Multiply The Fraction $\frac{1}{2}$ By $\frac{2}{3}$ And Reduce The Result To Its Lowest Terms.

Fraction multiplication is a fundamental arithmetic operation that involves combining two or more fractions. Understanding how to multiply fractions and simplify the result to its lowest terms is crucial for various mathematical concepts and real-world applications. In this comprehensive guide, we will delve into the process of multiplying fractions, reducing fractions to their simplest form, and explore the underlying principles that govern these operations. Our main focus will be on providing a clear, step-by-step explanation, accompanied by illustrative examples, to ensure a thorough understanding of the topic. We aim to provide an engaging learning experience that caters to both beginners and those looking to reinforce their existing knowledge.

To master the multiplication of fractions, you need to understand the basic structure of a fraction. A fraction consists of two parts: the numerator (the top number) and the denominator (the bottom number). The numerator represents the number of parts you have, while the denominator represents the total number of equal parts the whole is divided into. For example, in the fraction 1/2, the numerator (1) indicates that we have one part, and the denominator (2) indicates that the whole is divided into two equal parts. When multiplying fractions, we are essentially finding a fraction of a fraction. This means we are taking a portion of an already divided whole, further dividing it, and determining the resulting fraction.

The core principle of multiplying fractions is remarkably straightforward: multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator. This can be expressed as (a/b) * (c/d) = (a * c) / (b * d), where a, b, c, and d are integers, and b and d are not equal to zero. This rule applies regardless of whether the fractions have the same denominators or different denominators, which simplifies the multiplication process compared to addition or subtraction of fractions. However, the result obtained from this multiplication may not always be in its simplest form, which is why reducing fractions to their lowest terms is an essential subsequent step. Throughout this discussion, we will emphasize the importance of understanding this foundational rule and its application in various contexts, ensuring you can confidently multiply any given fractions.

To effectively multiply fractions, it's essential to follow a systematic approach that ensures accuracy and efficiency. This section provides a detailed, step-by-step guide on how to multiply fractions, focusing on clarity and ease of understanding. By breaking down the process into manageable steps, we aim to help you grasp the concept thoroughly and apply it confidently to various problems. Our guide covers both the multiplication step and the subsequent reduction to lowest terms, providing a comprehensive understanding of the entire process.

Step 1: Write down the fractions you want to multiply. Begin by identifying the fractions you need to multiply. For instance, consider the example provided: 1/2 * 2/3. Writing the fractions clearly is the first step toward solving the problem accurately. It sets the stage for the subsequent operations and ensures that you don't miss any details. Accuracy at this stage is crucial, as any error here will propagate through the rest of the calculation. Always double-check that you have written the fractions correctly before proceeding.

Step 2: Multiply the numerators. The next step is to multiply the numerators (the top numbers) of the fractions. In our example, the numerators are 1 and 2. Multiplying these gives us 1 * 2 = 2. This new number will become the numerator of the resulting fraction. Understanding this step is fundamental, as it directly applies the core principle of fraction multiplication. Remember, the numerator represents the number of parts we are considering, so multiplying them essentially combines these parts from both fractions.

Step 3: Multiply the denominators. Now, multiply the denominators (the bottom numbers) of the fractions. In our example, the denominators are 2 and 3. Multiplying these gives us 2 * 3 = 6. This new number will be the denominator of the resulting fraction. The denominator represents the total number of equal parts the whole is divided into, and multiplying the denominators together calculates the new total number of parts after the multiplication. This step is as critical as multiplying the numerators, as it completes the fraction multiplication process.

Step 4: Write the new fraction. After multiplying the numerators and denominators, write the new fraction by placing the product of the numerators over the product of the denominators. In our example, we have 2 (from 1 * 2) as the new numerator and 6 (from 2 * 3) as the new denominator. Thus, the new fraction is 2/6. This fraction represents the result of multiplying the original two fractions. However, it is not always in its simplest form, which leads us to the next crucial step: reducing the fraction to its lowest terms.

Step 5: Reduce the fraction to its lowest terms (if necessary). The final step is to reduce the fraction to its lowest terms, if possible. This means simplifying the fraction so that the numerator and denominator have no common factors other than 1. To do this, find the greatest common divisor (GCD) of the numerator and the denominator and divide both by it. In our example, the fraction is 2/6. The GCD of 2 and 6 is 2. Dividing both the numerator and the denominator by 2, we get 2 ÷ 2 = 1 and 6 ÷ 2 = 3. Therefore, the fraction 2/6 reduces to 1/3. This simplified fraction is equivalent to the original but is expressed in its simplest form, making it easier to understand and work with. Reducing fractions to their lowest terms is a crucial step in ensuring that the final answer is presented in the most concise and understandable way.

