Simplify Complex Fractions Step-by-Step Solution

Understanding Complex Fractions

Complex fractions, at their core, are fractions where either the numerator, the denominator, or both contain fractions themselves. This nested structure can appear intimidating, but the key to simplification lies in treating each part systematically and applying the basic rules of fraction arithmetic. To effectively simplify these complex fractions, it's essential to understand the structure and how the nested fractions interact. Recognizing the components – the main fraction bar, the numerator fraction, and the denominator fraction – is the first step in strategizing a solution. Our primary goal is to transform this intricate fraction into a simple fraction, where both the numerator and the denominator are free of any further fractional expressions. This involves several steps, such as finding common denominators, performing addition or subtraction, and finally, dividing fractions, which is the same as multiplying by the reciprocal. The ability to manipulate fractions accurately is a cornerstone of algebra, and mastering complex fractions reinforces this foundational skill. In the given expression, ${\frac{1-\frac{2}{x+1}}{1-\frac{1}{x}}}$ , we immediately observe the presence of fractions within the main fraction. This nesting is what defines it as a complex fraction and necessitates the application of specific simplification techniques. The challenge is to consolidate these individual fractions into single expressions, paving the way for the final simplification. The process is not just about arriving at the answer; it's also about enhancing your mathematical intuition and problem-solving capabilities. By understanding each step and the rationale behind it, you can approach similar problems with confidence and accuracy. This skill is invaluable not just in academics but also in various real-world applications where complex mathematical models need to be simplified for analysis and decision-making.

Step 1: Simplify the Numerator

The first key step to simplify the numerator of the complex fraction. Our focus here is on the expression ${1-\frac{2}{x+1}}$. To combine these terms, we need a common denominator. The number '1' can be expressed as a fraction with any denominator, so we'll rewrite '1' as ${\frac{x+1}{x+1}}$. This allows us to subtract the fractions directly, as they now share a common denominator. The process of finding a common denominator is a fundamental operation in fraction arithmetic. It's the linchpin that allows us to add or subtract fractions, as it ensures we're dealing with comparable units. In this case, the common denominator is ${x+1}$, which means we're expressing both terms of the numerator in terms of this denominator. The rewritten expression, ${\frac{x+1}{x+1} - \frac{2}{x+1}}$, sets the stage for a straightforward subtraction. We subtract the numerators while keeping the denominator constant. This step highlights the importance of understanding fraction equivalence and how changing the form of a number (like rewriting '1' as ${\frac{x+1}{x+1}}$) doesn't change its value but makes it compatible for operations with other fractions. The ability to identify and create common denominators is not just crucial for simplifying complex fractions but also for solving equations involving fractions and for a multitude of other algebraic manipulations. Once we've found the common denominator, the subtraction becomes a matter of simple arithmetic in the numerator. This methodical approach – breaking down the problem into manageable steps – is a hallmark of effective problem-solving in mathematics. It allows us to handle complex expressions with confidence and precision.

Now, subtract the numerators:

${\frac{x+1}{x+1} - \frac{2}{x+1} = \frac{(x+1) - 2}{x+1} = \frac{x-1}{x+1}}$

Step 2: Simplify the Denominator

Next, let's simplify the denominator of the complex fraction, which is ${1-\frac{1}{x}}$. Similar to the numerator, we need to find a common denominator to combine these terms. We can rewrite '1' as ${\frac{x}{x}}$, giving us a common denominator of 'x'. This transformation is crucial for the subsequent subtraction. The strategy here mirrors the approach we used for the numerator: expressing the whole number '1' as a fraction with the same denominator as the other term. This technique is a cornerstone of fraction arithmetic, enabling us to perform addition and subtraction across terms that initially appear incompatible. By rewriting '1' as ${\frac{x}{x}}$, we maintain the value while creating a common ground for the subtraction. This step underscores the flexibility and power of mathematical notation – the ability to represent the same quantity in different forms to suit our computational needs. The common denominator, 'x' in this case, acts as a bridge, allowing us to combine the terms seamlessly. This process highlights the importance of understanding the properties of fractions and how manipulating them correctly can simplify complex expressions. The skill of identifying and applying common denominators is not just limited to complex fractions; it's a fundamental tool in various mathematical contexts, including solving equations, working with rational expressions, and calculus. This methodical approach – converting terms to a common denominator before performing arithmetic operations – is a key principle in simplifying any expression involving fractions. The ability to master this technique builds a solid foundation for more advanced mathematical concepts and problem-solving strategies.

