Solving 150 ÷ (20 - 50) + 95 A Step-by-Step Guide
In the realm of mathematics, the order of operations is paramount to achieving accurate results. This principle, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which mathematical operations should be performed. In this comprehensive guide, we will delve into the step-by-step solution of the expression 150 ÷ (20 - 50) + 95, elucidating the application of PEMDAS and providing a clear understanding of the underlying concepts. This problem serves as an excellent example to reinforce your understanding of the order of operations and how it impacts mathematical calculations. Whether you're a student learning the basics or someone looking to brush up on your math skills, this detailed explanation will provide you with the knowledge and confidence to tackle similar problems.
Understanding the Order of Operations (PEMDAS/BODMAS)
Before we dive into the solution, it's crucial to grasp the fundamental principles of the order of operations. PEMDAS, or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) in some regions, is the cornerstone of mathematical problem-solving. This acronym dictates the precise sequence in which operations should be performed to arrive at the correct answer. Ignoring this order can lead to significantly different and incorrect results. The PEMDAS/BODMAS hierarchy ensures consistency and accuracy in mathematical calculations across various contexts and applications. Let's break down each component of PEMDAS:
- Parentheses/Brackets: Operations enclosed within parentheses or brackets take precedence. This means any calculations inside these groupings must be performed first. This step is crucial as it often dictates the initial steps in simplifying an expression. Think of parentheses as containers that need to be resolved before moving on to other operations.
- Exponents/Orders: Next in line are exponents or orders, which involve powers and roots. These operations indicate repeated multiplication or the inverse operation of finding the root of a number. Understanding exponents is vital in various fields, from scientific notation to financial calculations. For instance, squaring a number or finding the square root are common exponential operations.
- Multiplication and Division: Multiplication and division hold equal priority and are performed from left to right. This means you tackle these operations in the order they appear in the expression. If division comes before multiplication, you perform division first, and vice versa. This left-to-right rule is essential to avoid ambiguity and ensure the correct result.
- Addition and Subtraction: Similar to multiplication and division, addition and subtraction also have equal priority and are performed from left to right. Again, the order in which these operations appear dictates the sequence of calculation. This final step completes the simplification of the expression, leading to the final answer.
Understanding and applying PEMDAS/BODMAS correctly is not just a mathematical exercise; it's a fundamental skill that underpins many areas of science, engineering, and finance. It provides a structured approach to problem-solving, ensuring that calculations are performed in a logical and consistent manner. Without this framework, mathematical expressions could be interpreted in multiple ways, leading to confusion and errors.
Step-by-Step Solution of 150 ÷ (20 - 50) + 95
Now that we've established the importance of the order of operations, let's apply it to solve the expression 150 ÷ (20 - 50) + 95 step by step. By meticulously following PEMDAS, we can break down the problem into manageable parts and arrive at the correct solution. Each step is explained in detail to ensure clarity and understanding.
Step 1: Parentheses
The first step, according to PEMDAS, is to address the operation within the parentheses: (20 - 50). This is a subtraction operation, and performing it yields:
20 - 50 = -30
This result now replaces the original parentheses in the expression. The expression is simplified to:
150 ÷ (-30) + 95
By resolving the parentheses first, we've taken the crucial first step in simplifying the expression. This step highlights the importance of prioritizing operations within parentheses as they often influence the subsequent steps in the calculation.
Step 2: Division
Following PEMDAS, the next operation to perform is division. We have 150 ÷ (-30). Dividing 150 by -30 gives us:
150 ÷ (-30) = -5
Remember that dividing a positive number by a negative number results in a negative quotient. This result now replaces the division operation in the expression. The expression is further simplified to:
-5 + 95
This step demonstrates the next level of simplification by addressing the division operation. By following the order of operations, we're systematically reducing the complexity of the expression.
Step 3: Addition
The final step, according to PEMDAS, is addition. We now have -5 + 95. Adding -5 to 95 gives us:
-5 + 95 = 90
This is the final result of the expression. By performing the addition, we've completed the simplification process and arrived at the solution.
Therefore, the solution to the expression 150 ÷ (20 - 50) + 95 is 90.
This step-by-step solution clearly illustrates the power of PEMDAS in simplifying complex expressions. By adhering to the correct order of operations, we can confidently solve mathematical problems and arrive at accurate results.
