Solving Mixed Fraction Equations: 2 2/5 × 10/15 ÷ 6 2/3
Hey guys! Let's dive into solving this mixed fraction equation: 2 2/5 × 10/15 ÷ 6 2/3. This might look a little intimidating at first, but don't worry! We're going to break it down step by step so it’s super easy to understand. We'll cover everything from converting mixed fractions to improper fractions, to multiplying and dividing fractions like pros. So grab your pencils, and let’s get started!
Understanding Mixed Fractions
Before we jump into solving the equation, let's quickly recap what mixed fractions are and how to work with them. Mixed fractions, like 2 2/5 and 6 2/3, combine a whole number and a proper fraction. The key to working with these in equations is converting them into improper fractions. This makes multiplication and division way smoother. To convert a mixed fraction to an improper fraction, you multiply the whole number by the denominator of the fraction, add the numerator, and then place the result over the original denominator. For example, let’s convert 2 2/5 into an improper fraction.
First, multiply the whole number (2) by the denominator (5): 2 * 5 = 10. Then, add the numerator (2): 10 + 2 = 12. Finally, place this result over the original denominator (5), giving us 12/5. So, the mixed fraction 2 2/5 is equal to the improper fraction 12/5. We can do the same for 6 2/3. Multiply the whole number (6) by the denominator (3): 6 * 3 = 18. Then, add the numerator (2): 18 + 2 = 20. Place this over the original denominator (3), giving us 20/3. Thus, 6 2/3 is equal to 20/3. Now that we understand how to convert mixed fractions to improper fractions, we're well-prepared to tackle the original equation. Remember, this conversion is crucial because it allows us to perform multiplication and division much more easily. By changing mixed fractions into a single fraction, we eliminate the need to juggle whole numbers and fractions separately, simplifying the entire process. So, with this foundational knowledge in hand, let’s move on to the next step and see how we can apply this to solve our equation!
Converting Mixed Fractions to Improper Fractions
Alright, let’s put our knowledge into action! To solve the equation 2 2/5 × 10/15 ÷ 6 2/3, the first thing we need to do is convert the mixed fractions into improper fractions. We’ve already touched on this, but let's go through it step-by-step to make sure we’ve got it down pat. For the first mixed fraction, 2 2/5, we multiply the whole number (2) by the denominator (5), which gives us 10. Then, we add the numerator (2) to get 12. So, the improper fraction is 12/5. Easy peasy, right? Now, let's convert the second mixed fraction, 6 2/3, into an improper fraction. Multiply the whole number (6) by the denominator (3), which gives us 18. Add the numerator (2) to get 20. Therefore, the improper fraction is 20/3. Now that we’ve converted our mixed fractions, we can rewrite the original equation using improper fractions. Our equation 2 2/5 × 10/15 ÷ 6 2/3 now becomes 12/5 × 10/15 ÷ 20/3. See how much cleaner that looks? Converting to improper fractions is like laying the groundwork for solving the problem. It transforms the equation into a format that’s much easier to handle. Now, we can move forward with the multiplication and division operations without the complication of mixed numbers. So, with the equation in this new form, we're ready to tackle the next steps in solving it. Let’s head on over to multiplying and dividing these fractions and get closer to finding our final answer!
Multiplying Fractions
Now that we have our equation in terms of improper fractions: 12/5 × 10/15 ÷ 20/3, let’s tackle the multiplication part first. Remember, multiplying fractions is super straightforward. You simply multiply the numerators (the top numbers) together and then multiply the denominators (the bottom numbers) together. So, we're looking at 12/5 multiplied by 10/15. To multiply these, we multiply the numerators: 12 * 10 = 120. Then, we multiply the denominators: 5 * 15 = 75. This gives us the fraction 120/75. But hold on! We're not quite done yet. This fraction looks a bit bulky, and we always want to simplify our fractions to their simplest form. This means finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by that number. In this case, both 120 and 75 are divisible by 15. So, let’s divide both by 15. 120 ÷ 15 = 8 and 75 ÷ 15 = 5. This simplifies our fraction to 8/5. So, 12/5 multiplied by 10/15 equals 8/5. See how simplifying the fraction makes things cleaner and easier to work with? Now we’ve got a much more manageable number to carry forward in our equation. We've handled the multiplication part, and we’re one step closer to solving the entire problem. Next up, we’ll take on the division part. So get ready to divide, and let’s keep this momentum going! We're making great progress, guys!
Dividing Fractions
Okay, we've multiplied our first two fractions and simplified the result to 8/5. Now it's time to deal with the division part of our equation: 8/5 ÷ 20/3. Dividing fractions might seem a bit tricky, but there's a neat little trick that makes it super easy. Instead of dividing, we can multiply by the reciprocal of the second fraction. What does that mean? Well, the reciprocal of a fraction is just flipping it upside down. So, the reciprocal of 20/3 is 3/20. Now, instead of dividing 8/5 by 20/3, we multiply 8/5 by 3/20. Remember our multiplication rule? Multiply the numerators together and multiply the denominators together. So, 8 * 3 = 24, and 5 * 20 = 100. This gives us the fraction 24/100. But just like before, we need to simplify this fraction to its simplest form. Both 24 and 100 are divisible by 4. So, let’s divide both by 4. 24 ÷ 4 = 6, and 100 ÷ 4 = 25. This simplifies our fraction to 6/25. So, 8/5 divided by 20/3 equals 6/25. We’ve successfully navigated the division part of our equation! By flipping the second fraction and multiplying, we turned a division problem into a multiplication problem, which is much easier to handle. Now we have our final fraction, 6/25. This is the solution to our original equation: 2 2/5 × 10/15 ÷ 6 2/3. Let’s take a moment to appreciate how far we’ve come. We started with a seemingly complex problem involving mixed fractions, and we broke it down step by step. We converted mixed fractions to improper fractions, multiplied fractions, divided fractions by using reciprocals, and simplified our results along the way. Awesome job, guys! Now, let’s wrap up by summarizing our steps and solidifying our understanding.
Final Answer and Summary
Alright, we’ve reached the end of our journey! We started with the equation 2 2/5 × 10/15 ÷ 6 2/3 and, after a series of steps, we arrived at our final answer. Let’s quickly recap the steps we took to get there:
- Converting Mixed Fractions to Improper Fractions: We changed 2 2/5 to 12/5 and 6 2/3 to 20/3.
- Rewriting the Equation: Our equation became 12/5 × 10/15 ÷ 20/3.
- Multiplying Fractions: We multiplied 12/5 by 10/15, which gave us 120/75, and then simplified it to 8/5.
- Dividing Fractions: We divided 8/5 by 20/3 by multiplying 8/5 by the reciprocal of 20/3 (which is 3/20), resulting in 24/100. We then simplified this to 6/25.
So, our final answer is 6/25. Woohoo! You did it! Solving equations with mixed fractions might seem challenging at first, but by breaking them down into manageable steps, it becomes much easier. Remember, the key is to convert those mixed fractions into improper fractions, handle multiplication and division carefully, and always simplify your answers. This not only gives you the correct result but also keeps the numbers easier to work with throughout the process. Guys, I hope this step-by-step guide has helped you understand how to solve mixed fraction equations. Keep practicing, and you'll become a pro in no time! If you ever get stuck, just remember these steps, and you’ll be able to tackle any fraction problem that comes your way. Great job today, and keep up the fantastic work! We've nailed this problem, and you've added another valuable skill to your math toolkit. Keep shining, guys!
