Solving 7/16 × 4/3 × 1/2 A Step-by-Step Guide

Hey there, math enthusiasts! Today, we're diving deep into the world of fractions, specifically tackling the multiplication of three fractions: 7/16, 4/3, and 1/2. If you've ever felt a bit puzzled by fraction multiplication, don't worry; we're going to break it down step by step. This guide will not only help you solve this particular problem but also equip you with the skills to confidently handle similar questions in the future. Let's get started and conquer those fractions!

Understanding the Basics of Fraction Multiplication

Before we jump into the problem at hand, it's crucial to understand the fundamental concept of fraction multiplication. Guys, multiplying fractions is actually quite straightforward. Unlike addition or subtraction, where you need a common denominator, multiplication is all about multiplying straight across. This means you multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. It’s that simple! This method works regardless of the number of fractions you're dealing with, making it a powerful tool in your mathematical arsenal. The beauty of this method lies in its simplicity and directness. You don't need to find common denominators or perform any complex transformations. Just multiply the tops, multiply the bottoms, and you're on your way to the solution. Understanding this basic principle is the first step in mastering fraction multiplication and tackling more complex problems with ease.

When you approach a fraction multiplication problem, think of it as combining parts of parts. For example, if you're multiplying 1/2 by 1/3, you're essentially finding one-half of one-third. This visual representation can be incredibly helpful in grasping the concept and making the process more intuitive. Moreover, mastering fraction multiplication opens doors to various real-world applications. From cooking and baking to measuring and construction, fractions are an integral part of everyday life. Being able to confidently multiply fractions allows you to solve practical problems and make accurate calculations in a variety of contexts. So, let's solidify this foundation and move on to the problem at hand, where we'll apply this simple yet powerful principle to find the product of 7/16, 4/3, and 1/2. Remember, the key is to multiply straight across, and with a little practice, you'll become a fraction multiplication pro!

Step-by-Step Solution: Multiplying 7/16, 4/3, and 1/2

Okay, let's get down to business and solve this problem together. We need to find the product of 7/16, 4/3, and 1/2. Remember our golden rule: multiply straight across. First, we'll multiply the numerators: 7 × 4 × 1. Then, we'll multiply the denominators: 16 × 3 × 2. This gives us a new fraction, which we can then simplify if necessary. This step-by-step approach is crucial for tackling any fraction multiplication problem. By breaking it down into manageable parts, we avoid confusion and ensure accuracy. So, grab your pen and paper, and let's work through this together. Remember, the goal is not just to find the answer but also to understand the process. Once you grasp the method, you'll be able to apply it to a wide range of fraction multiplication problems.

Let's start with the numerators. Multiplying 7, 4, and 1 is straightforward: 7 × 4 = 28, and 28 × 1 = 28. So, our new numerator is 28. Now, let's move on to the denominators. Multiplying 16, 3, and 2 requires a bit more calculation: 16 × 3 = 48, and 48 × 2 = 96. So, our new denominator is 96. Now we have the fraction 28/96. But we're not done yet! The next crucial step is to simplify this fraction. Simplifying fractions means reducing them to their lowest terms, making them easier to work with and understand. This involves finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by it. So, let's dive into the simplification process and see how we can reduce 28/96 to its simplest form. This is where the real magic happens, transforming a seemingly complex fraction into a more manageable and elegant representation.

Simplifying the Resulting Fraction

Alright, we've arrived at the fraction 28/96. Great job, guys! Now, the next step is to simplify this fraction. Simplifying a fraction means reducing it to its lowest terms. To do this, we need to find the greatest common divisor (GCD) of both the numerator (28) and the denominator (96). The GCD is the largest number that divides evenly into both numbers. Once we find the GCD, we'll divide both the numerator and the denominator by it. This will give us the simplified fraction. This process of simplification is a cornerstone of fraction manipulation. It not only makes the fraction easier to understand and work with but also ensures that we're presenting the answer in its most concise and elegant form. Mastering simplification is key to becoming a fraction pro!

To find the GCD of 28 and 96, we can list the factors of each number. The factors of 28 are 1, 2, 4, 7, 14, and 28. The factors of 96 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, and 96. Looking at these lists, we can see that the greatest common factor is 4. So, the GCD of 28 and 96 is 4. Now, we divide both the numerator and the denominator by 4. 28 divided by 4 is 7, and 96 divided by 4 is 24. This gives us the simplified fraction 7/24. And there you have it! We've successfully simplified the fraction. This step not only makes our answer cleaner but also demonstrates a deeper understanding of fraction manipulation. Simplifying fractions is a valuable skill that will serve you well in various mathematical contexts. So, let's celebrate this small victory and move on to comparing our result with the given options.

Comparing the Simplified Fraction with the Options

Fantastic work, everyone! We've simplified the fraction 28/96 to 7/24. Now, the final step is to compare this simplified fraction with the options provided in the question. This is a crucial step because it ensures that we've not only solved the problem correctly but also that we're selecting the correct answer from the given choices. It's like putting the final piece of the puzzle in place. By comparing our result with the options, we can confidently confirm our solution and move on to the next challenge. This step reinforces the importance of accuracy and attention to detail in mathematics. So, let's carefully examine the options and see which one matches our simplified fraction of 7/24.

The options given are: A. 7/12, B. 21/32, C. 2^(13/48), and D. 7/24. By carefully comparing our simplified fraction, 7/24, with the options, we can clearly see that option D, 7/24, matches our result. This confirms that our calculations and simplification were accurate, and we've successfully solved the problem! It's always a satisfying moment when our hard work pays off and we arrive at the correct answer. This process of comparison not only validates our solution but also reinforces our understanding of the problem and the steps we took to solve it. So, let's give ourselves a pat on the back and celebrate this achievement. We've mastered the multiplication and simplification of fractions, and we're ready to tackle even more challenging problems in the future!

Conclusion: The Correct Answer and Key Takeaways

Woo-hoo! We've successfully navigated the world of fraction multiplication and simplification. The correct answer to the question is D. 7/24. You guys did an amazing job following along and working through the steps. This journey through fractions has not only helped us solve this specific problem but has also equipped us with valuable skills that we can apply to a wide range of mathematical challenges. Remember, the key to mastering fractions is understanding the basic principles and practicing regularly. With each problem you solve, you're building your confidence and solidifying your understanding. So, let's take a moment to recap the key takeaways from this exercise and see how we can apply them to future problems.

The main takeaways from this problem are: First, multiplying fractions involves multiplying the numerators and the denominators straight across. Second, simplifying fractions to their lowest terms is crucial for presenting the answer in its most concise form. Third, always double-check your work and compare your result with the given options to ensure accuracy. These principles are not only applicable to fraction multiplication but also to various other mathematical operations. By understanding and applying these concepts, you'll be well-equipped to tackle a wide range of mathematical challenges with confidence and precision. So, keep practicing, keep exploring, and keep pushing your mathematical boundaries. The world of numbers is vast and fascinating, and with each step you take, you're unlocking new possibilities and expanding your knowledge. Let's continue this journey of mathematical discovery and see what other exciting challenges we can conquer together!