| Exam Board | OCR |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2007 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Constraint derivation verification |
| Difficulty | Moderate -0.8 This is a straightforward constraint formulation question requiring only the translation of word statements into linear inequalities using given ratios. Part (i) is pure interpretation, part (ii) involves simple proportional reasoning identical to the given example, and parts (iii)-(iv) are direct transcriptions. No optimization solving, graphical work, or problem-solving insight is required—just systematic application of a standard recipe. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06b Slack variables: converting inequalities to equations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer: \(a =\) number of apple cakes; \(b =\) number of banana cakes; \(c =\) number of cherry cakes | B1, B1 | Identifying variables as "number of cakes". Indicating \(a\) as apple, \(b\) as banana and \(c\) as cherry. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer: \(4 \times 30 = 3 \times 40 = 4 \times 30 = 120\). \(\frac{a}{30} + \frac{b}{40} + \frac{c}{30} = 30 \times 40 \times 30\). \(4a + 3b + 4c \le 120\) or \(X = 4, Y = 3, Z = 4\) | M1, A1 | Any reasonable attempt. 4, 3 and 4. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer: \(a + b + c \ge 30\) (or \(a + b + c = 30\)); \(0 \le a \le 20\), \(0 \le b \le 25\), \(0 \le c \le 10\) (no need to say "all integer") | B1, M1, A1 | Constraint from total number of cakes correct. All three upper constraints correct. All three lower constraints correct also. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer: \(4a + 3b + 2c\) | B1 | Any multiple of this expression. |
**(i)**
Answer: $a =$ number of apple cakes; $b =$ number of banana cakes; $c =$ number of cherry cakes | B1, B1 | Identifying variables as "number of cakes". Indicating $a$ as apple, $b$ as banana and $c$ as cherry.
**(ii)**
Answer: $4 \times 30 = 3 \times 40 = 4 \times 30 = 120$. $\frac{a}{30} + \frac{b}{40} + \frac{c}{30} = 30 \times 40 \times 30$. $4a + 3b + 4c \le 120$ or $X = 4, Y = 3, Z = 4$ | M1, A1 | Any reasonable attempt. 4, 3 and 4.
**(iii)**
Answer: $a + b + c \ge 30$ (or $a + b + c = 30$); $0 \le a \le 20$, $0 \le b \le 25$, $0 \le c \le 10$ (no need to say "all integer") | B1, M1, A1 | Constraint from total number of cakes correct. All three upper constraints correct. All three lower constraints correct also.
**(iv)**
Answer: $4a + 3b + 2c$ | B1 | Any multiple of this expression.
**Total: 8**
---2 A baker can make apple cakes, banana cakes and cherry cakes.\\
The baker has exactly enough flour to make either 30 apple cakes or 20 banana cakes or 40 cherry cakes.
The baker has exactly enough sugar to make either 30 apple cakes or 40 banana cakes or 30 cherry cakes.
The baker has enough apples for 20 apple cakes, enough bananas for 25 banana cakes and enough cherries for 10 cherry cakes.
The baker has an order for 30 cakes.
The profit on each apple cake is 4 p , on each banana cake is 3 p and on each cherry cake is 2 p . The baker wants to maximise the profit on the order.\\
(i) The availability of flour leads to the constraint $4 a + 6 b + 3 c \leqslant 120$. Give the meaning of each of the variables $a , b$ and $c$ in this constraint.\\
(ii) Use the availability of sugar to give a second constraint of the form $X a + Y b + Z c \leqslant 120$, where $X , Y$ and $Z$ are numbers to be found.\\
(iii) Write down a constraint from the total order size. Write down constraints from the availability of apples, bananas and cherries.\\
(iv) Write down the objective function to be maximised.\\[0pt]
[You are not required to solve the resulting LP problem.]
\hfill \mbox{\textit{OCR D1 2007 Q2 [8]}}