Equivalent Expression For 100 + 20: A Math Problem

by Alex Johnson 51 views

Let's dive into a common math problem: finding the expression equivalent to 100 + 20. This type of question often appears in early algebra and arithmetic, testing your understanding of the distributive property and basic arithmetic operations. In this article, we'll break down the problem, examine the options, and arrive at the correct answer. We’ll also discuss why understanding these concepts is crucial for more advanced mathematics.

Understanding the Problem

At the heart of this question is the concept of equivalence in mathematical expressions. To put it simply, two expressions are equivalent if they yield the same result. Our task is to determine which of the given options produces the same value as 100 + 20. This requires us to evaluate the initial expression and then test each option to see if it matches. The initial expression, 100 + 20, is straightforward. Adding these two numbers together, we get 120. Therefore, we need to find an expression among the choices that also equals 120. This involves understanding order of operations (PEMDAS/BODMAS) and the distributive property, which are fundamental concepts in mathematics. The distributive property states that a(b + c) = ab + ac. We'll use this property to simplify some of the options and compare them with our target value of 120. Equivalence in mathematics isn't just about getting the same numerical answer; it’s also about understanding how different mathematical forms can represent the same value. This understanding is crucial in algebra, where simplifying expressions and solving equations often involve rewriting expressions in equivalent forms. Without a solid grasp of equivalence, manipulating equations becomes significantly harder. Therefore, mastering these foundational skills is not just about solving the problem at hand but also preparing for future mathematical challenges.

Analyzing the Options

Now, let's meticulously analyze each option provided to determine which one is equivalent to 100 + 20. We will evaluate each expression step-by-step, ensuring clarity and accuracy in our calculations. This process will not only help us identify the correct answer but also reinforce our understanding of mathematical operations and properties.

Option A: 5(20 + 4)

To evaluate this expression, we need to follow the order of operations, often remembered by the acronym PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). First, we address the operation within the parentheses: 20 + 4 = 24. Next, we multiply this result by 5: 5 * 24. To calculate 5 * 24, we can break it down if needed: 5 * 20 = 100 and 5 * 4 = 20. Adding these together, 100 + 20, gives us 120. Thus, 5(20 + 4) = 120. This option initially looks promising since it matches our target value. However, we need to evaluate all options to be certain.

Option B: 10(10 + 10)

Similar to Option A, we begin by simplifying the expression inside the parentheses: 10 + 10 = 20. Then, we multiply this result by 10: 10 * 20. This calculation is straightforward: 10 * 20 = 200. Therefore, 10(10 + 10) = 200. This value does not match our target of 120, so Option B is not the correct answer. Understanding why an option is incorrect is just as important as knowing the right answer. It reinforces the underlying mathematical principles and helps avoid similar mistakes in the future. In this case, the result of 200 clearly deviates from our required value, highlighting the importance of accurate calculation.

Option C: 10(10 + 3)

Following the same procedure, we first simplify the expression within the parentheses: 10 + 3 = 13. Then, we multiply this result by 10: 10 * 13. This calculation gives us 130. So, 10(10 + 3) = 130. This value also does not equal our target of 120, eliminating Option C as a possibility. The process of elimination is a valuable strategy in problem-solving, particularly in multiple-choice questions. By systematically ruling out incorrect options, we can narrow down the possibilities and increase our chances of selecting the correct answer. Furthermore, this method helps to identify specific areas where errors might occur, prompting a more focused review.

Option D: 5(20 + 3)

Again, we start by simplifying the expression inside the parentheses: 20 + 3 = 23. Next, we multiply this result by 5: 5 * 23. To calculate 5 * 23, we can break it down: 5 * 20 = 100 and 5 * 3 = 15. Adding these together, 100 + 15, gives us 115. Therefore, 5(20 + 3) = 115. This value does not match our target of 120, confirming that Option D is not the correct answer. Each calculation provides an opportunity to reinforce basic arithmetic skills. The ability to perform these calculations accurately and efficiently is essential for success in mathematics. Moreover, understanding the process behind each step enhances problem-solving confidence.

Determining the Correct Answer

After meticulously evaluating each option, we found that only one expression yields a result equivalent to 100 + 20, which equals 120. Let's recap our findings:

  • Option A: 5(20 + 4) = 5 * 24 = 120
  • Option B: 10(10 + 10) = 10 * 20 = 200
  • Option C: 10(10 + 3) = 10 * 13 = 130
  • Option D: 5(20 + 3) = 5 * 23 = 115

As we can see, Option A, 5(20 + 4), is the only expression that equals 120. Therefore, Option A is the correct answer. This methodical approach to solving the problem ensures accuracy and provides a clear understanding of the underlying mathematical principles. The confirmation of the correct answer is not just about finding the right solution but also about validating the entire process. It reinforces the understanding of the concepts involved and builds confidence in problem-solving abilities. Furthermore, it underscores the importance of careful calculation and attention to detail in mathematical problem-solving.

Why This Matters

Understanding how to find equivalent expressions is a cornerstone of mathematics. This concept isn't just limited to basic arithmetic; it's a fundamental skill that's essential for more advanced mathematical topics such as algebra, calculus, and beyond. In algebra, for instance, simplifying expressions and solving equations often involve rewriting expressions in equivalent forms. If you can't quickly recognize equivalent expressions, you'll struggle with these tasks. Moreover, the ability to manipulate mathematical expressions is crucial in real-world applications. From engineering to finance, professionals frequently need to simplify complex formulas and equations. Consider a scenario in engineering where you're calculating the force required to move an object. The initial formula might be complex, but by using your knowledge of equivalent expressions, you can simplify it to a more manageable form. This could save time and reduce the likelihood of errors. Similarly, in finance, calculating investment returns or loan payments often involves complex equations. Being able to simplify these equations makes the calculations easier and less prone to mistakes. The concept of equivalence also extends beyond numerical calculations. In logic and computer science, for example, understanding logical equivalence is critical for designing efficient algorithms and circuits. In summary, mastering the skill of identifying and manipulating equivalent expressions is an investment in your mathematical future. It's a skill that will pay dividends in your academic pursuits, professional career, and everyday life.

Conclusion

In conclusion, the expression equivalent to 100 + 20 is A. 5(20 + 4). We arrived at this answer by carefully evaluating each option and applying the order of operations. This problem highlights the importance of understanding basic arithmetic principles and the concept of equivalence in mathematics. Remember, the ability to manipulate and simplify expressions is a crucial skill that will serve you well in various fields. Keep practicing and building your mathematical foundation, and you'll be well-equipped to tackle more complex problems in the future.

For further exploration and practice on equivalent expressions, check out Khan Academy's section on equivalent algebraic expressions.