Hey guys! Today, we're diving deep into a fundamental concept in mathematics: equivalent expressions. Specifically, we're tackling the question of which expression is equivalent to 90 + 18. This might seem straightforward, but it's a fantastic opportunity to flex our mathematical muscles and understand the principles of factoring and the distributive property. So, grab your thinking caps, and let's get started!
Understanding Equivalent Expressions
Before we jump into solving the problem, let's make sure we're all on the same page about what equivalent expressions actually are. In simple terms, equivalent expressions are mathematical expressions that might look different but have the same value. Think of it like this: you can write the number '4' in many ways (2 + 2, 1 + 3, 2 * 2), but they all represent the same quantity. In algebra, we often manipulate expressions to find equivalent forms that are easier to work with or provide different insights into the problem. When dealing with equivalent expressions, we're essentially looking for different ways to represent the same numerical value or algebraic relationship. This is a crucial skill in mathematics, as it allows us to simplify complex problems, solve equations, and make connections between different mathematical concepts. For example, understanding equivalent expressions is essential when factoring polynomials, simplifying fractions, and solving systems of equations. The ability to recognize and manipulate equivalent expressions is like having a versatile tool in your mathematical toolkit, allowing you to approach problems from different angles and find the most efficient solution. So, keep this concept in mind as we explore the specific problem at hand, and you'll see how understanding equivalent expressions can make a world of difference.
When we talk about equivalent expressions, we're not just dealing with simple arithmetic; this concept extends to algebra and beyond. For instance, the expression x + x is equivalent to 2x, and (a + b)² is equivalent to a² + 2ab + b². Recognizing these equivalencies is key to simplifying algebraic equations and solving for unknowns. One of the most powerful tools for finding equivalent expressions is the distributive property, which we'll use later in this discussion. It allows us to multiply a single term by a group of terms inside parentheses, effectively expanding the expression. Conversely, we can use factoring, the reverse of distribution, to find a common factor among terms and rewrite the expression in a more compact form. Understanding these techniques is not only useful for academic math problems but also for real-world applications, such as budgeting, engineering, and computer programming. Being able to manipulate expressions and see different ways to represent the same quantity is a hallmark of mathematical fluency.
To truly grasp the concept, consider how equivalent expressions are used in everyday life. Imagine you're planning a party and need to buy drinks. If each guest will drink 2 cans of soda, and you're expecting 10 guests, you can express the total number of sodas needed as 2 * 10. This is equivalent to 10 + 10, or even 4 * 5. All these expressions represent the same number of sodas: 20. In a more complex scenario, think about calculating the total cost of items at a store. If you're buying 3 items that cost $5 each and 2 items that cost $10 each, you can express the total cost as 3 * 5 + 2 * 10. This expression is equivalent to 15 + 20, which simplifies to $35. These examples illustrate how equivalent expressions help us break down problems into manageable parts and find the most efficient way to solve them. As we move forward in mathematics, from algebra to calculus, the ability to recognize and manipulate equivalent expressions becomes even more critical. It's a foundational skill that underpins much of mathematical reasoning and problem-solving. Therefore, mastering this concept is an investment in your mathematical future.
Breaking Down the Problem: 90 + 18
Now, let's zoom in on our specific problem: finding an expression equivalent to 90 + 18. The first step is to simply add the two numbers together. 90 plus 18 is 108. So, our target number is 108. We need to find which of the given options also equals 108. This problem is a great example of how understanding basic arithmetic is crucial for more complex mathematical tasks. Before we can even think about factoring or using the distributive property, we need to know the value of the original expression. This highlights the importance of mastering the fundamentals of math. Addition, subtraction, multiplication, and division are the building blocks upon which all other mathematical concepts are built. Without a solid understanding of these basics, tackling more advanced topics becomes significantly more challenging. In this case, correctly adding 90 and 18 is the crucial first step that allows us to evaluate the given options and determine which one is equivalent. Think of it like building a house: you need a strong foundation before you can start adding walls and a roof. Similarly, in math, you need a strong foundation of basic arithmetic before you can tackle algebraic equations or calculus problems. So, always remember to double-check your basic calculations, as they are the key to unlocking more complex mathematical challenges. Once we have established that 90 + 18 = 108, we can confidently move on to analyzing the options and finding the equivalent expression.
With our target number firmly in mind, we can now approach the problem strategically. Instead of just randomly trying out each option, let's think about what makes an expression equivalent. As we discussed earlier, equivalent expressions are simply different ways of representing the same value. In this case, we're presented with expressions that involve multiplication and addition, which means we'll likely need to use the distributive property or factoring to find our match. This is where the fun begins! We're not just plugging in numbers; we're engaging in a bit of mathematical detective work. We're looking for clues, patterns, and relationships that will lead us to the correct answer. This kind of problem-solving is at the heart of mathematics. It's not just about memorizing formulas or following procedures; it's about thinking critically, analyzing information, and applying your knowledge in a creative way. By understanding the underlying principles, such as the distributive property and the concept of factoring, we can tackle a wide range of mathematical problems with confidence. So, let's put on our detective hats and start examining the options, keeping in mind that we're looking for an expression that, when simplified, gives us the magic number: 108.
