizes the importance of accurate calculation and adherence to the order of operations. Each step in the evaluation process plays a crucial role in determining the final result, and any deviation from the correct order can lead to an incorrect answer. By meticulously following PEMDAS, we can systematically evaluate expressions and confidently compare their values. This analysis demonstrates the significance of precision in mathematical operations and reinforces the need for a structured approach when solving problems.

Examining Option C: $1+10 \times 3$

Now, let's consider option C, the expression $1+10 \times 3$. Our goal remains to determine if this expression is equivalent to the original expression's value of $31$. To achieve this, we will consistently apply the order of operations, following PEMDAS. Adhering to PEMDAS is crucial for ensuring that we perform mathematical operations in the correct sequence, leading to accurate results. In this expression, we have both addition and multiplication. According to PEMDAS, multiplication must be performed before addition. Therefore, we first address the multiplication operation, which is $10 \times 3$. Multiplying $10$ by $3$ gives us $30$. The expression now simplifies to $1 + 30$. The next and final step is to perform the addition. Adding $1$ and $30$ results in $31$. Therefore, $1+10 \times 3 = 31$. Comparing this result, $31$, to the value of the original expression, which is also $31$, we find that option C is indeed equal to the original expression. This confirms that option C is a correct answer. This evaluation highlights the importance of correctly prioritizing operations according to PEMDAS. By performing the multiplication before the addition, we arrived at the correct value. This systematic approach not only ensures accuracy but also provides a clear and logical path to the solution. Understanding and applying the order of operations is fundamental to solving mathematical problems effectively.

Analyzing Option D: $9 \times 5+10$

Finally, we will analyze option D, which is the expression $9 \times 5+10$. As with the previous options, our objective is to determine if this expression evaluates to the same value as the original expression, which is $31$. To ensure accuracy, we will continue to apply the order of operations, PEMDAS. Consistent application of PEMDAS is key to correctly solving mathematical expressions and avoiding errors. In this expression, we have both multiplication and addition. According to PEMDAS, multiplication takes precedence over addition. Therefore, we begin by performing the multiplication operation, which is $9 \times 5$. Multiplying $9$ by $5$ gives us $45$. The expression now simplifies to $45 + 10$. The next and final step is to perform the addition. Adding $45$ and $10$ results in $55$. Therefore, $9 \times 5+10 = 55$. Comparing this result, $55$, to the value of the original expression, which is $31$, we can see that option D does not equal the original expression. This difference underscores the importance of following the order of operations meticulously. Each step in the evaluation process contributes to the final result, and an incorrect sequence of operations can lead to a wrong answer. By carefully applying PEMDAS, we can systematically evaluate expressions and confidently determine their values. This analysis reinforces the need for a structured approach to mathematical problem-solving, ensuring that we perform operations in the correct order and arrive at accurate solutions.

Conclusion: Mastering Order of Operations

In summary, to determine which expression is equal to $6+(2+3) \times 5$, we meticulously evaluated each option using the order of operations (PEMDAS). We found that the original expression evaluates to $31$. By systematically applying PEMDAS to each option, we were able to accurately compare their values. Option A, $5+4 \times(5-6)$, equals $1$, option B, $(4 \times 5)+3$, equals $23$, option C, $1+10 \times 3$, equals $31$, and option D, $9 \times 5+10$, equals $55$. Therefore, the only expression equal to the original expression is option C. This exercise demonstrates the critical importance of understanding and applying the order of operations in mathematics. PEMDAS serves as a reliable guide for correctly solving mathematical expressions, ensuring that we perform operations in the appropriate sequence. By mastering the order of operations, we can confidently tackle a wide range of mathematical problems and achieve accurate results. This skill is not only essential for academic success but also for various real-world applications where mathematical reasoning is required. The ability to correctly evaluate expressions is a fundamental aspect of mathematical literacy, empowering individuals to solve problems effectively and make informed decisions based on numerical data.