Problem 51 - Intermediate Algebra

Question

Simplify each complex fraction. $$ \frac{\frac{1}{a+1}+1}{\frac{3}{a-1}+1} $$

Step-by-Step Solution

Answer

The simplified form of the complex fraction is \( \frac{a-1}{a+1} \).

Step 1: Combine the Numerator

The numerator of the complex fraction is \( \frac{1}{a+1} + 1 \). To combine these terms, change '1' to \( \frac{a+1}{a+1} \) so that both terms have the same denominator. The numerator becomes: \[ \frac{1}{a+1} + \frac{a+1}{a+1} = \frac{1 + (a+1)}{a+1} = \frac{a+2}{a+1}. \]

Step 2: Combine the Denominator

The denominator of the complex fraction is \( \frac{3}{a-1} + 1 \). Similarly, change '1' to \( \frac{a-1}{a-1} \). The denominator becomes: \[ \frac{3}{a-1} + \frac{a-1}{a-1} = \frac{3 + (a-1)}{a-1} = \frac{a+2}{a-1}. \]

Step 3: Divide the Numerator by the Denominator

Now, you have a fraction \( \frac{\frac{a+2}{a+1}}{\frac{a+2}{a-1}} \). You divide by multiplying by the reciprocal: \[ \frac{a+2}{a+1} \times \frac{a-1}{a+2}. \]

Step 4: Simplify the Expression

In the multiplication \( \frac{a+2}{a+1} \times \frac{a-1}{a+2} \), the \( a+2 \) terms cancel out. Therefore, you are left with: \[ \frac{a-1}{a+1}. \]

Key Concepts

Simplification

Simplifying complex fractions involves reducing the expression into a simpler form that is easier to understand and work with. A complex fraction is a fraction where the numerator, the denominator, or both, are also fractions. This makes the original expression more complicated to solve. To simplify, you start by transforming these inner fractions within the numerator and denominator into a single fraction each.

• Identify the fractions within the numerator and denominator.
• Find a common denominator for each part.

In the given exercise, the numerator consists of the fraction \( \frac{1}{a+1} \) plus 1. To combine them, convert the whole number 1 to a fraction with the same denominator \( a+1 \). In the solution, it becomes \( \frac{a+1}{a+1} \). Now, add the fractions together to get \( \frac{a+2}{a+1} \). You will use a similar process for the denominator.

Numerator and Denominator

Understanding the roles of the numerator and denominator in a fraction is crucial when working with complex fractions. The numerator is the top part of a fraction, while the denominator is the bottom part. In our complex fraction, we need to carefully solve each separately to make the simplification possible.

• The numerator in this context is \( \frac{1}{a+1} + 1 \).
• The denominator is \( \frac{3}{a-1} + 1 \).

For each, you need to express the whole numbers as fractions with equal denominators to the existing fractions inside the complex fraction. This creates a common language across your numerical expression and helps in combining the terms. Once you have a single fraction for the numerator and one for the denominator, you can proceed to the next step of simplification.

Fraction Division

The division of fractions is an important skill when working with complex fractions. When you have simplified the numerator and denominator to single fractions, the next step is to divide one by the other. Fraction division is performed by multiplying by the reciprocal of the divisor.

• The reciprocal of a fraction is the fraction flipped upside down. For example, the reciprocal of \( \frac{a+2}{a-1} \) is \( \frac{a-1}{a+2} \).
• Use this reciprocal to multiply with the simplified numerator.

In this problem, after simplifying, you end up with \( \frac{\frac{a+2}{a+1}}{\frac{a+2}{a-1}} \). Convert this division into multiplication by using the reciprocal: \( \frac{a+2}{a+1} \times \frac{a-1}{a+2} \). This step often reveals opportunities to cancel terms, like the \( a+2 \) terms here, which allows you to simplify further to \( \frac{a-1}{a+1} \). This is the final simplified expression.