Calculating The Product Of 8/15, 6/5, And 1/3 A Step-by-Step Guide
Hey everyone! Today, we're diving into a fun math problem: calculating the product of three fractions: 8/15, 6/5, and 1/3. Don't worry, it's easier than it looks! We'll break it down step by step so you can follow along and master multiplying fractions. So, grab your pencils and let's get started!
Understanding Fraction Multiplication
Before we jump into the problem, let's quickly recap the basics of fraction multiplication. When you multiply fractions, you simply multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. That's it! No need to find common denominators like when you're adding or subtracting fractions. This fundamental principle makes fraction multiplication a straightforward process, whether you're dealing with two fractions or, in our case, three fractions. Understanding this concept is crucial for tackling more complex mathematical problems later on. When multiplying fractions, the order in which you multiply doesn't change the final answer. This is due to the commutative property of multiplication, which states that the order of numbers being multiplied does not affect the product. So, whether you multiply 8/15 by 6/5 first or start with 6/5 times 1/3, you will arrive at the same result. This flexibility can be useful when simplifying calculations, as you might find certain pairings of fractions easier to work with than others. Keep this in mind as we proceed through the problem, as it can help in making the calculations more manageable and less prone to errors. Also remember that the result of multiplying fractions can sometimes be simplified. This involves finding the greatest common divisor (GCD) of the numerator and the denominator and then dividing both by the GCD. Simplifying fractions makes them easier to understand and compare, and it's generally considered good practice to present your final answer in its simplest form. In our problem, we will address simplification as the last step to ensure our final answer is in its most concise form. Now that we've refreshed the basics, let's move on to the specific fractions we're working with and start the multiplication process.
Step-by-Step Calculation
Okay, let's get down to business! We need to find the product of 8/15, 6/5, and 1/3. Remember, we're just multiplying the numerators and the denominators. So, first, we'll multiply the numerators together: 8 * 6 * 1. Then, we'll multiply the denominators together: 15 * 5 * 3. This straightforward process allows us to combine the fractions into a single numerator and a single denominator, making the subsequent simplification easier to manage. This step is crucial because it sets the stage for reducing the fraction to its simplest form, which is our ultimate goal. By combining the numerators and denominators separately, we can clearly see the components that contribute to the final product, which helps in identifying common factors later on. Once we have the combined fraction, we can then look for opportunities to simplify by canceling out common factors between the numerator and the denominator, making the fraction easier to work with and understand. Don't worry about making mistakes at this stage; the important thing is to get the initial multiplication correct. Accuracy in this step is essential as it lays the foundation for the rest of the calculation. Now, let's calculate the products. 8 multiplied by 6 is 48, and 48 multiplied by 1 is still 48. So, our numerator is 48. For the denominators, 15 multiplied by 5 is 75, and 75 multiplied by 3 is 225. So, our denominator is 225. That means we have the fraction 48/225. But we're not done yet! We need to simplify this fraction to its simplest form. Simplifying fractions is a crucial skill in mathematics. It makes fractions easier to understand, compare, and work with in further calculations. A fraction is in its simplest form when the numerator and denominator have no common factors other than 1. Now, let's move on to simplifying the fraction 48/225.
Simplifying the Result
Alright, we've got 48/225, but let's make it even easier to work with. To simplify, we need to find the greatest common divisor (GCD) of 48 and 225. Finding the GCD might sound intimidating, but it's just the largest number that divides evenly into both 48 and 225. There are a couple of ways to find the GCD. One method is to list the factors of each number and see which is the largest they have in common. For 48, the factors are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. For 225, the factors are 1, 3, 5, 9, 15, 25, 45, 75, and 225. Looking at these lists, we can see that the greatest common factor is 3. Another method to find the GCD is to use the Euclidean algorithm, which involves successively dividing the larger number by the smaller number and then replacing the larger number with the remainder until the remainder is 0. The last non-zero remainder is the GCD. While this method might be more efficient for very large numbers, for our purposes, listing the factors works just fine. Once we've identified the GCD, we can use it to simplify the fraction. This involves dividing both the numerator and the denominator by the GCD. By doing so, we reduce the fraction to its simplest form, where the numerator and denominator have no common factors other than 1. This makes the fraction easier to understand and compare with other fractions. So, now that we know the GCD is 3, we're ready to simplify 48/225. This step is crucial for presenting our final answer in its most concise and understandable form. Let's get to it and see what the simplified fraction looks like!
Now, we divide both 48 and 225 by 3. 48 divided by 3 is 16, and 225 divided by 3 is 75. So, our simplified fraction is 16/75. We can double-check that 16 and 75 don't have any common factors other than 1, which means we've simplified it as much as possible. This final check is important to ensure that the fraction is indeed in its simplest form. We want to make sure that there are no more common factors between the numerator and the denominator that could be further divided out. By confirming this, we can be confident that our answer is presented in the most concise and understandable manner. Simplifying fractions to their lowest terms is not just a matter of mathematical correctness, but also a way of communicating the result clearly and effectively. A simplified fraction is easier to grasp and compare with other fractions, making it more useful in various mathematical contexts. So, taking the time to double-check the simplification is a worthwhile step in the problem-solving process. Now that we've simplified the fraction and confirmed that it's in its lowest terms, we can confidently move on to selecting the correct answer from the given options. We've done the hard work of calculation and simplification, so let's make sure we choose the right option.
The Final Answer
Okay, we've done the calculations and simplified the fraction. We found that the product of 8/15, 6/5, and 1/3 is 16/75. Now, let's look at the answer choices:
A) 48/30 B) 48/15 C) 16/75 D) 16/15
The correct answer is C) 16/75. You did it! We successfully calculated the product of the fractions and simplified the result. This process demonstrates the importance of understanding the basic rules of fraction multiplication and the technique of simplifying fractions. These skills are fundamental in mathematics and are used extensively in various areas, from algebra to calculus. Mastering these concepts will not only help you solve similar problems but also build a strong foundation for more advanced mathematical topics. Remember, the key to success in math is practice. The more you work through problems, the more comfortable and confident you'll become with the concepts and procedures involved. So, don't hesitate to try more examples and challenge yourself with different types of fraction problems. With consistent practice, you'll develop a strong understanding of fractions and their operations, making math a less daunting and more enjoyable subject. Keep up the great work, and you'll continue to excel in your mathematical journey! If you guys have any questions feel free to ask!
Conclusion
Great job, everyone! We've successfully navigated through this fraction multiplication problem. Remember, the key takeaways are to multiply numerators and denominators separately and then simplify the result. Multiplying fractions might seem tricky at first, but with practice, it becomes second nature. Keep practicing, and you'll become a fraction master in no time! And remember, math is not just about getting the right answer; it's also about understanding the process and building problem-solving skills. These skills are valuable not only in mathematics but also in many other areas of life. So, keep exploring, keep questioning, and keep learning. Math is a fascinating subject, and there's always something new to discover. As you continue your mathematical journey, you'll find that the concepts you learn build upon each other, creating a solid foundation for more advanced topics. So, embrace the challenges, celebrate your successes, and most importantly, enjoy the process of learning. Remember, every problem you solve is a step forward in your mathematical development. So, keep practicing, keep exploring, and never stop learning!
