Question

One kind of cake required $200$ g flour and $25$ g of fat, and another kind of cake required $100$ 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?

Answer 1

Let there be $x$ cakes of first kind and $y$ cakes of second kind. Therefore, $x\ge 0$ and $y\ge 0$
The given information can be complied in a table as follows.

	Flour (g)	Fat (g)
cakes of first kind, $x$	$200$	$25$
cakes of second kind, $y$	$100$	$50$
Availability	$5000$	$1000$

$\therefore$ $200x+100y\le 5000$
$\Rightarrow$ $2x+y\le 50$
$25x+50y\le 1000$
$\Rightarrow$ $x+2y\le 40$
Total numbers of cakes, $Z$, that can be made are, $Z=x+y$
The mathematical formulation of the given problem is
Maximise $Z=x+y.......\left(1\right)$
subject to the constraints,
$2x+y\le \mathrm{50........}\left(2\right)$
$x+2y\le \mathrm{40........}\left(3\right)$
$x,y\ge \mathrm{0............}\left(4\right)$
The feasible region determined by the system of constraints is as shown.
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.

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

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