Algebraic fractions
Algebra
- Higher
- Non-calculator
- 8 questions
- 29 marks
These are original practice questions, not a past paper. This site is independent and not affiliated with AQA or Pearson.
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Simplify $\dfrac{6x}{9}$.
2 marks
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Simplify $\dfrac{x^{2} - 9}{x - 3}$, for $x \ne 3$.
2 marks
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Simplify $\dfrac{3x + 6}{x + 2}$.
3 marks
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Write $\dfrac{2}{x} + \dfrac{3}{x}$ as a single fraction.
3 marks
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Write $\dfrac{3}{x + 1} + \dfrac{2}{x - 1}$ as a single fraction.
4 marks
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Simplify $\dfrac{x^{2} - 16}{x^{2} + 6x + 8}$.
4 marks
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Simplify fully $\dfrac{2x^{2} - 8}{x^{2} - x - 6} \times \dfrac{x - 3}{x + 2}$.
5 marks
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Solve $\dfrac{3}{x + 1} + \dfrac{1}{x - 1} = 1$.
6 marks
Mark scheme
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Answer. $\dfrac{2x}{3}$
- Divide numerator and denominator by $3$.
- $\dfrac{2x}{3}$
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Answer. $x + 3$
- $x^{2} - 9 = (x - 3)(x + 3)$
- Cancel $x - 3$ to leave $x + 3$.
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Answer. $3$
- $3x + 6 = 3(x + 2)$
- The fraction simplifies to $3$.
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Answer. $\dfrac{5}{x}$
- The denominators are the same.
- $\dfrac{5}{x}$
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Answer. $\dfrac{5x - 1}{x^{2} - 1}$
- Common denominator $(x + 1)(x - 1)$.
- Numerator $3(x - 1) + 2(x + 1) = 5x - 1$.
- $\dfrac{5x - 1}{x^{2} - 1}$
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Answer. $\dfrac{x - 4}{x + 2}$
- $(x - 4)(x + 4)$ over $(x + 2)(x + 4)$.
- Cancel $x + 4$ to leave $\dfrac{x - 4}{x + 2}$.
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Answer. $\dfrac{2(x - 2)}{x + 2}$
- $2x^{2} - 8 = 2(x - 2)(x + 2)$ and $x^{2} - x - 6 = (x - 3)(x + 2)$.
- The first fraction simplifies to $\dfrac{2(x - 2)}{x - 3}$.
- Multiplying by $\dfrac{x - 3}{x + 2}$ leaves $\dfrac{2(x - 2)}{x + 2}$.
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Answer. $x = 2 \pm \sqrt{3}$
- $\dfrac{3(x - 1) + (x + 1)}{x^{2} - 1} = 1$
- $4x - 2 = x^{2} - 1$
- $x^{2} - 4x + 1 = 0$
- $x = 2 \pm \sqrt{3}$