Simplifying (5/6) * (9/10) * (4/5): A Step-by-Step Guide

Hey guys! Let's dive into simplifying the fraction problem (5/6) * (9/10) * (4/5). Fractions might seem intimidating at first, but breaking them down step-by-step makes them super manageable. In this article, we'll explore a detailed, human-friendly approach to solving this problem, ensuring you understand each stage of the process. We will cover everything from the basic principles of fraction multiplication to simplifying the final result. This guide is designed to help you not just get the answer, but also understand the why behind each step. So, whether you're a student tackling homework or just looking to brush up on your math skills, let's get started and make fractions a breeze!

Understanding Fraction Multiplication

Before we tackle the main problem, let's quickly recap the basics of fraction multiplication. When you multiply fractions, you're essentially finding a fraction of a fraction. It's like cutting a cake into slices and then taking a portion of those slices. The rule is straightforward: you multiply the numerators (the top numbers) together to get the new numerator, and you multiply the denominators (the bottom numbers) together to get the new denominator. For example, if you have ${\frac{a}{b}}$ multiplied by ${\frac{c}{d}}$, the result is ${\frac{a \times c}{b \times d}}$. This simple rule forms the foundation for all fraction multiplication, and it’s crucial to understand it before moving on to more complex problems. By grasping this concept, you’ll be able to confidently handle various fraction-related calculations and applications. Remember, the key is to take it one step at a time, ensuring each multiplication is accurate and clearly understood. This foundational knowledge will not only help you solve this particular problem but also equip you with the skills to tackle any fraction multiplication challenge that comes your way. Let's keep this principle in mind as we move forward and apply it to our main question.

Multiplying Multiple Fractions

When dealing with multiple fractions, such as in our problem (5/6) * (9/10) * (4/5), the process is an extension of the basic rule. Instead of just multiplying two fractions, you multiply all the numerators together and all the denominators together. So, if you have fractions ${\frac{a}{b}}$, ${\frac{c}{d}}$, and ${\frac{e}{f}}$, the product would be ${\frac{a \times c \times e}{b \times d \times f}}$. This method scales seamlessly to any number of fractions, making it a versatile tool for various mathematical scenarios. The beauty of this approach lies in its consistency; no matter how many fractions you're dealing with, the principle remains the same. This straightforward method simplifies complex calculations and makes it easier to manage larger expressions. By understanding how to multiply multiple fractions, you can tackle more intricate problems with confidence and precision. Remember, the key is to keep track of all the numerators and denominators, ensuring each is included in the final product. This skill is invaluable in many areas of mathematics, from algebra to calculus, and mastering it will significantly enhance your problem-solving abilities.

Solving the Problem: (5/6) * (9/10) * (4/5)

Okay, let's get into the heart of the problem: ${\frac{5}{6} \times \frac{9}{10} \times \frac{4}{5}}$. Our first step is to multiply all the numerators together and then multiply all the denominators together, as we discussed. This means we'll have (5 * 9 * 4) as the new numerator and (6 * 10 * 5) as the new denominator. Performing these multiplications gives us ${\frac{180}{300}}$. So, we've successfully multiplied the fractions together, but we're not quite done yet. The next crucial step is to simplify this fraction. Simplifying fractions involves finding the greatest common factor (GCF) of the numerator and the denominator and then dividing both by that factor. This process reduces the fraction to its simplest form, making it easier to work with and understand. Simplifying is not just about getting to the final answer; it's about expressing the fraction in the most concise and clear way possible. Understanding this step is essential for mastering fraction manipulation and will be useful in many mathematical contexts. Let's move on to the simplification process to bring our fraction to its most manageable form.

Step-by-Step Multiplication

Let's break down the multiplication step-by-step to ensure we're on the same page. First, we multiply the numerators: 5 * 9 * 4. 5 multiplied by 9 gives us 45, and then 45 multiplied by 4 gives us 180. So, the new numerator is 180. Next, we multiply the denominators: 6 * 10 * 5. 6 multiplied by 10 is 60, and then 60 multiplied by 5 is 300. Therefore, the new denominator is 300. This gives us the fraction ${\frac{180}{300}}$. It’s important to perform these multiplications carefully, as accuracy at this stage is crucial for getting the correct final answer. Taking it one step at a time, ensuring each multiplication is correct, helps avoid errors and builds confidence in your calculations. This methodical approach not only helps in this particular problem but also establishes a good habit for tackling any mathematical problem. Remember, accuracy in the initial steps lays the foundation for a correct solution. Now that we have our initial fraction, ${\frac{180}{300}}$, let's move on to the next step: simplifying it.

