Learn on PengiAoPS: Introduction to Algebra (AMC 8 & 10)Chapter 4: More Variables

Lesson 4: Fractions

In this Grade 4 lesson from AoPS: Introduction to Algebra (AMC 8 & 10), students learn how to add and subtract algebraic fractions with multiple variables by finding a least common denominator. The lesson covers simplifying fractions through factoring and canceling common factors before combining them. Students practice these skills with variable expressions such as combining fractions with denominators like rs, 6x²y, and 2ab(a-1).

Section 1

Finding the Lowest Common Denominator

Property

The lowest common denominator (LCD) for two or more algebraic fractions is the simplest algebraic expression that is a multiple of each denominator. To find the LCD:

  1. Factor each denominator completely.
  2. For each factor, include the most copies of that factor that appears in any single denominator.
  3. Multiply together the factors of the LCD.

Examples

  • The LCD for 16a2b\frac{1}{6a^2b} and 59ab3\frac{5}{9ab^3} is 18a2b318a^2b^3. We need factors of 2⋅32⋅a2⋅b32 \cdot 3^2 \cdot a^2 \cdot b^3.

Section 2

To Add or Subtract Algebraic Fractions

Property

  1. Find the lowest common denominator (LCD).
  2. Build each fraction to an equivalent one with the LCD.
  3. Add or subtract the numerators, and keep the same denominator.
  4. Reduce if necessary.

Examples

Problem: Calculate 4x−3+2x+1\frac{4}{x-3} + \frac{2}{x+1}.

Step 1: Find the LCD.
The denominators are prime, so the LCD is (x−3)(x+1)(x-3)(x+1).

Lesson overview

Expand to review the lesson summary and core properties.

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Section 1

Finding the Lowest Common Denominator

Property

The lowest common denominator (LCD) for two or more algebraic fractions is the simplest algebraic expression that is a multiple of each denominator. To find the LCD:

  1. Factor each denominator completely.
  2. For each factor, include the most copies of that factor that appears in any single denominator.
  3. Multiply together the factors of the LCD.

Examples

  • The LCD for 16a2b\frac{1}{6a^2b} and 59ab3\frac{5}{9ab^3} is 18a2b318a^2b^3. We need factors of 2⋅32⋅a2⋅b32 \cdot 3^2 \cdot a^2 \cdot b^3.

Section 2

To Add or Subtract Algebraic Fractions

Property

  1. Find the lowest common denominator (LCD).
  2. Build each fraction to an equivalent one with the LCD.
  3. Add or subtract the numerators, and keep the same denominator.
  4. Reduce if necessary.

Examples

Problem: Calculate 4x−3+2x+1\frac{4}{x-3} + \frac{2}{x+1}.

Step 1: Find the LCD.
The denominators are prime, so the LCD is (x−3)(x+1)(x-3)(x+1).