One kind of cake requires 200g flour and 25g of fat,

Solution:

Let there be x cakes of first kind and y cakes of second kind. Therefore,

$x \geq 0$ and $y \geq 0$

The given information can be complied in a table as follows.

$\therefore 200 x+100 y \leq 5000$

$\Rightarrow 2 x+y \leq 50$

$25 x+50 y \leq 1000$

$\Rightarrow x+2 y \leq 40$

Total numbers of cakes, Z, that can be made are, Z = x + y

The mathematical formulation of the given problem is

Maximize $Z=x+y \ldots$

subject to the constraints,

$2 x+y \leq 50$                            (1)

$x+2 y \leq 40$                            (2)

$x, y \geq 0$                                (3)

The feasible region determined by the system of constraints is as follows.

The corner points are A (25, 0), B (20, 10), O (0, 0), and C (0, 20).

The values of Z at these corner points are as follows.

Thus, the maximum numbers of cakes that can be made are 30 (20 of one kind and 10 of the other kind).