How to Solve: Algebraic Fractions (Simplify, Solve Equations)

How to Solve: Algebraic Fractions (Simplify, Solve Equations)

For GCSE & A-Level Maths (Edexcel/AQA/OCR)

Introduction

"Algebraic fractions are worth 5–10 marks in your GCSE/A-Level exam—enough to push you up a grade. Master them, and you’ll solve equations faster, simplify complex expressions, and even tackle real-world problems like calculating medication doses or optimising business costs."

What You Need To Know First

1. Factorising quadratics (e.g., x² – 5x + 6 = (x – 2)(x – 3)).
2. Solving linear equations (e.g., 3x + 2 = 8 → x = 2).
3. Multiplying/dividing fractions (e.g., ½ × ⅔ = ⅓).

If you’re shaky on these, pause and revise them first.

Key Vocabulary

Formulas To Know

1. Simplifying fractions
   (a × c) / (b × c) = a / b MEMORISE THIS
   Cancel common factors in numerator/denominator.

2. Adding/subtracting fractions
   a/b ± c/d = (ad ± bc) / bd MEMORISE THIS
   Find a common denominator first.

3. Solving equations with fractions
   a/b = c → a = bc (if b ≠ 0) MEMORISE THIS
   Multiply both sides by the denominator to eliminate the fraction.

Step-by-Step Method

Part 1: Simplifying Algebraic Fractions

Step 1: Factorise the numerator and denominator. Step 2: Cancel any common factors (same brackets in top and bottom). Step 3: Write the simplified fraction. State restrictions (values that make the denominator zero).

Example: Simplify (x² – 9)/(x² – 3x). 1. Factorise: (x – 3)(x + 3) / x(x – 3). 2. Cancel (x – 3): (x + 3)/x. 3. Restriction: x ≠ 0, x ≠ 3.

Part 2: Solving Equations with Algebraic Fractions

Step 1: Identify restrictions (denominator ≠ 0). Step 2: Find a common denominator for all fractions. Step 3: Multiply every term by the common denominator to eliminate fractions. Step 4: Solve the resulting equation (linear/quadratic). Step 5: Check solutions against restrictions. Discard any that break them.

Example: Solve 2/x + 3/(x + 1) = 5. 1. Restrictions: x ≠ 0, x ≠ –1. 2. Common denominator: x(x + 1). 3. Multiply every term: 2(x + 1) + 3x = 5x(x + 1). 4. Expand: 2x + 2 + 3x = 5x² + 5x → 5x + 2 = 5x² + 5x.
Rearrange: 5x² – 2 = 0 → x² = 2/5 → x = ±√(2/5). 5. Check: x = √(2/5) and x = –√(2/5) don’t break restrictions. Valid solutions.

Worked Examples

Example 1 – Basic: Simplify

Simplify (6x² – 12x)/(3x). 1. Factorise numerator: 6x(x – 2)/3x. 2. Cancel 3x: 2(x – 2). 3. Restriction: x ≠ 0. Answer: 2x – 4 (or 2(x – 2)).

What we did and why: Factorised to cancel common terms. Always state restrictions!

Example 2 – Medium: Solve Equation

Solve 1/(x – 2) = 3/(x + 1). 1. Restrictions: x ≠ 2, x ≠ –1. 2. Cross-multiply: 1(x + 1) = 3(x – 2). 3. Expand: x + 1 = 3x – 6. 4. Solve: –2x = –7 → x = 3.5. 5. Check: 3.5 ≠ 2, –1. Valid. Answer: x = 3.5.

What we did and why: Cross-multiplied to eliminate fractions, then solved the linear equation. Always check restrictions!

Example 3 – Exam-Style: Disguised Problem

"The sum of a number and its reciprocal is 2.5. Find the number." 1. Let the number be x. Equation: x + 1/x = 2.5. 2. Restriction: x ≠ 0. 3. Multiply by x: x² + 1 = 2.5x. 4. Rearrange: x² – 2.5x + 1 = 0. 5. Solve quadratic: (2x – 1)(x – 2) = 0 → x = 0.5 or x = 2. 6. Check: Both valid. Answer: x = 0.5 or x = 2.

What we did and why: Translated words into an equation, eliminated fractions, and solved. Always verify solutions!

Common Mistakes

1. MISTAKE: Cancelling terms that aren’t factors.
   e.g., (x + 2)/(x + 3) → 2/3 (WRONG).
   WHY IT HAPPENS: Confusing addition with multiplication.
   CORRECT APPROACH: Only cancel factors (e.g., (x + 2)(x – 1)/(x + 2) = x – 1).

2. MISTAKE: Forgetting restrictions.
   e.g., Solving 1/(x – 1) = 2 → x = 1.5 but not stating x ≠ 1.
   WHY IT HAPPENS: Overlooking denominator rules.
   CORRECT APPROACH: Write restrictions first.

3. MISTAKE: Not multiplying every term by the common denominator.
   e.g., 2/x + 3 = 5 → 2 + 3 = 5x (WRONG).
   WHY IT HAPPENS: Skipping steps.
   CORRECT APPROACH: Multiply all terms by x: 2 + 3x = 5x.

4. MISTAKE: Assuming all solutions are valid.
   e.g., Solving 1/(x – 2) = 3 → x = 7/3 but not checking x ≠ 2.
   WHY IT HAPPENS: Rushing.
   CORRECT APPROACH: Always check restrictions.

5. MISTAKE: Expanding incorrectly after multiplying.
   e.g., x(x + 1) = x² + 1 (WRONG).
   WHY IT HAPPENS: Misapplying distributive law.
   CORRECT APPROACH: x(x + 1) = x² + x.

Exam Traps

1. Trap: Hidden restrictions.
   e.g., Question gives (x + 1)/(x² – 4) but doesn’t ask for restrictions.
   How to Spot it: Denominator has factors (e.g., x² – 4 = (x – 2)(x + 2)).
   How to Avoid it: Always write x ≠ 2, x ≠ –2 even if not asked.

2. Trap: Equations that simplify to 0 = 0 or 5 = 3.
   e.g., 1/(x – 1) = 1/(x – 1) → 0 = 0 (infinite solutions).
   How to Spot it: After simplifying, both sides are identical.
   How to Avoid it: Write "All real numbers except x = 1" (or "No solution" if false).

3. Trap: Quadratic denominators with no real solutions.
   e.g., 1/(x² + 1) = 2 → x² + 1 = 0.5 → x² = –0.5 (no real solutions).
   How to Spot it: Denominator is always positive (e.g., x² + 1).
   How to Avoid it: Check if the equation leads to x² = negative number.

1-Minute Recap

"Right, listen up—this is your 60-second cheat sheet for algebraic fractions: 1. Simplify: Factorise top and bottom, cancel common factors, state restrictions. 2. Solve equations: Find a common denominator, multiply every term, solve, check restrictions. 3. Watch out: Don’t cancel terms that aren’t factors, always write x ≠ [denominator zeros], and discard invalid solutions. 4. Exam traps: Hidden restrictions, infinite/no solutions, and quadratics with no real roots. You’ve got this. Now go smash those past papers!