Reducing fractions to their lowest terms, also known as simplifying fractions, is a fundamental skill in mathematics. A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1. This section will provide a detailed explanation of how to reduce fractions to their lowest terms, including the underlying principles and methods for finding the greatest common divisor (GCD). Understanding this process is essential for presenting fractions in their most concise and understandable form, making mathematical operations and comparisons easier.

To effectively reduce fractions, you must first grasp the concept of factors and the greatest common divisor (GCD). A factor of a number is an integer that divides evenly into that number. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. The greatest common divisor (GCD) of two or more numbers is the largest factor that they share. For instance, the GCD of 12 and 18 is 6, because 6 is the largest number that divides both 12 and 18 without leaving a remainder. Finding the GCD is crucial in reducing fractions, as it allows you to divide both the numerator and the denominator by the same number, thus simplifying the fraction while maintaining its value.

There are several methods for finding the GCD, but one of the most common and straightforward methods is listing the factors. To use this method, list all the factors of both the numerator and the denominator. Then, identify the largest factor that appears in both lists. This number is the GCD. For example, let's consider the fraction 24/36. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. Comparing these lists, we can see that the largest factor they have in common is 12. Therefore, the GCD of 24 and 36 is 12. Once you have found the GCD, you can use it to reduce the fraction to its lowest terms by dividing both the numerator and the denominator by the GCD. In the case of 24/36, we divide both 24 and 36 by 12, which gives us 2/3. This is the simplified form of the fraction.

Another method for finding the GCD is the Euclidean algorithm, which is particularly useful for larger numbers. The Euclidean algorithm involves repeatedly dividing the larger number by the smaller number and replacing the larger number with the remainder until the remainder is zero. The last non-zero remainder is the GCD. For example, let's find the GCD of 48 and 18 using the Euclidean algorithm. First, divide 48 by 18, which gives us a quotient of 2 and a remainder of 12. Then, replace 48 with 18 and 18 with 12, and repeat the process. Divide 18 by 12, which gives us a quotient of 1 and a remainder of 6. Replace 18 with 12 and 12 with 6. Divide 12 by 6, which gives us a quotient of 2 and a remainder of 0. Since the remainder is now 0, the last non-zero remainder (6) is the GCD of 48 and 18. Once you have the GCD, you can simplify the fraction as before by dividing both the numerator and the denominator by the GCD. Understanding and applying these methods for finding the GCD is crucial for effectively reducing fractions to their simplest form.

Now that we have a thorough understanding of multiplying fractions and reducing them to their lowest terms, let's apply these concepts to solve the problem presented. This section will walk through the step-by-step solution, reinforcing the methods discussed earlier and demonstrating how to arrive at the correct answer. By applying the principles in a practical context, we aim to solidify your understanding and build confidence in your ability to solve similar problems.

The problem asks us to multiply the fractions 1/2 and 2/3 and then reduce the result to its lowest terms. Following the steps outlined in Section 2, we begin by writing down the fractions we want to multiply: 1/2 * 2/3. This simple first step ensures that we have the problem clearly stated and are ready to proceed with the multiplication. Accuracy in this initial step is essential, as any error at this stage will impact the final result. Double-checking that the fractions are written correctly is a good practice to adopt.

Next, we multiply the numerators. The numerators in our fractions are 1 and 2. Multiplying these gives us 1 * 2 = 2. This product becomes the new numerator of our resulting fraction. Remember, the numerator represents the number of parts we have, so this step combines the parts from both fractions. Then, we multiply the denominators. The denominators in our fractions are 2 and 3. Multiplying these gives us 2 * 3 = 6. This product becomes the new denominator of our resulting fraction. The denominator represents the total number of equal parts the whole is divided into, and multiplying them together calculates the new total number of parts after the multiplication. Combining these results, we form the fraction 2/6.

However, the fraction 2/6 is not yet in its lowest terms. To simplify it, we need to find the greatest common divisor (GCD) of the numerator (2) and the denominator (6). The factors of 2 are 1 and 2, and the factors of 6 are 1, 2, 3, and 6. The largest factor that both numbers share is 2, so the GCD of 2 and 6 is 2. Now, we divide both the numerator and the denominator by the GCD. Dividing 2 by 2 gives us 1, and dividing 6 by 2 gives us 3. Therefore, the fraction 2/6 reduces to 1/3. This simplified fraction represents the same value as 2/6 but is expressed in its simplest form, making it the most appropriate answer.