Common Mistakes to Avoid
Understanding the order of operations is crucial, but it's equally important to be aware of common mistakes that can occur when applying PEMDAS/BODMAS. These errors often stem from a misunderstanding of the priority of operations or a simple oversight in the calculation process. Being mindful of these potential pitfalls can help you avoid mistakes and ensure accurate results. Let's explore some frequent errors:
-
Ignoring Parentheses: One of the most common mistakes is neglecting the parentheses or brackets. Remember, operations within parentheses must be performed first. Failing to do so can drastically alter the outcome of the calculation. Always prioritize parentheses to maintain the correct order.
-
Incorrect Order of Multiplication and Division: Multiplication and division have equal priority, and they should be performed from left to right. A common mistake is to perform multiplication before division, regardless of their order in the expression. This can lead to an incorrect result. Ensure you address these operations in the order they appear from left to right.
-
Incorrect Order of Addition and Subtraction: Similar to multiplication and division, addition and subtraction have equal priority and should be performed from left to right. Performing addition before subtraction (or vice versa) without considering their order in the expression is a common error. Always follow the left-to-right rule for these operations.
-
Misunderstanding Negative Signs: Negative signs can be tricky, especially when combined with other operations. For example, subtracting a negative number is the same as adding its positive counterpart. Misinterpreting negative signs can lead to significant errors. Pay close attention to the signs and their impact on the operations.
-
Rushing Through the Steps: Math problems, especially those involving multiple operations, require careful attention to detail. Rushing through the steps can lead to mistakes, even if you understand the underlying principles. Take your time, double-check your work, and ensure each step is performed correctly.
By being aware of these common mistakes, you can proactively avoid them and improve your accuracy in mathematical calculations. Consistent practice and attention to detail are key to mastering the order of operations and minimizing errors.
Practice Problems to Enhance Your Skills
To solidify your understanding of the order of operations and enhance your problem-solving skills, it's essential to engage in practice. Solving a variety of problems will expose you to different scenarios and help you apply PEMDAS/BODMAS effectively. Practice not only reinforces the rules but also builds your confidence in tackling complex mathematical expressions. Here are some practice problems to challenge yourself:
2 × (10 + 5) - 15 ÷ 336 ÷ (2 × 3) + 4^2 - 8100 - (40 ÷ 5) + 3 × 7(12 - 4) × 5 + 18 ÷ 25^2 - (3 × 8) + 24 ÷ 6
For each problem, remember to follow PEMDAS/BODMAS diligently. Start with the parentheses, then address exponents, followed by multiplication and division (from left to right), and finally, addition and subtraction (from left to right). Working through these problems step by step will help you internalize the order of operations and develop a systematic approach to problem-solving.
After solving each problem, it's beneficial to check your answers and review your steps. If you encounter any difficulties, revisit the explanations and examples provided earlier in this guide. Identifying and correcting your mistakes is a crucial part of the learning process. Consistent practice and self-assessment will significantly improve your mathematical proficiency.
Conclusion
In conclusion, mastering the order of operations is fundamental to success in mathematics and various other fields. The principle of PEMDAS/BODMAS provides a clear and consistent framework for solving mathematical expressions, ensuring accuracy and avoiding ambiguity. Throughout this guide, we've explored the importance of each step in PEMDAS, from parentheses to exponents, multiplication and division, and finally, addition and subtraction. We've also provided a detailed, step-by-step solution to the expression 150 ÷ (20 - 50) + 95, illustrating the practical application of these principles.
Furthermore, we've highlighted common mistakes to avoid, such as ignoring parentheses or misinterpreting the order of multiplication and division. By being aware of these potential pitfalls, you can proactively prevent errors and enhance your problem-solving skills. The practice problems offered in this guide provide an opportunity to reinforce your understanding and build confidence in tackling complex mathematical expressions.
Remember, the key to mastering the order of operations lies in consistent practice and a meticulous approach to problem-solving. By applying PEMDAS/BODMAS diligently, you can confidently navigate mathematical challenges and achieve accurate results. Whether you're a student learning the basics or a professional applying mathematical concepts in your field, a solid grasp of the order of operations is an invaluable asset.