Simplifying the Fraction

Now that we have ${\frac{180}{300}}$, the next step is to simplify this fraction to its lowest terms. Simplifying fractions makes them easier to understand and work with. To do this, we need to find the greatest common factor (GCF) of both the numerator (180) and the denominator (300). The GCF is the largest number that divides both 180 and 300 without leaving a remainder. There are a couple of ways to find the GCF, but one common method is to list the factors of each number and identify the largest one they have in common. Alternatively, you can use prime factorization to find the GCF. Once we've found the GCF, we divide both the numerator and the denominator by it. This process reduces the fraction to its simplest form. Simplifying fractions is a fundamental skill in mathematics, and it's essential for solving problems involving fractions efficiently. It not only helps in getting the final answer but also provides a clearer representation of the fraction's value. Let's find the GCF of 180 and 300 and simplify our fraction.

Finding the Greatest Common Factor (GCF)

To find the greatest common factor (GCF) of 180 and 300, let’s list their factors. The factors of 180 are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, and 180. The factors of 300 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, and 300. Looking at these lists, we can see that the largest number that appears in both is 60. Therefore, the GCF of 180 and 300 is 60. Another way to find the GCF is by using prime factorization, which involves breaking down each number into its prime factors. This method can be particularly helpful for larger numbers. However, for our purposes, listing the factors works well. Once we have the GCF, we can use it to simplify the fraction. Finding the GCF is a crucial step in simplifying fractions, and mastering this skill will greatly improve your ability to work with fractions effectively. Now that we know the GCF is 60, we can proceed to divide both the numerator and the denominator by it.

Dividing by the GCF

Now that we've determined that the GCF of 180 and 300 is 60, we can simplify the fraction ${\frac{180}{300}}$ by dividing both the numerator and the denominator by 60. So, we divide 180 by 60, which gives us 3, and we divide 300 by 60, which gives us 5. This means our simplified fraction is ${\frac{3}{5}}$. This is the fraction in its simplest form because 3 and 5 have no common factors other than 1. Dividing by the GCF ensures that we reduce the fraction as much as possible, making it easier to understand and use in further calculations. This step is the culmination of the simplification process, and it's essential to ensure accuracy in the division. Once the fraction is simplified, we have the final answer in its most concise form. Simplifying fractions is a key skill in mathematics, and mastering it will make many other mathematical tasks easier. So, our final simplified fraction for the problem (5/6) * (9/10) * (4/5) is ${\frac{3}{5}}$.

Final Answer and Conclusion

So, after multiplying the fractions ${\frac{5}{6}}$, ${\frac{9}{10}}$, and ${\frac{4}{5}}$ and then simplifying the result, we arrive at the final answer: ${\frac{3}{5}}$. This process involved multiplying the numerators together, multiplying the denominators together, and then simplifying the resulting fraction by finding the greatest common factor (GCF) and dividing both the numerator and denominator by it. Simplifying fractions is a crucial skill in mathematics, as it allows us to express fractions in their simplest and most understandable form. This not only makes the numbers easier to work with but also provides a clearer representation of the fraction’s value. By understanding and practicing these steps, you can confidently tackle any fraction multiplication problem and simplify the results effectively. Remember, the key is to break down the problem into manageable steps and ensure accuracy at each stage. Congratulations on solving this problem! You've now added another tool to your math toolkit.

Key Takeaways

Let's recap the key takeaways from solving this problem. First, when multiplying fractions, you multiply the numerators together and the denominators together. Second, simplifying fractions involves finding the greatest common factor (GCF) of the numerator and denominator and then dividing both by the GCF. Third, simplifying fractions is essential for expressing them in their simplest form, which makes them easier to understand and work with. These steps are fundamental to working with fractions and will be useful in many mathematical contexts. Mastering these concepts allows you to approach fraction problems with confidence and accuracy. Remember, practice makes perfect, so the more you work with fractions, the more comfortable you'll become. By understanding these key principles, you'll be well-equipped to tackle a wide range of fraction-related challenges.

Practice Problems

To solidify your understanding, here are a few practice problems you can try:

1. ${\frac{2}{3} \times \frac{6}{8} \times \frac{1}{2}}$
2. ${\frac{7}{10} \times \frac{5}{14} \times \frac{4}{5}}$
3. ${\frac{3}{4} \times \frac{8}{9} \times \frac{1}{6}}$

Try solving these problems using the steps we've discussed. Remember to multiply the fractions first and then simplify the result. Working through these examples will help reinforce your skills and build your confidence in handling fractions. Don't hesitate to review the steps outlined in this article if you need a refresher. Consistent practice is the key to mastering any mathematical concept, and fractions are no exception. Good luck, and happy calculating! By tackling these practice problems, you’ll not only strengthen your understanding but also develop a deeper appreciation for the elegance and utility of fractions in mathematics.

I hope this guide has been helpful in understanding how to solve the fraction problem (5/6) * (9/10) * (4/5). Keep practicing, and you'll become a fraction master in no time!