After performing the multiplication and simplification, we arrive at the final answer. This section will clearly state the solution and provide a concise explanation of why it is correct. By reiterating the steps and rationale, we aim to reinforce your understanding and ensure that you can confidently apply these concepts in future problems.

The correct answer to the problem "Multiply 1/2 * 2/3 and reduce to lowest terms" is D. 1/3. Let's recap the steps we took to arrive at this answer. First, we multiplied the numerators: 1 * 2 = 2. Then, we multiplied the denominators: 2 * 3 = 6. This gave us the fraction 2/6. However, this fraction is not in its lowest terms, so we proceeded to simplify it. To simplify 2/6, we found the greatest common divisor (GCD) of 2 and 6, which is 2. We then divided both the numerator and the denominator by the GCD: 2 ÷ 2 = 1 and 6 ÷ 2 = 3. This resulted in the simplified fraction 1/3.

The other options are incorrect because they do not follow the correct steps of fraction multiplication and simplification. Option A, 1/4, is incorrect because it likely arises from an error in either the multiplication or simplification process. Option B, 1/6, might result from dividing instead of multiplying, or from miscalculating the GCD. Option C, 4/3, is an improper fraction that is significantly larger than the expected result and likely stems from a misunderstanding of the multiplication process. By methodically following the steps of multiplying the numerators and denominators, and then reducing the resulting fraction to its lowest terms, we can confidently arrive at the correct answer: 1/3. This comprehensive approach ensures accuracy and demonstrates a clear understanding of the principles of fraction multiplication and simplification.

To further reinforce your understanding of multiplying fractions and reducing them to their lowest terms, it's essential to practice with a variety of problems. This section provides several practice questions that will allow you to apply the concepts you've learned. Working through these problems will help solidify your skills and build confidence in your ability to solve similar questions independently. Each problem offers an opportunity to practice the step-by-step process discussed earlier, ensuring a thorough grasp of the topic.

1. Multiply 3/4 * 2/5 and reduce to lowest terms.
2. Multiply 1/3 * 4/7 and reduce to lowest terms.
3. Multiply 5/8 * 2/3 and reduce to lowest terms.
4. Multiply 2/9 * 3/4 and reduce to lowest terms.
5. Multiply 7/10 * 5/6 and reduce to lowest terms.

These practice problems cover a range of scenarios, allowing you to apply the principles of fraction multiplication and simplification in different contexts. As you work through each problem, remember to follow the steps we've outlined: (1) Write down the fractions, (2) Multiply the numerators, (3) Multiply the denominators, (4) Write the new fraction, and (5) Reduce the fraction to its lowest terms if necessary. Pay close attention to identifying the greatest common divisor (GCD) when simplifying fractions, as this is a crucial step in arriving at the correct answer. By consistently applying these steps, you'll become more proficient in multiplying fractions and reducing them to their simplest form.

Working through these problems will not only enhance your skills but also help you identify any areas where you may need further clarification. If you encounter difficulties with a particular problem, revisit the explanations and examples provided in the earlier sections. Understanding the underlying principles is key to successfully multiplying fractions and simplifying them. Consistent practice and review will build a solid foundation in this fundamental mathematical skill.

In conclusion, mastering the multiplication of fractions and reducing them to their lowest terms is a crucial skill in mathematics. Throughout this guide, we have provided a comprehensive and step-by-step explanation of the process, ensuring that both beginners and those seeking to reinforce their knowledge can grasp the concepts effectively. We have explored the fundamental principles of fraction multiplication, the importance of simplifying fractions, and practical methods for finding the greatest common divisor (GCD). By following the outlined steps and engaging with the practice problems, you can confidently tackle fraction multiplication problems and present your answers in their simplest form.

The ability to multiply fractions and reduce them to their lowest terms is not only essential for mathematical problem-solving but also has numerous real-world applications. Whether you're calculating measurements, adjusting recipes, or working with proportions, a solid understanding of fractions is invaluable. This guide has aimed to provide you with the tools and knowledge necessary to excel in these areas. Remember, consistent practice is key to mastery. By regularly reviewing the concepts and working through different types of problems, you can further solidify your skills and develop a deeper understanding of fractions.

We encourage you to continue practicing and applying these concepts in various contexts. Mathematics is a subject that builds upon itself, and a strong foundation in fractions will undoubtedly benefit you in more advanced topics. If you encounter challenges along the way, don't hesitate to revisit this guide or seek additional resources. With dedication and practice, you can confidently navigate the world of fractions and mathematical operations, ensuring success in your academic and practical endeavors. The principles and methods discussed in this guide are designed to empower you with the skills needed to excel in mathematics and beyond.