Question 442 marksSection E

(a) How many bags of each type should the man buy to get maximum profit ?

OR

(b) Find the profit that the man can earn by selling all the bags.

Linear Programming
LPP — Optimal Solution by Corner-Point Method

Official answer

Optimise Z=100x+90yZ = 100x + 90y over the feasible region formed by the cost, storage and non-negativity constraints.


The LPP:


  • Maximise Z=100x+90yZ = 100x + 90y
  • subject to 3x+2y≤253x + 2y \le 25 (cost), x+y≤10x + y \le 10 (storage), x≥0x \ge 0, y≥0y \ge 0.

Corner points and Z-values:


  • (0,0)(0, 0): Z=0Z = 0
  • (25/3,0)≈(8.33,0)(25/3, 0) \approx (8.33, 0): Z≈833Z \approx 833
  • (5,5)(5, 5): Z=500+450=950Z = 500 + 450 = 950
  • (0,10)(0, 10): Z=900Z = 900

(a) Bags to buy for maximum profit:


  • Maximum Z occurs at (5,5)(5, 5), so buy 5 bags of rice and 5 bags of wheat.

(b) Maximum profit on selling all bags:


  • Maximum profit = Rs 950.
Answer keywords:corner point methodfeasible regionmaximise Z = 100x + 90yvertex (5, 5)5 bags rice 5 bags wheatmaximum profit Rs 9503x + 2y ≤ 25x + y ≤ 10

Marking scheme

  1. 1.1 mark: correctly identifying the corner points of the feasible region, including the vertex (5,5)(5, 5) from solving 3x+2y=253x + 2y = 25 and x+y=10x + y = 10.
  2. 2.1 mark: evaluating Z at the corners and concluding the maximum Z = Rs 950 at (5,5)(5, 5); for (a) state x=5,y=5x = 5, y = 5, for (b) state profit = Rs 950.
  3. 3.Full 2 marks for whichever alternative (a) or (b) is attempted with correct optimum.

Hint

List the corner points of the region bounded by 3x+2y≤253x + 2y \le 25, x+y≤10x + y \le 10, x,y≥0x, y \ge 0; the vertex (5,5)(5, 5) comes from solving the two lines simultaneously. Evaluate Z=100x+90yZ = 100x + 90y at each.

Quick oral answer

Evaluating Z=100x+90yZ = 100x + 90y at the corners, the maximum is Rs 950 at (5,5)(5, 5), so the man should buy 5 bags of rice and 5 bags of wheat to earn the maximum profit of Rs 950.

Common mistakes

  1. 1.Testing only the axis intercept (8.33,0)(8.33, 0) and reporting Rs 833 instead of checking the interior vertex (5,5)(5, 5) that gives Rs 950.
  2. 2.Arithmetic slip while solving 3x+2y=253x + 2y = 25 with x+y=10x + y = 10 — forgetting to substitute y=10−xy = 10 - x correctly gives a wrong vertex.
  3. 3.Overlooking (0,10)(0, 10) as a feasible corner because they assume the budget line is binding there, when in fact 2(10)=20≤252(10) = 20 \le 25.

Analysis and explanation

This part solves the LPP formulated in Q38(i)-(ii) by the corner-point (extreme-point) method.


Concept:


  • The feasible region is a bounded convex polygon; the optimal value of a linear objective always occurs at one of its corner (vertex) points, so we only need to evaluate Z there.
  • The key vertex (5,5)(5, 5) is the intersection of the two active lines 3x+2y=253x + 2y = 25 and x+y=10x + y = 10.

Working the intersection:


  • From x+y=10x + y = 10, y=10−xy = 10 - x; substituting, 3x+2(10−x)=25→x+20=25→x=5,y=53x + 2(10 - x) = 25 \to x + 20 = 25 \to x = 5, y = 5.

Exam trap:


  • Do not stop at the axis point (8.33,0)(8.33, 0); although it satisfies the budget, it gives only Rs 833, less than Rs 950. Every vertex must be tested.
  • Both alternatives (a) and (b) lead to the same optimum — (a) asks for the bag mix, (b) asks for the resulting profit; state x=5,y=5x = 5, y = 5 and Z = Rs 950 to be safe.

Real-world:


  • This is a classic budget-and-capacity trade-off a trader faces: an equal 5-5 split beats loading up on the higher-cost rice alone.

Interesting facts

The corner-point method rests on the Fundamental Theorem of Linear Programming: an optimum, if it exists over a bounded feasible region, is always attained at a vertex.

Here two constraints are simultaneously binding at the optimum (budget fully or partly used, storage fully used at 5+5=105+5 = 10 bags), which is typical of an efficient LPP solution.

At (5,5)(5, 5) the man spends 1800(5)+1200(5)1800(5) + 1200(5) = Rs 15,000 exactly and uses all 10 storage slots — both resources are fully utilised.

Question details

Q44 of 44 questions

Board
CBSE
Class
Class 12
Subject
Applied Mathematics
Year
2026
Section
Section E
Marks
2
Chapter
Linear Programming
Topic
LPP — Optimal Solution by Corner-Point Method
Bloom's level
apply
Syllabus topic
Unit 8: Linear Programming — Graphical Solution: Feasible Region, Corner Points and Optimal Value of the Objective Function
Author
CBSE
Reviewed by
Padhantu Academic Team

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Frequently asked questions

Why is the maximum profit found at (5,5)(5, 5) and not at the axis point (8.33,0)(8.33, 0)?

The corner-point method requires testing every vertex. At (8.33,0)(8.33, 0), Z=833Z = 833, but at (5,5)(5, 5), Z=500+450=950Z = 500 + 450 = 950, which is larger. Since the optimum of a linear objective on a bounded region lies at a vertex, (5,5)(5, 5) gives the true maximum profit of Rs 950.

How do I get the vertex (5,5)(5, 5)?

It is the intersection of the two binding lines 3x+2y=253x + 2y = 25 and x+y=10x + y = 10. Substituting y=10−xy = 10 - x into the first: 3x+2(10−x)=25→x+20=25→x=53x + 2(10 - x) = 25 \to x + 20 = 25 \to x = 5, so y=5y = 5.

Do parts (a) and (b) have different answers?

No — they describe the same optimum. Part (a) asks for the bag mix (5 rice, 5 wheat) and part (b) asks for the resulting profit (Rs 950). Solving the LPP once answers both.