Solve The Math Expression: [2/19×7/21]+[7/19 ×2/21]
In the realm of mathematics, fractions often present a unique challenge, especially when combined with operations like multiplication and addition. This article delves into the intricacies of solving the expression [2/19×7/21]+[7/19 ×2/21], providing a comprehensive guide to understanding the underlying principles and step-by-step solutions. We'll explore the fundamental concepts of fraction multiplication and addition, and how they intertwine to arrive at the final answer. Whether you're a student grappling with fraction problems or simply seeking to enhance your mathematical prowess, this exploration will equip you with the knowledge and skills to conquer similar challenges with confidence.
Understanding Fraction Multiplication
At its core, fraction multiplication involves combining parts of different wholes. To effectively tackle the expression at hand, a solid grasp of this concept is paramount. When multiplying fractions, we multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. This straightforward process forms the bedrock of fraction multiplication. Let's illustrate this with a simple example: 1/2 multiplied by 2/3. Multiplying the numerators (1 and 2) gives us 2, and multiplying the denominators (2 and 3) yields 6. Thus, 1/2 × 2/3 equals 2/6, which can then be simplified to 1/3.
Applying this principle to our expression, we first focus on the individual multiplication operations within the brackets. We have 2/19 multiplied by 7/21 and 7/19 multiplied by 2/21. Let's break down the first multiplication: 2/19 × 7/21. Multiplying the numerators (2 and 7) gives us 14, and multiplying the denominators (19 and 21) results in 399. Therefore, 2/19 × 7/21 equals 14/399. Similarly, for the second multiplication, 7/19 × 2/21, multiplying the numerators (7 and 2) gives us 14, and multiplying the denominators (19 and 21) again yields 399. Thus, 7/19 × 2/21 also equals 14/399. This foundational step showcases the elegance and simplicity of fraction multiplication.
Before proceeding further, it's crucial to consider simplification. Simplification involves reducing a fraction to its lowest terms by dividing both the numerator and denominator by their greatest common factor (GCF). In the case of 14/399, the GCF is 7. Dividing both 14 and 399 by 7, we get 2/57. Simplifying fractions not only makes them easier to work with but also provides a clearer representation of the fraction's value. However, for the sake of illustrating the subsequent addition process, we'll proceed with the unsimplified fractions (14/399) for now.
Mastering Fraction Addition
With the multiplication aspect addressed, we now turn our attention to fraction addition. Adding fractions requires a common denominator, meaning the fractions must have the same denominator before we can add their numerators. This common denominator serves as the foundation for combining the fractional parts accurately. In our expression, we have two fractions resulting from the multiplication operations: 14/399 and 14/399. Notice that they already share a common denominator of 399, which simplifies the addition process significantly.
To add fractions with a common denominator, we simply add their numerators and keep the denominator the same. In this case, we add the numerators 14 and 14, which gives us 28. The denominator remains 399. Therefore, 14/399 + 14/399 equals 28/399. This addition step highlights the importance of a common denominator in ensuring accurate results. If the fractions did not have a common denominator, we would need to find the least common multiple (LCM) of the denominators and convert the fractions accordingly before adding them.
Now that we have the result of the addition, 28/399, it's essential to simplify it to its lowest terms. To do this, we need to find the greatest common factor (GCF) of 28 and 399. The GCF of 28 and 399 is 7. Dividing both the numerator (28) and the denominator (399) by 7, we get 4/57. This simplification process is crucial for expressing the fraction in its most concise and understandable form.
Step-by-Step Solution: A Recap
To solidify your understanding, let's recap the step-by-step solution to the expression [2/19×7/21]+[7/19 ×2/21]:
- Multiply the fractions within the brackets: 2/19 × 7/21 = 14/399 and 7/19 × 2/21 = 14/399.
- Add the resulting fractions: 14/399 + 14/399 = 28/399.
- Simplify the final fraction: 28/399 simplified to 4/57.
Therefore, the solution to the expression [2/19×7/21]+[7/19 ×2/21] is 4/57. This step-by-step approach provides a clear and logical pathway to solving the problem, emphasizing the importance of each operation and its role in achieving the final answer.
Alternative Approaches and Insights
While the step-by-step method provides a solid foundation, let's explore some alternative approaches and insights that can further enhance your understanding of fraction manipulation. One such approach involves recognizing patterns and leveraging mathematical properties to simplify the process. In our expression, [2/19×7/21]+[7/19 ×2/21], we can observe that the terms within the brackets share a common factor.
