Adding Fractions 9/10 + 2/12 A Step By Step Guide

Introduction: Unlocking the Secrets of Fraction Addition

Hey guys! Ever felt like fractions are these mysterious puzzles that just won't fit together? Well, you're not alone! But guess what? Adding fractions can actually be super straightforward, and today, we're going to dive deep into one example: 9/10 + 2/12. This isn't just about getting the right answer; it's about understanding why we do what we do. So, buckle up, and let's embark on this fraction-filled adventure together! Our main goal here is to find the sum of these two fractions, and we'll break down each step to make sure you're a fraction-adding pro by the end. We'll start with the basics, like what fractions actually represent and why we can't just add the numerators and denominators straight away. Then, we'll tackle the crucial step of finding a common denominator – the secret ingredient to successful fraction addition. We'll explore different methods for finding this common ground, including the least common multiple (LCM), and show you how to apply it to our specific problem. Once we've got that sorted, we'll move on to rewriting the fractions with our shiny new common denominator. This is where the magic happens, as we transform our fractions into a form that's ready for addition. Finally, we'll add those fractions together, simplify the result (if necessary), and celebrate our fraction-adding victory! Throughout this journey, we'll be using real-world examples and analogies to help you visualize what's going on behind the scenes. Think of fractions as slices of a pizza – it makes the whole concept much more relatable, right? And remember, practice makes perfect. The more you work with fractions, the easier they'll become. So, let's get started and conquer those fractions, one step at a time! We'll make sure you understand every nook and cranny of the process, so you can confidently tackle any fraction addition problem that comes your way. Are you ready to become a fraction whiz? Let's do this!

Understanding Fractions: The Foundation of Addition

Before we jump into adding 9/10 and 2/12, let's rewind a bit and make sure we're all on the same page about what fractions actually mean. Think of a fraction as a way to represent a part of a whole. The bottom number, the denominator, tells us how many equal parts the whole is divided into. Imagine a pizza cut into 10 slices; the denominator is 10. The top number, the numerator, tells us how many of those parts we're talking about. If we have 9 slices, the numerator is 9, and we have 9/10 of the pizza. So, 9/10 simply means nine out of ten equal parts. Similarly, 2/12 means two out of twelve equal parts. Now, here's the tricky part: why can't we just add the numerators (9 + 2) and the denominators (10 + 12) directly? Well, imagine trying to add 9 slices from a pizza cut into 10 slices to 2 slices from a pizza cut into 12 slices. The slices are different sizes! It's like trying to add apples and oranges – they're just not the same. To add fractions, we need to make sure they have the same "slice size," or the same denominator. This is where the concept of a common denominator comes in. A common denominator is a number that both denominators can divide into evenly. It's like finding a common language for our fractions, so we can compare and combine them accurately. Think of it like this: if we want to add fractions representing portions of a cake, we need to make sure the cake is divided into the same number of slices in both cases. Only then can we meaningfully add the number of slices together. So, before we can add 9/10 and 2/12, we need to find a common denominator for 10 and 12. This is the key to unlocking the mystery of fraction addition. Once we have a common denominator, we can rewrite the fractions so that they both have the same denominator, and then we can simply add the numerators. This might sound a bit complicated now, but don't worry! We're going to break it down step by step, and you'll see how it all fits together. Understanding the why behind the process is just as important as knowing the how, so let's keep this foundation strong as we move forward. Remember, fractions are your friends, and with a little practice, you'll be adding them like a pro in no time!

Finding the Common Denominator: The Key to Addition

Okay, guys, now that we understand why we need a common denominator, let's get down to the nitty-gritty of how to find one for 9/10 + 2/12. This is a crucial step, so pay close attention! There are a couple of ways we can do this, but the most efficient method is to find the least common multiple (LCM) of the denominators. The LCM is the smallest number that both denominators divide into evenly. Think of it as the smallest "meeting point" for our denominators. So, for 10 and 12, we need to find the smallest number that both 10 and 12 can divide into without leaving a remainder. One way to find the LCM is to list out the multiples of each number until we find a common one. Multiples of 10 are: 10, 20, 30, 40, 50, 60, 70... Multiples of 12 are: 12, 24, 36, 48, 60, 72... See that? 60 appears in both lists! That means 60 is a common multiple of 10 and 12. But is it the least common multiple? In this case, yes! There's no smaller number that appears in both lists. Another way to find the LCM is by using prime factorization. Prime factorization is breaking down a number into its prime factors (numbers that are only divisible by 1 and themselves). The prime factorization of 10 is 2 x 5. The prime factorization of 12 is 2 x 2 x 3 (or 2² x 3). To find the LCM using prime factorization, we take the highest power of each prime factor that appears in either factorization. So, we have 2² (from 12), 3 (from 12), and 5 (from 10). Multiplying these together: 2² x 3 x 5 = 4 x 3 x 5 = 60. Voila! We get the same answer: 60. So, the least common multiple of 10 and 12 is 60. This means 60 will be our common denominator. Now, why is the LCM so important? Well, using the least common multiple keeps our numbers smaller and easier to work with. We could have used a larger common multiple, like 120 (which is also a common multiple of 10 and 12), but then our fractions would be bigger, and we'd have to simplify them more at the end. So, using the LCM is like taking the scenic route – it's the most efficient way to get to our destination. Now that we've found our common denominator, we're one big step closer to adding those fractions! In the next section, we'll learn how to rewrite our fractions with this new denominator, and then the real fun begins: adding them together!

