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12. One kind of cake requires 200g 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 cake 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.

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Vishal Baghel | Contributor-Level 10

4 months ago

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

x ≥ 0 and y ≥ 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

$\begin{array}{l} ∴ 200 x + 100 y \leq 5000 \\ \Rightarrow 2 x + y \leq 50 \\ 25 x + 50 y \leq 1000 \\ \Rightarrow x + 2 y \leq 40 \end{array}$

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

The mathematical formulation of the given problem is

Maximize $Z = x + y \dots (1)$

subject to the constraints,

$\begin{array}{l} 2 x + y \leq 50 . . . . . . . (2) \\ x + 2 y \leq 40 . . . . . . . (3) \\ x, y \geq 0 . . . . . . . . . . . . . . (4) \end{array}$

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).

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Given 2x + y – z = 3 . (i)

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12. One kind of cake requires 200g 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 cake 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.

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