Problem 75 - Algebra and Trigonometry

Question

Simplify the compound fractional expression. $$ 1-\frac{1}{1-\frac{1}{x}} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{-1}{x-1}\).

Step 1: Understand the Expression

The expression given is: \(1 - \frac{1}{1 - \frac{1}{x}}\). Our goal is to simplify this complex fraction by resolving the nested fractions step by step.

Step 2: Simplify the Inner Fraction

First, let's simplify the inner fraction \(\frac{1}{x}\) within the denominator. The expression for the denominator becomes \(1 - \frac{1}{x}\), which is \(\frac{x-1}{x}\) because \(1 = \frac{x}{x}\).

Step 3: Substitute Back into the Expression

Replace \(1 - \frac{1}{x}\) with \(\frac{x-1}{x}\) in the main expression. The expression now becomes: \(1 - \frac{1}{\frac{x-1}{x}}\).

Step 4: Simplify the Compound Fraction

To simplify the main expression \(1 - \frac{1}{\frac{x-1}{x}}\), multiply by the reciprocal: \(1 - \frac{x}{x-1}\).

Step 5: Combine into a Single Fraction

Rewrite the subtraction as a single fraction over a common denominator. The expression \(1 - \frac{x}{x-1}\) can be rewritten as: \[\frac{(x-1)-(x)}{x-1} = \frac{x-1-x}{x-1} = \frac{-1}{x-1}\].

Step 6: Simplify the Expression

The expression \(\frac{-1}{x-1}\) is the simplified form of the original compound fraction.

Key Concepts

Algebraic Fractions

Algebraic fractions are fractions where the numerator or the denominator, or both, include algebraic expressions, such as variables like \(x\). Understanding how to work with algebraic fractions is essential when dealing with equations or expressions that have varying parts.

When simplifying algebraic fractions, it helps to identify and isolate the variable terms. This prevents mistakes and ensures clarity in calculation. In the given exercise, recognizing that \(x\) appears as a variable fraction part helps in systematically deconstructing and solving the fraction.

Practice makes perfect with algebraic fractions. Regular exposure to these kinds of problems enhances intuition and problem-solving skills in simplifying and solving similar expressions in Algebra.

Nested Fractions

Nested fractions, also known as compound fractions or fractions within fractions, can look intimidating. In our problem, we have a compound fraction that requires careful untangling.

The key to solving nested fractions is to work from the inside out. Start by simplifying the innermost fraction first. This approach is similar to peeling away layers of an onion.

For example, in the expression \(1 - \frac{1}{1 - \frac{1}{x}}\), we begin by simplifying \(\frac{1}{x}\) which is inside the main fraction structure. Tackling it one step at a time prevents confusion and ensures that you don’t miss any critical simplification opportunities.

Fraction Simplification

Simplifying fractions involves rewriting them to the simplest possible form. This means breaking them down until they are as straightforward as they can be.

The critical step is finding common denominators or multiplying by reciprocal fractions to eliminate complex structures. For instance, when simplifying nested fractions, use the reciprocal method where needed, like transforming \(1 - \frac{1}{\frac{x-1}{x}}\) into \(1 - \frac{x}{x-1}\).

Keep an eye out for signs (positive or negative) as these can change the direction of simplification drastically. After getting a single fraction, further simplify by cancelling terms if possible, aiming for the most uncomplicated representation. In our example, the final expression simplifies to \(\frac{-1}{x-1}\).

Mathematical Problem Solving

Mathematical problem solving is all about strategically approaching math exercises. It involves clearly understanding the problem, planning a strategy to solve it, executing that strategy, and finally verifying the solution.

Let's dissect the given problem. First, clearly understand the expression and your goal: simplifying the fraction. Next, develop a plan – start from the innermost fraction. Execute the plan step-by-step, carefully calculating and simplifying at each stage.

Finally, check the result to ensure its accuracy. This could mean re-evaluating using alternate methods or double-checking calculations to confirm the final outcome is correct. Each of these steps is crucial in solving any mathematical problem successfully.