Simplifying (8/9 ÷ 24/5) × 2/3 A Step By Step Guide

Hey everyone! Let's break down this math problem together. We're going to simplify the expression (8/9 ÷ 24/5) × 2/3. Don't worry, it's not as scary as it looks! We'll go through each step carefully, so you'll totally get it. Math can be fun, especially when we tackle it step by step. Stick with me, and we'll conquer this problem together.

Understanding the Order of Operations

Before we dive into the specifics of this problem, it's super important to remember the order of operations. You might have heard of it as PEMDAS or BODMAS. This acronym helps us remember the correct sequence for solving math problems. It stands for:

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

Following this order ensures we get the right answer every time. In our problem, we have parentheses and then multiplication, so we'll address the division inside the parentheses first, and then the multiplication.

Why is the order of operations so important? Imagine if we did the multiplication before the division inside the parentheses. We'd end up with a completely different answer! The order of operations provides a standardized way to solve mathematical expressions, ensuring consistency and accuracy. It's like a recipe – if you don't follow the steps in the right order, your dish might not turn out so well!

Think of it like building a house. You need to lay the foundation before you can put up the walls, and you need the walls before you can put on the roof. Similarly, in math, certain operations need to be performed before others to ensure the final answer is correct. So, keep PEMDAS or BODMAS in mind as we move forward. It's our trusty guide in the world of mathematical expressions.

Step-by-Step Simplification

Okay, let's get down to business! Our expression is (8/9 ÷ 24/5) × 2/3. Remember, we need to tackle the division inside the parentheses first. When we divide fractions, we actually multiply by the reciprocal of the second fraction. The reciprocal is simply flipping the fraction – swapping the numerator and the denominator.

So, 24/5 becomes 5/24. Now, our division problem turns into a multiplication problem: 8/9 ÷ 24/5 becomes 8/9 × 5/24. This is a crucial step, so make sure you've got it! Dividing by a fraction is the same as multiplying by its inverse.

Now, we multiply the fractions: 8/9 × 5/24. To multiply fractions, we multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. So, 8 × 5 = 40 and 9 × 24 = 216. This gives us 40/216. But we're not done yet! We need to simplify this fraction.

Simplifying a fraction means reducing it to its lowest terms. We need to find the greatest common factor (GCF) of 40 and 216 and divide both the numerator and the denominator by it. The GCF of 40 and 216 is 8. So, we divide both 40 and 216 by 8: 40 ÷ 8 = 5 and 216 ÷ 8 = 27. This simplifies 40/216 to 5/27. Awesome! We've simplified the expression inside the parentheses.

Now, we have 5/27 × 2/3. This is the final step. Again, we multiply the numerators together and the denominators together: 5 × 2 = 10 and 27 × 3 = 81. So, we get 10/81. Can we simplify this fraction further? No, we can't. 10 and 81 have no common factors other than 1. Therefore, 10/81 is our final answer. Yay!

Common Mistakes to Avoid

In math, it's easy to make little mistakes that can throw off your whole answer. Let's talk about some common pitfalls to watch out for when simplifying expressions like this one.

One biggie is forgetting the order of operations. We already talked about how important PEMDAS/BODMAS is. If you multiply before dividing (inside the parentheses), you'll get the wrong answer. Always remember to tackle parentheses first, then exponents, then multiplication and division (from left to right), and finally addition and subtraction (from left to right). It's like a golden rule in math!

Another common mistake is messing up the reciprocal when dividing fractions. Remember, when you divide by a fraction, you multiply by its reciprocal. It's easy to forget to flip the second fraction, but that flip is crucial! Double-check that you've correctly found the reciprocal before multiplying.

Simplifying fractions is another area where mistakes can happen. Make sure you're finding the greatest common factor, not just any common factor. If you don't simplify completely, your answer will still be correct, but it won't be in its simplest form. It's like having a beautifully wrapped gift, but the bow is a little crooked. It's still a great gift, but it could be even better with a little extra attention to detail.

Finally, a simple but common error is making arithmetic mistakes in your multiplication or division. It's easy to slip up when you're multiplying or dividing larger numbers, especially under pressure. Take your time, double-check your work, and maybe even use a calculator to verify your calculations. Accuracy is key!

By being aware of these common mistakes, you can avoid them and boost your confidence in solving math problems. Remember, practice makes perfect, so keep at it!

Practice Problems

Alright, guys, let's put our knowledge to the test! Practice is key to mastering any math skill. Here are a few problems similar to the one we just solved. Try them out on your own, and then we'll discuss the solutions.

1. (6/7 ÷ 18/14) × 3/4
2. (10/11 ÷ 5/22) × 1/2
3. (4/5 ÷ 16/10) × 2/3

Take your time, remember the order of operations, and simplify your answers as much as possible. Don't be afraid to make mistakes – that's how we learn! Work through each step carefully, and you'll get there. Math is like a puzzle; each step is a piece that fits together to reveal the solution.

