ABC store can only sell a maximum of 100 packets of salt and 400 dozen eggs this month and decided to have two offers, 𝑥 and 𝑦. Offer 𝑥 is one bag of salt and one dozen eggs. This combination will sell for £30. Another offer is one packet of salt and six dozen eggs, which will sell for £50. XYZ store tries to sell the offers 𝑥 and 𝑦 as many as can maximise their sales. 

a) Formulate the objective function and constraints.

Let's formulate the problem step by step.

### Decision Variables:
Let:
- \( x \) be the number of Offer \( x \) sold (one packet of salt and one dozen eggs).
- \( y \) be the number of Offer \( y \) sold (one packet of salt and six dozen eggs).

### Objective Function:
The goal is to maximize the total sales revenue. The sales revenue is given by:
\[
\text{Maximize } Z = 30x + 50y
\]
Where:
- \( 30x \) is the revenue from selling \( x \) units of Offer \( x \).
- \( 50y \) is the revenue from selling \( y \) units of Offer \( y \).

### Constraints:
The constraints come from the limitations on the availability of salt and eggs.

#### 1. Salt Constraint:
- Offer \( x \) uses 1 packet of salt per sale.
- Offer \( y \) also uses 1 packet of salt per sale.
- The store can sell at most 100 packets of salt.

This gives us the constraint:
\[
x + y \leq 100
\]

#### 2. Eggs Constraint:
- Offer \( x \) uses 1 dozen eggs per sale.
- Offer \( y \) uses 6 dozen eggs per sale.
- The store can sell at most 400 dozen eggs.

This gives us the constraint:
\[
x + 6y \leq 400
\]

#### 3. Non-negativity Constraints:
The number of offers sold cannot be negative:
\[
x \geq 0, \quad y \geq 0
\]

### Summary:
The problem can be formulated as the following linear programming problem:

**Objective Function:**
\[
\text{Maximize } Z = 30x + 50y
\]

**Subject to:**
\[
\begin{aligned}
x + y & \leq 100 \quad \text{(Salt constraint)} \\
x + 6y & \leq 400 \quad \text{(Eggs constraint)} \\
x, y & \geq 0 \quad \text{(Non-negativity constraints)}
\end{aligned}
\]

This formulation can be used to determine the optimal values of \( x \) and \( y \) that will maximize the sales revenue.

Would you like more details on how to solve this or have any other questions?

### Relative Questions
1. Would you like to see the graphical solution of this problem?
2. Do you need help setting up the equations in a specific software tool?
3. Are there any other constraints you would like to consider?
4. Would you like an explanation of how to solve this using the Simplex method?
5. Do you want to analyze the sensitivity of the solution to changes in constraints?
6. Would you like to explore how changes in prices affect the optimal solution?
7. Do you want to know how to interpret the results in the context of the problem?
8. Are you interested in the dual of this linear programming problem?

**Tip:** When formulating a linear programming problem, always clearly define your decision variables, objective function, and constraints to avoid confusion.

Learn how to maximize sales revenue using offers involving salt and eggs at ABC store. Solve the linear programming problem step-by-step with constraints on salt and eggs quantities. Suitable for advanced high school students.

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