Problem 73 - Elementary Algebra

Question

For Problems \(73-76\), simplify each complex fraction. $$ 1-\frac{n}{1-\frac{1}{n}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{-(n^2 - n + 1)}{n-1} \).

Step 1: Simplifying the Inner Fraction

First, concentrate on the inner fraction \( \frac{1}{n} \). The entire expression is \( 1 - \frac{n}{1 - \frac{1}{n}} \), so focus on simplifying \( 1 - \frac{1}{n} \) which is in the denominator of the second term.

Step 2: Combining Terms in the Denominator

Express \( 1 - \frac{1}{n} \) with a common denominator: \( \frac{n}{n} - \frac{1}{n} = \frac{n-1}{n} \). This gives us a new denominator for the complex fraction.

Step 3: Rewriting the Complex Fraction

Now substitute the simplified denominator back into the original expression: \( 1 - \frac{n}{\frac{n-1}{n}} \).

Step 4: Simplify the Division

To simplify \( \frac{n}{\frac{n-1}{n}} \), multiply by the reciprocal: \( n \times \frac{n}{n-1} = \frac{n^2}{n-1} \).

Step 5: Simplify the Expression

The expression becomes \( 1 - \frac{n^2}{n-1} \). Rewrite as: \( \frac{n-1}{n-1} - \frac{n^2}{n-1} = \frac{n-1-n^2}{n-1} \).

Step 6: Final Simplification

Combine the terms in the numerator to get \( \frac{-(n^2 - n+1)}{n-1} \). This is the simplified form of the original complex fraction.

Key Concepts

Simplifying Fractions

When dealing with complex fractions, the main goal is to simplify them into a more manageable form. A complex fraction involves a fraction in the numerator, the denominator, or both.
In the example given, the complex fraction is represented as \(1 - \frac{n}{1 - \frac{1}{n}}\). Here, we see that part of this expression includes a fraction within a fraction.
When simplifying such fractions, it is often best to start with the innermost fraction. For instance, seeing \(1 - \frac{1}{n}\) within the denominator can be simplified first by finding a common denominator.

• Convert \(1\) to a fraction by rewriting it as \(\frac{n}{n}\).
• Subtract \(\frac{1}{n}\) from \(\frac{n}{n}\) to get \(\frac{n-1}{n}\).

Once the inner fraction is simplified, rewrite the larger fraction. This process transforms the problem step by step into a simpler expression.
Each layer of simplification brings you closer to your final answer.

Algebraic Expressions

Handling algebraic expressions requires understanding how to combine and rearrange variables and numbers.
In this exercise, expressions are simplified step by step through arithmetic and algebraic operations. For example, expressions like \(1 - \frac{n}{\frac{n-1}{n}}\) need careful attention to order and operations. Recall these tips:

• Replacing complex expressions with equivalent, simpler terms aids clarity.
• Find common denominators to facilitate subtracting and adding terms within the fractions.

In this context, understanding expressions as groups of terms is vital. Rewriting complex algebraic fractions in simpler terms often involves these algebraic manipulations, resulting in clearer, simplified forms.
Like a puzzle, every piece fits together when you approach it systematically.

Reciprocal Multiplication

Multiplying by the reciprocal transforms division into multiplication, simplifying complex fractions. For instance, consider the step \(\frac{n}{\frac{n-1}{n}}\). Rather than dividing by a fraction, multiply by its reciprocal.
The reciprocal of \(\frac{n-1}{n}\) is \(\frac{n}{n-1}\). By multiplying \(n\) by the reciprocal of \(\frac{n-1}{n}\), the expression becomes \(n \cdot \frac{n}{n-1} = \frac{n^2}{n-1}\).
This approach simplifies the division process and makes the computations more straightforward. When the reciprocal is applied correctly, the complex fraction unravels into simpler terms.

• Reciprocal multiplication is efficient and reduces errors
• Switch division into multiplication to make calculations simpler

Use this technique to eliminate the confusion that often accompanies dividing by fractions, making algebra problems less intimidating.