Solving Mathematical Expressions (b) {17 × (112 - 78)} ÷ 289 And (d) (700 ÷ 10) - {(12 × 8) ÷ (34 - 10)}
Introduction
In this article, we will delve into the step-by-step solutions of two mathematical expressions: (b) {17 × (112 - 78)} ÷ 289 and (d) (700 ÷ 10) - {(12 × 8) ÷ (34 - 10)}. These expressions involve a combination of arithmetic operations including multiplication, subtraction, division, and the use of parentheses and braces to define the order of operations. Understanding how to solve such expressions is fundamental to mastering basic arithmetic and algebra. We will break down each expression into manageable steps, ensuring clarity and accuracy in our calculations. This exploration will not only help in understanding the specific problems but also in developing a general approach to solving mathematical expressions of varying complexity.
Solving Expression (b): {17 × (112 - 78)} ÷ 289
Let's begin by dissecting the first expression: {17 × (112 - 78)} ÷ 289. To accurately solve this, we must adhere to the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This means we will first address the operations within the parentheses, then multiplication, and finally division.
Step 1: Parentheses
The initial step involves solving the operation within the parentheses: (112 - 78). This subtraction is straightforward:
112 - 78 = 34
So, the expression now simplifies to {17 × 34} ÷ 289.
Step 2: Multiplication
Next, we perform the multiplication within the braces: 17 × 34. This step requires multiplying these two numbers:
17 × 34 = 578
The expression is now further simplified to 578 ÷ 289.
Step 3: Division
The final operation is division: 578 ÷ 289. Dividing 578 by 289 gives us:
578 ÷ 289 = 2
Therefore, the solution to the expression {17 × (112 - 78)} ÷ 289 is 2. This methodical approach, focusing on each step individually, ensures accuracy and clarity in the solution process. Understanding and applying the order of operations is crucial in solving mathematical expressions correctly.
Solving Expression (d): (700 ÷ 10) - {(12 × 8) ÷ (34 - 10)}
Now, let's tackle the second expression: (700 ÷ 10) - {(12 × 8) ÷ (34 - 10)}. This expression is more complex, involving multiple operations and nested parentheses. Once again, we will follow the order of operations (PEMDAS) to ensure accurate calculation. This involves addressing parentheses first, followed by division and multiplication from left to right, and finally subtraction.
Step 1: Innermost Parentheses
We begin with the innermost parentheses: (34 - 10). Subtracting 10 from 34 is a simple operation:
34 - 10 = 24
The expression now becomes (700 ÷ 10) - {(12 × 8) ÷ 24}.
Step 2: Remaining Parentheses and Braces
Next, we address the remaining operations within the parentheses and braces. This involves two separate calculations. First, the division 700 ÷ 10:
700 ÷ 10 = 70
Then, within the braces, we have (12 × 8) ÷ 24. We first perform the multiplication:
12 × 8 = 96
Then, we divide the result by 24:
96 ÷ 24 = 4
The expression now simplifies to 70 - 4.
Step 3: Subtraction
The final step is subtraction: 70 - 4. Subtracting 4 from 70 is straightforward:
70 - 4 = 66
Thus, the solution to the expression (700 ÷ 10) - {(12 × 8) ÷ (34 - 10)} is 66. This detailed breakdown illustrates how adhering to the order of operations allows us to systematically solve complex mathematical expressions.
Importance of Order of Operations (PEMDAS/BODMAS)
Understanding and applying the correct order of operations is crucial in mathematics. The acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) and BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) serve as handy mnemonics for this order. Neglecting this order can lead to incorrect results, as the sequence in which operations are performed significantly impacts the outcome. For instance, in the expression (700 ÷ 10) - {(12 × 8) ÷ (34 - 10)}, if we were to subtract before dividing or multiplying, the result would be drastically different and incorrect. The consistent application of PEMDAS/BODMAS ensures that mathematical expressions are evaluated uniformly and accurately, making it a foundational concept in arithmetic and algebra.
In complex expressions, the presence of parentheses, brackets, and braces indicates priority levels, with operations inside these groupings performed before those outside. This hierarchical approach allows for the precise definition of mathematical relationships and ensures that calculations are performed in the intended sequence. Mastering the order of operations not only aids in solving equations but also in understanding mathematical logic and the structure of mathematical statements. This understanding is vital for more advanced mathematical studies and practical applications in various fields, including science, engineering, and finance.
Common Mistakes and How to Avoid Them
When solving mathematical expressions, several common mistakes can lead to incorrect answers. One of the most frequent errors is neglecting the order of operations (PEMDAS/BODMAS). For example, performing addition before multiplication or subtraction before division can drastically alter the outcome. Another common mistake is misinterpreting the role of parentheses and brackets, leading to operations being performed out of sequence. Sign errors, particularly with negative numbers, are also prevalent, especially when dealing with subtraction or multiplication of negative values.
To avoid these pitfalls, a systematic approach is essential. Always begin by identifying the operations within parentheses or brackets and working outwards. It's helpful to rewrite the expression after each step, simplifying it incrementally and reducing the chances of error. Double-checking each calculation and paying close attention to signs can also prevent mistakes. Practice is key to mastering the order of operations and developing a strong sense of mathematical accuracy. Regularly solving a variety of problems, from simple to complex, reinforces the correct application of PEMDAS/BODMAS and helps build confidence in your mathematical abilities. Additionally, using tools like calculators or online solvers can be beneficial for verifying answers, but it's crucial to understand the process rather than relying solely on these aids.
Practical Applications of These Concepts
The mathematical concepts discussed, such as the order of operations and solving complex expressions, are not confined to the classroom; they have widespread practical applications in various fields. In finance, for example, calculating interest, returns on investments, or budgeting requires a solid understanding of arithmetic operations and their correct order. Similarly, in engineering and physics, complex calculations involving multiple variables and operations are commonplace, and accurate application of PEMDAS/BODMAS is crucial for obtaining correct results. Computer programming also relies heavily on these concepts, as code often involves mathematical expressions that must be evaluated in a specific order to produce the desired output.
Beyond these professional applications, these mathematical skills are valuable in everyday life. From calculating discounts while shopping to determining cooking times based on recipe adjustments, the ability to perform arithmetic operations accurately is essential. Understanding these concepts also fosters critical thinking and problem-solving skills, which are transferable to various aspects of life. Whether it's planning a budget, managing finances, or simply solving a puzzle, the principles learned in mathematics provide a foundation for logical reasoning and decision-making. Thus, mastering these fundamental mathematical skills is not just an academic exercise but a practical necessity for navigating the complexities of the modern world.
Conclusion
In conclusion, we have meticulously solved two mathematical expressions: (b) {17 × (112 - 78)} ÷ 289 and (d) (700 ÷ 10) - {(12 × 8) ÷ (34 - 10)}. By adhering to the order of operations (PEMDAS/BODMAS), we systematically broke down each expression into manageable steps, ensuring accurate calculations. This process not only provides the solutions to these specific problems but also underscores the importance of a methodical approach to solving mathematical expressions in general. Understanding and applying the order of operations is a fundamental skill in mathematics, essential for both academic and practical applications.
The ability to solve complex expressions is not just about arriving at the correct answer; it's about developing a logical and structured way of thinking. The principles discussed here extend beyond arithmetic and algebra, influencing problem-solving skills in various domains. From financial planning to scientific research, the ability to dissect a problem into its constituent parts and address them systematically is invaluable. Therefore, mastering these mathematical concepts is an investment in one's overall analytical and problem-solving capabilities, contributing to success in both professional and personal endeavors.
