Solving 2 1/8 Times 2/3 A Step By Step Guide

Jul 6, 2025 by Admin 45 views

$Understanding the Multiplication of Mixed Numbers: A Detailed Guide to Solving $2 \frac{1}{8} \times \frac{2}{3}$$

Introduction

In the realm of mathematics, mastering the multiplication of mixed numbers and fractions is a crucial skill. This article delves into a step-by-step guide on how to solve the problem $2 \frac{1}{8} \times \frac{2}{3}$ . We will break down each stage, ensuring clarity and understanding for learners of all levels. Our primary focus will be on transforming mixed numbers into improper fractions, simplifying the multiplication process, and converting the final result back into a mixed number or a simplified fraction. This comprehensive exploration aims to equip you with the knowledge and confidence to tackle similar mathematical challenges effectively. Through detailed explanations and practical examples, we'll navigate the intricacies of fraction multiplication, highlighting key concepts and common pitfalls to avoid. Whether you're a student looking to improve your math skills or simply someone keen on refreshing your knowledge, this guide promises to be an invaluable resource. So, let’s embark on this mathematical journey together, unraveling the complexities of multiplying mixed numbers and fractions with ease and precision.

Converting Mixed Numbers to Improper Fractions

The first crucial step in solving $2 \frac{1}{8} \times \frac{2}{3}$ involves transforming the mixed number, $2 \frac{1}{8}$ , into an improper fraction. A mixed number combines a whole number and a fraction, while an improper fraction has a numerator larger than or equal to its denominator. The conversion process is essential because it simplifies the multiplication of fractions. To convert $2 \frac{1}{8}$ into an improper fraction, we follow a specific procedure: multiply the whole number (2) by the denominator of the fraction (8) and then add the numerator (1). This result becomes the new numerator, and the denominator remains the same. Mathematically, this can be represented as: (Whole Number × Denominator) + Numerator / Denominator. Applying this to our mixed number, we get (2 × 8) + 1 = 16 + 1 = 17. Therefore, the improper fraction equivalent of $2 \frac{1}{8}$ is $\frac{17}{8}$ . This conversion is a fundamental skill in dealing with mixed numbers in any arithmetic operation, be it addition, subtraction, multiplication, or division. Understanding this process not only aids in solving the problem at hand but also lays a strong foundation for more advanced mathematical concepts. Remember, the key is to ensure that the whole number part is properly integrated into the fractional representation, maintaining the overall value of the number. The ability to fluently convert between mixed numbers and improper fractions is a cornerstone of fraction manipulation, making mathematical problems easier to handle and solve. With this conversion complete, we can now proceed to the next stage of multiplying fractions, equipped with a clear and manageable form of the original mixed number.

Multiplying Fractions: $\frac{17}{8} \times \frac{2}{3}$

Now that we've converted the mixed number $2 \frac{1}{8}$ into the improper fraction $\frac{17}{8}$ , we can proceed with the multiplication: $\frac{17}{8} \times \frac{2}{3}$ . Multiplying fractions is a straightforward process that involves multiplying the numerators (the top numbers) and the denominators (the bottom numbers) separately. In this case, we multiply the numerators 17 and 2 to get 34, and we multiply the denominators 8 and 3 to get 24. This gives us the fraction $\frac{34}{24}$ . This step is a direct application of the fundamental rule of fraction multiplication: $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$ . It's crucial to ensure that you're multiplying the correct parts of the fractions – numerators with numerators and denominators with denominators. After performing the multiplication, we obtain a new fraction, which may or may not be in its simplest form. The fraction $\frac{34}{24}$ represents the direct result of the multiplication, but it's essential to check if it can be simplified further. Simplification involves finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by it. This process reduces the fraction to its lowest terms, making it easier to understand and work with. The ability to accurately multiply fractions is a foundational skill in mathematics, with applications extending beyond simple arithmetic into algebra, calculus, and various real-world scenarios. By mastering this step, you're building a solid base for more complex mathematical operations and problem-solving. So, with the product of the fractions calculated, we now move on to the crucial step of simplifying the result.

