Simplifying Fractions A Step-by-Step Guide To Solving 4/6 2/3 + 9/12

by Scholario Team 69 views

Hey guys! Ever get tripped up by fractions? Don't worry, you're not alone! Fractions can seem intimidating, but once you break them down, they're actually pretty straightforward. In this article, we're going to tackle a fraction problem that looks a little complex at first glance: 4/6 : 2/3 + 9/12. We'll go through each step together, so you'll not only get the answer but also understand why we're doing what we're doing. Think of this as your ultimate guide to simplifying fractions – no more fraction fear! We'll cover everything from simplifying individual fractions to understanding the order of operations (PEMDAS/BODMAS, anyone?). So, grab a pen and paper, and let's dive in!

Understanding the Problem: 4/6 : 2/3 + 9/12

Okay, before we jump into solving, let's make sure we really understand what the problem is asking. We have 4/6 : 2/3 + 9/12. Notice that we have a mix of operations here: division (the colon symbol ':') and addition. This means we need to remember our order of operations, often remembered by the acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). The important thing to remember is that division and multiplication have the same priority, and we work them from left to right. The same goes for addition and subtraction.

Now, let's look at the individual fractions. We have 4/6, 2/3, and 9/12. A key part of simplifying fraction problems is often simplifying the fractions themselves before we do any other operations. This makes the numbers smaller and easier to work with. So, we'll be looking for the greatest common factor (GCF) of the numerator (the top number) and the denominator (the bottom number) for each fraction. For example, 4/6 can be simplified because both 4 and 6 are divisible by 2. This initial assessment is crucial because it sets us up for a smoother solving process. We're not just blindly crunching numbers; we're strategizing to make the problem as manageable as possible. So, let's keep this in mind as we move on to the next step: simplifying those fractions!

Step 1: Simplifying Individual Fractions

Alright, let's get to the nitty-gritty of simplifying those fractions! As we discussed, simplifying fractions means finding the greatest common factor (GCF) of the numerator and denominator and then dividing both by that GCF. This process reduces the fraction to its simplest form, where the numerator and denominator have no common factors other than 1. Let's tackle each fraction in our problem, 4/6 : 2/3 + 9/12, one by one.

  • Simplifying 4/6: Look at the numbers 4 and 6. What's the biggest number that divides evenly into both? That's right, it's 2! So, we divide both the numerator (4) and the denominator (6) by 2. 4 divided by 2 is 2, and 6 divided by 2 is 3. So, 4/6 simplifies to 2/3. See? Not too scary!
  • Simplifying 2/3: Now, let's look at 2/3. Can we simplify this further? What's the GCF of 2 and 3? Well, the only number that divides evenly into both 2 and 3 is 1. When the GCF is 1, it means the fraction is already in its simplest form. So, 2/3 stays as 2/3. Nice and easy!
  • Simplifying 9/12: Finally, let's tackle 9/12. What's the biggest number that divides evenly into both 9 and 12? If you're thinking 3, you're on the right track! 9 divided by 3 is 3, and 12 divided by 3 is 4. So, 9/12 simplifies to 3/4.

Now that we've simplified each fraction, our problem looks a little different, and a little less intimidating: 2/3 : 2/3 + 3/4. See how simplifying first made things cleaner? This is why it's such a crucial step. Now, we're ready to move on to the next operation: division. Let's go!

Step 2: Dividing Fractions

Okay, guys, we've simplified our fractions, and now it's time to tackle the division part of our problem: 2/3 : 2/3 + 3/4. Remember our order of operations (PEMDAS/BODMAS)? Division comes before addition. So, we'll focus on the 2/3 : 2/3 part first. Now, how do we divide fractions? Here's the key: we don't actually divide. Instead, we multiply by the reciprocal.

What's a reciprocal, you ask? It's simply flipping the fraction! So, the reciprocal of 2/3 is 3/2. To divide fractions, we take the first fraction, change the division sign to a multiplication sign, and multiply by the reciprocal of the second fraction. Let's put this into action.

So, 2/3 : 2/3 becomes 2/3 * 3/2. Now we're multiplying fractions, which is much easier! To multiply fractions, we simply multiply the numerators together and the denominators together. So, (2 * 3) / (3 * 2) = 6/6. Ah, 6/6! That looks familiar, doesn't it? Any fraction where the numerator and denominator are the same is equal to 1. So, 2/3 : 2/3 = 1. We've knocked out the division part! Our problem now looks even simpler: 1 + 3/4. See how each step is making the problem more manageable? We're on the home stretch now. Let's move on to the final operation: addition.

Step 3: Adding Fractions

Alright, we've made it to the final step! We're now looking at the problem 1 + 3/4. This looks much less intimidating than our original problem, right? We've simplified, we've divided, and now we're adding. But we can't just add 1 and 3/4 as they are. We need to have a common denominator. Remember that to add (or subtract) fractions, they need to have the same denominator, the bottom number.

So, how do we add a whole number like 1 to a fraction like 3/4? We need to rewrite the whole number as a fraction with the same denominator as the other fraction. In this case, we want to rewrite 1 as a fraction with a denominator of 4. Think about it: any number divided by itself is 1. So, 4/4 is equal to 1! Now we can rewrite our problem as 4/4 + 3/4.

Now that we have a common denominator, adding is a breeze! We simply add the numerators and keep the denominator the same. So, 4/4 + 3/4 = (4 + 3) / 4 = 7/4. We've got our answer! But let's take it one step further. 7/4 is an improper fraction, meaning the numerator is bigger than the denominator. It's often good practice to convert improper fractions to mixed numbers. A mixed number has a whole number part and a fractional part. To convert 7/4 to a mixed number, we ask ourselves: how many times does 4 go into 7? It goes in once, with a remainder of 3. So, 7/4 is equal to 1 and 3/4 (often written as 1 3/4). And there we have it! We've successfully solved our problem.

Final Answer and Recap

Okay, let's recap what we've done and make sure we're crystal clear on the final answer. We started with the problem 4/6 : 2/3 + 9/12. Remember how daunting it looked at first? But we broke it down step-by-step, and now we've conquered it! Here's a quick rundown of our steps:

  1. Simplified Individual Fractions: We simplified 4/6 to 2/3 and 9/12 to 3/4. 2/3 was already in its simplest form.
  2. Divided Fractions: We tackled 2/3 : 2/3, which we solved by multiplying by the reciprocal, giving us 1.
  3. Added Fractions: We added 1 + 3/4 by rewriting 1 as 4/4, giving us 7/4, which we then converted to the mixed number 1 3/4.

So, the final answer to 4/6 : 2/3 + 9/12 is 1 3/4 (or 7/4). High five! You did it! See? Fractions don't have to be scary. By breaking them down into smaller, manageable steps and remembering the rules of order of operations and fraction manipulation, you can solve even the trickiest-looking problems. The key takeaways here are: simplify first, remember to multiply by the reciprocal when dividing, and always ensure a common denominator when adding or subtracting. Keep practicing, and you'll become a fraction master in no time!