Solving 125 × (42 - 17) + 36 A Step-by-Step Guide
Let's break down this mathematical problem step by step, guys! We've got a combination of multiplication, subtraction, and addition here, so it's crucial to follow the order of operations (PEMDAS/BODMAS) to get the correct answer. PEMDAS stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This mnemonic helps us remember the sequence in which we should perform mathematical operations.
Step 1: Solve the Parentheses
First things first, we need to tackle what's inside the parentheses: (42 - 17). This is a simple subtraction problem. When we subtract 17 from 42, we get 25. So, we can rewrite the expression as: 125 × 25 + 36. Simplifying the expression inside the parentheses is a fundamental step because it isolates the value that needs to be used in the subsequent operation. It's like setting the stage for the main act by clearing away any distractions. This initial step ensures that we're working with the core numbers that will contribute to the final answer.
Step 2: Multiplication
Now that we've dealt with the parentheses, the next operation in line is multiplication. We have 125 × 25. To solve this, we can multiply 125 by 25. This might seem a bit daunting at first, but let's break it down. Multiplying 125 by 20 gives us 2500 (since 125 × 2 = 250, then add a zero). Then, we multiply 125 by 5, which gives us 625. Adding these two results together, 2500 + 625, gives us 3125. So, 125 × 25 = 3125. Performing multiplication accurately is crucial because it forms the backbone of this calculation. It transforms the initial numbers into a significant intermediate value that carries weight in the final result. A mistake in the multiplication step can throw off the entire calculation, emphasizing the need for precision and careful execution.
Step 3: Addition
We're almost there! The final step is addition. We have 3125 + 36. This is a straightforward addition problem. Adding 36 to 3125 is relatively simple. We can think of it as adding 30 to 3125, which gives us 3155, and then adding the remaining 6, which gives us 3161. So, 3125 + 36 = 3161. Addition is the concluding step that brings all the intermediate results together to form the final answer. It's the culmination of all the previous operations, and its accuracy depends on the correct execution of each preceding step. Getting the addition right ensures that the final result is a true reflection of the initial expression.
Final Result
Therefore, the final answer to the calculation 125 × (42 - 17) + 36 is 3161. By following the order of operations and breaking down the problem into smaller, manageable steps, we've successfully solved it! Remember, PEMDAS/BODMAS is your friend in these situations. Understanding the order of operations is paramount in mathematics. It's the universal language that ensures everyone arrives at the same answer when solving a problem. Without this order, mathematical expressions would be open to interpretation, leading to chaos and inconsistency. By adhering to PEMDAS/BODMAS, we maintain clarity and precision in our calculations, ensuring that each step logically follows the previous one, ultimately leading to the correct solution. This structured approach not only helps in solving complex problems but also builds a strong foundation for more advanced mathematical concepts.
To fully grasp how we arrived at the solution, let's delve into the key mathematical concepts we employed. These concepts are the building blocks of not just this problem, but also a wide array of mathematical challenges. Understanding them thoroughly will empower you to tackle similar problems with confidence and accuracy.
Order of Operations (PEMDAS/BODMAS)
The cornerstone of this calculation is the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) in the United States, or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) in other parts of the world. This set of rules dictates the sequence in which mathematical operations should be performed to ensure consistency and accuracy in calculations. It's like a mathematical grammar, guiding us on how to structure and solve expressions. The order of operations is not just a set of rules; it's a fundamental principle that underpins mathematical logic. It ensures that mathematical expressions have a single, unambiguous interpretation. Without it, the same expression could yield different results depending on the order in which operations are performed. This would lead to confusion and invalidate the very purpose of mathematics as a precise and consistent language.
Parentheses/Brackets
Operations enclosed within parentheses or brackets are always performed first. This is because parentheses indicate a priority, signaling that the enclosed expression should be treated as a single entity. They act like a container, isolating a particular calculation that needs to be resolved before interacting with the rest of the expression. Think of parentheses as a way of creating a mini-problem within the larger problem. Solving what's inside the parentheses first is like solving a puzzle piece before fitting it into the larger picture. It simplifies the overall expression by reducing the complexity of the calculation step by step. This prioritization is crucial for maintaining the integrity of the mathematical statement and ensuring that the final answer is logically derived.