Simplifying the Improper Fraction

After multiplying $\frac{17}{8}$ by $\frac{2}{3}$ , we obtained the improper fraction $\frac{34}{24}$ . Simplifying this fraction is a vital step in presenting the answer in its most concise and understandable form. To simplify a fraction, we need to find the greatest common divisor (GCD) of the numerator (34) and the denominator (24) and then divide both by this GCD. The GCD is the largest number that divides both 34 and 24 without leaving a remainder. To find the GCD, we can list the factors of each number: Factors of 34: 1, 2, 17, 34 Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. The largest number that appears in both lists is 2, so the GCD of 34 and 24 is 2. Now, we divide both the numerator and the denominator by the GCD: $\frac{34 ÷ 2}{24 ÷ 2} = \frac{17}{12}$ . This process of dividing by the GCD is known as simplifying or reducing the fraction to its lowest terms. The resulting fraction, $\frac{17}{12}$ , is an improper fraction because the numerator (17) is greater than the denominator (12). While $\frac{17}{12}$ is a simplified fraction, it is often preferable to express the final answer as a mixed number, especially when dealing with practical applications. The ability to simplify fractions is a fundamental skill in mathematics, ensuring that answers are presented in their most manageable form. It not only makes the numbers easier to work with but also aids in comparing fractions and understanding their relative values. Simplifying fractions is a crucial step in problem-solving, ensuring clarity and precision in mathematical calculations. Now that we've simplified the improper fraction, we move on to the final step of converting it back into a mixed number for a more intuitive representation of the result.

Converting the Improper Fraction Back to a Mixed Number

Having simplified the improper fraction $\frac{34}{24}$ to $\frac{17}{12}$ , our final step is to convert this improper fraction back into a mixed number. This conversion provides a more intuitive understanding of the quantity, as it separates the whole number part from the fractional part. To convert $\frac{17}{12}$ into a mixed number, we divide the numerator (17) by the denominator (12). The quotient represents the whole number part of the mixed number, and the remainder becomes the numerator of the fractional part, while the denominator remains the same. When we divide 17 by 12, we get a quotient of 1 and a remainder of 5. This means that $\frac{17}{12}$ can be expressed as 1 whole number and a fraction with a numerator of 5 and a denominator of 12. Therefore, the mixed number is $1 \frac{5}{12}$ . This conversion process effectively separates the whole units contained within the improper fraction from the remaining fractional part. The mixed number $1 \frac{5}{12}$ is a clear and concise way to represent the result of our initial multiplication problem, $2 \frac{1}{8} \times \frac{2}{3}$ . It provides an immediate sense of the magnitude of the answer, making it easier to visualize and apply in practical contexts. Converting improper fractions to mixed numbers is a fundamental skill in mathematics, allowing for a more intuitive understanding and interpretation of fractional quantities. This final step completes our journey through the multiplication of a mixed number and a fraction, demonstrating the importance of each step in achieving an accurate and meaningful result.

Conclusion

In conclusion, we have successfully navigated the process of multiplying a mixed number and a fraction, specifically $2 \frac{1}{8} \times \frac{2}{3}$ . We began by converting the mixed number into an improper fraction, $\frac{17}{8}$ , which allowed us to perform the multiplication more easily. We then multiplied the fractions, resulting in $\frac{34}{24}$ , which we simplified to $\frac{17}{12}$ by finding the greatest common divisor. Finally, we converted the improper fraction $\frac{17}{12}$ back into a mixed number, $1 \frac{5}{12}$ , to provide a clear and intuitive representation of the final answer. This step-by-step approach highlights the importance of each stage in the process, from converting mixed numbers to simplifying fractions. Mastering these skills is crucial for success in mathematics, as they form the foundation for more advanced topics. Understanding the concepts behind each step, such as why we convert mixed numbers and how to simplify fractions, is just as important as being able to perform the calculations. The ability to confidently multiply mixed numbers and fractions is a valuable asset, applicable in various real-world scenarios, from cooking and baking to construction and engineering. By practicing these steps and understanding the underlying principles, you can build a strong foundation in mathematics and tackle more complex problems with confidence. This comprehensive guide has hopefully equipped you with the knowledge and skills necessary to solve similar problems and further your mathematical journey.