1. 1
   Two workers' combined task rates are represented by algebraic fractions.

   (a)

   Two workers share a task. Worker A completes $\frac{x}{3}$ of the task per hour and Worker B completes $\frac{x-4}{2}$ of the task per hour. Simplify $\frac{x}{3} + \frac{x-4}{2}$ into a single fraction.

   [1 mark]

   • A$\frac{5x-12}{6}$
   • B$\frac{2x-4}{5}$
   • C$\frac{5x-4}{6}$
   • D$\frac{5x-12}{5}$

   (b)

   Worker C's rate is $\frac{3a}{4}$ and Worker D's rate is $\frac{9a}{10}$. Simplify $\frac{3a}{4} \times \frac{9a}{10}$.

   [1 mark]

   • A$\frac{12a^2}{40}$
   • B$\frac{27a^2}{40}$
   • C$\frac{27a}{40}$
   • D$\frac{27a^2}{14}$

   (c)

   Worker E divides their rate $\frac{5x}{6}$ by Worker F's rate $\frac{x}{3}$. Simplify $\frac{5x}{6} \div \frac{x}{3}$.

   [1 mark]

   • A$\frac{5x^2}{2}$
   • B$\frac{5}{18}$
   • C$\frac{15}{6}$
   • D$\frac{5}{2}$

   Total for question 1: 3 marks

2. 2
   A chef scales a recipe, combining ingredient quantities expressed as algebraic fractions.

   (a)

   A chef scales a recipe using the fractions $\frac{2x}{5}$ cups of flour and $\frac{x+1}{3}$ cups of sugar. Combine these into a single fraction by calculating $\frac{2x}{5} + \frac{x+1}{3}$.

   [2 marks]

   (b)

   The chef then needs to simplify the ratio of two ingredient fractions, $\frac{4x}{9} \div \frac{2x}{3}$. Simplify this expression fully.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A student practises simplifying rational algebraic expressions by factorising numerators and denominators.

   (a)

   A student simplifies the rational expression $\frac{x^2 - 2x}{x^2 - 5x + 6}$. Factorise both the numerator and denominator, then simplify the expression fully.

   [3 marks]

   (b)

   The student then simplifies $\frac{x^2 - 9}{x^2 + x - 6}$. Factorise both the numerator and denominator, then simplify the expression fully.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A plumber solves equations involving fractional flow rates with linear denominators.

   (a)

   A plumber calculates pipe flow rates and must solve the fractional equation $\frac{5}{x} + \frac{3}{2} = \frac{7}{x}$ for $x$, where $x \neq 0$. Show full working.

   [4 marks]

   (b)

   The plumber then must solve $\frac{4}{x+1} = \frac{2}{x-1}$ for $x$, where $x \neq -1$ and $x \neq 1$. Show full working.

   [4 marks]

   (c)

   Finally, the plumber must solve $\frac{3}{x} + \frac{2}{x+2} = 1$ for $x$, where $x \neq 0$ and $x \neq -2$. Show full working, including any necessary rearrangement into a quadratic equation.

   [5 marks]

   Total for question 4: 13 marks

End of questions