Evaluate The Mathematical Expression (6/7 - 3/7) × (14/23) ÷ (6/-23).
In the realm of mathematics, evaluating expressions is a fundamental skill. This article will delve into the step-by-step process of evaluating the expression (6/7 - 3/7) × (14/23) ÷ (6/-23), providing a clear and concise explanation for each operation. This exploration will not only demonstrate the correct methodology but also highlight the underlying mathematical principles involved. Understanding these principles is crucial for mastering more complex mathematical problems in the future. We will break down the expression into manageable parts, ensuring that every step is easily comprehensible, even for those who may find mathematics challenging. So, let's embark on this mathematical journey together and unravel the intricacies of this expression.
Step 1: Simplify the Expression Inside the Parentheses
Our initial focus is on simplifying the expression within the parentheses: (6/7 - 3/7). This involves subtracting two fractions that share a common denominator, which makes the process straightforward. When subtracting fractions with the same denominator, we simply subtract the numerators and keep the denominator the same. In this case, we have 6/7 minus 3/7. Subtracting the numerators, 6 minus 3, gives us 3. Therefore, the result of the subtraction is 3/7. This seemingly simple step is a crucial building block for the rest of the calculation. By simplifying the expression inside the parentheses first, we reduce the complexity of the overall expression and make it easier to manage. This principle of simplifying within parentheses first is a cornerstone of the order of operations in mathematics, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Mastering this step is essential for tackling more complex mathematical problems involving multiple operations. Furthermore, understanding fraction arithmetic is fundamental to many areas of mathematics, including algebra, calculus, and statistics. A solid grasp of these basic principles will pave the way for success in advanced mathematical studies.
Step 2: Multiplication
Now that we've simplified the expression inside the parentheses to 3/7, we can move on to the multiplication operation. The next part of our expression is (3/7) × (14/23). Multiplying fractions involves multiplying the numerators together and the denominators together. So, we multiply 3 by 14, which equals 42. Then, we multiply 7 by 23, which equals 161. This gives us the fraction 42/161. However, this is not the final answer, as we can simplify this fraction further. Simplifying fractions is an essential step in mathematical problem-solving, as it allows us to express the result in its simplest form. To simplify 42/161, we need to find the greatest common divisor (GCD) of 42 and 161. The GCD is the largest number that divides both 42 and 161 without leaving a remainder. In this case, the GCD is 7. Dividing both the numerator and the denominator by 7, we get 6/23. This simplified fraction is much easier to work with in subsequent calculations. The ability to multiply and simplify fractions is a fundamental skill in mathematics, with applications in various fields, including engineering, finance, and computer science. Understanding how to perform these operations accurately and efficiently is crucial for success in these areas.
Step 3: Division
The final step in evaluating our expression is division. We now have to divide (6/23) by (6/-23). Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. So, the reciprocal of (6/-23) is (-23/6). Therefore, our division problem becomes a multiplication problem: (6/23) × (-23/6). Now we multiply the numerators: 6 multiplied by -23 equals -138. Then, we multiply the denominators: 23 multiplied by 6 equals 138. This gives us the fraction -138/138. This fraction can be simplified further. Since the numerator and the denominator are the same magnitude but with opposite signs, the fraction simplifies to -1. This is our final answer. Understanding the concept of reciprocals and how they are used in division is essential for mastering fraction arithmetic. Division is the inverse operation of multiplication, and understanding this relationship is crucial for solving a wide range of mathematical problems. The ability to divide fractions confidently and accurately is a valuable skill that will serve you well in various mathematical contexts.
Final Answer
After meticulously following each step, we have successfully evaluated the expression (6/7 - 3/7) × (14/23) ÷ (6/-23). We began by simplifying the expression within the parentheses, then performed the multiplication, and finally executed the division. Through these steps, we arrived at the final answer: -1. This result underscores the importance of adhering to the order of operations and the fundamental principles of fraction arithmetic. The journey of evaluating this expression has not only provided us with a numerical answer but has also reinforced our understanding of essential mathematical concepts. Mastering these concepts is crucial for tackling more complex mathematical problems and for applying mathematical principles in various real-world scenarios. The ability to confidently and accurately evaluate mathematical expressions is a valuable skill that will benefit you in both academic and professional pursuits. So, keep practicing, keep exploring, and continue to build your mathematical foundation.