MathsISC Class 12 · Linear Programming

Formulating Linear Programming Problems from Words

Q: How do I decide whether a constraint is ≤ or ≥?

Limits on resources ("available", "at most", "cannot exceed") give ≤. Requirements ("at least", "minimum", "must meet") give ≥.

Q: Do I always have to solve after formulating?

Only if asked. Some questions say "formulate only", but most ISC questions ask for the formulation and the graphical solution together.

Q: What if the answer is not a whole number?

Linear programming allows fractional values, which is one of its limitations. ISC problems are usually set so that the optimal corner has whole-number coordinates; if not, comment on it.

Q: Are advantages and limitations of LPP in the ISC syllabus?

Yes. The 2027 syllabus lists the terminology, advantages, limitations, application areas and types of LP problems along with formulation.

Q: How many marks does formulation carry?

In the 4-mark LPP questions of 2024 and 2025, writing the objective function and constraints was the first step of the marking, before the graph and corner points.

Updated 15 October 2026 · Fact-checked

To formulate an LPP, define decision variables (x, y) for the quantities you choose, write the objective function Z to be maximised or minimised in terms of them, translate every limit in the question into a linear inequality, and add the non-negativity restrictions $x \ge 0$, $y \ge 0$. The formulation then feeds the graphical method.

Understand Formulating Linear Programming Problems

Why formulation matters. In ISC, the linear programming question is a word problem: in 2024 a shopkeeper buying sacks of rice and wheat, in 2025 two kinds of books on a library shelf. Most of the marks depend on turning the words into a correct model; the 2025 examiners reported that most candidates wrote the objective function wrongly, and in both years candidates could not graph the inequalities correctly.

Terminology.

Decision variables: the quantities the decision-maker controls, such as the number of chairs $x$ and stools $y$ .
Objective function: the linear expression $Z = a x + b y$ to be maximised (profit, output, reach) or minimised (cost, time, waste).
Constraints: linear inequalities expressing limited resources or requirements: $\leq$ for "at most", "cannot exceed", "available"; $\geq$ for "at least", "minimum requirement".
Non-negativity restrictions: $x \geq 0$ , $y \geq 0$ , because counts and amounts cannot be negative.
Optimisation: finding the feasible point that gives the best value of Z.

Four steps to formulate.
1. Read the question and identify what is being decided. Write "Let x = number of … and y = number of …", with units.
2. Identify the goal and write Z with the per-unit profit or cost.
3. Make a small table: rows for the two products, columns for each resource, and a final row for the limit. Each column gives one constraint.
4. Add $x, y \geq 0$ .

Translating phrases.

Words	Inequality
at most 40 in all	$x + y \leq 40$
at least 10 of the first	$x \geq 10$
x at least twice y	$x \geq 2 y$
y not more than half of the total	$y \leq \frac{x + y}{2}$ , i.e. $y \leq x$
costs ₹300 and ₹1200, budget ₹6000	$300 x + 1200 y \leq 6000$

Simplify by dividing by common factors ( $300 x + 1200 y \leq 6000$ becomes $x + 4 y \leq 20$ ); this makes graphing easier and is accepted.

Units must match. If one product takes 20 minutes and the limit is 10 hours, convert the limit to 600 minutes first. Weights in grams and kilograms, money in rupees and thousands: convert before writing the inequality.

Redundant constraints. A constraint that is automatically satisfied once the others hold does not change the feasible region. It is still part of the formulation, but it produces no corner.

Types of LPP.

Manufacturing (product mix): how many of each product to make to maximise profit with limited machine hours, labour or material.
Diet (mixture): how much of each food to mix to meet nutrient requirements at minimum cost; constraints are " $\geq$ ".
Allocation and investment: dividing a fixed budget, land or time between two uses.
Transportation: deciding quantities sent from sources to destinations at minimum cost (in two variables only).

