Complex Fractions

Overview

A complex fraction is a fraction where the numerator, denominator, or both are themselves fractions. The goal in simplifying complex fractions is to rewrite them as a simple fraction. This can be done by finding a common denominator or multiplying both the numerator and denominator by the least common denominator (LCD) of the fractions within the complex fraction.

Steps to Simplify a Complex Fraction

  1. Identify the fractions within the numerator and denominator.
  2. Find the least common denominator (LCD) of the fractions in the numerator and denominator.
  3. Multiply both the numerator and denominator by the LCD to eliminate the fractions.
  4. Simplify the resulting expression.

Example 1: Simplifying a Complex Fraction

Consider the following complex fraction:

\( \frac{\frac{1}{x} + \frac{2}{y}}{\frac{3}{x} - \frac{4}{y}} \)

Step 1: Find the least common denominator (LCD) for the terms in the numerator and denominator. The LCD of \( x \) and \( y \) is \( xy \).

Step 2: Multiply both the numerator and denominator by \( xy \) to eliminate the fractions:

\( \frac{xy \cdot \left(\frac{1}{x} + \frac{2}{y}\right)}{xy \cdot \left(\frac{3}{x} - \frac{4}{y}\right)} \)

Step 3: Simplify both the numerator and the denominator:

\( \frac{y + 2x}{3y - 4x} \)

Thus, the simplified form of the complex fraction is:

\( \frac{y + 2x}{3y - 4x} \)

Example 2: Complex Fraction with a Fraction in the Denominator

Consider the following complex fraction:

\( \frac{\frac{1}{x}}{\frac{2}{y} + \frac{3}{z}} \)

Step 1: Find the least common denominator (LCD) for the terms in the denominator. The LCD of \( y \) and \( z \) is \( yz \).

Step 2: Multiply both the numerator and denominator by \( yz \) to eliminate the fractions:

\( \frac{yz \cdot \frac{1}{x}}{yz \cdot \left(\frac{2}{y} + \frac{3}{z}\right)} \)

Step 3: Simplify both the numerator and the denominator:

\( \frac{yz}{x} \div \left(2z + 3y\right) \)

Thus, the simplified form of the complex fraction is:

\( \frac{yz}{x(2z + 3y)} \)

Practice Questions

  1. Question 1: Simplify the following complex fraction:

    \( \frac{\frac{2}{x} + \frac{3}{y}}{\frac{4}{x} - \frac{5}{y}} \)

    Solution

    Step 1: Find the LCD of \( x \) and \( y \), which is \( xy \).

    Step 2: Multiply both the numerator and denominator by \( xy \):

    \( \frac{xy \cdot \left(\frac{2}{x} + \frac{3}{y}\right)}{xy \cdot \left(\frac{4}{x} - \frac{5}{y}\right)} \)

    Step 3: Simplify both the numerator and denominator:

    \( \frac{2y + 3x}{4y - 5x} \)

  2. Question 2: Simplify the following complex fraction:

    \( \frac{\frac{3}{x}}{\frac{2}{y} + \frac{4}{z}} \)

    Solution

    Step 1: Find the LCD of \( y \) and \( z \), which is \( yz \).

    Step 2: Multiply both the numerator and denominator by \( yz \):

    \( \frac{yz \cdot \frac{3}{x}}{yz \cdot \left(\frac{2}{y} + \frac{4}{z}\right)} \)

    Step 3: Simplify both the numerator and denominator:

    \( \frac{3yz}{x(2z + 4y)} \)

  3. Question 3: Simplify the following complex fraction:

    \( \frac{\frac{5}{x} + \frac{6}{y}}{\frac{7}{x} - \frac{8}{y}} \)

    Solution

    Step 1: Find the LCD of \( x \) and \( y \), which is \( xy \).

    Step 2: Multiply both the numerator and denominator by \( xy \):

    \( \frac{xy \cdot \left(\frac{5}{x} + \frac{6}{y}\right)}{xy \cdot \left(\frac{7}{x} - \frac{8}{y}\right)} \)

    Step 3: Simplify both the numerator and denominator:

    \( \frac{5y + 6x}{7y - 8x} \)

  4. Question 4: Simplify the following complex fraction:

    \( \frac{\frac{1}{x}}{\frac{2}{y} + \frac{3}{z}} \)

    Solution

    Step 1: Find the LCD of \( y \) and \( z \), which is \( yz \).

    Step 2: Multiply both the numerator and denominator by \( yz \):

    \( \frac{yz \cdot \frac{1}{x}}{yz \cdot \left(\frac{2}{y} + \frac{3}{z}\right)} \)

    Step 3: Simplify both the numerator and denominator:

    \( \frac{yz}{x(2z + 3y)} \)

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