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.

(30) Let $x$ be the number of cakes of the first kind and $y$ be the number of cakes of the second kind.
Therefore,$x \geq 0$ and $y \geq 0$.
The given information can be summarized in the following table:

Type of Cake	Flour $(g)$	Fat $(g)$
First kind $(x)$	$200$	$25$
Second kind $(y)$	$100$	$50$
Availability	$5000$	$1000$

Constraints:
$200x + 100y \leq 5000 \Rightarrow 2x + y \leq 50$
$25x + 50y \leq 1000 \Rightarrow x + 2y \leq 40$
Objective function: Maximize $Z = x + y$.
The feasible region is determined by the constraints $2x + y \leq 50$,$x + 2y \leq 40$,$x \geq 0$,and $y \geq 0$. The corner points of the feasible region are $O(0, 0)$,$A(25, 0)$,$B(20, 10)$,and $C(0, 20)$.
Evaluating $Z = x + y$ at corner points:

Corner Point	$Z = x + y$
$O(0, 0)$	$0$
$A(25, 0)$	$25$
$B(20, 10)$	$30$ (Maximum)
$C(0, 20)$	$20$

Thus,the maximum number of cakes that can be made is $30$ ($20$ of the first kind and $10$ of the second kind).