Advantages. LPP gives the best use of scarce resources, turns a vague planning question into precise numbers, shows which constraints are binding, and helps managers compare plans objectively.

Limitations. The objective and constraints must be linear; it assumes certainty (known prices, times, demand); the variables may come out fractional where only whole numbers make sense; it handles a single objective; and real problems may have many variables, beyond the graphical method.

Formulating Linear Programming Problems: formulas to remember

General form: $Optimise Z = a x + b y$; Profit per unit × units, or cost per unit × units.
Resource constraint: $a_{1} x + b_{1} y \leq c_{1}$; "At most", "available", "cannot exceed".
Requirement constraint: $a_{2} x + b_{2} y \geq c_{2}$; "At least", "minimum".
Ratio condition: $x \geq k y ⟺ x - k y \geq 0$; "x at least k times y".
Non-negativity: $x \geq 0, y \geq 0$; Always written.

How to solve Formulating Linear Programming Problems questions

Use a resource table: it turns a paragraph into constraints almost mechanically.

1Underline the decision ("how many…", "how much…") and define x and y with units.
2Underline the goal (maximise profit, minimise cost) and write Z.
3Draw a table with the products as rows and each resource as a column; put the limit under each column.
4Write one inequality per column, choosing ≤ or ≥ from the wording, and convert units to match.
5Add any extra conditions (minimum numbers, ratios) and the non-negativity restrictions.
6Simplify each inequality by common factors and check by testing a sensible point.

Formulating Linear Programming Problems in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

A redundant constraint

Suppose a student makes handmade cards (x) and bookmarks (y) to sell. A card takes 20 minutes and a bookmark 10 minutes, and she has 10 hours. Materials cost ₹15 per card and ₹5 per bookmark, from a budget of ₹600. Profit is ₹25 per card and ₹10 per bookmark.

Formulate the LPP, spot the redundant constraint and find the best plan.

Try it first, then show the solutionSolution

Time: $20 x + 10 y \leq 600$ , i.e. $2 x + y \leq 60$ . Money: $15 x + 5 y \leq 600$ , i.e. $3 x + y \leq 120$ . Maximise $Z = 25 x + 10 y$ , $x, y \geq 0$ .
If $2 x + y \leq 60$ with $x \geq 0$ , then $3 x + y = (2 x + y) + x \leq 60 + 30 = 90 < 120$ (as $x \leq 30$ ). So the money constraint is redundant.
Corners (0, 0), (30, 0), (0, 60): Z = 0, 750, 600.

Answer: Make 30 cards and no bookmarks for ₹750; time, not money, is the real limit.

Memory hook: One table column = one constraint; a redundant one never makes a corner.

A diet-type (minimum cost) formulation

Imagine a pet owner mixing two foods. Food A costs ₹40 per kg and gives 2 units of protein and 1 unit of fat per kg; food B costs ₹30 per kg and gives 1 unit of protein and 1 unit of fat per kg. The pet needs at least 8 units of protein and 6 units of fat a week.

Formulate the LPP for the cheapest weekly mix.

Try it first, then show the solutionSolution

Let x kg of A and y kg of B. Minimise $Z = 40 x + 30 y$ .
Protein: $2 x + y \geq 8$ ; fat: $x + y \geq 6$ ; $x, y \geq 0$ .

Answer: Minimise $Z = 40 x + 30 y$ subject to $2 x + y \geq 8$ , $x + y \geq 6$ , $x, y \geq 0$ .

Memory hook: Diet problems: minimise cost, and every nutrient is an "at least".

Tricks to remember Formulating Linear Programming Problems

Short memory aids. Say each one aloud once, then use it on a question the same day.

V-O-C-N
Variables, Objective, Constraints, Non-negativity.
Use it when: The order of every formulation.
Columns are constraints
In the resource table, read down each column for one inequality.
Use it when: Writing constraints quickly.
Most ≤, least ≥
"At most" points down (≤); "at least" points up (≥).
Use it when: Choosing the sign.
Same units first
Hours with hours, kg with kg, before writing a single inequality.
Use it when: Avoiding scale errors.

