Common Factor Calculation: Step-by-Step Solutions
Hey guys! Let's dive into some math problems today, specifically focusing on how to use the common factor method to simplify calculations. We've got three expressions to break down, so grab your pencils and let's get started! We'll be tackling these expressions step-by-step, making sure everyone understands the process. It's all about finding that common thread and pulling it out to make the math easier.
b) 437 * 109 - 437 * 54 + 437 * 203
In this first expression, our main focus is on identifying the common factor. When you look at 437 * 109 - 437 * 54 + 437 * 203, what stands out? It's pretty clear, right? The number 437 appears in each term. This is our common factor, and it’s the key to simplifying this expression. The beauty of using the common factor method is that it allows us to transform a series of multiplications and additions/subtractions into a single multiplication, which is often much easier to handle.
So, how do we actually use this common factor? Well, we factor it out! Imagine we're pulling the 437 out of each term and placing it in front of a set of parentheses. Inside the parentheses, we'll put what's left over from each term. This gives us: 437 * (109 - 54 + 203). See how we've essentially reversed the distributive property? Instead of multiplying 437 by each number inside the parentheses, we're factoring it out.
Now, the expression looks much simpler. We just need to deal with the numbers inside the parentheses: 109 - 54 + 203. Let's tackle this step-by-step. First, 109 - 54 equals 55. So now we have 55 + 203. Adding those together, we get 258. Great! We've simplified the expression inside the parentheses to a single number.
Our expression now looks like this: 437 * 258. This is a much easier calculation than what we started with! Now, we just need to multiply 437 by 258. You can use a calculator, do it by hand, or use your favorite multiplication method. The result is 112746. So, the final answer for this expression is 112746. We took a seemingly complex calculation and broke it down into manageable steps by identifying and using the common factor. That’s the power of this method!
d) 2011 * 5 + 2011 * 7 + 2011 * 49 + 2011 * 39
Alright, let's move on to the next one! This expression, 2011 * 5 + 2011 * 7 + 2011 * 49 + 2011 * 39, might look a bit intimidating at first glance, but don't worry, we've got this. Just like before, our primary goal here is to spot the common factor. Take a good look – what number is present in each term? You guessed it: 2011. This is the key to simplifying this expression, and we're going to use it to our advantage. Remember, the common factor method is all about making complex calculations easier by identifying and extracting the shared element.
Now that we've identified 2011 as the common factor, let's factor it out. We're essentially reversing the distributive property again. We pull the 2011 out front and create a set of parentheses containing the leftovers from each term. This gives us: 2011 * (5 + 7 + 49 + 39). Notice how we've transformed a series of multiplications and additions into a single multiplication combined with an addition problem inside the parentheses. This is a huge step in simplifying the expression.
Next up, we need to simplify the expression inside the parentheses: 5 + 7 + 49 + 39. This is just a matter of adding the numbers together. Let's do it step-by-step to avoid any errors. 5 + 7 equals 12. Then, 12 + 49 equals 61. Finally, 61 + 39 equals 100. Awesome! We've simplified the expression inside the parentheses down to a nice, round number: 100.
Our expression now looks like this: 2011 * 100. This is a much simpler calculation than the original one, right? Multiplying by 100 is super easy – we just add two zeros to the end of the number. So, 2011 * 100 equals 201100. That's it! The final answer for this expression is 201100. We successfully used the common factor method to simplify a seemingly complex calculation and arrive at the solution. See how powerful this method can be?
f) 13 * 17 + 13 * 25 - 42 * 11 - 2 * 41
Okay, let's tackle the last one! This expression, 13 * 17 + 13 * 25 - 42 * 11 - 2 * 41, looks a bit different from the previous ones. At first glance, it might not seem like there's an obvious common factor across all terms. But don't worry, we can still use the common factor method, we might just need to do a little rearranging and thinking outside the box. Our main goal remains the same: to identify and extract any shared factors to simplify the calculation. Sometimes, it's about grouping terms strategically to reveal those hidden commonalities.
Let's start by looking at the first two terms: 13 * 17 + 13 * 25. Here, we can clearly see that 13 is a common factor. So, let's factor it out of these two terms: 13 * (17 + 25). This is a good start! Now, let's simplify the expression inside the parentheses: 17 + 25 equals 42. So, this part of the expression becomes 13 * 42. Great!
Now, let's look at the remaining terms: - 42 * 11 - 2 * 41. Hmmm, there doesn't seem to be an immediate common factor here. However, notice the 42 in the first part of this expression. We also ended up with 13 * 42 from the first two terms. This suggests we might be able to manipulate the second part of the expression to create a common factor of 42. Let's rewrite the last term, 2 * 41, as 82. This doesn't directly help us with a common factor of 42, but it's important to explore different avenues.
Let's go back and rewrite the entire expression with our simplified first part: 13 * 42 - 42 * 11 - 82. Now, we can see a common factor of 42 in the first two terms! Let's factor it out: 42 * (13 - 11) - 82. This is progress! Now, simplify the expression inside the parentheses: 13 - 11 equals 2. So, we have 42 * 2 - 82.
Now, let's calculate 42 * 2, which equals 84. So, our expression becomes 84 - 82. This is a simple subtraction! 84 - 82 equals 2. Therefore, the final answer for this expression is 2. We had to do a little more work on this one, rearranging and simplifying in stages, but we still successfully used the common factor method (along with some basic arithmetic) to arrive at the solution. This problem highlights that sometimes finding the common factor requires a bit of creative manipulation and a step-by-step approach.
Conclusion
So, there you have it! We've tackled three different expressions using the common factor method. Remember, the key is to identify the number that appears in each term and then factor it out. This transforms a complex series of calculations into a simpler one. Keep practicing, and you'll become a pro at spotting those common factors and simplifying expressions like a boss! You guys did great! Math can be fun when you break it down step-by-step. Keep exploring and keep learning!