Math help from the Learning Centre

Learning Objectives

• Simplify complex fractions by identifying and applying common strategies.

• Perform addition, subtraction, multiplication, and division with complex fractions.

• Apply the procedure of clearing fractions in equations to solve fractional equations.

• Develop conceptual understanding by comparing different methods and recognizing key differences.

A complex fraction is a fraction where the numerator, the denominator, or both contain fractions themselves.

Examples:

\[\frac{\frac{3}{4}}{\frac{5}{6}}, \frac{2+\frac{1}{3}}{5-\frac{2}{7}}\]

Identify the Structure - Similarities and Differences

Explore similarities between:

• Simple fractions (e.g., \(\frac{3}{4}\) ) and complex fractions (fractions inside numerator and/or denominator).
• Operations on complex fractions and regular fractions.

Activity:

Look at these two fractions and describe what is similar and what is different about them.

Fraction A: \(\frac{7}{9}\)

Fraction B: \(\frac{\frac{3}{5}}{4}\)

Simplifying Complex Fractions 

Method A: Finding the Least Common Denominator (LCD) in numerator and denominator, rewrite, then simplify.

Example:

Simplfy

\[\frac{\frac{2}{3}+\frac{1}{6}}{\frac{5}{4}-\frac{1}{2}}\]

• Find the LCD of numerator fractions: 6
• Find the LCD of denominator fractions: 4
• Rewrite numerator: \(\frac{4}{6}+\frac{1}{6}=\frac{5}{6}\)
• Rewrite denominator: \(\frac{5}{4} - \frac{2}{4}=\frac{3}{4}\)
• Now the complex fraction is \(\frac{\frac{5}{6}}{\frac{3}{4}}=\frac{5}{6}\times \frac{4}{3}=\frac{20}{18}=\frac{10}{9}\)

Method B: Multiply numerator and denominator by the LCD of all small fractions to clear the inner fractions at once.

• LCD of 3, 6, 4, 2 is 12
• Multiply top and bottom by 12:

\[\frac{\left(\frac{2}{3}+\frac{1}{6}\right)\times 12}{\left(\frac{5}{4}-\frac{1}{2}\right)\times 12}=\frac{(8+2)}{(15-6)}=\frac{10}{9}\]

Perform Operations with Complex Fractions 

Practice Problems:

1. Simplify: \[\frac{\frac{1}{2}}{\frac{3}{4}}\]
2. Simplify: \[\frac{3+\frac{2}{5}}{1-\frac{1}{10}}\]
3. Multiply and simplify: \[\frac{\frac{2}{3}}{\frac{5}{6}}\times\frac{3}{4}\]

Clearing Fractions in Equations to Solve Fractional Equations

When an equation has fractions, you cna clear fractions by multiplying every term by the LCD of all fractions in the equation.

Example:

Solve for \(x\):\[\frac{2}{3}x+\frac{1}{4}=\frac{5}{6}\]

1. Find LCD of denominators 3, 4, 6: 12
2. Multiply both sides of the equation by 12: \[12\times\left(\frac{2}{3}x+\frac{1}{4}\right)=12\times\frac{5}{6}\]
3. Simplify: \[12\times\frac{2}{3}x=8x, 12\times\frac{1}{4}=3, 12 \times\frac{5}{6}=10\]
   • Equation becomes: \(8x+3=10\)
4. Solve for \(x\): \[8x=7, x=\frac{7}{8}\]

Solve these fractional Equations

1. \(\frac{1}{2}x-\frac{3}{4}=\frac{5}{6}\)
2. \(\frac{2x}{5}+\frac{1}{3}=\frac{4}{3}\)
3. \(\frac{x+1}{4}=\frac{3x-2}{6}\)

• How is multiplying numerator and denominator by LCD (Method B) similar or different from rewriting each fraction separately (Method A)?

• How does clearing fractions in equations relate to simplifying complex fractions?

• Which method do you find easier and why? How does this help your understanding?

• Complex fractions can be simplified by rewriting all parts with a common denominator or by clearing denominators directly.

• Multiplying numerator and denominator by the LCD is a powerful way to clear complex fractions quickly.

• Clearing fractions in equations simplifies solving fractional equations.

• Noticing similarities and differences between methods deepens understanding and helps choose the best approach.

Designed by Matthew Cheung. This work is licensed under a Creative Commons Attribution 4.0 International License.