Formulating Linear Programming Problems: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: A budget constraint · 1 mark · MCQBoard level

A dealer has ₹6000 to buy x radios at ₹300 each and y speakers at ₹1200 each. The budget constraint is:
(a) $x + 4 y \geq 20$
(b) $4 x + y \leq 20$
(c) $x + 4 y \leq 20$
(d) $300 x + 1200 y \geq 6000$

Try it first, then show the solutionSolution

$300 x + 1200 y \leq 6000$ ; divide by 300.

Answer: (c).

Example 2: Translating conditions · 2 marks · Short answerBoard level

Write inequalities for: a hall can seat at most 40 people in total; at least 10 must be adults (x); the number of adults must be at least twice the number of children (y).

Try it first, then show the solutionSolution

$x + y \leq 40$ ; $x \geq 10$ . (1 mark)
$x \geq 2 y$ , i.e. $x - 2 y \geq 0$ ; and $x, y \geq 0$ . (1 mark)

Answer: $x + y \leq 40$ , $x \geq 10$ , $x - 2 y \geq 0$ , $x, y \geq 0$ .

Example 3: Product mix with three constraints · 5 marks · Long answerHarder board level

Suppose a workshop makes chairs and stools. A chair needs 2 h of cutting, 1 h of assembly and 1 h of polishing; a stool needs 1 h of cutting, 3 h of assembly and 1 h of polishing. Each week 40 h of cutting, 60 h of assembly and 24 h of polishing are available. Profit is ₹300 per chair and ₹200 per stool. Formulate the LPP and solve it graphically.

Try it first, then show the solutionSolution

Let x chairs and y stools. Maximise $Z = 300 x + 200 y$ . (1 mark)
Cutting $2 x + y \leq 40$ ; assembly $x + 3 y \leq 60$ ; polishing $x + y \leq 24$ ; $x, y \geq 0$ . (1 mark)
Intersections: $2 x + y = 40$ with $x + y = 24$ gives (16, 8); $x + y = 24$ with $x + 3 y = 60$ gives (6, 18). Both satisfy the third constraint. (1 mark)
Corners (0, 0), (20, 0), (16, 8), (6, 18), (0, 20); Z = 0, 6000, 6400, 5400, 4000. (1 mark)
Maximum profit ₹6400 with 16 chairs and 8 stools. (1 mark)

Answer: 16 chairs and 8 stools; ₹6400.

Example 4: An investment problem · 4 marks · Long answerHarder board level

Suppose an investor has ₹1,00,000 to split between scheme A (8% a year) and scheme B (10% a year). At least ₹20,000 must go into A and at most ₹50,000 into B. Formulate and solve to maximise the annual return.

Try it first, then show the solutionSolution

Let ₹x in A, ₹y in B. Maximise $Z = 0.08 x + 0.10 y$ . (1 mark)
$x + y \leq 100000$ , $x \geq 20000$ , $y \leq 50000$ , $y \geq 0$ . (1 mark)
Corners (20000, 0), (100000, 0), (50000, 50000), (20000, 50000); Z = 1600, 8000, 9000, 6600. (1 mark)
Best: ₹50,000 in each scheme, return ₹9000. (1 mark)

Answer: ₹50,000 in A and ₹50,000 in B; ₹9000 a year.

Example 5: Ratio and proportion conditions · 3 marks · Short answerHOTS / competency

Write the constraints: a farmer plants x hectares of wheat and y hectares of mustard on at most 30 hectares; mustard must not exceed half of the total area planted; wheat must be at least 5 hectares; and the area of mustard must be at least one-fifth of the area of wheat.

