A goldsmith makes rings and chains. A ring needs 2 g of gold and a chain needs 6 g, and 48 g of gold is available. He can make at most 14 pieces in all, and the number of chains must not be more than the number of rings. The profit is Rs. 500 on a ring and Rs. 1200 on a chain. The maximum profit is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → Rs. 10500
Let him make x rings and y chains. Then
Maximise \(Z = 500x + 1200y\)
subject to
\(2x + 6y \le 48\), that is, \(x + 3y \le 24\)
\(x + y \le 14\)
\(y \le x\)
\(x \ge 0, \; y \ge 0\)
Meeting points of the boundary lines:
\(x + 3y = 24\) and \(x + y = 14\): \(y = 5, \; x = 9\)
\(x + 3y = 24\) and \(y = x\): \(x = y = 6\)
The corner points are (0, 0), (14, 0), (9, 5) and (6, 6). Values of Z:
\(Z(14, 0) = 7000\)
\(Z(9, 5) = 4500 + 6000 = 10500\)
\(Z(6, 6) = 3000 + 7200 = 10200\)
So the maximum profit is Rs. 10500, with 9 rings and 5 chains.
Why the other options are incorrect
Option (1) Rs. 7000 — Incorrect. This is the profit from 14 rings only, at (14, 0).
Option (2) Rs. 9600 — Incorrect. This is the profit from 8 chains only, but then the chains are more than the rings.
Option (3) Rs. 10200 — Incorrect. This is the value at (6, 6), which is less than at (9, 5).