One kind of cake requires 200 g of flour and 25 g of fat and another kind of cake requires 100 g of flour and 50 g of fat. Find the maximum number of cakes which can be made from 5 kg of flour and 1 kg of fat assuming that there is no shortage of the other ingredients used in making the cakes.

. : Let x be the number of cakes of first kind and y be the number of cakes of second kind.

Clearly, $x \ge 0,y \ge 0$. We make the following table from the given data :

Since the maximum number of cakes can be made from 5 kg of flour and 1 kg of fat,

we have the constraints
$200x + 100y \le 5000$

$25x + 50y \le 1000$

Total number of cakes to be made is $Z = x + y$

Hence, the mathematical formulation of the problem is :

Maximize $Z = x + y$ ...(1)

subject to the constraints
$2x + y \le 50$ ... (2)

$x + 2y \le 40$ ... (3)

$x,y \ge 0$ ... (4)

${l_1}:2x + y = 50;$ ${l_2}:x + 2y = 40$

Let us graph the inequalities (2) to (4).

The feasible region determined by the system is shown in the graph.

Here, we observe that the feasible region is bounded.

Let us evaluate Z at the comer points A(25, 0), E(20, 10) and B(0, 20).

Hence, maximum number of cakes is 30 when 20 cakes of one

kind and 10 cakes of another kind can be made with given ingredients.