Simplifying Fractions A Step-by-Step Guide To (8/9 ÷ 24/5) × 2/3

by BRAINLY IN FTUNILA 65 views
Iklan Headers

Hey guys! Math can sometimes look like a scary monster, especially when we see fractions flying around. But don't worry, we're going to break down this problem – (8/9 ÷ 24/5) × 2/3 – into super easy steps. Think of it as a delicious cake; we'll slice it up one piece at a time until it's all gone! So, let's grab our math tools and dive right in. We’ll make sure you not only understand how to solve this specific problem but also feel confident tackling similar ones in the future. Remember, practice makes perfect, and with a little guidance, you’ll be a fraction-busting pro in no time! Let's get started and turn that math monster into a math friend.

Understanding the Order of Operations

Before we even touch those fractions, let's quickly chat about something called the order of operations. You might have heard of PEMDAS or BODMAS. It’s like a secret code that tells us the right way to solve math problems. PEMDAS stands for:

  • Parentheses
  • Exponents
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)

BODMAS is just a slightly different version:

  • Brackets
  • Orders
  • Division and Multiplication (from left to right)
  • Addition and Subtraction (from left to right)

Basically, it's the same thing! This order is super important because if we don't follow it, we might end up with the wrong answer. In our problem, (8/9 ÷ 24/5) × 2/3, we see parentheses and then division and multiplication. So, we'll tackle the parentheses first, then handle the division, and finally, do the multiplication. Ignoring this order would be like trying to build a house starting with the roof – it just won't work! So, keep PEMDAS/BODMAS in mind as our trusty guide throughout this problem. It's the key to unlocking the correct answer and making sure our math journey is smooth and successful. Mastering this order is the first step to conquering any math problem, no matter how intimidating it might seem at first. With PEMDAS/BODMAS in our toolkit, we're ready to face those fractions head-on!

Step 1: Solving the Division Inside the Parentheses (8/9 ÷ 24/5)

Okay, the first stop on our math adventure is the parentheses: (8/9 ÷ 24/5). Inside, we have a division problem involving two fractions. Now, dividing fractions might seem a little tricky at first, but there’s a cool trick we can use: "Keep, Change, Flip." Sounds like a dance move, right? But it’s actually a super helpful way to divide fractions.

Here’s what it means:

  • Keep: We keep the first fraction exactly as it is. So, 8/9 stays 8/9.
  • Change: We change the division sign (÷) to a multiplication sign (×).
  • Flip: We flip the second fraction, which means we swap the numerator (the top number) and the denominator (the bottom number). So, 24/5 becomes 5/24.

So, our division problem, 8/9 ÷ 24/5, now transforms into a multiplication problem: 8/9 × 5/24. See? Much friendlier already! Now, before we jump into multiplying, let’s see if we can simplify things a bit. This is where the magic of canceling comes in. We look for common factors between the numerators and the denominators. In this case, we notice that 8 and 24 share a common factor of 8. We can divide both 8 and 24 by 8:

  • 8 ÷ 8 = 1
  • 24 ÷ 8 = 3

So, our problem now looks like this: 1/9 × 5/3. We've made the numbers smaller and easier to work with, which reduces the chance of making mistakes later on. This step is all about making our lives easier and the math cleaner. By simplifying before we multiply, we're setting ourselves up for success. Remember, good mathematicians are also efficient mathematicians! They look for ways to make the process smoother and less complicated. Canceling common factors is a key skill in fraction manipulation, and mastering it will make you a fraction superstar. Now, let's move on to the next part: multiplying these simplified fractions.

Step 2: Multiplying the Simplified Fractions (1/9 × 5/3)

Alright, we've transformed our division problem into a multiplication problem, and we've even simplified the fractions. Now comes the fun part: multiplying! Multiplying fractions is actually quite straightforward. We simply multiply the numerators (the top numbers) together and then multiply the denominators (the bottom numbers) together. So, we have 1/9 × 5/3. Let's break it down:

  • Multiply the numerators: 1 × 5 = 5
  • Multiply the denominators: 9 × 3 = 27

Therefore, 1/9 × 5/3 = 5/27. Ta-da! We've successfully multiplied our fractions. This resulting fraction, 5/27, represents the solution to the division problem we tackled inside the parentheses in the first step. It's like we've solved a mini-puzzle within the bigger puzzle. Now, it's important to take a moment and see if we can simplify this fraction further. Can we divide both the numerator (5) and the denominator (27) by the same number? In this case, the answer is no. 5 is a prime number (only divisible by 1 and itself), and 27 is not divisible by 5. So, 5/27 is in its simplest form. We've done a great job so far! We've navigated the division, simplified, and multiplied. Now we have a concrete result for the part of the problem that was inside the parentheses. This methodical approach, taking it one step at a time, is what makes tackling complex math problems manageable and even enjoyable. We're building our way towards the final solution, and each step we complete brings us closer to the finish line. Let's keep that momentum going as we move on to the next step in our journey!

