A person wants to invest an amount of Rs. 75,000. He has two options A and B yielding 8% and 9% return respectively on the invested amount. He plans to invest at least Rs. 15,000 in Plan A and at least Rs. 25,000 in Plan B. Also he wants that his investment in Plan A is less than or equal to his investment in Plan B. Which of the following options describes the given LPP to maximize the return (where x and y are investments in Plan A and Plan B respectively)?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → maximize Z = 0.08x + 0.09y, \(x \ge 15000\), \(y \ge 25000\), \(x + y \le 75000\), \(x \le y\), \(x, y \ge 0\)
Each sentence of the question gives one part of the LPP. The return is 8% of x plus 9% of y:
\(Z = 0.08x + 0.09y\)
“At least” means ≥:
\(x \ge 15000\)
\(y \ge 25000\)
He has only Rs. 75,000, so the total cannot exceed it:
\(x + y \le 75000\)
Investment in Plan A is less than or equal to that in Plan B:
\(x \le y\)
with \(x, y \ge 0\). Only Option (4) has all these constraints.
Why the other options are incorrect
Option (1) maximize Z = 0.08x + 0.09y, \(x \ge 15000\), \(y \ge 25000\), \(x + y \ge 75000\), \(x \le y\), \(x, y \ge 0\) — Incorrect. \(x + y \ge 75000\) would allow investing more than the Rs. 75,000 he has.
Option (2) maximize Z = 0.08x + 0.09y, \(x \ge 15000\), \(y \le 25000\), \(x + y \ge 75000\), \(x \le y\), \(x, y \ge 0\) — Incorrect. \(y \le 25000\) says at most Rs. 25,000 in Plan B, not at least; the total constraint is also reversed.
Option (3) maximize Z = 0.08x + 0.09y, \(x \ge 15000\), \(y \ge 25000\), \(x + y \le 75000\), \(x \ge y\), \(x, y \ge 0\) — Incorrect. \(x \ge y\) puts more in Plan A than in Plan B, the opposite of his condition.