Solve 14/15 ÷ (-2/3) And Represent The Solution On A Number Line.

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In the realm of mathematics, mastering the division of fractions is a fundamental skill that paves the way for more advanced concepts. This article delves into the intricacies of solving the expression 14/15 ÷ (-2/3), providing a comprehensive, step-by-step guide to arrive at the solution. Furthermore, we will explore how to represent this mathematical operation on a number line, offering a visual understanding of the process. Understanding these concepts is not just crucial for academic success but also finds applications in various real-world scenarios, from finance and economics to engineering and everyday problem-solving. This detailed explanation will ensure that even those who find fractions daunting can grasp the core principles and apply them effectively.

Unraveling the Division of Fractions

When faced with the task of dividing one fraction by another, the key lies in understanding the concept of reciprocals. Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is obtained by simply swapping the numerator and the denominator. For instance, the reciprocal of -2/3 is -3/2. This principle forms the bedrock of fraction division and simplifies the process significantly. By converting the division problem into a multiplication problem, we can leverage our understanding of fraction multiplication, which is often more straightforward. This fundamental transformation is crucial for tackling the problem at hand and serves as a cornerstone for more complex mathematical operations involving fractions. Mastering this concept not only aids in solving immediate problems but also builds a solid foundation for future mathematical endeavors.

Step-by-Step Solution

Let's tackle the problem 14/15 ÷ (-2/3) step by step to ensure clarity and understanding.

  1. Rewrite the division as multiplication by the reciprocal: As we've established, dividing by a fraction is equivalent to multiplying by its reciprocal. Thus, we rewrite the expression as 14/15 × (-3/2).
  2. Multiply the numerators and the denominators: Now, we multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. This gives us (14 × -3) / (15 × 2).
  3. Simplify the resulting fraction: Performing the multiplication, we get -42/30. This fraction can be simplified by finding the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 42 and 30 is 6. Dividing both the numerator and the denominator by 6, we get -7/5.
  4. Express as a mixed number (optional): While -7/5 is a perfectly valid answer, it can also be expressed as a mixed number for better understanding of its magnitude. -7/5 is equal to -1 and 2/5. This means it is one whole unit less than zero, with an additional 2/5 portion.

By following these steps meticulously, we arrive at the solution: 14/15 ÷ (-2/3) = -7/5 or -1 2/5. This systematic approach ensures accuracy and helps in building confidence when dealing with fraction division problems. Each step is grounded in fundamental mathematical principles, making the process both logical and efficient. The ability to break down a complex problem into smaller, manageable steps is a crucial skill in mathematics and beyond.

Representing on the Number Line

Visualizing mathematical operations on a number line can significantly enhance understanding, especially when dealing with fractions. A number line provides a graphical representation of numbers and their relationships, making it easier to grasp the concept of division and its outcome. In the case of 14/15 ÷ (-2/3), representing the solution on a number line involves marking the fractions and illustrating the division process. This visual aid is particularly helpful for those who benefit from a spatial understanding of mathematical concepts. By seeing the fractions laid out on a line, the division operation becomes more concrete and intuitive.

Steps to Represent on the Number Line

Let's break down how to represent -7/5 (or -1 2/5) on a number line, visualizing the solution we obtained earlier.

  1. Draw the number line: Begin by drawing a horizontal line. Mark the zero point in the middle. Extend the line to both the positive and negative sides. It’s important to have sufficient space on both sides to accommodate the numbers we need to represent.
  2. Mark the integers: Mark the integers (whole numbers) clearly on the number line. For our purpose, we need to mark at least -2, -1, 0, and 1, as our result lies between -1 and -2. Accurate spacing between the integers is crucial for a clear representation.
  3. Divide the line into fractions: Since we are dealing with fifths (-7/5), divide the space between each integer into five equal parts. This division allows us to accurately represent fractions with a denominator of 5. Each of these segments represents 1/5 of the unit interval.
  4. Locate -7/5: Starting from zero, move seven segments to the left (negative direction). This point represents -7/5, which is equivalent to -1 2/5. This placement visually demonstrates the value of the fraction in relation to the integers.
  5. Mark the point: Clearly mark the point on the number line that corresponds to -7/5. You can label it as -7/5 or -1 2/5, or both, for clarity. This marked point serves as the visual representation of the solution to our division problem.

