| Exam Board | OCR MEI |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2006 |
| Session | January |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Formulation from word problem |
| Difficulty | Moderate -0.5 This is a standard D1 simulation question requiring students to assign random number ranges to probabilities and complete a table. While it involves multiple parts, each step follows routine procedures taught in Decision Maths 1 with no novel problem-solving required. The calculations are straightforward once the rules are established, making it slightly easier than average for A-level. |
| Time taken (mins) | 1 | 1.5 | 2 | 2.5 | 3 |
| Probability | \(\frac { 1 } { 7 }\) | \(\frac { 2 } { 7 }\) | \(\frac { 2 } { 7 }\) | \(\frac { 1 } { 7 }\) | \(\frac { 1 } { 7 }\) |
| Answer | Marks |
|---|---|
| (i) 12.5 kg butter requires 250 g of butter; 10 kg requires 3 kg of sugar | B1 B1; B1 B1 |
| Answer | Marks |
|---|---|
| Graph shows axes labelled and scaled; butter line; sugar line; shading of feasible region; optimal point at \((9, 4)\); solution: Make 9 kg toffee and 4 kg fudge | B1; B1; B1; B1 axes labelled and scaled; B1 butter line; B1 sugar line; B1 shading; B1 max \(x+y\) + solution |
| (iii) 12.5 kg of toffee and no fudge – either by comparing 68.75 with 67.50 with 45, or by gradient argument. Toffee price must decrease by £0.36, or to £15.14. | M1; A1; B1 B1 |
**(i)** 12.5 kg butter requires 250 g of butter; 10 kg requires 3 kg of sugar | B1 B1; B1 B1 |
**(ii)** Variables: Let $x$ = kg of toffee made; Let $y$ = kg of fudge made
Objective: $\text{Max } x + y$
Constraints: $100x + 150y \leq 1500$; $800x + 700y \leq 10000$
Graph shows axes labelled and scaled; butter line; sugar line; shading of feasible region; optimal point at $(9, 4)$; solution: Make 9 kg toffee and 4 kg fudge | B1; B1; B1; B1 axes labelled and scaled; B1 butter line; B1 sugar line; B1 shading; B1 max $x+y$ + solution |
**(iii)** 12.5 kg of toffee and no fudge – either by comparing 68.75 with 67.50 with 45, or by gradient argument. Toffee price must decrease by £0.36, or to £15.14. | M1; A1; B1 B1 |
---4 Table 4 shows the butter and sugar content in two recipes. The first recipe is for 1 kg of toffee and the second is for 1 kg of fudge.
\begin{table}[h]
\section*{Table 6.1}
(ii) Specify an efficient rule for using one-digit random numbers to simulate the time taken at the till by customers purchasing fuel.
Table 6.2 shows the distribution of time taken at the till by customers who are not buying fuel.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | }
\hline
Time taken (mins) & 1 & 1.5 & 2 & 2.5 & 3 \\
\hline
Probability & $\frac { 1 } { 7 }$ & $\frac { 2 } { 7 }$ & $\frac { 2 } { 7 }$ & $\frac { 1 } { 7 }$ & $\frac { 1 } { 7 }$ \\
\hline
\end{tabular}
\end{center}
\section*{Table 6.2}
(iii) Specify an efficient rule for using two-digit random numbers to simulate the time taken at the till by customers not buying fuel.
What is the advantage in using two-digit random numbers instead of one-digit random numbers in this part of the question?
The table in the insert shows a partially completed simulation study of 10 customers arriving at the till.\\
(iv) Complete the table using the random numbers which are provided.\\
(v) Calculate the mean total time spent queuing and paying.
\end{table}
\hfill \mbox{\textit{OCR MEI D1 2006 Q4 [16]}}