Step 3: Multiplying the Result by 2/3 (5/27 × 2/3)

Okay, we're on the home stretch! We've conquered the parentheses and found that (8/9 ÷ 24/5) simplifies to 5/27. Now, we need to take that result and multiply it by the final fraction in our original problem: 2/3. So, we're looking at 5/27 × 2/3. Just like before, multiplying fractions involves multiplying the numerators together and the denominators together.

Let's do it:

  • Multiply the numerators: 5 × 2 = 10
  • Multiply the denominators: 27 × 3 = 81

So, 5/27 × 2/3 = 10/81. We're almost there! We've multiplied the fractions, and now we have our final result. But before we celebrate, there's one last crucial step: simplifying the fraction. We need to check if 10/81 can be reduced to its simplest form. This means looking for a common factor that divides both 10 and 81. Let's think about the factors of 10: 1, 2, 5, and 10. Now let's consider the factors of 81: 1, 3, 9, 27, and 81. Do we see any common factors other than 1? Nope! This means that 10 and 81 don't share any common factors (other than 1), so the fraction 10/81 is already in its simplest form. We've done it! We've successfully solved the entire problem, step by step. We navigated the order of operations, conquered division and multiplication of fractions, and simplified our final answer. This is a fantastic achievement, and you should feel proud of your hard work and perseverance. Remember, breaking down complex problems into smaller, manageable steps is the key to success in math and in life. Now, let's write down our final answer and celebrate our victory!

Final Answer: 10/81

Drumroll, please! After all our hard work, careful calculations, and step-by-step problem-solving, we've arrived at the final answer: 10/81. This fraction represents the simplified solution to the original problem, (8/9 ÷ 24/5) × 2/3. We started with a seemingly complex expression involving fractions, division, multiplication, and parentheses. But by following the order of operations (PEMDAS/BODMAS), breaking the problem down into smaller, manageable steps, and applying our knowledge of fraction manipulation, we were able to navigate through it successfully. Remember, each step was important: simplifying fractions before multiplying, converting division into multiplication using the “Keep, Change, Flip” method, and always checking if the final answer can be simplified further. These are the skills that will serve you well in your math journey and beyond. Getting to the final answer is not just about getting the right number; it’s about the process of learning, understanding, and applying mathematical concepts. It's about building confidence in your problem-solving abilities and developing a positive attitude towards math. So, take a moment to appreciate your accomplishment. You tackled a challenging problem and emerged victorious! And now, you have another tool in your math toolkit to help you conquer future challenges. Congratulations on your success! Keep practicing, keep exploring, and keep having fun with math!

Practice Problems

Now that we've conquered this problem together, the best way to solidify your understanding is to practice! Here are a few similar problems you can try on your own. Remember to follow the same steps we used: the order of operations, simplifying fractions, and the “Keep, Change, Flip” method for division. Don't be afraid to make mistakes – they're part of the learning process. The key is to learn from them and keep practicing. Working through these problems will not only reinforce your understanding of fraction operations but also build your confidence in tackling more complex math challenges. So, grab a pencil and paper, and let's put your new skills to the test! Remember, practice makes perfect, and the more you work with fractions, the easier they will become. You've got this!

  1. (1/2 ÷ 3/4) × 5/6
  2. (2/5 ÷ 8/10) × 1/3
  3. (7/9 ÷ 14/3) × 2/5

These practice problems offer a great opportunity to apply what you've learned in a new context. Each problem is structured similarly to the example we worked through, so you can use the same step-by-step approach. If you get stuck, don't hesitate to revisit the explanations and examples we discussed earlier. Remember, the goal is not just to get the right answer, but to understand the process and develop your problem-solving skills. So, take your time, work through each step carefully, and celebrate your progress along the way. With each problem you solve, you're building your mathematical muscles and becoming more confident in your ability to handle fractions and other math challenges. Happy practicing!

Repair Input Keyword

Simplify the expression (8/9 ÷ 24/5) × 2/3. What is the solution?

SEO Title

Simplifying Fractions A Step-by-Step Guide to (8/9 ÷ 24/5) × 2/3