Write the constraints that relate the total cost of both types of bags.

Linear Programming

LPP Formulation — Cost Constraint

Official Answer

The cost of the bags bought cannot exceed the Rs 15,000 available.

Cost (budget) constraint:

Cost of rice = Rs 1,800 per bag $\times x = 1800x$ .
Cost of wheat = Rs 1,200 per bag $\times y = 1200y$ .
Total cost ≤ available money: $1800x + 1200y \le 15000$ .

In simplest form (dividing by 600):

$3x + 2y \le 25$ , with $x \ge 0$ , $y \ge 0$ .

cost constraintbudget constraint1800x + 1200y ≤ 150003x + 2y ≤ 25less than or equallinear programming constraintnon-negativity

Marking Scheme

11 mark: writing the cost constraint $1800x + 1200y \le 15000$ (or the reduced form $3x + 2y \le 25$ ).
2Accept either the full or simplified inequality; deduct if written as an equality or with the wrong inequality direction.

Hint

Multiply each variable by its cost per bag, keep the total within Rs 15,000, and use '≤' (not '=').

Quick Oral Answer

The money spent — Rs 1,800 per rice bag and Rs 1,200 per wheat bag — must stay within Rs 15,000, so $1800x + 1200y \le 15000$ , which simplifies to $3x + 2y \le 25$ .

Analysis & Explanation

This part supplies the budget (money) constraint of the LPP.

Concept:

A constraint is an inequality that a feasible solution must satisfy. The spending on rice plus wheat must stay within the Rs 15,000 budget, giving a '≤' inequality.

Exam trap:

Do not write it as an equation (= 15000); the man need not spend every rupee, so it is '≤'.
The storage limit $x + y \le 10$ is a separate constraint — the question here asks only for the cost relation, so keep them distinct.

Simplification:

Reducing $1800x + 1200y \le 15000$ to $3x + 2y \le 25$ keeps the arithmetic light when you later find corner points, and either form earns full marks.

Common Mistakes

1Writing the constraint as an equation $1800x + 1200y = 15000$ instead of an inequality with '≤'.
2Reversing the inequality sign to '≥', which would wrongly force spending of at least Rs 15,000.
3Mixing in the storage limit ( $x + y \le 10$ ) when only the cost relation is asked.

Interesting Facts

In LPP terminology this is a 'resource constraint' — money is the limited resource being allocated between two competing goods.

Dividing $1800x + 1200y \le 15000$ by 600 is valid because dividing an inequality by a positive number preserves its direction — a small but frequently misapplied algebra rule.

The feasible region of this problem is bounded by the budget line, the storage line and the two axes, forming a convex polygon whose corners hold the optimum.

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Frequently Asked Questions

Why is the cost constraint an inequality and not an equation?

The man has up to Rs 15,000; he is not required to spend all of it. His total spending $1800x + 1200y$ must be less than or equal to 15000, so the correct relation is $1800x + 1200y \le 15000$ .

Is $3x + 2y \le 25$ an acceptable answer?

Yes. Dividing $1800x + 1200y \le 15000$ by 600 gives $3x + 2y \le 25$ . Since 600 is positive, the inequality direction is unchanged, so both forms are equivalent and earn full marks.