A person has at most Rs. 1,00,000 to invest in two schemes A and B, which give annual returns of 8% and 10% respectively. He wants to invest at least Rs. 30,000 in scheme A, and the amount in scheme B should not exceed 1.5 times the amount in scheme A. The maximum annual return he can get is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → Rs. 9200
Let Rs. x thousand be invested in A and Rs. y thousand in B. Then the problem is:
Maximise \(Z = 0.08x + 0.10y\) (in thousand rupees)
subject to \(x + y \le 100\), \(x \ge 30\), \(y \le 1.5x\), \(y \ge 0\)
The line \(y = 1.5x\) meets \(x + y = 100\) where
\(2.5x = 100\)
\(x = 40, \; y = 60\)
It meets \(x = 30\) at (30, 45). The bounded region has corner points (30, 0), (100, 0), (40, 60) and (30, 45). Values of Z (in thousand rupees):
\(Z(30, 0) = 2.4\)
\(Z(100, 0) = 8\)
\(Z(40, 60) = 3.2 + 6 = 9.2\)
\(Z(30, 45) = 2.4 + 4.5 = 6.9\)
The maximum return is Rs. 9200, with Rs. 40,000 in A and Rs. 60,000 in B.
Why the other options are incorrect
Option (1) Rs. 8000 — Incorrect. This is the return with all Rs. 1,00,000 in A, which is less than Rs. 9200.
Option (3) Rs. 9600 — Incorrect. This needs Rs. 20,000 in A and Rs. 80,000 in B, which breaks the condition of at least Rs. 30,000 in A.
Option (4) Rs. 10000 — Incorrect. This is all the money in B, which breaks the condition of at least Rs. 30,000 in A.