Remember, our goal is not just to find the answer but also to understand the process of finding it. This is crucial for developing mathematical fluency. When you understand the underlying concepts, you can apply them to different situations and solve problems you've never seen before. Think of it like learning a language: you don't just memorize phrases; you learn the grammar and vocabulary so you can construct your own sentences. Similarly, in math, you don't just memorize formulas; you learn the principles so you can apply them to a variety of problems. This approach not only helps you succeed in your math classes but also develops your critical thinking and problem-solving skills, which are valuable in all aspects of life. So, as we work through this problem, pay attention to the reasoning behind each step. Ask yourself why we're doing what we're doing. This will deepen your understanding and make you a more confident and capable mathematician. Now, with our focus on both the answer and the process, let's dive into the options and see which one leads us to 108.
Evaluating the Options
Let's examine each option one by one to see which one equals 108. This is where our understanding of the distributive property comes into play. The distributive property states that a( b + c) = a * b* + a * c*. We'll use this property to simplify each option and see if it matches our target number. This process of systematically evaluating each option is a common and effective problem-solving strategy in mathematics. It's like conducting an experiment: you have a hypothesis (a possible solution), and you test it to see if it holds true. If it doesn't, you move on to the next hypothesis. This methodical approach helps us avoid making hasty judgments and ensures that we consider all possibilities. In this case, we're not just guessing; we're using a logical and structured method to arrive at the correct answer. This skill of systematic evaluation is not only useful in math but also in other areas of life, such as science, engineering, and even everyday decision-making. By breaking down a problem into smaller parts and evaluating each part individually, we can make more informed choices and avoid errors. So, let's apply this approach to our problem and carefully examine each option to see which one is the equivalent expression for 90 + 18.
Option A: 3(87 + 6). To evaluate this, we first add the numbers inside the parentheses: 87 + 6 = 93. Then, we multiply 3 by 93: 3 * 93 = 279. This is clearly not equal to 108, so Option A is incorrect. This is a good example of how a seemingly small mistake in addition can lead to a completely wrong answer. It underscores the importance of careful calculation and double-checking your work. In mathematics, precision is key. Even a minor error can have a cascading effect, leading to an incorrect final result. This is why it's always a good idea to take your time, show your work, and review your steps. In this case, if we had made a mistake in adding 87 and 6, we would have ended up with the wrong value and potentially chosen the incorrect option. By carefully following the order of operations and paying attention to detail, we can minimize the risk of errors and increase our chances of success. So, let's carry this lesson forward as we evaluate the remaining options, ensuring that we approach each step with care and precision. The fact that Option A doesn't match our target number is valuable information in itself. It eliminates one possibility and allows us to focus our attention on the remaining options.
Option B: 3(30 + 5). Again, we start by adding the numbers inside the parentheses: 30 + 5 = 35. Then, we multiply 3 by 35: 3 * 35 = 105. This is close to 108, but not quite there, so Option B is also incorrect. This highlights the importance of not jumping to conclusions. Just because an answer is close doesn't mean it's correct. In mathematics, it's crucial to be precise and accurate. A small difference can be the difference between a right and wrong answer. This is particularly true in fields like engineering and physics, where even minor errors can have significant consequences. So, it's essential to develop a habit of double-checking your work and verifying that your answer is exactly what the problem asks for. In this case, the fact that 105 is close to 108 might tempt some people to choose Option B without further consideration. However, by adhering to a systematic approach and carefully evaluating each option, we can avoid this trap and ensure that we select the correct answer. The fact that Option B is close to the correct answer might also give us a clue about the structure of the correct expression. It suggests that the numbers inside the parentheses should add up to a value that, when multiplied by 3, is close to 108. This can help us narrow down our search and focus our attention on options that are more likely to be correct.
Option C: 3(30 + 6). Let's follow the same steps: 30 + 6 = 36. Now, we multiply 3 by 36: 3 * 36 = 108. Bingo! This matches our target number, so Option C is the correct answer. This is a great feeling, isn't it? It's the satisfaction of having solved a puzzle, of having applied your knowledge and skills to arrive at the correct solution. But the learning doesn't stop here. Now that we've found the answer, let's take a moment to reflect on why this option worked. We can see that 30 + 6 is a way of breaking down the numbers 90 and 18 into multiples of 3. 90 is 3 times 30, and 18 is 3 times 6. This is the essence of factoring: finding common factors and using them to rewrite an expression in a different form. Understanding this underlying principle will not only help you solve similar problems in the future but also deepen your understanding of mathematical relationships. So, let's not just celebrate the victory; let's also take the time to analyze the strategy and the concepts involved. This is the key to becoming a truly proficient mathematician.
