| Exam Board | Edexcel |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Formulation from word problem |
| Difficulty | Moderate -0.8 This is a straightforward linear programming formulation question requiring translation of word constraints into inequalities. The constraints are clearly stated, the variables are defined, and students only need to apply standard techniques (e.g., '85% of tickets are standard' becomes x ≤ 0.85(x+y)). No problem-solving, optimization, or graphical work is required—just mechanical translation, making it easier than average. |
| Spec | 7.06a LP formulation: variables, constraints, objective function |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Maximise \(P = 5x + 8y\) | B1 | |
| Subject to: \(x + y \leq 450\); \(\frac{17}{20}(x + y) \geq x\), simplifies to \(3x \leq 17y\); \(x \geq 3y\); \((x, y \geq 0)\) | B1, M1 A1, M1 A1 | (6) |
| 6 marks |
| Answer/Working | Marks | Guidance |
|---|---|---|
| **Maximise** $P = 5x + 8y$ | B1 | |
| **Subject to:** $x + y \leq 450$; $\frac{17}{20}(x + y) \geq x$, simplifies to $3x \leq 17y$; $x \geq 3y$; $(x, y \geq 0)$ | B1, M1 A1, M1 A1 | (6) |
| | | **6 marks** |
**Notes for Question 7:**
- 1B1: CAO – expression correct and 'maximise' (accept 500x + 800y)
- 2B1: CAO ($x + y \leq 450$)
- 1M1: Correct method – must see $\frac{85}{100}(x + y) \mathbf{\square} x$ (or equivalent) where $\mathbf{\square}$ is any inequality or equals. The bracket must be present or implied by later working
- 1A1: CAO – simplified – answer must have integer coefficients ($3x \leq 17y$)
- 2M1: Correct method – one of $x \mathbf{\square} 3y$ or $3x \mathbf{\square} y$ (or equivalent) where $\mathbf{\square}$ is any inequality or equals
- 2A1: CAO – answer must have integer coefficients ($x \geq 3y$)7. A theatre company is planning to sell two types of ticket, standard and premier. The theatre company has completed some market research and has used this to form the following constraints.
\begin{itemize}
\item They will sell at most 450 tickets.
\item They will sell at least three times as many standard tickets as premier tickets.
\item At most $85 \%$ of all the tickets sold will be standard.
\end{itemize}
The theatre wants to maximise its profit.
The profit on each standard ticket sold is $\pounds 5$ and the profit on each premier ticket sold is $\pounds 8$\\
Let $x$ represent the number of standard tickets sold and $y$ represent the number of premier tickets sold.
Formulate this as a linear programming problem, stating the objective and listing the constraints as simplified inequalities with integer coefficients.
You should not attempt to solve the problem.\\
(Total 6 marks)\\
\hfill \mbox{\textit{Edexcel D1 2016 Q7 [6]}}