Algebraic Fractions Worksheet

Algebraic Fractions are fractions in which the numerator and the denominator contain algebraic expressions (expressions involving variables, constants, and operations such as addition, subtraction, multiplication, and division). They are similar to numerical fractions but involve variables.

In this article, we will learn how to solve the question of Algebraic fractions.

What are Algebraic Fractions?

Algebraic fractions are a fundamental concept in algebra and are used extensively in solving equations, simplifying expressions, and calculus. Understanding how to manipulate and simplify these fractions is essential for solving more complex algebraic problems.

• Numerator: The top part of an algebraic fraction, which is an algebraic expression.
• Denominator: The bottom part of an algebraic fraction, also an algebraic expression.

The general form of an algebraic fraction is:

P(x)/Q(x)​

where P(x) and Q(x) are polynomials or other algebraic expressions.

Examples

1. \frac{x + 2}{x - 1}
2. \frac{3x^2 - 4x + 5}{2x + 1}
3. \frac{5}{x^2 - 9}

How to solve Algebraic Fractions?

Here’s a general steps to approach problems involving algebraic fractions:

Simplifying Algebraic Fractions

• Factorize the numerator and the denominator completely.
• Cancel common factors that appear in both the numerator and the denominator

Finding a Common Denominator

• Find the least common denominator (LCD): The LCD is the least common multiple of the denominators of the fractions.
• Rewrite each fraction with the LCD as the new denominator.
• Combine the fractions by adding or subtracting the numerators.
• Simplify the resulting fraction, if possible.

Solving Equations Involving Algebraic Fractions

1. Find a common denominator and multiply every term by it to eliminate the denominators.
2. Simplify the resulting equation and solve for the variable.
3. Check for extraneous solutions: Values that make any original denominator zero should be excluded from the solution set.

Addition and Subtraction of Algebraic Fractions

Finding a Common Denominator

• Purpose: To add or subtract algebraic fractions, they must have the same denominator.
• Process:
  1. Identify the denominators of the fractions.
  2. Find the least common multiple (LCM) of these denominators. The LCM is the smallest expression that each denominator can divide into.
  3. Rewrite each fraction with the common denominator by multiplying the numerator and denominator of each fraction by whatever is needed to achieve the common denominator.

Simplifying the Result

• Combine the numerators over the common denominator.
• Simplify the numerator if possible by combining like terms.
• Reduce the fraction to its simplest form by canceling common factors in the numerator and the denominator.

Multiplication and Division of Algebraic Fractions

Cross-Multiplication (For Division)

• Purpose: To simplify the division of two algebraic fractions.
• Process:
  1. Invert the second fraction (i.e., take its reciprocal).
  2. Multiply the first fraction by this reciprocal.

Simplifying Before Multiplying or Dividing

• Factorize the numerators and denominators of the fractions if possible.
• Cancel any common factors between the numerators and denominators before performing the multiplication or division.
• Multiply the numerators together and the denominators together after simplification.
• Simplify the resulting fraction by canceling any common factors.

Example Problems

Addition and Subtraction Example

Ex 1: \frac{3}{x} + \frac{2}{x+2}

Solution:

Common Denominator: x(x+2)

Rewrite Fractions: \frac{3(x+2)}{x(x+2)} + \frac{2x}{x(x+2)} = \frac{3x + 6 + 2x}{x(x+2)} = \frac{5x + 6}{x(x+2)}

Simplified Result: \frac{5x + 6}{x(x+2)}

Multiplication Example

Ex 2: \frac{3x}{4y} \times \frac{2y^2}{9x}\

Simplify: \frac{3x \cdot 2y^2}{4y \cdot 9x} = \frac{6xy^2}{36xy} = \frac{y}{6}

Division Example

Ex 3: \frac{3x}{4y} \div \frac{2y^2}{9x}

Invert and Multiply: \frac{3x}{4y} \times \frac{9x}{2y^2} = \frac{3x \cdot 9x}{4y \cdot 2y^2} = \frac{27x^2}{8y^3}

Applications of Algebraic Fractions: Practical Real-World Problems

Mixture Problems: Calculate the concentration of a resulting mixture from different solution concentrations.

Rate Problems: Determine speeds and compare travel times for different distances.

Work Problems: Combine work rates of multiple workers to find the total work rate.

Proportion Problems: Adjust recipe ingredients for different serving sizes.