Try it first, then show the solutionSolution

$x + y \leq 30$ and $x \geq 5$ . (1 mark)
$y \leq \frac{x + y}{2}$ simplifies to $y \leq x$ . (1 mark)
$y \geq \frac{x}{5}$ , i.e. $x - 5 y \leq 0$ ; and $y \geq 0$ . (1 mark)

Answer: $x + y \leq 30$ , $x \geq 5$ , $y \leq x$ , $x - 5 y \leq 0$ , $y \geq 0$ .

Example 6: A delivery plan · 4 marks · Case-basedHOTS / competency

Suppose a cyclist delivers small parcels (2 kg, fee ₹40) and large parcels (5 kg, fee ₹90). The bike can carry at most 42 kg and at most 12 parcels per trip.
(i) Formulate the LPP to maximise the fee per trip. (2 marks)
(ii) Solve it graphically. (2 marks)

Try it first, then show the solutionSolution

(i) Let x small, y large. Maximise $Z = 40 x + 90 y$ subject to $2 x + 5 y \leq 42$ , $x + y \leq 12$ , $x, y \geq 0$ .
(ii) Intersection: $2 (12 - y) + 5 y = 42$ gives $y = 6$ , $x = 6$ .
Corners (0, 0), (12, 0), (6, 6), (0, 8.4); Z = 0, 480, 780, 756.
Maximum ₹780 with 6 small and 6 large parcels (a whole-number corner).

Answer: 6 small and 6 large parcels; ₹780.

Where students lose marks in Formulating Linear Programming Problems

The slips examiners see most often, and what to write instead.

Mistake: Wrong objective function
The ISC 2025 examiners reported candidates who wrote the objective wrongly from the problem statement.
Do this: Ask "what is being maximised, and how much does one unit of each variable add?"
Usually costs: 1 mark plus all follow-through.
Mistake: Constraint signs reversed
"At most" written as ≥ (or vice versa) shades the wrong side and gives a wrong region.
Do this: Read the phrase aloud and test a point that obviously should be allowed.
Usually costs: 1 to 2 marks.
Mistake: Mixed units
Writing $20 x + 10 y \leq 10$ with minutes on the left and hours on the right.
Wrong working
$20 x + 10 y \leq 10$
Right working
$20 x + 10 y \leq 600$ (10 h = 600 min).
Do this: Convert limits to the same units as the coefficients.
Usually costs: 1 mark.
Mistake: Forgetting non-negativity
The restrictions $x, y \geq 0$ are part of the formulation and carry credit.
Do this: Always end the constraint list with them.
Usually costs: ½ to 1 mark.

Exam tips

Begin with "Let x = … and y = …" in words; the examiner looks for it.
Show the resource table if the question is long; it organises the data and earns method marks.
Simplify inequalities before graphing; the corner points are easier to compute.
If the variables are counts, comment on whether the optimal corner has whole-number values.

