Lesson 1.3: Algebraic Fractions and Rearranging Formulae

Introduction

In this lesson, we will delve into the fascinating world of algebraic fractions and the important skill of rearranging formulae. By the end of this lesson, you, students, will have developed a solid understanding of how to simplify, add, subtract, multiply, and divide algebraic fractions. You will also learn how to rearrange formulae, changing the subject when necessary, and accurately substitute values into these formulas. The knowledge gained here will serve as a crucial foundation for more advanced topics in mathematics.

Learning Objectives

• Simplify, add, subtract, multiply, and divide algebraic fractions.
• Rearrange formulae to change the subject, particularly when the subject appears only once.
• Substitute values into formulae accurately.
• Simplify compound algebraic fractions correctly.
• Rearrange a formula to make a specified variable the subject.

Understanding Algebraic Fractions

Algebraic fractions involve variables in the numerator, the denominator, or both. Just like numerical fractions, they can be simplified, and operations such as addition, subtraction, multiplication, and division can be performed.

Simplifying Algebraic Fractions

To simplify an algebraic fraction, we factor both the numerator and denominator and then cancel out any common factors.

Example 1: Simplifying an Algebraic Fraction

Consider the algebraic fraction:

$$\frac{x^2 - 9}{x^2 - 3x}$$

Step 1: Factor the numerator and the denominator:

• Numerator: $x^2 - 9$ can be factored as $(x - 3)(x + 3)$.
• Denominator: $x^2 - 3x = x(x - 3)$.

Thus, the fraction becomes:

$$\frac{(x - 3)(x + 3)}{x(x - 3)}$$

Step 2: Cancel out the common factor $(x - 3)$:

$$\frac{x + 3}{x}$$

Therefore, the simplified form of the original fraction is:

$$\frac{x + 3}{x}$$

Adding and Subtracting Algebraic Fractions

To add or subtract algebraic fractions, we need to have a common denominator. If the denominators are not the same, we must find the least common denominator (LCD).

Example 2: Adding Algebraic Fractions

Consider the fractions:

$$\frac{2}{x} + \frac{3}{x^2}$$

Step 1: Determine the LCD, which is $x^2$.

Step 2: Convert each fraction:

• The first fraction becomes:

$$\frac{2 \cdot x}{x \cdot x} = \frac{2x}{x^2}$$

• The second fraction is already in terms of the LCD:

$$\frac{3}{x^2}$$

Combine the fractions:

$$\frac{2x + 3}{x^2}$$

Thus, the sum is:

$$\frac{2x + 3}{x^2}$$

Multiplying and Dividing Algebraic Fractions

For multiplication and division of fractions, we use the following rules:

1. To multiply, multiply the numerators and denominators directly.
2. For division, multiply by the reciprocal of the second fraction.

Example 3: Multiplying Algebraic Fractions

Let’s multiply:

$$\frac{3x}{4} \cdot \frac{8}{x^2}$$

Step 1: Multiply the numerators and the denominators:

$$\frac{3x \cdot 8}{4 \cdot x^2} = \frac{24x}{4x^2}$$

Step 2: Simplify:

$$\frac{24}{4} \cdot \frac{x}{x^2} = 6 \cdot \frac{1}{x} = \frac{6}{x}$$

The result is:

$$\frac{6}{x}$$

Example 4: Dividing Algebraic Fractions

Consider:

$$\frac{5x^2}{6} \div \frac{3}{2x}$$

Step 1: Rewrite the division as multiplication by the reciprocal:

$$\frac{5x^2}{6} \cdot \frac{2x}{3}$$

Step 2: Multiply:

$$\frac{5x^2 \cdot 2x}{6 \cdot 3} = \frac{10x^3}{18}$$

Step 3: Simplify:

$$\frac{10}{18} \cdot \frac{x^3}{1} = \frac{5}{9} x^3$$

Thus, the result is:

$$\frac{5x^3}{9}$$

Rearranging Formulae

Rearranging formulae involves changing which variable is the subject of the equation. When the subject appears once in the formula, this is often straightforward but can require careful manipulation of the other terms.

Steps for Rearranging Formulae

1. Identify the target variable you need to make the subject.
2. Use inverse operations to isolate the target variable.
3. Perform the same operations on both sides of the equation.

Example 5: Rearranging a Formula

Consider the following formula for the area $A$ of a rectangle:

$$A = l \cdot w$$

We want to rearrange this formula to make $w$ the subject.

Step 1: Divide both sides by $l$:

$$w = \frac{A}{l}$$

Thus, $w$ is now isolated and the rearranged formula is:

$$w = \frac{A}{l}$$

Compound Algebraic Fractions

A compound fraction is a fraction that contains another fraction in either the numerator or the denominator. Simplifying compound algebraic fractions involves simplifying the numerator and denominator separately then combining them.

Example 6: Simplifying a Compound Algebraic Fraction

Given:

$$\frac{\frac{1}{x} + \frac{1}{y}}{\frac{1}{z}}$$

Step 1: Find a common denominator for the numerator:

$$\frac{\frac{y + x}{xy}}{\frac{1}{z}}$$

Step 2: Flip the denominator and multiply:

$$\frac{y + x}{xy} \cdot z = \frac{z(y + x)}{xy}

$$

Thus, the simplified form is:

$$\frac{z(y + x)}{xy}$$

Conclusion

In this lesson, students, you have learned to master the manipulation of algebraic fractions and the important technique of rearranging formulae. Understanding how to effectively handle these concepts sets a solid foundation for further studies in algebra and beyond. Practice is key to gaining fluency, so be sure to work through plenty of exercises to reinforce your understanding.

Study Notes

• Algebraic Fractions: Fractions that contain variables in the numerator and/or denominator.
• Simplification: Factor both numerator and denominator; cancel common factors.
• Addition/Subtraction: Find a common denominator before combining.
• Multiplication/Division: Multiply numerators and denominators, use reciprocal for division.
• Rearranging Formulae: Use inverse operations to isolate the desired variable.
• Compound Fractions: Simplify by addressing the numerator and denominator separately.