Solve the following linear programming problem graphically:
Maximize \( z = x + y \), subject to constraints:
\[ 2x + 5y \leq 100, \quad 8x + 5y \leq 200, \quad x \geq 0, \quad y \geq 0. \]
Step 1: Plot the constraints
The given constraints are: \( 2x + 5y = 100 \): The line passes through \( (50, 0) \) and \( (0, 20) \).
\( 8x + 5y = 200 \): The line passes through \( (25, 0) \) and \( (0, 40) \).
\( x \geq 0 \) and \( y \geq 0 \): Restricts the feasible region to the first quadrant.
The feasible region is the intersection of all these constraints, forming a polygon bounded by the vertices.
Step 2: Find corner points
The corner points of the feasible region are determined by solving the equations pairwise: \( O(0, 0) \): Intersection of \( x = 0 \) and \( y = 0 \).
\( A(25, 0) \): Intersection of \( 8x + 5y = 200 \) and \( y = 0 \).
\( B\left(\frac{50}{3}, \frac{40}{3}\right) \): Intersection of \( 2x + 5y = 100 \) and \( 8x + 5y = 200 \).
\( C(0, 20) \): Intersection of \( 2x + 5y = 100 \) and \( x = 0 \).
Step 3: Evaluate the objective function at corner points
Substitute the coordinates of the vertices into \( z = x + y \): \[ \begin{array}{|c|c|} \hline \text{Corner Point} & \text{Value of } z = x + y \\ \hline O(0, 0) & 0 + 0 = 0 \\ A(25, 0) & 25 + 0 = 25 \\ B\left(\frac{50}{3}, \frac{40}{3}\right) & \frac{50}{3} + \frac{40}{3} = \frac{90}{3} = 30 \\ C(0, 20) & 0 + 20 = 20 \\ \hline \end{array} \]
Step 4: Find the maximum value of \( z \)
The maximum value of \( z \) is \( 30 \), which occurs at \( B\left(\frac{50}{3}, \frac{40}{3}\right) \).
Conclusion: The maximum value of \( z \) is \( 30 \) when \( x = \frac{50}{3} \) and \( y = \frac{40}{3} \).
For a Linear Programming Problem, find min \( Z = 5x + 3y \) (where \( Z \) is the objective function) for the feasible region shaded in the given figure.
Rupal, Shanu and Trisha were partners in a firm sharing profits and losses in the ratio of 4:3:1. Their Balance Sheet as at 31st March, 2024 was as follows:
(i) Trisha's share of profit was entirely taken by Shanu.
(ii) Fixed assets were found to be undervalued by Rs 2,40,000.
(iii) Stock was revalued at Rs 2,00,000.
(iv) Goodwill of the firm was valued at Rs 8,00,000 on Trisha's retirement.
(v) The total capital of the new firm was fixed at Rs 16,00,000 which was adjusted according to the new profit sharing ratio of the partners. For this necessary cash was paid off or brought in by the partners as the case may be.
Prepare Revaluation Account and Partners' Capital Accounts.