Algebraic Fractions Masterclass

1. Simplifying Algebraic Fractions

To simplify a fraction, you need to cancel common factors from the numerator (top) and denominator (bottom).

The Golden Rule: You can only cancel things that are being multiplied. You cannot cancel terms that are added or subtracted!

❌ Wrong: \(\frac{x^2 + 5}{x}\) → Cancel the \(x\)'s to get \(x + 5\). (This is illegal math!)
✅ Right: Factorise first, then cancel.
Method:
  1. Factorise the numerator completely.
  2. Factorise the denominator completely.
  3. Cancel any matching brackets or terms.
Example: Simplify \(\frac{x^2 + 5x}{x^2 - 25}\)
Step 1: Factorise Top Take out \(x\): \(x(x + 5)\)
Step 2: Factorise Bottom Difference of Two Squares: \((x + 5)(x - 5)\)
Step 3: Cancel \(\frac{x(x + 5)}{(x + 5)(x - 5)}\) (The \((x+5)\) brackets cancel out)
Answer: \(\frac{x}{x - 5}\)

2. Multiplying & Dividing

This works exactly like normal fractions.

Example: \(\frac{3x}{4} \div \frac{9x^2}{2}\)
Step 1: Keep, Change, Flip \(\frac{3x}{4} \times \frac{2}{9x^2}\)
Step 2: Multiply Across Top: \(3x \times 2 = 6x\)
Bottom: \(4 \times 9x^2 = 36x^2\)
Step 3: Simplify \(\frac{6x}{36x^2} = \frac{1}{6x}\)
Answer: \(\frac{1}{6x}\)

3. Adding & Subtracting

You must have a Common Denominator.

The "Cross-Multiply" Method (for two fractions): $$ \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} $$ 1. Multiply diagonally to get the new numerator.
2. Multiply the two bottoms to get the new denominator.
3. Simplify the result.
Example: \(\frac{3}{x} + \frac{2}{x + 1}\)
Step 1: Find Common Denominator Multiply bottoms: \(x(x + 1)\)
Step 2: Cross Multiply Top becomes: \(3(x + 1) + 2(x)\)
Step 3: Expand and Simplify Top \(3x + 3 + 2x = 5x + 3\)
Answer: \(\frac{5x + 3}{x(x + 1)}\)

Practice Questions & Solutions

Try these yourself, then click to reveal the solution.

Q1: Simplify \(\frac{15x^3}{5x}\)
Divide the numbers: \(15 \div 5 = 3\)
Divide the powers: \(x^3 \div x = x^2\)
Answer: \(3x^2\)
Q2: Simplify \(\frac{4x + 12}{x + 3}\)
Factorise the top: \(4(x + 3)\)
The fraction is now: \(\frac{4(x + 3)}{x + 3}\)
Cancel the \((x+3)\).
Answer: 4
Q3: Simplify \(\frac{x^2 + 3x + 2}{x + 1}\)
Factorise the quadratic on top.
Multiplies to 2, Adds to 3? (2 and 1).
Top becomes: \((x + 2)(x + 1)\)
Cancel the \((x+1)\).
Answer: \(x + 2\)
Q4: Simplify \(\frac{x^2 - 16}{2x + 8}\)
Top (Difference of Squares): \((x + 4)(x - 4)\)
Bottom (Common Factor): \(2(x + 4)\)
Cancel \((x+4)\).
Answer: \(\frac{x - 4}{2}\)
Q5: Calculate \(\frac{2}{x} \times \frac{x^2}{4}\)
Multiply tops: \(2x^2\)
Multiply bottoms: \(4x\)
Simplify \(\frac{2x^2}{4x}\):
\(2/4\) becomes \(1/2\). \(x^2/x\) becomes \(x\).
Answer: \(\frac{x}{2}\)
Q6: Calculate \(\frac{x}{3} + \frac{x}{5}\)
Common denominator is \(3 \times 5 = 15\).
Cross multiply: \(5(x) + 3(x) = 5x + 3x = 8x\)
Answer: \(\frac{8x}{15}\)
Q7: Calculate \(\frac{4}{x} - \frac{2}{x^2}\)
Common denominator needs to be \(x^2\).
Multiply the first fraction (top and bottom) by \(x\): \(\frac{4x}{x^2}\).
Now subtract: \(\frac{4x}{x^2} - \frac{2}{x^2}\)
Answer: \(\frac{4x - 2}{x^2}\)
Q8: Add \(\frac{3}{x+1} + \frac{1}{x-2}\)
Denominator: \((x+1)(x-2)\)
Cross multiply top: \(3(x-2) + 1(x+1)\)
Expand top: \(3x - 6 + x + 1\)
Simplify top: \(4x - 5\)
Answer: \(\frac{4x - 5}{(x+1)(x-2)}\)
Q9: Divide \(\frac{x+3}{5} \div \frac{x^2-9}{10}\)
Keep, Change, Flip:
\(\frac{x+3}{5} \times \frac{10}{x^2-9}\)
Factorise the new bottom right: \(x^2-9 = (x+3)(x-3)\)
Expression: \(\frac{x+3}{5} \times \frac{10}{(x+3)(x-3)}\)
Cancel \((x+3)\) and simplify \(10/5 = 2\).
Answer: \(\frac{2}{x-3}\)
Q10: Challenge - Subtract \(\frac{5}{x-1} - \frac{2}{x+3}\)
Denominator: \((x-1)(x+3)\)
Cross multiply top: \(5(x+3) - 2(x-1)\)
Watch the negative!
Expand top: \(5x + 15 - 2x + 2\) (Note: \(-2 \times -1 = +2\))
Simplify top: \(3x + 17\)
Answer: \(\frac{3x + 17}{(x-1)(x+3)}\)