Solving 39 ÷ (3+10) - 3 × 7 A Step-by-Step Guide

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In the realm of mathematics, tackling arithmetic expressions is a fundamental skill. This article delves into the process of solving the expression 39÷(3+10)−3×739 \div (3+10) - 3 \times 7, providing a step-by-step guide to ensure clarity and accuracy. Understanding the order of operations is crucial in simplifying mathematical expressions. This principle, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which operations should be performed. PEMDAS ensures that mathematical expressions are evaluated consistently, leading to a unique and correct solution. When faced with an arithmetic expression, the first step involves identifying and simplifying any expressions within parentheses. Parentheses act as grouping symbols, indicating that the operations enclosed within them should be performed before any other operations in the expression. This hierarchical approach ensures that complex expressions are broken down into manageable parts, making the simplification process more straightforward. Once parentheses have been addressed, the next step is to evaluate any exponents present in the expression. Exponents represent repeated multiplication, and their evaluation is crucial in determining the overall value of the expression. Failing to address exponents in the correct order can lead to significant errors in the final result. With parentheses and exponents out of the way, the focus shifts to multiplication and division. These operations hold equal precedence and are performed from left to right in the expression. This left-to-right approach ensures consistency and avoids ambiguity in the evaluation process. It's important to remember that multiplication and division are inverse operations, and their order can significantly impact the final result. Finally, the expression is simplified by performing addition and subtraction operations. Similar to multiplication and division, addition and subtraction hold equal precedence and are performed from left to right. This final step combines the results of the previous operations to arrive at the simplified value of the expression. By adhering to the order of operations, we ensure that the expression is evaluated accurately and consistently. The acronym PEMDAS serves as a valuable mnemonic device for remembering the correct order: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

Step 1: Simplify the Expression Inside Parentheses

The first step in solving this expression is to simplify the expression within the parentheses. In this case, we have (3+10)(3 + 10), which is a straightforward addition operation. The main focus in this initial step is to isolate and resolve the terms enclosed within the parentheses, adhering to the order of operations which prioritizes parenthetical content. When simplifying expressions with parentheses, it's crucial to remember that the operations inside the parentheses take precedence over any other operations in the expression. This rule is a cornerstone of mathematical conventions and ensures consistency in calculations. By addressing the parentheses first, we effectively create a simplified value that can then be used in subsequent calculations. This approach breaks down the complex expression into smaller, more manageable components, making the overall solution process more transparent and less prone to errors. The ability to simplify expressions within parentheses is a fundamental skill in mathematics. It's not only applicable to basic arithmetic but also extends to more complex algebraic equations and formulas. Mastering this skill allows for a systematic and organized approach to problem-solving, ensuring accuracy and efficiency. In the context of this particular expression, simplifying (3+10)(3 + 10) involves a simple addition operation. Adding the two numbers together yields the result 13. This value then replaces the original expression within parentheses, transforming the overall expression into a simpler form. Once the expression within parentheses is simplified, we can proceed to the next step in the order of operations. This typically involves addressing exponents, multiplication, division, addition, and subtraction, in that specific order. However, in this particular case, there are no exponents, so we move directly to multiplication and division. By following this structured approach, we can ensure that we are adhering to mathematical conventions and arriving at the correct solution. Remember, the key to simplifying expressions is to break them down into smaller, more manageable steps. By focusing on one operation at a time and following the order of operations, we can confidently tackle even the most complex mathematical problems. The simplification of expressions within parentheses is a prime example of this principle, allowing us to transform the original expression into a more workable format.

Step 2: Perform the Division

Having simplified the expression within the parentheses, we now turn our attention to the division operation. The original expression 39÷(3+10)−3×739 \div (3+10) - 3 \times 7 has been transformed to 39÷13−3×739 \div 13 - 3 \times 7. Following the order of operations (PEMDAS), we address division and multiplication before subtraction. This step focuses specifically on the division part of the equation, which is 39÷1339 \div 13. Division is one of the four basic arithmetic operations, and it is the inverse of multiplication. In this instance, we are dividing 39 by 13, which means we are determining how many times 13 fits into 39. Understanding the concept of division is crucial for solving various mathematical problems, ranging from simple arithmetic to complex algebraic equations. Division is used in numerous real-world scenarios, such as splitting costs among friends, calculating unit prices, or determining the number of items that can be made from a given quantity of materials. The ability to perform division accurately is an essential skill for everyday life. When performing division, it's helpful to have a solid understanding of multiplication tables. Knowing the multiples of a number makes it easier to determine how many times it goes into another number. In this case, we can recall that 13 multiplied by 3 equals 39. Therefore, 39÷13=339 \div 13 = 3. This result is a whole number, which simplifies the calculation. However, it's important to remember that division doesn't always result in whole numbers. Sometimes, the result is a fraction or a decimal. In such cases, it's crucial to perform the division carefully and accurately. Now that we have performed the division operation, we have further simplified the expression. The original expression has now become 3−3×73 - 3 \times 7. We have successfully reduced the complexity of the equation by addressing the division component. The next step in the order of operations is to perform multiplication, which we will address in the subsequent section. By systematically addressing each operation in the correct order, we can ensure that we arrive at the correct solution. The division step is a critical part of the overall solution process, and its accurate execution is essential for obtaining the final answer. Understanding the principles of division and practicing its application will enhance your mathematical skills and your ability to solve a wide range of problems.