Common Mistakes and How to Avoid Them

Misunderstanding the Difference between Terms and Factors

• Terms are parts of an expression that are added or subtracted.
• Factors are parts of an expression that are multiplied.
• Example: In the expression 3x+2, 3x and 2 are terms. In the expression 3x⋅2, 3 and x and 2 are factors.

How to Avoid:

• Always look for multiplication signs to identify factors.
• When simplifying, combine like terms (terms with the same variable and exponent) but factor common factors in a multiplication expression.

Incorrectly Canceling Terms

• Mistake: Canceling terms instead of factors.
  • Incorrect: 2x+4/2≠x+2
  • Correct: 2(x+2)/2=x+2

How to Avoid:

• Only cancel factors, not terms.
• Factor the numerator and denominator completely before canceling.

Solved problems: Algebraic Fractions

Problem 1: Simplify \frac{6x^2}{9x}.

Solution:

Factorize: 6x²=6x⋅x and 9x=9⋅x.

The fraction becomes \frac{6x \cdot x}{9 \cdot x}.

Cancel the common factor x: \frac{6x \cdot x}{9 \cdot x} = \frac{6}{9}x = \frac{2}{3}.

Problem 2: Add \frac{2}{x+1} + \frac{3}{x-1}​.

Solution:

Find the LCD: (x+1)(x−1).

Rewrite each fraction: \frac{2}{x+1} = \frac{2(x-1)}{(x+1)(x-1)} = \frac{2x-2}{(x+1)(x-1)}

=\frac{3}{x-1} = \frac{3(x+1)}{(x+1)(x-1)} = \frac{3x+3}{(x+1)(x-1)}

Add the fractions: \frac{2x-2}{(x+1)(x-1)} + \frac{3x+3}{(x+1)(x-1)} = \frac{(2x-2) + (3x+3)}{(x+1)(x-1)}

= \frac{5x+1}{(x+1)(x-1)}

Problem 3: Solve \frac{x}{x+2} = \frac{3}{x-2}.

Solution:

Find the common denominator: (x+2)(x−2).

Multiply every term by the common denominator: (x+2)(x-2) \cdot \frac{x}{x+2} = (x+2)(x-2) \cdot(x+2)(x-2) \cdot \frac{x}{x+2} = (x+2)(x-2) \cdot \frac{3}{x-2}

Simplify: x(x−2)=3(x+2)

x²−2x=3x+6

Solve the quadratic equation: x²−2x−3x−6=0

x²−5x−6=0

(x−6)(x+1)=0

x = 6 \quad \text{or} \quad x = -1

Check for extraneous solutions:

x=−2 and x=2 are not in the domain, but the solutions are x=6 and x=−1, so they are valid.

The solution is x=6 and x=−1.

Problem 4: Divide and simplify: x+42x−5÷x+1x2−9\frac{x + 4}{2x - 5} \div \frac{x + 1}{x^2 - 9}2x−5x+4​÷x2−9x+1​

Solution:

Take the reciprocal of the second fraction and multiply: \\frac{x + 4}{2x - 5} \cdot \frac{x^2 - 9}{x + 1}

Factor where possible:\frac{x + 4}{2x - 5} \cdot \frac{(x - 3)(x + 3)}{x + 1}

Final Answer: \frac{(x + 4)(x - 3)(x + 3)}{(2x - 5)(x + 1)}

Problem 5: Solve: \frac{x - 2}{x + 1} = \frac{2}{x - 1}​

Solution:

Cross-multiply to eliminate the fractions:(x−2)(x−1)=2(x+1)

Expand both sides:x²−x−2x+2=2x+2

Combine like terms and bring all terms to one side:x²−3x+2=2x+2

x² - 5x = 0

Factor the quadratic:x(x−5)=0

Solve for x: x=0 or x=5

Check for extraneous solutions:
Substitute back into the original equation. The value x=0 is valid, and x=5 is valid as well. No extraneous solutions are present.

Final Answer: x=0 and x=5

Worksheet: Algebraic Fractions

Download the Worksheet from here - Algebraic Fractions Worksheet

Conclusion

Working with algebraic fractions involves various operations such as simplification, addition, subtraction, multiplication, and division. The key is to ensure all fractions are fully simplified and that any potential restrictions on the variables are noted to avoid undefined expressions. When solving equations involving algebraic fractions, it’s important to consider extraneous solutions by substituting potential solutions back into the original equation. This careful checking ensures the validity of the solutions.