Practice questions on Formulating Linear Programming Problems

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
A lorry can carry at most 6 tonnes. Sack A weighs 50 kg and sack B 75 kg. With x sacks of A and y of B, the constraint is:
(a) $2 x + 3 y \leq 240$
(b) $50 x + 75 y \leq 6$
(c) $3 x + 2 y \leq 240$
(d) $2 x + 3 y \geq 240$
Hint
6 tonnes = 6000 kg.
Answer
(a).
Q2 · 2 marks · Short answerBoard level
State two limitations of linear programming.
Hint
Think linearity and certainty.
Answer
Any two: relationships must be linear; data are assumed certain; solutions may be fractional; only one objective; large problems need other methods.
Q3 · 4 marks · Long answerHarder board level
Suppose a tailor has 60 m of cloth and 60 hours. A shirt needs 2 m and 3 h; a pair of trousers 3 m and 2 h. Profit: ₹100 per shirt, ₹120 per pair of trousers. Formulate and solve.
Hint
Lines meet at (12, 12).
Answer
Max $Z = 100 x + 120 y$ , $2 x + 3 y \leq 60$ , $3 x + 2 y \leq 60$ , $x, y \geq 0$ ; corners (0, 0), (20, 0), (12, 12), (0, 20); Z = 0, 2000, 2640, 2400; 12 shirts and 12 trousers, ₹2640.
Q4 · 1 mark · Assertion-reasonHarder board level
Assertion (A): The restrictions $x \geq 0$ , $y \geq 0$ are included in an LPP because the decision variables are quantities that cannot be negative.
Reason (R): Every LPP has a bounded feasible region.
(a) Both true, R explains A. (b) Both true, R does not explain A. (c) A true, R false. (d) A false, R true.
Hint
Diet problems often have unbounded regions.
Answer
(c).
Q5 · 2 marks · Short answerHarder board level
Write the constraints: the number of tables (x) must be at least one-third of the number of chairs (y); at most 20 items in all; at least 4 tables.
Hint
Clear the fraction.
Answer
$3 x - y \geq 0$ , $x + y \leq 20$ , $x \geq 4$ , $y \geq 0$ .
Q6 · 4 marks · Long answerHOTS / competency
Suppose a gardener needs at least 10 kg of nitrogen and 11 kg of phosphate. A bag of brand A (₹300) gives 2 kg N and 1 kg P; a bag of brand B (₹250) gives 1 kg N and 2 kg P. Formulate and find the cheapest choice.
Hint
Corners (0, 10), (3, 4), (11, 0).
Answer
Min $Z = 300 x + 250 y$ , $2 x + y \geq 10$ , $x + 2 y \geq 11$ , $x, y \geq 0$ . Z = 2500, 1900, 3300; the half-plane $300 x + 250 y < 1900$ has no feasible point, so the minimum ₹1900 is at 3 bags of A and 4 of B.
Q7 · 4 marks · Case-basedHOTS / competency
Case study: Suppose a company buys x radio spots at ₹2000 each (reach 3000 people) and y online adverts at ₹1000 each (reach 1000 people) with a budget of ₹20,000. It wants at least 4 radio spots and no more than 12 online adverts.
(i) Formulate the LPP to maximise reach. (2 marks)
(ii) Find the corner points. (1 mark)
(iii) Find the best plan. (1 mark)
Hint
Budget: $2 x + y \leq 20$ .
Answer
(i) Max $Z = 3000 x + 1000 y$ , $2 x + y \leq 20$ , $x \geq 4$ , $0 \leq y \leq 12$ . (ii) (4, 0), (10, 0), (4, 12). (iii) Z = 12 000, 30 000, 24 000: 10 radio spots, no online adverts, reach 30 000.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.What are decision variables?Show answer
The quantities whose values the decision-maker chooses.
2.Write "x is at least three times y" as an inequality.Show answer
$x - 3 y \geq 0$ .
3.Name two types of LPP.Show answer
Any two: manufacturing (product mix), diet, allocation/investment, transportation.
4.Simplify $250 x + 500 y \leq 10000$ .Show answer
$x + 2 y \leq 40$ .
5.State one advantage of linear programming.Show answer
It finds the best use of limited resources (or makes planning objective and precise).

Formulating Linear Programming Problems: frequently asked questions

How do I decide whether a constraint is ≤ or ≥?

Limits on resources ("available", "at most", "cannot exceed") give ≤. Requirements ("at least", "minimum", "must meet") give ≥.

Do I always have to solve after formulating?

Only if asked. Some questions say "formulate only", but most ISC questions ask for the formulation and the graphical solution together.

What if the answer is not a whole number?

Linear programming allows fractional values, which is one of its limitations. ISC problems are usually set so that the optimal corner has whole-number coordinates; if not, comment on it.

Are advantages and limitations of LPP in the ISC syllabus?

Yes. The 2027 syllabus lists the terminology, advantages, limitations, application areas and types of LP problems along with formulation.

How many marks does formulation carry?

In the 4-mark LPP questions of 2024 and 2025, writing the objective function and constraints was the first step of the marking, before the graph and corner points.