Step 3: Perform the Multiplication

Following the order of operations, having completed the division, we now move on to the multiplication operation. Our expression currently stands at 3−3×73 - 3 \times 7. In this step, we will address the multiplication component, which is 3×73 \times 7. Multiplication, a fundamental arithmetic operation, represents repeated addition. In this context, 3×73 \times 7 signifies adding 3 to itself 7 times, or vice versa. The ability to perform multiplication accurately and efficiently is crucial for solving a wide range of mathematical problems. Multiplication is used extensively in various real-world applications, from calculating areas and volumes to determining the cost of multiple items. Mastering multiplication tables is a cornerstone of mathematical proficiency. Knowing the products of single-digit numbers allows for faster and more accurate calculations. In this specific case, we are multiplying 3 by 7. Recalling the multiplication table, we know that 3×7=213 \times 7 = 21. This result is a whole number, which simplifies the subsequent calculation. However, it's important to remember that multiplication can also involve decimals and fractions. Understanding how to multiply these types of numbers is essential for a comprehensive grasp of mathematics. Now that we have performed the multiplication operation, our expression has been further simplified to 3−213 - 21. We have successfully reduced the complexity of the equation by addressing the multiplication component. The final step in the order of operations is to perform the subtraction, which we will address in the following section. By systematically addressing each operation in the correct order, we ensure that we arrive at the accurate solution. The multiplication step is a vital part of the overall solution process, and its correct execution is crucial for obtaining the final answer. A strong understanding of multiplication principles and consistent practice will enhance your mathematical abilities and your capacity to tackle a diverse array of problems. In this instance, performing the multiplication operation has brought us one step closer to the final solution, demonstrating the importance of following the order of operations in mathematical calculations.

Step 4: Perform the Subtraction

With the division and multiplication operations completed, the final step in solving the expression is to perform the subtraction. Our expression has been simplified to 3−213 - 21. Subtraction, another fundamental arithmetic operation, involves finding the difference between two numbers. In this case, we are subtracting 21 from 3. Understanding the concept of subtraction is essential for solving a wide range of mathematical problems, from simple arithmetic to more complex algebraic equations. Subtraction is used extensively in everyday life, such as calculating change, measuring distances, and comparing quantities. The ability to perform subtraction accurately is a crucial skill for both academic and practical applications. When performing subtraction, it's important to consider the signs of the numbers involved. In this instance, we are subtracting a larger number (21) from a smaller number (3). This will result in a negative number. To perform the subtraction, we can think of it as adding a negative number: 3+(−21)3 + (-21). Alternatively, we can find the difference between the absolute values of the numbers and then apply the sign of the larger number. The absolute value of 3 is 3, and the absolute value of 21 is 21. The difference between 21 and 3 is 18. Since 21 is larger and has a negative sign, the result will be -18. Therefore, 3−21=−183 - 21 = -18. This result is a negative integer, which demonstrates that the final answer can be a negative number. Now that we have performed the subtraction operation, we have arrived at the final solution for the expression. The original expression 39÷(3+10)−3×739 \div (3+10) - 3 \times 7 has been simplified to -18. We have successfully solved the problem by following the order of operations and performing each operation accurately. The subtraction step is a critical part of the overall solution process, and its correct execution is essential for obtaining the final answer. A strong understanding of subtraction principles and consistent practice will enhance your mathematical abilities and your capacity to tackle a diverse array of problems. In this case, performing the subtraction operation has brought us to the ultimate solution, demonstrating the importance of understanding and applying the order of operations in mathematical calculations.

Final Answer

By following the order of operations meticulously, we have successfully simplified the expression 39÷(3+10)−3×739 \div (3+10) - 3 \times 7. We began by addressing the expression within parentheses, then proceeded with division, followed by multiplication, and finally, subtraction. Each step was executed with careful attention to detail, ensuring accuracy and consistency in our calculations. The initial step involved simplifying the expression within the parentheses: (3+10)(3 + 10). This straightforward addition resulted in 13, transforming the original expression into 39÷13−3×739 \div 13 - 3 \times 7. This simplification paved the way for the subsequent operations, making the overall process more manageable. Next, we addressed the division operation: 39÷1339 \div 13. This division yielded the result 3, further simplifying the expression to 3−3×73 - 3 \times 7. At each stage, we adhered to the order of operations, ensuring that the mathematical conventions were followed correctly. Following the division, we tackled the multiplication operation: 3×73 \times 7. This multiplication resulted in 21, transforming the expression into 3−213 - 21. The systematic approach, guided by the order of operations, allowed us to break down the complex expression into smaller, more manageable components. Finally, we performed the subtraction operation: 3−213 - 21. This subtraction resulted in -18, which is the final answer to the expression. The negative result underscores the importance of considering the signs of numbers when performing mathematical operations. Through this step-by-step process, we have demonstrated the power of the order of operations in simplifying complex arithmetic expressions. By adhering to this principle, we can ensure accuracy and consistency in our calculations, leading to the correct solution. The final answer, -18, represents the culmination of our efforts, showcasing the successful application of mathematical principles. In conclusion, the solution to the expression 39÷(3+10)−3×739 \div (3+10) - 3 \times 7 is -18. This result highlights the significance of following the order of operations and performing each step with precision. By mastering these fundamental mathematical skills, we can confidently tackle a wide range of arithmetic problems.