Now, subtract the fractions:

${\frac{x}{x} - \frac{1}{x} = \frac{x-1}{x}}$

Step 3: Divide the Simplified Numerator by the Simplified Denominator

Now that we've simplified the numerator and the denominator, we're ready to divide the two resulting fractions. This step involves dividing ${\frac{x-1}{x+1}}$ by ${\frac{x-1}{x}}$. Dividing by a fraction is the same as multiplying by its reciprocal. This principle is a cornerstone of fraction arithmetic and is crucial for simplifying complex fractions. The act of dividing by a fraction can be intuitively understood as asking how many times the divisor (the fraction we're dividing by) fits into the dividend (the fraction being divided). However, mathematically, it's more efficient to multiply by the reciprocal. The reciprocal of a fraction is simply the fraction flipped – the numerator becomes the denominator, and vice versa. This transformation turns division into multiplication, an operation that is often easier to handle. The rule of "invert and multiply" is not just a procedural trick; it's rooted in the fundamental properties of fractions and their relationship to division. Applying this rule allows us to convert a complex division problem into a simpler multiplication problem. This step underscores the importance of understanding the inverse relationship between multiplication and division and how this relationship extends to fractions. Mastering this concept is essential for simplifying not only complex fractions but also for solving equations involving fractions and for various other mathematical manipulations. The ability to fluently switch between division and multiplication by using reciprocals is a valuable skill that enhances mathematical problem-solving efficiency and accuracy.

So, we multiply ${\frac{x-1}{x+1}}$ by the reciprocal of ${\frac{x-1}{x}}$, which is ${\frac{x}{x-1}}$:

${\frac{x-1}{x+1} \div \frac{x-1}{x} = \frac{x-1}{x+1} \cdot \frac{x}{x-1}}$

Step 4: Simplify the Resulting Expression

Finally, we simplify the resulting expression after performing the multiplication. We have ${\frac{x-1}{x+1} \cdot \frac{x}{x-1}}$. Notice that the term ${(x-1)}$ appears in both the numerator and the denominator, allowing us to cancel it out. This cancellation is a crucial step in simplifying algebraic expressions. The ability to identify and cancel common factors is a fundamental skill in algebra. It's based on the principle that any non-zero number divided by itself equals one, and multiplying or dividing by one doesn't change the value of an expression. In this case, the common factor ${(x-1)}$ can be canceled because it appears as a multiplicative term in both the numerator and the denominator. This simplification process not only reduces the complexity of the expression but also reveals its underlying structure more clearly. Recognizing opportunities for cancellation is a key aspect of algebraic manipulation and is essential for solving equations, simplifying rational expressions, and various other mathematical tasks. This step underscores the importance of factoring and understanding the multiplicative relationships within an expression. By canceling common factors, we're essentially stripping away unnecessary complexity and revealing the simplest form of the expression. This skill is not just valuable for academic mathematics; it's also crucial in many real-world applications where mathematical models need to be simplified for analysis and interpretation. The ability to streamline expressions through cancellation enhances both efficiency and accuracy in mathematical problem-solving.

After canceling the ${(x-1)}$ terms, we are left with:

${\frac{\cancel{(x-1)}}{x+1} \cdot \frac{x}{\cancel{(x-1)}} = \frac{x}{x+1}}$

Final Answer

Therefore, the final answer to the simplified complex fraction ${\frac{1-\frac{2}{x+1}}{1-\frac{1}{x}}}$ is ${\frac{x}{x+1}}$. This result showcases the power of systematic simplification and the importance of understanding fundamental fraction operations. The journey from a complex nested fraction to the simplified form ${\frac{x}{x+1}}$ illustrates the elegance of mathematical manipulation. By methodically addressing each component of the complex fraction – simplifying the numerator, simplifying the denominator, dividing the simplified fractions, and canceling common factors – we arrive at a concise and easily understandable result. This process not only provides the solution but also reinforces crucial mathematical skills, including finding common denominators, performing fraction arithmetic, and recognizing opportunities for simplification. The final answer, ${\frac{x}{x+1}}$, is not just a numerical result; it's a testament to the power of structured problem-solving and a deeper understanding of algebraic principles. This ability to simplify complex expressions is invaluable in various mathematical contexts, from solving equations to analyzing functions and beyond. Mastering these techniques equips you with the confidence and competence to tackle more advanced mathematical challenges. The journey of simplification is as important as the destination, fostering critical thinking and a methodical approach to problem-solving that extends far beyond the realm of fractions.

Thus, the correct answer is:

C. ${\frac{x}{x+1}}$