Learn on PengiAoPS: Introduction to Algebra (AMC 8 & 10)Chapter 2: x Marks the Spot

Lesson 4: Fractions

In this Grade 4 lesson from AoPS Introduction to Algebra, students learn how to simplify fractions involving variable expressions by factoring numerators and denominators to cancel common factors. The lesson also covers adding and subtracting algebraic fractions by finding a common denominator, including cases where denominators contain polynomial expressions like s+2. These skills are developed through AMC-style problems drawn from Chapter 2 of the AoPS curriculum.

Section 1

Reducing Algebraic Fractions

Property

Fundamental Principle of Fractions: We can multiply or divide the numerator and denominator of a fraction by the same nonzero factor, and the new fraction will be equivalent to the old one.

\frac{a \cdot c}{b \cdot c} = \frac{a}{b} \quad \text{if} \quad b, c \neq 0

To reduce an algebraic fraction:

Factor the numerator and the denominator.
Divide the numerator and denominator by any common factors.

Caution: We can cancel common factors, but we cannot cancel common terms.

Examples

To reduce $\frac{12x^5y^2}{8x^3y^3}$ , we find the common factor $4x^3y^2$ . Factoring gives $\frac{3x^2 \cdot 4x^3y^2}{2y \cdot 4x^3y^2}$ , which simplifies to $\frac{3x^2}{2y}$ .
The fraction $\frac{x+4}{x+8}$ cannot be reduced. The $x$ is a term, not a factor, so it cannot be canceled.
To reduce $\frac{7x+14}{21}$ , first factor the numerator and denominator: $\frac{7(x+2)}{7(3)}$ . Canceling the common factor of 7 leaves $\frac{x+2}{3}$ .

Explanation

To simplify an algebraic fraction, you must first factor the top and bottom completely. Then, you can cancel out identical factors. Remember, you can only cancel parts that are multiplied, not parts that are added or subtracted.

Section 2

Canceling Factors vs. Terms

Property

We can cancel common factors (expressions that are multiplied together), but not common terms (expressions that are added or subtracted).

Examples

Correct: $\frac{5(x+2)}{5} = x+2$ because 5 is a factor. Incorrect: $\frac{5x+2}{5} \neq x+2$ because 5 is not a factor of the entire numerator.

In the fraction $\frac{x+8}{y+8}$ , you cannot cancel the 8s. They are terms being added, not factors being multiplied.

Section 3

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:

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

Examples

The LCD for $\frac{1}{6a^2b}$ and $\frac{5}{9ab^3}$ is $18a^2b^3$ . We need factors of $2 \cdot 3^2 \cdot a^2 \cdot b^3$ .

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Reducing Algebraic Fractions

Property

Fundamental Principle of Fractions: We can multiply or divide the numerator and denominator of a fraction by the same nonzero factor, and the new fraction will be equivalent to the old one.

\frac{a \cdot c}{b \cdot c} = \frac{a}{b} \quad \text{if} \quad b, c \neq 0

To reduce an algebraic fraction:

Factor the numerator and the denominator.
Divide the numerator and denominator by any common factors.

Caution: We can cancel common factors, but we cannot cancel common terms.

Examples

To reduce $\frac{12x^5y^2}{8x^3y^3}$ , we find the common factor $4x^3y^2$ . Factoring gives $\frac{3x^2 \cdot 4x^3y^2}{2y \cdot 4x^3y^2}$ , which simplifies to $\frac{3x^2}{2y}$ .
The fraction $\frac{x+4}{x+8}$ cannot be reduced. The $x$ is a term, not a factor, so it cannot be canceled.
To reduce $\frac{7x+14}{21}$ , first factor the numerator and denominator: $\frac{7(x+2)}{7(3)}$ . Canceling the common factor of 7 leaves $\frac{x+2}{3}$ .

Explanation

Section 2

Canceling Factors vs. Terms

Property

We can cancel common factors (expressions that are multiplied together), but not common terms (expressions that are added or subtracted).

Examples

Correct: $\frac{5(x+2)}{5} = x+2$ because 5 is a factor. Incorrect: $\frac{5x+2}{5} \neq x+2$ because 5 is not a factor of the entire numerator.

In the fraction $\frac{x+8}{y+8}$ , you cannot cancel the 8s. They are terms being added, not factors being multiplied.

Section 3

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:

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

Examples

The LCD for $\frac{1}{6a^2b}$ and $\frac{5}{9ab^3}$ is $18a^2b^3$ . We need factors of $2 \cdot 3^2 \cdot a^2 \cdot b^3$ .

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Reducing Algebraic Fractions

Property

Fundamental Principle of Fractions: We can multiply or divide the numerator and denominator of a fraction by the same nonzero factor, and the new fraction will be equivalent to the old one.

\frac{a \cdot c}{b \cdot c} = \frac{a}{b} \quad \text{if} \quad b, c \neq 0

To reduce an algebraic fraction:

Factor the numerator and the denominator.
Divide the numerator and denominator by any common factors.

Caution: We can cancel common factors, but we cannot cancel common terms.

Examples

To reduce $\frac{12x^5y^2}{8x^3y^3}$ , we find the common factor $4x^3y^2$ . Factoring gives $\frac{3x^2 \cdot 4x^3y^2}{2y \cdot 4x^3y^2}$ , which simplifies to $\frac{3x^2}{2y}$ .
The fraction $\frac{x+4}{x+8}$ cannot be reduced. The $x$ is a term, not a factor, so it cannot be canceled.
To reduce $\frac{7x+14}{21}$ , first factor the numerator and denominator: $\frac{7(x+2)}{7(3)}$ . Canceling the common factor of 7 leaves $\frac{x+2}{3}$ .

Explanation

Section 2

Canceling Factors vs. Terms

Property

We can cancel common factors (expressions that are multiplied together), but not common terms (expressions that are added or subtracted).

Examples

Correct: $\frac{5(x+2)}{5} = x+2$ because 5 is a factor. Incorrect: $\frac{5x+2}{5} \neq x+2$ because 5 is not a factor of the entire numerator.

In the fraction $\frac{x+8}{y+8}$ , you cannot cancel the 8s. They are terms being added, not factors being multiplied.

Section 3

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:

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

Examples

The LCD for $\frac{1}{6a^2b}$ and $\frac{5}{9ab^3}$ is $18a^2b^3$ . We need factors of $2 \cdot 3^2 \cdot a^2 \cdot b^3$ .