A café sells three types of cake: banana, chocolate and walnut. The café gives 12% discount on total cost to any customer who buys more than 12 cakes. The number of each type of cake bought by Alice, Betty and Charlotte, together with the total amount paid, are shown in the following table.

	Banana	Chocolate	Walnut	Amount paid ($)
Alice	5	3	2	52
Betty	3	4	5	68
Charlotte	4	8	2	66

Determine the original selling price of each type of cake.

1. Determine whether each customer qualifies for the discount based on the number of cakes bought.

Total cakes bought by each customer:

• Alice: $5 + 3 + 2 = 10 \le 12$ (no discount)
• Betty: $3 + 4 + 5 = 12 \le 12$ (no discount)
• Charlotte: $4 + 8 + 2 = 14 > 12$ ($12\%$ discount)

2. Define variables and set up the system of linear equations.

Let $x$, $y$, and $z$ be the original selling prices of banana, chocolate, and walnut cakes respectively.

From Alice's purchase:
$5x + 3y + 2z = 52$

From Betty's purchase:
$3x + 4y + 5z = 68$

From Charlotte's purchase (with a $12\%$ discount, paying $88\%$):
$0.88(4x + 8y + 2z) = 66$
$4x + 8y + 2z = \frac{66}{0.88} = 75$

3. Solve the system of equations.

Solving the system of linear equations (using a graphing calculator):
$x = 4.5, \quad y = 5.5, \quad z = 6.5$

The original selling prices are $4.50 for a banana cake, $5.50 for a chocolate cake, and $6.50 for a walnut cake.