Simplifying Algebraic Fractions

Basic Algebra in Full | Lesson 7

🧠 What You’ll Learn

In this lesson, you’ll learn how fractions and algebraic fractions work from the ground up, including how to simplify, combine, and manipulate them correctly using clear mathematical reasoning and connected algebra skills.

You’ll learn how to:

• Understand what fractions represent and identify numerators, denominators, and fraction bars
• Recognize the main types of fractions: proper, improper, and mixed fractions
• Convert mixed fractions into improper fractions correctly
• Multiply and divide fractions using cancellation and reciprocals
• Add and subtract fractions with the same denominator
• Find common denominators using the LCM method
• Use denominator multiplication as an alternative method for adding and subtracting fractions
• Compare different fraction methods and understand when each is most useful
• Understand what algebraic fractions are and how variables behave inside fractions
• Distinguish correctly between variables in the numerator and denominator
• Simplify algebraic fractions using multiplication and division principles
• Understand when cancelling terms is valid and when it is mathematically incorrect
• Apply operator ownership and operator families when simplifying expressions
• Add and subtract algebraic fractions with variables in the denominator
• Work through multi-step algebraic fraction problems involving brackets, variables, and distributive expansion
• Combine fraction skills with earlier algebra concepts to solve more advanced expressions confidently

By the end, you’ll be able to confidently simplify, combine, and solve arithmetic and algebraic fractions while understanding the mathematical logic behind every step.

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🧩Practice Exercises

These exercises help you build from foundation to advanced understanding.

You can:

  • Start at your level
  • Or work through all levels step by step

✍️ Attempt exercises before checking answers.

🎯 Levels

Each level matches a stage of learning and shows
what you should be able to understand and solve at that point.

🟢 Foundation → You’re learning this topic for the first time
🟠 Developing → You understand the basics and are building on them
⚪ Confident → You’re ready for deeper, multi-step problems
⚫ Advanced → You want to challenge your understanding further

📘 Not sure where you fit? → View Level Guide

💡 Each set of exercises is designed to match what students at that stage are typically expected to handle.

Start with your level, then try the next one when you feel ready.

  1. Simplify the mixed fraction $2\frac{1}{4}$ into an improper fraction.
  2. Simplify: $\frac{3}{5}\times\frac{2}{9}$
  3. Simplify: $\frac{4}{7}\div\frac{2}{3}$
  4. Simplify: $\frac{5}{8}+\frac{1}{8}$
  5. Simplify: $\frac{2}{3}+\frac{1}{6}$
Show Answers
  1. $\frac{9}{4}$
  2. $\frac{2}{15}$
  3. $\frac{6}{7}$
  4. $\frac{3}{4}$
  5. $\frac{5}{6}$

  1. Simplify: $\frac{3}{4}-\frac{1}{6}$
  2. Simplify: $\frac{1}{2}+\frac{1}{3}+\frac{1}{6}$
  3. Simplify: $\frac{2}{x}+\frac{1}{x}$
  4. Simplify: $\frac{3}{2x}-\frac{1}{4x}$
  5. Simplify: $\frac{2}{a}+\frac{3}{b}$
Show Answers
  1. $\frac{7}{12}$
  2. $1$
  3. $\frac{3}{x}$
  4. $\frac{5}{4x}$
  5. $\frac{2b+3a}{ab}$

  1. Simplify: $\frac{4x}{5}+\frac{6x}{5}-\frac{3x}{5}$
  2. Simplify: $\frac{5}{x}+\frac{2}{3x}$
  3. Simplify: $\frac{3}{a-1}+\frac{2}{a}$
  4. Simplify: $\frac{x^2+4x}{x}$
  5. Simplify: $\frac{1}{2}\left(6x-\frac{x}{2}+4\right)$
Show Answers
  1. $\frac{7x}{5}$
  2. $\frac{17}{3x}$
  3. $\frac{5a-2}{a(a-1)}$
  4. $x+4$
  5. $\frac{11x}{4}+2$

  1. Simplify: $\frac{3}{4}\left(8x-\frac{2x}{3}+4\right)-\frac{x}{2}$
  2. Simplify: $\frac{5}{6a}-\frac{1}{4a}+\frac{7}{12a}$
  3. Simplify: $\frac{2}{x}+\frac{3}{2x}-\frac{1}{4x}$
  4. Simplify: $\frac{1}{3}\left(9x-\frac{3}{2}(2x-8)\right)+\frac{1}{4}\left(12-\frac{x}{2}\right)$
  5. Simplify: $\frac{2x^2+6x}{2x}-\frac{x}{4}$
Show Answers
  1. $\frac{35x+24}{8}$
  2. $\frac{7}{6a}$
  3. $\frac{13}{4x}$
  4. $\frac{11x+120}{24}$
  5. $3+\frac{3x}{4}$

🔁 Continue Learning

⏮️ Back to Basic Algebra in Full
⏭️ Next Lesson → Lesson 8: Order of Operations (BODMAS/PEMDAS)

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