Solving 150 ÷ (20 - 50) + 95 A Step-by-Step Guide

In this article, we will delve into the step-by-step solution of the mathematical expression 150 ÷ (20 - 50) + 95. This problem serves as an excellent example to illustrate the importance of following the correct order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Neglecting this order can lead to incorrect results, highlighting the necessity of a systematic approach. Understanding the order of operations is not just a mathematical necessity; it's a fundamental skill that applies to various aspects of problem-solving and logical reasoning in everyday life. From managing finances to planning projects, the ability to prioritize and execute tasks in a specific order is crucial. Therefore, mastering the PEMDAS principle in mathematics equips individuals with a valuable tool that extends far beyond the classroom. This article aims to break down the expression into manageable steps, ensuring a clear and concise understanding of the solution process. By meticulously examining each operation, we will not only arrive at the correct answer but also reinforce the underlying principles of mathematical computation. The ultimate goal is to empower readers with the confidence and skills to tackle similar problems independently, fostering a deeper appreciation for the elegance and precision of mathematics. As we progress through the solution, we will emphasize the significance of each step, explaining the rationale behind the chosen order of operations. This approach will not only provide a correct answer but also enhance the reader's understanding of the fundamental concepts governing mathematical calculations. So, let's embark on this mathematical journey together and unravel the solution to the expression 150 ÷ (20 - 50) + 95, step by step.

Step 1: Parentheses First (20 - 50)

The first step in solving the expression 150 ÷ (20 - 50) + 95 is to address the operation within the parentheses. According to the order of operations, PEMDAS, parentheses take precedence over all other operations. This means we must calculate the value of (20 - 50) before proceeding with any other calculations. Inside the parentheses, we have a subtraction operation. Subtracting 50 from 20 results in a negative number. This concept of negative numbers is a fundamental aspect of mathematics and is crucial for understanding various mathematical principles. When subtracting a larger number from a smaller number, the result will always be negative. In this case, 20 - 50 = -30. The negative sign indicates that the result is less than zero on the number line. This understanding of negative numbers is not only essential for solving this particular problem but also for tackling more complex mathematical equations and real-world scenarios. For instance, in finance, negative numbers represent debts or losses, while in science, they can indicate temperatures below zero or altitudes below sea level. Therefore, grasping the concept of negative numbers is a vital skill for both mathematical proficiency and practical application. Now that we have calculated the value within the parentheses, we can substitute it back into the original expression. This gives us a new expression: 150 ÷ (-30) + 95. The parentheses are now resolved, and we can move on to the next step in the order of operations. This step-by-step approach ensures that we are following the correct procedure and minimizing the chances of errors. By breaking down the problem into smaller, manageable steps, we can gain a clearer understanding of the solution process and reinforce our grasp of the fundamental principles of mathematics. So, let's proceed to the next step and continue unraveling the solution to this mathematical expression.

Step 2: Division 150 ÷ (-30)

With the parentheses resolved, the next operation in the expression 150 ÷ (-30) + 95 according to PEMDAS is division. We now need to perform the division 150 ÷ (-30). This step involves dividing a positive number by a negative number, which is an important concept to understand in arithmetic. When dividing a positive number by a negative number, the result will always be a negative number. This is because division can be thought of as the inverse operation of multiplication, and the rules for multiplying positive and negative numbers also apply to division. Specifically, a positive number multiplied by a negative number yields a negative result, and similarly, a positive number divided by a negative number yields a negative result. To perform the division, we divide the absolute values of the numbers first. The absolute value of 150 is 150, and the absolute value of -30 is 30. So, we divide 150 by 30, which equals 5. However, since we are dividing a positive number by a negative number, the result will be negative. Therefore, 150 ÷ (-30) = -5. This result is crucial for the next step in solving the expression. It's essential to remember the rules for dividing positive and negative numbers to avoid errors in calculations. Understanding these rules is not only important for solving mathematical problems but also for applying mathematics in various real-world scenarios. For example, in physics, dividing a positive displacement by a negative time interval would give a negative velocity, indicating movement in the opposite direction. Now that we have completed the division, we can substitute the result back into the expression. This gives us -5 + 95. We have simplified the expression further, and only one operation remains. This methodical approach ensures that we are following the correct order of operations and moving closer to the final answer. So, let's proceed to the final step and complete the calculation.

Step 3: Addition -5 + 95

After performing the division, we are left with the expression -5 + 95. According to PEMDAS, addition is the final operation we need to perform in this problem. This step involves adding a negative number to a positive number. Adding a negative number is equivalent to subtracting the absolute value of the negative number from the positive number. In this case, we are adding -5 to 95, which is the same as subtracting 5 from 95. This concept is fundamental to understanding how positive and negative numbers interact in mathematical operations. Visualizing a number line can be helpful in understanding this concept. Starting at -5 on the number line, adding 95 means moving 95 units to the right. This will result in a positive number because we are moving past zero and into the positive side of the number line. Alternatively, starting at 95 on the number line, adding -5 means moving 5 units to the left, which is the same as subtracting 5. To perform the addition, we subtract the absolute value of -5 (which is 5) from 95. So, 95 - 5 = 90. Therefore, -5 + 95 = 90. This is the final result of the expression. We have successfully navigated through all the steps of the order of operations, ensuring that we arrived at the correct answer. This step-by-step approach not only provides the solution but also reinforces the importance of following the correct procedure in mathematical calculations. Understanding how to add positive and negative numbers is crucial for various mathematical applications, from basic arithmetic to more advanced algebraic equations. It is also essential for solving real-world problems involving finances, temperatures, and other quantities that can be represented by positive and negative numbers. Now that we have completed the final step, we can confidently state the solution to the expression 150 ÷ (20 - 50) + 95.

Final Answer: 90

In conclusion, by meticulously following the order of operations (PEMDAS), we have successfully solved the mathematical expression 150 ÷ (20 - 50) + 95. The step-by-step breakdown of the problem allowed us to understand each operation and its significance in arriving at the final answer. We began by addressing the parentheses, calculating (20 - 50) = -30. This initial step set the stage for the subsequent operations. Understanding the importance of parentheses in mathematical expressions is crucial, as they dictate the priority of calculations. Next, we performed the division, 150 ÷ (-30) = -5. This step highlighted the rules for dividing positive and negative numbers, reinforcing the concept that a positive number divided by a negative number results in a negative number. The division step further simplified the expression, bringing us closer to the final solution. Finally, we completed the addition, -5 + 95 = 90. This final step involved adding a negative number to a positive number, which is equivalent to subtracting the absolute value of the negative number from the positive number. The addition operation brought us to the final result, which is 90. Therefore, the solution to the expression 150 ÷ (20 - 50) + 95 is 90. This result demonstrates the power of following the order of operations and the importance of understanding the rules governing mathematical calculations. By mastering these fundamental principles, individuals can confidently tackle a wide range of mathematical problems and apply these skills in various real-world scenarios. The ability to solve mathematical expressions accurately is not only essential for academic success but also for critical thinking and problem-solving in everyday life. This exercise serves as a valuable reminder of the significance of mathematical literacy and its role in empowering individuals to navigate the complexities of the world around them. We hope this detailed explanation has provided a clear and comprehensive understanding of the solution process, fostering a deeper appreciation for the elegance and precision of mathematics.