Equivalent Expressions Explained Finding The Expression Equivalent To 100 + 50

In the realm of mathematics, understanding equivalent expressions is a fundamental concept. An equivalent expression is simply a different way of writing the same mathematical idea. It's like saying the same thing in different words – the meaning stays the same, even if the presentation is different. This concept is especially important when we start dealing with more complex algebraic manipulations and problem-solving. In this article, we will explore this concept by dissecting the expression 100 + 50 and determining which of the given options is equivalent to it. We will delve into the arithmetic operations involved, break down each option, and provide a clear, step-by-step explanation to ensure a thorough understanding. The objective is not just to find the correct answer but also to grasp the underlying principles of equivalent expressions, which is crucial for success in higher-level mathematics.

Breaking Down the Original Expression: 100 + 50

Before we dive into the multiple-choice options, let’s first establish a solid understanding of the expression we are trying to match: 100 + 50. This is a straightforward addition problem. We are simply adding two numbers together. The sum of 100 and 50 is, without any doubt, 150. This result, 150, serves as our target. Any expression that simplifies to 150 is equivalent to our original expression. This simple calculation forms the foundation for our analysis. Now, we can proceed to examine each option and see if it simplifies to the same value. We'll be using the order of operations (PEMDAS/BODMAS) to ensure we simplify each expression correctly. Remember, the order of operations dictates the sequence in which we perform mathematical operations: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Mastering this order is critical for accurate simplification and comparing expressions effectively. Let's move on to analyzing the first option.

Evaluating Option A: 2(50 + 24)

Now, let's turn our attention to the first option, A. 2(50 + 24). To determine if this expression is equivalent to 100 + 50, we need to simplify it using the order of operations. The first step, according to PEMDAS/BODMAS, is to address the parentheses. Inside the parentheses, we have the addition operation: 50 + 24. Performing this addition, we get 74. So, the expression now becomes 2(74). The next operation is multiplication. We are multiplying 2 by 74. When we multiply 2 by 74, we get 148. Therefore, the simplified value of the expression 2(50 + 24) is 148. Comparing this result to our target value of 150, we can see that 148 is not equal to 150. This means that option A is not equivalent to the original expression. Although it's close, the difference of 2 is enough to disqualify it. Understanding this process of simplification is key to mastering equivalent expressions. Now, let’s proceed to examine the next option and see if it matches our target.

Analyzing Option B: 2(50 + 48)

Moving on to option B, we have the expression 2(50 + 48). Similar to option A, we will simplify this using the order of operations. We begin by looking inside the parentheses. Here, we have the addition 50 + 48. Adding these two numbers together, we get 98. So, the expression now becomes 2(98). The next step is to perform the multiplication. We need to multiply 2 by 98. When we multiply 2 by 98, the result is 196. So, the simplified value of the expression 2(50 + 48) is 196. Comparing this result to our target value of 150, it is clear that 196 is significantly larger than 150. This indicates that option B is not equivalent to the original expression. The large difference between 196 and 150 further reinforces the importance of accurate calculations when dealing with equivalent expressions. Now, let’s continue our analysis by examining the next option and see if it matches our target value of 150.

Dissecting Option C: 25(4 + 50)

Let's consider option C, which presents the expression 25(4 + 50). Following the established pattern, we will simplify this expression using the order of operations. Our first focus is on the parentheses. Inside the parentheses, we have the addition 4 + 50. Performing this addition, we get 54. So, the expression now becomes 25(54). The next operation is multiplication. We are multiplying 25 by 54. This multiplication yields a result of 1350. Therefore, the simplified value of the expression 25(4 + 50) is 1350. When we compare this result to our target value of 150, we find a substantial difference. 1350 is significantly larger than 150. This clearly indicates that option C is not equivalent to the original expression. The large magnitude of the result underscores the critical role each number plays in determining the final value of an expression. Now, we have only one option left to consider. Let’s analyze option D and see if it matches our target value of 150.

Evaluating Option D: 5(20 + 10)

Finally, we arrive at option D, which presents the expression 5(20 + 10). As with the previous options, we will simplify this expression following the order of operations. Our first step is to address the parentheses. Inside the parentheses, we have the addition operation: 20 + 10. Performing this addition gives us 30. So, the expression now becomes 5(30). The next operation is multiplication. We are multiplying 5 by 30. When we multiply 5 by 30, we get 150. Therefore, the simplified value of the expression 5(20 + 10) is 150. Now, we compare this result to our target value, which is also 150. We see that the simplified value of option D, 150, is exactly equal to our target value. This means that option D is equivalent to the original expression, 100 + 50. This confirms that D. 5(20 + 10) is the correct answer. We have successfully identified the equivalent expression by systematically simplifying each option and comparing the results.

Conclusion: The Power of Equivalent Expressions

In conclusion, the expression equivalent to 100 + 50 is D. 5(20 + 10). We arrived at this answer by systematically simplifying each of the given options using the order of operations and comparing the results to our target value of 150. This exercise highlights the importance of understanding equivalent expressions in mathematics. Equivalent expressions are different ways of representing the same mathematical value. Being able to identify and manipulate equivalent expressions is a crucial skill for success in algebra and beyond. It allows us to solve problems in different ways, simplify complex equations, and gain a deeper understanding of mathematical relationships. Mastering the order of operations is fundamental to this process. By following the correct sequence of steps, we can accurately simplify expressions and avoid errors. This detailed exploration not only provides the answer to the specific question but also reinforces the broader concept of equivalent expressions and their significance in mathematics. Understanding these principles will undoubtedly be beneficial as you continue your mathematical journey.