Option D: 3(30 + 15). Just to be thorough, let's check this one too: 30 + 15 = 45. Then, 3 * 45 = 135. This is not equal to 108, so Option D is incorrect. Checking all the options, even after finding the correct answer, is a good habit to develop. It helps to reinforce your understanding and ensures that you haven't made any errors. It's like proofreading your writing: you might think you've caught all the mistakes, but a second look can often reveal errors you missed the first time. Similarly, in math, checking your work can help you catch calculation errors or logical fallacies. It also gives you an opportunity to consolidate your understanding of the concepts involved. In this case, by checking Option D, we've further confirmed that Option C is indeed the correct answer. We've also reinforced our understanding of how the distributive property works and how to evaluate expressions involving parentheses and multiplication. This thoroughness and attention to detail are hallmarks of a strong mathematician. So, let's always strive to be meticulous and diligent in our problem-solving, checking our work and verifying our answers whenever possible.
The Answer and Why It Works
So, the expression equivalent to 90 + 18 is C. 3(30 + 6). This works because we can distribute the 3 across the terms inside the parentheses: 3 * 30 + 3 * 6 = 90 + 18. This demonstrates the power of the distributive property and how it allows us to rewrite expressions in equivalent forms. Understanding the distributive property is like having a mathematical Swiss Army knife; it's a versatile tool that can be used in a wide variety of situations. It's not just about multiplying a number by a sum; it's about understanding how multiplication interacts with addition and subtraction. This understanding is crucial for simplifying algebraic expressions, solving equations, and even tackling more advanced topics like calculus. Think of it as the foundation for many other mathematical concepts. When you master the distributive property, you unlock a whole new level of mathematical fluency. You can see how expressions can be manipulated and transformed, and you can use this knowledge to solve problems more efficiently. So, it's worth taking the time to truly understand this property and practice applying it in different contexts. In this case, the distributive property is the key to unlocking the equivalence between 3(30 + 6) and 90 + 18. It allows us to see that these two expressions, though they look different, represent the same value. This is the essence of equivalent expressions, and it's a concept that is fundamental to all of mathematics.
Moreover, this problem also subtly introduces the idea of factoring. Factoring is the reverse of distribution; it's the process of finding common factors and using them to rewrite an expression as a product. In this case, we can see that 3 is a common factor of both 90 and 18. We can factor out the 3 to get 3(30 + 6). This is a powerful technique that is used extensively in algebra and beyond. Factoring allows us to simplify expressions, solve equations, and reveal hidden relationships between numbers and variables. It's like taking a complex puzzle and breaking it down into smaller, more manageable pieces. When you factor an expression, you're essentially finding its building blocks, the fundamental components that make it up. This can give you a deeper understanding of the expression and how it behaves. Factoring is not just a mathematical trick; it's a way of seeing structure and order in what might seem like a chaotic mess. So, as you continue your mathematical journey, pay close attention to factoring techniques. They will serve you well in a wide range of contexts. In this problem, the connection between distribution and factoring is clear: distribution allows us to expand an expression, while factoring allows us to condense it. Both are valuable tools in our mathematical toolkit.
By understanding the distributive property and the concept of factoring, we've not only solved this specific problem but also gained a deeper appreciation for the underlying principles of mathematics. This is the real value of learning math: it's not just about getting the right answer; it's about developing your critical thinking skills and your ability to see patterns and relationships. These skills are transferable to all areas of life, from problem-solving at work to making informed decisions in your personal life. Mathematics is a powerful tool for understanding the world around us, and the more you learn, the more empowered you become. So, don't just memorize formulas; strive to understand the why behind them. Ask questions, explore different approaches, and challenge yourself to think critically. This is the path to mathematical mastery. And remember, even seemingly simple problems like this one can offer valuable insights into fundamental concepts. By breaking down the problem, evaluating the options, and reflecting on the solution, we've learned more than just the answer; we've deepened our understanding of equivalent expressions, the distributive property, and the process of factoring. This is the kind of learning that will stick with you and help you succeed in your mathematical journey.
Key Takeaways
- Equivalent expressions are different ways of writing the same value.
- The distributive property is a powerful tool for simplifying expressions.
- Factoring is the reverse of distribution and can help us find equivalent expressions.
- Systematically evaluating options is a good problem-solving strategy.
By mastering these concepts, you'll be well-equipped to tackle similar problems and excel in your mathematical studies. Keep practicing, keep exploring, and never stop asking questions. Math is a journey, and every problem you solve is a step forward!
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Equivalent Expressions Guide Mastering 90+18 Problem