Rewriting Fractions: Preparing for the Sum

Alright, team, we've found our common denominator: 60! Now comes the exciting part: rewriting 9/10 and 2/12 so they both have this denominator. This might seem a little tricky at first, but trust me, it's totally doable. The key is to remember that we can multiply a fraction by a form of 1 (like 2/2 or 5/5) without changing its value. Think of it like this: if you have half a pizza, and you cut each slice in half, you still have half the pizza, just with more slices. The amount of pizza hasn't changed, just the way it's divided. So, how do we figure out what to multiply each fraction by to get a denominator of 60? Let's start with 9/10. We need to find a number that we can multiply 10 by to get 60. What do you think? That's right, it's 6! (10 x 6 = 60). So, we're going to multiply 9/10 by 6/6 (which is just a fancy way of writing 1). Remember, we need to multiply both the numerator and the denominator by the same number to keep the fraction equivalent. So, 9/10 multiplied by 6/6 is (9 x 6) / (10 x 6), which equals 54/60. Awesome! We've rewritten 9/10 as 54/60. Now, let's tackle 2/12. What do we need to multiply 12 by to get 60? If you said 5, you're on fire! (12 x 5 = 60). So, we'll multiply 2/12 by 5/5. This gives us (2 x 5) / (12 x 5), which equals 10/60. Fantastic! We've rewritten 2/12 as 10/60. Now, look what we've done! We've transformed our original problem, 9/10 + 2/12, into something much easier: 54/60 + 10/60. Both fractions now have the same denominator, so we're finally ready to add them together. This step of rewriting the fractions is super important because it ensures that we're adding equal-sized pieces. We're no longer trying to add slices from pizzas cut into different numbers of slices; we're now adding slices from pizzas cut into the same number of slices (60, in this case). Think of it like building with LEGOs: you can only connect bricks that have the same size connectors. Rewriting the fractions with a common denominator is like making sure all our LEGO bricks have the same connectors, so we can build something awesome! So, give yourselves a pat on the back – you've mastered another key step in fraction addition. We're almost there! In the next section, we'll finally add those fractions together and see what we get.

Adding and Simplifying: The Final Steps

Okay, guys, the moment we've been waiting for! We've got our fractions rewritten with a common denominator: 54/60 + 10/60. Now, the actual addition is the easy part. When fractions have the same denominator, all we need to do is add the numerators and keep the denominator the same. It's like saying, "If I have 54 slices of a pizza cut into 60 slices, and I add 10 more slices, how many slices do I have in total?" The pizza is still cut into 60 slices, so our denominator stays as 60. We just add the numerators: 54 + 10 = 64. So, 54/60 + 10/60 = 64/60. We did it! We've added the fractions! But hold on, we're not quite finished yet. Our answer, 64/60, is an improper fraction. An improper fraction is a fraction where the numerator is greater than the denominator. This means we have more than one whole. While 64/60 is a correct answer, it's often best to simplify it and express it as a mixed number (a whole number and a fraction). To do this, we need to figure out how many times 60 goes into 64. It goes in once, with a remainder of 4. This means we have 1 whole (60/60) and 4 extra slices (4/60). So, we can write 64/60 as the mixed number 1 4/60. But we're still not quite done! The fraction part of our mixed number, 4/60, can be simplified further. We need to find the greatest common factor (GCF) of 4 and 60. The GCF is the largest number that divides into both numbers evenly. The factors of 4 are 1, 2, and 4. The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. The greatest common factor of 4 and 60 is 4. So, we can divide both the numerator and the denominator of 4/60 by 4 to simplify it. 4 divided by 4 is 1, and 60 divided by 4 is 15. So, 4/60 simplifies to 1/15. Now, we can rewrite our mixed number as 1 1/15. And that's it! We've added the fractions and simplified the result completely. Our final answer is 1 1/15. High five! You've successfully navigated the world of fraction addition. Remember, simplifying fractions is like putting the finishing touches on a masterpiece – it makes your answer as clear and concise as possible. So, always check if your answer can be simplified, whether it's an improper fraction or a fraction within a mixed number. You've come a long way in this fraction-adding journey. You've learned about the meaning of fractions, the importance of common denominators, how to rewrite fractions, how to add them, and how to simplify the result. You're well on your way to becoming a fraction master! Now, go out there and conquer those fractions with confidence!

Conclusion: Celebrating Your Fraction Mastery

Wow, guys, we did it! We successfully tackled the fraction addition problem 9/10 + 2/12, and we didn't just get the answer, we understood why we did what we did. That's the real victory here. We started by understanding what fractions represent – parts of a whole – and why we need a common denominator to add them. We then learned how to find the least common multiple (LCM), the most efficient common denominator, and how to rewrite fractions with that denominator. We added the rewritten fractions together and, finally, simplified our answer to its simplest form. That final answer, 1 1/15, represents the sum of 9/10 and 2/12 in the clearest and most concise way possible. But this journey was about more than just one problem. It was about building a solid foundation for understanding fractions and how they work. You now have the tools and knowledge to tackle a wide range of fraction addition problems. You understand the importance of finding a common denominator, rewriting fractions, and simplifying your answers. You've also learned about improper fractions and mixed numbers and how to convert between them. This knowledge will not only help you in your math classes but also in real-life situations where fractions pop up all the time – from cooking and baking to measuring and calculating proportions. Remember, practice makes perfect. The more you work with fractions, the more comfortable and confident you'll become. So, don't be afraid to challenge yourself with different problems and explore the fascinating world of fractions. Think of this article as your starting point, your guide on this journey. Revisit it whenever you need a refresher, and keep building on the skills you've learned today. And most importantly, remember that math can be fun! Fractions might have seemed intimidating at first, but you've shown that you can conquer them with a little understanding and effort. So, celebrate your achievement, give yourself a pat on the back, and get ready for the next mathematical adventure! You've got this!