Exponents/Orders
Next in line are exponents or orders (powers and roots). These operations involve raising a number to a certain power or finding the root of a number. Exponents represent repeated multiplication, while roots represent the inverse operation of exponentiation. Understanding exponents and orders is essential for dealing with quantities that grow or shrink rapidly. They appear in various fields, from calculating compound interest to modeling population growth. Mastering exponents and roots allows us to express and manipulate large and small quantities efficiently. They add a layer of sophistication to our mathematical toolkit, enabling us to tackle problems that involve exponential relationships.
Multiplication and Division
Multiplication and division are performed next, from left to right. These operations are inverse to each other; multiplication combines quantities, while division separates them. They are fundamental to arithmetic and are used extensively in everyday calculations. The left-to-right rule is important when both multiplication and division are present in the same expression. It ensures that the calculation is performed in the correct sequence, just like reading a sentence from left to right. This consistency is crucial for arriving at the correct answer and avoiding ambiguity.
Addition and Subtraction
Finally, addition and subtraction are performed, also from left to right. These are the basic operations of arithmetic, representing the combining and taking away of quantities. Like multiplication and division, addition and subtraction are inverse operations. They are the foundation upon which more complex mathematical concepts are built. The left-to-right rule applies here as well, ensuring that the calculation is performed in a consistent manner. Addition and subtraction are not just about numbers; they also represent real-world scenarios such as combining resources or tracking expenses. Mastering these operations is essential for both mathematical proficiency and practical life skills.
Basic Arithmetic Operations
This problem also reinforces the basic arithmetic operations: addition, subtraction, multiplication, and division. These are the fundamental building blocks of mathematics, and a solid understanding of these operations is crucial for tackling more complex problems. Each operation serves a unique purpose in the world of mathematics:
Addition
Addition is the process of combining two or more numbers to find their total sum. It's the most basic of arithmetic operations and is used in countless everyday situations, from calculating grocery bills to measuring ingredients for a recipe. The symbol for addition is the plus sign (+). Addition is not just about counting; it's about understanding how quantities combine and grow. It's the foundation for more advanced concepts such as algebra and calculus. Mastering addition is essential for building a strong mathematical foundation.
Subtraction
Subtraction is the process of finding the difference between two numbers. It's the inverse operation of addition and is used to determine how much is left after taking away a certain amount. The symbol for subtraction is the minus sign (-). Subtraction is a crucial skill for managing finances, calculating change, and solving problems that involve reducing quantities. It helps us understand the relationship between numbers and how they change relative to each other. Just like addition, subtraction is a cornerstone of mathematical literacy.
Multiplication
Multiplication is a shorthand way of adding the same number multiple times. It's a fundamental operation for scaling quantities and calculating areas and volumes. The symbol for multiplication is often represented by a times sign (×) or a dot (·). Multiplication allows us to perform repeated addition efficiently. It's used in a wide range of applications, from calculating the total cost of multiple items to determining the distance traveled at a constant speed. Understanding multiplication is essential for solving problems involving ratios, proportions, and exponential growth.
Division
Division is the process of splitting a number into equal parts. It's the inverse operation of multiplication and is used to determine how many times one number fits into another. The symbol for division is often represented by a division sign (÷) or a fraction bar (/). Division helps us understand how quantities can be shared or distributed equally. It's crucial for solving problems involving fractions, percentages, and ratios. Mastering division is essential for developing a comprehensive understanding of arithmetic.
In conclusion, solving the problem 125 × (42 - 17) + 36 = 3161 involves applying the order of operations (PEMDAS/BODMAS) and a firm grasp of basic arithmetic. These skills are fundamental to mathematical literacy and problem-solving. By understanding and practicing these concepts, you'll be well-equipped to tackle a wide range of mathematical challenges. Remember, mathematics is not just about memorizing formulas; it's about developing a logical and systematic approach to problem-solving. So, keep practicing, keep exploring, and keep pushing your mathematical boundaries!
