| Exam Board | Edexcel |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2018 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Three-variable constraint reduction |
| Difficulty | Standard +0.3 This is a straightforward linear programming formulation question requiring translation of verbal constraints into inequalities and basic optimization with two variables fixed. Part (a) involves routine constraint writing (non-vanilla < 200, ratio constraint, percentage constraint), and part (b) reduces to single-variable optimization once x and z are given. The three-variable setup and percentage constraint add minor complexity above the most basic LP questions, but this remains a standard textbook exercise requiring no novel insight. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06c Working with constraints: algebra and ad hoc methods |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Maximise \(0.75x + 1.2y + 1.45z\) | B1 | Expression correct (or \(75x+120y+145z\)) together with 'maximise' or 'max' |
| Subject to \(x + z < 200\) | B1 | CAO (\(x+z<200\)) |
| \(5y \geq 2x\) | M1 A1 | Correct method: \(5y \square 2x\); answer must have integer coefficients with like terms collected |
| \(\frac{3}{4}(x+y+z) \geq y \Rightarrow 3x+3z \geq y\) | M1 A1 (6) | \(\frac{3}{4}(x+y+z) \square y\); bracket must be present or implied; answer must have integer coefficients |
| \((x, y, z \geq 0)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(x=100, z=25\) leading to | M1 | Substituting \(x=100\) and \(z=25\) into inequalities, obtaining either two values of \(y\) or two inequalities for \(y\) |
| \(40 \leq y \leq 375\) | A1 | CAO - \(40 \leq y \leq 375\) (oe) or \(y=40\) and \(y=375\) |
| Minimum profit (£)\(159.25\), Maximum profit (£)\(561.25\) | A1 (3) | CAO on min and max profit; condone lack of or incorrect units |
## Question 7:
### Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Maximise $0.75x + 1.2y + 1.45z$ | B1 | Expression correct (or $75x+120y+145z$) **together with** 'maximise' or 'max' |
| Subject to $x + z < 200$ | B1 | CAO ($x+z<200$) |
| $5y \geq 2x$ | M1 A1 | Correct method: $5y \square 2x$; answer must have integer coefficients with like terms collected |
| $\frac{3}{4}(x+y+z) \geq y \Rightarrow 3x+3z \geq y$ | M1 A1 **(6)** | $\frac{3}{4}(x+y+z) \square y$; bracket must be present or implied; answer must have integer coefficients |
| $(x, y, z \geq 0)$ | | |
### Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $x=100, z=25$ leading to | M1 | Substituting $x=100$ **and** $z=25$ into inequalities, obtaining either two values of $y$ or two inequalities for $y$ |
| $40 \leq y \leq 375$ | A1 | CAO - $40 \leq y \leq 375$ (oe) or $y=40$ **and** $y=375$ |
| Minimum profit (£)$159.25$, Maximum profit (£)$561.25$ | A1 **(3)** | CAO on min and max profit; condone lack of or incorrect units |
---7. Emily is planning to sell three types of milkshake, strawberry, vanilla and chocolate.
Emily has completed some market research and has used this to form the following constraints on the number of milkshakes that she will sell each week.
\begin{itemize}
\item She will sell fewer than 200 non-vanilla milkshakes in total.
\item She will sell at most 2.5 times as many strawberry milkshakes as vanilla milkshakes.
\item At most, $75 \%$ of the milkshakes that she will sell will be vanilla.
\end{itemize}
The profit on each strawberry milkshake sold is $\pounds 0.75$, the profit on each vanilla milkshake sold is $\pounds 1.20$ and the profit on each chocolate milkshake sold is $\pounds 1.45$
Emily wants to maximise her profit.\\
Let $x$ represent the number of strawberry milkshakes sold, let $y$ represent the number of vanilla milkshakes sold and let $z$ represent the number of chocolate milkshakes sold.
\begin{enumerate}[label=(\alph*)]
\item Formulate this as a linear programming problem, stating the objective and listing the constraints as simplified inequalities with integer coefficients.
In week 1, Emily sells 100 strawberry milkshakes and 25 chocolate milkshakes.
\item Calculate the maximum possible profit and minimum possible profit, in pounds, for the sale of all milkshakes in week 1. You must show your working.
\end{enumerate}
\hfill \mbox{\textit{Edexcel D1 2018 Q7 [9]}}