Notice that both terms have 2/19 and 7/21 as factors, just in different orders. This allows us to use the commutative property of multiplication, which states that the order in which we multiply numbers does not affect the result (a × b = b × a). By recognizing and applying such properties, we can often streamline the solution process. In this case, we can rewrite the expression as (2/19 × 7/21) + (2/19 × 7/21).
Now, we have two identical terms being added together. This is equivalent to multiplying the term by 2. So, the expression becomes 2 × (2/19 × 7/21). This simplification significantly reduces the number of steps required to solve the problem. This alternative approach demonstrates the power of pattern recognition and strategic simplification in mathematics.
Next, we can multiply the fractions within the parentheses: 2/19 × 7/21 = 14/399, as we calculated earlier. Now, we have 2 × (14/399). Multiplying a fraction by a whole number involves multiplying the numerator by the whole number and keeping the denominator the same. Thus, 2 × (14/399) equals 28/399.
Finally, we simplify the fraction 28/399 to its lowest terms, which, as we determined previously, is 4/57. This alternative approach, while seemingly different, arrives at the same final answer: 4/57. This reaffirms the flexibility and interconnectedness of mathematical concepts and problem-solving strategies.
Real-World Applications of Fraction Operations
The manipulation of fractions extends far beyond the classroom, finding practical applications in numerous real-world scenarios. Understanding fraction multiplication and addition is essential in fields such as cooking, construction, finance, and even music. These applications highlight the tangible relevance and importance of mastering fraction operations.
In cooking, recipes often call for ingredients in fractional amounts. For instance, a recipe might require 1/2 cup of flour, 1/4 teaspoon of salt, and 3/4 cup of sugar. Adjusting the recipe to serve a different number of people requires multiplying or adding these fractional amounts. A baker who wants to double the recipe needs to multiply each fraction by 2, while someone who wants to halve the recipe needs to multiply each fraction by 1/2. This demonstrates how fraction multiplication is crucial for scaling recipes accurately.
In construction, fractions are used extensively in measurements and calculations. Architects and builders need to work with fractional dimensions when designing and constructing buildings. For example, a wall might be 8 1/2 feet long, and a window might be 3 1/4 feet wide. Determining how many windows can fit on the wall or calculating the total amount of material needed often involves adding and subtracting fractions. This underscores the importance of fraction operations in ensuring precise and accurate construction.
In finance, fractions play a vital role in calculating interest rates, investment returns, and debt payments. Interest rates are often expressed as percentages, which are essentially fractions out of 100. Calculating the interest earned on a savings account or the monthly payment on a loan involves multiplying fractions. This highlights the significance of fraction operations in financial planning and decision-making.
Even in music, fractions are fundamental to understanding rhythm and timing. Musical notes have fractional durations, such as a quarter note, half note, or eighth note. The relationships between these note durations are expressed as fractions. A musician needs to understand these fractional relationships to play music accurately and maintain a consistent tempo. This illustrates the pervasive nature of fractions across diverse disciplines.
Conclusion: Embracing the Power of Fractions
In conclusion, the expression [2/19×7/21]+[7/19 ×2/21] serves as a valuable case study for understanding fraction multiplication and addition. By breaking down the problem into smaller, manageable steps, we've demonstrated how to confidently navigate these operations. From mastering the fundamentals of multiplying numerators and denominators to recognizing the importance of common denominators in addition, we've explored the core principles that govern fraction manipulation. This journey into the world of fractions has equipped you with the tools and knowledge to tackle similar challenges with greater ease and accuracy.
Moreover, we've delved into alternative approaches and insights, showcasing the power of pattern recognition and strategic simplification in mathematics. By recognizing common factors and leveraging mathematical properties, we can often streamline the solution process and arrive at the answer more efficiently. This exploration of alternative methods underscores the flexibility and interconnectedness of mathematical concepts.
Finally, we've highlighted the real-world applications of fraction operations, demonstrating their relevance in diverse fields such as cooking, construction, finance, and music. These examples serve as a reminder that the mathematical skills we acquire in the classroom have tangible value in our daily lives and professional endeavors. This understanding of the practical applications of fractions reinforces the importance of mastering these concepts.
As you continue your mathematical journey, embrace the power of fractions and their ability to unlock a deeper understanding of the world around you. With a solid foundation in fraction operations, you'll be well-equipped to tackle more complex mathematical challenges and appreciate the beauty and elegance of this fundamental mathematical concept. This mastery of fractions will empower you to confidently navigate a wide range of problems and applications.