By following these steps, you can effectively represent the solution of 14/15 ÷ (-2/3) on a number line. This visual representation not only reinforces the understanding of the numerical solution but also helps in developing a stronger intuition for how fractions and division work together. The number line becomes a powerful tool for visualizing mathematical concepts, making them more accessible and comprehensible.

Real-World Applications and Economic Relevance

The division of fractions is not merely an abstract mathematical concept; it has practical applications in numerous real-world scenarios, many of which are directly relevant to economics. Understanding how to manipulate fractions is essential for tasks ranging from budgeting and financial analysis to resource allocation and economic modeling. In economics, fractions often represent proportions, ratios, and rates, which are fundamental to understanding economic indicators and trends. The ability to perform these calculations accurately is crucial for making informed decisions in both personal and professional contexts.

Applications in Economics

  1. Financial Analysis: In finance, fractions are used to calculate investment returns, interest rates, and debt-to-equity ratios. For example, understanding how to divide fractions is crucial when calculating the yield on a bond or determining the profitability of an investment. Financial analysts rely heavily on these calculations to assess the financial health of companies and make investment recommendations. Accurate handling of fractions ensures precise financial analysis, which is vital for sound investment strategies.
  2. Budgeting: Both individuals and organizations use fractions when creating and managing budgets. Allocating a portion of income to various expenses, such as housing, food, and savings, often involves fractional calculations. Understanding how to divide fractions helps in determining how to distribute resources effectively. For instance, calculating what fraction of a monthly income should be allocated to rent requires a solid grasp of fraction division. Effective budgeting is a cornerstone of financial stability, and proficiency in fraction manipulation is a key component.
  3. Resource Allocation: In economics, resource allocation involves distributing scarce resources among competing uses. This often requires dividing resources into fractional parts based on various criteria, such as demand, efficiency, or equity. Governments and businesses alike use fractional calculations to allocate budgets, personnel, and materials. For example, a company might need to divide its marketing budget among different advertising channels, each receiving a fraction of the total budget. Efficient resource allocation is crucial for economic productivity and sustainability.
  4. Economic Modeling: Economists use mathematical models to analyze and predict economic phenomena. Fractions are frequently used in these models to represent economic variables and relationships, such as the marginal propensity to consume or the unemployment rate. Understanding how to divide fractions is essential for building and interpreting these models. Economic models provide insights into complex economic systems, and accurate mathematical operations, including fraction division, are critical for the reliability of these models.

Practical Examples

Consider a scenario where a company needs to divide a budget of $100,000 among three departments: marketing, research and development (R&D), and operations. If marketing is allocated 1/4 of the budget, R&D is allocated 2/5, and operations receive the remainder, calculating the exact amount each department receives requires division and subtraction of fractions. Similarly, if an investor wants to calculate the return on an investment that has yielded 3 1/2% interest, understanding fraction division is essential to determine the actual monetary gain. These examples highlight the direct relevance of fraction division in practical financial and economic situations. Mastery of these calculations empowers individuals and organizations to make informed decisions, contributing to financial success and economic stability.

Conclusion

In summary, solving 14/15 ÷ (-2/3) involves understanding the principles of fraction division, which entails multiplying by the reciprocal. The step-by-step solution leads us to -7/5 or -1 2/5. Representing this on a number line provides a visual confirmation of the result. The real-world applications, particularly in economics, underscore the importance of mastering this mathematical skill. From financial analysis to resource allocation, the ability to divide fractions accurately is crucial for informed decision-making. By grasping these concepts, individuals can enhance their problem-solving abilities and navigate various practical scenarios with confidence. The combination of theoretical understanding and practical application makes the division of fractions a fundamental skill in mathematics and economics alike. This knowledge empowers individuals to engage with complex problems and contribute effectively to both personal and professional endeavors.