Solving $3 \times 7 + 4 - 6 \div 3 - 7 + 45 \div 5 \times 4 + 49$ A Step-by-Step Guide

In the realm of mathematics, expressions often appear as puzzles waiting to be solved. This article delves into the step-by-step solution of the numerical expression $3 \times 7 + 4 - 6 \div 3 - 7 + 45 \div 5 \times 4 + 49$. We will meticulously break down the problem, applying the order of operations, also known as PEMDAS/BODMAS, to arrive at the correct answer. Understanding the order of operations is crucial in mathematics, as it ensures that complex expressions are evaluated consistently, leading to a unique and accurate solution.

Unraveling the Order of Operations: PEMDAS/BODMAS

The acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) serves as a guide to the correct sequence of operations in a mathematical expression. This hierarchical order dictates that operations within parentheses or brackets are performed first, followed by exponents or orders, then multiplication and division (from left to right), and finally, addition and subtraction (from left to right). By adhering to this order, we can systematically simplify complex expressions, avoiding ambiguity and ensuring accuracy.

Step 1: Multiplication and Division

Our mathematical journey begins with the multiplication and division operations, which hold equal precedence and are performed from left to right. Examining the expression $3 \times 7 + 4 - 6 \div 3 - 7 + 45 \div 5 \times 4 + 49$, we encounter several multiplication and division operations. Let's tackle them one by one:

• $3 \times 7 = 21$: The first multiplication yields 21, setting the stage for the next operation.
• $6 \div 3 = 2$: Next, we perform the division, obtaining 2.
• $45 \div 5 = 9$: The subsequent division results in 9.
• $9 \times 4 = 36$: Finally, we multiply 9 by 4, arriving at 36.

Now, our expression has been simplified to $21 + 4 - 2 - 7 + 36 + 49$, a form that is more manageable for the remaining operations.

Step 2: Addition and Subtraction

With the multiplication and division operations resolved, we move on to addition and subtraction, which also share equal precedence and are performed from left to right. Starting from the left, we systematically add and subtract the numbers:

• $21 + 4 = 25$: The initial addition yields 25.
• $25 - 2 = 23$: Subtracting 2 from 25, we get 23.
• $23 - 7 = 16$: Continuing the subtraction, we obtain 16.
• $16 + 36 = 52$: Adding 36 to 16, we arrive at 52.
• $52 + 49 = 101$: Finally, adding 49 to 52, we reach the grand total of 101.

Therefore, the solution to the mathematical expression $3 \times 7 + 4 - 6 \div 3 - 7 + 45 \div 5 \times 4 + 49$ is 101. By meticulously following the order of operations, we have successfully navigated the complexities of the expression and arrived at the correct answer.

Importance of Order of Operations

The order of operations is not merely a convention; it is a fundamental principle in mathematics that ensures consistency and accuracy in calculations. Without a standardized order, the same expression could yield multiple results, leading to confusion and errors. Consider the expression $3 + 4 \times 2$. If we perform the addition first, we get $7 \times 2 = 14$. However, if we adhere to PEMDAS/BODMAS and perform the multiplication first, we get $3 + 8 = 11$, which is the correct answer. This simple example illustrates the critical role of the order of operations in obtaining accurate results.

The order of operations is not only essential for solving mathematical expressions but also forms the bedrock of more advanced mathematical concepts. In algebra, calculus, and other higher-level mathematics, the order of operations is used extensively to simplify equations, solve problems, and develop mathematical models. A solid grasp of PEMDAS/BODMAS is therefore crucial for success in mathematics and related fields.

Real-World Applications of Mathematical Expressions

Mathematical expressions, like the one we solved in this article, are not confined to the realm of textbooks and classrooms. They permeate various aspects of our daily lives, from calculating expenses to planning budgets. Understanding how to solve these expressions can empower us to make informed decisions and navigate real-world scenarios with confidence.

For instance, imagine you are planning a road trip and need to calculate the total cost of fuel. You know the distance of the trip, the fuel efficiency of your car, and the price of gasoline. You can use a mathematical expression to combine these factors and determine the total fuel cost. Similarly, when calculating monthly expenses, you can use mathematical expressions to add up various costs, such as rent, utilities, and groceries, to arrive at a total expenditure.

In the business world, mathematical expressions are used extensively for financial analysis, project management, and resource allocation. Businesses use spreadsheets, which are essentially grids of mathematical expressions, to track income, expenses, and profits. They also use mathematical models to forecast sales, optimize production, and make strategic decisions. The ability to understand and manipulate mathematical expressions is therefore a valuable asset in various professional settings.

Conclusion: Mastering Mathematical Expressions

The journey through the expression $3 \times 7 + 4 - 6 \div 3 - 7 + 45 \div 5 \times 4 + 49$ has highlighted the importance of the order of operations in mathematics. By adhering to PEMDAS/BODMAS, we were able to systematically simplify the expression and arrive at the correct answer, 101. This exercise underscores the critical role of order of operations in ensuring accuracy and consistency in mathematical calculations.

The ability to solve mathematical expressions is not just an academic skill; it is a valuable tool that can be applied in various real-world scenarios. From calculating personal finances to making informed business decisions, the power of mathematical expressions is undeniable. By mastering the order of operations and practicing problem-solving, we can unlock the full potential of mathematics and enhance our ability to navigate the complexities of the world around us.

Therefore, embrace the challenge of mathematical expressions, and delve into the world of numbers and symbols. With each problem solved, your mathematical skills will strengthen, and your confidence in tackling complex challenges will grow. Remember, mathematics is not just about numbers; it is about logical thinking, problem-solving, and the pursuit of knowledge.