9 A factory can make three different kinds of balloon pack: gold, silver and bronze. Each pack contains three different types of balloon: \(A , B\) and \(C\). Each gold pack has 2 type \(A\) balloons, 3 type \(B\) balloons and 6 type \(C\) balloons.
Each silver pack has 3 type \(A\) balloons, 4 type \(B\) balloons and 2 type \(C\) balloons.
Each bronze pack has 5 type \(A\) balloons, 3 type \(B\) balloons and 2 type \(C\) balloons.
Every hour, the maximum number of each type of balloon available is 400 type \(A\), 400 type \(B\) and 400 type \(C\). Every hour, the factory must pack at least 1000 balloons.
Every hour, the factory must pack more type \(A\) balloons than type \(B\) balloons.
Every hour, the factory must ensure that no more than \(40 \%\) of the total balloons packed are type \(C\) balloons. Every hour, the factory makes \(x\) gold, \(y\) silver and \(z\) bronze packs.
Formulate the above situation as 6 inequalities, in addition to \(x \geqslant 0 , y \geqslant 0 , z \geqslant 0\), simplifying your answers.
(8 marks)

**Question 9:**

| Inequality | Working | Mark |
|---|---|---|
| **Type A:** $2x + 3y + 5z \leq 400$ | A balloons constraint | B1 |
| **Type B:** $3x + 4y + 3z \leq 400$ | B balloons constraint | B1 |
| **Type C:** $6x + 2y + 2z \leq 400$ | C balloons constraint | B1 |
| **Total ≥ 1000:** $11x + 9y + 10z \geq 1000$ | At least 1000 balloons packed | B1 |
| **More A than B:** $2x + 3y + 5z > 3x + 4y + 3z$ → $2z > x + y$ | A balloons exceed B balloons | B1 |
| **C ≤ 40%:** $6x + 2y + 2z \leq \frac{2}{5}(11x + 9y + 10z)$ → $30x + 10y + 10z \leq 22x + 18y + 20z$ → $8x \leq 8y + 10z$ → $4x \leq 4y + 5z$ | No more than 40% type C | B1 |

9 A factory can make three different kinds of balloon pack: gold, silver and bronze. Each pack contains three different types of balloon: $A , B$ and $C$.

Each gold pack has 2 type $A$ balloons, 3 type $B$ balloons and 6 type $C$ balloons.\\
Each silver pack has 3 type $A$ balloons, 4 type $B$ balloons and 2 type $C$ balloons.\\
Each bronze pack has 5 type $A$ balloons, 3 type $B$ balloons and 2 type $C$ balloons.\\
Every hour, the maximum number of each type of balloon available is 400 type $A$, 400 type $B$ and 400 type $C$.

Every hour, the factory must pack at least 1000 balloons.\\
Every hour, the factory must pack more type $A$ balloons than type $B$ balloons.\\
Every hour, the factory must ensure that no more than $40 \%$ of the total balloons packed are type $C$ balloons.

Every hour, the factory makes $x$ gold, $y$ silver and $z$ bronze packs.\\
Formulate the above situation as 6 inequalities, in addition to $x \geqslant 0 , y \geqslant 0 , z \geqslant 0$, simplifying your answers.\\
(8 marks)

\hfill \mbox{\textit{AQA D1 2013 Q9 [8]}}