Problem 2 - Prealgebra and Introductory Algebra

Question

Exercises 1 to 3 are the examples of complex fractions given at the beginning of Objective 11.4A. By what fraction would you multiply each complex fraction in order to simplify it? $$\frac{4+\frac{1}{x}}{3+\frac{2}{x}}$$

Step-by-Step Solution

Answer

The fraction to multiply by is \( \frac{x}{2} \). The simplified complex fraction is \( \frac{8x + 1}{3x + 2} \)

Step 1: Identifying the fraction in the denominator

In our complex fraction, the fraction in the denominator is \( \frac{2}{x} \)

Step 2: Calculating the reciprocal

The reciprocal of a fraction is obtained by swapping its numerator and denominator. Therefore, the reciprocal of \( \frac{2}{x} \) is \( \frac{x}{2} \)

Step 3: Multiplying the complex fraction

Multiply the complex fraction by the reciprocal of the denominator's fraction. The result is: \( \frac{4+\frac{1}{x}}{3+\frac{2}{x}} \times \frac{x}{2} = \frac{2x \left(4+\frac{1}{x}\right)}{x \left(3+\frac{2}{x}\right)} = \frac{8x + 1}{3x + 2} \)

Step 4: Simplifying the result

Check if the result can be further simplified. The fraction \( \frac{8x + 1}{3x + 2} \) cannot be simplified further, hence, this is the final result.

Key Concepts

Simplifying Complex Fractions

Understanding how to simplify complex fractions is a fundamental skill in algebra. A complex fraction is a fraction where the numerator, the denominator, or both, are themselves fractions. The goal of simplification is to make these expressions easier to work with and to interpret.

To simplify a complex fraction such as \(\frac{4+\frac{1}{x}}{3+\frac{2}{x}}\), we can follow a systematic approach. First, identify the least common multiple (LCM) of the denominators in the complex fraction, or alternatively, find a fraction that can simplify the expression when multiplied. In this case, multiplying the entire complex fraction by the reciprocal of the fraction in the denominator is a smart move.

Finding Reciprocal to Simplify

By multiplying the complex fraction by the reciprocal of the smaller fraction in the denominator (\(\frac{x}{2}\)), we transform the complex fraction into a simpler form without changing its value. This crucial step helps eliminate the smaller fractions, allowing us to combine like terms and reduce the complex fraction to a standard fraction.

Reciprocal of a Fraction

The reciprocal of a fraction is simply switching its numerator and denominator. If we have a fraction \(\frac{a}{b}\), its reciprocal would be \(\frac{b}{a}\). It's important to note that the reciprocal of a fraction is not just a formal step; it has practical applications, especially when simplifying complex fractions.

In practice, when multiplying a fraction by its reciprocal, the result is 1 (\(\frac{a}{b} \times \frac{b}{a} = 1\)). When dealing with complex fractions, finding the reciprocal of a particular part of the expression, typically the denominator, is often the key to simplification. As shown in the exercise, the reciprocal of \(\frac{2}{x}\) is \(\frac{x}{2}\), a critical step in the simplification process.

Multiplying Fractions

Multiplying fractions is another foundational concept in mathematics. To multiply fractions, you simply multiply the numerators together and the denominators together. For example, if you're multiplying \(\frac{a}{b}\) by \(\frac{c}{d}\), the result would be \(\frac{a \cdot c}{b \cdot d}\).

However, when multiplying complex fractions by the reciprocal of one of their components, as described in the previous sections, we are deploying this strategy for the purpose of simplification. After multiplication, it's always a good habit to check whether the fraction can be reduced further by cancelling any common factors between the numerator and the denominator. In the provided exercise, after multiplication, we obtain the simplified fraction \(\frac{8x + 1}{3x + 2}\), with no common factors to cancel out, hence it's already in its simplest form.