1. 1
   A tutor teaches algebraic fraction simplification using three worked examples for a revision worksheet: $\frac{2x^2+3x}{4x^2-9}$, $\frac{x^2-1}{x^2+4x+3}$, and adding $\frac{2}{x+1}+\frac{3}{x-2}$.

   (a)

   Simplify $\frac{2x^2+3x}{4x^2-9}$.

   [1 mark]

   • A$\frac{x}{2x-3}$
   • B$\frac{x}{2x+3}$
   • C$\frac{2x}{2x-3}$
   • D$\frac{x+3}{2x-3}$

   (b)

   Simplify $\frac{x^2-1}{x^2+4x+3}$.

   [1 mark]

   • A$\frac{x-1}{x-3}$
   • B$\frac{x+1}{x+3}$
   • C$\frac{x-1}{x+1}$
   • D$\frac{x-1}{x+3}$

   (c)

   Express $\frac{2}{x+1}+\frac{3}{x-2}$ as a single fraction.

   [1 mark]

   • A$\frac{5}{(x+1)(x-2)}$
   • B$\frac{5x+1}{(x+1)(x-2)}$
   • C$\frac{5x-1}{(x+1)(x-2)}$
   • D$\frac{x+5}{(x+1)(x-2)}$

   Total for question 1: 3 marks

2. 2
   A statistics tutor prepares revision fractions for students. The first task is $\frac{3x+1}{x+2}-\frac{x-2}{x-1}$. The second task simplifies $\frac{x^2+5x+6}{x^2-4}$.

   (a)

   Simplify $\frac{3x+1}{x+2}-\frac{x-2}{x-1}$, expressing your answer as a single fraction.

   [2 marks]

   (b)

   Simplify $\frac{x^2+5x+6}{x^2-4}$ fully.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A group hires a shared taxi; the fare per passenger is $\frac{60}{n}$ dollars for $n$ passengers, and $\frac{60}{n+1}$ if one more joins. Separately, the group compares this to a fraction $\frac{x^2-3x}{x^2-9}$ representing a discount calculation.

   (a)

   A shared taxi fare per passenger is modelled as $\frac{60}{n}$ dollars, where $n$ is the number of passengers. When one extra passenger joins, the fare per person becomes $\frac{60}{n+1}$. Write the difference in fare per passenger, $\frac{60}{n}-\frac{60}{n+1}$, as a single fraction, showing all steps.

   [3 marks]

   (b)

   The group's discount calculation is modelled by $\frac{x^2-3x}{x^2-9}$. Simplify this expression fully, showing your factorisation clearly.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   Two pipes fill a tank: pipe A alone takes $x$ hours and pipe B alone takes $(x+3)$ hours. Their combined filling rate is $\frac{1}{x}+\frac{1}{x+3}$ of the tank per hour. A third pipe's relative rate is compared using the fraction $\frac{x^2+x-6}{x^2-4}$, and a fourth comparison requires dividing $\frac{2x}{x+1} \div \frac{4x^2}{x^2-1}$.

   (a)

   A pipe-filling problem states that pipe A alone fills a tank in $x$ hours, so it fills $\frac{1}{x}$ of the tank per hour, and pipe B alone takes $(x+3)$ hours, filling $\frac{1}{x+3}$ of the tank per hour. Write the combined rate $\frac{1}{x}+\frac{1}{x+3}$ as a single fraction, showing all algebraic steps and simplifying the numerator fully.

   [4 marks]

   (b)

   A third pipe's relative rate is modelled by $\frac{x^2+x-6}{x^2-4}$. Simplify this fraction fully, showing your factorisation of both numerator and denominator.

   [4 marks]

   (c)

   A fourth comparison requires calculating $\frac{2x}{x+1} \div \frac{4x^2}{x^2-1}$. Simplify this expression fully, showing every step including turning the division into a multiplication.

   [5 marks]

   Total for question 4: 13 marks

End of questions