After you've given these problems a try, we can walk through the solutions together. We'll break down each step, just like we did with the original problem. This will help solidify your understanding and give you even more confidence in your math abilities. So, grab a pencil and paper, and let's get practicing! You've got this!

Solutions and Explanations

Okay, let's review the solutions to those practice problems. Remember, the goal is not just to get the right answer, but to understand why it's the right answer. So, we'll break down each step and explain the reasoning behind it.

Problem 1: (6/7 ÷ 18/14) × 3/4

First, we tackle the division inside the parentheses. Remember, dividing by a fraction is the same as multiplying by its reciprocal. So, 6/7 ÷ 18/14 becomes 6/7 × 14/18. Now we multiply: (6 × 14) / (7 × 18) = 84/126. This fraction can be simplified. The greatest common factor of 84 and 126 is 42. Dividing both numerator and denominator by 42, we get 2/3. Great! Now we have 2/3 × 3/4.

Next, we multiply the fractions: (2 × 3) / (3 × 4) = 6/12. This fraction can also be simplified. The greatest common factor of 6 and 12 is 6. Dividing both numerator and denominator by 6, we get 1/2. So, the final answer for problem 1 is 1/2. How did you do?

Problem 2: (10/11 ÷ 5/22) × 1/2

Again, we start with the division inside the parentheses. 10/11 ÷ 5/22 becomes 10/11 × 22/5. Multiplying, we get (10 × 22) / (11 × 5) = 220/55. This fraction simplifies nicely! The greatest common factor of 220 and 55 is 55. Dividing both numerator and denominator by 55, we get 4/1 or simply 4. Excellent! Now we have 4 × 1/2.

Multiplying 4 by 1/2 is the same as finding half of 4, which is 2. So, the final answer for problem 2 is 2. Did you get it right?

Problem 3: (4/5 ÷ 16/10) × 2/3

Let's tackle that division first! 4/5 ÷ 16/10 becomes 4/5 × 10/16. Multiplying, we get (4 × 10) / (5 × 16) = 40/80. This fraction can be simplified. The greatest common factor of 40 and 80 is 40. Dividing both numerator and denominator by 40, we get 1/2. Fantastic! Now we have 1/2 × 2/3.

Finally, we multiply: (1 × 2) / (2 × 3) = 2/6. This fraction can also be simplified. The greatest common factor of 2 and 6 is 2. Dividing both numerator and denominator by 2, we get 1/3. So, the final answer for problem 3 is 1/3. Well done!

How did you do on the practice problems? If you got them all right, congratulations! You've got a solid understanding of simplifying expressions with fractions. If you made a few mistakes, don't worry! That's part of the learning process. Review the steps, identify where you went wrong, and try again. With practice, you'll become a fraction-simplifying pro!

Real-World Applications

You might be thinking, "Okay, this is cool, but when am I ever going to use this in real life?" Well, you might be surprised! Simplifying expressions with fractions comes in handy in all sorts of situations.

Think about cooking. Recipes often use fractions to represent ingredients. If you want to double or halve a recipe, you'll need to multiply or divide fractions. Understanding how to simplify those fractions can help you measure ingredients accurately and avoid kitchen disasters. No one wants a cake that's twice as salty!

Another example is in construction or carpentry. When building something, you often need to measure lengths and cut materials to specific sizes. These measurements might involve fractions, and simplifying those fractions can help you make accurate cuts and ensure your project comes together perfectly. Imagine building a bookshelf where the shelves are all slightly different lengths because you didn't simplify your fractions correctly!

Fractions are also used in financial calculations. For example, if you're figuring out how much of your paycheck to save or how to split a bill with friends, you might need to work with fractions. Simplifying those fractions can help you make smart financial decisions and avoid overspending. Saving money is always a good idea!

Even in everyday situations like telling time, we use fractions. Half an hour, a quarter of an hour – these are all fractions of an hour. Understanding fractions can help you manage your time effectively and stay on schedule. Being on time is a valuable skill!

So, the next time you're faced with a real-world problem involving fractions, remember the skills we've discussed here. Simplifying expressions might seem like an abstract concept, but it's a powerful tool that can help you in many practical situations. Math is everywhere!

Conclusion

Well, guys, we've reached the end of our journey through simplifying the expression (8/9 ÷ 24/5) × 2/3. We've covered a lot of ground, from understanding the order of operations to tackling practice problems and exploring real-world applications. Hopefully, you now feel more confident in your ability to simplify expressions with fractions.

Remember, the key to success in math is practice. The more you practice, the more comfortable you'll become with the concepts and the more easily you'll be able to apply them. Don't be afraid to make mistakes – they're a natural part of the learning process. Just learn from them and keep moving forward.

Math can be challenging, but it's also incredibly rewarding. The ability to solve problems and think critically is a valuable skill that will serve you well in all areas of life. So, keep practicing, keep learning, and keep exploring the wonderful world of mathematics. You've got this!

If you have any questions or want to explore other math topics, feel free to ask. Math is a vast and fascinating subject, and there's always something new to discover. Keep your curiosity alive, and never stop learning! Thanks for joining me on this math adventure!