IGCSE · Practice questions · Algebraic fractions

Factorise, then combine.

Extended. Six original Cambridge-style questions on algebraic fractions: simplifying, multiplying, dividing, adding and subtracting.

Original questions All questions on this page are original work, written in the Cambridge IGCSE style. They are not from past papers. They test the same concepts and skills the syllabus rewards.

The method

Same rules as number fractions.

Simplify: factorise top and bottom fully, then cancel common factors.
Multiply: tops times tops, bottoms times bottoms, cancelling first.
Divide: flip the second fraction and multiply.
Add/subtract: common denominator; numerators in brackets.
Finish: factorise the final numerator to see if anything else cancels.

Recall

[3 marks]

Simplify (a) \(\dfrac{15a^{2}b}{25ab^{3}}\) (b) \(\dfrac{4x + 12}{x^{2} + 3x}\).

(a) \(\dfrac{3a}{5b^{2}}\) ✓
(b) \(\dfrac{4(x + 3)}{x(x + 3)} = \dfrac{4}{x}\) ✓

Calculation

[3 marks]

Simplify \(\dfrac{x^{2} - x - 6}{x^{2} - 4}\).

Top: \((x - 3)(x + 2)\). Bottom: \((x - 2)(x + 2)\). ✓
Cancel \((x + 2)\): \(\dfrac{x - 3}{x - 2}\) ✓

Calculation

[2 marks]

Simplify \(\dfrac{4x}{15} \div \dfrac{8x^{2}}{5}\).

\(\dfrac{4x}{15} \times \dfrac{5}{8x^{2}} = \dfrac{20x}{120x^{2}}\) ✓
\(= \dfrac{1}{6x}\) ✓

Calculation

[3 marks]

Write \(\dfrac{2x - 1}{3} - \dfrac{x - 4}{5}\) as a single fraction in its simplest form.

\(\dfrac{5(2x - 1) - 3(x - 4)}{15}\) ✓
\(= \dfrac{10x - 5 - 3x + 12}{15} = \dfrac{7x + 7}{15}\) ✓
\(= \dfrac{7(x + 1)}{15}\) ✓

Calculation

[3 marks]

Write \(\dfrac{2}{x + 3} + \dfrac{5}{x - 4}\) as a single fraction in its simplest form.

\(\dfrac{2(x - 4) + 5(x + 3)}{(x + 3)(x - 4)}\) ✓
\(= \dfrac{7x + 7}{(x + 3)(x - 4)} = \dfrac{7(x + 1)}{(x + 3)(x - 4)}\) ✓

Proof

[4 marks]

(a) Show that \(\dfrac{1}{x} - \dfrac{1}{x + 1} = \dfrac{1}{x(x + 1)}\).

(b) Hence, without a calculator, find the exact value of \(\dfrac{1}{1 \times 2} + \dfrac{1}{2 \times 3} + \dfrac{1}{3 \times 4} + \dots + \dfrac{1}{9 \times 10}\).

(a) \(\dfrac{(x + 1) - x}{x(x + 1)} = \dfrac{1}{x(x + 1)}\) ✓
(b) Each term is \(\dfrac{1}{n} - \dfrac{1}{n + 1}\): \(\left(1 - \dfrac{1}{2}\right) + \left(\dfrac{1}{2} - \dfrac{1}{3}\right) + \dots + \left(\dfrac{1}{9} - \dfrac{1}{10}\right)\) ✓
Everything cancels except \(1 - \dfrac{1}{10}\) ✓
\(= \dfrac{9}{10}\) ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.